diff --git a/data_all_eng_slimpj/shuffled/split2/finalzkxi b/data_all_eng_slimpj/shuffled/split2/finalzkxi new file mode 100644 index 0000000000000000000000000000000000000000..1af5e811f643c0e67bf8b721f8f6a8dfc39b29f2 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzkxi @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction} \n\nEdge devices such as smartphones, remote sensors and smart home appliances\ngenerate massive amounts of data \\citep{wang2018smart, cao2017deepmood, shi2016promise}. In recent years, Federated Learning (FL)\nhas emerged as a technique to train models on this data while\npreserving privacy \\citep{FedAvg,FedProx}.\n\nIn FL, we have a single server that is connected to many clients. Each\nclient stores a local dataset that it does not want to share with the server\nbecause of privacy concerns or law enforcement \\citep{voigt2017eu}. The server wants to train a model on all local\ndatasets. To this end, it initializes the model and sends it to a random\nsubset of clients. Each client trains the model on its local dataset and\nsends the trained model back to the server. The server accumulates all\ntrained models into an updated model for the next iteration and repeats the\nprocess for several rounds until some termination criterion is met. This\nprocedure enables the server to train a model without accessing any local\ndatasets. \n\nToday's neural network models often have millions or even billions\n\\citep{gpt3} of parameters, which makes high communication costs a\nconcern in FL. In fact, \\citet{carbon} suggest that communication between clients\nand server may account for over 70\\% of energy consumption in FL. Reducing\ncommunication in FL is an attractive area of research because it lowers\nbandwidth requirements, energy consumption and training time.\n\nCommunication in FL occurs in two phases: Sending parameters from the\nserver to clients (\\emph{downlink}) and sending updated parameters from\nclients to the server (\\emph{uplink}). Uplink bandwidth usually imposes a\ntighter bottleneck than downlink bandwidth. This has several reasons. For\none, the average global mobile upload bandwidth is currently less than one\nfourth of the download bandwidth \\citep{speedtest}. For another, FL downlink\ncommunication sends the same parameters to each client. Broadcasting\nparameters is usually more efficient than the accumulation of parameters\nfrom different clients that is required for uplink communication \\citep{LFL,\nFedPAQ}. For these reasons, we seek to compress uplink communication.\n\n\\begin{figure}[h] \n \\begin{subfigure}[b]{0.50\\textwidth}\n \\def18em{18em}\n \\input{samplequant_complex_static.pdf_tex}\n \\caption{Static quantization.}\n \\label{fig:static_quantization}\n \\end{subfigure} \n \\hfill\n \\begin{subfigure}[b]{0.50\\textwidth}\n \\def18em{18em}\n \\input{samplequant_complex_dynamic.pdf_tex}\n \\label{fig:dynamic_quantization}\n \\caption{Client-adaptive quantization.}\n \\end{subfigure} \n \\caption{Static quantization vs. client-adaptive quantization when\n accumulating parameters $p_A$ and $p_B$. (a): Static quantization uses the\n same quantization level for $p_A$ and $p_B$. (b) Client-adaptive\n quantization uses a slightly higher quantization level for $p_B$ because\n $p_B$ is weighted more heavily. This allows us to use a significantly lower quantization level $q_A$ for $p_A$ while keeping the quantization error measure\n $\\mathrm{E}_{p_A,p_B}\\left[\\mathrm{Var}\\left(\\quantizer(p)\\right)\\right]$\n roughly constant. Since communication is approximately proportional to\n $q_A + q_B$, client-adaptive quantization communicates less data.}\n \\label{fig:clientdynamicquant}\n \\vspace*{-2mm}\n\\end{figure}\n\n\n\\begin{wrapfigure}{r}{0.49\\textwidth}\n \\vspace*{-5mm}\n \\begin{tikzpicture}[scale=0.9]\n \\fill [magenta, opacity=0.2] (0,2.1) rectangle (6, 3.7);\n \\fill [teal, opacity=0.2] (0,1.1) rectangle (6, 2.1);\n \\fill [brown, opacity=0.2] (0,0) rectangle (6, 1.1);\n \\draw [<->, >=stealth] (0,4) -- (0,0) -- (6.3,0);\n \\node [left] at (0,2) {Loss};\n \\draw [magenta, thick](0,3.7) .. controls (0.1,1.3) and (0.3,1.3) .. (5.5, 1.4) node [above, magenta] {\\small q=1};\n \\draw [teal, thick,](0,3.7) .. controls (0.6,0.7) and (1.6,0.7) .. (5.5, 0.85) node [above, teal] {\\small q=2};\n \\draw [brown, thick](0,3.7) .. controls (1.8,0.45) and (3.8,0.45) .. (5.5, 0.45) node [below, brown] {\\small q=4};\n \\begin{scope}\n \\clip (0, 2.1) rectangle (6, 4);\n \\draw [black, very thick, dotted, line cap=round, dash pattern=on 0pt off 2.5\\pgflinewidth](0,3.7) .. controls (0.1,1.3) and (0.3,1.3) .. (5.5, 1.4);\n \\end{scope}\n \\begin{scope}[shift={(-1.02,0.00)}]\n \\clip (0, 0) rectangle (6, 1.1);\n \\draw [brown, dotted, very thick, line cap=round, dash pattern=on 0pt off 2.5\\pgflinewidth](0,4) .. controls (1.8,0.45) and (3.8,0.45) .. (5.5, 0.45);\n \\end{scope}\n \\begin{scope}[shift={(-0.245,-0.00)}]\n \\clip (0, 1.065) rectangle (6, 2.1);\n \\draw [teal, very thick, dotted, line cap=round, dash pattern=on 0pt off 2.5\\pgflinewidth](0,4) .. controls (0.6,0.7) and (1.6,0.7) .. (5.5, 0.85);\n \\end{scope}\n \\node [below] at (2, 0.7) {\\small adaptive q};\n\n \\begin{scope}[shift={(0,-1.5)}]\n \\begin{axis}[%\n ,xlabel=Communication\n ,xlabel near ticks\n ,ylabel near ticks\n ,ylabel=q\n ,axis x line = bottom,axis y line = left\n ,ytick={0, 1,2,4}\n ,xtick=\\empty\n ,ymax=4.5\n ,ymin=0\n ,width=3in\n ,height=1.2in\n ]\n \\addplot+[const plot, no marks, thick] coordinates {(0,1) (0.15,1) (0.15,2) (0.7,2) (0.7,4) (3,4)};\n \\end{axis}\n \\end{scope}\n\n \\end{tikzpicture} \n \\caption{Time-adaptive quantization. A small quantization level (q)\n decreases the loss with less communication than a large q, but converges\n to a higher loss. This motivates an adaptive quantization strategy that uses a small q as long as \n it is beneficial and then switches over to a large q. We generalize this idea into an algorithm\n that monotonically increases q based on the training loss.}\n \\label{fig:timedynamicquant}\n\\end{wrapfigure}\n\nA large class of compression algorithms for FL apply some lossy quantizer $\\quantizer$, optionally\nfollowed by a lossless compression stage. $\\quantizer$ usually provides a ``quantization level''\nhyperparameter $q$ to control the coarseness of quantization (e.g. the number of bins\nfor fixed-point quantization). When $q$ is kept constant during training, we speak of\n\\emph{static quantization}. When $q$ changes, we speak of \\emph{adaptive quantization}. Adaptive\nquantization can exploit asymmetries in the FL framework to minimize communication. One such\nasymmetry lies in FL's training time, where we observe that early training rounds can use a lower\n$q$ without affecting convergence. \\Cref{fig:timedynamicquant} illustrates how \\emph{time-adaptive\nquantization} leverages this phenomenon to minimize communication. Another asymmetry lies in FL's\nclient space, because most FL algorithms weight client contributions to the global model\nproportional to their local dataset sizes. \\Cref{fig:clientdynamicquant} illustrates how\n\\emph{client-adaptive quantization} can minimize the quantization error. Intuitively, FL clients\nwith greater weighting should have a greater communication budget and our proposed client-adaptive\nquantization achieves this in a principled way. To this end, we introduce the expected variance of\nan accumulation of quantized parameters, $\\mathbb{E}[\\mathrm{Var}(\\sum\\quantizer(p))]$, as a measure of the\nquantization error. Our client-adaptive quantization algorithm then assigns clients minimal\nquantization levels, subject to a fixed $\\mathbb{E}[\\mathrm{Var}(\\sum\\quantizer(p))]$. This lowers the amount of\ndata communicated from clients to the server, without increasing the quantization error.\n\nDAdaQuant (Doubly Adaptive Quantization) combines time- and client-adaptive quantization with an adaptation\nof the QSGD fixed-point quantization algorithm to achieve state-of-the-art\nFL uplink compression.\nIn this paper, we make the following contributions:\n\n\\begin{itemize}[noitemsep,leftmargin=*] \n \\item We introduce the concept of client-adaptive quantization and develop algorithms for time-\n and client-adaptive quantization that are computationally efficient, empirically Pareto optimal\n and compatible with arbitrary FL quantizers. Our client-adaptive quantization is provably optimal\n for stochastic fixed-point quantizers. \n \n \\item We create Federated QSGD as an adaptation of the stochastic\n fixed-point quantizer QSGD that works with FL. Federated QSGD outperforms\n all other quantizers, establishing a strong baseline for FL compression with\n static quantization.\n\n \\item We combine time- and client-adaptive quantization into DAdaQuant. We\n demonstrate DAdaQuant's state-of-the-art compression by empirically\n comparing it against several competitive FL compression\n algorithms.\n\\end{itemize}\n\n\n\\section{Related Work}\n\nFL research has explored several approaches to reduce communication. We\nidentify three general directions.\n\nFirst, there is a growing interest of\ninvestigating FL algorithms that can converge in fewer rounds. FedAvg\n\\citep{FedAvg} achieves this with prolonged local training, while FOLB\n\\citep{folb} speeds up convergence through a more principled client\nsampling. Since communication is proportional to the number of training\nrounds, these algorithms effectively reduce communication.\n\nSecondly, communication can be reduced by reducing the model\nsize because the model size is proportional to the amount of training communication.\nPruneFL \\citep{fedprune} progressively prunes the model over the course of\ntraining, while AFD \\citep{feddropout} only trains submodels on clients.\n\nThirdly, it is possible to directly compress FL training communication. FL\ncompression algorithms typically apply techniques like top-k sparsification\n\\citep{fedzip, fetchsgd} or quantization \\citep{FedPAQ, uveqfed} to\nparameter updates, optionally followed by lossless compression. Our work\napplies to quantization-based compression algorithms. It is partially based\non QSGD \\citep{QSGD}, which combines lossy fixed-point quantization with a\nlossless compression algorithm to compress gradients communicated in\ndistributed training. DAdaQuant adapts QSGD into Federated QSGD, which works\nwith Federated Learning. DAdaQuant also draws inspiration from FedPAQ\n\\citep{FedPAQ}, the first FL framework to use lossy compression based on\nmodel parameter update quantization. However, FedPAQ does not explore the\nadvantages of additional lossless compression or adaptive quantization.\nUVeQFed \\citep{uveqfed} is an FL compression algorithm that generalizes\nscalar quantization to vector quantization and subsequently employs lossless\ncompression with arithmetic coding. Like FedPAQ, UVeQFed also limits itself\nto a single static quantization level.\n\nFaster convergence, model size reduction and communication compression are\northogonal techniques, so they can be combined for further communication\nsavings. For this paper, we limit the scope of empirical comparisons to\nquantization-based FL compression algorithms.\n\nFor quantization-based compression for model training, prior works have\ndemonstrated that DNNs can be successfully trained in low-precision\n\\citep{banner2018scalable,gupta2015deep,sun2019hybrid}. There are also\nseveral adaptive quantization algorithms for training neural networks in a\nnon-distributed setting. \\Citet{adaparams} use different quantization levels\nfor different parameters of a neural network. FracTrain \\citep{FracTrain}\nintroduced multi-dimensional adaptive quantization by developing\ntime-adaptive quantization and combining it with parameter-adaptive\nquantization. However, FracTrain uses the current loss to decide on the\nquantization level. FL generally can only compute local client losses that\nare too noisy to be practical for FracTrain. AdaQuantFL introduces\ntime-adaptive quantization to FL, but requires the global loss\n\\citep{adaquantfl}. To compute the global loss, AdaQuantFL has to\ncommunicate with every client each round. We show in\n\\Cref{sec:experimentsresults} that this quickly becomes impractical as the\nnumber of clients grows. DAdaQuant's time-adaptive quantization overcomes\nthis issue without compromising on the underlying FL communication. In\naddition, to the best of our knowledge, DAdaQuant is the first algorithm to\nuse client-adaptive quantization.\n\n\\section{The DAdaQuant method}\n\n\\subsection{Federated Learning}\n\nFederated Learning assumes a client-server topology with a set ${\\mathbb{C}} =\n\\{c_i|i \\in \\{1,2...N\\}\\}$ of $N$ clients that are connected to a single server. Each client\n$c_k$ has a local dataset $D_k$ from the local data distribution\n$\\mathcal{D}_k$.\nGiven a model $M$ with parameters ${\\bm{p}}$, a loss function\n$f_{{\\bm{p}}}(d\\in D_k)$ and the local loss $F_k({\\bm{p}}) =\n\\frac{1}{|D_k|}\\sum_{d \\in D_k} f_{\\bm{p}}(d)$, FL seeks to minimize the global\nloss $G({\\bm{p}}) = \\sum_{k=1}^{N} \\frac{|D_k|}{\\sum_l|D_l|}F_k({\\bm{p}})\\label{eq:flobjective}$.\n\n\n\\subsection{Federated Averaging (FedAvg)}\n\nDAdaQuant makes only minimal assumptions about the FL algorithm. Crucially,\nDAdaquant can complement FedAvg \\citep{FedAvg}, which is representative of a\nlarge class of FL algorithms.\n\nFedAvg trains the model $M$ over several rounds. In each round $t$, FedAvg\nsends the model parameters ${\\bm{p}}_t$ to a random subset ${\\mathbb{S}}_t$ of $K$ clients\nwho then optimize their local objectives $F_k({\\bm{p}}_t)$ and send the updated\nmodel parameters ${\\bm{p}}_{t+1}^k$ back to the server. The server accumulates\nall parameters into the new global model ${\\bm{p}}_{t+1} = \\sum_{k\\in{\\mathbb{S}}_t}\n\\frac{|D_k|}{\\sum_j |D_j|} {\\bm{p}}_{t+1}^{k}$\\; and starts the next round.\n\\Cref{alg:fedavgdadaquant} lists FedAvg in detail. For our experiments, we\nuse the FedProx \\citep{FedProx} adaptation of FedAvg. FedProx improves the\nconvergence of FedAvg by adding the proximal term\n$\\frac{\\mu}{2}\\|{\\bm{p}}_{t+1}^k - {\\bm{p}}_t\\|^2$ to the local objective\n$F_k({\\bm{p}}_{t+1}^k)$ in \\Cref{alg:fedavgobjective} of\n\\Cref{alg:fedavgdadaquant}.\n\n\\subsection{Quantization with Federated QSGD}\n\\label{sec:qsgd}\n\nWhile DAdaQuant can be applied to any quantizer with a configurable\nquantization level, it is optimized for fixed-point quantization. We\nintroduce Federated QSGD as a competitive fixed-point quantizer on top of\nwhich DAdaQuant is applied. \n\nIn general, fixed-point quantization uses a quantizer $\\quantizer_q$ with\nquantization level $q$ that splits $\\mathbb{R}_{\\geq0}$ and $\\mathbb{R}_{\\leq0}$ into $q$ intervals each.\n$\\quantizer_q(p)$ then returns the sign of $p$ and $|p|$ rounded to one of\nthe endpoints of its encompassing interval. $\\quantizer_q({\\bm{p}})$ quantizes the\nvector ${\\bm{p}}$ elementwise.\n\nWe design DAdaQuant's quantization stage based on QSGD, an efficient\nfixed-point quantizer for state-of-the-art gradient compression.\nQSGD quantizes a vector ${\\bm{p}}$ in\nthree steps:\n\\begin{enumerate}[noitemsep]\n \\item Quantize ${\\bm{p}}$ as $\\quantizer_q(\\frac{{\\bm{p}}}{||{\\bm{p}}||_2})$ into $q$ bins in $[0,1]$, storing signs and $||{\\bm{p}}||_2$ separately. (\\emph{lossy})\n \\item Encode the resulting integers with 0 run-length encoding. (\\emph{lossless})\n \\item Encode the resulting integers with Elias $\\omega$ coding. (\\emph{lossless})\n\\end{enumerate}\n\nQSGD has been designed specifically for quantizing gradients. This makes it\nnot directly applicable to parameter compression. To overcome this\nlimitation, we apply difference coding to uplink compression, first\nintroduced to FL by FedPAQ. Each client $c_k$ applies $\\quantizer_q$ to the\n\\emph{parameter updates} ${\\bm{p}}^k_{t+1}-{\\bm{p}}_t$ (cf.\n\\Cref{alg:fedavgclientreturn} of \\Cref{alg:fedavgdadaquant}) and sends them to the server. The server\nkeeps track of the previous parameters ${\\bm{p}}_t$ and accumulates the quantized\nparameter updates into the new parameters as ${\\bm{p}}_{t+1} = {\\bm{p}}_t + \\sum_{k\\in{\\mathbb{S}}_t}\n\\frac{|D_k|}{\\sum_l |D_l|} \\quantizer_q({\\bm{p}}^k_{t+1}-{\\bm{p}}_t)$ (cf.\n\\Cref{alg:fedavgaccumulate} of \\Cref{alg:fedavgdadaquant}). We find that QSGD works well with parameter\nupdates, which can be regarded as an accumulation of gradients over several\ntraining steps. We call this adaptation of QSGD \\emph{Federated QSGD}.\n\n\\subsection{Time-adaptive quantization}\n\nTime-adaptive quantization uses a different quantization level $q_t$ for\neach round $t$ of FL training. DAdaQuant chooses $q_t$ to minimize\ncommunication costs without sacrificing accuracy. To this end, we find that\nlower quantization levels suffice to initially reduce the loss, while partly\ntrained models require higher quantization levels to further improve (as illustrated in \\Cref{fig:timedynamicquant}).\nFracTrain is built on similar observations for non-distributed training. \nTherefore, we design DAdaQuant to mimic FracTrain in monotonically\nincreasing $q_t$ as a function of $t$ and using the training loss to inform\nincreases in $q_t$.\n\nWhen $q$ is too low, FL converges prematurely. Like FracTrain, DAdaQuant\nmonitors the FL loss and increases $q$ when it converges. Unlike FracTrain,\nthere is no single centralized loss function to evaluate and unlike\nAdaQuantFL, we do not assume availability of global training loss\n$G({\\bm{p}}_t)$. Instead, we estimate $G({\\bm{p}}_t)$ as the average\nlocal loss $\\hat{G}_t = \\sum_{k\\in{\\mathbb{S}}_t} \\frac{|D_k|}{\\sum_l\n|D_l|}F_k({\\bm{p}}_t)$ where ${\\mathbb{S}}_t$ is the set of clients sampled at round $t$.\nSince ${\\mathbb{S}}_t$ typically consists of only a small fraction of all clients,\n$\\hat{G}_t$ is a very noisy estimate of $G({\\bm{p}}_t)$. This makes it\nunsuitable for convergence detection. Instead, DAdaQuant tracks a running\naverage loss $\\doublehat{{G}}_{t} = \\psi \\doublehat{{G}}_{t-1} + (1-\\psi) \\hat{G}_{t}$.\n\nWe initialize $q_1 = q_\\text{min}$ for some $q_\\text{min} \\in {\\mathbb{N}}$.\nDAdaQuant determines training to converge whenever $\\doublehat{{G}}_{t} \\geq\n\\doublehat{{G}}_{t+1-\\phi}$ for some $\\phi \\in {\\mathbb{N}}$ that specifies the number of\nrounds across which we compare $\\doublehat{{G}}$. On convergence, DAdaQuant sets\n$q_{t} = 2q_{t-1}$ and keeps the quantization level fixed for at least\n$\\phi$ rounds to enable reductions in $G$ to manifest in $\\doublehat{{G}}$.\nEventually, the training loss converges regardless of the quantization\nlevel. To avoid unconstrained quantization increases on convergence, we\nlimit the quantization level to $q_\\text{max}$.\n\nThe following equation\nsummarizes DAdaQuant's time-adaptive quantization:\n$$\nq_{t} \\longleftarrow\n\\begin{cases}\n q_{\\text{min}} & t = 0 \\\\\n 2q_{t-1} & t > 0 \\text{ and } \\doublehat{{G}}_{t-1} \\geq \\doublehat{{G}}_{t-\\phi} \\text{ and } t > \\phi \\text{ and } 2q_{t-1} < q_{\\text{max}} \\text{ and } q_{t-1} = q_{t-\\phi} \\\\\n q_{t-1} & \\text{else}\n\\end{cases}\n$$\n\n\\begin{figure}[]\\small\\centering\n\\begin{subtable}[]{0.492\\textwidth}\n\\begin{tabular}{l|l|lllll}\n\\multicolumn{2}{r|}{Round} & \\multicolumn{1}{l|}{1} & \\multicolumn{1}{l|}{2} & \\multicolumn{1}{l|}{3} & \\multicolumn{1}{l|}{4} & \\multicolumn{1}{l|}{5} \\\\ \\hline\nClient & Samples & \\multicolumn{5}{c}{Quantization level} \\\\ \\hline\nA & 1 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{} \\\\\nB & 2 & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{} \\\\\nC & 3 & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{8}} \\\\\nD & 4 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{8}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{8}}\n\\end{tabular}\n\\caption{Static quantization.}\n\\end{subtable}\\hfill\n\\begin{subtable}[h]{0.492\\textwidth}\n\\begin{tabular}{l|l|lllll}\n\\multicolumn{2}{r|}{Round} & \\multicolumn{1}{l|}{1} & \\multicolumn{1}{l|}{2} & \\multicolumn{1}{l|}{3} & \\multicolumn{1}{l|}{4} & \\multicolumn{1}{l|}{5} \\\\ \\hline\nClient & Samples & \\multicolumn{5}{c}{Quantization level} \\\\ \\hline\nA & 1 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{4}} & \\multicolumn{1}{l|}{} \\\\\nB & 2 & \\multicolumn{1}{l|}{\\gradient{1}} & \\multicolumn{1}{l|}{\\gradient{2}} & \\multicolumn{1}{l|}{\\gradient{2}} & \\multicolumn{1}{l|}{\\gradient{4}} & \\multicolumn{1}{l|}{} \\\\\nC & 3 & \\multicolumn{1}{l|}{\\gradient{1}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{2}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{8}} \\\\\nD & 4 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{2}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{8}}\n\\end{tabular}\n\\caption{Time-adaptive quantization.}\n\\end{subtable}\n\n \\begin{subtable}[h]{0.492\\textwidth}\n \\begin{tabular}{l|l|lllll}\n \\multicolumn{2}{r|}{Round} & \\multicolumn{1}{l|}{1} & \\multicolumn{1}{l|}{2} & \\multicolumn{1}{l|}{3} & \\multicolumn{1}{l|}{4} & \\multicolumn{1}{l|}{5} \\\\ \\hline\n Client & Samples & \\multicolumn{5}{c}{Quantization level} \\\\ \\hline\n A & 1 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{6}} & \\multicolumn{1}{l|}{} \\\\\n B & 2 & \\multicolumn{1}{l|}{\\gradient{7}} & \\multicolumn{1}{l|}{\\gradient{6}} & \\multicolumn{1}{l|}{\\gradient{7}} & \\multicolumn{1}{l|}{\\gradient{9}} & \\multicolumn{1}{l|}{} \\\\\n C & 3 & \\multicolumn{1}{l|}{\\gradient{9}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{9}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{7}} \\\\\n D & 4 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{9}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{9}}\n \\end{tabular}\\hspace{0.47em}\n \\caption{Client-adaptive quantization.}\n \\end{subtable}\n \\begin{subtable}[h]{0.492\\textwidth}\n \\begin{tabular}{l|l|lllll}\n \\multicolumn{2}{r|}{Round} & \\multicolumn{1}{l|}{1} & \\multicolumn{1}{l|}{2} & \\multicolumn{1}{l|}{3} & \\multicolumn{1}{l|}{4} & \\multicolumn{1}{l|}{5} \\\\ \\hline\n Client & Samples & \\multicolumn{5}{c}{Quantization level} \\\\ \\hline\n A & 1 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{3}} & \\multicolumn{1}{l|}{} \\\\\n B & 2 & \\multicolumn{1}{l|}{\\gradient{1}} & \\multicolumn{1}{l|}{\\gradient{1}} & \\multicolumn{1}{l|}{\\gradient{2}} & \\multicolumn{1}{l|}{\\gradient{5}} & \\multicolumn{1}{l|}{} \\\\\n C & 3 & \\multicolumn{1}{l|}{\\gradient{1}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{2}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{7}} \\\\\n D & 4 & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{1}} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{\\gradient{9}}\n \\end{tabular}\n \\caption{Time-adaptive and client-adaptive quantization.}\n \\end{subtable}\n \\caption{Exemplary quantization level assignment for 4 FL clients that train over 5 rounds. Each round, two clients get sampled for training.}\n \\label{fig:quantexample}\n\\end{figure}\n\n\\subsection{Client-adaptive quantization}\n\nFL algorithms typically accumulate each parameter $p_i$ over all clients\ninto a weighted average $p = \\sum_{i=1}^K{w_ip_i}$ (see \\Cref{alg:fedavgdadaquant}).\nQuantized FL communicates and accumulates quantized parameters\n$\\quantizer_q(p) = \\sum_{i=1}^K{w_i\\quantizer_q(p_i)}$ where $q$ is the\nquantization level. We define the quantization error $e^q_p$ as $e^q_p = |p\n- \\quantizer_q(p)|$. We observe that $\\mathbb{E}_{p_1\\ldots\np_K}[\\mathrm{Var}(\\quantizer_q(p))]$ is a useful statistic of the quantization error\nbecause it strongly correlates with the loss added by quantization. For\na stochastic, unbiased fixed-point compressor like Federated QSGD, $\\mathbb{E}_{p_1\\ldots\np_K}[\\mathrm{Var}(\\quantizer_q(p))]$ equals $\\mathbb{E}_{p_1\\ldots p_K}[\\mathrm{Var}(e^q_p)]$ and\ncan be evaluated analytically.\n\nWe observe in our experiments that communication cost per client is roughly\na linear function of Federated QSGD's quantization level $q$. This means that the\ncommunication cost per round is proportional to $Q = Kq$. We call $Q$ the\ncommunication budget and use it as a proxy measure of communication cost.\n\nClient-adaptive quantization dynamically adjusts the quantization level of each client. This means\nthat even within a single round, each client $c_k$ can be assigned a different quantization level $q_k$. The\ncommunication budget of client-adaptive quantization is then $Q = \\sum_{k=1}^K{q_k}$ and\n$\\quantizer_q(p)$ generalizes to $\\quantizer_{q_1\\ldots q_K}(p) = \\sum_{i=1}^K{w_i\\quantizer_{q_i}(p_i)}$. We devise an algorithm that chooses $q_k$ to minimize $Q$ subject to $\\mathbb{E}_{p_1\\ldots\np_K}[\\mathrm{Var}(e^{q_1\\ldots q_K}_p)] = \\mathbb{E}_{p_1\\ldots p_K}[\\mathrm{Var}(e^q_p)]$ for a given $q$. Thus, our algorithm effectively minimizes\ncommunication costs while maintaining a quantization error similar to static quantization. \\Cref{theorem:q} provides us with an analytical formula for quantization\nlevels $q_1\\ldots q_K$. \n\n\\begin{theorem}{Given parameters $p_1\\ldots p_k\\sim\\mathcal{U}[-t, t]$ and quantization level $q$, $\\min_{q_1\\ldots q_K}\\sum_{i=1}^K{q_i}$\n subject to $\\mathbb{E}_{p_1\\ldots p_K}[\\mathrm{Var}(e^{q_1\\ldots q_K}_p)] = \\mathbb{E}_{p_1\\ldots p_K}[\\mathrm{Var}(e^q_p)]$ is minimized by $q_i = \\sqrt{\\frac{a}{b}}\\times w_i^{2\/3}$ where $a = {\\sum_{j=1}^K w_j^{2\/3}}$ and $b = {\\sum_{j=1}^K \\frac{w_j^2}{q^2}}$.}\n\\label{theorem:q}\n\\end{theorem}\n\nDAdaQuant applies \\Cref{theorem:q} to lower communication costs while\nmaintaining the same loss as static quantization does with a fixed $q$. To\nensure that quantization levels are natural numbers, DAdaQuant approximates\nthe optimal real-valued solution as $q_i = \\max(1,\n\\text{round}(\\sqrt{\\frac{a}{b}}\\times w_i^{2\/3}))$. \\Cref{sec:proofs} gives a detailed proof\nof \\Cref{theorem:q}. To the best of our knowledge, DAdaQuant is the first algorithm\nto use client-adaptive quantization.\n\n\\begin{algorithm}[H]\n \\SetAlgoLined\n \\Fn(){\\FServer{}} {\n Initialize $w_i = \\frac{|D_i|}{\\sum_j |D_j|}$ for all $i\\in[1,\\ldots,N]$\\;\n \\For(){$t = 0,\\dots,T-1$}{\n \n Choose ${\\mathbb{S}}_t \\subset {\\mathbb{C}}$ with $|{\\mathbb{S}}_t| = K$, including each $c_k \\in {\\mathbb{C}}$ with uniform probability\\;\n \\colorbox{brown!40}{\n $q_{t} \\longleftarrow\n \\begin{cases}\n q_{\\text{min}} & t = 0 \\\\\n 2q_{t-1} & t > 0 \\text{ and } \\doublehat{{G}}_{t-1} \\geq \\doublehat{{G}}_{t-\\phi} \\text{ and } t > \\phi \\text{ and } q_{t} \\leq q_{\\text{max}} \\text{ and } q_{t-1} = q_{t-\\phi} \\\\\n q_{t-1} & \\text{else}\n \\end{cases}$}\\;\n \\For(in parallel){$c_k \\in {\\mathbb{S}}_t$}{\n \\colorbox{magenta!40}{\n $q_t^k \\longleftarrow\n \\sqrt{\\sum_{j=1}^K w_j^{2\/3}\/\\sum_{j=1}^K \\frac{w_j^2}{q^2}}\n $\n }\\;\n \n $Send(c_k, {\\bm{p}}_t, ${$q_t^k$}$)$\\; \n \n $Receive(c_k, {\\bm{p}}_{t+1}^k,${$\\hat{G}_t^k$}$)$\\;\n }\n \n ${\\bm{p}}_{t+1} \\longleftarrow \\sum_{k\\in{\\mathbb{S}}_t} w_k {\\bm{p}}_{t+1}^{k}$\\;\n \\label{alg:fedavgaccumulate}\n \\colorbox{brown!40}{\n $\\hat{G}_{t} \\longleftarrow \\sum_{k\\in{\\mathbb{S}}_t} w_k \\hat{G}_t^k$\n }\\;\n \\colorbox{brown!40}{\n $\\doublehat{{G}}_{t} \\longleftarrow \n \\begin{cases}\n \\hat{G}_0 & t = 0 \\\\\n \\psi \\doublehat{{G}}_{t-1} + (1-\\psi) \\hat{G}_{t} & \\textrm{else} \\\\\n \\end{cases}$\n }\\;\n } \n }\n \\Fn(){\\FClient{$c_k$}} {\n $Receive(\\textrm{Server}, {\\bm{p}}_t,$\\,{$q_t^k$}$)$\\;\n \\colorbox{brown!40}{$\\hat{G}_t^k \\longleftarrow F_k({\\bm{p}}_{t})$}\\;\n \n ${\\bm{p}}_{t+1}^k \\longleftarrow$ $F_k({\\bm{p}}_{t+1}^k)$ trained with SGD for $E$ epochs with learning rate $\\eta$\\;\n \\label{alg:fedavgobjective}\n $Send(\\textrm{Server},\\,$\\colorbox{teal!40}{$\\quantizer_{q_t^k}({\\bm{p}}_{t+1}^k)$}$, ${$\\hat{G}_t^k$}$)$\\;\n \\label{alg:fedavgclientreturn}\n }\n \\caption{The FedAvg and DAdaQuant algorithms. The uncolored lines list FedAvg. Adding the colored lines creates DAdaQuant. \\textrect{teal} --- quantization, \\textrect{magenta} --- client-adaptive quantization, \\textrect{brown} --- time-adaptive quantization.}\n \\label{alg:fedavgdadaquant}\n\\end{algorithm}\n\n\\subsection{Doubly-adaptive quantization (DAdaQuant)}\n\nDAdaQuant combines the time-adaptive and client-adaptive quantization\nalgorithms described in the previous sections. At each round $t$,\ntime-adaptive quantization determines a preliminary quantization level\n$q_t$. Client-adaptive quantization then finds the client quantization\nlevels $q_t^k, k \\in \\{1, \\ldots, K\\}$ that minimize $\\sum_{i=1}^K{q_i}$\nsubject to $\\mathbb{E}_{p_1\\ldots p_K}[\\mathrm{Var}(e^{q_1\\ldots q_K}_p)] = \\mathbb{E}_{p_1\\ldots\np_K}[\\mathrm{Var}(e^q_p)]$. \\Cref{alg:fedavgdadaquant} lists DAdaQuant in detail.\n\\Cref{fig:quantexample} gives an example of how our time-adaptive,\nclient-adaptive and doubly-adaptive quantization algorithms set quantization levels.\n\n\\Citet{FedPAQ} prove the convergence of FL with quantization for convex\nand non-convex cases as long as the quantizer $\\quantizer$ is (1) unbiased\nand (2) has a bounded variance. These convergence results extend to\nDAdaQuant when combined with any quantizer that satisfies (1) and (2) for\nDAdaQuant's minimum quantization level $q=1$. Crucially, this includes\nFederated QSGD. \n\nWe highlight DAdaQuant's low overhead and general applicability. The\ncomputational overhead is dominated by an additional evaluation epoch per\nround per client to compute $\\doublehat{{G}}_t$, which is negligible when training\nfor many epochs per round. DAdaQuant can compliment any FL algorithm that\ntrains models over several rounds and accumulates a weighted average of\nclient parameters. Most FL algorithms, including FedAvg, follow this design.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Experimental details}\n\n\\textbf{Evaluation} We use DAdaQuant with Federated QSGD to train\ndifferent models with FedProx on different datasets for a fixed number of rounds.\nWe monitor the test loss and accuracy at fixed intervals and measure\nuplink communication at every round across all devices.\n\n\\textbf{Models \\& datasets} We select a broad and diverse set of five\nmodels and datasets to demonstrate the general applicability of DAdaQuant.\nTo this end, we use DAdaQuant to train a linear model, CNNs and LSTMs of\nvarying complexity on a federated synthetic dataset (\\dataset{Synthetic}),\nas well as two federated image datasets (\\dataset{FEMNIST} and\n\\dataset{CelebA}) and two federated natural language datasets\n(\\dataset{Sent140} and \\dataset{Shakespeare}) from the LEAF \\citep{LEAF}\nproject for standardized FL research. We refer to\n\\Cref{sec:models_datasets_detailed} for more information on the\nmodels, datasets, training objectives and implementation.\n\n\\textbf{System heterogeneity}\nIn practice, FL has to cope with clients that have different compute\ncapabilities. We follow \\citet{FedProx} and simulate this \\emph{system\nheterogeneity} by randomly reducing the number of epochs to $E'$ for a\nrandom subset ${\\mathbb{S}}_t' \\subset {\\mathbb{S}}_t$ of clients at each round $t$, where\n$E'$ is sampled from $[1, \\ldots, E]$ and $|{\\mathbb{S}}_t'| = 0.9K$. \n\n\\textbf{Baselines}\nWe compare DAdaQuant against competing quantization-based algorithms for FL\nparameter compression, namely Federated QSGD, FedPAQ \\citep{FedPAQ}, GZip\nwith fixed-point quantization (FxPQ + GZip), UVeQFed \\citep{uveqfed} and\nFP8. Federated QSGD (see \\cref{sec:qsgd}) is our most important baseline\nbecause it outperforms the other algorithms. FedPAQ only applies fixed-point\nquantization, which is equivalent to Federated QSGD without lossless\ncompression. Similarly, FxPQ + GZip is equivalent to Federated QSGD with\nGzip for its lossless compression stages. UVeQFed generalizes scalar\nquantization to vector quantization, followed by arithmetic coding. We apply\nUVeQFed with the optimal hyperparameters reported by its authors. FP8\n\\citep{FP8} is a floating-point quantizer that uses an 8-bit floating-point\nformat designed for storing neural network gradients. We also evaluate all\nexperiments without compression to establish an accuracy benchmark.\n\n\\textbf{Hyperparameters} With the exception of \\dataset{CelebA}, all our datasets\nand models are also used by \\citeauthor{FedProx}. We therefore adopt most of\nthe hyperparameters from \\citeauthor{FedProx} and use LEAF's hyperparameters for \\dataset{CelebA} \\cite{LEAF}.\nFor all experiments, we sample 10 clients each round. We train\n\\dataset{Synthetic}, \\dataset{FEMNIST} and \\dataset{CelebA} for 500 rounds\neach. We train \\dataset{Sent140} for 1000 rounds due to slow convergence and\n\\dataset{Shakespeare} for 50 rounds due to rapid convergence. We use batch\nsize 10, learning rates 0.01, 0.003, 0.3, 0.8, 0.1 and $\\mu$s (FedProx's\nproximal term coefficient) 1, 1, 1, 0.001, 0 for \\dataset{Synthetic},\n\\dataset{FEMNIST}, \\dataset{Sent140}, \\dataset{Shakespeare},\n\\dataset{CelebA} respectively. We randomly split the local datasets into\n80\\% training set and 20\\% test set.\n\nTo select the quantization level $q$ for static quantization with Federated\nQSGD, FedPAQ and FxPQ + GZip, we run a gridsearch over $q = 1, 2, 4, 8,\n\\ldots$ and choose for each dataset the lowest $q$ for which Federated QSGD\nexceeds uncompressed training in accuracy. We set UVeQFed's ``coding rate''\nhyperparameter $R=4$, which is the lowest value for which UVeQFed achieves\nnegligible accuracy differences compared to uncompressed training.\nWe set\nthe remaining hyperparameters of UVeQFed to the optimal values reported by\nits authors. \\Cref{sec:uveqfed} shows further experiments that compare against\nUVeQFed with $R$ chosen to maximize its compression factor.\n\nFor DAdaQuant's time-adaptive quantization, we set $\\psi$ to 0.9, $\\phi$ to\n${1\/10}^{th}$ of the number of rounds and $q_\\textrm{max}$ to the\nquantization level $q$ for each experiment. For \\dataset{Synthetic} and\n\\dataset{FEMNIST}, we set $q_\\textrm{min}$ to 1. We find that\n\\dataset{Sent140}, \\dataset{Shakespeare} and \\dataset{CelebA} require a high\nquantization level to achieve top accuracies and\/or converge in few rounds.\nThis prevents time-adaptive quantization from increasing the quantization\nlevel quickly enough, resulting in prolonged low-precision training that\nhurts model performance. To counter this effect, we set $q_\\textrm{min}$ to\n$q_\\textrm{max}\/2$. This effectively results in binary time-adaptive\nquantization with an initial low-precision phase with $q =\nq_\\textrm{max}\/2$, followed by a high-precision phase with $q =\nq_\\textrm{max}$.\n\n\\subsection{Results}\n\\label{sec:experimentsresults}\n\n\\begin{wrapfigure}[18]{r}{0.40\\textwidth}\n \\vspace*{-1.7cm} \n \\begin{center}\n \n \\scalebox{0.75}{\n \\input{anc\/adaquantfl.pgf}}\n \\end{center}\n \\caption{Comparison of AdaQuantFL and DAdaQuant. We plot the total\n client$\\rightarrow$server communication required to train an MLR model on\n synthetic datasets with 10, 100, 200 and 400 clients. AdaQuantFL's\n communication increases linearly with the number of clients because it\n trains the model on all clients at each round. In contrast, DAdaQuant's\n communication does not change with the number of clients.} \n \\label{fig:adaquantfl} \n\\end{wrapfigure}\n\nWe repeat the main experiments three times and report average results and\ntheir standard deviation (where applicable). \\Cref{tab:results} shows the\nhighest accuracy and total communication for each experiment.\n\\Cref{fig:pareto} plots the maximum accuracy achieved for any given amount\nof communication.\n\n \n\\begin{table}[]\n \\scriptsize\n \n \\centering\n \\begin{tabular}{ll@{\\hskip -0mm}ll@{\\hskip 2mm}ll@{\\hskip 2mm}l}\n \\multicolumn{1}{l|}{} & \\multicolumn{2}{c|}{\\textbf{Synthetic}} & \\multicolumn{2}{c|}{\\textbf{FEMNIST}} & \\multicolumn{2}{c}{\\textbf{Sent140}} \\\\ \\hline\n \\rowcolor[HTML]{EFEFEF} \n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{Uncompressed}} & {$78.3\\pm 0.3$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$12.2$\\,MB}} & {$77.7\\pm 0.4$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$\\!\\!\\!\\!\\!132.1$\\,GB}} & {$69.7\\pm 0.5$} & {$43.9$\\,GB} \\\\\n \\multicolumn{1}{l|}{\\textbf{Federated QSGD}} & {$-0.1\\pm 0.1$} & \\multicolumn{1}{l|}{{$17\\times$}} & {$+0.7\\pm 0.5$} & \\multicolumn{1}{l|}{{$\\!\\!\\!\\!\\!2809\\times$}} & {$-0.0\\pm 0.5$} & {$90\\times$} \\\\\n \\rowcolor[HTML]{EFEFEF} \n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{FP8}} & {$\\bm{+0.1\\pm 0.4}$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$4.0\\times$ ($0.23\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-0.1\\pm 0.4$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$\\!\\!\\!\\!\\!4.0\\times$ ($0.00\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-0.2\\pm 0.5$} & {$4.0\\times$ ($0.04\\!\\times\\!\\!\\!\\!\\!\\times$)} \\\\\n \\multicolumn{1}{l|}{\\textbf{FedPAQ (FxPQ)}} & {$-0.1\\pm 0.1$} & \\multicolumn{1}{l|}{{$6.4\\times$ ($0.37\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$+0.7\\pm 0.5$} & \\multicolumn{1}{l|}{{$\\!\\!\\!\\!\\!11\\times$ ($0.00\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-0.0\\pm 0.5$} & {$4.0\\times$ ($0.04\\!\\times\\!\\!\\!\\!\\!\\times$)} \\\\\n \\rowcolor[HTML]{EFEFEF} \n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{\\color[HTML]{333333} \\textbf{FxPQ + GZip}}} & {\\color[HTML]{333333} {$-0.1\\pm 0.1$}} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{\\color[HTML]{333333} {$14\\times$ ($0.82\\!\\times\\!\\!\\!\\!\\!\\times$)}}} & {\\color[HTML]{333333} {$+0.6\\pm 0.2$}} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{\\color[HTML]{333333} {$\\!\\!\\!\\!\\!1557\\times$ ($0.55\\!\\times\\!\\!\\!\\!\\!\\times$)}}} & {\\color[HTML]{333333} {$-0.0\\pm 0.6$}} & {\\color[HTML]{333333} {$71\\times$ ($0.79\\!\\times\\!\\!\\!\\!\\!\\times$)}} \\\\\n \\multicolumn{1}{l|}{\\textbf{UVeQFed}} & {$-0.5\\pm 0.2$} & \\multicolumn{1}{l|}{{$0.6\\times$ ($0.03\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-2.8\\pm 0.5$} & \\multicolumn{1}{l|}{{$\\!\\!\\!\\!\\!12\\times$ ($0.00\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$+0.0\\pm 0.2$} & {$15\\times$ ($0.16\\!\\times\\!\\!\\!\\!\\!\\times$)} \\\\ \\hline\n \\rowcolor[HTML]{EFEFEF} \n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{DAdaQuant}} & {$-0.2\\pm 0.4$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$\\bm{48\\times}$ ($\\bm{2.81\\!\\times\\!\\!\\!\\!\\!\\times}$)}} & {$+0.7\\pm 0.1$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$\\!\\!\\!\\!\\!\\bm{4772\\times}$ ($\\bm{1.70\\!\\times\\!\\!\\!\\!\\!\\times$})}} & {$-0.1\\pm 0.4$} & {$\\bm{108\\times}$ ($\\bm{1.19\\!\\times\\!\\!\\!\\!\\!\\times}$)} \\\\\n \\multicolumn{1}{l|}{\\textbf{DAdaQuant$_{\\text{time}}$}} & {$-0.1\\pm 0.5$} & \\multicolumn{1}{l|}{{$37\\times$ ($2.16\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$\\bm{+0.8\\pm 0.2}$} & \\multicolumn{1}{l|}{{$\\!\\!\\!\\!\\!4518\\times$ ($1.61\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-0.1\\pm 0.6$} & {$93\\times$ ($1.03\\!\\times\\!\\!\\!\\!\\!\\times$)} \\\\\n \\rowcolor[HTML]{EFEFEF} \n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{DAdaQuant$_{\\text{clients}}$}} & {$+0.0\\pm 0.3$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$26\\times$ ($1.51\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$+0.7\\pm 0.4$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$\\!\\!\\!\\!\\!3017\\times$ ($1.07\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$\\bm{+0.1\\pm 0.6}$} & {$105\\times$ ($1.16\\!\\times\\!\\!\\!\\!\\!\\times$)} \\\\\n & & & & & & \\\\\n \\multicolumn{1}{l|}{} & \\multicolumn{2}{c|}{\\textbf{Shakespeare}} & \\multicolumn{2}{c}{\\textbf{Celeba}} & & \\\\ \\hhline{-----}\n \n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{Uncompressed}} & \\cellcolor[HTML]{EFEFEF}{$\\bm{49.9\\pm 0.3}$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$267.0$\\,MB}} & \\cellcolor[HTML]{EFEFEF}{$90.4\\pm 0.0$} & \\cellcolor[HTML]{EFEFEF}{$12.6$\\,GB} & & \\\\\n \\multicolumn{1}{l|}{\\textbf{Federated QSGD}} & {$-0.5\\pm 0.6$} & \\multicolumn{1}{l|}{{$9.5\\times$}} & {$-0.1\\pm 0.1$} & {$648\\times$} & & \\\\\n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{FP8}} & \\cellcolor[HTML]{EFEFEF}{$-0.2\\pm 0.4$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$4.0\\times$ ($0.42\\!\\times\\!\\!\\!\\!\\!\\times$)}} & \\cellcolor[HTML]{EFEFEF}{$\\bm{+0.0\\pm 0.1}$} & \\cellcolor[HTML]{EFEFEF}{$4.0\\times$ ($0.01\\!\\times\\!\\!\\!\\!\\!\\times$)} & & \\\\\n \\multicolumn{1}{l|}{\\textbf{FedPAQ (FxPQ)}} & {$-0.5\\pm 0.6$} & \\multicolumn{1}{l|}{{$3.2\\times$ ($0.34\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-0.1\\pm 0.1$} & {$6.4\\times$ ($0.01\\!\\times\\!\\!\\!\\!\\!\\times$)} & & \\\\\n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{FxPQ + GZip}} & \\cellcolor[HTML]{EFEFEF}{$-0.5\\pm 0.6$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$9.3\\times$ ($0.97\\!\\times\\!\\!\\!\\!\\!\\times$)}} & \\cellcolor[HTML]{EFEFEF}{$-0.1\\pm 0.2$} & \\cellcolor[HTML]{EFEFEF}{$494\\times$ ($0.76\\!\\times\\!\\!\\!\\!\\!\\times$)} & & \\\\\n \\multicolumn{1}{l|}{\\textbf{UVeQFed}} & {$-0.0\\pm 0.4$} & \\multicolumn{1}{l|}{{$7.9\\times$ ($0.83\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-0.4\\pm 0.3$} & {$31\\times$ ($0.05\\!\\times\\!\\!\\!\\!\\!\\times$)} & & \\\\ \\hhline{-----}\n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{DAdaQuant}} & \\cellcolor[HTML]{EFEFEF}{$-0.6\\pm 0.5$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$\\bm{21\\times}$ ($\\bm{2.21\\!\\times\\!\\!\\!\\!\\!\\times}$)}} & \\cellcolor[HTML]{EFEFEF}{$-0.1\\pm 0.1$} & \\cellcolor[HTML]{EFEFEF}{$\\bm{775\\times}$ ($\\bm{1.20\\!\\times\\!\\!\\!\\!\\!\\times}$)} & & \\\\\n \\multicolumn{1}{l|}{\\textbf{DAdaQuant$_{\\text{time}}$}} & {$-0.5\\pm 0.5$} & \\multicolumn{1}{l|}{{$12\\times$ ($1.29\\!\\times\\!\\!\\!\\!\\!\\times$)}} & {$-0.1\\pm 0.2$} & {$716\\times$ ($1.10\\!\\times\\!\\!\\!\\!\\!\\times$)} & & \\\\\n \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}\\textbf{DAdaQuant$_{\\text{clients}}$}} & \\cellcolor[HTML]{EFEFEF}{$-0.4\\pm 0.5$} & \\multicolumn{1}{l|}{\\cellcolor[HTML]{EFEFEF}{$16\\times$ ($1.67\\!\\times\\!\\!\\!\\!\\!\\times$)}} & \\cellcolor[HTML]{EFEFEF}{$-0.1\\pm 0.0$} & \\cellcolor[HTML]{EFEFEF}{$700\\times$ ($1.08\\!\\times\\!\\!\\!\\!\\!\\times$)} & & \n \\end{tabular}\n \n \\caption{Top-1 test accuracies and total client$\\rightarrow$server communication of all baselines, DAdaQuant,\n DAdaQuant$_\\textrm{time}$ and DAdaQuant$_\\textrm{clients}$. Entry $x \\pm y\\,\\,\\,\\, p\\!\\times (q\\!\\times\\!\\!\\!\\!\\times)$ denotes an accuracy difference of x\\% w.r.t. the uncompressed\n accuracy with a standard deviation of y\\%, a compression factor of $p$ w.r.t. the uncompressed\n communication and a compression factor of $q$ w.r.t. Federated QSGD.\n \n }\n \\label{tab:results}\n\\end{table}\n\n\\paragraph{Baselines}\n\n\\Cref{tab:results} shows that the accuracy of most experiments lies within\nthe margin of error of the uncompressed experiments. This reiterates the\nviability of quantization-based compression algorithms for communication\nreduction in FL.\nFor all experiments, Federated QSGD achieves a significantly higher\ncompression factor than the other baselines. The authors of FedPAQ and\nUVeQFed also compare their methods against QSGD and report them as superior.\nHowever, FedPAQ is compared against ``unfederated'' QSGD that communicates\ngradients after each local training step and UVeQFed is \ncompared against QSGD without its lossless compression stages.\n\n\\paragraph{Time-adaptive quantization} The purely time-adaptive version of\nDAdaQuant, DAdaQuant$_\\textrm{time}$, universally outperforms Federated QSGD\nand the other baselines in \\Cref{tab:results}, achieving comparable accuracies\nwhile lowering communication costs. DAdaQuant$_\\textrm{time}$\nperforms particularly well on \\dataset{Synthetic} and \\dataset{FEMNIST},\nwhere it starts from the lowest possible quantization level $q=1$. However,\nbinary time-adaptive quantization still measurably improves over QSGD for\n\\dataset{Sent140}, \\dataset{Shakespeare} and \\dataset{Celeba}.\n\n\\Cref{fig:adaquantfl} provides empirical evidence that AdaQuantFL's communication scales linearly\nwith the number of clients. As a result, AdaQuantFL is prohibitively expensive for datasets with\nthousands of clients such as \\dataset{Celeba} and \\dataset{Sent140}. DAdaQuant does not face this\nproblem because its communication is unaffected by the number of clients. \n\n\\begin{figure}\n \\vspace*{-0.7cm}\n \\begin{center} \n \\hspace*{-0.5cm}\n \\scalebox{0.75}{\n \\input{anc\/pareto_paper_full.pgf}}\n \\end{center}\n \\vspace*{-0.3cm} \n \\caption{Communication-accuracy trade-off curves for training on\n \\dataset{FEMNIST} with Federated QSGD and DAdaQuant. We plot the average highest\n accuracies achieved up to any given amount of client$\\rightarrow$server communication.\n \\Cref{sec:paretofull} shows curves for all datasets, with similar results.}\n \\label{fig:pareto}\n\\end{figure}\n\n\\paragraph{Client-adaptive quantization}\nThe purely time-adaptive version of DAdaQuant, DAdaQuant$_\\textrm{clients}$,\nalso universally outperforms Federated QSGD and the other baselines in\n\\Cref{tab:results}, achieving similar accuracies while lowering\ncommunication costs. Unsurprisingly, the performance of\nDAdaQuant$_\\textrm{clients}$ is correlated with the coefficient of variation\n$c_v = \\frac{\\sigma}{\\mu}$ of the numbers of samples in the local datasets\nwith mean $\\mu$ and standard deviation $\\sigma$: \\dataset{Synthetic}\n($c_v=3.3$) and \\dataset{Shakespeare} ($c_v=1.7$) achieve significantly\nhigher compression factors than \\dataset{Sent140} ($c_v=0.3$),\n\\dataset{FEMNIST} ($c_v=0.4$) and \\dataset{Celeba} ($c_v=0.3$).\n\n\\paragraph{DAdaQuant}\nDAdaQuant outperforms DAdaQuant$_\\textrm{time}$ and\nDAdaQuant$_\\textrm{clients}$ in communication while achieving similar\naccuracies. The compression factors of DAdaQuant are roughly multiplicative\nin those of DAdaQuant$_\\textrm{clients}$ and DAdaQuant$_\\textrm{time}$. This\ndemonstrates that we can effectively combine time- and client-adaptive\nquantization for maximal communication savings.\n\n\n\\paragraph{Pareto optimality} \\Cref{fig:pareto} shows that DAdaQuant\nachieves a higher accuracy than the strongest baseline, Federated QSGD, for\nany fixed accuracy. This means that DAdaQuant is Pareto optimal for the\ndatasets we have explored.\n \n\\section{Conclusion}\n\nWe introduced DAdaQuant as a computationally efficient and robust algorithm\nto boost the performance of quantization-based FL compression algorithms. We\nshowed intuitively and mathematically how DAdaQuant's dynamic adjustment of\nthe quantization level across time and clients minimize\nclient$\\rightarrow$server communication while maintaining convergence speed.\nOur experiments establish DAdaQuant as nearly universally superior over\nstatic quantizers, achieving state-of-the-art compression factors when\napplied to Federated QSGD. The communication savings of DAdaQuant\neffectively lower FL bandwidth usage, energy consumption and training time.\nFuture work may apply and adapt DAdaQuant to new quantizers, further pushing\nthe state of the art in FL uplink compression.\n \n\\section{Reproducibility Statement}\n\nOur submission includes a repository with the source code for DAdaQuant and\nfor the experiments presented in this paper. All the datasets used in our\nexperiments are publicly available. Any post-processing steps of the\ndatasets are described in \\Cref{sec:models_datasets_detailed}. To facilitate\nthe reproduction of our results, we have bundled all our source code,\ndependencies and datasets into a Docker image. The repository submitted with\nthis paper contains instructions on how to use this Docker image and\nreproduce all plots and tables in this paper.\n\n\\section{Ethics Statement}\n\nFL trains models on private client datasets in a privacy-preserving manner.\nHowever, FL does not completely eliminate privacy concerns, because the\ntransmitted model updates and the learned model parameters may expose the\nprivate client data from which they are derived. Our work does not directly\ntarget privacy concerns in FL. With that said, it is worth noting that\nDAdaQuant does not expose any client data that is not already exposed\nthrough standard FL training algorithms. In fact, DAdaQuant reduces the\namount of exposed data through lossy compression of the model\nupdates. We therefore believe that DAdaQuant is free of ethical\ncomplications. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nGlobal symmetries are instrumental in organizing our understanding of phases of matter. The celebrated Landau paradiagm classifies phases according to broken symmetries, which also determines the universality classes of transitions between phases.\nSymmetry principles become even more powerful from the point of view of long wavelength, low-energy physics, as the renormalization group fixed points (i.e. IR) often embody more symmetries than the microscopic lattice model (i.e. UV), which is the phenomenon of emergent symmetry~\\cite{Moessner2001,Isakov2003,Senthil2004,YYHe2016,NvsenMa2019}. A common example is the emergence of continuous space-time symmetries in the field-theoretical description of a continuous phase transition~\\cite{altland2010condensed}. It is even plausible that a critical point is determined up to finite choices by its full emergent symmetry, which is the basic philosophy (or educated guess) behind the conformal bootstrap program~\\cite{Poland2019}.\n\nModern developments in quantum many-body physics have significantly broadened the scope of quantum phases beyond the Landau classification~\\cite{Wen2019}. For these exotic phases, more general notions of global symmetry are called for to completely characterize the phases and the associated phase transitions. Intuitively, these ``beyond Landau'' phases do not have local order parameters. Instead, non-local observables are often needed to characterize them. For a well-known example, confined and deconfined phases of a gauge theory are distinguished by the behavior of the expectation value of Wilson loop operators~\\cite{fradkin2013field,Gregor2011}. To incorporate such extended observables into the symmetry framework, higher-form symmetries~\\cite{Nussinov2006, Nussinov2009, Gaiotto_2015}, and more generally algebraic symmetries~\\cite{ji2019categorical, kong2020algebraic} have been introduced. These are symmetries whose charged objects are spatially extended, e.g. strings and membranes. In other words, their symmetry transformations only act nontrivially on extended objects. Most notably, spontaneous breaking of such higher symmetries can lead to highly entangled phases, such as topological order~\\cite{Gaiotto_2015}. Therefore, even though topologically ordered phases are often said to be beyond the Landau paradiagm, they can actually be understood within a similar conceptual framework once higher symmetries are included. In addition, just as the usual global symmetries, higher-form symmetries can have quantum anomalies~\\cite{Gaiotto_2015}, which lead to strong non-perturbative constraints on low-energy dynamics~\\cite{Gaiotto2017}.\n\nIn this work, we make use of the prototypical continuous quantum phase transition, the Ising transition, to elucidate the functionality of the higher-form symmetry. The motivation to re-examine the well-understood Ising transition is the following: in addition to the defining 0-form $\\mathbb{Z}_2$ symmetry, the topological requirement that $\\mathbb{Z}_2$ domain walls must be closed (in the absence of spatial boundary) can be equivalently formulated as having an unbreakable $\\mathbb{Z}_2$ $(D-1)$-form symmetry, where $D$ is the spatial dimension. Gapped phase on either side of the transition spontaneously breaks one and only one of the two symmetries. Therefore to correctly determine the full emergent internal symmetry in the Ising CFT, the $\\mathbb{Z}_2$ higher-form symmetry should be taken into account. For $D=2$, the $1$-form symmetry manifests more clearly in the dual formulation~\\cite{Wegner1971}, namely as the confinement-deconfinement transition of a $\\mathbb{Z}_2$ gauge theory, which will shed light on higher-form symmetry breaking transitions in a concrete setting.\n\nA basic question about a global symmetry is whether it is broken spontaneously or not in the ground state. For clarity, let us focus on the $D=2$ case. It is well-known that the Ising symmetric, or ``quantum disordered'' phase, spontaneously breaks the higher-form symmetry, and the opposite in the Ising symmetry-breaking phase. The fate at the critical point remains unclear to date. To diagonose higher-form symmetry breaking, we compute the ground state expectation value of the ``order parameter'' for the higher-form symmetry -- commonly known as the disorder operator in literature~\\cite{KadanoffPRB1971,Fradkin2016,XCWu2020,YCWang2021,XCWu2021}, which creates a domain wall in the Ising system. Spontaneous breaking of the $\\mathbb{Z}_2$ $1$-form symmetry is signified by the perimeter law for the disorder operator. In the dual formulation, the corresponding object is the Wilson loop operator. Through large-scale QMC simulations, we find numerically that at the transition, the disorder operator defined on a rectangular region scales as $l^s e^{-a_1l}$, where $l$ is the perimeter of the region, and $s>0$ is a universal constant. We thus conclude that the 1-form symmetry is spontaneously broken at the (2+1)d Ising transition, and it remains so in the disordered phase of the model. This is in stark contrast with the $D=1$ case, where the disorder operator has a power-law decay. \n\nTo corroborate the numerical results, we consider generally disorder operator corresponding to a 0-form $\\mathbb{Z}_2$ symmetry in a free scalar theory in $D$ dimensions, which is a stable fixed point for $D\\geq 3$. We show that for the kind of $\\mathbb{Z}_2$ symmetry in this case, the disorder operator can be related to the 2nd Renyi entropy. Therefore, the disorder operator also obeys a ``perimeter'' (i.e. volume of the boundary) scaling, with possibly multiplicative power-law correction. Whether the higher-form symmetry is broken or not is determined by the subleading power-law corrections. We also discuss other free theories, such as a Fermi liquid, where the decay of the disorder operator is in between the ``perimeter'' and the ``area'' laws, and therefore no higher-form symmetry breaking.\n\nThe rest of the paper is organized as follows. In Sec.~\\ref{sec:ii} we review higher-form symmetry and its spontaneous breaking, and its relevancy in conventional phases. We also consider higher-form symmetry breaking in free and interacting conformal field theories. In Sec.~\\ref{sec:iii} we specialize to the setting of quantum Ising model in (2+1)d and define the disorder operator. Sec.~\\ref{sec:iv} presents the main numerical results from quantum Monte Carlo simulations, which reveal the key evidence of the 1-form symmetry breaking at the $(2+1)$d Ising transition. Sec.~\\ref{sec:v} outlines a few immediate directions about the higher-form symmetry breaking and their measurements in unbiased numerical treatments in other quantum many-body systems.\n\n\\section{Generalized global symmetry}\n\\label{sec:ii}\nConsider a quantum many-body system in $D$ spatial dimensions. Global symmetries are unitary transformations which commute with the Hamiltonian. Typically the symmetry transformation is defined over the entire system, and charges of the global symmetry are carried by particle-like objects.\n\nAn important generalization of global symmetry is the higher-form symmetry~\\cite{Gaiotto_2015}. For an integer $p\\geq 0$, $p$-form symmetry transformations act nontrivially on $p$-dimensional objects. In other words, ``charges'' of $p$-form symmetry are carried by extended objects. In this language, the usual global symmetry is 0-form as the particle-like object is of 0-dimension. $p$-form symmetry transformations themselves are unitary operators supported on each codimension-$p$ (i.e. spatial dimension $(D-p)$) closed submanifold $M_{D-p}$. In particular, it means that there are infinitely many symmetry transformations in the thermodynamic limit. In this work we will only consider discrete, Abelian higher-form symmetry, so for each submanifold $M_{D-p}$ the associated unitary operators form a finite Abelian group $G$.\nPhysically, higher-form symmetry means that the certain $p$-dimensional objects are charged under the group $G$, and the quantum numbers they carry constrain the processes of creation, annihilation and splitting etc. In particular, these extended objects are ``unbreakable'', i.e. they are always closed and can not end on $(p-1)$-dimensional objects.\n\nFor a concrete example, let us consider (2+1)$d$ $\\mathbb{Z}_2$ gauge theory definend on a square lattice. Each edge of the lattice is associated with a $\\mathbb{Z}_2$ gauge field (i.e. a qubit), subject to the Gauss's law at each site $v$:\n\\begin{equation}\n\t\\prod_{e\\ni v}\\tau_e^x=1.\n\t\\label{eqn:gauss}\n\\end{equation}\nHere $e$ runs over edges ending on $v$.\n\nThe divergence-free condition implies that there are no electric charges in the gauge theory. In other words, all $\\mathbb{Z}_2$ electric field lines must form loops. An electric loop can be created by applying the following operator along any closed path $\\gamma$ on the lattice:\n\\begin{equation}\n\tW_e(\\gamma)=\\prod_{e\\in \\gamma}\\tau_e^z.\n\t\\label{}\n\\end{equation}\nThe corresponding $\\mathbb{Z}_2$ 1-form symmetry operator is defined as\n\\begin{equation}\n\tW_m(\\gamma^\\star)=\\prod_{e\\perp \\gamma^\\star} \\tau_e^x\n\t\\label{eqn:Wm}\n\\end{equation}\nfor any closed path $\\gamma^\\star$ on the dual lattice. Here the subscript $m$ in $W_m$ indicates that this is actually the string operator for $\\mathbb{Z}_2$ flux excitations. In field theory parlance, $W_e$ is the Wilson operator of the $\\mathbb{Z}_2$ gauge theory, and $W_m$ is the corresponding Gukov-Witten operator~\\cite{gukov2008rigid}. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figure\/braiding}\n\t\\caption{(a) 0-form symmetry charge is a point-like object, measured by the symmetry transformation defined on the entire system (i.e. at a fixed time slice) (b) 1-form symmetry charge is a loop (the solid line), measured by the symmetry transformation defined on a loop as well when the two are linked. }\n\t\\label{fig:braiding}\n\\end{figure}\n\nWe notice that the $W_m(\\gamma^\\star)$ operator is in fact the product of Gauss's law term $\\prod_{v\\in e}\\tau_e^x$ for all $v$ in the region enclosed by $\\gamma^\\star$. In other words, the smallest possible $\\gamma^\\star$ is a loop around one vertex $v$, and the fact tht $W_m(\\gamma^\\star)$ is conserved by the dynamics means that the gauge charge at site $v$ must be conserved (mod 2) as well. Therefore, the $\\mathbb{Z}_2$ gauge theory with electric 1-form symmetry is one with completely static charges, including the case with no charges at all. For applications in relativistic quantum field theories, it is usually further required that the 1-form symmetry transformation is ``topological'', i.e. not affected by local deformation of the loop $\\gamma^\\star$, which is equivalent to the absence of gauge charge as given in Eq. \\eqref{eqn:gauss}.\n\nIt is instructive to consider how the 1-form charge of an electric loop can be measured. This is most clearly done in space-time: to measure a $p$-dimensional charge, one ``wraps'' around the charge by a $(D-p)$-dimensional symmetry operator. Appying the symmetry transformation is equivalent to shrinking the symmetry operator, and in $(D+1)$ spacetime, because of the linking the two must collide, and the non-commutativity (e.g. between $W_e$ and $W_m$) measures the charge value. We illustrate the process for $p=0$ (Fig. \\ref{fig:braiding}(a)) and $p=1$ (Fig. \\ref{fig:braiding}(b)), in three-dimensional space-time.\n\nNow consider the following Hamiltonian of Ising gauge theory:\n\\begin{equation}\n\tH=-J\\sum_e\\tau_e^x - K\\sum_p \\prod_{e\\in \\partial p}\\tau_e^z,\n\t\\label{}\n\\end{equation}\nwhere $J,K > 0$. When $J\\ll K$, the ground state is in the deconfined phase, which can be viewed as an equal-weight superposition of all closed $\\mathbb{Z}_2$ electric loops. In this phase, the $\\mathbb{Z}_2$ 1-form symmetry is spontaneously broken. When $J\\gg K$, the ground state is a product state with $\\tau_e^x=1$ everywhere, and the 1-form symmetry is preserved. This is the confined phase. Similar to the usual boson condensation, the expectation value of the electric loop creation operator $W_e(\\gamma)$ can be used to characterize the 1-form symmetry breaking phase, which obeys perimeter law in the deconfined phase.\n\nThis example shows that higher-forms symmetry naturally arises in gauge theories. In condensed matter applications, gauge theories are usually emergent~\\cite{Senthil2004,NvsenMa2018}, which means that dynamical gauge charges are inevitably present and the electric 1-form symmetry is explicitly broken. Even under such circumstances, at energy scales well below the electric charge gap, the theory still has an emergent 1-form symmetry~\\cite{WenPRB2019}.\n\nLet us now discuss more generally the spontaneous breaking of higher-form symmetry~\\cite{Gaiotto_2015, Lake2018, Hofman2018}. We will assume that the symmetry group is discrete. For a $p$-form symmetry, a charged object is created by an extended operator $W(C)$ defined on a $p$-dimensional manifold $C$. When the symmetry is unbroken, we have\n\\begin{equation}\n\t\\langle W(C)\\rangle \\sim e^{-t_{p+1} \\mathrm{Area}(C)},\n\t\\label{eq:eq5}\n\\end{equation}\nwhere $\\mathrm{Area}(C)$ is the volume of a minimal $(p+1)$-dimensional manifold whose boundary is $C$. $t_{p+1}$ can be understood as the ``tension'' of the $(p+1)$-dimensional manifold. This generalizes the exponential decay of charged local operator for the 0-form case. On the other hand, when the symmetry is spontaneously broken,\n\\begin{equation}\n\t\\langle W(C)\\rangle \\sim e^{-t_p \\mathrm{Perimeter}(C)},\n\t\\label{}\n\\end{equation}\nwhere $\\mathrm{Perimeter}(C)$ denotes the ``volume'' of $C$ itself. Importantly the expectation value only depends locally on $C$, which is the analog of the factorization of the correlation function of local order parameter $\\langle O(x)O^\\dag(y)\\rangle \\approx \\langle O(x)\\rangle \\langle O^\\dag(y)\\rangle$ for $0$-form symmetry. One can then redefine the operator $W(C)$ to remove the perimeter scaling and in that case $\\langle W(C)\\rangle$ would approach a constant in the limit of large $C$~\\cite{HastingsPRB2005}. At critical point, however, subleading corrections become important, which will be examined below. \n\nThe $\\mathbb{Z}_2$ gauge theory is famously dual to a quantum Ising model~\\cite{Kogut1979}. In fact, more generally, there is a duality transformation which relates a system with global $\\mathbb{Z}_2$ 0-form symmetry (in the $\\mathbb{Z}_2$ even sector) to one with global $\\mathbb{Z}_2$ $(D-1)$-form symmetry, a generalization of the Kramers-Wannier duality in (1+1)d. \n\nLet us now review the duality in (2+1)d. The dual Ising spins are defined on plaquettes, whose centers form the dual lattice. For a given edge $e$ of the original lattice, denote the two adjacent plaquettes by $p$ and $q$, as shown in the figure below:\n\\begin{center}\n\\begin{tikzpicture}\n\t\\draw[thick] (-1.3, 0) -- (1.3, 0);\n\t\\draw[thick] (-1.3, 1) -- (1.3, 1);\n\t\\draw[thick] (-1.3,-1) -- (1.3,-1);\n\t\\draw[thick, dashed] (-1.3,-0.5) -- (1.3,-0.5);\n\t\\draw[thick, dashed] (-1.3,0.5) -- (1.3,0.5);\n\t\\draw[thick] (-1, -1.3) -- (-1, 1.3);\n\t\\draw[thick] (0, -1.3) -- (0, 1.3);\n\t\\draw[thick] (1, -1.3) -- (1, 1.3);\n\t\\draw[thick, dashed] (0.5, -1.3) -- (0.5, 1.3);\n\t\\draw[thick, dashed] (-0.5, -1.3) -- (-0.5, 1.3);\n\t\\filldraw (-0.5, -0.5) circle (1.5pt);\n\t\\filldraw (0.5, -0.5) circle (1.5pt);\n\t\\filldraw (-0.5, 0.5) circle (1.5pt);\n\t\\filldraw (0.5, 0.5) circle (1.5pt);\n\t\\node at (-0.35, 0.3) {$p$};\n\t\\node at (0.61, 0.3) {$q$};\n\t\\node at (0.13, 0.7) {$e$};\n\\end{tikzpicture}\n\\end{center}\n\nThe duality map is defined as follows:\n\\begin{equation}\n\t\\sigma_{p}^z\\sigma_{q}^z \\leftrightarrow \\tau_e^x, \\sigma^x_{p} \\leftrightarrow \\prod_{e\\in \\partial p}\\tau_e^z.\n\t\\label{}\n\\end{equation}\nNote that the expression automatically ensures $\\prod_p\\sigma^x_p=1$ in a closed system, so the dual spin system has a $\\mathbb{Z}_2$ 0-form symmetry generated by $S=\\prod_p \\sigma_p^x$, and the map can only be done in the $\\mathbb{Z}_2$ even sector with $S=1$~\\footnote{In a sense the $\\mathbb{Z}_2$ symmetry is gauged. In fact one way to derive the duality is to first gauge the $\\mathbb{Z}_2$ symmetry and then perform gauge transformations to eliminate the Ising matter.}. Conversely, the mapping also implies $\\prod_{v\\in e}\\tau_x^e=1$, and in fact $W_m(\\gamma^\\star)=1$ for any $\\gamma^*$, i.e. the $\\mathbb{Z}_2$ 1-form symmetry is strictly enforced.\n\nIn the dual model, the electric field line of the $\\mathbb{Z}_2$ gauge theory becomes the domain walls separating regions with opposite Ising magnetizations. Therefore, a Wilson loop $W_e(\\gamma)$ maps to\n\\begin{equation}\n\tX_M=\\prod_{p\\in M} \\sigma_p^x,\n\t\\label{eqn:Xdef}\n\\end{equation}\nwhere $\\partial M=\\gamma$, i.e. $M$ is the region enclosed by $\\gamma$. Physically $X_M$ flips all the Ising spins in the region $M$, thus creating a domain wall along the boundary $\\gamma$. It is called the disorder operator for the Ising system, which will be the focus of our study below.\n\nUnder the duality map, the Hamiltonian becomes\n\\begin{equation}\n\tH=-J\\sum_{\\langle pq\\rangle}\\sigma_p^z\\sigma_q^z - K\\sum_p \\sigma_p^x.\n\t\\label{eqn:TFI}\n\\end{equation}\n\nThe phases of the gauge theory can be readily understood in the dual representation. For $K\\gg J$, the $\\mathbb{Z}_2$ gauge theory is in the deconfined phase, which means that the ground state contains arbitrarily large electric loops. For the dual Ising model, the ground state is disordered, with all $\\sigma_p^x=1$. If we work in the $\\sigma^z$ eigenbasis (which is natural to discuss symmetry breaking), the ground state wavefunction is given by\n\\begin{equation}\n\t\\ket{\\psi_{K=\\infty}}\\propto \\prod_p \\frac{1+\\sigma_p^x}{2}\\ket{\\uparrow\\uparrow\\cdots\\uparrow}.\n\t\\label{}\n\\end{equation}\nNamely we pick any basis state and apply the ground state projector. Expanding out the projector, one can see that the wavefunction is an equal superposition of all domain wall configurations, i.e. a condensation of domain walls. Since the domain walls carry $\\mathbb{Z}_2$ 1-form charges, the condensation breaks the 1-form symmetry spontaneously, much like the Bose condensation spontaneously breaks the conservation of particle numbers\n\nIn the other limit $K\\ll J$, the gauge theory is confined. Correspondingly, the dual Ising model is in the ferromagnetically ordered phase: there are two degenerate ground states $\\ket{\\uparrow\\cdots\\uparrow}$ and $\\ket{\\downarrow\\cdots\\downarrow}$. There are no domain walls at all in the limit $K\\rightarrow 0$. When a small but finite $K\/J$ is turned on, quantum fluctuations create domain walls on top of the fully polarized ground states, but these domain walls are small and sparse.\n\n\n\n\\subsection{Non-invertible anomaly and gapless states}\nA notable feature of the duality map is that on either side, only one of two symmetries, the $\\mathbb{Z}_2$ 0-form and the $\\mathbb{Z}_2$ 1-form symmetries, is faithfully represented (in the sense that the symmetry transformation is implemented by a nontrivial operator, even though the duality is supposed to work only in the symmetric sector). The other symmetry transformation is mapped to the identity at the operator level. Physically, only one of them is an explicit global symmetry, while the other one appears as a global constraint (e.g. on the Ising side, domain walls of the 0-form global symmetry are codimension-1 closed manifolds, which is the manifestation that they are charged under a $(D-1)$-form symmetry). \n\nA closely related fact is that the ordered phase for one symmetry is necessarily the disordered phase of the other, and any non-degenerate gapped phase must break one and only one of the two symmetries. This has been proven rigorously in one spatial dimension~\\cite{Levin2019}, and is believed to hold in general dimensions as well.\n\nIt is clear from these results that these two symmetries can not be considered as completely independent. Recently, Ref. [\\onlinecite{JiPRR2019}] proposed that the precise relation between the two dual symmetries is captured by the notion of a non-invertible quantum anomaly. Intuively, the meaning of the non-invertible anomaly in the context of the $\\mathbb{Z}_2$ Ising model can be understood as follows: the charge of the $\\mathbb{Z}_2$ 0-form symmetry is an Ising spin flip, while the charge of the $\\mathbb{Z}_2$ 1-form symmetry is an Ising domain wall. These two objects have nontrivial mutual ``braiding'', in the sense that when an Ising charge is moved across a domain wall, it picks up a minus sign due to the Ising symmetry transformation applied to one side of the domain wall. In other words, the charge of the 1-form symmetry is actually a flux loop of the 0-form symmetry. Ref. [\\onlinecite{JiPRR2019}] suggested that two symmetries whose charged objects braid nontrivially with each other can not be realized faithfully in a local Hilbert space. If locality is insisted, then the only option is to realize the $D$ spatial dimensional system as the boundary of a $\\mathbb{Z}_2$ toric code model in $(D+1)$ spatial dimension. In this case, the charged objects are in fact bulk topological excitations brought to the boundary. The nontrivial braiding statistics between the two kinds of charges reflects the topological order in the bulk. Such an anomaly is fundamentally different from more familiar 't Hooft anomaly realized on the boundary of a symmetry-protected topological phase (which is an invertible state). We refer to Ref. [\\onlinecite{JiPRR2019}] for more thorough discussions of the non-invertible anomaly.\n\n\n\n\n Since any gapped state must break one of the two symmetries, it is a very natural question to ask whether there are gapless states that preserve both symmetries. An obvious candidate for such a gapless state is the symmetry-breaking continuous transition. At the transition, the two-point correlation function of the Ising order parameter decays algebraically with the distance, implying that the $\\mathbb{Z}_2$ 0-form symmetry is indeed unbroken. For the dual $(D-1)$-form symmetry, the Kramers-Wannier duality maps the disorder operator, which is a string operator in the Ising basis, to the two-point correlator of the Ising order parameter. Therefore the expectation value of the disorder operator also exhibits power-law correlation, and the dual $0$-form symmetry is preserved. Therefore the Ising conformal field theory in (1+1)d indeed provides an example of symmetric gapless state with non-invertible anomaly~\\cite{JiPRR2019}. But for the case of $D>1$, the situation is far from clear and that is what we will address in this paper. First we analyze the expectation value of the disorder operator in a free field theory.\n\n\\subsection{Scaling of disorder operator in field theory}\nWe now discuss the scaling form of the disorder operator at or near the critical point from a field-theoretical point of view. The natural starting point is the Gaussian fixed point, i.e. a free scalar theory, described by the following Hamiltonian\n\\begin{equation}\n\t{H}[\\phi]=\\int\\mathrm{d}^D\\mathbf{r}\\,\\left[\\frac{\\pi^2}{2}+\\frac{1}{2}(\\nabla \\phi)^2\\right].\n\t\\label{eqn:freeboson1}\n\\end{equation}\nThe real scalar $\\phi$ can be thought of as the coarse grained Ising order parameter, and $\\pi$ is the conjugate momentum of the real scalar $\\phi$. The $\\mathbb{Z}_2$ symmetry acts as $\\phi\\rightarrow -\\phi$. The disorder operator $X_M$ is basically defined as the continuum version of Eq. \\eqref{eqn:Xdef}, where the $\\mathbb{Z}_2$ symmetry is applied to a finite region $M$.\n\nInterestingly, for the free theory the expectation value of the disorder operator can be related to another well-studied quantity, the 2nd Renyi entanglement entropy $S_2$. More precisely, for a region $M$, we have\n\\begin{equation}\n\te^{-S_2(M)}=\\langle X_M\\rangle.\n\t\\label{eqn:S2=X}\n\\end{equation}\nHere $S_2(M)$ is the 2nd Renyi entropy of the region $M$.\n\nTo see why this is the case, recall that the 2nd Renyi entropy $S_2$ for a region $M$ of a quantum state $\\ket{\\Psi}$ is given by\n\\begin{equation}\n\te^{-S_2(M)}=\\Tr \\rho_M^2,\n\t\\label{}\n\\end{equation}\nwhere $\\rho_M$ is the reduced density matrix for the region $M$, obtained from tracing out the degrees of freedom in the complement $\\ol{M}$: $\\rho_M= \\Tr_{\\ol{M}} \\ket{\\Psi}\\bra{\\Psi}$. In the following we denote the ground wave functional of the state $\\ket{\\Psi}$ by $\\Psi(\\phi)$:\n\\begin{equation}\n\t\\ket{\\Psi}=\\int D\\phi\\, \\Psi(\\phi)\\ket{\\phi}.\n\t\\label{eqn:wfn}\n\\end{equation}\n\nThe Renyi entropy can be calculated with a replica trick, which we now review in the Hamiltonian formalism. Consider two identical copies of the system, in the state $\\ket{\\Psi}\\otimes\\ket{\\Psi}$. In the field theory example, the fields in the two copies are denoted by $\\phi^{(1)}$ and $\\phi^{(2)}$, respectively. We denote the basis state with a given field configuration $\\phi^{(i)}$ in the $i$-th copy by $\\ket{\\phi^{(i)}_M,\\phi^{(i)}_{\\ol{M}}}$, where $\\phi^{(i)}_M$ is the field configuration restricted to $M$ and similarly $\\phi^{(i)}_{\\ol{M}}$ for the complement of $M$. Since the two copies are completely identical, there is a swap symmetry $R$ acting between the two copies $R: \\phi^{(1)}\\leftrightarrow \\phi^{(2)}$. $R_M$ then swaps the field configurations only within the region $M$:\n\\begin{equation}\n\tR_M\\ket{\\phi^{(1)}_M, \\phi^{(1)}_{\\ol{M}}}\\otimes\\ket{\\phi^{(2)}_M, \\phi^{(2)}_{\\ol{M}}}=\n\t\\ket{\\phi^{(2)}_M, \\phi^{(1)}_{\\ol{M}}}\\otimes\\ket{\\phi^{(1)}_M, \\phi^{(2)}_{\\ol{M}}}.\n\t\\label{}\n\\end{equation}\n\nThe expectation of $R_M$ on the replicated ground state $\\ket{\\Psi}\\otimes\\ket{\\Psi}$ is then given by\n\\begin{equation}\n\t\\begin{split}\n\t(\\bra{\\Psi}&\\otimes\\bra{\\Psi})R_M(\\ket{\\Psi}\\otimes\\ket{\\Psi})\\\\\n\t&=\\int \\prod_{i=1,2}D\\phi^{(i)}_M D\\phi^{(i)}_{\\ol{M}}\\,\\Psi(\\phi^{(1)}_M,\\phi^{(1)}_{\\ol{M}})\\Psi^*(\\phi^{(2)}_M,\\phi^{(1)}_{\\ol{M}})\\\\\n\t&\\quad\\quad\\Psi(\\phi^{(2)}_M,\\phi^{(2)}_{\\ol{M}})\\Psi^*(\\phi^{(1)}_M,\\phi^{(2)}_{\\ol{M}})\\\\\n\t&=\\int D\\phi^{(1)}_M D\\phi^{(2)}_{M}\\,\\rho_M(\\phi^{(1)}_M, \\phi^{(2)}_M)\\rho_M(\\phi^{(2)}_M, \\phi^{(1)}_M)\\\\\n\t&=\\Tr \\rho_M^2.\n\t\\end{split}\n\t\\label{}\n\\end{equation}\nTherefore the Renyi entropy is the expectation value of the disorder operator for the replica symmetry.\n\nFor a free theory, we rotate the basis to $\\phi_\\pm = \\frac{1}{\\sqrt{2}}(\\phi^{(1)}\\pm \\phi^{(2)})$. In the new basis, the swap symmetry operator becomes:\n\\begin{equation}\n\tR:\\phi_\\pm \\rightarrow \\pm \\phi_\\pm.\n\t\\label{}\n\\end{equation}\nIt is straightforward to check that the Hamiltonian of the replica takes essentially the same form in the new basis:\n\\begin{equation}\n\tH[\\phi^{(1)}]+H[\\phi^{(2)}]=H[\\phi_+]+H[\\phi_-].\n\t\\label{}\n\\end{equation}\nThe ground state again is factorized: $\\ket{\\Psi}\\otimes\\ket{\\Psi}=\\ket{\\Psi}_+\\otimes\\ket{\\Psi}_-$, where $\\ket{\\Psi}_\\pm$ is the state of the $\\phi_\\pm$ field, with the same wave functional as $\\phi$: $\\braket{\\phi_\\pm |\\Psi}_\\pm = \\Psi(\\phi_\\pm)$ as defined in Eq. \\eqref{eqn:wfn}.\n\nWe can now compute the expectation value of $R_M$:\n\\begin{equation}\n\t(\\bra{\\Psi}_+\\otimes\\bra{\\Psi}_-)R_M(\\ket{\\Psi}_+\\otimes\\ket{\\Psi}_-) =\\braket{X_M}.\n\t\\label{}\n\\end{equation}\nwhere we used the fact that $R$ acts as the identity on $\\phi_+$. For $\\phi_-$, $R_M$ is nothing but the disorder operator $X_M$. \n\nThe 2nd Renyi entropy of a free scalar has been well-studied~\\cite{Casini2006,Casini2009, Dowker2015, ElvangPLB2015, Helmes2016, Bueno2019, Berthiere2018} and we summarize the results below.\n\nIt is important to distinguish the case where the boundary is smooth and those with sharp corners on the boundary.\n\nFirst consider a smooth boundary. For a sphere of radius $R$, in $D=1,2,3$ we have:\n\\begin{equation}\n\tS_2=\n\t\\begin{cases}\n\t\t\\frac{1}{6}\\ln R & D=1\\\\\n\t\ta_1\\frac{R}{\\epsilon} -\\gamma & D=2\\\\\n\t\ta_2\\left(\\frac{R}{\\epsilon}\\right)^2-\\frac{1}{192}\\ln \\frac{R}{\\epsilon} & D=3\n\t\\end{cases}.\n\t\\label{eqn:S2scaling}\n\\end{equation}\nHere $\\epsilon$ is a short-distance cutoff, e.g. the lattice spacing, $a_1, a_2$ non-universal coefficients and $\\gamma$ a universal constant. For a more general smooth entangling boundary, in 2D the same form holds although the constant correction $\\gamma$ depends on shape of the region. In 3D, it is known that the coefficient of the logarithmic divergent part of the Renyi entropy can be determined entirely from the local geometric data (e.g. curvature) of the surface in a general CFT~\\cite{Solodukhin2008, Fursaev2012}. \n\nIf the boundary has sharp corners then there are additional divergent terms in the entropy. The prototypical case is $D=2$ when the entangling region has sharp corners. In that case\n\\begin{equation}\n\tS_2=a_1\\frac{l}{\\epsilon}-s\\ln \\frac{l}{\\epsilon},\n\t\\label{eq:eq16}\n\\end{equation}\nwhere $l$ is the perimeter of the entangling region and $s$ is an universal function that only depends on the opening angles of the corners.\nFor real free scalar, the coefficient of the logarithmic correction is $s\\approx 0.0260$ for a square region (so four $\\pi\/2$ corners, as those in Fig.~\\ref{fig:fig1})~\\cite{Casini2009, Helmes2016}. \n\nQualitatively, it is important that for $D=2,3$ the leading term in $S_2$ always obeys an ``perimeter'' law, i.e. it only depends on the ``area'' (length in 2D) of the entangling boundary. If instead we view $S_2$ as the disorder operator for the $\\mathbb{Z}_2$ replica symmetry, the non-universal, cutoff-dependent perimeter term can be removed by redefining the disorder operator locally along the boundary, and the remaining term is universal. For $D=2$, the subleading term is either a \\emph{negative} constant when the boundary is smooth, or a $\\ln l$ correction with a \\emph{negative} coefficient. So according to the relation Eq.~\\eqref{eqn:S2=X}, the disorder parameter $\\braket{X_M}$, after renormalizing away the perimeter term, does not decrease with the size of $M$, and therefore the corresponding $(D-1)$-form symmetry is spontaneously broken. This is consistent with the fact that the replica symmetry itself must be preserved as there is no coupling between the two copies.\n\nAlthough the free Gaussian theory is unstable against quartic interactions below the upper critical dimension, and the actual critical theory is the interacting Wilson-Fisher fixed point, results from the free theory can still provide useful insights. It is well-known that for $D=1$, for $M$ an interval of length $R$ the disorder operator $\\braket{X_M}\\sim R^{-1\/4}$, the same power-law decay as that of the Ising order parameter due to Kramers-Wannier duality. For $D=2$, we will resort to numerical simulations below to address the question.\n\n Notice that the relation between $\\langle X\\rangle$ and $S_2$ essentially holds for all free theories, including free fermions. For example, the disorder operator associated with the fermion parity symmetry is also equal to $S_2$. Interestingly, for a Fermi liquid, it is well-known that $\\ln \\langle X\\rangle = - S_2\\sim -l^{D-1}\\ln l$~\\cite{Gioev2006, Wolf2006}, where here $l$ is the linear size of the region. This is an example of a gapless state where the $(D-1)$-form symmetry is preserved. Similar results hold for non-interacting bosonic systems with ``Bose surface''~\\cite{LaiPRL2013}, an example of which in 2D is given by the exciton Bose liquid~\\cite{ParamekantiPRB2002, TayPRL2010}:\n \\begin{equation}\n\t H=\\int\\mathrm{d}^2\\mathbf{r}\\,\\left[\\frac{\\pi^2}{2}+\\kappa (\\partial_x\\partial_y \\phi)^2\\right].\n\t \\label{}\n \\end{equation}\n\\newline\nIn other words, to preserve both the $0$-form symmetry and the dual $(D-1)$-form symmetry, it is necessary to have a surface of gapless modes in the momentum space.\n\nWhile analytical results discussed in this work are limited to free theories, we conjecture that similar scaling relations hold for interacting CFTs as well. To see why this is plausible, we notice that the entanglement Hamiltonian of a CFT is algebraically ``localized'' near the boundary of the subsystem~\\cite{Casini2011}, which suggests that even for a non-local observable, such as the disorder operator, the major contribution is expected to come from the boundary, and hence a perimeter law scaling. We leave a more systematic study along these lines for future work. In Sec. \\ref{sec:iv} we numerically confirm our conjecture for the Ising CFT in (2+1)d.\n\nWe now briefly discuss what happens if a small mass is turned on in Eq. \\eqref{eqn:freeboson1}. Suppose we are in a gapped phase, and denote by $\\xi$ the correlation length. In general, we expect that $S_2$ obeys an perimeter scaling in the gapped phase, namely the leading term in $S_2$ is given by $a\\frac{R}{\\epsilon}$. In 2D for a disk entangling region of radius $R$, we have~\\cite{Metlitski2009EE}\n\\begin{equation}\n\tS_2= a_{c}\\frac{R}{\\xi} + f\\left( \\frac{R}{\\xi} \\right).\n\t\\label{eq:eq17}\n\\end{equation}\nHere $a_{c}$ is the value of $a$ at the critical point (which was denoted by $a_1$ in Eq. \\eqref{eqn:S2scaling}). The function $f(x)$ satisfies\n\\begin{equation}\n\tf(x)\\rightarrow\n\t\\begin{cases}\n\t\trx & x\\rightarrow \\infty\\\\\n\t\t-\\gamma_c & x\\rightarrow 0\n\t\\end{cases}.\n\t\\label{}\n\\end{equation}\nHere $r$ is an universal constant (once the definition of $\\xi$ is fixed). Suppose the transition is tuned by an external parameter $g$ and the critical point is reached at $g_c$. Since $\\xi\\sim (g-g_c)^{-\\nu}$ where $\\nu$ is the correlation length exponent, one finds that\n\\begin{equation}\n\ta-a_c \\sim (g-g_c)^{\\nu},\n\t\\label{eq:eq19}\n\\end{equation}\n \n\n\\section{Order and disorder in Ising spin models}\n\\label{sec:iii}\n\nIn the following we study $1$-form symmetry breaking in the transverse field Ising (TFI) model which gives rise to the $(2+1)$d Ising transition. We have reviewed the connection with the $\\mathbb{Z}_2$ gauge thory in Sec. \\ref{sec:ii}, as well as the 1-form symmetry in the Ising spin system. We will now focus more on the quantitative aspects of the TFI model. Even though the TFI model and the $\\mathbb{Z}_2$ lattice gauge theory are equivalent by the duality map, we choose to work with the TFI model here because the numerical simulation is more straightforward.\n\n We will now consider a square lattice with one Ising spin per site, and the global Ising symmetry is generated by $S=\\prod_\\mb{r}\\sigma_\\mb{r}^x$. There are, generally speaking, two phases: a ``disordered'' phase, where the Ising symmetry is preserved by the ground state~\\footnote{We note that there are in fact two distinct types of Ising-disordered phases in 2D, one trivial paramagnet and the other one a nontrivial Ising symmetry-protected topological phase.}, and an ordered phase where the ground states spontaneously break the symmetry. They are separated by a quantum phase transition, described by a conformal field theory with $\\mathbb{Z}_2$ symmetry. It is well-understood how to characterize the Ising symmetry breaking (and its absence) in the three cases: consider the two-point correlation function of the order parameter $\\sigma^z_\\mb{r}$. The asympotic forms of the correlation function $\\langle \\sigma_\\mb{r}^z\\sigma_\\mb{r'}^z\\rangle$ for large $|\\mb{r}-\\mb{r}'|$ distinguish the three cases:\n\\begin{equation}\n\t\\langle \\sigma_\\mb{r}^z\\sigma_\\mb{r'}^z\\rangle\\sim\n\t\\begin{cases}\n\t\te^{-\\frac{|\\mb{r}-\\mb{r}'|}{\\xi}} & \\text{disordered}\\\\\n\t\t\\frac{1}{|\\mb{r}-\\mb{r}'|^{2\\Delta}} & \\text{critical}\\\\\n\t\t\\text{const.} & \\text{ordered}\n\t\\end{cases}.\n\t\\label{}\n\\end{equation}\nIn both the disordered phase and the quantum critical point, the Ising symmetry is preserved because of the absence of long-range order. The prototypical lattice model that displays all these features is the TFI model defined on a square lattice:\n\\begin{equation}\n\tH=-\\sum_{\\langle \\mb{r}\\mb{r'}\\rangle}\\sigma_\\mb{r}^z\\sigma_{\\mb{r}'}^z - h\\sum_\\mb{r} \\sigma_\\mb{r}^x, h\\geq 0.\n\t\\label{eq:eq6}\n\\end{equation}\nNote that this is the same as Eq. \\eqref{eqn:TFI}, but we have set $J=1$ and renamed $K$ by $h$, to align with the standard convention in literature. The model is in the ordered (disordered) phase for $h\\ll 1$ ($h\\gg 1$). The precise location of the critical point varies with dimension, $h_c=1$ in $D=1$ and $h_c=3.044$ in $D=2$~\\cite{Bloete2002,ZiHongLiu2019}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.72\\linewidth]{.\/figure\/fig1.pdf}\n\t\\caption{Disorder operator $X$ applied on regions with different shapes: (a) $M$ is a square region with size $R\\times R$ and perimeter $l$. (b) $M$ is a rectangular region with size $R\\times 2R$.}\n\t\\label{fig:fig1}\n\\end{figure}\n\n We will be interested in the disorder operator:\n\\begin{equation}\n\tX_{ M}=\\prod_{\\mb{r}\\in M}\\sigma_\\mb{r}^x,\n\t\\label{eq:eq8}\n\\end{equation}\nwhere $M$ is a rectangle region in the lattice, illustrated in Fig.~\\ref{fig:fig1}.\nIn Ref.~\\onlinecite{ji2019categorical} this operator is called the patch symmetry operator.\n\n When $X_M$ is applied to e.g. $\\ket{\\uparrow\\cdots\\uparrow}$, a domain wall is created along the boundary of the region $M$. These operators are charged under the dual $\\mathbb{Z}_2$ 1-form symmetry. One can easily see that $\\bra{\\psi_{h=\\infty}}X_M\\ket{\\psi_{h=\\infty}}=1$, and $\\bra{\\psi_{h=0}}X_M\\ket{\\psi_{h=0}}=0$. More generally,\n\\begin{equation}\n\t\\bra{\\psi}X_M\\ket{\\psi} \\sim\n\t\\begin{cases}\n\t\te^{-al_M} & h>h_c\\\\\n\t\te^{-bA_M} & hh_c$}\n\nFirst we present results in the disordered phase $h>h_c$. As shown in Eq.~\\eqref{eq:eq9}, we expect that the disorder operator obeys a perimeter law scaling, and for $h\\gg h_c$ the coefficient is given in Eq.~\\eqref{eq:eq10}.\n\nFig.~\\ref{fig:fig2} shows the QMC-obtained $\\ln\\langle X_M \\rangle$ as a function of $l$ for different values of $h$. The temperature is taken to be $\\beta=10$, and we have checked that the results already converge for this value of $\\beta$. We observe a clear linear scaling, and the inset shows that for large field $h\\gg h_c$, the slopes of the $\\ln\\langle X_M \\rangle$ are indeed given by $1\/8h^{2}$ asympototically.\n\n\\begin{figure}[htp!]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{.\/figure\/fig3.pdf}\n\t\\caption{$\\ln(a_{c} - a)$ versus $\\ln(h-h_{c})$ in the disordered phase for $L=24$ when $h$ is approaching the critical point. The fitted slope (red line) is $0.63\\pm 0.02$, consistent with the correlation length exponent of the $(2+1)$d Ising transition, as expected in Eq.~\\eqref{eq:eq19}.}\n\t\\label{fig:fig3}\n\\end{figure}\n\nNow we consider the other limit, when $h$ is approaching the critical point $h_c$ from the disordered side. To test the scaling given in Eq.~\\eqref{eq:eq19}, we measure the disorder operator and find the slope $a$ by a linear fit. Fig.~\\ref{fig:fig3} shows $a_c-a$ as a funtion of $h-h_c$ in a log-log plot. A clear power law manifests in the data, and the exponent is found to be $\\nu=0.63(2)$. Considering the finite-size effect, the result agrees very well with the 3D Ising correlation length exponent.\n\n\\subsection{Critical point $h=h_c$}\n The central question to be addressed is whether the $\\mathbb{Z}_2$ 1-form symmetry is spontanously broken at the critical point. To this end, we measure the disorder operator $\\langle X \\rangle$ at $h=h_c$ and scale the inverse temperature $\\beta=L$ in these simulations. We have also checked that finite-$\\beta$ effect is negligible in our calculations.\n\nFig.~\\ref{fig:fig4} shows $\\ln\\langle X_{M} \\rangle$ as a funtion of the perimeter $l$, where $M$ is taken to be a square region, as illustrated in Fig.~\\ref{fig:fig1} (a). Results for different system sizes $L=8,16,24,32,40$ are presented and it is clear that the finite-size effect is negligible. The data clearly demonstrates a linear scaling as in Eq.~\\eqref{eq:eq16} and the slope $a_1$ quickly converges to $0.0394 \\pm 0.0004$. \n\n\n\n\\begin{figure}[htp!]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{.\/figure\/fig4.pdf}\n\t\\caption{$-\\ln(\\langle X \\rangle)$ versus $l$ at the critical point. We use the relation of Eq.~\\eqref{eq:eq16} to fit the data and the fitted curve of the data upto $L=40$ is $-\\ln(\\langle X \\rangle)=(0.0394 \\pm 0.0004)l-(0.0267 \\pm 0.005)\\ln(l)-(0.0158 \\pm 0.008)$.}\n\t\\label{fig:fig4}\n\\end{figure}\n\nAs we have explained, the boundary of $M$ generally contributes to the disorder operator a term proportional to the perimeter. To detect 1-form symmetry breaking, we need to check whether $\\langle X\\rangle$ depends on the area or not. For this purpose, we consider rectangular regions with different aspect ratios: one with $1:1$ (Fig.~\\ref{fig:fig1} (a)) and the other with $1:2$ (Fig.~\\ref{fig:fig1} (b)), and present the results of $\\langle X \\rangle$ at the $h=h_c$ together in Fig.~\\ref{fig:fig5}. It can be seen that the two sets of data basically fall on the same curve, indicating that the disorder parameter only depends on the perimeter.\n\n\\begin{figure}[htp!]\n\\centering\n\\includegraphics[width=\\columnwidth]{.\/figure\/fig5.pdf}\n\\caption{$-\\ln \\langle X_M\\rangle$ versus $l$ at the phase transition point for $M$ with the shape $R\\times R$ (already shown in Fig.~\\ref{fig:fig4}) and $R \\times 2R$, for system size $L=32$. The blue line represents the fitted curve of the data for $R\\times 2R$ using the relation specified in Eq.~\\eqref{eq:eq16}. The fitted result of $R\\times 2R$ is $-\\ln\\langle X \\rangle=(0.0397 \\pm 0.0002)l-(0.0279 \\pm 0.003)\\ln(l)-(0.0192 \\pm 0.006)$ and for $R\\times R$ at $L=32$ the result is $-\\ln\\langle X \\rangle=(0.0399 \\pm 0.0003)l-(0.0272 \\pm 0.004)\\ln(l)-(0.0162 \\pm 0.005)$. The coefficients are indistiguishable within errorbars.}\n\\label{fig:fig5}\n\n\\end{figure}\n\nGiven the relation between $\\langle X_M\\rangle$ and the Renyi entropy in the free theory, let us examine possible corner contributions to $\\langle X_M\\rangle$, which is parameterized in the coefficient $s$ of Eq.~\\eqref{eq:eq16}. We fit the data points in Fig.~\\ref{fig:fig5} to Eq.~\\eqref{eq:eq16}, which yields $s=0.0272 \\pm 0.004$, close to the free value. We perform the same fit for data points with aspect ratio $1:2$ and obtain essentially the same results ($s=0.0279\\pm 0.003$). The agreement between the fitting results for regions with different aspect ratios again lends strong support for the perimeter dependence of $\\langle X_M\\rangle$ even beyond the leading order, and consequently the 1-form symmetry breaking at the $(2+1)$d Ising CFT. \n\n\nThe convergence of the coefficients $a_1$, $s$ and $a_0$ versus the linear system size $L$ is given in Fig.~\\ref{fig:fig7} in Appendix~\\ref{Sec:appB3}.\n\n\\subsection{Ordered phase $h1$. The most challenging case is $D=2$ where the transition is described by the interacting Wilson-Fisher fixed point, and we exploit large-scale quantum Monte Carlo simulations. We use the disorder operator of the Ising system to probe the breaking of the dual higher-form symmetry. We find numerically that at the critical point of the 2D quantum Ising model, the one-form disorder operator exhibits sponatenous symmetry breaking as in the disordered phase, whereas in the ordered phase, the one-form symmetry is intact. \n\nThe disorder operator is intimately related to a line defect (also called a twist operator) in a Ising CFT, around which the spin operator sees an anti-periodic boundary condition. In fact, a line defect is nothing but the boundary of a disorder operator. It is believed that in general such a line defect can flow to a conformal one at low energy, which is indeed consistent with a perimeter law scaling for the expectation value of the disorder operator~\\footnote{We are grateful for Shu-Heng Shao for discussions on this point.}. Local properties of disorder line defects have been previously investigated in Ref. [\\onlinecite{Billo2013}] and [\\onlinecite{Gaiotto2013}]. It will be interesting to understand the relation between the local properties with the universal corner contributions to the disorder operator~\\cite{Bueno2015}.\n\nOur findings, besides elucidating the physics of quantum Ising systems from a new angle, provides a working example of higher-form symmetry at practical use. \nSimilar physical systems can be studied, for example, the disordered operator constructed in this work is readily generalized to the $(2+1)$d XY transition and can be measured with unbiased QMC simulations. Another important direction is to study other higher-form symmetry breaking transitions, such as 1-form symmetry breaking transition in 3D systems. It would also be interesting to investigate the ultility of the disorder operator in the topological Ising paramagnetic phase.\nMore applications in quantum lattice models are awaiting to be explored, and will certainly lead to new insight for a new framework that unifies our understanding of the exotic quantum phases and transitions going beyond the Landau paradigm and those within. \n\n{\\it Note added.-} We would like to draw the reader's attention to few closely related recent works by X.-C. Wu, C.-M. Jian and C. Xu~\\cite{XCWu2020,XCWu2021} and by some of the present author on scaling of disorder oeprator at $(2+1)$d U(1) quantum criticality~\\cite{YCWang2021}. \n\n\\section*{Acknowledgement}\nJRZ, ZY and ZYM thank the enjoyable discussions with Yan-cheng Wang and Yang Qi and acknowledge the support from the RGC of Hong Kong SAR of China\n(Grant Nos. 17303019 and 17301420), MOST through the\nNational Key Research and Development Program (Grant\nNo. 2016YFA0300502) and the Strategic Priority Research\nProgram of the Chinese Academy of Sciences (Grant No.\nXDB33000000). We are grateful for Xiao-Gang Wen, Shu-Heng Shao and William William-Krempa for helpful comments. MC would like to thank Zhen Bi, Wenjie Ji and Chao-Ming Jian for enlightening discussions and acknowledges support from NSF (DMR-1846109). We thank the Computational Initiative at the\nFaculty of Science and the Information Technology Services at the University of Hong Kong, and the\nTianhe-1A, Tianhe-2 and Tianhe-3 prototype platforms at the\nNational Supercomputer Centers in Tianjin and Guangzhou for\ntheir technical support and generous allocation of CPU time.\n\n\n\\section{Free scalar field}\n\\label{sec:app1}\nIn this appendix, we calculate the disorder operator in a Gaussian fixed point and prove that its expectation value is intimately related to the evaluation of the 2nd Renyi entropy.\n\nConsider a free scalar field in $d$ spatial dimensions:\n\\begin{equation}\n\tH=\\frac{1}{2}\\int\\mathrm{d}^d\\mb{x}\\, \\big[\\pi(\\mb{x})^2 + (\\nabla \\phi(\\mb{x}))^2\\big].\n\t\\label{}\n\\end{equation}\nThe theory has a $\\mathbb{Z}_2$ symmetry $X: \\phi\\rightarrow -\\phi$.\n\nTo calculate the disorder operator, let us regularize the theory on a lattice:\n\\begin{equation}\n\tH=\\frac{1}{2}\\sum_i \\pi_i^2 + \\frac{1}{2}\\sum_{ij}\\phi_iK_{ij}\\phi_j.\n\t\\label{}\n\\end{equation}\nHere $i, j, \\dots$ label lattice sites. Define $W= \\sqrt{K}$. In the $\\phi$ basis, the ground state wavefunction is given by\n\\begin{equation}\n\t\\Psi(\\phi)=\\left(\\det \\frac{W}{\\pi}\\right)^{1\/4} e^{-\\frac{1}{2}\\phi^\\mathsf{T} W \\phi}.\n\t\\label{}\n\\end{equation}\nThe reduced density is then\n\\begin{equation}\n\t\\rho(\\phi, \\phi')=\\Psi^*(\\phi)\\Psi(\\phi')=\\sqrt{\\det\\frac{W}{\\pi}} e^{-\\frac{1}{2}(\\phi^\\mathsf{T} W \\phi + {\\phi'}^\\mathsf{T} W\\phi')}.\n\t\\label{}\n\\end{equation}\n\nConsider a region $M$, and represent the covariance matrix $W$ as\n\\begin{equation}\n\tW =\n\t\\begin{pmatrix}\n\t\tA & B \\\\\n\t\tB^\\mathsf{T} & C\n\t\\end{pmatrix},\n\t\\label{}\n\\end{equation}\nwhere the block $B$ ($C$) are the restriction of the matrix $W$ to sites inside (outside) $M$.\n\\begin{eqnarray}\n\t\\rho_M(\\phi_\\mathrm{i}, \\phi_\\mathrm{i}') &=& \\int\\cal{D}\\phi_\\mathrm{o}\\,\\rho(\\{\\phi_\\mathrm{i}, \\phi_\\mathrm{o}\\},\\{\\phi_\\mathrm{i}', \\phi_\\mathrm{o}\\}) \\nonumber\\\\\n\t&=&\\sqrt{\\det\\frac{W}{\\pi}} e^{-\\frac{1}{2}(\\phi_\\mathrm{i}^\\mathsf{T} A\\phi_\\mathrm{i} + {\\phi_\\mathrm{i}'}^\\mathsf{T} A\\phi_\\mathrm{i}')}\\int\\cal{D}\\phi_\\mathrm{o}\\,e^{-\\phi_\\mathrm{o}^\\mathsf{T} C\\phi_\\mathrm{o} + (\\phi_\\mathrm{i}+\\phi_\\mathrm{i}')^\\mathsf{T} B\\phi_\\mathrm{o}} \\nonumber\\\\\n\t&=& \\sqrt{\\det\\frac{W}{\\pi}} e^{-\\frac{1}{2}(\\phi_\\mathrm{i}^\\mathsf{T} A\\phi_\\mathrm{i} + {\\phi_\\mathrm{i}'}^\\mathsf{T} A\\phi_\\mathrm{i}')} \\frac{1}{\\sqrt{\\det \\frac{C}{\\pi}}}e^{\\frac{1}{4}(\\phi_\\mathrm{i}+\\phi_\\mathrm{i}')^\\mathsf{T} B^\\mathsf{T} C^{-1}B (\\phi_\\mathrm{i}+\\phi_\\mathrm{i}')}.\n\t\\label{}\n\\end{eqnarray}\nWe use the identity that\n\\begin{equation}\n\t\\begin{split}\n\t\\det W &= \\det (A-B^\\mathsf{T} C^{-1}B) \\det C\\\\\n\t\\end{split}\n\t\\label{}\n\\end{equation}\n\nThe reduced density matrix\n\\begin{equation}\n\t\\rho_M(\\phi, \\phi')= \\sqrt{\\det \\frac{A-B^\\mathsf{T} C^{-1}B}{\\pi}}\\,e^{-\\frac{1}{2}(\\phi^\\mathsf{T} A\\phi + {\\phi'}^\\mathsf{T} A\\phi')+\\frac{1}{4}(\\phi+\\phi')^\\mathsf{T} B^\\mathsf{T} C^{-1}B (\\phi+\\phi')}\n\t\\label{}\n\\end{equation}\n Now we can calculate the disorder operator:\n\\begin{eqnarray}\n\t\\langle X_M\\rangle &=& \\Tr (\\rho_M X) \\nonumber\\\\\n\t&=& \\int \\mathcal{D}\\phi\\mathcal{D}\\phi'\\, \\langle \\phi|\\rho_M|\\phi'\\rangle\\langle \\phi'|U|\\phi\\rangle\\nonumber\\\\\n\t&=& \\int \\mathcal{D}\\phi\\, \\langle \\phi|\\rho_M|\\mathrm{-}\\phi\\rangle \\nonumber\\\\\n\n\t&=& \\sqrt{\\frac{\\det (A-B^\\mathsf{T} C^{-1}B)}{\\det A}}\\nonumber\\\\\n\t&=&\\sqrt{\\det(\\mathds{1}-A^{-1\/2}B^\\mathsf{T} C^{-1}B A^{-1\/2})}.\n\t\\label{}\n\\end{eqnarray}\nThe second Renyi entropy\n\\begin{eqnarray}\n\t\te^{-S_2} &=& \\Tr \\rho_M^2 \\nonumber\\\\\n\t&=&\\int\\cal{D}\\phi\\cal{D}\\phi'\\, \\rho_M^2(\\phi, \\phi')\\nonumber\\\\\n\t&=& \\det \\frac{A-B^\\mathsf{T} C^{-1}B}{\\pi}\\int\\cal{D}\\phi\\cal{D}\\phi' \\,e^{-\\phi^\\mathsf{T} A\\phi - {\\phi'}^\\mathsf{T} A\\phi'+\\frac{1}{2}(\\phi+\\phi')^\\mathsf{T} B^\\mathsf{T} C^{-1}B (\\phi+\\phi')}\\nonumber\\\\\n\t&=&\\det \\frac{A-B^\\mathsf{T} C^{-1}B}{\\pi}\\int\\cal{D}\\phi_+\\cal{D}\\phi_- \\, e^{- \\phi_+^\\mathsf{T} A \\phi_+ - \\phi_-^\\mathsf{T} A\\phi_- + \\phi_+^\\mathsf{T} B^\\mathsf{T} C^{-1}B \\phi_+} \\nonumber\\\\\n\t&=& \\sqrt{\\det(\\mathds{1}-A^{-1\/2}B^\\mathsf{T} C^{-1}B A^{-1\/2})}.\n\t\\label{}\n\\end{eqnarray}\nHere $\\phi_\\pm = \\frac{\\phi\\pm \\phi'}{\\sqrt{2}}$. Thus we have found $e^{-S_2}=\\langle X_M\\rangle$.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Classification accuracy across four days}\\label{appen:classification}\n The detailed results of Naive and CTR model trained under different hourly data in four days are shown in Table~\\ref{tab:res:classification:2018-02-12} \\ref{tab:res:classification:2018-04-05} \\ref{tab:res:classification:2018-08-26} \\ref{tab:res:classification:2018-10-23}, respectively. First, across all days, CTR models always outperform the baseline model. Moreover, the performance decay is up to $3.74\\%$ for Naive model (in Apr. 5) and up to $1.71\\%$ for CTR model (CTR$_{12}$ in Aug. 26) along the day. CTR models have much smaller performance variations with the fluctuation of loads and renewables along the four days.\n Second, the CTR model scales well with more hours data included. The stark performance decay can only be observed with more than 12 hours data included.\n\n \\begin{table*}[!hb]\n \\centering\n \\caption{Classification accuracy of Naive and CTR models (Feb. 12)}\n \\resizebox{.7\\columnwidth}{!}{%\n \\label{tab:res:classification:2018-02-12}\n \\begin{tabular}{lrrrrrrrrr}\n \\toprule\n \\multicolumn{1}{c}{Hour} & Naive & CTR$_1$ & CTR$_2$ & CTR$_3$ & CTR$_4$ & CTR$_6$ & CTR$_8$ & CTR$_{12}$ & CTR$_{24}$ \\\\\n \\midrule\n 03:00 - 04:00 & 98.37 & 99.24 & 99.24 & 99.29 & 99.24 & \\textbf{99.31} & 99.26 & 99.17 & 98.94 \\\\\n 04:00 - 05:00 & 98.79 & 99.36 & 99.38 & 99.40 & 99.32 & \\textbf{99.42} & 99.37 & 99.30 & 99.11 \\\\\n 05:00 - 06:00 & 98.74 & 99.68 & 99.70 & \\textbf{99.71} & 99.69 & \\textbf{99.71} & 99.70 & 99.65 & 99.55 \\\\\n 06:00 - 07:00 & 98.60 & 99.85 & \\textbf{99.87} & 99.86 & \\textbf{99.87} & 99.86 & 99.86 & 99.84 & 99.77 \\\\\n 07:00 - 08:00 & 98.14 & \\textbf{99.88} & \\textbf{99.88} & 99.87 & 99.85 & 99.87 & 99.85 & 99.85 & 99.80 \\\\\n 08:00 - 09:00 & 98.66 & \\textbf{99.97} & \\textbf{99.97} & \\textbf{99.97} & \\textbf{99.97} & \\textbf{99.97} & 99.96 & 99.96 & 99.95 \\\\\n 09:00 - 10:00 & 99.27 & 99.96 & 99.96 & 99.96 & \\textbf{99.97} & 99.96 & 99.96 & 99.95 & 99.94 \\\\\n 10:00 - 11:00 & 99.04 & \\textbf{99.80} & \\textbf{99.80} & \\textbf{99.80} & 99.79 & 99.77 & 99.77 & 99.74 & 99.70 \\\\\n 11:00 - 12:00 & 97.64 & 99.24 & \\textbf{99.38} & \\textbf{99.38} & 99.34 & 99.29 & 99.31 & 99.21 & 99.08 \\\\\n 12:00 - 13:00 & 97.96 & \\textbf{99.37} & 99.36 & 99.34 & 99.31 & 99.30 & 99.31 & 99.19 & 99.04 \\\\\n 13:00 - 14:00 & 97.97 & 99.22 & \\textbf{99.26} & 99.25 & 99.21 & 99.21 & 99.18 & 99.05 & 98.91 \\\\\n 14:00 - 15:00 & 97.25 & 99.29 & \\textbf{99.32} & 99.30 & 99.29 & 99.29 & 99.27 & 99.13 & 98.95 \\\\\n 15:00 - 16:00 & 98.37 & \\textbf{99.25} & 99.23 & \\textbf{99.25} & 99.18 & 99.19 & 99.16 & 99.06 & 98.94 \\\\\n 16:00 - 17:00 & 98.71 & 99.52 & 99.53 & \\textbf{99.54} & 99.53 & 99.51 & 99.45 & 99.43 & 99.34 \\\\\n 17:00 - 18:00 & 98.12 & \\textbf{99.69} & \\textbf{99.69} & 99.68 & \\textbf{99.69} & 99.65 & 99.62 & 99.59 & 99.53 \\\\\n 18:00 - 19:00 & 98.93 & \\textbf{99.88} & \\textbf{99.88} & \\textbf{99.88} & \\textbf{99.88} & 99.82 & 99.83 & 99.82 & 99.78 \\\\\n 19:00 - 20:00 & 99.17 & 99.87 & \\textbf{99.88} & \\textbf{99.88} & \\textbf{99.88} & 99.84 & 99.85 & 99.83 & 99.81 \\\\\n 20:00 - 21:00 & 99.03 & 99.68 & 99.64 & \\textbf{99.69} & 99.59 & 99.58 & 99.59 & 99.58 & 99.54 \\\\\n 21:00 - 22:00 & 97.83 & \\textbf{99.47} & \\textbf{99.47} & 99.39 & 99.38 & 99.31 & 99.38 & 99.33 & 99.22 \\\\\n 22:00 - 23:00 & 96.11 & 99.45 & \\textbf{99.47} & 99.42 & 99.37 & 99.32 & 99.36 & 99.30 & 99.23 \\\\\n 23:00 - 24:00 & 96.82 & \\textbf{99.02} & 98.96 & 98.89 & 98.82 & 98.76 & 98.81 & 98.71 & 98.60 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\end{table*}\n \\begin{table*}[!h]\n \\centering\n \\caption{Classification accuracy of Naive and CTR models (Apr. 05)}\n \\resizebox{.7\\columnwidth}{!}{%\n \\label{tab:res:classification:2018-04-05}\n \\begin{tabular}{lrrrrrrrrr}\n \\toprule\n \\multicolumn{1}{c}{Hour} & Naive & CTR$_1$ & CTR$_2$ & CTR$_3$ & CTR$_4$ & CTR$_6$ & CTR$_8$ & CTR$_{12}$ & CTR$_{24}$ \\\\\n \\midrule\n 03:00 - 04:00 & 97.95 & \\textbf{99.24} & \\textbf{99.24} & \\textbf{99.24} & \\textbf{99.24} & \\textbf{99.24} & 99.21 & 99.11 & 98.95 \\\\\n 04:00 - 05:00 & 98.29 & \\textbf{99.34} & 99.33 & 99.33 & 99.31 & 99.33 & 99.31 & 99.23 & 99.08 \\\\\n 05:00 - 06:00 & 98.43 & 99.44 & 99.45 & \\textbf{99.46} & 99.44 & \\textbf{99.46} & 99.44 & 99.36 & 99.19 \\\\\n 06:00 - 07:00 & 99.02 & 99.66 & \\textbf{99.69} & \\textbf{99.69} & \\textbf{99.69} & 99.60 & \\textbf{99.69} & 99.65 & 99.55 \\\\\n 07:00 - 08:00 & 98.20 & \\textbf{99.56} & 99.54 & 99.55 & 99.53 & 99.46 & 99.53 & 99.49 & 99.35 \\\\\n 08:00 - 09:00 & 98.22 & 99.31 & \\textbf{99.34} & 99.30 & 99.32 & 99.24 & 99.23 & 99.24 & 99.11 \\\\\n 09:00 - 10:00 & 98.05 & 99.27 & 99.27 & 99.27 & \\textbf{99.28} & 99.15 & 99.16 & 99.17 & 98.97 \\\\\n 10:00 - 11:00 & 98.40 & 99.26 & 99.29 & \\textbf{99.30} & \\textbf{99.30} & 99.21 & 99.23 & 99.23 & 99.08 \\\\\n 11:00 - 12:00 & 97.68 & 98.94 & 98.97 & 98.98 & \\textbf{99.00} & 98.86 & 98.89 & 98.88 & 98.64 \\\\\n 12:00 - 13:00 & 97.11 & 98.46 & \\textbf{98.51} & 98.45 & 98.44 & 98.22 & 98.42 & 98.10 & 98.05 \\\\\n 13:00 - 14:00 & 97.36 & \\textbf{98.83} & 98.80 & 98.77 & 98.78 & 98.56 & 98.65 & 98.45 & 98.32 \\\\\n 14:00 - 15:00 & 96.64 & 98.94 & 98.97 & 98.98 & \\textbf{99.00} & 98.82 & 98.89 & 98.71 & 98.64 \\\\\n 15:00 - 16:00 & 96.55 & 99.10 & \\textbf{99.21} & 99.14 & \\textbf{99.21} & 99.01 & 99.07 & 98.93 & 98.81 \\\\\n 16:00 - 17:00 & 97.36 & \\textbf{98.93} & 98.71 & 98.81 & 98.56 & 98.60 & 98.54 & 98.52 & 98.38 \\\\\n 17:00 - 18:00 & 97.03 & \\textbf{98.72} & 98.59 & 98.67 & 98.51 & 98.46 & 98.47 & 98.42 & 98.31 \\\\\n 18:00 - 19:00 & 98.01 & 99.15 & \\textbf{99.19} & \\textbf{99.19} & 99.02 & 99.11 & 99.02 & 98.95 & 98.88 \\\\\n 19:00 - 20:00 & 98.72 & 99.33 & 99.39 & \\textbf{99.40} & 99.21 & 99.33 & 99.27 & 99.20 & 99.15 \\\\\n 20:00 - 21:00 & 98.87 & 99.37 & 99.41 & \\textbf{99.42} & 99.38 & 99.38 & 99.31 & 99.24 & 99.21 \\\\\n 21:00 - 22:00 & 98.04 & 99.41 & \\textbf{99.43} & 99.34 & 99.39 & 99.35 & 99.30 & 99.26 & 99.22 \\\\\n 22:00 - 23:00 & 95.28 & 99.22 & 99.23 & 99.23 & \\textbf{99.25} & 99.17 & 99.07 & 99.00 & 98.87 \\\\\n 23:00 - 24:00 & 98.37 & \\textbf{99.39} & 99.36 & 99.35 & 99.37 & 99.33 & 99.25 & 99.22 & 99.16 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\end{table*}\n \n \\vspace{10cm}\n \n \\begin{table*}[!h]\n \\centering\n \\caption{Classification accuracy of Naive and CTR models (Aug. 26)}\n \\resizebox{.7\\columnwidth}{!}{%\n \\label{tab:res:classification:2018-08-26}\n \\begin{tabular}{lrrrrrrrrr}\n \\toprule\n \\multicolumn{1}{c}{Hour} & Naive & CTR$_1$ & CTR$_2$ & CTR$_3$ & CTR$_4$ & CTR$_6$ & CTR$_8$ & CTR$_{12}$ & CTR$_{24}$ \\\\\n \\midrule\n 03:00 - 04:00 & 99.73 & \\textbf{99.95} & 99.94 & 99.92 & \\textbf{99.95} & 99.92 & 99.91 & 99.90 & 99.88 \\\\\n 04:00 - 05:00 & 99.73 & \\textbf{99.92} & 99.91 & 99.89 & 99.88 & 99.89 & 99.88 & 99.88 & 99.85 \\\\\n 05:00 - 06:00 & 99.66 & 99.88 & \\textbf{99.90} & 99.88 & 99.87 & 99.88 & 99.87 & 99.86 & 99.84 \\\\\n 06:00 - 07:00 & 97.93 & 99.65 & \\textbf{99.67} & 99.66 & \\textbf{99.67} & 99.62 & 99.65 & 99.61 & 99.50 \\\\\n 07:00 - 08:00 & 98.94 & \\textbf{99.63} & \\textbf{99.63} & \\textbf{99.63} & \\textbf{99.63} & 99.60 & 99.60 & 99.59 & 99.51 \\\\\n 08:00 - 09:00 & 98.94 & \\textbf{99.62} & \\textbf{99.62} & \\textbf{99.62} & 99.59 & 99.58 & 99.46 & 99.58 & 99.49 \\\\\n 09:00 - 10:00 & 98.87 & 99.52 & \\textbf{99.55} & 99.49 & 99.52 & 99.51 & 99.39 & 99.52 & 99.41 \\\\\n 10:00 - 11:00 & 98.91 & \\textbf{99.59} & 99.58 & 99.57 & 99.58 & 99.57 & 99.50 & 99.58 & 99.50 \\\\\n 11:00 - 12:00 & 97.98 & 99.15 & 99.18 & \\textbf{99.19} & \\textbf{99.19} & 99.17 & 99.15 & 99.15 & 99.10 \\\\\n 12:00 - 13:00 & 98.08 & 99.03 & \\textbf{99.07} & 99.06 & \\textbf{99.07} & 98.95 & 98.83 & 98.44 & 98.82 \\\\\n 13:00 - 14:00 & 98.31 & 98.71 & 98.87 & 98.97 & \\textbf{98.98} & 98.81 & 98.64 & 98.35 & 98.65 \\\\\n 14:00 - 15:00 & 98.05 & 98.68 & 98.84 & 98.93 & \\textbf{98.97} & 98.90 & 98.56 & 98.20 & 98.60 \\\\\n 15:00 - 16:00 & 98.15 & 98.77 & 98.90 & 98.89 & \\textbf{98.98} & 98.90 & 98.67 & 98.29 & 98.68 \\\\\n 16:00 - 17:00 & 97.85 & 98.75 & 98.75 & 98.88 & 98.62 & \\textbf{98.94} & 98.32 & 98.19 & 98.66 \\\\\n 17:00 - 18:00 & 98.96 & 99.32 & 99.36 & \\textbf{99.38} & \\textbf{99.38} & 99.31 & 99.37 & 99.11 & 99.31 \\\\\n 18:00 - 19:00 & 98.77 & 99.42 & \\textbf{99.46} & 99.44 & 99.43 & 99.36 & 99.39 & 99.10 & 99.28 \\\\\n 19:00 - 20:00 & 98.68 & \\textbf{99.64} & \\textbf{99.64} & 99.63 & 99.61 & 99.59 & 99.58 & 99.42 & 99.53 \\\\\n 20:00 - 21:00 & 98.42 & \\textbf{99.55} & 99.50 & 99.50 & 99.53 & 99.46 & 99.46 & 99.25 & 99.38 \\\\\n 21:00 - 22:00 & 97.96 & 99.46 & 99.44 & 99.45 & \\textbf{99.47} & 99.35 & 99.36 & 99.06 & 99.25 \\\\\n 22:00 - 23:00 & 98.63 & 99.59 & 99.61 & 99.59 & \\textbf{99.62} & 99.52 & 99.53 & 99.31 & 99.46 \\\\\n 23:00 - 24:00 & 98.89 & 99.63 & \\textbf{99.64} & 99.61 & 99.62 & 99.52 & 99.54 & 99.29 & 99.46 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\end{table*}\n \\begin{table*}[!h]\n \\centering\n \\caption{Classification accuracy of Naive and CTR models (Oct. 23)}\n \\resizebox{.7\\columnwidth}{!}{%\n \\label{tab:res:classification:2018-10-23}\n \\begin{tabular}{lrrrrrrrrr}\n \\toprule\n \\multicolumn{1}{c}{Hour} & Naive & CTR$_1$ & CTR$_2$ & CTR$_3$ & CTR$_4$ & CTR$_6$ & CTR$_8$ & CTR$_{12}$ & CTR$_{24}$ \\\\\n \\midrule\n 03:00 - 04:00 & 99.46 & \\textbf{99.75} & \\textbf{99.75} & 99.71 & \\textbf{99.75} & 99.70 & 99.71 & 99.64 & 99.60 \\\\\n 04:00 - 05:00 & 98.62 & 99.55 & \\textbf{99.56} & 99.50 & 99.54 & 99.50 & 99.54 & 99.39 & 99.22 \\\\\n 05:00 - 06:00 & 98.95 & 99.57 & \\textbf{99.60} & 99.55 & 99.58 & 99.55 & \\textbf{99.60} & 99.50 & 99.32 \\\\\n 06:00 - 07:00 & 99.16 & 99.55 & \\textbf{99.56} & 99.53 & 99.55 & 99.42 & 99.55 & 99.42 & 99.29 \\\\\n 07:00 - 08:00 & 98.43 & 99.31 & \\textbf{99.34} & 99.30 & 99.25 & 99.09 & 99.27 & 99.01 & 98.74 \\\\\n 08:00 - 09:00 & 97.02 & 98.99 & 98.99 & \\textbf{99.01} & 98.90 & 98.80 & 98.70 & 98.67 & 98.41 \\\\\n 09:00 - 10:00 & 97.14 & 99.03 & \\textbf{99.05} & 98.90 & 98.91 & 98.83 & 98.71 & 98.65 & 98.33 \\\\\n 10:00 - 11:00 & 97.66 & \\textbf{98.85} & 98.67 & 98.75 & 98.77 & 98.67 & 98.55 & 98.50 & 98.15 \\\\\n 11:00 - 12:00 & 97.74 & \\textbf{98.75} & 98.62 & 98.71 & 98.68 & 98.65 & 98.60 & 98.48 & 98.24 \\\\\n 12:00 - 13:00 & 97.34 & 98.61 & \\textbf{98.70} & 98.65 & 98.52 & 98.41 & 98.51 & 98.26 & 98.05 \\\\\n 13:00 - 14:00 & 98.43 & 98.81 & 98.84 & \\textbf{98.86} & 98.71 & 98.68 & 98.66 & 98.55 & 98.40 \\\\\n 14:00 - 15:00 & 98.33 & 98.76 & \\textbf{98.86} & 98.82 & 98.70 & 98.62 & 98.63 & 98.52 & 98.35 \\\\\n 15:00 - 16:00 & 97.00 & 98.64 & \\textbf{98.83} & 98.76 & 98.63 & 98.57 & 98.52 & 98.33 & 98.07 \\\\\n 16:00 - 17:00 & 98.47 & 99.31 & 99.33 & \\textbf{99.36} & 99.31 & 99.15 & 99.28 & 99.08 & 98.99 \\\\\n 17:00 - 18:00 & 99.13 & \\textbf{99.47} & \\textbf{99.47} & \\textbf{99.47} & \\textbf{99.47} & 99.33 & 99.46 & 99.32 & 99.25 \\\\\n 18:00 - 19:00 & 99.13 & 99.57 & \\textbf{99.58} & 99.57 & 99.54 & 99.55 & 99.54 & 99.43 & 99.36 \\\\\n 19:00 - 20:00 & 98.20 & 99.35 & \\textbf{99.42} & \\textbf{99.42} & 99.34 & 99.38 & 99.36 & 99.21 & 99.06 \\\\\n 20:00 - 21:00 & 98.78 & 99.48 & 99.51 & \\textbf{99.52} & 99.47 & 99.49 & 99.47 & 99.34 & 99.20 \\\\\n 21:00 - 22:00 & 97.76 & \\textbf{99.54} & 99.53 & 99.52 & 99.52 & 99.52 & 99.50 & 99.40 & 99.18 \\\\\n 22:00 - 23:00 & 98.69 & \\textbf{99.59} & \\textbf{99.59} & 99.58 & 99.58 & 99.57 & 99.56 & 99.46 & 99.40 \\\\\n 23:00 - 24:00 & 99.01 & 99.57 & \\textbf{99.58} & 99.55 & 99.55 & 99.53 & 99.53 & 99.43 & 99.34 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\end{table*}\n\n\n\\newpage\n\\subsection{Hourly data distribution}\\label{appen:data_dist}\nThe distribution shifts between different hours are measured using the hamming distance of commitment decisions and the energy distance of the input features, shown in Figure~\\ref{fig:heat_map_commitment_decisions} and \\ref{fig:heat_map_energy_distance}, respectively. Each entry indicates the distance between the corresponding hours. Therefore, the diagonal entries are always 0 since the distances between the same dataset are $0$. Interestingly, the block-wise patterns can be observed for energy distances, which is more obvious for hamming distances. Within the block which consists of several consecutive hours, the distribution shift tends to be small. Cross the block, the distribution shift largely. This, to some degree, explains why the CTR models scale well when the span of the combined dataset is less than certain hours, from the data distribution perspective.\n\n\\begin{figure*}[!h]\n\\centering\n\\includegraphics[width=0.30\\linewidth]{img\/heatmap_2018-02-12.pdf}\n\\includegraphics[width=0.30\\linewidth]{img\/heatmap_2018-04-05.pdf}\\\\\n\\includegraphics[width=0.30\\linewidth]{img\/heatmap_2018-08-26.pdf}\n\\includegraphics[width=0.30\\linewidth]{img\/heatmap_2018-10-23.pdf}\n\\caption{Heat map of commitment decisions across hours}\n\\label{fig:heat_map_commitment_decisions}\n\\end{figure*}\n\n\n\\begin{figure*}[!h]\n\\centering\n\\includegraphics[width=0.30\\linewidth]{img\/all_ed_heatmap_2018-02-12.pdf}\n\\includegraphics[width=0.30\\linewidth]{img\/all_ed_heatmap_2018-04-05.pdf} \\\\\n\\includegraphics[width=0.30\\linewidth]{img\/all_ed_heatmap_2018-08-26.pdf}\n\\includegraphics[width=0.30\\linewidth]{img\/all_ed_heatmap_2018-10-23.pdf}\n\\caption{Heat map of the energy distances of hourly distribution across hours}\n\\label{fig:heat_map_energy_distance}\n\\end{figure*}\n\n\n\n\\section{Conclusion}\n\nThe paper presented several challenges that may hamper the use of ML\nmodels in power systems operations, including the variability of\nelectricity demand and renewable production, the variations in\nproduction costs, and the combinatorial structure of commitment\ndecisions. To address these challenges, the paper proposed a novel ML\npipeline that leverages day-ahead forecasts and\nthe TSO's knowledge of commitment decisions several hours before they\ntake place. The proposed pipeline consists of two main phases: 1)\nday-ahead data preparation and training; and 2) real-time predictions,\nwhich fits naturally within the operations of a TSO. Moreover, informed\nby the behavior of real-time markets, the paper proposed a novel\nClassification-Then-Regression (CTR) approach that leverages deep neural\nnetworks based on Latent Surgical Interventions to capture generator\ncommitments and generators operating at their limits. Computational\nexperiments that replicated MISO's operational pipeline on a real,\nlarge-scale transmission system demonstrated the feasibility of the\napproach. In particular, the results show that optimization proxies\nbased on ML models have the potential to provide operators with new\ntools for real-time risk monitoring.\n\nSeveral extensions are possible, which will be the topic of future\nwork. The integration, during training, of Lagrangian-based penalties\nhas the potential to further improve the performance of the neural\nnetwork models. Such methods have been shown to improve the\nfeasibility of the predictions with respect to the original\nconstraints. Moreover, once trained, the proposed classifier may be\nused to also accelerate the SCED resolution, by providing hints as to\nwhich variables should be fixed at their minimum or maximum limit.\nFinally, the proposed CTR model is not limited to SCED models, and may\nbe applied to any optimization problems.\n\n\\section*{Acknowledgments}\n\nThis research is partly funded by NSF Awards 1912244\nand 2112533, and ARPA-E Perform Award AR0001136.\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe \\textit{Security-Constrained Economic Dispatch} (SCED) is a fundamental optimization model for Transmission System Operators (TSOs) to clear real-time energy markets while ensuring reliable operations of power grids \\cite{conejo2018power}.\nIn the US, TSOs like MISO and PJM execute a SCED every five minutes, which means that the optimization problem must be solved in an even tighter time frame, i.e., well under a minute \\cite{Chen2018_MarketClearingSoftware}.\nSecurity constraints, which enforce robustness against the loss of any individual component, render SCED models particularly challenging for large systems \\cite{Chiang2015_SolvingSCOPF,Wang2016_SolvingCorrectiveSCOPF,velloso2021exact} unless only a subset of contingencies is considered.\nWith more distributed resources and increased operational uncertainty, such computational bottlenecks will only become more critical \\cite{Chen2018_MarketClearingSoftware}.\n\nThis paper is motivated by the growing share of renewable generation, especially wind, in the MISO system, which calls for risk-aware market-clearing algorithms.\nOne particular challenge is the desire to perform risk analysis in real time, by solving a large number of scenarios for load and renewable production \\cite{werho2021_ScenarioGenerationWind}.\nHowever, systematically solving many SCED instances is not practical given the tight constraints of real-time operations. To overcome this computational challenge, this paper proposes to learn an optimization proxy for SCED, i.e., a Machine Learning (ML) model that can predict an optimal solution for SCED, within acceptable numerical tolerances and in milliseconds.\n\nIt is important to emphasize that the present goal is not to replace optimization-based market-clearing tools.\nInstead, the proposed optimization proxy provides operators with an additional tool for risk assessment.\nThis allows to quickly evaluate how the system would behave under various scenarios, without the need to run costly optimizations. In particular, because these predictions are combined into aggregated risk metrics, small prediction errors on individual instances are acceptable.\n\n\\subsection{Related Literature and Challenges}\n\\label{sec:intro:motivation}\n\n The combination of ML and optimization for power systems has attracted increased attention in recent years. A first thread \\cite{pan2019deepopf, fioretto2020predicting, chatzos2020high, lei2020data, chatzos2021spatial, velloso2020combining, zamzam2020learning, owerko2020optimal} uses ML models to predict an optimal solution to the Optimal Power Flow (OPF) problem.\n Pan et al. \\cite{pan2019deepopf} train a Deep Neural Network (DNN) model to predict solutions to the security-constrained DC-OPFs, and report results on systems with no more than 300 buses. One common limitation of ML models, also noted in \\cite{pan2019deepopf}, is that predictions are not guaranteed to satisfy the original problem's constraints. To address this limitation, recent efforts \\cite{fioretto2020predicting, chatzos2020high} integrate Lagrangian duality into the training of DNNs in order to capture the physical and operational constraints of AC-OPFs. A similar approach is followed in Velloso et al. \\cite{velloso2020combining} in the context of preventive security-constrained DC-OPF. Namely, a DNN is trained using Lagrangian duality techniques, then used as a proxy for the time-consuming master problem in a column-and-constraint generation algorithm. More recently, Chatzos et al. \\cite{chatzos2021spatial} embed the Lagrangian duality framework in a two-stage learning that exploits a regional decomposition of the power network, enabling a more efficient distributed training.\n \n \n Another research thread is the integration of ML models within optimization algorithms, in order to improve runtime performance. For instance, a number of papers (e.g., \\cite{deka2019learning,misra2021learning,xavier2021learning, yang2020fast,guha2019machine}) try to identify active constraints in order to reduce the problem complexity. In \\cite{xavier2021learning}, the authors investigate learning feasible solutions to the unit commitment problem. Venzka el al. \\cite{venzke2020neural} use a DNN model to learn the feasible region of a dynamic SCOPF and transform the resulting DNN into a mixed-inter linear program.\n \n Existing research papers in ML for power systems typically suffer from two limitations. On the one hand, most papers report numerical results on small academic test systems, which are one to two orders of magnitude smaller than real transmission systems. This is especially concerning, as higher-dimensional data has an adverse impact on convergence and accuracy of machine-learning algorithms.\n On the other hand, almost all papers rely on artificially-generated data whose distribution does not capture the variability found in actual operations. For instance, only changes in load are considered, typically without capturing spatio-temporal correlations. Other sources of uncertainty, such as renewable production, are not considered, nor is the variability of economic bids.\n Finally, changes in commitment decisions throughout the day are rarely addressed, although they introduce non-trivial, combinatorial, distribution shifts.\n\n\\subsection{Contributions and Outline}\n\\label{sec:intro:contributions}\n\n \n To address the above limitations, the paper proposes a novel ML pipeline that is grounded in the structure of real-world market operations. The approach leverages the TSO's forward knowledge of 1) commitment decisions and 2) day-ahead forecasts for load and renewable productions. Indeed, both are available by the time the day-ahead market has been executed. Furthermore, the paper presents an in-depth analysis of the real-time market behavior, which informs a novel ML architecture to better capture the nature of actual operations.\n \n \\begin{figure}[!t]\n \\centering\n \\includegraphics[width=0.95\\columnwidth]{img\/ML_pipelines-crop.pdf}\n \\caption{The proposed machine learning pipeline.}\n \\label{fig:learning_pipeline}\n \\end{figure}\n \n The proposed learning pipeline is depicted in Figure~\\ref{fig:learning_pipeline}. First, commitment decisions and day-ahead forecasts for load and renewable generation are gathered, following the clearing of the day-ahead market. Second, this information is used to generate a training dataset of SCED instances, using classical data augmentation techniques. Third, specialized ML models are trained, ideally one for each hour of the operating day, thereby alleviating the combinatorial explosion of commitment decisions: each ML model only needs to consider one set of commitments and focus on load\/renewable scenarios around the forecasts for that hour. Fourth, throughout the operating day, the trained models are used in real time to evaluate a large number of scenarios. The entire data-generation and training procedure is completed in a few hours.\n \n \n The rest of the paper is organized as follows. Section \\ref{sec:opt_pipeline} describes the interplay between day-ahead and real-time markets in the MISO system, and gives an overview of the real-time SCED for MISO. Section \\ref{sec:methodology} analyzes the behavior of the real-time market solutions, proposes a combined Classification-Then-Regression architecture, and presents the overall ML pipeline. Numerical experiments on a real-life system are reported in Section \\ref{sec:results}\n\n\\section{The Learning Methodology}\n\\label{sec:methodology}\n\nThis section reviews the learning methodology for the RT-SCED. First,\nSection \\ref{sec:pattern_dispatch} presents an analysis of the\nbehavior of optimal SCED solutions in MISO's optimization pipeline.\nThese patterns motivate a novel ML architecture, which is described in\nSection \\ref{sec:ml_model}, followed by an overview of the proposed ML\npipeline in Section \\ref{sec:ML_pipeline}. Further details on the\ndata are given in Section \\ref{sec:results}, as wells as in\n\\cite{PSCC2022-data}.\n\n\\subsection{Pattern Analysis of Optimal SCED solutions} \n\\label{sec:pattern_dispatch}\n \\begin{figure}[!t]\n \\centering\n \\includegraphics[width=0.95\\columnwidth]{img\/daily_commitments.png}\n \\vspace{0.2cm}\n \\caption{Distribution of unique hourly commitments per day, for each month of the year 2018. Higher values indicate higher variability in commitment decisions.}\n \\label{fig:daily_commitment}\n \\end{figure}\n \n The system at hand is the French transmission system, whose network topology is provided by the French TSO, RTE.\n It contains $6{,}708$ buses, $8{,}965$ transmission lines and transformers, and $1{,}890$ individual generators, $1{,}177$ of which are wind and solar generators and $713$ are conventional units.\n The MISO optimization pipeline, described in Section \\ref{sec:opt_pipeline}, is replicated on this system for the entire year 2018, yielding $365$ DA-SCUC instances and about $100{,}000$ RT-SCED instances.\n Relevant statistics are reported next.\n \nFirst, unsurprisingly, commitment decisions display a high variability; for example, in 2018, a total of $5{,}380$ different hourly commitments were recorded across the total $8{,}760$ hours of the year, where each ``hourly commitment\" is a binary vector of size\n713 that contains the (hourly) commitment status of conventional generators. \nThe intra-day variability of commitment decisions follows a seasonal pattern, which is illustrated in Figure \\ref{fig:daily_commitment}. \nNamely, for each month of the year, Figure \\ref{fig:daily_commitment} displays the distribution, across every day of the month, of the number of unique hourly commitments over a day. The higher values correspond to the higher variability in commitment decisions, while the lower values indicate that the commitment decisions are stable throughout the day. Typically, the variability of commitment decisions is lower in summer and higher in winter; this behavior is expected since more generators are online in winter, which naturally tends to yield more diverse commitments.\nIndeed, in June 2018, all the days have at least 5 and at most 19 different hourly commitments, with $50\\%$ of days having less than 7 and $50\\%$ having more than 8. In contrast, in January 2018, except for two outliers, every day has at least 19 different commitments, and 16 days had a different commitment decision every hour. \nOverall this combinatorial explosion of commitment decisions has an adverse effect on ML models, since it creates distribution shifts on unseen commitments that is detrimental to the performance.\n \n \\begin{figure}[!t]\n \\centering\n \\includegraphics[width=.5\\columnwidth]{img\/01_2018_gen_at_bounds_non_renew-crop.pdf} \n \\includegraphics[width=.438\\columnwidth]{img\/08_2018_gen_at_bounds_non_renew-crop.pdf} \n \\vspace{0.2cm}\n \\caption{Proportion of generators at their maximum and minimum limits in January (left) and August (right).}\n \\label{fig:dispatch_pattern}\n \\end{figure}\n \nSecond, an analysis of RT-SCED solutions reveals, also unsurprisingly, that a majority of generators are dispatched to either their minimum or maximum limit; such limits include economic offer data and ramping constraints. Detailed statistics are reported in Figure \\ref{fig:dispatch_pattern} for January and August 2018: each plot quantifies, across all RT-SCED instances for that month, the proportion of generators dispatched at their minimum (at\\_min) or maximum (at\\_max) limit, or neither (non\\_tight); these statistics exclude the renewable generators and the generators for which the minimum and maximum limit are identical. In January, the median proportion of generators being dispatched at their minimum (resp., maximum) limit is close to $40\\%$ (resp., $20\\%$), while in August, these values are around $45\\%$ and $20\\%$. \nThe variability is also visibly higher in winter, echoing the previous observations for\ncommitment decisions.\n\n\\subsection{First Classify, then Perform Regression}\n\\label{sec:ml_model}\n \\begin{figure*}[!t]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{img\/Models.pdf}\n \\vspace{0.2cm}\n \\caption{The Proposed CTR model architecture. \\textbf{Left}: Overall structure of the Classification-Then-Regression (CTR) model where the outputs of the classifier are used to inform the subsequent regressor. Thus the regressor only needs to predict the dispatches of non-tight generators; \\textbf{Right}: The Latent Surgical Intervention (LSI) block for each classifier and regressor in the CTR model in which a gate operator takes the binary indicator to filter out some representation from fully connected layers. The result is then added into the representations element-wisely. }\n \\label{fig:ML_model}\n \\end{figure*}\n\n The previous analysis indicates that, given the knowledge of which generators are dispatched to their minimum or maximum limit, only a small number of generators need to be predicted.\n This suggests a \\textbf{Classification-Then-Regression} (CTR) approach, first screening the active generators to identify those at their bounds, and then applying a regression to predict the dispatch of the remaining generators.\n The overall architecture is depicted in Figure~\\ref{fig:ML_model}.\n The input of the learning task is a dataset $\\mathcal{D} = \\{(\\mathbf{x}_i, \\mathbf{y}_i, \\mathbf{p}_i)\\}_{i=1}^N$, where $\\mathbf{x}_i$, $\\mathbf{y}_i$, $\\mathbf{p}_i$ represent the $i^{th}$ observation of the system state, the status indicators of the generators (i.e., whether each generator is at its maximum\/minimum limits), and the optimal dispatches, respectively.\n\n \\begin{table}[!t]\n \\centering\n \\caption{Input features of DNN model}\n \\label{tab: ML_features}\n \\begin{tabular}{ccc}\n \\toprule\n Feature & Size & Source\\\\\n \\midrule\n Loads & $L$ & Load forecasts\\\\\n Cost of generators & $G$ & Bids\\\\\n Cost of reserves & $2G$ & Bids\\\\\n Previous solution & $G$ & SCED\\\\\n Commitment decisions & $G$ & SCUC\\\\\n Reserve Commitments & $G$ & SCUC\\\\\n Generator min\/max limits & $2G$ & Renewable forecasts\\\\\n Line losses factor & $2B+1$ & System\\\\\n \\bottomrule\n \\end{tabular}\n \\end{table}\n\nThe CTR model is composed of two modules; classifier and\nregressor. First, the classifier component aims at classifying whether\neach generator is at its minimum or maximum limit. It considers the\nstatus of the power network with $B$ buses, $G$ generators, and $L$\nloads as its inputs. Specifically, the classifier, parameterized by\n$\\mathbf{w}_1$, is a mapping $f_{\\mathbf{w}_1}: \\mathbb{R}^{d}\n\\xrightarrow{} \\{0, 1\\}^{2G}$, where $d$ is the dimension of the input\nfeatures. The overall input features describe the state of the power\nsystem, detailed in Table~\\ref{tab: ML_features}. Also, it outputs the\nbinary vector of $2G$ meaning that, for each generator, there are two\nclassification choices: one for determining whether it is dispatched at its\nmaximum limit and one for determining whether it is dispatched\nat its minimum limit. In the experiments presented in the subsequent\nsection, the dimension $d$ of the input space can be as large as\n$24{,}403$ in the RTE system. The optimal trainable parameters\n$\\mathbf{w^*_1}$ in the classifier are obtained through minimizing the\nloss function as follows:\n \n \\begin{align}\n \\mathbf{w^*_1} = \\argmin_{\\mathbf{w_1}} \\frac{1}{N}\\sum_{i=1}^N \\mathcal{L}_c(\\mathbf{y}_i, f_{\\mathbf{w}_1}(\\mathbf{x}_i)), \\label{eq: cls_loss}\n \\end{align}\n where $\\mathcal{L}_c$ denotes the cross entropy loss, i.e., \n \\begin{align}\n \\mathcal{L}_c(\\mathbf{y}_i, \\hat{\\mathbf{y}}_i) = -\\sum_{j=1}^{2G} \\mathbf{y}_{i,j}\\log(\\hat{\\mathbf{y}}_{i,j}) + (1-\\mathbf{y}_{i,j})\\log(1-\\hat{\\mathbf{y}}_{i,j}).\n \\end{align}\n\n The second architectural component, the regressor that is parameterized by $\\mathbf{w}_2$, is a mapping $f_{\\mathbf{w}_2}: \\mathbb{R}^{d+2G} \\xrightarrow{} \\mathbb{R}^{G}$.\n The additional $2G$ features in the input of the regressor come from the outputs of the classifier.\n Given the trained classifier $f_{\\mathbf{w}^*_1}$, the optimal trainable parameters of the regressor $\\mathbf{w^*_2}$ are obtained by minimizing the loss function $\\mathcal{L}_r$ over all training instances as \n \\begin{align}\n \\label{eq: reg_loss}\n \\mathbf{w^*_2} = \\argmin_{\\mathbf{w_2}} \\frac{1}{N}\\sum_{i=1}^N \\mathcal{L}_r \\left( \\mathbf{p}_i, f_{\\mathbf{w}_2}\\left(\\mathbf{x}_i, f_{\\mathbf{w}^*_1}(\\mathbf{x}_i)\\right) \\right), \n \\end{align}\n where $\\mathcal{L}_r$ is the mean absolute error (MAE) loss, i.e., $\\mathcal{L}_r(\\mathbf{p}, \\hat{\\mathbf{p}}) = \\|\\mathbf{p} - \\hat{\\mathbf{p}}\\|_1$.\n \nThe CTR architecture features a deep neural network\n(DNN). Specifically, it uses a Latent Surgical Intervention (LSI)\nnetwork \\cite{donnot2018latent} as its building block. LSI is a\nvariant of residual neural networks \\cite{he2016identity} that is\naugmented by binary interventions. As illustrated in Figure\n\\ref{fig:ML_model}, the LSI block exploits, via gate operators, the\nbinary information coming from commitment decisions and the\nclassifier. As mentioned earlier, because the variability in SCED\nsolutions primarily comes from the combinatorial nature of the\ncommitment decisions, it is crucial to design the DNN architecture to\nuse commitment decisions as the input of the model. Thus, the\nLSI-based CTR model makes it possible to learn the generator dispatch\nfrom various commitment decisions, thereby allowing the proposed\napproach to generalize to the cases where the models are trained over\nmultiple commitment decisions.\n \n\\subsection{Machine Learning Pipeline} \n\\label{sec:ML_pipeline}\n\nThe high variability of SCED instances in real operations, in terms of\ncommitment decisions and forecasts of loads and renewable energy\nsources, makes it extremely challenging to learn the SCED optimization.\nTo mitigate this significant variability, this paper proposes a\nlearning pipeline that closely follows MISO's optimization pipeline.\nThe machine-learning pipeline is depicted in\nFigure~\\ref{fig:learning_pipeline} and consists of three phases: data\npreparation, training, and prediction.\n \n \\paragraph{Data Preparation} At 12pm of day $D-1$, the TSO outputs the commitment decisions of generators for the next 24 hours and Monte-Carlo scenarios for day $D$.\n Each scenario consists of 15 minute-level forecasts for load demands and renewable generators.\n Since the SCED is solved every 5 minutes throughout a day, the ML pipeline first linearly interpolates the forecasts at 5 minute level.\n Further data augmentation may be performed, if necessary, by perturbing load and renewable generations following the strategy described in \\cite{chatzos2020high}.\n The data is then used as input to SCED optimization models, which generate the various optimal dispatches.\n The entire dataset is then divided into subsets, each of which spans one or a few hours, based on computational considerations. Indeed, the goal is to strike the proper balance between the accuracy of the models and the training time, since the models have to be available before midnight. This data preparation process yields the input instances to the learning task described previously.\n \n\\paragraph{Training}\n\nFirst, each subset is split into the traditional\ntraining\/validation\/test instances. The CTR models are trained on\ntraining instances in sequence while the validation dataset is used\nfor hyperparameter tuning and the test dataset is used for reporting\nthe performance of the machine learning models. This training step\ntakes place in parallel for each subset.\n\n\\paragraph{Prediction} Starting from midnight on day $D$, at each time step, the corresponding ML model takes the latest system state as input and predicts the optimal dispatch of SCED models in real-time.\n\n\\section{Overview of MISO's Market-Clearing Pipeline} \n\\label{sec:opt_pipeline}\n\nThis section describes the interplay between MISO's day-ahead and real-time markets; the reader is referred to \\cite{BPM_002} for a detailed overview of MISO's energy and reserve markets.\n\n\\subsection{MISO Optimization Pipeline}\n\nThe day-ahead market consists of two phases, and is executed every day\nat 10am. First, a day-ahead security-constrained unit commitment\n(DA-SCUC) is executed: it outputs the commitment and regulation status\nof each generator for every hour of the following day. Then, a\nday-ahead SCED is executed to compute day-ahead prices and settle the\nmarket. The results of the day-ahead market clearing are then posted\nonline at approximately 1pm, i.e., there is a delay of several hours\nbefore the commitment decisions take effect. While MISO may commit\nadditional units during the operating day, through out-of-market\nreliability studies, in practice, $99\\%$ of commitment decisions are\ndecided in the day-ahead market\n\\cite{Chen2018_MarketClearingSoftware}. Accordingly, for simplicity\nand without loss of generality, this paper assumes that commitment\ndecisions from the DA-SCUC are not modified afterwards.\n\nThen, throughout the operating day, the real-time market is executed\nevery 5 minutes. This real-time SCED (RT-SCED) adjusts the dispatch\nof every generator in response to variations in load and renewable\nproduction, and maximizes economic benefit. Despite its short-time\nhorizon, the RT-SCED must still account for uncertainty in load and\nrenewable production. In the MISO system, this uncertainty mainly\nstems from the intermittency of wind farms, and from increasing\nvariability in load. The latter is caused, in part, by the growing\nnumber of behind-the-meter distributed energy resources (DERs) such as\nresidential storage and rooftop solar. Therefore, operators\ncontinuously monitor the state of the power grid, and may take\npreventive and\/or corrective actions to ensure safe and reliable\noperations.\n\n\\subsection{RT-SCED Formulation}\n\nThe RT-SCED used by MISO is a DC-based linear programming formulation that co-optimizes energy and reserves \\cite{MISO2009_SCED}.\nThe reader is referred to \\cite{BPM002_D} for the full mathematical formulation; only its core elements are described here.\nThe computation of market-clearing prices is beyond the scope of this paper, and is therefore not discussed here.\n\nThe RT-SCED model comprises, for each generator, one variable for energy dispatch and up to four categories of reserves: regulating, spinning, online supplemental, and offline supplemental.\nIn addition, each generator is subject to ramping constraints and individual limits on energy and reserve dispatch.\nEnergy production costs are modeled as piece-wise linear convex functions.\nEach market participant submits its own production costs and reserve prices via MISO's market portal, and may submit different offers for each hour of the day.\nFor intermittent generators such as solar and wind, the latest forecast is used in lieu of binding offers.\n\nAt the system level, line losses are estimated in real-time from\nMISO's state estimator, and incorporated in the formulation using a\nloss factor approach as in \\cite{FERC2017_MarginalLossCalculation}.\nTransmission constraints are modeled using PTDF matrices, where flow\nsensitivities with respect to power injections and withdrawals are\nprovided by an external tool. Reserves are dispatched on individual\ngenerators, in order to meet zonal and market-wide minimum\nrequirements. Additional constraints ensure that, in a contingency\nevent, reserves may be deployed without tripping transmission lines.\nPower balance, reserve requirements, and transmission limits are soft,\ni.e., they may be violated, albeit at a reasonably high cost.\n\nIn summary, the RT-SCED receives the following inputs: the commitment decisions from DA-SCUC, the most recent forecast for load and renewable production, the economic limits and production costs of the generators, the current state estimation, and the transmission constraints and reserve requirements. The RT-SCED produces as outputs the active power and reserve dispatch for each generator.\n\n\\section{Experimental Results}\n\\label{sec:results}\n\n\\subsection{Test Cases}\n\\label{sec:data_setup}\n\nThe experiments replicate MISO's operations on the French transmission\nsystem, for four representative days in 2018, namely, February\n12\\textsuperscript{th}, April 5\\textsuperscript{th}, August\n26\\textsuperscript{th}, and October 23\\textsuperscript{rd}. These\nfour days were selected at random to represent annual seasonality.\nFor each operating day, 2000 Monte-Carlo scenarios for total load and\nrenewable power production are sampled in a day-ahead fashion, i.e.,\nscenarios are produced at noon of the previous day. These scenarios\nare illustrated in Figure \\ref{fig:res:scenarios}, which displays 200\nscenarios for solar production, wind production, total load, and total\nnet load, respectively. The total net load in this figure represents\nthe amount of power production expected to be generated by the\nconventional generators, which is identical to the total load minus\nthe renewable generation.\n \nThe forecasting models for load consumption and renewable power\ngeneration in this paper used Long Short-Term Memory (LSTM) neural\nnetworks \\cite{hochreiter1997long} and the scenarios are generated by\na MC-dropout approach \\cite{gal2016dropout}. Other forecasting methods\nand scenario generation approaches could be used instead, without\nchanging the methodology.\n\n \n \\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\columnwidth]{img\/sced_learning_solar.png}\n \\includegraphics[width=0.48\\columnwidth]{img\/sced_learning_wind.png}\\\\\n \\includegraphics[width=0.48\\columnwidth]{img\/sced_learning_load.png}\n \\includegraphics[width=0.48\\columnwidth]{img\/sced_learning_net_load.png}\\\\\n \\caption{Day-ahead scenarios for solar output (top-left), wind output (top-right), total load (bottom-left) and net load (bottom-right). 200 Monte-Carlo scenarios are depicted.}\n \\label{fig:res:scenarios}\n \\end{figure}\n\nDay-ahead commitment decisions are obtained by solving a DA-SCUC problem and recording its solution. \nThe SCUC formulation used in the present experiment follows\nMISO's DA-SCUC described in \\cite{BPM_002,BPM_002_B}. \nAgain, the proposed ML pipeline is agnostic to the SCUC formulation itself and how it is solved, as it only requires the resulting commitments.\n\nFinally, for each scenario and each day, 288 RT-SCED instances are\nsolved (one every 5 minutes), yielding a total of $576{,}000$\ninstances. This initial dataset is then divided into 24 hourly\ndatasets, each containing $24{,}000$ SCED instances, since commitment\ndecisions are hourly. To avoid information leakage, the data is split\nbetween training\/validation\/testing instances as follows: $85\\%$ of\nscenarios are used for training, $7.5\\%$ for validation, and $7.5\\%$\nfor testing. All the reported results are in terms of the testing\ninstances.\n \nThe optimization problems (SCUC and SCED) are formulated in the JuMP\nmodeling language \\cite{DunningHuchetteLubin2017_JuMP} and solved with\nGurobi 9.1 \\cite{gurobi}, using Linux machines with dual Intel Xeon\n6226@2.7GHz CPUs on the PACE Phoenix cluster \\cite{PACE}; the entire\ndata generation phase is performed in less than 2 hours. The proposed\nCTR model is implemented using PyTorch \\cite{NEURIPS2019_9015} and trained using the Adam optimizer \\cite{kingma2014adam} with a learning rate of 5e-4.\nA grid search is performed to choose the hyperparameters, i.e., the\ndimension of the hidden layers (taken in the set \\{64, 128, 256\\}),\nand the number of layers (from the set \\{2, 3, 4, 5, 6\\}) for the\nfully connected layers in the LSI block. A leaky ReLU with leakiness\nof $\\alpha=0.01$ is used as the activation function in the LSI\nblock. An early stopping criterion with 20 epochs is used for training the classifier and regressor models in order to prevent overfitting: when the loss values (Eq. \\ref{eq: cls_loss} and Eq. \\ref{eq: reg_loss}) on the validation dataset do not decrease for 20 consecutive epochs, the training process is terminated. Training is performed using Tesla V100-PCIE GPUs with 16GBs\nHBM2 RAM, on machines with Intel CPU cores at 2.1GHz.\n\n\\subsection{Baselines}\n\nThe proposed CTR models are evaluated against the following baselines:\nNaive CTR, Naive Regression (Naive Reg), and Regression (Reg). The\nnaive baselines replicate the behavior of the previous SCED solution,\ni.e., they use the dispatch solution obtained 5 minutes earlier. The\nnaive baselines are motivated by the fact that, if the system does not\nchange much between two consecutive intervals, then the SCED solution\nshould not be changed much either. The Naive CTR uses the same\napproach as the CTR models: it first predicts whether a given\ngenerator is at its minimum (resp., maximum) limit if it was at its\nminimum (resp. maximum) limit in the previous dispatch; then, for the\nremaining generators, it predicts the active dispatch using the\nregressor. The naive baselines are expected to perform worse when\nlarge fluctuations in load and renewable production are observed,\nwhich typically occurs in the morning and evening. Note that these\ntimes of the day also display the largest ramping needs, and are among\nthe most critical for reliability. The effectiveness of the CTR\narchitecture is also demonstrated by comparing it to Reg for the optimal active dispatch, i.e., by omitting the\nclassification step from the CTR.\n\n\\subsection{Optimal Dispatch Prediction Errors}\n \n \\begin{table}[!t]\n \\centering\n \\caption{Average Classification Accuracy (\\%) of the CTR Classifier and Naive CTR on 4 Representative Days in 2018.}\n \\label{tab:res:classification_overall}\n \\begin{tabular}{lccccc}\n \\toprule\n & \\multicolumn{4}{c}{Dates} & \\\\\n \\cmidrule{2-5}\n Methods & Feb. 12 & Apr. 5 & Aug. 26 & Oct. 23 & Avg. \\\\\n \\midrule\n Naive classifier & 98.26 & 97.79 & 98.64 & 98.31 & 98.25 \\\\\n CTR & 99.56 & 99.18 & 99.40 & 99.24 & 99.35 \\\\\n \n \n \n \n \n \n \n \\bottomrule\n \\end{tabular}\n \\end{table}\n \n \n Table \\ref{tab:res:classification_overall} reports the overall\n classification accuracy of the CTR classifier and the Naive CTR across\n four representative days in 2018. Surprisingly, the naive classifier\n is a strong baseline, with an accuracy that ranges from $97.79\\%$ in\n the Spring to $98.64\\%$ in the Summer. The CTR classifier always\n improves on the baseline, by around one percentage point on average\n and by up to $1.40$ percentage point in the Spring. More detailed results\n about the classifiers are given in Appendix~\\ref{appen:classification}.\n \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1\\columnwidth]{img\/MAE_along_days_CTR_1_no_sub-crop.pdf}\n \\caption{Mean Absolute Error (MAE) Over Time on Feb. 12 (top-left), Apr. 5 (top-right), Aug. 26 (bottom-left), and Oct. 23 (bottom-right).}\n \\label{fig:MAE_along_time}\n\\end{figure}\n\n\\begin{table}[t]\n \\centering\n \\caption{Mean Absolute Error (MW) by Generator Size$^{\\dagger}$.}\n \\label{tab:res:disaggerated_dsipatch}\n \\resizebox{0.95\\columnwidth}{!}{\n \\begin{tabular}{lrrrrrrrrrrrr}\n \\toprule\n Date & Method & Small & Medium & Large & All \\\\\n \\midrule\n \\multirow{4}{*}{Feb. 12} & Naive Reg & 0.122 & 0.465 & 1.602 & 0.374 \\\\\n & Naive CTR & 0.084 & 0.333 & 1.128 & 0.262 \\\\\n & Reg & 0.057 & 0.188 & 0.654 & 0.153 \\\\\n & CTR & \\textbf{0.043} & \\textbf{0.141} & \\textbf{0.535} & \\textbf{0.117} \\\\\n \\midrule\n \\multirow{4}{*}{Apr. 05} & Naive Reg & 0.242 & 0.345 & 7.480 & 0.772 \\\\\n & Naive CTR & 0.197 & 0.220 & 4.374 & 0.463 \\\\\n & Reg & 0.149 & 0.152 & 2.553 & 0.282 \\\\\n & CTR & \\textbf{0.105} & \\textbf{0.110} & \\textbf{2.291} & \\textbf{0.241} \\\\\n \\midrule\n \\multirow{4}{*}{Aug. 26} & Naive Reg & 0.097 & 0.256 & 7.447 & 0.637 \\\\\n & Naive CTR & 0.080 & 0.149 & 4.045 & 0.352 \\\\\n & Reg & 0.054 & 0.124 & 2.454 & 0.218 \\\\\n & CTR & \\textbf{0.034} & \\textbf{0.064} & \\textbf{2.220} & \\textbf{0.190} \\\\\n \\midrule\n \\multirow{4}{*}{Oct. 23} & Naive Reg & 0.176 & 0.535 & 8.425 & 0.778 \\\\\n & Naive CTR & 0.140 & 0.341 & 4.596 & 0.434 \\\\\n & Reg & 0.106 & 0.192 & 2.724 & 0.263 \\\\\n & CTR & \\textbf{0.076} & \\textbf{0.145} & \\textbf{2.525} & \\textbf{0.235} \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\\\\n $^{\\dagger}$Small: 0-10MW; Medium: 10-100MW; Large: $>$100MW\\\\\n\\end{table}\n\nFigure~\\ref{fig:MAE_along_time} reports the mean absolute error (MAE)\nfor active power dispatch of the different models for each of the\nconsidered days. Given a ground-truth dispatch $p_i^g$\nand the predicted dispatch $\\hat{p}^g_i$, the MAE is\ndefined as\n\\begin{align*}\n MAE = \\frac{1}{N} \\frac{1}{G} \\sum_{i=1}^{N} \\sum_{g=1}^{G} \\|p^g_i - \\hat{p}^g_i\\|,\n\\end{align*}\nwhere $N$ is the number of instances in the test dataset and $G$ is\nthe number of generators for each instance. As shown in\nFigure~\\ref{fig:MAE_along_time}, the MAEs of CTR are always lower than\nthose of Reg, demonstrating the benefits of the classifier. The MAEs\nof Naive CTR are always lower than those of Naive Reg, showing that\neven a naive classifier has benefits. Moreover, the methods using DNNs\nfor classification and regression, i.e., CTR and Reg, are always\nbetter than their naive counterparts, demonstrating the value of deep\nlearning. Note that the performance of the naive methods fluctuates\nduring the day, mainly due to the variability of the commitments. The\nnaive methods are not robust with respect to changes in commitments,\ncontrary to the proposed CTR approach.\n\nTo investigate how the machine-learning models perform for different\ngenerator types, Table~\\ref{tab:res:disaggerated_dsipatch}\nreports their behavior for different generator sizes. The generators\nare clustered into three groups based on their actual active\ndispatches, and the MAE is reported for each group separately.\nSmall-size generators have a capacity between $0$ and $10$MW,\nmedium-size generators have a capacity between $10$ and $100$MW, and\nlarge-size generators have a capacity above $100$MW. The last column\nreport the MAEs across all\ngenerators. Table~\\ref{tab:res:disaggerated_dsipatch} further confirms\nthat the CTR models consistently outperform the corresponding\nregressors. These improvements are most notable for small and medium\ngenerators which are expected to have higher variability: the MAEs are\ndecreased by up to $37.04\\%$ across small generators and $48.39\\%$\nacross medium generators (in Aug. 26).\nMoreover, across four days, the CTR model achieves a Mean Average Percentage Error (MAPE) of $0.59\\%$ and $0.34\\%$ for medium and large generators.\n\n\n\\subsection{Solving Time vs Inference Time}\nSolving the SCED using traditional optimization tools takes, on\naverage, $15.93$ seconds. Actual computing times fluctuate throughout\nthe day, with increased load and congestion leading to higher\ncomputing times. In contrast, evaluating the optimization proxy for a\nbatch of 288 instances takes an average $1.5$ milliseconds. In other\nwords, roughly $200{,}000$ scenarios may be evaluated in less than one\nsecond. This represents an improvement of 4 orders of magnitude, even\nunder the assumption that several hundred SCED instances can be solved\nin parallel.\n\\section{Sensitivity Analysis}\n\\label{sec:results2}\n\n\\subsection{Motivations and Experiment Settings}\n\nHow many CTR models need to be trained is an important question to\nponder in practice. Preparing a single CTR model for 24 hours on Day\n$D$ would be most convenient. However, as described in\nSection~\\ref{sec:pattern_dispatch}, the variability in commitments is\nnotoriously harmful to the prediction accuracy and training time of\nthe CTR model. However, successive hours during a day may only have a\nfew differences in commitments and hence a single CTR model may be\nsufficient for predicting the associated SCED optimizations.\n \nTo answer that question, the original instance data is used to produce\na variety of datasets, containing data for consecutive 2, 3, 4, 6, 8,\n12, and 24 hours. For instance, the 8 hours dataset contains three\nsets of instances, each grouping SCED data for 8 successive\nhours. Three models are then trained using instances covering their 8\nhours of data. All models use the termination criterion presented in Section \\ref{sec:results}. \nThe various CTR models for 1, 2, ..., 24 hours are\nthen compared for the 24 hours of the day, using the models\nappropriate for each hour. It is expected that the quality of the\nprediction will deteriorate with a coarser granularity, since there\nwill be a larger variability in commitments and net loads. However,\nthe LSI architecture of the CTR model and the availability of more\ninstances during training may compensate for this increase in\nvariability. In the following, CTR models trained on $i$ hours are\ndenoted as CTR$_i$.\n\n\\subsection{Results}\n \n\n \\begin{figure*}[t]\n \\centering\n \\includegraphics[width=1.8\\columnwidth]{img\/MAE_CTR_days-crop.pdf}\n \\caption{MAEs of CTR models for representative days: the CTR model\n scales well when trained over multiple hours. The performance\n degradation becomes significant only when aggregating 6 hours\n or more.}\n \\label{fig:MAE_multiple_hours}\n \\end{figure*}\n\n\n\nFigure~\\ref{fig:MAE_multiple_hours} illustrates the MAE values of the\nvarious CTR models. The performance of the CTR model remains strong\neven when aggregating up to 4 successive hours. As more hours are\naggregated, the performance starts to degrade. To get more insight on\nthe performance of the various CTR models, it is useful to consider\nthe energy distance \\cite{rizzo2016energy} between the empirical\ndistributions of the testing and training instances. Recall that, given\ntwo empirical distributions $\\{x_i\\}_{i=1}^N$ and $\\{y_i\\}_{i=1}^M$,\ntheir energy distance is given by\n \\begin{align*}\n \\mathcal{E}(\\mathcal{X},\\mathcal{Y}) &= \\frac{2}{NM} \\sum_{i=1}^{N} \\sum_{j=1}^{M} \\|x_i - y_j\\| \\\\\n &- \\frac{1}{N^2} \\sum_{i=1}^{N} \\sum_{j=1}^{N} \\|x_i - x_j\\| \\\\\n &- \\frac{1}{M^2} \\sum_{i=1}^{M} \\sum_{j=1}^{M} \\|y_i - y_j\\|.\n \\end{align*}\n \n \\begin{figure}[t]\n \\centering\n \\includegraphics[width=1\\columnwidth]{img\/energy_scatter_MAE_scaled-crop.pdf}\n \\caption{Scaled MAE value of the CTR models vs. energy distance between the test datasets and the training datasets for given hour's period on Feb. 12 (top-left), Apr. 5 (top-right), Aug. 26 (bottom-left), and Oct. 23 (bottom-right).}\n \\label{fig:MAE_energy_distances}\n \\end{figure}\n\nIt is thus interesting to study the relationship between the MAE of a\nCTR model and the energy distance between its testing and training\ndatasets. Figure~\\ref{fig:MAE_energy_distances} reports the\nrelationship between the scaled MAE and the energy distance, where the\nscaled MAE is the MAE value divided by the best MAE value across all\nexperiments. Each dot captures the scaled MAE value (y-axis) for a CTR\nmodel and the associated energy distance (x-axis) between its training\nand testing datasets.\n\nObserve first that the energy distance increases as more hourly data\nare combined into the training set. In\nFigure~\\ref{fig:MAE_energy_distances}, the dots from CTR$_i$ tend to\nhave smaller energy distances than those of CTR$_j$ for $j > i$. The\nscaled MAE also increases as the energy distance grows. More\ninterestingly, Figure~\\ref{fig:MAE_energy_distances} shows that the\nMAE value only increases slightly as the energy distance becomes\nlarger. Specifically, an increase of $10^2$ in energy distance\nproduces an increase of only 10\\% in MAE (corresponding to a scaled MAE\nof 1.1). These results confirm that the CTR models scale well when\nfacing a reasonable amount of variability in commitment decisions and\nnet loads. However, when the energy distance goes over a certain\nthreshold, then the performance of CTR models starts to degrade\nsignificantly even in presence of more data and more training time.\nNote that the energy distance between the training and testing\ndatasets can be computed before the training process. Hence, the\nproper granularity for the CTR model can be chosen to obtain the\ndesired accuracy. This quantification shows the potential of CTR to\nperform well with more complex components involved in the MISO\npipeline such as the Look Ahead Commitment (LAC).\n \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=.49\\columnwidth]{img\/reg_training_time-crop.pdf}\n \\includegraphics[width=.47\\columnwidth]{img\/cls_training_time-crop.pdf}\n \\caption{Training Times of CTR$_i$ Models: Regression (Left) and Classification (Right) }\n \\label{fig:training_time}\n \\end{figure}\n \nFigure~\\ref{fig:training_time} reports the training times of the CTR models. The average training time increases as more hourly data are included. The regression takes more training time than the classification task. Section \\ref{sec:ML_pipeline} indicated that the training process must complete within about 12 hours after accumulating the data instances at 12pm on day $D-1$.\nThis is clearly possible for all the CTR$_i$ models with $i \\leq 4$. Consider CTR$_4$ for instance. Figure~\\ref{fig:training_time} shows that it can be trained for a 4-hour block within 10 hours in the worst case (4 hours for classification and 6 hours for regression). As a result, all 6 CRT$_4$ models can easily be trained in parallel within the required time frame. ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nUltra-hot Jupiters allow us to advance our understanding of planetary migration and orbital stability (\\citealt{Delrez2016}), and they offer great prospects for atmospheric characterization (\\citealt{Parmentier2018}). Their high temperature (typically higher than about 2000\\,K) simplifies the atmospheric chemistry by dissociating molecular species into their atomic constituents (\\citealt{Lothringer2018}). Multiple atoms and ions could thus be detected in the atmospheric limb of ultra-hot Jupiters using high-resolution transmission spectroscopy (e.g., \\citealt{Hoeijmakers2018, Hoeijmakers2019} for the prototypical ultra-hot Jupiter KELT-9b). These planets are also interesting candidates for probing evaporation and the effect of photo-ionization on the upper atmospheric structure. Their extreme irradiation by the host star causes the hydrodynamical expansion of their upper atmosphere, allowing metals to escape and be detected in the near-ultraviolet after they are ionized in the exosphere (\\citealt{Fossati2010}, \\citealt{Haswell2012}, \\citealt{Sing2019}). \n\nInterestingly, several ultra-hot Jupiters were found on highly misaligned orbits (e.g. WASP-12b, \\citealt{Albrecht2012}; WASP-33b, \\citealt{Cameron2010}; WASP-121b, \\citealt{Delrez2016}), suggesting dynamical migration processes induced by gravitational interactions with companions, rather than disk migration (e.g. \\citealt{Nagasawa2008}, \\citealt{Fabrycky2007}, \\citealt{Guillochon2011}). At such close distances to their stars, ultra-hot Jupiters are subjected to strong tidal interactions that determine their final orbital evolution. Precisely measuring the orbital architecture of ultra-hot Jupiters and monitoring its evolution is thus of particular importance to determine their migration history and their potential decay into the star. The occultation of a rotating star by a transiting planet removes the light of the hidden photosphere from the observed stellar lines (the so-called Rossiter-McLaughlin effect, or RM effect, \\citealt{Holt1893}; \\citealt{Rossiter1924}; \\citealt{McLaughlin1924}). Different techniques have been developed to analyze the radial velocity (RV) anomaly induced by the distortion of the stellar absorption lines (e.g., \\citealt{ohta2005}, \\citealt{gimenez2006}, \\citealt{hirano2011b}, \\citealt{boue2013}), to model their profile while accounting for the planet occultation (e.g., \\citealt{Cameron2010}, \\citealt{Gandolfi2012}, \\citealt{Crouzet2017}), or to isolate the local stellar lines from the planet-occulted regions (e.g., \\citealt{Cegla2016}, \\citealt{Bourrier_2018_Nat}). These techniques enable deducing the trajectory of the planet across the stellar disk, and thus inferring the projected or true 3D alignment between the spins of the planetary orbit and the stellar rotation. \\\\\n\nThe ultra-hot Jupiter WASP-121b (\\citealt{Delrez2016}) is a good candidate for both atmospheric and orbital architecture studies (Table~\\ref{tab:sys_prop}). This super-inflated gas giant transits a bright F6-type star (V = 10.4), favoring optical transmission spectroscopy measurements. Its near-polar orbit at the edge of the Roche limit ($P$ = 1.27\\,days) makes WASP-121b subject to strong tidal interactions with the star (\\citealt{Delrez2016}) and an intense atmospheric escape. The increase in transit depth of WASP-121b toward near-UV wavelengths (\\citealt{Evans2018}, \\citealt{Salz2019_NUV_WASP121b}) was recently shown to arise from iron and magnesium atoms that escape into the exosphere (\\citealt{Sing2019}), which confirms the hydrodynamical evaporation of WASP-121b and opens new avenues to link the structure and composition of the lower and upper atmosphere.\n\nIn the present study we investigate the atmosphere of WASP-121b, and refine the properties of its planetary system. In Sect.~\\ref{sec:RV_fit}, we reanalyze long-term RV and activity indexes of the system. Sect.~\\ref{sec:reloaded RM} exploits transit spectroscopy of WASP-121b obtained with the High Accuracy Radial velocity Planet Searcher (HARPS), combined with simultaneous EulerCam photometry, to analyze the orbital architecture of WASP-121b and its star. In Sect.~\\ref{sec:atmo_struc} we characterize the atmospheric structure of the planet at the limb, using a new method to isolate the signal of the planetary atmosphere from the occulted stellar lines. We conclude the study in Sect.~\\ref{sec:conclu}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Radial velocity monitoring of WASP-121}\n\\label{sec:RV_fit}\n\n\\subsection{Planet-induced motion}\n\nWe analyzed RV data points of WASP-121 obtained with the Coralie (\\citealt{baranne1996}, \\citealt{Queloz2000}) and HARPS (\\citealt{Mayor2003}) spectrographs to revise the semi-amplitude of the stellar reflex motion and the mass of WASP-121b (the complete RV dataset is shown in Fig.~\\ref{fig:RV_ana_appendix_nobin}). RV data were analyzed with the Data and Analysis Center for Exoplanets web platform (DACE\\footnote{https:\/\/dace.unige.ch}). We excluded datapoints obtained during four planet transits (one observed with Coralie, the other three with HARPS) and binned the remaining data in time by 0.25 day separately for each instrument (to mitigate short-term stellar signals and to avoid favorring HARPS datapoints). The processed data (Fig.~\\ref{fig:RV_fit}) were fit with the Keplerian model described in \\citet{Delisle2016}. It was combined with the activity detrending described in \\citet{Delisle2018}, which adds a term that is linearly correlated with the bisector of the cross-correlation functions (CCFs). The model was fit to the data using a Markov chain Monte Carlo (MCMC) algorithm (\\citealt{Diaz2014,Diaz2016}) with Gaussian priors on the period, time of mid-transit, eccentricity, and periastron argument derived from photometry obtained with the Transiting Exoplanet Survey Satellite (TESS) by \\citet{Bourrier2019}. Results are given in Table~\\ref{fig:RV_ana_appendix}. The mass we derive for WASP-121b is consistent with that of \\citet{Delrez2016}. We kept the values of the properties that have been derived from the TESS photometry as our final estimates for the revised planetary properties (Table~\\ref{tab:sys_prop}), because the fit to the RV data did not improve their precision, nor changed their values significantly.\\\\\n\n\n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0.cm 0.cm 0.cm 0cm,clip=true,width=\\columnwidth]{WASP-121_rvs_binned}\n\\caption[]{Radial velocities of WASP-121b, phase-folded with the orbital period and detrended from stellar activity. Coralie data points have been obtained before (red) and after (blue) the fiber upgrade (\\citealt{Segransan2010}). HARPS datapoints are shown in gold. Our best-fit Keplerian model to the out-of-transit data is shown as a solid black line (datapoints obtained in the frame of transit observations were binned, see text). The first and fourth transit contacts are indicated by dash-dotted vertical black lines. }\n\\label{fig:RV_fit}\n\\end{figure}\n\\end{center}\n\n\n\n\n\\subsection{Stellar rotation}\n\\label{sec:Prot}\n\nAfter the contribution of WASP-121b was removed, the periodogram of the RV residuals reveals three significant signals at periods of 0.89, 1.13, and $\\sim$8.4 days. These signals are also visible when periodograms of the bisector span (BIS SPAN) and the full-width at half-maximum (FWHM) time-series measured on the CCF are analyzed. They arise from magnetically active regions at the surface of fast-rotating WASP-121, and are all aliases of one another. We show in Sect.~\\ref{sec:results_RM} that the signal at $\\sim$8.4 days must be an alias because of the high measured stellar projected rotational velocity. We then used the technique proposed by \\citet{Dawson2010} to determine whether the signals at 0.89 or 1.13 days directly trace the rotational modulation of WASP-121. To distinguish the real signal and aliases, \\citet{Dawson2010} proposed simulating data with the same time sampling and injecting the signals that are to be tested as being real or aliases. For each injected signal, a comparison of the period and phase of all the aliases created by the observational sampling is then performed between the simulated and real data set. Using this technique, \\citet{Dawson2010} were able to show that the period originally derived for planet 55\\,Cnc\\,e (\\citealt{McArthur2004}, \\citealt{Fischer2008}) was an alias of the real signal.\\\\\nHere, we extend the approach of \\citet{Dawson2010} by performing 100 simulations for each injected signal, taking different configurations for the noise into account. We also analyze the rotational signal using the RVs, the BIS SPAN and the FWHM time-series. For each real or alias signal in the real or simulated data, we calculate the area below each peak and its phase (Fig.\\,\\ref{fig:1}). The area is defined as the sum of the power for all frequencies that lie 5 bins away from the frequency corresponding to the maximum power of the peak. Finally, the sum of both the absolute phase and area differences are calculated for each of the 100 simulations on the RVs, the BIS SPAN and the FWHM. These sums are given in Table\\,\\ref{table:rotation_period}. Overall, we observe smaller differences between the real and simulated data when the 1.13-day signal is considered compared to the signal at 0.89 day. We therefore propose that the 1.13-day signal traces the rotational modulation of WASP-121. \\\\\n\n\n\n\\begin{table}[tbh]\n\\caption{\n\\label{table:rotation_period}\nArea and phase differences, in arbitrary units, for the 0.89- and 1.13-day signals seen in the RV, BIS SPAN, and FWHM time-series periodograms. Bold numbers highlight the lower difference values when the two signals are compared.}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\hline\\hline \n & Period [d] \t\t\t\t\t& Area difference & Phase difference \\\\\n\\hline\nRV & \\begin{tabular}{@{}c@{}}0.89 \\\\ 1.13\\end{tabular} & \\begin{tabular}{@{}c@{}}922 \\\\ \\bf{696}\\end{tabular} & \\begin{tabular}{@{}c@{}}1629 \\\\ \\bf{1347}\\end{tabular} \\\\\n\\hline\nBIS SPAN & \\begin{tabular}{@{}c@{}}0.89 \\\\ 1.13\\end{tabular} & \\begin{tabular}{@{}c@{}}1923 \\\\ \\bf{690}\\end{tabular} & \\begin{tabular}{@{}c@{}}\\bf{447} \\\\ 647\\end{tabular} \\\\\n\\hline\nFWHM & \\begin{tabular}{@{}c@{}}0.89 \\\\ 1.13\\end{tabular} & \\begin{tabular}{@{}c@{}}1975 \\\\ \\bf{1202}\\end{tabular} & \\begin{tabular}{@{}c@{}}\\bf{2608} \\\\ 2820\\end{tabular} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{figure*}[]\n\\center\n\\includegraphics[angle=0,width=16cm]{gls_with_alias_period.pdf}\n\\caption[]{Comparison between the real and alias signals in the RV data of WASP-121 corrected for the planet signal (red shadow) and in the simulated data (black). The top panel corresponds to the 0.89-day signal, the bottom panel to the 1.13-day signal. These plots correspond to one realization of noise out of 100 different trials. Real signals are shown in black (panels A3 and B2), yearly aliases ($\\pm$0.0027 days$^{-1}$) in green (panels A3 and B2), daily aliases ($\\pm$1, $\\pm$1.0027 and $\\pm$1.0056 days$^{-1}$) in blue (panels A1, A4, B1, and B4), and 2-day aliases ($\\pm$2.0027 and $\\pm$2.0055 days$^{-1}$) in purple (panels A2, A5, B3, and B5). Arrows at the top of each peak show the phase of each signal for the real (red) and the simulated data (black).}\n\\label{fig:1}\n\\end{figure*}\n\n\n\n\nWe also ran a periodogram analysis on the residuals between the TESS photometry and the best-fit model derived by \\citet{Bourrier2019}. The two strongest peaks are measured at periods of 1.16 and 1.37 days. The first signal corresponds well to the rotational modulation identified in the RV of WASP-121, and likely originates in the same active regions at the surface of the star. WASP-121 was observed over two TESS orbits. We cut each of them in half, and ran independent periodogram analyses on the four resulting segments. The stronger 1.37-day signal is only present in the second TESS orbit, with similar power in its two halves (Fig.~\\ref{fig:TESS:residualspg}). Our best interpretation is that WASP-121 rotates differentially, with the 1.37-day signal arising from active regions located at higher latitudes (and thus rotating slower) than those responsible for the 1.13-day signal. These high-latitude regions would have developed rapidly around epoch $\\sim 1502$ and lasted at least for the rest of the TESS observations. The possibility for differential rotation is investigated in more detail in Sect.~\\ref{sec:fit_RM}. \\\\\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[trim=1.5cm 0.8cm 1.cm 0.5cm,clip=true,width=\\columnwidth]{PAPER_pg_res_Orb2}\n\\caption{\\label{fig:TESS:residualspg} Lomb-Scargle periodogram of the residuals between TESS photometry in orbit 22 and the best-fit model for WASP-121b. The orange dashed line indicates the rotational modulation identified in the RVs at 1.13\\,d. The peak indicated by the red dashed line at 1.37\\,d is only present in this orbit, and likely traces a transient active region at higher stellar latitudes.}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table*} \n\\caption[]{Properties of the WASP-121 system}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{c|c|c|c|c}\n \\hline\n \\hline\n Parameter & Symbol & Value & Unit & Origin \\\\\n \\hline \n\\textit{Stellar properties} & & & & \\\\ \n \\hline \nMass\t\t& $M_{\\star}$ & 1.358$\\stackrel{+0.075}{_{-0.084}}$ & M$_{\\odot}$ & \\citealt{Delrez2016} \\\\\nRadius\t\t& $R_{\\star}$ & 1.458$\\pm$0.030 & R$_{\\odot}$ & \\citealt{Delrez2016} \\\\ \nDensity\t\t& $\\rho_{\\star}$ & 0.434$\\pm$0.038 & $\\rho_{\\odot}$ & \\citealt{Delrez2016}$^{\\dagger}$ \\\\\nLimb-darkening coefficients & $u_{1}$ & 0.364$\\stackrel{+0.034}{_{-0.030}}$ & & EulerCam \\\\\n\t\t\t& $u_{2}$ & 0.146$\\stackrel{+0.066}{_{-0.049}}$ & & EulerCam \\\\\nInclination & $i_\\mathrm{\\star}^{\\rm North}$ & 8.1$\\stackrel{+3.0}{_{-2.6}}$ & deg & RM \\\\\n & $i_\\mathrm{\\star}^{\\rm South}$ & 171.9$\\stackrel{+2.5}{_{-3.4}}$ & deg & RM \\\\\nEquatorial velocity & $v_\\mathrm{eq}$ & [65.28 - 120] & km\\,s$^{-1}$ & RM \\\\\n \\hline\n \\hline \n\\textit{Planetary properties} & & & \\\\ \n \\hline\nTransit epoch\t& $T_{0}$ & 2458119.72074$\\pm$0.00017 & BJD$_\\mathrm{TDB}$ & TESS \\\\ \nOrbital period & $P$ & 1.27492504$^{+1.5\\times 10^{-7}}_{-1.4\\times 10^{-7}}$ & d & (TESS+EulerCam) \\\\ \nScaled semi-major axis & $a_\\mathrm{p}\/R_{\\star}$ & 3.8131$^{+0.0075}_{-0.0060}$ & & (TESS+EulerCam) \\\\\nSemi-major axis & $a_\\mathrm{p}$ & 0.02596$^{+0.00043}_{-0.00063}$ & au & (TESS+EulerCam)$^{\\dagger}$ \\\\\t\t\t\nEccentricity & $e$ & [0 - 0.0032] & & TESS \\\\\nArgument of periastron & $\\omega$ & 10$\\pm$10 & deg & TESS \\\\\nOrbital inclination \t& $i_\\mathrm{p}$ & 88.49$\\pm$0.16 & deg & (TESS+RM) \\\\ \nImpact parameter & $b$ & 0.10$\\pm$0.01 & & (TESS+RM)$^{\\dagger}$ \\\\ \nTransit durations & $T_\\mathrm{14}$ & 2.9053$\\stackrel{+0.0065}{_{-0.0059}}$ & h & TESS$^{\\dagger}$ \\\\\n\t\t\t\t& $T_\\mathrm{23}$ & 2.2605$\\stackrel{+0.0055}{_{-0.0053}}$ & h & TESS$^{\\dagger}$ \\\\\nPlanet-to-star radius ratio & $R^\\mathrm{T}_\\mathrm{p}\/R_{\\star}$ & 0.12355$\\stackrel{+0.00033}{_{-0.00029}}$ & & TESS \n\\\\ \n \t\t\t\t\t\t & $R^\\mathrm{E}_\\mathrm{p}\/R_{\\star}$ & 0.12534$\\stackrel{+0.00043}{_{-0.00060}}$ & & EulerCam\n\\\\ \nRadius & $R^\\mathrm{T}_\\mathrm{p}$ & 1.753$\\pm$0.036 & $R_\\mathrm{Jup}$ & TESS$^{\\dagger}$ \\\\\n & $R^\\mathrm{E}_\\mathrm{p}$ & 1.773$\\stackrel{+0.041}{_{-0.033}}$ & $R_\\mathrm{Jup}$ & EulerCam$^{\\dagger}$ \\\\\nStellar reflex velocity\t& $K$ & 177.0$\\stackrel{+8.5}{_{-8.1}}$ & m\\,s$^{-1}$ & RV \\\\\nMass & $M_\\mathrm{p}$ & 1.157$\\pm$0.070 & $M_\\mathrm{Jup}$ & RV$^{\\dagger}$ \\\\\nDensity & $\\rho_\\mathrm{p}$ & 0.266$\\stackrel{+0.024}{_{-0.022}}$ & g\\,cm$^{-3}$ & (TESS+RV)$^{\\dagger}$ \\\\\nSurface gravity & $g_\\mathrm{p}$ & 9.33$\\stackrel{+0.71}{_{-0.67}}$ & m\\,s$^{-2}$ & (TESS+RV)$^{\\dagger}$ \\\\\nSky-projected obliquity & $\\lambda$ & 87.20$\\stackrel{+0.41}{_{-0.45}}$ & deg & RM \\\\\n3D obliquity & $\\psi^{\\rm North}$ & 88.1$\\pm$0.25 & deg & RM$^{\\dagger}$\\\\\n\t\t\t & $\\psi^{\\rm South}$ & 91.11$\\pm$0.20 & deg & RM$^{\\dagger}$\\\\\t \t\t\t \n \\hline\n \\end{tabular}\n \\begin{tablenotes}[para,flushleft]\n Notes: Values in square brackets indicate the 1$\\sigma$ confidence intervals for the equatorial velocity and eccentricity, whose probability distributions peak at the lower boundary values for these parameters. The 3$\\sigma$ confidence intervals for these parameters are [65.28 - 295]\\,km\\,s$^{-1}$ and [0 - 0.0078]. Properties with TESS origin are reported from \\citet{Bourrier2019}, or revised when combined with other datasets. Coefficients $u_1$ and $u_2$ are associated with a quadratic limb-darkening law. The daggers indicate derived parameters. Planetary density and surface gravity were calculated using the lowest planet-to-star radius ratio (from TESS). There are two possible solutions for the stellar inclination and 3D obliquity of WASP-121b, marked as \\textit{North} or \\textit{South} depending on which pole of the star is visible. \\\\\n \\end{tablenotes}\n \\end{threeparttable}\n\\label{tab:sys_prop}\n\\end{table*}\n\n\n\n\\section{Reloaded Rossiter-McLaughlin analysis}\n\\label{sec:reloaded RM}\n\n\n\\subsection{HARPS observations of WASP-121}\n\\label{sec:HARPS_data}\n\nWe studied the orbital architecture of WASP-121b and the properties of its host star by analyzing three transit observations obtained with the HARPS echelle spectrograph (HEARTS survey, ESO program 100.C-0750; PI: D. Ehrenreich). Three visits were scheduled on 31 December 2017 (Visit 1), 9 January 2018 (Visit 2), and 14 January 2018 (Visit 3). They lasted between 6.6 and 8.1\\,h, covering the full duration of the transit ($\\sim$2.9\\,h) with sufficient baseline on each side to determine the unocculted stellar properties (Table~\\ref{tab:log}).\n\nObservations were reduced with the HARPS (version 3.8) Data Reduction Software, yielding spectra with resolving power 115,000 and covering the region 380-690 nm. The reduction includes a correction of the color effect due to variability of the extinction caused by Earth's atmosphere during the transit (e.g., \\citealt{bourrier2014b}, \\citealt{Bourrier_2018_Nat}). The spectrum of a hot F-type star such as WASP-121 contains far fewer absorption lines than later-type stars. Including these absent lines into the mask would reduce the contrast of the CCF and the precision on their derived properties. Furthermore, the fast rotation of WASP-121 broadens the stellar lines, blending lines that are isolated in the spectrum of colder stars. A single mask line needs to be associated with unresolved stellar lines contributing to the same blended line to avoid introducing correlated information into the CCF. We thus computed CCFs for each spectral order using a custom mask specific to WASP-121 (this mask is available in electronic form at the CDS). All measured 1D spectra were averaged and smoothed with a 0.09\\,\\AA\\, moving average. The continuum was estimated by running a local maximum detection algorithm on the spectrum, using an alpha-shape algorithm to remove unreliable local maxima and applying a cubic interpolation on the remaining maxima to extrapolate the continuum on the full wavelength grid. The stellar lines to be included in our custom mask were then defined as a local minimum surrounded by two local maxima. A first estimate of the detectable lines and their position was made by running a local minimum detection algorithm on the stellar spectrum. Positions were then derived more accurately as the minimum of a parabola fit around the line minimum in a window of $\\pm$3\\,km\\,s$^{-1}$. We discarded lines with windows smaller than 5 pixels, lines with derived centers farther away than 0.03\\,\\AA\\, from the local minimum, and shallow lines with relative flux difference between the local minimum and highest local maxima smaller than 0.05. Last, we generated a synthetic telluric spectrum with Molecfit \\citep{Smette2015}, and removed mask lines for which the core of a neighboring telluric line (with depth ratios with the mask line higher than 2\\%) entered the region defined by the two local maxima of the mask line for at least one Earth barycentric RV of the spectrum. The final mask is composed of 1828 lines. Their weights were set to the relative flux difference between the stellar lines local minimum and the average of their two local maxima (\\citealt{pepe2002}). \n\nBecause the CCFs generated with the HARPS DRS are oversampled with a step of 0.25\\,km\\,s$^{-1}$, and have a pixel width of about 0.8\\,km\\,s$^{-1}$, we kept one in four points in all CCFs prior to their analysis (\\citealt{Cegla2017}). Here we note that by construction, our custom mask lines are at rest in the stellar rest frame. The null velocity in the CCFs calculated with this mask thus directly corresponds to the stellar rest velocity. \\\\\n\n\n\n\\begin{table} \n\\caption[]{Log of WASP-121 b HARPS transit observations}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{c|c|c|c}\n \\hline\n \\hline\n Visits & 1 & 2 & 3 \\\\\n \\hline\n Date (start) & 31-12-17 & 09-01-18 & 14-01-18 \\\\\n Number of exposures & 35 & 55 & 50 \\\\\n Exposure duration (s) & 570-720 & 500-600 & 500-660 \\\\ \n Exposure S\/N (550\\,nm) & 26-61 & 21-43 & 32-49 \\\\\n \\hline\n \\end{tabular}\n \\begin{tablenotes}[para,flushleft]\n Notes: The S\/N is given per pixel.\\\\\n \\end{tablenotes}\n \\end{threeparttable}\n\\label{tab:log}\n\\end{table}\n\n\n\n\n\n\\subsection{Simultaneous EulerCam photometry}\n\\label{sec:LC_fit}\n\nThe reloaded RM technique requires knowledge of the transit light curve in the spectral band of the CCFs (\\citealt{Cegla2016}). Measuring the transit simultaneously in photometry and spectroscopy further allows us to determine occulted spots and plages along the chord that is transited by the planet. We therefore obtained simultaneous photometry throughout the transits in Visits 1 and 2 using EulerCam at the 1.2m Euler telescope at La Silla. Observations were carried out using an r'-Gunn filter to match the HARPS wavelength band as closely as possible, and we applied a slight defocus to the telescope to improve the target point spread function (PSF) sampling and the observation efficiency. After standard image correction procedures, we extracted relative aperture photometry, iteratively selecting a set of stable reference stars and testing a number of extraction apertures. For details on EulerCam and the relevant observation and data reduction procedures, see \\citet{Lendl2012}. We combined the new broadband photometry with two archival EulerCam light curves observed in r' band on 2014 January 19 and 23 (Fig.~\\ref{fig:EULER_LC_all}, \\citealt{Delrez2016}). The four light curves are shown in Fig.~\\ref{fig:EULER_LC_all}, and they are available in electronic form at the CDS. \n\nThe increased scatter during the transit on 2014 January 23 is caused by the passage of a cloud. The light curve obtained in Visit 1 shows a much shallower transit than the light curves that were obtained during the other epochs. We did not find any large variation (beyond the mmag level) in the overall stellar brightness between Visits 1 and 2, as are created, for example, by changing star spot coverage, which would translate into an offset in the measured transit depth. No variations in transit depth similar to that of Visit 1 are found in any of the 17 transits observed with TESS (\\citealt{Bourrier2019}). We lack a convincing physical explanation for this anomaly, and suggest that instrumental effects linked to image saturation are likely the origin of this variation. Indeed, the target saturated the detector near the transit center in Visit 1, and the data are therefore likely affected by detector nonlinearity at high flux levels. The light curve from Visit 1 was therefore excluded from further analyses. \\\\\n\nWe made use of the MCMC code described in \\citet{Lendl2017} to fit the EulerCam data. We assumed a uniform prior distribution for the star-to-planet radius ratio (i.e., this parameter was fit without any a priori constraints), and placed normal prior distributions on the impact parameter, the transit duration, the mid-transit time, and the planetary period. These priors were centered on the values derived in \\citet{Bourrier2019}, and their width corresponds to the respective 1\\,$\\sigma$ uncertainties. We used the routines of \\citet{Espinoza2015} to compute quadratic limb-darkening coefficients, using a wide ($\\sigma_{prior}=0.1$) normal prior distribution centered on these values in our analysis. Our code allows for the use of parametric baseline models (see, e.g., \\citealt{Gillon2010}), and we find that the light curves of 2014 January 19 and 23, and 2018 January 14 are best fit by models of the form $p(t^2)+p(\\mathit{xy}^1) + p(\\mathit{FWHM}^1)$, $p(t^1)+ p(\\mathit{FWHM}^2)$, and $p(t^2)+p(\\mathit{xy}^1)+p(\\mathit{FWHM}^2)$, respectively, where $p(\\mathit{i}^n)$ refers to a polynomial of order $n$ in parameter $i$. The parameters are the time $t$, coordinate shifts $xy$, and stellar $\\mathit{FWHM}$. System properties specific to the EulerCam passband are given in Table~\\ref{tab:sys_prop}. The baseline between the EulerCam observations is long, as transits are separated by several years, and therefore these transits improved the precision on the orbital period and semi-major axis compared to the TESS fit. We updated their values accordingly.\\\\\n\nWe compared our measurement for R$_\\mathrm{p}\/$R$_{*}$ (0.1253$\\stackrel{+0.0004}{_{-0.0006}}$ from EulerCam in 619-710\\,\\AA\\,) and that of \\citet{Bourrier2019} (0.1236$\\pm{0.0003}$ from TESS in 600-1000\\,\\AA\\,) with those of \\citet{Evans2018} obtained with the G430 and G750 HST\/WFC3 grisms, averaging their measurements within the EulerCam and TESS respective passbands. The \\citet{Evans2018} results yield R$_\\mathrm{p}\/$R$_{*}$ = 0.12238$\\pm$0.00036 (EulerCam) and 0.12244$\\pm$0.00021 (TESS), which is significantly lower than the EulerCam and TESS measurements by 0.003 (5.3$\\sigma$) and 0.001 (2.7$\\sigma$), respectively. \\citet{Evans2018} previously noted that their measurements were lower than values obtained using ground-based photometry in the $B$, $r'$, and $z'$ bandpass (\\citealt{Delrez2016}) and proposed that these discrepancies could arise from systematics in the latter measurements. Interestingly, our planet-to-star radius ratios are consistent with those obtained by \\citet{Delrez2016} in the bands that overlap with those of EulerCam (0.12521$\\pm$0.0007 in 555-670\\,\\AA\\,) and TESS (0.12298$\\pm$0.0012 in 836-943\\,\\AA\\,). The good agreement between ground- and space-based measurements might suggest that the reduction procedure or systematics specific to the HST data might have offset the transit depths derived by \\citet{Evans2018}.\\\\\n\n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0.cm 2.cm 0.cm 0cm,clip=true,width=\\columnwidth]{EULER_LC_all}\n\\caption[]{Transit light curves of WASP-121b obtained with EulerCam, offset by 0.02 for visibility. Best-fit models fitted to the 2014 and 2018 data are shown in red. They include a common transit model and a detrending model specific to each visit. The abnormal shape of the 2017 light curve is likely due to instrumental effects. For this epoch we only overplot the transit model.}\n\\label{fig:EULER_LC_all}\n\\end{figure}\n\\end{center}\n\n\n\n\n\n\n\n\\subsection{Analysis of the local stellar CCFs}\n\\label{sec:extra}\n\nThe HARPS CCFs (heareafter CCF$_\\mathrm{DI}$) originate from starlight integrated over the disk of WASP-121. We used the reloaded RM technique (\\citealt{Cegla2016}, see also \\citealt{Bourrier2017_WASP8,Bourrier_2018_Nat}) to isolate the local CCF (hereafter CCF$_\\mathrm{loc}$) from the regions of the photosphere that are occulted by WASP-121b. The CCF$_\\mathrm{DI}$ calculated in the stellar rest frame were first corrected for the stellar Keplerian motion induced by WASP-121b. We identified the CCF$_\\mathrm{DI}$ obtained outside of the transit, taking care to exclude those that even partially overlapped with the transit window, and coadded them to build a ``master-out'' CCF$_\\mathrm{DI}$ in each night, which corresponds to the unocculted star. The continua of the master-out and individual CCF$_\\mathrm{DI}$ outside of the transit were normalized to the same continuum at unity, while in-transit CCF$_\\mathrm{DI}$ were scaled to reflect the planetary disk absorption. This scaling was made using the theoretical transit light curve derived from the fit to the EulerCam data (Sect.~\\ref{sec:LC_fit}), whose spectral range is closer to that of HARPS than that of TESS. \n\nThe CCF$_\\mathrm{loc}$ associated with the planet-occulted regions were retrieved by subtracting the scaled in-transit CCF$_\\mathrm{DI}$ from the master-out in each night. The local stellar line profiles from the planet-occulted regions of the photosphere are clearly visible in Fig.~\\ref{fig:2D_maps}. They are always redshifted, and this redshift slightly increases along the transit chord. WASP-121b therefore always transits that hemisphere of the star that rotates away from us, with a transit chord farther from the projected stellar spin axis at egress than at ingress. This preliminary analysis implies that the sky-projected obliquity $\\lambda$ must be slightly lower than 90$^{\\circ}$, in contrast to the value of 102.2$\\pm$5.5$^{\\circ}$ (using the same convention as in the present study) derived by \\citealt{Delrez2016} from a classical velocimetric analysis of the RM effect in CORALIE data.\n\n\n\\begin{center}\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[trim=0cm 0cm 0cm 0cm,clip=true,width=\\columnwidth]{2D_maps_col}\n\\caption[]{Maps of the residuals between the scaled CCF$_\\mathrm{DI}$ and their master-out in each visit. Residuals are colored as a function of their flux, and plotted as a function of RV in the stellar rest frame (in abscissa) and orbital phase (in ordinate). The vertical dashed black line indicates the stellar rest velocity. Horizontal dotted lines are the transit contacts. In-transit residuals correspond to the CCF$_\\mathrm{loc}$, and show the average local stellar line profile (recognizable by a lower flux in the CCF$_\\mathrm{loc}$ cores) from the planet-occulted regions of the stellar disk. For comparison, the spectroscopic width of the disk-integrated stellar lines is about 14\\,km\\,s$^{-1}$. Black crosses with error bars indicate the centroids of the detected stellar line profile. The slanted dashed black line tracks the orbital trajectory of the planet.}\n\\label{fig:2D_maps}\n\\end{figure}\n\\end{center}\n\n\nThe RV centroids of the CCF$_\\mathrm{loc}$ can generally be derived from a simple Gaussian fit. The CCFs generated with our custom mask for WASP-121, however, show side lobes that would limit the precision of CCF properties derived with a Gaussian fit. Therefore, we used the double-Gaussian model introduced by \\citet{Bourrier_2018_Nat} for the M dwarf GJ\\,436, which consists of the sum of a Gaussian function representing the CCF continuum and side lobes, and an inverted Gaussian function representing the CCF core. As illustrated in Fig.~\\ref{fig:CCF_DI_fit}, the double-Gaussian model reproduces the entire CCF of WASP-121 well and thus exploits the full information contained in its profile. \n\nWe performed a preliminary fit to the CCF$_\\mathrm{loc}$ using a double-Gaussian model where the FWHM ratio, contrast ratio, and centroid difference between the core and lobe components were set to the best-fit values for the nightly master-out CCF$_\\mathrm{DI}$ (as in \\citealt{Bourrier_2018_Nat}). The local average stellar line is considered detected if the amplitude of the model CCF$_\\mathrm{loc}$ (defined as the flux difference between the minimum of the model CCF$_\\mathrm{loc}$ and its continuum) is three times larger than the dispersion in the continuum of the observed CCF$_\\mathrm{loc}$. This led us to discard a few CCF$_\\mathrm{loc}$ that were located very near the stellar limb, where the lower flux and partial occultation by the planet yield very low S\/Ns ratios. The remaining CCF$_\\mathrm{loc}$ were shifted to the same rest velocity and averaged on each night to create a master local CCF$_\\mathrm{loc}$ (e.g., \\citealt{Wyttenbach2017}). The comparison with the master CCF$_\\mathrm{DI}$ (Fig.~\\ref{fig:Compa_Models_Out_Loc}) clearly shows the effect of rotational broadening; the local average stellar line is far narrower and deeper than the disk-integrated line. Both CCFs show sidelobes, which are well fit with a double-Gaussian model but with different properties. In the master CCF$_\\mathrm{loc}$ the lobe component is broader, and more redshifted relative to the core component, than in the master CCF$_\\mathrm{DI}$. The final fit to the CCF$_\\mathrm{loc}$ in individual exposures was performed with a double-Gaussian model where the core and lobe components were linked as in the nightly master CCF$_\\mathrm{loc}$. Flux errors assigned to the CCF$_\\mathrm{loc}$ were set to the standard deviation in their continuum flux, and the uncertainties on the derived parameters were set to the 1\\,$\\sigma$ statistical errors from a Levenberg-Marquardt least-squares minimisation. The local stellar surface RVs were defined as the derived centroids of the CCF$_\\mathrm{loc}$ core component.\\\\\n\n\n\n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0cm 0cm 0cm 0cm,clip=true,width=\\columnwidth]{CCF_DI_fit.pdf}\n\\caption[]{Typical CCF$_\\mathrm{DI}$ integrated over the disk of WASP-121 (blue points, obtained during one of the out-of-transit exposures in Visit 2). The solid black profile is the best-fit double-Gaussian model to the measured CCF. The dashed black profiles show the individual Gaussian components to this model, which yields a low dispersion on the fit residual (bottom panel). The blue shaded regions indicate the velocity ranges used to define the CCF continuum.}\n\\label{fig:CCF_DI_fit}\n\\end{figure}\n\\end{center}\n\n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[trim=1.5cm 0cm 1cm 0cm,clip=true,width=\\columnwidth]{Compa_Models_Out_Loc.pdf}\n\\caption[]{Master-out CCF$_\\mathrm{DI}$ (magenta) and master local CCF$_\\mathrm{loc}$ (blue), binned over the three visits and normalized to the same continuum. The dashed and dotted black profiles show the best-fit models to the master-out and master-local, respectively. They are based on the same double-Gaussian model, but with different correlations between the properties of the lobe and core Gaussian components.}\n\\label{fig:Compa_Models_Out_Loc}\n\\end{figure}\n\\end{center}\n\n\n\n\n\n\n\n\\subsection{Analysis of the stellar rotation and orbital architecture}\n\\label{sec:fit_RM}\n\n\\subsubsection{Model and prior constraints}\n\\label{sec:priors_RM}\n\nDespite some variability, the local RVs follow a similar trend in the three visits (Fig.~\\ref{fig:RV_local}). They become more redshifted along the transit chord and remain always positive, confirming the preliminary interpretation performed in Sect.~\\ref{sec:extra} of a near-polar orbit only crossing the redshifted stellar hemisphere. The orbital architecture of the system and the properties of the velocity field of the stellar photosphere can be derived from the fit to the local RVs using the reloaded RM model (\\citealt{Cegla2016}; see their Figure 3 for the definitions of the coordinate system and angle conventions), which calculates brightness-weighted theoretical RVs averaged over each planet-occulted region. In previous reloaded RM studies (\\citealt{Cegla2016}, \\citealt{Bourrier2017_WASP8, Bourrier_2018_Nat}) the model was fit to the data by varying the stellar projected rotational velocity $v_{\\rm eq}\\sin i_{*}$ (and in some cases, the differential rotation or convective motions of the stellar photosphere) and the sky-projected obliquity $\\lambda$. The latter parameter alone thus controlled the model planet trajectory, and the coordinates of the occulted regions. The near-polar orbit of WASP-121b, however, results in $v_{\\rm eq}\\sin i_{*}$ being strongly degenerate with the planet impact parameter (see Appendix~\\ref{apn:polar_orb}), which remains poorly determined because of the uncertainty on the orbital inclination $i_\\mathrm{p}$ (\\citealt{Bourrier2019}). Interestingly, an impact parameter close to zero would require that the planet cross the projected stellar spin axis (where local RVs are zero), which is incompatible with the transit of a single stellar hemisphere indicated by the positive local RVs series. This means that more stringent constraints can be derived on the orbital inclination from the fit to the local RVs, and we therefore modified the reloaded RM model to include $i_\\mathrm{p}$ as a free parameter. The scaled semi-major axis of WASP-121b has less influence on the local RVs and is much better determined than $i_\\mathrm{p}$. After checking that $a_\\mathrm{p}\/R_{*}$ could not be better constrained through the fit, we therefore fixed it to its nominal value. Similarly, the other orbital properties and the ephemeris of WASP-121b are known to a much higher precision through transit photometry and velocimetry than could be obtained via the fit to the local RVs, and they were accordingly fixed to their nominal values. The planet-to-star radius ratio and the stellar limb-darkening coefficients cannot be retrieved from the fit to the local RVs because absolute flux levels are lost in HARPS ground-based data. We note that the measured local RVs do not depend on our choice for $i_\\mathrm{p}$ because the photometric scaling of the CCFs was performed directly with the transit light-curve model fit to the simultaneous EulerCam data (Sect.~\\ref{sec:extra}). \\\\\n\n\n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0cm 0.cm 0cm 0cm,clip=true,width=\\columnwidth]{PAPER_WASP121b_RV_stsurf_phase.pdf}\n\\caption[]{Radial velocities of the stellar surface regions occulted by WASP-121b as a function of orbital phase. Horizontal bars show the exposure durations. The black curve is the best-fit reloaded RM model (indistinguishable between the low- and high- $i_{*}$ solutions) to the three visits. Dashed vertical lines are the transit contacts. The horizontal dashed line highlights the stellar rest velocity, which is found along the projected stellar spin axis. \\textbf{Upper panel}: Local RVs in individual Visits 1 (red), 2 (gold), and 3 (blue). \\textbf{Bottom panel}: Local RVs derived from the CCF$_\\mathrm{loc}$ binned over the three visits, shown separately for the sake of clarity.}\n\\label{fig:RV_local}\n\\end{figure}\n\\end{center}\n\n\nAdditional constraints can be set from the independent measurements of stellar line broadening and the stellar rotational period. \\citet{Delrez2016} derived a spectroscopic value $v_{\\rm eq}\\sin i_{*\/\\rm spec}$ = 13.56$\\stackrel{+0.69}{_{-0.68}}$\\,km\\,s$^{-1}$ from the fit to stellar \\ion{Fe}{i} lines in CORALIE spectra. A similar estimate can be derived from the comparison between the HARPS master-out CCF$_\\mathrm{DI}$ and master-local CCF$_\\mathrm{loc}$. Under the assumption that CCF$_\\mathrm{loc}$ measured along the transit chord are representative of the entire stellar disk, the observed master-out was fit by tiling a model star with the master-local CCF$_\\mathrm{loc}$, weighted by the limb-darkening law derived from the EulerCam photometry, and shifted in RV position by the solid rotation of the photosphere, which was let free to vary. We obtain a good fit for $v_{\\rm eq}\\sin i_{*} \\sim $13.9\\,km\\,s$^{-1}$ (Fig.~\\ref{fig:Fit_Mout_Mloc}), suggesting that the local average stellar line profile does not change substantially across the stellar disk within the precision of HARPS, and that $v_{\\rm eq}\\sin i_{*\/\\rm spec}$ can be used as a prior for the stellar projected rotational velocity. \n\nAnalysis of ground-based spectroscopy and TESS photometry of WASP-121 (Sect.~\\ref{sec:RV_fit}) revealed a persistent rotational modulation at 1.13 days, and a transient modulation at 1.37 days. We understand these results as an indication of differential rotation, with the equator of WASP-121 rotating at least as fast as the latitudes probed by the 1.13 days signal, and the transient signal arising from higher latitudes that rotate more slowly. This sets a prior on the stellar equatorial velocity $v_{\\rm eq}\\geqslant$ 65.28\\,km\\,s$^{-1}$.\n\nThe three local RV series were simultaneously fit with the updated RM model. We assumed a solar-like differential rotation law P$(\\theta_\\mathrm{lat})$ = P$_\\mathrm{eq}\/(1 - \\alpha$\\,sin$^{2}(\\theta_\\mathrm{lat}))$, where $\\theta_\\mathrm{lat}$ is the stellar latitude and $\\alpha = 1 - $P$_\\mathrm{eq}\/$P$_\\mathrm{pole}$ is the relative differential rotation rate (\\citealt{Cegla2016}). We accounted for the blur caused by the planetary motion during a HARPS exposure by oversampling the transit chord between the planetary position at the beginning and end of each exposure (\\citealt{Bourrier2017_WASP8}). We sampled the posterior distributions of the model parameters using \\textit{emcee} MCMC (Foreman2013), as in \\citet{Bourrier_2018_Nat}. Jump parameters for the MCMC are the stellar equatorial velocity $v_{\\rm eq}$, the cosine of the stellar inclination cos$(i_{*})$, the sky-projected obliquity $\\lambda$, the orbital inclination $i_\\mathrm{p}$, and the differential rotation rate $\\alpha$. We set a uniform prior on $v_{\\rm eq}$ and a Gaussian prior on $v_{\\rm eq}\\sin i_{*\/\\rm spec}$, following the above discussion. The posterior distribution from the fit to TESS photometry was used as prior for $i_\\mathrm{p}$ (\\citealt{Bourrier2019}). Uniform priors were set on the other parameters over their definition range: [-1 ; 1] for cos$(i_{*})$, [-180 ; 180]$^{\\circ}$ for $\\lambda$, and [-1 ; 1] for $\\alpha$. \\\\\n\n\n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[trim=1cm 0cm 1cm 0.5cm,clip=true,width=\\columnwidth]{PAPER_WASP121b_scr_Mout_Mloc_binned_HARPS-binned.pdf}\n\\caption[]{Master-out CCF$_\\mathrm{DI}$ (thick magenta line) and its best fit (black line) obtained by tiling a model star with the limb-darkened master local CCF$_\\mathrm{loc}$, used as a proxy for the specific stellar intensity profile. The best-fit value for the projected stellar rotational velocity, which controls the spectral position of this profile over the model stellar disk, is in good agreement with its measured spectroscopic value and yields a good fit to the observed master-out CCF$_\\mathrm{DI}$.}\n\\label{fig:Fit_Mout_Mloc}\n\\end{figure}\n\\end{center}\n\n\n\n\n\n\\subsubsection{Results}\n\\label{sec:results_RM}\n\nPosterior probability distributions are shown in Figs.~\\ref{fig:PD_lowi} and \\ref{fig:PD_highi}. Best-fit values for the model parameters were set to the median of their distributions, and are given in Table~\\ref{tab:sys_prop}. Some of the parameter distributions are asymmetrical, and we therefore chose to define their $1\\sigma$ uncertainties using the highest density intervals, which contain 68.3\\% of the posterior distribution mass such that no point outside the interval has a higher density than any point within it. The probability distributions show unique solutions for all model parameters, except for the stellar inclination. While we find that WASP-121 is highly inclined, the data do not allow us to distinguish whether the south pole ($i_\\mathrm{*}$ = 171.9$\\stackrel{+2.5}{_{-3.4}}^{\\circ}$) or the north pole ($i_\\mathrm{*}$ = 8.1$\\stackrel{+3.0}{_{-2.6}}^{\\circ}$) is visible. Both scenarios yield similar $\\chi^{2}$ of 111 for 43 degrees of freedom. The relatively high reduced $\\chi^{2}$ (2.6) is caused by the dispersion of the local RV measurements between the three nights. Deviations from the nominal best-fit model beyond the photon noise are present in all nights and in all phases of the transit, suggesting that variability in the local photospheric properties of this active star could be the origin of these variations. The noise in individual CCF$_\\mathrm{loc}$ prevents us from searching for variations in their bisector span. No clear correlations were found between the local RVs and the FWHM or contrast of the CCF$_\\mathrm{loc}$, with the EulerCam photometry, or with the Ca\\,II$_\\mathrm{HK}$, H$\\alpha$, and Na activity indexes. We show the best-fit model for the local stellar RVs in Fig.~\\ref{fig:RV_local} and the orbital architecture corresponding to the visible north pole in Fig.~\\ref{fig:disque}. \n\nThe stellar equatorial rotation remains poorly constrained, with a highest density interval of [65.28 - 120]\\,km\\,s$^{-1}$ that corresponds to rotation periods between [0.61 - 1.13]\\,days. The probability distribution for $v_\\mathrm{eq}$ nonetheless favors low velocities, which suggests that the persistent 1.13-day signal measured in photometry and ground-based data arises from active regions close to the stellar equator. We cannot confirm the differential rotation of WASP-121, with $\\alpha$ = 0.08$\\stackrel{+0.11}{_{-0.13}}$, but this result excludes high differential rotation rates and is consistent within 1$\\sigma$ with the observed rotational modulations. Indeed, the constraints $P_\\mathrm{eq}\\leqslant$ 1.13\\,days and $P_\\mathrm{pole}\\geqslant$ 1.34\\,days imply $\\alpha\\geqslant$ 0.16. These results are also consistent with measurements obtained for Kepler stars by \\citet{Balona2016}, who showed that $|\\alpha|$ ranges between 0 and 0.2 for stars with rotation periods on the order of 1 day (see their Figure 9). We note that even in the case of differential rotation, the signal measured at $\\sim$8.4\\,days (Sect.~\\ref{sec:Prot}) cannot trace the rotational modulation of a high-latitude region because the lowest $\\alpha$ required would be 0.87 at the stellar poles. Measurements of the local surface RVs at higher S\/N, for instance, with the ESPRESSO spectrograph, will be crucial in assessing the differential rotation of WASP-121.\\\\\n\nThe orbit of WASP-121b is almost but not exactly edge-on ($i_\\mathrm{p}$ = 88.49$\\pm$0.16$^{\\circ}$) and polar ($\\lambda$ = 87.20$\\stackrel{+0.41}{_{-0.45}}^{\\circ}$). We substantially improved the precision on these properties compared to previous studies, and find that $\\lambda$ is 15$^{\\circ}$ lower (3$\\sigma$) than the value derived by \\citet{Delrez2016} (we converted their spin-orbit angle $\\beta$ = 257.8$^{\\circ}$ in the same frame as our study). We combined the probability distributions of $i_\\mathrm{p}$, $\\lambda$, and $i_{*}$ to derive the 3D obliquity of the system, $\\psi$ = arccos(sin\\,$i_{*}$ cos\\,$\\lambda$ sin\\,$i_\\mathrm{p}$ + cos\\,$i_{*}$ cos\\,$i_\\mathrm{p}$), and measure $\\psi^{\\rm South}$ = 91.11$\\pm$0.20$^{\\circ}$ (stellar south pole visible) or $\\psi^{\\rm North}$ = 88.1$\\pm$0.25$^{\\circ}$ (north pole visible). We note that our result for the obliquity does not change the conclusion by \\citet{Delrez2016} that WASP-121b is on a highly misaligned orbit, and that it likely underwent strong dynamical interactions with a third companion, possibly an outer planet, during the life of the system (1.5$\\pm$1.0\\,Gyr). The dynamical evolution of WASP-121b is now controlled by tidal interactions with the star, leading to a gradual decrease in the obliquity and semi-major axis of the planet and to its eventual disruption (\\citealt{Delrez2016}). Even with a strong tidal dissipation, however, it would take millions of years to decrease the obliquity by one degree (\\citealt{Delrez2016}), and our value for the semi-major axis is not significantly lower than that of \\citealt{Delrez2016}. This mechanism therefore cannot explain the difference between our measurement for $\\lambda$ and that of \\citet{Delrez2016}, which could be due to a bias induced by their use of the classical RM technique (\\citealt{Cegla2016a}). An interesting alternative might be the nodal precession of the orbit, however, as is the case for the ultra-hot Jupiter WASP-33b (\\citealt{Johnson_WASP33}). The uncertainties on the orbital inclination and obliquity from \\citet{Delrez2016} prevent us from measuring a clear variation in the argument of the ascending node, with a decrease of -0.95$\\pm$0.64$^{\\circ}$ in about three years (from the end of 2014 to the end of 2017). Interestingly this decrease would correspond to a stellar gravitational quadrupole moment of 9.0$\\times$10$^{-4}$ (for $\\Psi_\\mathrm{North}$) or -1.5$\\times$10$^{-4}$ (for $\\Psi_\\mathrm{South}$), however, calculated with the equation in \\citet{Barnes2013}. A negative moment is excluded by the expected oblateness of WASP-121, but the former solution is on the same order as moments estimated for the early-type fast-rotating star WASP-33 (\\citealt{Johnson_WASP33,Johnson_WASP33_erratum}, \\citealt{Iorio2016}).\\\\\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0.5cm 3cm 0.5cm 1cm,clip=true,width=\\columnwidth]{WASP121b_st_disk_ross_lowistar.pdf}\n\\caption[]{Projection of WASP-121 in the plane of sky for the best-fit scenario where the north pole of the star is visible. The stellar spin axis is displayed as a black arrow extending from the north pole. The stellar equator is represented as a black line, solid when visible and dashed when hidden from view. The stellar disk is colored as a function of its RV field. The normal to the orbital plane is shown as a green arrow, and the orbital trajectory is displayed as a green curve. The black disk is WASP-121b, to scale.}\n\\label{fig:disque}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Distinguishing the planetary and stellar atmospheres}\n\\label{sec:atmo_struc}\n\nAtmospheres of ultra-hot Jupiters were recently found to contain atomic metal species (e.g., \\citealt{Hoeijmakers2018, Hoeijmakers2019}), which are prevented from condensing by the high temperatures (e.g., \\citealt{Visscher2010}, \\citealt{Wakeford2017}). As an ultra-hot Jupiter, WASP-121b receives extreme amounts of stellar radiation and likely undergoes atmospheric escape orders of magnitude larger than for hot Jupiters (\\citealt{Salz2019_NUV_WASP121b}). Magnesium and iron ions were recently detected in its exosphere through their near-UV absorption lines (\\citealt{Sing2019}), consistent with the marginally larger transit depth measured in broadband near-UV by \\citet{Salz2019_NUV_WASP121b}. These metallic species likely become photoionized within the exosphere after being carried upward by the hydrodynamically expanding upper atmosphere. They could be present in their neutral form in the atmosphere of WASP-121b, and yield strong absorption in optical lines. The custom mask we built to define the CCFs of WASP-121 is based on the stellar spectral absorption lines (Sect.~\\ref{sec:HARPS_data}), most of which arise from iron in the stellar atmosphere. Indeed, cross-matching our mask with the VALD database (\\citealt{Piskunov1995,Kupka2000,Ryabchikova2015}) shows that of their 989 lines they have in common, more than half (570) arise from neutral iron. The second most frequent species is neutral nickel, with 67 lines. This means that if the atmospheric limb of WASP-121b contains atomic iron, we would expect its average signature to be superimposed on the stellar CCF measured during transit. We present here a summary of the technique that we devised to search for and extract the atmospheric absorption signal of an exoplanet, based on the reloaded RM approach. It will be fully described in a forthcoming paper. \\\\\n\n\\subsection{Method}\n\nThe in-transit CCF$_\\mathrm{loc}$ extracted in Sect.~\\ref{sec:extra} corresponds to the specific intensity spectrum of the occulted stellar region, multiplied by the wavelength-dependant occulting area of the planet. The intensity spectrum contains the cross-correlated absorption line from the stellar photosphere, centered at the RV of the occulted region. The occulting area is the sum of the continuum level set by the opaque planetary layers (here averaged over the HARPS band) and the equivalent surface of the atmospheric limb. If the planet contains species that absorb the CCF mask lines, this surface corresponds to the cross-correlated absorption line from the planetary atmosphere, centered at the orbital RV of the planet. When the stellar and planetary absorption lines follow sufficiently different tracks in RV-phase space, as is the case with WASP-121b (Fig.~\\ref{fig:2D_maps}), it is possible to distinguish their individual contributions from the CCF$_\\mathrm{loc}$.\\\\\n\n\\begin{enumerate}\n\\item The first step consists of subtracting the stellar light that is occulted by the planetary continuum from the CCF$_\\mathrm{loc}$. To do this, we used the master CCF$_\\mathrm{loc}$, assuming that it is representative of the individual CCF$_\\mathrm{loc}$ along the transit chord (see Sect.~\\ref{sec:extra}). The master was rescaled to the correct photometric level using the best-fit EulerCam transit model (Sect.~\\ref{sec:LC_fit}), which accounts for the limb-darkening and planetary continuum associated with each exposure. The rescaled master was then shifted to the RV of the planet-occulted regions, calculated with the best-fit model for the local stellar surface RVs (Sect.~\\ref{sec:results_RM}). These operations yield the CCF of the product between the local stellar spectra and the transmission spectrum of the atmospheric limb in each exposure.\n\n\\item The second step consists of dividing these CCFs by the master CCF$_\\mathrm{loc}$, rescaled and shifted as described in the first step, to isolate the cross-correlated absorption line of the atmospheric limb, or CCF$_\\mathrm{atm}$. The scaling was made using the total surface of the star rather than the surface associated with the planetaty continuum, to obtain CCF$_\\mathrm{atm}$ in classical units of absorption relative to the stellar surface. The RV-phase maps of the CCF$_\\mathrm{atm}$ from WASP-121b reveal a bright streak aligned with the orbital trajectory of the planet, which is visible only during transit, and is therefore consistent with absorption by metals in the atmosphere of WASP-121b. \n\n\\item The third and last step consists of shifting all CCF$_\\mathrm{atm}$ into the planet rest frame, and averaging them over exposures where the entire planet occults the star. We calculated the theoretical RV track of the planet in the stellar rest frame using the orbital properties of the planet listed in Table~\\ref{tab:sys_prop}. Ingress and egress are excluded because they probe a smaller fraction of the planetary atmosphere that varies in time. Observing WASP-121b with higher-sensitivity spectrographs such as ESPRESSO might allow studying the shape of the planetary signal during ingress\/egress, and possibly resolving longitudinal variations in the planetary atmosphere. We note that we analyzed the three HARPS visits binned together, because of the small amplitude of the planetary signal, and so that the master CCF$_\\mathrm{loc}$ could be determined with a high SNR. The low dispersion of residuals outside of the planetary track in Fig.~\\ref{fig:2D_atmo_maps} confirms that, within the precision of the HARPS data, the master CCF$_\\mathrm{loc}$ is representative of the stellar line along the transit chord. \n\\end{enumerate}\nThe interest of this approach is that it allows us to directly use the local stellar lines that are measured along the transit chord to correct for the bias of the atmospheric signal induced by the RM effect (e.g., \\citealt{Louden2015}, \\citealt{Casasayas2017,Casasayas2018,Casasayas2019}). One caveat is that step 2 divides the CCF of the product between planetary and stellar lines by the CCF of the stellar lines. Unless all dominant planetary or stellar lines in the CCF mask keep the same profile, this division does not fully remove the contribution of the stellar lines in exposures where they overlap with the planetary lines. We will address this caveat in the forthcoming paper.\\\\\n\nWe performed a preliminary analysis to identify the velocity range that is absorbed by the planetary atmosphere in each exposure. We then carried out the reloaded RM analysis again (Sect.~\\ref{sec:reloaded RM}), excluding these planet-absorbed ranges from the fits to the CCF$_\\mathrm{loc}$ and from the construction of the masters CCF$_\\mathrm{out}$ and CCF$_\\mathrm{loc}$. In four exposures (from phase -0.007 to 0.018) the RVs of the transit chord and planetary orbit are too close to fit the uncontaminated local stellar line and retrieve its centroid (see Fig.~\\ref{fig:2D_maps}). We fit the remaining RVs as in Sect.~\\ref{sec:fit_RM} and found no significant changes in the properties derived in Sect.\\ref{sec:results_RM}. The contamination from the planet likely does not bias the local stellar RVs beyond the precision of the HARPS data, and the contaminated phase range likely has less influence on the RV model than if it were closer to ingress or egress. Future studies of WASP-121b and similar planets using higher-precision spectrographs should nonetheless take special care with planet-contaminated exposures. The final extraction of the planetary signal was performed using the local RV model derived in Sect.\\ref{sec:results_RM} and the new uncontaminated master stellar CCF$_\\mathrm{loc}$. Fig.~\\ref{fig:2D_atmo_maps} shows the final RV-phase map of the CCF$_\\mathrm{atm}$. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0cm 0cm 0.cm 0cm,clip=true,width=\\columnwidth]{binned_HARPS-binned_WASP121b_CCF_res_abs_PAPER.pdf}\n\\caption[]{Map of the atmospheric CCF$_\\mathrm{atm}$ binned over the three visits, colored as a function of absorption, and plotted as a function of RV in the stellar rest frame (in abscissa) and orbital phase (in ordinate). Horizontal dotted lines are the transit contacts. The bright streak is the absorption signature from the planetary atmosphere. It follows the track of the planetary orbital motion (solid green curve), but with a slight blueshift.}\n\\label{fig:2D_atmo_maps}\n\\end{figure}\n\n\n\n\\subsection{Results}\n\nThe master atmospheric signal, shown in Fig.~\\ref{fig:master_atm}, is well fitt with a Gaussian profile. Errors on the master CCF$_\\mathrm{atm}$ were set to the dispersion in its continuum. We measure a significant blueshift of -5.2$\\pm$0.5\\,km\\,s$^{-1}$ in the planetary rest frame, and an FWHM of 14.5$\\pm$1.2\\,km\\,s$^{-1}$. Correcting this width for the HARPS LSF broadening (2.61\\,km\\,s$^{-1}$) and for the blurr induced by the planet motion during an exposure\\footnote{The blur does not quadratically broaden the Gaussian profile of the atmospheric signal. We therefore shifted Gaussian profiles at the rate of the planetary motion during a 690\\,s long exposure and compared their average with the measured signal.} yields a net FWHM of 12.9$\\pm$1.2\\,km\\,s$^{-1}$ for the atmospheric signal. Thermal broadening contributes negligibly to the measured width (FWHM$_\\mathrm{thermal}\\sim$1.5\\,km\\,s$^{-1}$ at 2800\\,K). If WASP-121b is tidally locked, then its atmosphere rotates in the stellar rest frame with the same angular velocity as the planet orbits the star (5.7$\\times$10$^{-5}$\\,rad\\,s$^{-1}$). Accounting for the orbital inclination (but assuming that the planet is not inclined with respect to its orbital plane), we obtain a projected rotational RV of 7.15\\,km\\,s$^{-1}$ for atmospheric layers close to the planet surface, corresponding to an FWHM of 11.9\\,km\\,s$^{-1}$. Planetary rotation (in the stellar rest frame) therefore likely accounts for most of the atmospheric broadening, especially if the measured signal arises from higher altitude where the planetary rotation induces a higher velocity.\\\\\n\nThe measured blueshift could trace fast winds going from the dayside to the nightside along both terminators, as predicted for atmospheric circulation in the lower atmospheric layers of hot Jupiters (\\citealt{Showman2013}). In this scenario the hotspot is expected to be located at the substellar point, as is indeed measured in the TESS phase curve of WASP-121b (\\citealt{Bourrier2019}). However, it might then be expected that heat is efficiently restributed through the fast day- to nightside winds, whereas the phase curve revealed a strong temperature contrast. This might indicate that the iron signal arises from different layers than those probed by the TESS photometry. It has been proposed (\\citealt{Beatty2019,Keating2019}) that the nightsides of most hot Jupiters are covered with clouds of similar composition, which would form at temperatures of about 1100\\,K. With an irradiation temperature of $\\sim$3310\\,K, WASP-121\\,b is in a regime where such clouds are not yet predicted to disperse (\\citealt{Keating2019}). The HARPS measurements might therefore probe absorption signals from layers at lower altitudes than are probed by TESS, where fast day- to nightside winds homogenize temperature longitudinally. Meanwhile, the TESS phase curve could trace emission from high-altitude clouds on the nightside (T$_\\mathrm{night} <$ 2200\\,K at 1$\\sigma$), which would hide the emission from the deeper, hotter regions probed on the dayside (T$_\\mathrm{day}$ = 2870\\,K). Alternatively, the measured blueshift could trace an anisotropic expansion of the upper atmospheric layers, for example, due to the asymmetrical irradiation of the dayside atmosphere (\\citealt{Guo2013}) or its compression by stellar wind and radiation. Interestingly, a stronger but marginal blueshift was measured in the metal species escaping WASP-121b (\\citealt{Sing2019}), supporting the idea that the atmospheric layers are increasingly blueshifted as their altitude increases. We note that varying the stellar mass within its 3$\\sigma$ uncertainties, thus affecting the planet orbital velocity track (e.g., \\citealt{Hoeijmakers2019}), does not change the measured blueshift within its uncertainty. \n\n\n\n\n\nWe do not have the precision required to study individual HARPS exposures, but we analyzed the shape and position of the planetary signal averaged over the first half, and then the second half, of the transit (ingress and egress excluded). We found that the absorption signal maintains the same FWHM (13.2$\\pm$1.1\\,km\\,s$^{-1}$ and 13.2$\\pm$2.0\\,km\\,s$^{-1}$, respectively) but becomes more blueshifted (from -3.82$\\pm$0.48 to -6.63$\\pm$0.86\\,km\\,s$^{-1}$). Interestingly, blueshifted absorption signals whose shift increases during transit have been observed in the near-IR helium lines of extended planetary atmospheres (\\citealt{Allart2018}, \\citealt{Nortmann2018}, \\citealt{Salz2018}). It is unclear whether these features trace material that escapes from WASP-121b and is blown away by the stellar wind or radiation pressure, as no absorption is observed before or after the transits and the absorption profile shows no strong asymmetries. The atmospheric circulation may show strong spatial asymmetries, and the atmospheric limb probes regions with different speeds as the tidally locked planet rotates during transit. Three-dimensional simulations of the planetary atmosphere and more precise observations are required to explore the origin of the measured blueshift.\\\\\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0.5cm 0cm 0.5cm 0.5cm,clip=true,width=\\columnwidth]{MplCCF_res_absCCF.pdf}\n\\caption[]{Master atmospheric CCF$_\\mathrm{atm}$ averaged over the full in-transit exposures. The absorption signal from the planetary atmosphere is clearly detected and well approximated by a Gaussian profile (dashed black profile) with a significant blueshift with respect to the planetary rest velocity (vertical dotted black line). }\n\\label{fig:master_atm}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclu}\n\n\nThe ultra-hot Jupiter WASP-121b, transiting a bright F-type star on a near-polar orbit, offers great opportunities to investigate the dynamical and atmospheric properties of giant planets in extreme gravitational and energetic conditions. \\\\\n\nWe combined RV measurements with EulerCam and published TESS photometry to revise the orbital and bulk properties of the planet. Three HARPS transit observations of WASP-121b were then used to refine the orbital architecture of the system. We applied the reloaded RM method to isolate the properties of the stellar photosphere along the transit chord, using a custom mask to compute the CCF of WASP-121, and simultaneous EulerCam photometry to rescale them to their absolute flux level. Analysis of the local RVs from the planet-occulted regions confirms the near-polar orbit of WASP-121b, which leads to a strong degeneracy between impact parameter and stellar rotational velocity. We thus improved the reloaded RM model to include the orbital inclination and semi-major axis in the fit to the local RVs. We further derived independent constraints on the stellar rotation period by analyzing the activity indexes of the star, and by comparing the shapes of the local and disk-integrated stellar lines. This allowed us to derive the stellar inclination, orbital inclination, and 3D obliquity to a high precision (Table~\\ref{tab:sys_prop}), and to exclude high differental rotation rates for WASP-121. These measurements will be helpful in constraining studies of WASP-121b past and future dynamical evolution. We encourage follow-up transit observations of the planet to monitor a possible evolution of the obliquity and impact parameter that would result from the nodal precession of the orbit.\\\\\n\nThe custom mask used to calculate the CCFs of WASP-121 was built from the stellar lines, most of which arise from iron transitions. The presence of iron is also expected in the atmosphere of ultra-hot Jupiters because the high temperatures prevent it from condensing. As a result, we developed a new method for removing the contribution of the stellar lines from the local CCFs of the planet-occulted regions and isolating the contribution from the planetary atmosphere. This method is based on the possibility of directly deriving from the data the local stellar lines, uncontaminated by the planet, which is possible when the orbital trajectory of the planet and its transit chord across the stellar surface are sufficiently separated in RV-phase space. The application of this method to the HARPS observations of WASP-121b binned over three transits revealed the absorption CCF of iron in the planet atmospheric limb. The width of the signal is consistent with the rotation of WASP-121b, if it is tidally locked. The absorption signal is blueshifted in the planetary rest frame, increasing from -3.82$\\pm$0.48 during the first half of the transit to -6.63$\\pm$0.86\\,km\\,s$^{-1}$ in the second half. This is reminiscent of the effect seen for the ultra-hot gas giant WASP-76\\,b (Ehrenreich et al. 2020). These features could arise from day- to nightside winds along both terminators or from the upward winds of an anisotropically expanding atmosphere, combined with the different regions probed by the atmospheric limb as the planet rotates during transit. Observations at higher spectral resolution and with a better sensitivity, for instance, with the ESPRESSO spectrograph, will enable refining the shape of the signal and its temporal evolution. Similar measurements at other wavelengths, searching for species located in different layers than iron, would furthermore allow us to map the full dynamical structure of the WASP-121b atmosphere.\\\\\n\nLike their colder relatives, ultra-hot Jupiters display a wide range of orbital architectures (from aligned, such as WASP-19b, \\citealt{TregloanReed2013} to nearly polar, such as WASP-121b). Ground-based instruments with high resolving power (e.g., HARPS and ESPRESSO in the visible; CARMENES, SPIRou, and NIRPS in the infrared), will make it possible to investigate in details their dynamical properties and to carry out transmission and emission spectroscopy of their atmosphere, allowing us to identify precisely the signatures of their atomic and molecular components and characterize their 3D atmospheric flows.\\\\ \n\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgements}\nWe thank the referee for their fair and useful review of our study. We thank J.B. Delisle for his advice in correcting for activity in the RV measurements and N. Hara for his help in statistical matters. V.B. and R.A acknowledge support by the Swiss National Science Foundation (SNSF) in the frame of the National Centre for Competence in Research ``PlanetS''. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (project Four Aces, grant agreement No 724427; project Exo-Atmos, grant agreement no. 679633). This publication made use of the Data \\& Analysis Center for Exoplanets (DACE), which is a facility based at the University of Geneva (CH) dedicated to extrasolar planets data visualisation, exchange and analysis. DACE is a platform of the PlanetS NCCR, federating the Swiss expertise in Exoplanet research. The DACE platform is available at https:\/\/dace.unige.ch. N.A-D. acknowledges the support of FONDECYT project 3180063. This work has made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsubsection*{\\bibname}}\n\n\\usepackage{pifont\n\\newcommand{\\cmark}{\\ding{51}}%\n\\newcommand{\\xmark}{\\ding{55}}%\n\n\\usepackage{fullpage}\n\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc} \n\\usepackage{hyperref}\n\\usepackage{booktabs} \n\\usepackage{amsfonts,amsmath,amsthm,amssymb} \n\\usepackage{nicefrac} \n\\usepackage{microtype} \n\\usepackage{graphicx}\n\n\\usepackage{hyperref} \n\\usepackage{url} \n\\usepackage{booktabs} \n\\usepackage{amsfonts,amsmath,amsthm} \n\\usepackage{nicefrac} \n\\usepackage{microtype} \n\\usepackage{graphicx}\n\\usepackage{todonotes}\n\\usepackage{bm}\n\\usepackage{soul}\n\\usepackage{mathrsfs}\n\\usepackage[capposition=bottom]{floatrow}\n\n\\input{math_macros}\n\n\\setlength\\columnsep{0.30in}\n\n\n\n\\title{Harmonizable mixture kernels with variational Fourier features}\n\n\\author{ Zheyang Shen \\qquad Markus Heinonen \\qquad Samuel Kaski \\\\ Helsinki Institute for Information Technology HIIT \\\\ Aalto University}\n\n\n\\begin{document}\n\\twocolumn[\\maketitle]\n\n\\begin{abstract}\nThe expressive power of Gaussian processes depends heavily on the choice of kernel. In this work we propose the novel harmonizable mixture kernel (HMK), a family of expressive, interpretable, non-stationary kernels derived from mixture models on the generalized spectral representation. As a theoretically sound treatment of non-stationary kernels, HMK supports harmonizable covariances, a wide subset of kernels including all stationary and many non-stationary covariances. We also propose variational Fourier features, an inter-domain sparse GP inference framework that offers a representative set of `inducing frequencies'. We show that harmonizable mixture kernels interpolate between local patterns, and that variational Fourier features offers a robust kernel learning framework for the new kernel family. \n\\end{abstract}\n\n\\section{INTRODUCTION}\n\nKernel methods are one of the cornerstones of machine learning and pattern recognition. Kernels, as a measure of similarity between two objects, depart from common linear hypotheses by allowing for complex nonlinear patterns \\citep{vapnik2013nature}. In a Bayesian framework, kernels are interpreted probabilistically as covariance functions of random processes, such as for the Gaussian processes (GP) in Bayesian nonparametrics.\nAs rich distributions over functions, GPs serve as an intuitive nonparametric inference paradigm, with well-defined posterior distributions. \\par\nThe kernel of a GP encodes the prior knowledge of the underlying function. \nThe \\emph{squared exponential} (SE) kernel is a common choice which, however, can only model global monotonic covariance patterns, while generalisations have explored local monotonicities \\citep{gibbs1998bayesian, paciorek2004nonstationary}.\nIn contrast, expressive kernels can learn hidden representations of the data \\citep{wilson2013gaussian}.\\par\nThe two main approaches to construct expressive kernels are composition of simple kernel functions \\citep{archambeau2011multiple, durrande2016detecting, gonen2011multiple, rasmussen2006, sun2018differentiable}, and modelling of the spectral representation of the kernel \\citep{wilson2013gaussian, samo2015generalized, remes2017non}. In the compositional approach kernels are composed of simpler kernels, whose choice often remains ad-hoc.\n\\par\nThe spectral representation approach proposed by \\citet{quia2010sparse}, and extended by \\citet{wilson2013gaussian}, constructs \\emph{stationary} kernels as the Fourier transform of a Gaussian mixture, with theoretical support from the Bochner's theorem. Stationary kernels are unsuitable for large-scale datasets that are typically rife with locally-varying patterns \\citep{samo2016string}. \\citet{remes2017non} proposed a practical \\emph{non-stationary} spectral kernel generalisation based on Gaussian process frequency functions, but with explicitly unclear theoretical foundations. An earlier technical report studied a non-stationary spectral kernel family derived via the generalised Fourier transform \\citep{samo2015generalized}. \\citet{samo2017advances} expanded the analysis into non-stationary continuous bounded kernels. \\par\nThe cubic time complexity of GP models significantly hinders their scalability. Sparse Gaussian process models \\citep{herbrich2003fast, snelson2006sparse, titsias2009variational,hensman2015scalable} scale GP models with variational inference on pseudo-input points as a concise representation of the input data. Inter-domain Gaussian processes generalize sparse GP models by linearly transforming the original GP and computing cross-covariances, thus putting the inducing points on the transformed domain \\citep{lazaro2009inter}.\n\\begin{table*}[t]\n \\centering\n \\resizebox{\\textwidth}{!}{\n \\begin{tabular}{lcccr}\n Kernel & Harmonizable & Non-stationary & Spectral inference & Reference \\\\\n \\hline\n SE: squared exponential & \\cmark & \\xmark & \\cmark & \\citet{rasmussen2006} \\\\\n SS: sparse spectral & \\cmark & \\xmark &\\cmark & \\citet{quia2010sparse} \\\\\n SM: spectral mixture & \\cmark & \\xmark & \\cmark & \\citet{wilson2013gaussian} \\\\\n GSK: generalised spectral kernel & \\cmark & \\cmark & \\xmark &\\citet{samo2017advances}\\\\\n GSM: generalised spectral mixture &\\bf{?} & \\cmark & \\xmark &\\citet{remes2017non} \\\\\n HMK: harmonizable mixture kernel & \\cmark & \\cmark & \\cmark & current work\n \\end{tabular}\n }\n \\caption{Overview of proposed spectral kernels. The SE, SS and SM kernels are stationary with scalable spectral inference paradigms \\citep{lazaro2009inter, quia2010sparse, gal2015improving}. The GSM kernel is theoretically poorly defined with unknown harmonizable properties. HMK is well-defined with variational Fourier features as spectral inference.}\n \\label{tab:spkernels}\n\\end{table*}\n\n\nIn this paper we propose a theoretically sound treatment of non-stationary kernels, with main contributions:\n\\begin{itemize}\n \\item We present a detailed analysis of \\textit{harmonizability}, a concept mainly existent in statistics literature. Harmonizable kernels are non-stationary kernels interpretable with their \\emph{generalized} spectral representations, similar to stationary ones.\n \n \\item We propose practical \\emph{harmonizable mixture kernels} (HMK), a class of kernels dense in the set of harmonizable covariances with a mixture generalized spectral distribution.\n \\item We propose \\emph{variational Fourier features}, an inter-domain GP inference framework for GPs equipped with HMK. Functions drawn from such GP priors have a well-defined Fourier transform, a desirable property not found in stationary GPs.\n \n \n \n\\end{itemize}\n\n\n\n\n\n\\section{HARMONIZABLE KERNELS}\n\nIn this section we introduce \\emph{harmonizability}, a generalization of stationarity largely unknown to the field of machine learning. We first define harmonizable kernel, and then analyze two existing special cases of harmonizable kernels, stationary and locally stationary kernels. We present a theorem demonstrating the expressiveness of previous stationary spectral kernels. Finally, we introduce Wigner transform as a tool to interpret and analyze these kernels.\\par\nThroughout the discussion in the paper, we consider complex-valued kernels with vectorial input $k(\\mathbf{x}, \\mathbf{x}'): \\mathbb{R}^D\\times \\mathbb{R}^D\\mapsto\\mathbb{C}$, and we denote vectors from the input (data) domain with symbols $\\mathbf{x}, \\mathbf{x}', \\boldsymbol{\\tau}, \\bf{t}$, while we denote frequencies with symbols $\\boldsymbol{\\xi}, \\boldsymbol{\\omega}$.\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{plots\/fig1_op4-crop.pdf}\n \\caption{Comparison of Gaussian, SS, SM, GSK, GSM and HM kernels (columns) with respect to the kernel, Wigner distribution, and the generalized spectral density including real and imaginary part (rows).}\n \\label{fig:fig1}\n\\end{figure*}\n\n\\subsection{Harmonizable kernel definition}\n\nA harmonizable kernel \\citep{kakihara1985, yaglom1987correlation, loeve1994probability} is a kernel with a \\emph{generalized spectral distribution} defined by a generalized Fourier transform:\n\\begin{definition}\nA complex-valued bounded continuous kernel $k: \\mathbb{R}^D \\times\\mathbb{R}^D\\mapsto \\mathbb{C}$ is \\emph{harmonizable} when it can be represented as\n\\begin{align}\n k(\\mathbf{x},\\mathbf{x}') &= \\int_{\\mathbb{R}^D\\times\\mathbb{R}^D} e^{2i\\pi(\\boldsymbol{\\omega}^\\top \\mathbf{x}-\\boldsymbol{\\xi}^\\top \\mathbf{x}')}\\mu_{\\Psi_k}(\\text{d}\\boldsymbol{\\omega}, \\text{d}\\boldsymbol{\\xi}),\n\\end{align}\nwhere $\\mu_{\\Psi_k}$ is the Lebesgue-Stieltjes measure associated to some positive definite function $\\Psi_k(\\boldsymbol{\\omega}, \\boldsymbol{\\xi})$ with bounded variations.\n\\end{definition}\n\nHarmonizability is a property shared by kernels and random processes with such kernels. The positive definite measure induced by function $\\Psi_k$ is defined as the generalized spectral distribution of the kernel, and when $\\mu_{\\Psi_k}$ is twice differentiable, the derivative $S_k(\\boldsymbol{\\omega}, \\boldsymbol{\\xi}) = \\dfrac{\\partial^2\\Psi_k}{\\partial\\boldsymbol{\\omega}\\partial\\boldsymbol{\\xi}}$ is defined as \\emph{generalized spectral density} (GSD).\\par\nHarmonizable kernel is a very general class in the sense that it contains a large portion of bounded, continuous kernels (See Table \\ref{tab:spkernels}) with only a handful of (somewhat pathological) exceptions \\citep{yaglom1987correlation}.\n\n\n\\subsection{Comparison with Bochner's theorem}\nStationary kernels are kernels whose value only depends on the distance $\\boldsymbol{\\tau}=\\mathbf{x}-\\mathbf{x}'$, and therefore is invariant to translation of the input. Bochner's theorem \\citep{bochner1959lectures, stein2012interpolation} expresses similar relation between finite measures and kernels:\n\\begin{theorem}\n(Bochner) A complex-valued function $k: \\mathbb{R}^D\\times\\mathbb{R}^D\\mapsto\\mathbb{C}$ is the covariance function of a weakly stationary mean square continuous complex-valued random process on $\\mathbb{R}^D$ if and only if it can be represented as \\begin{align}\n k(\\boldsymbol{\\tau}) &= \\int_{\\mathbb{R}^D} e^{2i\\pi\\boldsymbol{\\omega}^\\top\\boldsymbol{\\tau}} \\psi_k(\\text{d}\\boldsymbol{\\omega}).\n\\end{align}\nwhere $\\psi_k$ is a positive finite measure.\n\\end{theorem}\nBochner's theorem draws duality between the space of finite measures to the space of stationary kernels. The \\emph{spectral distribution} $\\psi_k$ of a stationary kernel is the finite measure induced by a Fourier transform. And when $\\psi_k$ is absolutely continuous with respect to the Lebesgue measure, its density is called \\emph{spectral density} (SD), $S_k(\\boldsymbol{\\omega})=\\dfrac{\\d{\\psi_k(\\boldsymbol{\\omega})}}{\\d{\\boldsymbol{\\omega}}}$.\\par\nHarmonizable kernels include stationary kernels as a special case. When the mass of the measure $\\mu_\\Psi$ is concentrated on the diagonal $\\boldsymbol{\\omega}=\\boldsymbol{\\xi}$, the generalized inverse Fourier transform devolves into an inverse Fourier transform with respect to $\\boldsymbol{\\tau}=\\mathbf{x}-\\mathbf{x}'$, and therefore recovers the exact form in Bochner's theorem.\n\nA key distinction between the two spectral distributions is that the spectral distribution is a nonnegative finite measure, but the generalized spectral distribution is a complex-valued measure with subsets assigned to complex numbers. Even with a real-valued harmonizable kernel, $\\Psi_k$ can be complex-valued.\n\n\\subsection{Stationary spectral kernels}\nThe perspective of viewing the spectral distribution as a normalized probability measure makes it possible to construct expressive stationary kernels by modeling their spectral distributions. Notable examples include the sparse spectrum (SS) kernel \\citep{quia2010sparse}, and spectral mixture (SM) kernel \\citep{wilson2013gaussian},\n\\begin{align}\n k_{SS}(\\boldsymbol{\\tau}) &= \\sum_{q=1}^Q \\alpha_q\\cos(2\\pi\\boldsymbol{\\omega}_q^\\top\\boldsymbol{\\tau}),\\\\\n k_{SM}(\\boldsymbol{\\tau}) &= \\sum_{q=1}^Q \\alpha_qe^{-2\\pi^2\\tau^\\top{\\boldsymbol{\\Sigma}}_q\\tau}\\cos(2\\pi\\boldsymbol{\\omega}_q^\\top\\boldsymbol{\\tau}),\n\\end{align}\nwith number of components $Q \\in \\mathbb{N}_+$, the component weights (amplitudes) $\\alpha_q \\in \\mathbb{R}_+$, the (mean) frequencies $\\boldsymbol{\\omega}_q\\in\\mathbb{R}_+^D$, and the frequency covariances ${\\boldsymbol{\\Sigma}}_q \\succeq \\mathbf{0}$.\nHere we prove a theorem demonstrating the expressiveness of the above two kernels.\n\\begin{theorem}\nLet $h$ be a complex-valued positive definite, continuous and integrable function. Then the family of \\emph{generalized spectral kernels}\n\\begin{align}\n k_{GS}(\\boldsymbol{\\tau}) &= \\sum_{q=1}^Q \\alpha_q h(\\boldsymbol{\\tau}\\circ\\boldsymbol{\\gamma}_q)e^{2i\\pi\\boldsymbol{\\omega}_q^\\top\\boldsymbol{\\tau}},\n\\end{align}\nis dense in the family of stationary, complex-valued kernels with respect to pointwise convergence of functions. Here $\\circ$ denotes the Hadamard product, $\\alpha_q\\in\\mathbb{R}_+$, $\\boldsymbol{\\omega}_k\\in\\mathbb{R}^D$, $\\boldsymbol{\\gamma}_k\\in\\mathbb{R}^{D}_+$, $Q\\in\\mathbb{N}_+$.\n\\end{theorem}\n\\begin{proofskch}\nWe know that discrete measures are dense in the Banach space of finite measures. Therefore, the complex extension of sparse spectrum kernel\n $k_{SS}(\\boldsymbol{\\tau}) = \\sum_{k=1}^K \\alpha_k e^{2i\\pi\\boldsymbol{\\omega}_k^\\top\\boldsymbol{\\tau}}$ is dense in stationary kernels.\\par\nFor each $q$, the function $\\dfrac{\\alpha_q}{h(0)} h(\\boldsymbol{\\tau}\\circ\\boldsymbol{\\gamma}_q)e^{2i\\pi\\boldsymbol{\\omega}_k^\\top\\boldsymbol{\\tau}}$ converges to $\\alpha_q e^{2i\\pi\\boldsymbol{\\omega}_q^\\top\\boldsymbol{\\tau}}$ pointwise as $\\boldsymbol{\\gamma}_q\\downarrow \\mathbf{0}$. Therefore, the proposed kernel form is dense in the set of sparse spectrum kernels, and by extension, stationary kernels.\nSee Section 1 in supplementary materials for a more detailed proof.\n\\end{proofskch}\\par\nWe strengthen the claim of \\citet{samo2015generalized} by adding a constraint $\\alpha_k > 0$ that restricts the family of functions to only valid PSD kernels \\citep{samo2017advances}. The spectral distribution of $k_{GS}$ is\n\\begin{align}\n \\psi_{k_{GS}}(\\boldsymbol{\\xi}) &= \\sum_{q=1}^Q \\dfrac{\\alpha_q}{\\prod_{d=1}^D\\gamma_{kd}}\\psi_h((\\boldsymbol{\\xi}-\\boldsymbol{\\omega}_k)\\oslash\\boldsymbol{\\gamma}_k),\n\\end{align}\nwith $\\oslash$ denoting elementwise division of vectors. A real-valued kernel can be obtained by averaging a complex kernel with its complex conjugate, which induces a symmetry on the spectral distribution, $\\psi_k(\\boldsymbol{\\xi}) = \\psi_k(-\\boldsymbol{\\xi})$. For instance, the SM kernel has the symmetric Gaussian mixture spectral distribution \n\\begin{align}\n \\psi_{k_{SM}}(\\boldsymbol{\\xi}) &= \\dfrac{1}{2}\\sum_{q=1}^Q\\alpha_q(\\mathcal{N}(\\boldsymbol{\\xi}|\\boldsymbol{\\omega}_q, {\\boldsymbol{\\Sigma}}_q)+\\mathcal{N}(\\boldsymbol{\\xi}|-\\boldsymbol{\\omega}_q, {\\boldsymbol{\\Sigma}}_q)). \n\\end{align}\n\n\\subsection{Locally stationary kernels}\n\nAs a generalization of stationary kernels, the locally stationary kernels \\citep{silverman1957locally} are a simple yet unexplored concept in machine learning. A locally stationary kernel is a stationary kernel multiplied by a sliding power factor:\n\\begin{align}\n k_{LS}(\\mathbf{x},\\mathbf{x}') &= k_1\\left(\\dfrac{\\mathbf{x}+\\mathbf{x}'}{2}\\right)k_2(\\mathbf{x}-\\mathbf{x}').\n\\end{align}\nwhere $k_1: \\mathbb{R}^D\\mapsto\\mathbb{R}_{\\geq 0}$ is an arbitrary nonnegative function, and $k_2:\\mathbb{R}^D\\mapsto\\mathbb{C}$ is a stationary kernel. $k_1$ is a function of the \\emph{centroid} between $\\mathbf{x}$ and $\\mathbf{x}'$, describing the scale of covariance on a global structure, while $k_2$ as a stationary covariance describes the local structure \\citep{genton2001classes}. It is straightforward to see that locally stationary kernels reduce into stationary kernels when $k_1$ is constant.\n\nIntegrable locally stationary kernels are of particular interest because they are harmonizable with a GSD. Consider a locally stationary Gaussian kernel (LSG) defined as a SE kernel multiplied by a Gaussian density on the centroid $\\widetilde{\\mathbf{x}} = (\\mathbf{x}+\\mathbf{x}')\/2$. Its GSD can be obtained using the generalized Wiener-Khintchin relations \\citep{silverman1957locally}.\n\\begin{align}\n k_{\\text{LSG}}(\\mathbf{x}, \\mathbf{x}') &= e^{-2\\pi^2\\widetilde{\\mathbf{x}}^\\top{\\boldsymbol{\\Sigma}}_1\\widetilde{\\mathbf{x}}}e^{-2\\pi^2\\boldsymbol{\\tau}^\\top{\\boldsymbol{\\Sigma}}_2\\boldsymbol{\\tau}},\\\\\n S_{k_{\\text{LSG}}}(\\boldsymbol{\\omega}, \\boldsymbol{\\xi}) &= \\mathcal{N}\\left(\\left.\\dfrac{\\boldsymbol{\\omega}+\\boldsymbol{\\xi}}{2}\\right\\vert 0, {\\boldsymbol{\\Sigma}}_2\\right)\\mathcal{N}\\left(\\left.\\boldsymbol{\\omega}-\\boldsymbol{\\xi}\\right\\vert 0, {\\boldsymbol{\\Sigma}}_1\\right).\n\\end{align}\n\n\\subsection{Interpreting spectral kernels}\n\nWhile the spectral distribution of a stationary kernel can be easily interpreted as a `spectrum', the analogy does not apply to harmonizable kernels. In this section, we introduce the Wigner transform \\citep{flandrin1998time} which adds interpretability to kernels with spectral representations.\n\\begin{definition}\nThe \\emph{Wigner distribution function} (WDF) of a kernel $k(\\cdot,\\cdot):\\mathbb{R}^D\\times\\mathbb{R}^D\\mapsto\\mathbb{C}$ is defined as $W_k:\\mathbb{R}^D\\times\\mathbb{R}^D\\mapsto\\mathbb{R}$:\n\\begin{align}\n W_k(\\mathbf{x}, \\boldsymbol{\\omega}) &= \\int_{\\mathbb{R}^D} k\\left(\\mathbf{x}+\\dfrac{\\boldsymbol{\\tau}}{2}, \\mathbf{x}-\\dfrac{\\boldsymbol{\\tau}}{2}\\right)e^{-2i\\pi\\boldsymbol{\\omega}^\\top\\boldsymbol{\\tau}} \\d{\\boldsymbol{\\tau}}.\n\\end{align}\n\\end{definition}\n\nThe Wigner transform first changes the kernel form $k$ into a function of the centroid of the input: $(\\mathbf{x}+\\mathbf{x}')\/2$ and the lag $\\mathbf{x}-\\mathbf{x}'$, and then takes the Fourier transform of the lag. The Wigner distribution functions are fully equivalent to non-stationary kernels. Given the domain of WDF, we can view WDF as a `spectrogram' demonstrating the relation between input and frequency. Converting an arbitrary kernel into its Wigner distribution sheds light into the frequency structure of the kernel (See Figure \\ref{fig:fig1}).\\par\nThe WDFs of locally stationary kernels adhere to the intuitive notion of local stationarity where frequencies remain constant at a local scale. Take locally stationary Gaussian kernel $k_{\\text{LSG}}$ as an example:\n\\begin{align}\n W_{k_{\\text{LSG}}}(\\mathbf{x},\\boldsymbol{\\omega}) &= \\mathcal{N}(\\boldsymbol{\\omega}|\\mathbf{0}, {\\boldsymbol{\\Sigma}}_2) e^{-2\\pi^2\\mathbf{x}^\\top{\\boldsymbol{\\Sigma}}_1\\mathbf{x}}.\n\\end{align}\n\n\\section{HARMONIZABLE MIXTURE KERNEL}\nIn this section we propose a novel \\emph{harmonizable mixture kernel}, a family of kernels dense in harmonizable covariance functions. We present the kernel in an intentionally concise form, and refer the reader to the Section 2 in the Supplements for a full derivation.\n\n\\subsection{Kernel form and spectral representations}\n\nThe \\emph{harmonizable mixture kernel} (HMK) is defined with an additive structure:\n\\begin{align}\n k_{\\text{HM}}(\\mathbf{x},\\mathbf{x}')&=\\sum_{p=1}^P k_p(\\mathbf{x}-\\mathbf{x}_p, \\mathbf{x}'-\\mathbf{x}_p),\\\\\n k_p(\\mathbf{x}, \\mathbf{x}') &= k_{\\text{LSG}}(\\mathbf{x}\\circ\\boldsymbol{\\gamma}_p, \\mathbf{x}'\\circ\\boldsymbol{\\gamma}_p)\\phi_p(\\mathbf{x})^\\top\\mathbf{B}_p\\phi_p(\\mathbf{x}'),\n\\end{align}\nwhere $P\\in\\mathbb{N}_+$ is the number of centers, $\\left(\\phi_p(\\mathbf{x})\\right)_{q=1}^{Q_p}=e^{2i\\pi\\boldsymbol{\\mu}_{pq}^\\top\\mathbf{x}}$ are sinusoidal feature maps, $\\mathbf{B}_p\\succeq\\mathbf{0}_{Q_p}$ are spectral amplitudes, $\\boldsymbol{\\gamma}_p\\in\\mathbb{R}^D_+$ are input scalings, $\\mathbf{x}_p\\in\\mathbb{R}^D$ are input shifts, and $\\boldsymbol{\\mu}_{pq}\\in\\mathbb{R}^D$ are frequencies. It is easy to verify $k_{\\text{HM}}$ as a valid kernel, for each $k_p$ is a product of an LSG kernel and an inner product with finite basis expansion of sinusoidal functions.\\par\n\nHMKs have closed form spectral representations such as \\emph{generalized spectral density} (See Section 2 in the Supplement for detailed derivation):\n\\begin{align}\n S_{k_{\\text{HM}}}(\\boldsymbol{\\omega}, \\boldsymbol{\\xi}) &= \\sum_{p=1}^P S_{k_p}(\\boldsymbol{\\omega}, \\boldsymbol{\\xi})e^{-2i\\pi\\mathbf{x}_p^\\top(\\boldsymbol{\\omega}-\\boldsymbol{\\xi})},\\\\\n S_{k_p}(\\boldsymbol{\\omega}, \\boldsymbol{\\xi}) &= \\dfrac{1}{\\prod_{d=1}^D\\gamma_{pd}^2}\\sum_{1\\leq i, j \\leq Q_p} b_{pij}S_{pij}(\\boldsymbol{\\omega}, \\boldsymbol{\\xi}),\\\\\n S_{pij}(\\boldsymbol{\\omega}, \\boldsymbol{\\xi})&=S_{k_{\\text{LSG}}}((\\boldsymbol{\\omega}-\\boldsymbol{\\mu}_{pi})\\oslash\\boldsymbol{\\gamma}_p, (\\boldsymbol{\\xi}-\\boldsymbol{\\mu}_{pj})\\oslash\\boldsymbol{\\gamma}_p).\n\\end{align}\nThe \\emph{Wigner distribution function} can be obtained in a similar fashion\n\\begin{align}\n W_{k_{\\text{HM}}}(\\mathbf{x},\\boldsymbol{\\omega})&=\\sum_{p=1}^P W_{k_p}(\\mathbf{x}-\\mathbf{x}_p, \\boldsymbol{\\omega}),\\\\\n W_{k_p}(\\mathbf{x},\\omega) &= \\dfrac{1}{\\prod_{d=1}^D\\gamma_{pd}}\\sum_{1\\leq i,j\\leq Q_p} W_{pij}(\\mathbf{x},\\boldsymbol{\\omega}),\\\\\n W_{pij}(\\mathbf{x},\\boldsymbol{\\omega}) &= W_{k_{\\text{LSG}}}\\left(\\mathbf{x}\\circ\\boldsymbol{\\gamma}_p, \\left(\\boldsymbol{\\omega}-(\\boldsymbol{\\mu}_{pi}+\\boldsymbol{\\mu}_{pj})\/2\\right)\\oslash\\boldsymbol{\\gamma}_p\\right)\\notag\\\\\n &\\times\\cos(2\\pi(\\boldsymbol{\\mu}_{pi}-\\boldsymbol{\\mu}_{pj})^\\top\\mathbf{x}).\n\\end{align}\nThe kernel form, GSD and WDF both take a normal density form. It is straightforward to see $S_{k_{\\text{HM}}}$ is PSD, and that $k_{\\text{Hm}}(-\\mathbf{x}, -\\mathbf{x}')$ is the GSD of $S_{k_{\\text{HM}}}$. A real-valued kernel $k_r$ is obtained by averaging with its complex conjugate: $W_{k_r}(\\mathbf{x},\\boldsymbol{\\omega})=W_{k_r}(\\mathbf{x},-\\boldsymbol{\\omega})$, $S_{k_r}(\\boldsymbol{\\omega}, \\boldsymbol{\\xi}) = S_{k_r}(-\\boldsymbol{\\omega}, -\\boldsymbol{\\xi})$.\n\n\\subsection{Expressiveness of HMK}\nSimilar to the construction of \\emph{generalized spectral kernels}, we can construct a generalized version $k_h$ where $k_{\\text{LSG}}$ is replaced by $k_{\\text{LS}}$, a locally stationary kernel with a GSD.\n\\begin{theorem} Given a continuous, integrable kernel $k_{\\text{LS}}$ with a valid \\emph{generalized spectral density}, the harmonizable mixture kernel\n\\begin{align}\n k_h(\\mathbf{x},\\mathbf{x}')&=\\sum_{p=1}^P k_p(\\mathbf{x}-\\mathbf{x}_p, \\mathbf{x}'-\\mathbf{x}_p),\\\\\n k_p(\\mathbf{x}, \\mathbf{x}')&=k_{\\text{LS}}(\\mathbf{x}\\circ\\boldsymbol{\\gamma}_p,\\mathbf{x}'\\circ\\boldsymbol{\\gamma}_p)\\phi_p(\\mathbf{x})^\\top\\mathbf{B}_p\\phi_p(\\mathbf{x}'),\n\\end{align}\nis dense in the family of harmonizable covariances with respect to pointwise convergence of functions. Here $P\\in\\mathbb{N}_+$, $(\\phi_p(\\mathbf{x}))_q=e^{2i\\pi\\boldsymbol{\\mu}_{pq}^\\top\\mathbf{x}}$, $q=1,\\hdots, Q_p$, $\\boldsymbol{\\gamma}_p\\in\\mathbb{R}_+^D$, $\\mathbf{x}_p\\in\\mathbb{R}^D$, $\\boldsymbol{\\mu}_{pq}\\in\\mathbb{R}^D$, $\\mathbf{B}_p$ as positive definite Hermitian matrices.\n\\end{theorem}\n\\begin{proof}\nSee Section 3 in the supplementary materials.\n\\end{proof}\n\n\\section{VARIATIONAL FOURIER FEATURES}\n\nIn this section we propose variational inference for the harmonizable kernels applied in Gaussian process models.\n\nWe assume a dataset of $n$ input $X = \\{\\mathbf{x}_i\\}_{i=1}^n$ and output $\\mathbf{y} = \\{ y_i\\} \\in \\mathbb{R}^n$ observations from some function $f(\\mathbf{x})$ with a Gaussian observation model:\n\\begin{align}\ny = f(\\mathbf{x}) + \\varepsilon, \\qquad \\varepsilon \\sim \\mathcal{N}(0, \\sigma_y^2). \\label{eq:noise}\n\\end{align}\n\n\\subsection{Gaussian processes}\n\nGaussian processes (GP) are a family of Bayesian models that characterise distributions of functions \\citep{rasmussen2006}. We assume a zero-mean Gaussian process prior on a latent function $f(\\mathbf{x}) \\in \\mathbb{R}$ over vector inputs $\\mathbf{x} \\in \\mathbb{R}^D$\n\\begin{align}\nf(\\mathbf{x}) &\\sim \\mathcal{GP}( 0, K(\\mathbf{x},\\mathbf{x}')),\n\\end{align}\nwhich defines a priori distribution over function values $f(\\mathbf{x})$ with mean $\\mathbb{E}[ f(\\mathbf{x})] = 0$ and covariance \n\\begin{align}\n\\cov[ f(\\mathbf{x}), f(\\mathbf{x}')] &= K(\\mathbf{x},\\mathbf{x}').\n\\end{align}\nA GP prior specifies that for any collection of $n$ inputs $X$, the corresponding function values $\\mathbf{f} = ( f(\\mathbf{x}_1), \\ldots, f(\\mathbf{x}_n))^\\top \\in \\mathbb{R}^n$ are coupled by following a multivariate normal distribution \n$\\mathbf{f} \\sim \\mathcal{N}(\\mathbf{0}, \\mathbf{K}_{ff}),$\nwhere $\\mathbf{K}_{ff} = (K(\\mathbf{x}_i, \\mathbf{x}_j))_{i,j=1}^n \\in \\mathbb{R}^{n \\times n}$ is the kernel matrix over input pairs. The key property of GP's is that output predictions $f(\\mathbf{x})$ and $f(\\mathbf{x}')$ correlate according to how similar are their inputs $\\mathbf{x}$ and $\\mathbf{x}'$ as defined by the kernel $K(\\mathbf{x},\\mathbf{x}') \\in \\mathbb{R}$. \n\n\n\n\n\n\\subsection{Variational inference with inducing features}\n\n\nIn this section, we introduce variational inference of sparse GPs in an inter-domain setting. Consider a GP prior $f(\\mathbf{x})\\sim\\mathcal{GP}(0, k)$, and a valid linear transform $\\mathcal{L}$ projecting $f$ to another GP $\\mathcal{L}_f(\\mathbf{z})\\sim\\mathcal{GP}(0, k')$. \n\n\nWe begin by \\emph{augmenting} the Gaussian process with $m < n$ inducing variables $u_j = \\mathcal{L}_f(\\mathbf{z}_j)$ using a Gaussian model. $\\mathbf{z}_j$ are \\emph{inducing features} placed on the domain of $\\mathcal{L}_f(\\mathbf{z})$, with prior $p(\\u) = \\mathcal{N}( \\u | \\mathbf{0}, \\mathbf{K}_{uu})$ and a conditional model \\citep{hensman2015scalable}\n\\begin{align}\n p(\\mathbf{f} | \\u) &= \\mathcal{N}( \\mathbf{A} \\u, \\mathbf{K}_{ff} - \\mathbf{A} \\mathbf{K}_{uu} \\mathbf{A}^\\dag), \\label{eq:interp}\n\\end{align}\nwhere $\\mathbf{A} = \\mathbf{K}_{fu} \\mathbf{K}_{uu}^{-1}$, and $\\mathbf{A}^\\dag$ denotes the Hermitian transpose of $\\mathbf{A}$ allowing for complex GPs.\nThe kernel $\\mathbf{K}_{uu}$ is between the $m \\times m$ inducing variables\nand the kernel $\\mathbf{K}_{fu}$ is the cross covariance of $\\mathcal{L}$, $\\left(\\mathbf{K}_{fu}\\right)_{is} = \\cov(f(\\mathbf{x}_i), \\mathcal{L}_f(\\mathbf{z}_s))$. Next, we define a variational approximation $q(\\u) = \\mathcal{N}( \\u | \\mathbf{m}, \\mathbf{S})$ with the Gaussian interpolation model \\eqref{eq:interp},\n\\begin{align}\n q(\\mathbf{f})\n &= \\mathcal{N}( \\mathbf{f} | \\mathbf{A} \\mathbf{m}, \\mathbf{K}_{ff} - \\mathbf{A} (\\mathbf{S} - \\mathbf{K}_{uu}) \\mathbf{A}^\\dag),\n\\end{align}\nwith free variational mean $\\mathbf{m} \\in \\mathbb{R}^m$ and variational covariance $\\mathbf{S} \\in \\mathbb{R}^{m \\times m}$ to be optimised. Finally, variational inference \\citep{blei2016} describes an evidence lower bound (ELBO) of augmented Gaussian processes as\n\\begin{align}\n \\hspace{-2.5mm}\\log p(\\mathbf{y}) & \\ge \\sum_{i=1}^n \\mathbb{E}_{q(f_i)} \\log p(y_i | f_i) - \\mathrm{KL}[ q(\\u) || p(\\u)]. \\label{eq:elbo}\n\\end{align}\n\n\n\\subsection{Fourier transform of a harmonizable GP}\n\n\nIn this section, we compute cross-covariances between a GP and the Fourier transform of the GP. Consider a GP prior $f\\sim\\mathcal{GP}(0,k)$ where the kernel $k$ is harmonizable with a GSD $S_k$ and where $\\hat{f}$ is the Fourier transform of $f$,\n\\begin{align}\n \\hat{f}(\\boldsymbol{\\omega}) &\\triangleq \\int_{\\mathbb{R}^D} f(\\mathbf{x})e^{-2i\\pi\\boldsymbol{\\omega}^\\top \\mathbf{x}}\\d{\\mathbf{x}}.\n\\end{align}\nThe validity of this setting is easily verified because $f$ is square integrable on expectation,\n\\begin{align}\n \\mathbb{E}\\left\\{\\int_{\\mathbb{R}^D} |f(\\mathbf{x})|^2\\d{\\mathbf{x}}\\right\\} &= \\int_{\\mathbb{R}^D} k(\\mathbf{x},\\mathbf{x}) \\d{\\mathbf{x}} < \\infty.\n\\end{align}\n\nWe can therefore derive the cross-covariances\n\\begin{align}\n \\cov(\\hat{f}(\\boldsymbol{\\omega}), f(\\mathbf{x}))\n &= \\int_{\\mathbb{R}^D} k(\\t,\\mathbf{x}) e^{-2 i \\pi\\boldsymbol{\\omega}^\\top \\t} \\d{\\t} \\\\\n \\cov(\\hat{f}(\\boldsymbol{\\omega}), \\hat{f}(\\boldsymbol{\\xi}))\n \n &= S_k(\\boldsymbol{\\omega}, \\boldsymbol{\\xi}).\n\\end{align}\nThe above derivation is valid for any harmonizable kernel with a GSD. The Fourier transform of $\\mathcal{GP}(0,k)$ is a complex-valued GP with kernel $S_k$, which correlates to the original GP.\\par\n \n\nFor harmonizable, integrable kernel $k$, we can construct an inter-domain sparse GP model defined in 4.2 by plugging in $\\mathcal{L}_f = \\hat{f}$.\n\n\\begin{figure*}[t]\n\\begin{center}\n \\includegraphics[width=\\textwidth, clip=true]{plots\/banana1.pdf}\n\\end{center}\n \\caption{Sparse GP classification with the banana dataset. The model is learned by an HMK with $P=4$ components, and thus 2 inducing frequencies for each component constitute a total of $2 \\times 4$ inducing frequencies.}\n \\label{figure:gpc}\n\\end{figure*}\n\n\\subsection{Variational Fourier features of the harmonizable mixture kernel}\nHMK belongs to the kernel family discussed in 4.3, but we can utilize the additive structure of an HMK $k_{HM} = \\sum_{p=1}^P k_p(\\mathbf{x}-\\mathbf{x}_p, \\mathbf{x}'-\\mathbf{x}_p)$. A GP with kernel $k_{HM}$ can be decomposed into $P$ independent GPs:\n\\begin{align}\n f(\\mathbf{x}) &= \\sum_{p=1}^P f_p(\\mathbf{x}-\\mathbf{x}_p),\\\\\n f_p(\\mathbf{x}) &\\sim \\mathcal{GP}(0, k_p(\\mathbf{x}, \\mathbf{x}')).\n\\end{align}\nGiven this formulation, we can derive \\emph{variational Fourier features} with inducing frequencies conditioned on one $f_p$. For the $p^{th}$ component, we have $m_p$ inducing frequencies $(\\boldsymbol{\\omega}_{p1}, \\ldots, \\boldsymbol{\\omega}_{pm_p})$ and $m_p$ inducing values $(u_{p1}, \\cdots, u_{pm_p})$. We can compute inter-domain covariances in a similar fashion:\n\\begin{align}\n \\mathbf{K}_{fu}(\\boldsymbol{\\omega}_{qj}, \\mathbf{x}) &\\triangleq \\cov(f(x), u_{qj}) \\label{eq:kfu} \\\\\n &= \\sum_{p=1}^P\\cov(f_p(\\mathbf{x}-\\mathbf{x}_p), u_{qj}) \\notag \\\\\n &= \\cov(f_q(\\mathbf{x}-\\mathbf{x}_q), \\hat{f}_q(\\boldsymbol{\\omega}_{qj})). \\notag\n\\end{align}\nSimilarly, we compute entries of the matrix $K_{uu}$\n\\begin{align}\n \\mathbf{K}_{uu}(\\boldsymbol{\\omega}_{pi}, \\boldsymbol{\\omega}_{qj}) \\triangleq \\cov(u_{pi}, u_{qj}) &= \\begin{cases}\n S_p(\\boldsymbol{\\omega}_{pi}, \\boldsymbol{\\omega}_{qj}), p=q,\\\\\n 0, p\\neq q.\n \\end{cases} \\label{eq:kuu}\n\\end{align}\nThe matrix $\\mathbf{K}_{uu}$ allows for a block diagonal structure, which allows for faster matrix inversion. The variational Fourier features are then completed by plugging in entries in $\\mathbf{K}_{fu}$ \\eqref{eq:kfu} and $\\mathbf{K}_{uu}$ \\eqref{eq:kuu} into the evidence lower bound \\eqref{eq:elbo}.\\par\n\\subsection{Connection to previous work}\nIn this section we demonstrate that an inter-domain stationary GP with windowed Fourier transform \\citep{lazaro2009inter} is equivalent to a rescaled VFF with a tweaked kernel. GPs with stationary kernels do not have valid Fourier transform, therefore, previous attempts of using Fourier transforms of GPs have been accompanied by a window function:\n\\begin{align}\n \\mathcal{L}_f(\\boldsymbol{\\omega}) &= \\int_{\\mathbb{R}^D} f(\\mathbf{x}) w(\\mathbf{x}) e^{-2i\\pi\\boldsymbol{\\omega}^\\top \\mathbf{x}} \\d{\\mathbf{x}}.\n\\end{align}\nThe windowing function $w(\\mathbf{x})$ can be a soft Gaussian window $w(\\mathbf{x}) = \\mathcal{N}(\\mathbf{x}|\\boldsymbol{\\mu},{\\boldsymbol{\\Sigma}})$ \\citep{lazaro2009inter} or a hard interval window $w(x)=\\mathbb{I}_{[a\\leq x\\leq b]}e^{2i\\pi a}$ \\citep{hensman2017variational}. The windowing approach shares the caveat of a blurred version of the frequency space, caused by an inaccurate Fourier transform\\citep{lazaro2009inter}.\\par\nConsider $f\\sim\\mathcal{GP}(0, k)$ where $k$ is a stationary kernel, and $w(\\mathbf{x}) = \\mathcal{N}(x|\\mu,{\\boldsymbol{\\Sigma}})$, we see that $g(\\mathbf{x}) = w(\\mathbf{x})f(\\mathbf{x}) \\sim\\mathcal{GP}(0, w(\\mathbf{x})w(\\mathbf{x}')k(\\mathbf{x}-\\mathbf{x}'))$. It is easy to verify that the kernel of $g(\\mathbf{x})$ is locally stationary. There exist the following relations of cross-covariances:\n\\begin{align}\n \\cov(f(\\mathbf{x}), \\mathcal{L}_f(\\boldsymbol{\\omega})) &= \\dfrac{\\cov(g(\\mathbf{x}), \\hat{g}(\\boldsymbol{\\omega}))}{w(\\mathbf{x})},\\\\\n \\cov(\\mathcal{L}_f(\\boldsymbol{\\omega}), \\mathcal{L}_f(\\boldsymbol{\\xi})) &= \\cov(\\hat{g}(\\boldsymbol{\\omega}), \\hat{g}(\\boldsymbol{\\xi})).\n\\end{align}\nTherefore, windowed inter-domain GPs are equivalent to rescaled GPs with a tweaked kernel.\n\\section{EXPERIMENTS}\nIn this section, we experiment with the harmonizable mixture kernels for kernel recovery, GP classification and regression. We use a simplied version of the harmonizable kernel where the two matrices of the locally stationary $k_{\\text{LSG}}$ are diagonals: ${\\boldsymbol{\\Sigma}}_1=\\mbox{diag}(\\sigma_d^2)$, ${\\boldsymbol{\\Sigma}}_2=\\lambda^2 I$. See Section 6 in the supplement for more detailed information.\n\\subsection{Kernel recovery}\nWe demonstrate the expressiveness of HMK by using it to recover certain non-stationary kernels. We choose the non-stationary \\emph{generalized spectral mixture kernel} (GSM) \\citep{remes2017non} and the covariance function of a time-inverted fractional Brownian motion (IFBM):\n\n\\resizebox{1.00\\columnwidth}{!}{\n\\begin{tabular}{l}\n $k_{\\text{GSM}}(x,x') = w(x)w(x') k_{\\text{Gibbs}}(x, x')\\cos(2\\pi(\\mu(x)x-\\mu(x')x')), $ \\\\\n $k_{\\text{Gibbs}}(x, x') = \\sqrt{\\dfrac{2l(x)l(x')}{l(x)^2+l(x')^2}}\\exp\\left(-\\dfrac{(x-x')^2}{l(x)^2+l(x')^2}\\right), $ \\\\\n $k_{\\text{IFBM}}(t,s) = \\dfrac{1}{2}\\left(\\dfrac{1}{t^{2h}}+\\dfrac{1}{s^{2h}} - \\left\\vert\\dfrac{1}{t}-\\dfrac{1}{s}\\right\\vert^{2h}\\right),$ \\\\\n\\end{tabular}\n}\n\nwhere $s, t \\in (0.1, 1.1]$ and $x, x' \\in [-1, 1]$. \nThe hyperparameters of $k_{\\text{HM}}$ are randomly initialized, and optimized with stochastic gradient descent.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\columnwidth, clip=true, ]{plots\/approx4-crop.pdf}\n \\caption{Kernel recovery experiment with true kernels (left) against SM kernel approximations (right).}\n \\label{fig:ifbm_rec}\n\\end{figure}\n\nBoth kernels can be recovered almost perfectly with mean squared errors of $0.0033$ and $0.0008$. The result indicates that we can use the GSD and the Wigner distribution of the approximating HM kernel to interpret the GSM kernel (see Section 5 in supplementary materials). \n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\columnwidth, clip=true]{plots\/solar4-crop.pdf}\n \\caption{Sparse GP regression with solar irradiance dataset.}\n \\label{figure:gpr}\n\\end{figure}\n\n\\subsection{GP classification with banana dataset}\n\nIn this section, we show the effectiveness of variational Fourier fetures in GP classification with HMK. We use an HMK with $P=4$ components to classify the banana dataset, and compare SVGP with inducing points (IP) \\citep{hensman2015scalable} and SVGP with variational Fourier features (VFF). The model parameters are learned by alternating optimization rounds of natural gradients for the variational parameters, and Adam optimizer for the other parameters \\citep{salimbeni2018natural}.\n\nFigure \\ref{figure:gpc} shows the decision boundaries of the two methods over the number of inducing points.\nFor both variants, we experiment with model complexities from 6 to 24 inducing points in IP, and from 2 to 8 inducing frequencies for each component of HMK in the VFF. The centers of HMK (red triangles) spread to support the data distribution. The IP method is slightly more complex compared to VFF at the same parameter counts in terms of nonzero entries in the variational parameters.\n\nThe VFF method recovers roughly the correct decision boundary even with a small number of inducing frequencies, while converging faster to the decision boundaries as the number of inducing frequencies increases.\n\n\\subsection{GP regression with solar irradiance}\n\nIn this section, we demonstrate the effectiveness of HMK in interpolation for the non-stationary solar irradiance dataset. We run sparse GP regression with squared exponential, spectral mixture and harmonizable mixture kernels, and show the predicted mean, and 95\\% confidence intervals for each model (See Figure \\ref{figure:gpc}).\n\nWe use sparse GP regression proposed in \\citep{titsias2009variational} with 50 inducing points marked at the x axis. The SE kernel can not estimate the periodic pattern and overestimates the signal smoothness. The SM kernel fits the training data well, but misidentifies frequencies on the first and fourth interval of the test set.\n\n\nFor sparse GP with HMK, we use the same framework where the variational lower bound is adjusted for VFF. The model extrapolates better for the added flexibility of nonstationarity, and the inducing frequencies aggregate near the learned frequencies. Both first and last test intervals are well fitted. The Wigner distribution with inducing frequencies of the optimised HM kernel is shown in Figure \\ref{figure:gpc}d.\n\n\n\n\n\n\n\\section{CONCLUSION}\nIn this paper, we extend the generalization of Gaussian processes by proposing harmonizable mixture kernel, a non-stationary kernel spanning the wide class of harmonizable covariances. Such kernels can be used as an expressive tool for GP models. We also proposed variational Fourier features, an inter-domain inference framework used as drop-in replacements for sparse GPs. This work bridges previous research on spectral representation of kernels and sparse Gaussian processes.\\par\nDespite its expressiveness, one may brand the parametric form of HMK as not fully Bayesian, since it contradicts the nonparametric nature of GPs. A fully Bayesian approach would be to place a nonparametric prior over harmonizable mixture kernels, representing the uncertainty of the kernel form \\citep{shah2014student}. \n\n\\bibliographystyle{plainnat}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaabf b/data_all_eng_slimpj/shuffled/split2/finalzzaabf new file mode 100644 index 0000000000000000000000000000000000000000..f5eee472f1e7aab594a86ed80067b04bb85d6598 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzaabf @@ -0,0 +1,5 @@ +{"text":"\\section*{Dedication}\n\nThis thesis is dedicated to my parents, Sonia and Oscar, and my brother John Paul.\nTheir love and support are everything to me, and this milestone is as much a celebration\nof accomplishments as it is a testament to the strength of family.\n\\clearpage\n\\acknowledgements\n\nThese past five years have been an incredible privilege for, if nothing else,\nthe extraordinary individuals with whom I met and worked.\nDay in and day out, my group members challenge, encourage, and support\nme, and I am thankful for their companionship throughout this journey:\nDavid Hicks,\nDemet Usanmaz,\nEric Gossett,\nPinku Nath,\nMarco Esters,\nRico Friedrich,\nPranab Sarker,\nDenise Ford,\nPauline Colinet,\nCamilo Calderon,\nCarlo De Santo,\nGeena Gomez,\nHarvey Shi,\nAllison Stelling,\nYoav Lederer,\nLuis Agapito,\nand\nManuela Damian.\n\nI am fortunate to have so many mentors and guides who, in various\ncapacities, have opened my eyes to new fields and fueled my scientific curiosity:\nCormac Toher,\nFrisco Rose,\nOlexandr Isayev,\nKesong Yang,\nStefano Sanvito,\nAmir Natan,\nMichael Mehl,\nPatrick McGuire,\nJes\\'{u}s Carrete,\nNatalio Mingo,\nMatthias Scheffler,\nClaudia Draxl,\nValentin Stanev,\nIchiro Takeuchi,\nand\nOhad Levy.\n\nI am incredibly appreciative of my professors, teachers,\nand advisors from Cornell University and Bloomfield High School\nwho helped me construct the vision in which this milestone is achievable:\nSara Xayarath Hern\\'{a}ndez,\nJoel Brock,\nErnest Fontes,\nKenneth Card,\nDaniel Di Domenico,\nMarian Connolly,\nBrian Miller,\nLou Cappello,\nand\nManuela Gonnella.\n\nAbove all, I am especially grateful to my PhD Advisor, Stefano Curtarolo,\nfor the opportunity to discover my passion in this field.\nI have been blessed with many in my life who believe in me, but few as\nemphatically as Stefano.\n``\\textit{Non ducor, duco}''.\n\nFinally, I acknowledge support from the National Science Foundation Graduate Research Fellowship under Grant No. DGF1106401.\n\n\\clearpage\n\\tableofcontents\n\\listoffigures\n\\listoftables\n\\chapters\n\\chapter{Introduction}\n\n\\begin{center}\n``\\textit{Nihil est in intellectu quod non sit prius in sensu}''\\footnote{``Nothing is in the intellect that was not first in the senses.''} \\\\\n--- Thomas Aquinas's \\textit{Quaestiones Disputatae de Veritate}, \\\\ quaestio 2, articulus 3, argumentum 19.\n\\end{center}\n\nMaterials discovery drives technological innovation, spanning\nthe stones and simple metals that forged the first tools to the semiconductors that power today's computers.\nHistorically, these advancements follow from intuition and\nserendipity~\\cite{curtarolo:art94,curtarolo:art124,MGI,Norman_RPP_2016,Eberhart_NMat_2004}.\nAs such, major breakthroughs --- which are few and far between --- are seldom predictable.\nFortunately, ``\\textit{big data}'' is powering a paradigm shift:\nmaterials informatics.\nIntegration of data-centric approaches in an otherwise \\textit{a posteriori} field promises to\nbridge the widening gap between observation and understanding,\naccelerating the pace of technology.\nMore importantly, data-driven modeling --- offering predictions grounded in empirical evidence ---\nmay finally break with tradition, enabling control over discovery and\nachieving rational materials design.\n\nWielding data to accelerate innovation is not a new idea,\nsince it constitutes standard practice in biology~\\cite{Reichhardt_Nature_1999,Luscombe_MIM_2001}\nand chemistry~\\cite{Brown_Chemoinformatics_1998,Gasteiger_Chemoinformatics_2003}.\nYet its adoption in materials science has been slow, as it was first introduced in the early 2000's~\\cite{curtarolo:art13}.\nThis delay can be attributed to the ongoing development of standard \\nobreak\\mbox{\\it ab-initio}\\\npackages~\\cite{kresse_vasp,VASP4_2,vasp_cms1996,vasp_prb1996,quantum_espresso_2009,gonze:abinit,Blum_CPC2009_AIM},\nparticularly to better address calculation of the exchange correlation energy~\\cite{PBE,Perdew_SCAN_PRL_2015}.\nNevertheless, the impact of density functional theory ({\\small DFT}) on computational materials science cannot be understated~\\cite{nmatHT},\noffering a reasonable compromise between cost and accuracy~\\cite{Haas_PRB_2009}.\nThe success of these implementations has stimulated the rapid development\nof automated frameworks and corresponding data repositories,\nincluding {\\small AFLOW}\\ (\\underline{A}utomatic \\underline{Flow} for Materials Discovery)~\\cite{aflowPAPER,curtarolo:art110,curtarolo:art85,curtarolo:art63,aflowBZ,curtarolo:art57,curtarolo:art53,curtarolo:art49,monsterPGM,aflowANRL,aflowPI},\nNovel Materials Discovery Laboratory~\\cite{nomad},\nMaterials Project~\\cite{APL_Mater_Jain2013},\nOpen Quantum Materials Database~\\cite{Saal_JOM_2013},\nComputational Materials Repository~\\cite{cmr_repository},\nand Automated Interactive Infrastructure and Database for Computational Science~\\cite{Pizzi_AiiDA_2016}.\nThese house an abundance of materials data.\nFor instance, the {\\small AFLOW}\\ framework, described in Section~\\ref{sec:aflow_chp},\nhas characterized more than 2 million compounds, each by about\n100 different properties accessible via the {\\sf \\AFLOW.org}\\ online database~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nInvestigations employing this data have not only led to advancements in modeling\nelectronics~\\cite{nmatTI,curtarolo:art94,curtarolo:art124,ceder:nature_1998},\nthermoelectrics~\\cite{curtarolo:art96,curtarolo:art114,curtarolo:art115,curtarolo:art119,curtarolo:art120,curtarolo:art125,curtarolo:art129},\nsuperalloys~\\cite{curtarolo:art113},\nand metallic glasses~\\cite{curtarolo:art112},\nbut also to the synthesis of two new magnets --- the first\ndiscovered by computational approaches~\\cite{curtarolo:art109}.\n\nFurther advancements are contingent on continued development and expansion of these materials repositories.\nNew entries are generated both by\n\\textbf{i.} calculating the properties of previously observed compounds\nfrom sources such as the Inorganic Crystal Structure Database~\\cite{ICSD} ({\\small ICSD}),\nand\n\\textbf{ii.} decorating structure prototypes~\\cite{aflowANRL,curtarolo:art130}.\nConsidering all possible crystals of different arrangements and decorations~\\cite{curtarolo:art124,Walsh_NChem_2015},\nthe analysis of existing structures --- a small subset --- is a critical first-step in determining fruitful directions for exploration.\nFor example, Section~\\ref{sec:art130} presents a general overview of the structure types appearing in an important\nclass of the solid compounds, \\nobreak\\mbox{\\it i.e.}, binary and ternary compounds of the 6A column oxides, sulfides, and selenides.\nIt contains an in-depth statistical analysis of these compounds, including the prevalence of various structure types,\ntheir symmetry properties, compositions, stoichiometries and unit cell sizes.\nResults reveal that these compound families include preferred stoichiometries and structure\ntypes that may reflect both their specific chemistry and research bias in the\navailable empirical data.\nDetection of non-overlapping gaps and missing stoichiometries in such\npopulations will guide subsequent studies: structures are avoided in the event that they are chemically\nunfavorable, or targeted to complement existing measurements.\n\nWith materials of interest identified, accurate computation of their properties demands\na set of reliable calculation parameters\/thresholds~\\cite{curtarolo:art104}.\nThese inputs need to be understood by researchers, and should be reported by the originators\nto ensure reproducibility and enable collaborative database expansion.\nAs described in Section~\\ref{sec:art104}, the {\\small AFLOW}\\ Standard defines these parameters\nfor high-throughput electronic structure calculations of crystals --- the basis for all {\\small AFLOW}\\ characterizations.\nStandard values are established for reciprocal space grid density,\nplane wave basis set kinetic energy cut-off, exchange-correlation\nfunctionals, pseudopotentials, {\\small DFT}$+U$ parameters, and convergence criteria.\n\nExploration of more complex properties~\\cite{curtarolo:art96,curtarolo:art115} and materials~\\cite{curtarolo:art110,curtarolo:art112}\ntypically warrants advanced (and expensive) characterization techniques~\\cite{Hedin_GW_1965,GW,ScUJ,Malashevich_GW_TiO_PRB2014,Patrick_GW_TiO2_JPCM2012}.\nFortunately, state-of-the-art workflows~\\cite{curtarolo:art96,curtarolo:art110,curtarolo:art115} and\ncareful descriptor development~\\cite{curtarolo:art112} have\nenabled experimentally-validated modeling within a {\\small DFT}\\ framework.\nFor instance, a thorough description of thermomechanical properties\nrequires difficult and time-consuming experiments.\nThis limits the availability of data:\none of the main obstacles for\nthe development of effective accelerated materials design strategies.\nSection~\\ref{sec:art115} introduces an automated, integrated workflow with robust error-correction\nwithin the {\\small AFLOW}\\ framework {that combines} the newly devised\n``Automatic Elasticity Library'' with the previously implemented {\\small GIBBS}\\ method~\\cite{curtarolo:art96}.\nThe former extracts the mechanical properties from several automatic self-consistent stress-strain calculations,\nwhile the latter employs those mechanical properties to evaluate the thermodynamics within the Debye model.\n{The} thermomechanical {workflow} is benchmarked against a set of\n74 experimentally characterized systems to pinpoint a\nrobust computational methodology for the evaluation of bulk and shear moduli,\nPoisson ratios, Debye temperatures, Gr{\\\"u}neisen parameters, and thermal conductivities of a wide variety of materials.\nThe effect of different choices of equations of state {and exchange-correlation functionals}\nis examined and the optimum combination of properties for the\nLeibfried-Schl{\\\"o}mann prediction of thermal conductivity is identified,\nleading to improved agreement with experimental results compared to the {\\small GIBBS}-only approach.\nThe {\\small AEL}-{\\small AGL}\\ framework has been applied to the {\\sf \\AFLOW.org}\\ data repositories to compute the thermomechanical properties\nof over 5,000 unique materials.\n\nSimilar to thermomechanical characterizations,\ndescriptions of thermodynamic stability and\nstructural\/chemical disorder are also resolved through an analysis of aggregate sets of \\nobreak\\mbox{\\it ab-initio}\\ calculations.\n\\textit{A priori} prediction of phase stability\nrequires\nknowledge of all energetically-competing structures at formation conditions.\nLarge materials repositories\noffer a path to prediction through the construction of\n\\nobreak\\mbox{\\it ab-initio}\\ phase diagrams, \\nobreak\\mbox{\\it i.e.}, the convex hull\nat a given temperature\/pressure.\nHowever, limited access to relevant data and software infrastructure has\nrendered thermodynamic characterizations largely peripheral,\ndespite their continued success in dictating synthesizability.\nIn Section~\\ref{sec:art146}, a new module is presented for autonomous thermodynamic stability analysis\nimplemented within {\\small AFLOW}.\nPowered by the {\\small AFLUX}\\ Search-{\\small API}, {\\small \\AFLOWHULLtitle}\\ leverages data of more than\n2 million compounds characterized in the {\\sf \\AFLOW.org}\\ repository,\nand can be employed locally from any {\\small UNIX}-like computer.\nThis module integrates a range of functionality:\nthe identification of stable phases and equivalent structures, phase coexistence,\nmeasures for robust stability, and determination of decomposition reactions.\nAs a proof-of-concept, thermodynamic characterizations have been performed\nfor more than 1,300 binary and ternary systems, enabling the identification of several\ncandidate phases for synthesis based on their relative stability criterion --- including\n17\\ promising $C15_{b}$-type structures and two half-Heuslers.\nIn addition to a full report included herein, an interactive online web application\nhas been developed, showcasing the results of the analysis, and is\nlocated at {\\sf aflow.org\/aflow-chull}.\n\nThe convex hull construction has fueled the generation of\nnovel descriptors for glass forming ability~\\cite{curtarolo:art112} and,\nmore generally, modeling structurally disordered systems.\nStatistical methods are employed to address chemically disordered\nstructures, where system-wide properties are resolved through an analysis of\nrepresentative ordered supercells~\\cite{curtarolo:art109}.\nIncorporating the effects of disorder is a necessary, albeit difficult, step in materials modeling.\nNot only is disorder intrinsic to all materials,\nbut it also offers a route to enhanced and even otherwise inaccessible functionality,\nas demonstrated by its ubiquity in technological applications.\nProminent examples include glasses~\\cite{kelton1991crystal,kelton2010nucleation,kelton1998new},\nsuperalloys~\\cite{Donachie_ASM_2002},\nfuel cells~\\cite{Xie_ACatB_2015},\nhigh-temperature superconductors~\\cite{Bednorz_ZPBCM_1986,Maeno_Nature_1994},\nand low thermal conductivity thermoelectrics~\\cite{Winter_JACerS_2007}.\n\nPredicting material properties of chemically disordered systems remains a\nformidable challenge in rational materials design.\nA proper analysis of such systems by means of a supercell approach requires\nconsideration of all possible superstructures, which can be a time-consuming process.\nOn the contrary, the use of quasirandom-approximants, while\ncomputational effective, implicitly bias the analysis toward disordered states with the lowest site correlations.\nIn Section~\\ref{sec:art110}, a novel framework is proposed\nto investigate stoichiometrically driven trends of disordered systems\n(\\nobreak\\mbox{\\it i.e.}, having partial occupation and\/or disorder in the atomic sites).\nAt the heart of the approach is the identification and analysis of unique supercells of a\nvirtually equivalent stoichiometry to the disordered material.\nBoltzmann statistics are employed to resolve system-wide properties at a high-throughput level.\nTo maximize efficiency and accessibility, this method has been integrated within {\\small AFLOW}.\nAs proof of concept, the approach is applied to three systems of interest,\na zinc chalcogenide (ZnS$_{1-x}$Se$_x$),\na wide-gap oxide semiconductor (Mg$_{x}$Zn$_{1-x}$O),\nand an iron alloy (Fe$_{1-x}$Cu$_{x}$)\nat various stoichiometries.\nThese systems exhibit properties that are highly tunable as a function of composition,\ncharacterized by optical bowing and linear ferromagnetic behavior.\nNot only are these qualities\npredicted, but additional insight into underlying physical mechanisms is revealed.\n\nThe aforementioned frameworks --- offering characterizations of thermomechanical\nand thermodynamic properties, as well as resolving features of disordered systems ---\nhave both benefited from and stimulated the development of the {\\sf \\AFLOW.org}\\ repository.\nThe combination of plentiful and diverse materials data~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}\nand its programmatic accessibility~\\cite{aflowAPI,aflux} also\njustify the application of data-mining techniques.\nThese methods can\nresolve subtle trends and correlations among materials and their\nproperties~\\cite{curtarolo:art94,Ghiringhelli_PRL_2015,curtarolo:art124,curtarolo:art129,curtarolo:art135},\nas well as motivate the formulation of novel property descriptors~\\cite{curtarolo:art112,curtarolo:art139}.\nIn fact, materials data generated by automated frameworks are\nconducive to such approaches,\nwhere strict standardizations of calculation parameters~\\cite{curtarolo:art104} not only ensure\nreproducibility, but also a minimum accuracy threshold.\nErrors from approximations or choice in parameters can therefore be treated as systematic,\nwhich are easily identified and rectified by \\underline{m}achine \\underline{l}earning ({\\small ML}) algorithms.\nModels have been generated for predicting electronic~\\cite{curtarolo:art124},\nthermomechanical~\\cite{curtarolo:art124,deJong_SR_2016} and vibrational~\\cite{curtarolo:art120,curtarolo:art129} properties,\nas well as the thermodynamic stability of both ordered~\\cite{Ghiringhelli_PRL_2015}\nand disordered~\\cite{Ward_ML_GFA_NPGCompMat_2016} phases.\nIn Section~\\ref{sec:art124}, data from the {\\small AFLOW}\\ repository for \\nobreak\\mbox{\\it ab-initio}\\ calculations\nis combined with Quantitative Materials Structure-Property Relationship models to predict important properties:\nmetal\/insulator classification, band gap energy, bulk\/shear moduli, Debye temperature, and heat capacities.\nThe prediction's accuracy compares well with the quality of the training data for virtually any\nstoichiometric inorganic crystalline material, reciprocating the available thermomechanical experimental data.\nThe universality of the approach is attributed to the construction of the descriptors: Property-Labeled Materials Fragments.\nThe representations require only minimal structural input allowing straightforward implementations of simple heuristic design rules.\n\n{\\small ML}\\ approaches are expected to become indispensable in two specific scenarios, prediction\nof complex properties and screening of large sets of materials.\nFor example, feature-importance analyses have informed on the interactions that elicit high-temperature\nsuperconductivity~\\cite{curtarolo:art94,curtarolo:art137}, an\nelusive phenomenon in which the driving mechanisms are still contested.\nSuperconductivity has been the focus of enormous research efforts since its discovery more than a century ago.\nYet, some features\nremain poorly understood; mainly the connection\nbetween superconductivity and chemical\/structural properties of materials.\nTo bridge the gap, several machine learning schemes are developed in Sections~\\ref{sec:art094} and \\ref{sec:art137}\nto model the critical temperatures $\\left(T_{\\mathrm{c}}\\right)$ of\nknown superconductors available via the SuperCon database.\nAs expected, these analyses suggest distinct mechanisms are responsible for driving superconductivity in\ndifferent classes of materials.\nHowever, they also hint at very complex physical interactions.\nFortunately, {\\small ML}\\ algorithms like random forests~\\cite{randomforests} are\ncapable of extracting very complicated functional relationships.\nIn the case of predicting $T_{\\mathrm{c}}$, these ``black-box'' models are\nquite valuable as\nfew alternative practical modeling schemes exist.\n\nIn Section~\\ref{sec:art094}, novel analytical approaches are introduced based on structural and electronic materials fingerprints\nand applied to predict the $T_{\\mathrm{c}}$\nof known superconductors.\nThe framework is employed to \\textbf{i.} query large databases of materials using similarity concepts,\n\\textbf{ii.} map the connectivity of materials space (\\nobreak\\mbox{\\it i.e.}, as materials cartograms)\nfor rapidly identifying regions with unique organizations\/properties,\nand \\textbf{iii.} develop predictive Quantitative Materials Structure-Property Relationship models for guiding materials design.\nThe materials fingerprinting and cartography approaches are\neffective computational tools to analyze,\nvisualize, model, and design new materials.\n\nSuperconductors are revisited in a much more in-depth study presented in Section~\\ref{sec:art137},\nleveraging the full set of the $12,000+$ materials in the SuperCon database.\nMaterials are first divided into two classes based on their $T_{\\mathrm{c}}$ values,\nabove and below $10$~K,\nand a classification model predicting this label is trained.\nThe model uses coarse-grained features based only on the chemical compositions.\nIt shows strong predictive power, with out-of-sample accuracy of about $92\\%$.\nSeparate regression models\nare developed to predict the values of $T_{\\mathrm{c}}$ for cuprate, iron-based, and low-$T_{c}$ compounds.\nThese models also demonstrate good performance,\nwith learned predictors offering\ninsights into the mechanisms behind superconductivity in different families of materials.\nTo improve the accuracy and interpretability of these models,\nnew features are incorporated using materials data\nfrom the {\\sf \\AFLOW.org}\\ repository.\nTo find potential new superconductors, the classification and regression models are combined into a single integrated pipeline\nand employed to search the entire Inorganic Crystallographic Structure Database ({\\small ICSD}).\nMore than 30 non-cuprate and non-iron-based oxides are selected as candidate materials.\n\nBeyond superconductors, {\\small ML}\\ models are created to predict properties of thermoelectrics (Section~\\ref{sec:art120})\nand permanent magnets (Section~\\ref{sec:art109}).\nThermoelectric materials generate an electric voltage when subjected to a temperature gradient, or\nconversely create\na temperature gradient when a voltage is applied~\\cite{snyder_complex_2008, nolas_thermoelectrics:_2001}.\nWith no moving parts and their resulting scalability, thermoelectrics\nhave potential applications in power generation for spacecraft,\nenergy recovery from waste heat in automotive and industrial facilities~\\cite{bell_cooling_2008, disalvo99},\nand spot cooling for nanoelectronics using the Peltier cooling effect~\\cite{bell_cooling_2008, disalvo99}.\nHowever, most of the available thermoelectric materials have low efficiency, only converting a few percent\nof the available thermal energy into electricity.\nTherefore, a major goal of thermoelectrics research is to develop new materials that have\nhigher thermoelectric efficiency as determined by a figure of merit~\\cite{snyder_complex_2008, nolas_thermoelectrics:_2001}.\nThe metric is dependent on quantities such as the Seebeck coefficient and electrical\/thermal conductivities.\nOne promising path to optimizing the figure of method is to\nminimize the lattice thermal conductivity.\n\nIn Section~\\ref{sec:art120}, the thermal conductivity $\\left(\\kappa\\right)$ is analyzed for\nsemiconducting oxides and fluorides with cubic perovskite structures.\nUsing finite-temperature phonon calculations and {\\small ML}\\ methods,\nthe mechanical stability of about 400 structures is resolved at 0~K, 300~K, and 1000~K.\nOf these, 92 compounds are determined to be mechanically stable at high temperatures\n--- including 36 not mentioned in the literature so far --- for which $\\kappa$ is calculated.\nSeveral trends are revealed, including\n\\textbf{i.} $\\kappa$ generally being smaller in fluorides than in oxides,\nlargely due to the lower ionic charge,\nand \\textbf{ii.} $\\kappa$ decreasing more\nslowly than the usual $T^{-1}$ behavior for most cubic perovskites.\nAnalyses expose the simple structural descriptors that correlate with $|\\kappa|$.\nThis set is also screened for materials exhibiting negative thermal expansion.\nThe study highlights a general strategy coupling force constants calculations with an iterative {\\small ML}\\ scheme\nto accelerate the discovery of mechanically stable compounds at high temperatures.\n\nThe role of {\\small ML}\\ models in predicting magnetic properties is of particular significance,\nas their \\textit{a priori} predictions were validated with the discovery of two new magnets.\nMagnetic materials underpin modern technologies, ranging from data storage to energy conversion and contactless sensing.\nHowever, the development of a new high-performance magnet is a long and often unpredictable process, and only\nabout two dozen feature in mainstream applications.\nIn Section~\\ref{sec:art109}, a systematic pathway is described to the discovery of novel\nmagnetic materials.\nBased on an extensive electronic structure library of Heusler alloys containing 236,115 compounds,\nalloys displaying magnetic order are selected, and it is determined\nwhether they can be fabricated at thermodynamic equilibrium.\nSpecifically, a full stability analysis is carried out for intermetallic Heusler alloys made only of transition metals.\nAmong the possible 36,540 candidates, 248 are found to be thermodynamically stable but only 20 are magnetic.\nThe magnetic ordering temperature, $T_\\mathrm{C}$, has then been estimated by a regression\ncalibrated on the experimental $T_\\mathrm{C}$ of about 60 known compounds.\nAs a final validation, the synthesis is attempted for a few of the predicted compounds,\nand two new magnets are produced.\nOne, Co$_2$MnTi, displays a remarkably high $T_\\mathrm{C}$ in perfect agreement with\nthe predictions, while the other, Mn$_2$PtPd, is an antiferromagnet.\nThis work paves the way for large-scale design of novel magnetic materials at unprecedented speed.\n\nOverall, data-driven approaches have extended materials modeling capabilities within a {\\small DFT}\\ framework.\nDescriptors for thermodynamic stability and formation\/features of disordered materials\nare accessible through analyses of ensembles of ordered structures,\nstimulating the development of large materials repositories.\nTo match the growth of these databases, insight-extraction must also be automated.\n{\\small ML}\\ methods are employed to reveal structure-property relationships and expose similarities among materials.\nUltimately, the power in {\\small ML}\\ lies in the speed of its predictions, which out-paces\n{\\small DFT}\\ calculations by orders of magnitude~\\cite{Isayev_ChemSci_2017}.\nEfforts to explore the full materials space through brute-force {\\small DFT}\\\ncalculations are impractical;\nstudies conservatively enumerate the size\nof possible hypothetical structures to be as large as 10$^{100}$~\\cite{Walsh_NChem_2015}.\nGiven that the number of currently characterized materials pales in comparison\nto the true potential diversity, methods --- like those presented here ---\nto filter\/screen the most interesting candidate materials\nwill play an integral role in future materials discovery workflows.\n\\clearpage\n\\chapter{The Automatic Flow Framework for Materials Discovery}\nMaterials informatics requires large repositories of materials data to identify trends in and correlations between materials properties,\nas well as for training machine learning models.\nSuch patterns lead to the formulation of descriptors that guide rational materials design.\nGenerating large databases of computational materials properties requires robust, integrated, automated frameworks~\\cite{nmatHT}.\nBuilt-in error correction and standardized parameter sets enable the production and analysis of data without direct intervention from human researchers.\nCurrent examples of such frameworks include\n{\\small AFLOW}\\ (\\underline{A}utomatic \\underline{{\\small FLOW}})~\\cite{aflowPAPER, aflowBZ, aflowlibPAPER, aflowAPI, curtarolo:art104, aflowlib.org, aflow_fleet_chapter, aflowPI, paoflow},\nMaterials Project~\\cite{materialsproject.org, APL_Mater_Jain2013, CMS_Ong2012b, Mathew_Atomate_CMS_2017},\n{\\small OQMD}\\ (\\underline{O}pen \\underline{Q}uantum \\underline{M}aterials \\underline{D}atabase)~\\cite{Saal_JOM_2013, Kirklin_AdEM_2013, Kirklin_ActaMat_2016},\nthe Computational Materials Repository~\\cite{cmr_repository} and its associated scripting interface {\\small ASE} (\\underline{A}tomic \\underline{S}imulation \\underline{E}nvironment)~\\cite{ase},\n{\\small AiiDA}\\ (\\underline{A}utomated \\underline{I}nteractive \\underline{I}nfrastructure and \\underline{Da}tabase for Computational Science)~\\cite{aiida.net, Pizzi_AiiDA_2016, Mounet_AiiDA2D_NNano_2018},\nand the Open Materials Database at \\verb|httk.openmaterialsdb.se| with its associated \\underline{H}igh-\\underline{T}hroughput \\underline{T}ool\\underline{k}it ({\\small HTTK}).\nOther computational materials science resources include the aggregated repository maintained by the \\underline{No}vel \\underline{Ma}terials \\underline{D}iscovery ({NOMAD}) Laboratory~\\cite{nomad},\nthe Materials Mine database available at \\verb|www.materials-mine.com|,\nand the \\underline{T}heoretical \\underline{C}rystallography \\underline{O}pen \\underline{D}atabase ({\\small TCOD})~\\cite{Merkys_TCOD_2017}.\nFor this data to be consumable by automated machine learning algorithms,\nit must be organized in programmatically accessible repositories~\\cite{aflowlibPAPER, aflowAPI, aflowlib.org, materialsproject.org, APL_Mater_Jain2013, Saal_JOM_2013, nomad}.\nThese frameworks also contain modules that combine and analyze data from various calculations to predict complex thermomechanical phenomena, such as lattice thermal conductivity and mechanical stability.\n\nComputational strategies have already had success in predicting materials for\napplications including photovoltaics~\\cite{YuZunger2012_PRL},\nwater-splitters~\\cite{CastelliJacobsen2012_EnEnvSci},\ncarbon capture and gas storage~\\cite{LinSmit2012_NMAT_carbon_capture, Alapati_JPCC_2012},\nnuclear detection and scintillators~\\cite{Derenzo:2011io, Ortiz09, aflowSCINT, curtarolo:art46},\ntopological insulators~\\cite{nmatTI, Lin_NatMat_HalfHeuslers_2010},\npiezoelectrics~\\cite{Armiento_PRB_2011, Vanderbilt_Piezoelectrics_PRL2012},\nthermoelectric materials~\\cite{curtarolo:art68, madsen2006, aflowKAPPA, curtarolo:art85},\ncatalysis~\\cite{Norskov09},\nand battery cathode materials~\\cite{Hautier-JMC2011, Hautier-ChemMater2011, Mueller-ChemMater2011}.\nMore recently, computational materials data has been combined with machine learning approaches\nto predict electronic and thermomechanical properties~\\cite{curtarolo:art124, deJong_SR_2016},\nand to identify superconducting materials~\\cite{curtarolo:art94}.\nDescriptors are also being constructed to describe the formation of disordered\nmaterials, and have recently been used to predict the glass forming\nability of binary alloy systems~\\cite{curtarolo:art112}.\nThese successes demonstrate that\naccelerated materials design can be achieved by combining structured data sets generated\nusing autonomous computational methods with intelligently formulated descriptors and machine learning.\n\n\\section{Automated computational materials \\texorpdfstring{design \\\\ frameworks}{design frameworks}}\n\\label{sec:aflow_chp}\n\nRapid generation of materials data relies on automated frameworks such as\n{\\small AFLOW}~\\cite{aflowPAPER, aflowBZ, aflowlibPAPER, aflowAPI, curtarolo:art104},\nMaterials Project's \\verb|pymatgen|~\\cite{CMS_Ong2012b} and \\verb|atomate|~\\cite{Mathew_Atomate_CMS_2017},\n{\\small OQMD}~\\cite{Saal_JOM_2013, Kirklin_AdEM_2013, Kirklin_ActaMat_2016},\n{\\small ASE}~\\cite{ase}, and {\\small AiiDA}~\\cite{Pizzi_AiiDA_2016}.\nThe general automated workflow is illustrated in Figure~\\ref{fig:aflow_chp:materials_design_workflow}.\nThese frameworks begin by creating the input files required by the electronic structure\ncodes that perform the quantum-mechanics level calculations, where the initial geometry is\ngenerated by decorating structural prototypes (Figure~\\ref{fig:aflow_chp:materials_design_workflow}(a, b)).\nThey execute and monitor these calculations, reading any error messages written to the\noutput files and diagnosing calculation failures.\nDepending on the nature of the errors, these frameworks are equipped with a catalog of prescribed solutions ---\nenabling them to adjust the appropriate parameters and restart the calculations (Figure~\\ref{fig:aflow_chp:materials_design_workflow}(c)).\nAt the end of a successful calculation, the frameworks parse the output files to extract the relevant materials data\nsuch as total energy, electronic band gap, and relaxed cell volume.\nFinally, the calculated properties are organized and formatted for entry into machine-accessible, searchable and sortable databases.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig001}\n\\mycaption[Computational materials data generation workflow.]\n{({\\bf a}) Crystallographic prototypes are extracted from databases such as the\n{\\small ICSD}\\ or the NRL crystal structure library, or generated by enumeration algorithms.\nThe illustrated examples are for the rocksalt, zincblende, wurtzite, Heusler, anti-Heusler and\nhalf-Heusler structures.\n({\\bf b}) New candidate materials are generated by decorating the atomic sites with different elements.\n({\\bf c}) Automated {\\small DFT}\\ calculations are used to optimize the geometric structure and calculate energetic, electronic,\nthermal, and elastic properties.\nCalculations are monitored to detect errors.\nThe input parameters are adjusted to compensate for the problem and the calculation is re-run.\nResults are formatted and added to an online data repository to facilitate programmatic access.\n({\\bf d}) Calculated data is used to plot the convex hull phase diagrams for each alloy system to identify stable compounds.}\n\\label{fig:aflow_chp:materials_design_workflow}\n\\end{figure}\n\nIn addition to running and managing the quantum-mechanics level calculations, the frameworks also\nmaintain a broad selection of post-processing libraries for extracting additional properties,\nsuch as calculating x-ray diffraction (XRD) spectra from relaxed atomic coordinates, and the\nformation enthalpies for the convex hull analysis to identify stable compounds (Figure~\\ref{fig:aflow_chp:materials_design_workflow}(d)).\nResults from calculations of distorted structures can be combined to calculate\nthermal and elastic properties~\\cite{aflowPAPER, curtarolo:art96, curtarolo:art100, curtarolo:art115},\nand results from different compositions and structural phases can be amalgamated to generate thermodynamic phase diagrams.\n\n\\subsection{Generating and using databases for materials discovery}\n\nA major aim of high-throughput computational materials science is to identify new, thermodynamically stable compounds.\nThis requires the generation of new materials structures, which have not been previously reported in the literature,\nto populate the databases. The accuracy of analyses involving sets of structures, such as that used to determine thermodynamic stability,\nis contingent on sufficient exploration of the full range of possibilities. Therefore, autonomous materials design frameworks\nsuch as {\\small AFLOW}\\ use crystallographic prototypes to generate new materials entries consistently and reproducibly.\n\nCrystallographic prototypes are the basic building blocks used to generate the wide range of materials entries involved in\ncomputational materials discovery.\nThese prototypes are based on \\textbf{i.} structures commonly observed in nature~\\cite{ICSD, navy_crystal_prototypes, aflowANRL},\nsuch as the rocksalt, zincblende, wurtzite or Heusler structures illustrated in Figure~\\ref{fig:aflow_chp:materials_design_workflow}(b),\nas well as \\textbf{ii.} hypothetical structures, such as those enumerated by the methods described in References~\\onlinecite{enum1, enum2}.\nThe {\\small AFLOW}\\ Library of Crystallographic Prototypes~\\cite{aflowANRL} is also available online at \\url{aflow.org\/CrystalDatabase\/}, where\nusers can choose from hundreds of crystal prototypes with adjustable parameters, and which can be decorated to generate new input\nstructures for materials science calculations.\n\nNew materials are then generated by decorating the various atomic sites in the crystallographic prototype with different elements.\nThese decorated prototypes serve as the structural input for \\nobreak\\mbox{\\it ab-initio}\\ calculations.\nA full relaxation of the geometries and energy determination follows, from which phase diagrams for stability analyses can be constructed.\nThe resulting materials data are then stored in an online data repository for future consideration.\n\nThe phase diagram of a given alloy system can be approximated by considering the low-temperature limit in\nwhich the behavior of the system is dictated by the ground state~\\cite{monster, monsterPGM}.\nIn compositional space, the lower-half convex hull defines the minimum energy surface and the\nground-state configurations of the system.\nAll non-ground-state stoichiometries are unstable, with the decomposition described by the\nhull facet directly below it.\nIn the case of a binary system, the facet is a tie-line as illustrated in Figure~\\ref{fig:aflow_chp:convex_hulls}(a).\nThe energy gained from this decomposition is geometrically represented by the (vertical-)distance of the\ncompound from the facet and quantifies the excitation energy involved in forming this compound.\nWhile the minimum energy surface changes at finite temperature (favoring disordered structures),\nthe $T=0$~K excitation energy serves as a reasonable descriptor for relative thermodynamic\nstability~\\cite{curtarolo:art113}.\nThis analysis generates valuable information such as ground-state structures,\nexcitation energies, and phase coexistence for storage in the\nonline data repository.\nThis stability data can be visualized and displayed by online modules,\nsuch as those developed by {\\small AFLOW}~\\cite{curtarolo:art113}, the Materials Project~\\cite{Ong_ChemMat_2008},\nand the {\\small OQMD}~\\cite{Akbarzadeh2007, Kirklin_AdEM_2013}.\nAn example visualization from {\\small AFLOW}\\ is shown in Figure~\\ref{fig:aflow_chp:convex_hulls}(b).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.8\\linewidth]{fig002}\n\\mycaption[Convex hull phase diagrams for multicomponent alloys systems.]\n{({\\bf a}) Schematic illustrating construction of convex hull for a general\nbinary alloy system $A_{x}B_{1-x}$. Ground state structures are depicted as red points, with the minimum energy\nsurface outlined with blue lines. The minimum energy surface is formed by\nconnecting the lowest energy structures with tie lines which form a convex hull.\nUnstable structures are shown in green, with the decomposition reaction indicated\nby orange arrows, and the decomposition energy indicated in purple.\n({\\bf b}) Example ternary convex hulls as generated by {\\small AFLOW}.}\n\\label{fig:aflow_chp:convex_hulls}\n\\end{figure}\n\nConvex hull phase diagrams have been used to discover new thermodynamically\nstable compounds in a wide range of alloy systems, including hafnium~\\cite{curtarolo:art49, curtarolo:art51},\nrhodium~\\cite{curtarolo:art53}, rhenium~\\cite{curtarolo:art63}, ruthenium~\\cite{curtarolo:art67}, and technetium ~\\cite{curtarolo:art70}\nwith various transition metals, as well as the Co-Pt system~\\cite{curtarolo:art66}. Magnesium alloy systems such as the lightweight\nLi-Mg system~\\cite{curtarolo:art55} and 34 other Mg-based systems~\\cite{curtarolo:art54} have also been investigated.\nThis approach has also been used to calculate the solubility of elements in titanium alloys~\\cite{curtarolo:art47}, to study the effect of hydrogen\non phase separation in iron-vanadium~\\cite{curtarolo:art74}, and to find new superhard tungsten nitride compounds~\\cite{curtarolo:art90}.\nThe data has been employed to generate structure maps for hcp metals~\\cite{curtarolo:art57},\nas well as to search for new stable compounds with the Pt$_8$Ti phase~\\cite{curtarolo:art56},\nand with the $L1_1$ and $L1_3$ crystal structures~\\cite{curtarolo:art71}.\nNote that even if a structure does not lie on the ground state convex hull, this does not rule out its existence.\nIt may be synthesizable under specific temperature and pressure conditions, and then be metastable under ambient\nconditions.\n\n\\subsection{Standardized protocols for automated data generation}\n\nStandard calculation protocols and parameters sets~\\cite{curtarolo:art104} are essential to\nthe identification of trends and correlations among materials properties.\nThe workhorse method for calculating quantum-mechanically resolved materials properties\nis \\underline{d}ensity \\underline{f}unctional \\underline{t}heory ({\\small DFT}).\n{\\small DFT}\\ is based on the Hohenberg-Kohn theorem~\\cite{Hohenberg_PR_1964}, which proves that for a ground state system,\nthe potential energy is a unique functional of the density: $V (\\mathbf{r}) = V(\\rho(\\mathbf{r}))$.\nThis allows for the charge density $\\rho(\\mathbf{r})$ to be used as the central variable for the calculations\nrather than the many-body wave function $\\Psi(\\mathbf{r}_{1}, \\mathbf{r}_{2}, ..., \\mathbf{r}_{N})$,\ndramatically reducing the number of degrees of freedom in the calculation.\n\nThe Kohn-Sham equations~\\cite{DFT} map the $n$ coupled equations for the system of $n$ interacting particles\nonto a system of $n$ independent equations for $n$ non-interacting particles:\n\\begin{equation}\n\\label{eq:aflow_chp:kohnshameqns}\n\\left[ -\\frac{\\hbar^2}{2m} \\nabla^2 + V_s (\\mathbf{r}) \\right] \\phi_i (\\mathbf{r}) = \\varepsilon_i \\phi_i(\\mathbf{r}),\n\\end{equation}\nwhere $\\phi_i(\\mathbf{r})$ are the non-interacting Kohn-Sham eigenfunctions and $\\varepsilon_i$ are their eigenenergies.\n$V_s (\\mathbf{r})$ is the Kohn-Sham potential:\n\\begin{equation}\n\\label{eq:aflow_chp:kohnshampotential}\n V_s (\\mathbf{r}) = V(\\mathbf{r}) + \\int e^2 \\frac{\\rho_s (\\mathbf{r}^{\\prime})}{|\\mathbf{r} - \\mathbf{r}^{\\prime}|}\n d^3 \\mathbf{r}^{\\prime} + V_{\\substack{\\scalebox{0.6}{XC}}}\\left[\\rho_s(\\mathbf{r})\\right],\n\\end{equation}\nwhere $V(\\mathbf{r})$ is the external potential\n(which includes influences of the nuclei, applied fields, and the core electrons when pseudopotentials are used),\nthe second term is the direct Coulomb potential, and $V_{\\substack{\\scalebox{0.6}{XC}}}\\left[\\rho_s(\\mathbf{r})\\right]$ is the exchange-correlation term.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig003}\n\\mycaption{Standardized paths in reciprocal space for calculation of the electronic band\nstructures for the 25 different lattice types~\\cite{aflowBZ}.}\n\\label{fig:aflow_chp:band_structure_paths}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig004}\n\\mycaption[Side-by-side visualization of the crystal structure and Brillouin Zone using Jmol~\\cite{Jmol_Hanson,Jmol}.]\n{(\\textbf{a}) The structure highlighted is Ag$_{3}$KS$_{2}$ ({\\small ICSD}\\ \\#73581): \\url{http:\/\/aflow.org\/material.php?id=Ag6K2S4_ICSD_73581}.\n(\\textbf{b}) The {\\small AFLOW}\\ Standard path of high-symmetry \\textbf{k}-points is illustrated in the Brillouin Zone~\\cite{aflowBZ}.}\n\\label{fig:aflow_fleet:jmol_bz}\n\\end{figure}\n\nThe mapping onto a system of $n$ non-interacting particles comes at the cost of introducing the\nexchange-correlation potential $V_{\\substack{\\scalebox{0.6}{XC}}}\\left[\\rho_s(\\mathbf{r})\\right]$, the exact form of which is unknown and must be approximated.\nThe simplest approximation is the \\underline{l}ocal \\underline{d}ensity \\underline{a}pproximation ({\\small LDA})~\\cite{Perdew_prb_1981},\nin which the magnitude of the exchange-correlation energy at a particular point in space is\nassumed to be proportional to the magnitude of the density at that point in space.\nDespite its simplicity, {\\small LDA}\\ produces realistic results for atomic structure, elastic and vibrational properties\nfor a wide range of systems. However, it tends to overestimate the binding energies of materials, even\nputting crystal bulk phases in the wrong energetic order~\\cite{Zupan_LDAperformance_PRB_1998}.\nBeyond {\\small LDA}\\ is the \\underline{G}eneralized \\underline{G}radient \\underline{A}pproximation ({\\small GGA}), in which the exchange correlation term\nis a functional of the charge density and its gradient at each point in space.\nThere are several forms of {\\small GGA}\\, including those developed by Perdew, Burke and Ernzerhof ({\\small PBE}~\\cite{PBE}), or by Lee, Yang and Parr ({\\small LYP}~\\cite{LYP_1988}).\nA more recent development is the meta-{\\small GGA}\\ \\underline{S}trongly \\underline{C}onstrained and \\underline{A}ppropriately \\underline{N}ormed ({\\small SCAN})\nfunctional~\\cite{Perdew_SCAN_PRL_2015}, which satisfies all 17 known exact constraints on meta-{\\small GGA}\\ functionals.\n\nThe major limitations of {\\small LDA}\\ and {\\small GGA}\\ include their inability to adequately describe systems with strongly correlated or localized electrons,\ndue to the local and semilocal nature of the functionals.\nTreatments include the Hubbard $U$ corrections~\\cite{LiechDFTU, Dudarev_dftu}, self-interaction corrections~\\cite{Perdew_prb_1981}\nand hybrid functionals such as Becke's 3-parameter modification of {\\small LYP} ({\\small B3LYP}~\\cite{B3LYP_1993}), and that of Heyd, Scuseria and Ernzerhof ({\\small Heyd2003}~\\cite{Heyd2003}).\n\nWithin the context of \\nobreak\\mbox{\\it ab-initio}\\ structure prediction calculations, {\\small GGA}-{\\small PBE}\\ is the usual standard since it tends to produce\naccurate geometries and lattice constants~\\cite{monster}.\nFor accounting for strong correlation effects, the {\\small DFT}$+U$ method~\\cite{LiechDFTU, Dudarev_dftu}\nis often favored in large-scale automated database generation due to its low computational overhead.\nHowever, the traditional {\\small DFT}$+U$ procedure requires the addition of an empirical factor to the potential~\\cite{LiechDFTU, Dudarev_dftu}.\nRecently, methods have been implemented to calculate the $U$ parameter self-consistently from first-principles, such as the ACBN0 functional~\\cite{curtarolo:art93}.\n\n{\\small DFT}\\ also suffers from an inadequate description of excited\/unoccupied states, as the theory\nis fundamentally based on the ground state.\nExtensions for describing excited states include time-dependent {\\small DFT}\\ ({\\small TDDFT})~\\cite{Hedin_GW_1965} and the GW correction~\\cite{GW}.\nHowever, these methods are typically much more expensive than standard {\\small DFT}, and are not generally considered for large scale database generation.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig005}\n\\mycaption[Example band structure and density of states images automatically generated and\nserved through the {\\sf \\AFLOW.org}\\ data repository.]\n{The structure highlighted is AlCo$_{2}$Fe ({\\small ICSD}\\ \\#57607): \\url{http:\/\/aflow.org\/material.php?id=Al1Co2Fe1_ICSD_57607}.\nThe results of the spin-polarized calculation are differentiated by: color on the band structure plot\n(black\/red for majority\/minority spin), and sign on the density of states plot (positive\/negative for majority\/minority spin).\nThe band structure is calculated following the {\\small AFLOW}\\ Standard path of high-symmetry \\textbf{k}-points~\\cite{aflowBZ}.}\n\\label{fig:aflow_fleet:bs_plot}\n\\end{figure}\n\nAt the technical implementation level, there are\nmany {\\small DFT}\\ software packages available, including\n{\\small VASP}~\\cite{kresse_vasp, vasp_prb1996, vasp_cms1996, kresse_vasp_paw},\n\\QUANTUMESPRESSO~\\cite{qe, Giannozzi:2017io}, {\\small ABINIT}~\\cite{gonze:abinit, abinit_2009},\n{\\small FHI-AIMS}~\\cite{Blum_CPC2009_AIM}, {\\small SIESTA}~\\cite{Soler2002SIESTA} and {\\small GAUSSIAN}~\\cite{Gaussian_2009}.\nThese codes are generally distinguished by the choice of basis set.\nThere are two principle types of basis sets: plane waves, which take the form $\\psi (\\mathbf{r}) = \\sum e^{i \\mathbf{k}\\cdot\\mathbf{r}}$,\nand local orbitals, formed by a sum over functions $\\phi_a (\\mathbf{r})$ localized at particular points in space, such as\ngaussians or numerical atomic orbitals~\\cite{Hehre_self_consistent_molecular_orbit_JCP1969}.\nPlane wave based packages include {\\small VASP}, \\QUANTUMESPRESSO\\ and {\\small ABINIT}, and are generally better suited to periodic systems such as bulk inorganic materials.\nLocal orbital based packages include {\\small FHI-AIMS}, {\\small SIESTA}\\ and {\\small GAUSSIAN}, and are generally better suited to non-periodic systems such as organic molecules.\nIn the field of automated computational materials science, plane wave codes such as {\\small VASP}\\ are generally preferred:\nit is straightforward to automatically and systematically generate well-converged basis sets\nsince there is only a single parameter to adjust, namely the cut-off energy determining the number\nof plane waves in the basis set.\nLocal orbital basis sets tend to have far more independently adjustable degrees of freedom,\nsuch as the number of basis orbitals per atomic orbital as well as their respective cut-off radii,\nmaking the automated generation of reliable basis sets more difficult.\nTherefore, a typical standardized protocol for automated materials science calculations~\\cite{curtarolo:art104} relies on\nthe {\\small VASP}\\ software package with a basis set cut-off energy higher than that recommended by the {\\small VASP}\\ potential files,\nin combination with the {\\small PBE}\\ formulation of {\\small GGA}.\n\nFinally, it is necessary to automate the generation of the \\textbf{k}-point grid and pathways in\nreciprocal space used for the calculation of forces, energies and the electronic band structure.\nIn general, {\\small DFT}\\ codes use standardized methods such as the Monkhorst-Pack scheme~\\cite{MonkhorstPack} to generate reciprocal lattice \\textbf{k}-point grids,\nalthough optimized grids have been calculated for different lattice types and are available online~\\cite{Wisesa_Kgrids_PRB_2016}.\nOptimizing \\textbf{k}-point grid density is a computationally expensive process that is difficult to automate,\nso instead standardized grid densities based on the concept of\n``\\underline{$k$}-\\underline{p}oints \\underline{p}er \\underline{r}eciprocal \\underline{a}tom'' ({\\small KPPRA}) are used.\nThe {\\small KPPRA}\\ value is chosen to be sufficiently large to ensure convergence for all systems.\nTypical recommended values used for {\\small KPPRA}\\ range from 6,000 to 10,000~\\cite{curtarolo:art104},\nso that a material with two atoms in the calculation cell will have a \\textbf{k}-point mesh of at least 3,000 to 5,000 points.\nStandardized directions in reciprocal space have also been defined for the calculation of the\nband structure as illustrated in Figure~\\ref{fig:aflow_chp:band_structure_paths}~\\cite{aflowBZ} and Figure~\\ref{fig:aflow_fleet:jmol_bz}.\nThese paths are optimized to include all of the high-symmetry points of the lattice.\nA standard band structure plot as generated by {\\small AFLOW}\\ is illustrated in Figure~\\ref{fig:aflow_fleet:bs_plot}.\n\n\\subsection{Integrated calculation of materials properties}\n\\label{subsec:aflow_chp:thermomechanical}\n\nAutomated frameworks such as {\\small AFLOW}\\ combine the computational analysis of properties including symmetry, electronic structure,\nelasticity, and thermal behavior into integrated workflows.\nCrystal symmetry information is used to find the primitive cell to reduce the size of {\\small DFT}\\ calculations,\nto determine the appropriate paths in reciprocal space for electronic band structure calculations (see Figure~\\ref{fig:aflow_chp:band_structure_paths}~\\cite{aflowBZ}),\nand to determine the set of inequivalent distortions for phonon and elasticity calculations.\nThermal and elastic properties of materials are important for predicting the thermodynamic and mechanical stability\nof structural phases~\\cite{Greaves_Poisson_NMat_2011, Poirier_Earth_Interior_2000, Mouhat_Elastic_PRB_2014, curtarolo:art106}\nand assessing their importance for a variety of applications.\nElastic properties such as the shear and bulk moduli are important for predicting the hardness\nof materials~\\cite{Chen_hardness_Intermetallics_2011, Teter_Hardness_MRS_1998},\nand thus their resistance to wear and distortion.\nElasticity tensors can be used to predict the properties of composite\nmaterials~\\cite{Hashin_Multiphase_JMPS_1963, Zohdi_Polycrystalline_IJNME_2001}.\nThey are also important in geophysics for modeling the propagation of seismic waves\nin order to investigate the mineral composition of geological\nformations~\\cite{Poirier_Earth_Interior_2000, Anderson_Elastic_RGP_1968, Karki_Elastic_RGP_2001}.\nThe lattice thermal conductivity $\\left(\\kappa_{\\substack{\\scalebox{0.6}{L}}}\\right)$ is a crucial\ndesign parameter in a wide range of important\ntechnologies, such as the development of new thermoelectric\nmaterials~\\cite{zebarjadi_perspectives_2012,aflowKAPPA,Garrity_thermoelectrics_PRB_2016},\nheat sink materials for thermal management in electronic devices~\\cite{Yeh_2002},\nand rewritable phase-change memories~\\cite{Wright_tnano_2011}.\nHigh thermal conductivity materials, which typically have a zincblende or diamond-like structure, are essential\nin microelectronic and nanoelectronic devices for achieving\nefficient heat removal~\\cite{Watari_MRS_2001}, and have\nbeen intensively studied for the past few decades~\\cite{Slack_1987}.\nLow thermal conductivity materials constitute\nthe basis of a new generation of thermoelectric materials and thermal\nbarrier coatings~\\cite{Snyder_jmatchem_2011}.\n\nThe calculation of thermal and elastic properties offer an excellent example of the power of\nintegrated computational materials design frameworks.\nWith a single input file, these frameworks can automatically set-up and run calculations of\ndifferent distorted cells, and combine the resulting energies and\nforces to calculate thermal and mechanical properties.\n\n\\subsubsection{Autonomous symmetry analysis}\n\nCritical to any analysis of crystals is the accurate determination of the symmetry profile.\nFor example, symmetry serves to\n\\textbf{i.} validate the forms of the elastic constants\nand compliance tensors, where the crystal symmetry dictates equivalence or absence\nof specific tensor elements~\\cite{nye_symmetry, curtarolo:art100, Mouhat_Elastic_PRB_2014}, and\n\\textbf{ii.} reduce the number of \\nobreak\\mbox{\\it ab-initio}\\ calculations needed for phonon\ncalculations, where, in the case of the finite-displacement method, equivalent\natoms and distortion directions are identified through factor group and site symmetry\nanalyses~\\cite{Maradudin1971}.\n\nAutonomous workflows for elasticity and vibrational characterizations\ntherefore require a correspondingly robust symmetry analysis.\nUnfortunately, standard symmetry packages~\\cite{stokes_findsym,Stokes_FROZSL_Ferroelectrics_1995,platon_2003,spglib},\ncatering to different objectives, depend on tolerance-tuning to\novercome numerical instabilities and atypical data --- emanating from\nfinite temperature measurements and uncertainty in experimentally reported observations.\nThese tolerances are responsible for validating mappings and identifying isometries,\nsuch as the $n$-fold operator depicted in Figure~\\ref{fig:aflow_chp:sym}(a).\nSome standard packages define separate tolerances for space, angle~\\cite{spglib},\nand even operation type~\\cite{stokes_findsym,Stokes_FROZSL_Ferroelectrics_1995,platon_2003}\n(\\nobreak\\mbox{\\it e.g.}, rotation \\nobreak\\mbox{\\it vs.}\\ inversion).\nEach parameter introduces a factorial expansion of unique inputs, which can result in\ndistinct symmetry profiles as illustrated in Figure~\\ref{fig:aflow_chp:sym}(b).\nBy varying the spatial tolerance $\\epsilon$, four different space groups can be observed\nfor AgBr ({\\small ICSD}\\ \\#56551\\footnote{{h}ttp:\/\/www.aflow.org\/material.php?id=56551}), if one is found at all.\nGaps in the range, where no consistent symmetry profile can be resolved, are\nparticularly problematic in automated frameworks, triggering critical failures in subsequent analyses.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig006}\n\\mycaption[Challenges in autonomous symmetry analysis.]\n{(\\textbf{a}) An illustration of a general $n$-fold symmetry operation.\n(\\textbf{b}) Possible space group determinations with mapping tolerance $\\epsilon$ for AgBr ({\\small ICSD}\\ \\#56551).\n(\\textbf{c}) Warping of mapping tolerance sphere with a transformation from cartesian to fractional basis.}\n\\label{fig:aflow_chp:sym}\n\\end{figure}\n\nCell shape can also complicate mapping determinations.\nAnisotropies in the cell, such as skewness of lattice vectors, translate\nto distortions of fractional and reciprocal spaces.\nA uniform tolerance sphere in cartesian space, inside which points are considered mapped,\ngenerally warps to a sheared spheroid, as depicted in Figure~\\ref{fig:aflow_chp:sym}(c).\nHence, distances in these spaces are direction-dependent, compromising the integrity\nof rapid minimum-image determinations~\\cite{hloucha_minimumimage_1998} and generally warranting\nprohibitively expensive algorithms~\\cite{curtarolo:art135}.\nSuch failures can result in incommensurate symmetry profiles, where the real space\nlattice profile (\\nobreak\\mbox{\\it e.g.}, bcc) does not match that of the reciprocal space (fcc).\n\nThe new {\\small AFLOW-SYM}\\ module~\\cite{curtarolo:art135} within {\\small AFLOW}\\ offers careful treatment of tolerances, with extensive\nvalidation schemes, to mitigate the aforementioned challenges.\nAlthough a user-defined tolerance input is still available, {\\small AFLOW}\\ defaults to one of two pre-defined\ntolerances, namely \\texttt{tight} (standard) and \\texttt{loose}.\nShould any discrepancies occur, these defaults are the starting values of a large tolerance scan,\nas shown in Figure~\\ref{fig:aflow_chp:sym}(b).\nA number of validation schemes have been incorporated to catch such discrepancies.\nThese checks are consistent with crystallographic group theory principles, validating operation\ntypes and cardinalities~\\cite{tables_crystallography}.\nFrom considerations of different extreme cell shapes, a heuristic threshold has been defined\nto classify scenarios where mapping failures are likely to occur --- based on skewness and mapping tolerance.\nWhen benchmarked against standard packages for over 54,000 structures in the Inorganic Crystal Structure Database,\n{\\small AFLOW-SYM}\\ consistently resolves\nthe symmetry characterization most compatible with experimental observations~\\cite{curtarolo:art135}.\n\nAlong with accuracy, {\\small AFLOW-SYM}\\ delivers a wealth of symmetry properties and representations\nto satisfy injection into any analysis or workflow.\nThe full set of operators --- including that of the point-, factor-, crystallographic point-, space groups,\nand site symmetries --- are provided in matrix, axis-angle, matrix generator, and quaternion representations in\nboth cartesian and fractional coordinates.\nA span of characterizations, organized by degree of symmetry-breaking, are available, including\nthose of the lattice, superlattice, crystal, and crystal-spin.\nSpace group and Wyckoff positions are also resolved.\nThe full dataset is made available in both plain-text and {\\small JSON}\\ formats.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig007}\n\\mycaption[Calculation of thermomechanical properties.]\n{({\\bf a}) {\\small AEL}\\ applies a set of independent normal and shear strains to the crystal structure to obtain\nthe elastic constants.\n({\\bf b}) {\\small AGL}\\ applies a set of isotropic strains to the unit cell to obtain\nenergy \\nobreak\\mbox{\\it vs.}\\ volume data, which is fitted by a polynomial in order to\ncalculate the bulk modulus as a function of volume, $B_{\\mathrm S} (V)$.\n$B_{\\mathrm S} (V)$ is then used to calculate the Debye temperature as a function of volume and thus\nthe vibrational free energy as a function of temperature.\nThe Gibbs free energy as a function of volume is then minimized for each pressure and temperature\npoint to obtain the equilibrium volume and other thermomechanical properties.\n({\\bf c}) {\\small APL}\\ obtains the harmonic \\underline{i}nteratomic \\underline{f}orce \\underline{c}onstants ({\\small IFC}{}s) from supercell calculations where inequivalent\natoms are displaced in inequivalent directions, and then the changes in the forces on the other atoms are calculated.\nThe {\\small IFC}{}s are then used to construct the dynamical matrix, which is diagonalized to obtain the phonon eigenmodes.\n{\\small AAPL}\\ calculates three-phonon scattering effects by performing supercell calculations where pairs of inequivalent atoms are displaced in inequivalent directions, and\n the changes in the forces on the other atoms in the supercell are calculated to obtain the third-order anharmonic {\\small IFC}{}s.}\n\\label{fig:aflow_chp:thermomechanical}\n\\end{figure}\n\n\\subsubsection{Harmonic phonons}\n\nThermal properties can be obtained by directly\ncalculating the phonon dispersion from the dynamical matrix of {\\small IFC}{}s.\nThe approach is implemented within the {\\small \\underline{A}FLOW} \\underline{P}honon \\underline{L}ibrary\n({\\small APL})~\\cite{aflowPAPER}.\nThe {\\small IFC}{}s are determined from a set of supercell calculations in which the atoms are\ndisplaced from their equilibrium positions~\\cite{Maradudin1971} as shown in Figure~\\ref{fig:aflow_chp:thermomechanical}(c).\n\nThe {\\small IFC}{}s derive from a Taylor expansion of the potential energy, $V$, of the crystal\nabout the atoms' equilibrium positions:\n\\begin{multline}\n V=\\left.V\\right|_{\\mathbf{r}(i,t)=0,\\forall i}+\n \\sum_{i,\\alpha}\\left.\\frac{\\partial V}{\\partial r(i,t)^{\\alpha}}\\right|_{\\mathbf{r}(i,t)=0,\\forall i} r(i,t)^{\\alpha} \\\\+\n \\frac{1}{2}\\sum_{\\substack{i,\\alpha,\\\\ j,\\beta}}\\left.\\frac{\\partial^2V}\n{\\partial r(i,t)^{\\alpha}\\partial r(j,t)^{\\beta}}\\right|_{\\mathbf{r}(i,t)=0,\\forall i}\n r(i,t)^{\\alpha}r(j,t)^{\\beta}+\n\\ldots,\n\\label{eq:aflow_chp:PE_harmonic}\n\\end{multline}\nwhere $r(i,t)^{\\alpha}$ is the $\\alpha$-cartesian component ($\\alpha=x,y,z$) of the time-dependent atomic displacement\n$\\mathbf{r}(t)$ of the $i^{\\mathrm{th}}$ atom about its equilibrium position,\n$\\left.V\\right|_{\\mathbf{r}(i,t)=0,\\forall i}$ is the potential energy of the crystal in its equilibrium configuration,\n$\\left.\\partial V\/\\partial r(i,t)^{\\alpha}\\right|_{\\mathbf{r}(i,t)=0,\\forall i}$\nis the negative of the force acting in the $\\alpha$ direction on atom $i$ in the equilibrium configuration\n(zero by definition), and\n$\\left.\\partial^2V\/\\partial r(i,t)^{\\alpha}\\partial r(j,t)^{\\beta}\\right|_{\\mathbf{r}(i,t)=0,\\forall i}$\nconstitute the {\\small IFC}{}s $\\phi(i,j)_{\\alpha,\\beta}$.\nTo first approximation, $\\phi(i,j)_{\\alpha,\\beta}$ is the negative of the force exerted\nin the $\\alpha$ direction on atom $i$ when atom\n$j$ is displaced in the $\\beta$ direction with all other atoms maintaining their equilibrium positions,\nas shown in Figure~\\ref{fig:aflow_chp:thermomechanical}(c).\nAll higher order terms are neglected in the harmonic approximation.\n\nCorrespondingly, the equations of motion of the lattice are as follows:\n\\begin{equation}\n M(i)\\ddot{r}(i,t)^{\\alpha}=\n-\\sum_{j,\\beta}\\phi(i,j)_{\\alpha,\\beta}\nr(j,t)^{\\beta}\\quad\\forall i,\\alpha,\n\\label{eq:aflow_chp:eom_harmonic}\n\\end{equation}\nand can be solved by a plane wave solution of the form\n\\begin{equation}\nr(i,t)^{\\alpha}=\\frac{v(i)^{\\alpha}}{\\sqrt{M(i)}} e^{\\mathrm{i}\\left(\\mathbf{q}\\cdot\\mathbf{R}_{l} - \\omega t \\right)},\n\\end{equation}\nwhere $v(i)^{\\alpha}$ form the phonon eigenvectors (polarization vector),\n$M(i)$ is the mass of the $i^{\\mathrm{th}}$ atom,\n$\\mathbf{q}$ is the wave vector,\n$\\mathbf{R}_{l}$ is the position of lattice point $l$,\nand $\\omega$ form the phonon eigenvalues (frequencies).\nThe approach is nearly identical to that taken for electrons in a periodic potential (Bloch waves)~\\cite{ashcroft_mermin}.\nPlugging this solution into the equations of motion (Equation~\\ref{eq:aflow_chp:eom_harmonic}) yields the following set of linear equations:\n\\begin{equation}\n\\omega^{2}v(i)^{\\alpha}=\n \\sum_{j,\\beta}D_{i,j}^{\\alpha,\\beta}(\\mathbf{q})\n v(j)^{\\beta}\\quad\\forall i,\\alpha,\n\\end{equation}\nwhere the dynamical matrix $D_{i,j}^{\\alpha,\\beta}(\\mathbf{q})$ is defined as\n\\begin{equation}\nD_{i,j}^{\\alpha,\\beta}(\\mathbf{q})=\n \\sum_{l}\\frac{\\phi(i,j)_{\\alpha,\\beta}}{\\sqrt{M(i)M(j)}} e^{-\\mathrm{i}\\mathbf{q}\\cdot\\left(\\mathbf{R}_{l}-\\mathbf{R}_{0}\\right)}.\n\\end{equation}\nThe problem can be equivalently represented by a standard eigenvalue equation:\n\\begin{equation}\n\\omega^{2}\n\\begin{bmatrix}\n \\\\\n\\mathbf{v} \\\\\n ~\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n & & \\\\\n & \\mathbf{D}(\\mathbf{q}) & \\\\\n & &\n\\end{bmatrix}\n\\begin{bmatrix}\n \\\\\n\\mathbf{v} \\\\\n ~\n\\end{bmatrix},\n\\label{eq:aflow_chp:dyn_eigen}\n\\end{equation}\nwhere\nthe dynamical matrix and phonon eigenvectors have dimensions $\\left(3 n_{\\mathrm{a}} \\times 3 n_{\\mathrm{a}}\\right)$\nand $\\left(3 n_{\\mathrm{a}} \\times 1 \\right)$, respectively, and $n_{\\mathrm{a}}$ is the number of atoms in the cell.\nHence, Equation~\\ref{eq:aflow_chp:dyn_eigen} has $\\lambda=3 n_{\\mathrm{a}}$ solutions\/modes referred to as branches.\nIn practice, Equation~\\ref{eq:aflow_chp:dyn_eigen} is solved for discrete sets of $\\mathbf{q}$-points to compute\nthe phonon density of states (grid over all possible $\\mathbf{q}$) and dispersion\n(along the high-symmetry paths of the lattice~\\cite{aflowBZ}).\nThus, the phonon eigenvalues and eigenvectors are appropriately denoted $\\omega_{\\lambda}(\\mathbf{q})$ and\n$\\mathbf{v}_{\\lambda}(\\mathbf{q})$, respectively.\n\nSimilar to the electronic Hamiltonian, the dynamical matrix is Hermitian, \\nobreak\\mbox{\\it i.e.},\n$\\mathbf{D}(\\mathbf{q})=\\mathbf{D}^{*}(\\mathbf{q})$.\nThus $\\omega_{\\lambda}^{2}(\\mathbf{q})$ must also be real, so $\\omega_{\\lambda}(\\mathbf{q})$ can either be real or purely imaginary.\nHowever, a purely imaginary frequency corresponds to vibrational motion of the lattice that increases exponentially in time.\nTherefore, imaginary frequencies, or those corresponding to soft modes, indicate the structure is dynamically unstable.\nIn the case of a symmetric, high-temperature phase, soft modes suggest there exists a lower symmetry structure\nstable at $T=0$~K.\nTemperature effects on phonon frequencies can be modeled with\n\\begin{equation}\n\\widetilde{\\omega}_{\\lambda}^{2}(\\mathbf{q},T)=\\omega_{\\lambda}^{2}(\\mathbf{q},T=0)+\\eta T^2,\n\\end{equation}\nwhere $\\eta$ is positive in general.\nThe two structures, the symmetric and the stable, differ by the distortion\ncorresponding to this ``frozen'' (non-vibrating) mode.\nUpon heating, the temperature term increases until the frequency reaches zero, and a phase transition occurs from\nthe stable structure to the symmetric~\\cite{Dove_LatDynam_1993}.\n\nIn practice, soft modes~\\cite{Parlinski_Phonon_Software} may indicate:\n\\textbf{i.} the structure is dynamically unstable at $T$,\n\\textbf{ii.} the symmetry of the structure is lower than that considered, perhaps due to magnetism,\n\\textbf{iii.} strong electronic correlations, or\n\\textbf{iv.} long range interactions play a significant role, and a larger supercell should be considered.\n\nWith the phonon density of states computed, the following thermal properties can be calculated:\nthe internal vibrational energy\n\\begin{equation}\n U_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x},T)=\\int_{0}^{\\infty} \\left( \\frac{1}{2} + \\frac{1}{e^{\\left(\\beta \\hbar \\omega\\right)}-1} \\right) \\hbar \\omega g(\\mathbf{x};\\omega) d\\omega,\n\\end{equation}\nthe vibrational component of the free energy $F_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x}; T)$\n\\begin{equation}\nF_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x}; T) \\!=\\!\\! \\int_0^{\\infty} \\!\\!\\left[\\frac{\\hbar \\omega}{2} \\!+\\!\n\\frac{1}{\\beta} \\ \\mathrm{log}\\!\\left(1\\!-\\!{\\mathrm e}^{- \\beta \\hbar \\omega }\\right)\\!\\right]\\!g(\\mathbf{x}; \\omega) d\\omega,\n\\label{eq:aflow_chp:fvib}\n\\end{equation}\nthe vibration entropy\n\\begin{equation}\nS_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x},T)=\\frac{U_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x},T)-F_{\\substack{\\scalebox{0.6}{vib}}}(\\mathbf{x}; T)}{T},\n\\end{equation}\nand the isochoric specific heat\n\\begin{equation}\nC_{{\\substack{\\scalebox{0.6}{V}}}, {\\substack{\\scalebox{0.6}{vib}}}}(\\mathbf{x},T)=\\int_{0}^{\\infty} \\frac{ k_{\\substack{\\scalebox{0.6}{B}}} \\left(\\beta \\hbar \\omega \\right)^2 g(\\mathbf{x};\\omega)}{\\left(1-e^{-\\left(\\beta \\hbar \\omega\\right)}\\right) \\left(e^{\\left(\\beta \\hbar \\omega\\right)}-1\\right) } d\\omega.\n\\end{equation}\n\n\\subsubsection{Quasi-harmonic phonons}\n\nThe harmonic approximation does not describe phonon-phonon scattering, and so cannot be used to\ncalculate properties such as thermal conductivity or thermal expansion.\nTo obtain these properties, either the quasi-harmonic approximation can be used,\nor a full calculation of the higher order anharmonic {\\small IFC}{}s can be performed.\nThe quasi-harmonic approximation is the less computationally demanding of these two methods,\nand compares harmonic calculations of phonon properties at different volumes to predict anharmonic properties.\nThe different volume calculations can be in the form of harmonic phonon calculations as described\nabove~\\cite{curtarolo:art114, curtarolo:art119},\nor simple static primitive cell calculations~\\cite{Blanco_CPC_GIBBS_2004, curtarolo:art96}.\nThe \\underline{Q}uasi-\\underline{H}armonic \\underline{A}pproximation\nis implemented within {\\small APL}\\ and referred to as {\\small QHA-APL}~\\cite{curtarolo:art96}.\nIn the case of the quasi-harmonic phonon calculations, the anharmonicity of the system is described by\nthe mode-resolved Gr{\\\"u}neisen parameters, which are given by the change in the phonon frequencies as a function of volume\n\\begin{equation}\n\\label{eq:aflow_chp:gamma_micro}\n\\gamma_{\\lambda}(\\mathbf{q}) = - \\frac{V}{\\omega_{\\lambda}(\\mathbf{q})} \\frac{\\partial \\omega_{\\lambda}(\\mathbf{q})}{\\partial V},\n\\end{equation}\nwhere $\\gamma_{\\lambda}(\\mathbf{q})$ is the parameter for the wave vector $\\mathbf{q}$ and the $\\lambda^{\\rm{th}}$ mode of the phonon dispersion.\nThe average of the $\\gamma_{\\lambda}(\\mathbf{q})$ values, weighted by the specific heat capacity of each mode $C_{{\\substack{\\scalebox{0.6}{V}}},\\lambda}(\\mathbf{q})$, gives the average\nGr{\\\"u}neisen parameter:\n\\begin{equation}\n\\label{eq:aflow_chp:gamma_ave}\n\\gamma = \\frac{\\sum_{\\lambda,\\mathbf{q}} \\gamma_{\\lambda}(\\mathbf{q}) C_{{\\substack{\\scalebox{0.6}{V}}},\\lambda}(\\mathbf{q})}{C_{\\substack{\\scalebox{0.6}{V}}}}.\n\\end{equation}\nThe specific heat capacity, Debye temperature and Gr{\\\"u}neisen parameter can then be combined to\ncalculate other properties such as the specific heat capacity at constant pressure $C_{\\rm p}$,\nthe thermal coefficient of expansion $\\alpha$, and the lattice thermal\nconductivity $\\kappa_{\\substack{\\scalebox{0.6}{L}}}$~\\cite{curtarolo:art119}, using similar expressions to those described in Section~\\ref{sec:art115}.\n\n\\subsubsection{Anharmonic phonons}\n\nThe full calculation of the anharmonic {\\small IFC}{}s requires performing supercell calculations in which pairs of\ninequivalent atoms are displaced in all pairs of\ninequivalent directions~\\cite{Broido2007, Wu_PRB_2012, ward_ab_2009, ward_intrinsic_2010, Zhang_JACS_2012, Li_PRB_2012, Lindsay_PRL_2013, Lindsay_PRB_2013, Li_ShengBTE_CPC_2014, curtarolo:art125}\nas illustrated in Figure~\\ref{fig:aflow_chp:thermomechanical}(c).\nThe third order anharmonic {\\small IFC}{}s can then be obtained by calculating the change in the forces on\nall of the other atoms due to these displacements.\nThis method has been implemented in the form of a fully\nautomated integrated workflow in the {\\small AFLOW}\\ framework,\nwhere it is referred to as the {\\small \\underline{A}FLOW} \\underline{A}nharmonic\n\\underline{P}honon \\underline{L}ibrary ({\\small AAPL})~\\cite{curtarolo:art125}.\nThis approach can provide very accurate results for the lattice thermal conductivity when combined\nwith accurate electronic structure methods\n\\cite{curtarolo:art125},\nbut quickly becomes very expensive for systems with multiple\ninequivalent atoms or low symmetry.\nTherefore, simpler methods such as the quasi-harmonic Debye model\ntend to be used for initial rapid screening~\\cite{curtarolo:art96, curtarolo:art115}, while\nthe more accurate and expensive methods are used for characterizing systems\nthat are promising candidates for specific engineering applications.\n\n\\subsection{Online data repositories}\n\nRendering the massive quantities of data generated using automated \\nobreak\\mbox{\\it ab-initio}\\ frameworks available\nfor other researchers requires going beyond the conventional methods\nfor the dissemination of scientific results in the form of journal articles.\nInstead, this data is typically made available in online data repositories, which can usually be accessed both\nmanually via interactive web portals, and programmatically via an \\underline{a}pplication \\underline{p}rogramming \\underline{i}nterface ({\\small API}).\n\n\\subsubsection{Computational materials data web portals}\n\nMost computational data repositories include an interactive web portal front end that enables manual data access.\nThese web portals usually include online applications to facilitate data retrieval and analysis.\nThe front page of the {\\small AFLOW}\\ data repository is displayed in Figure~\\ref{fig:aflow_chp:aflow_web_apps}(a).\nThe main features include a search bar where information such as\n{\\small ICSD}\\ reference number, {\\small \\underline{A}FLOW} \\underline{u}nique \\underline{id}entifier ({\\small AUID}) or the chemical formula can be entered\nin order to retrieve specific materials entries.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig008}\n\\mycaption[{\\small AFLOW}\\ web applications.]\n{({\\bf a}) Front page of the {\\small AFLOW}\\ online data repository, highlighting the link to\n({\\bf b}) the {\\small AFLOW}\\ advanced search application, which facilitates complex search\nqueries including filtering by chemical composition and materials properties and\n({\\bf c}) the {\\small AFLOW}\\ interactive convex hull generator, showing the 3D hull for the Pt-Sc-Zn ternary alloy system.}\n\\label{fig:aflow_chp:aflow_web_apps}\n\\end{figure}\n\nBelow are buttons linking to several different online applications such as the advanced search functionality,\nconvex hull phase diagram generators, machine learning applications~\\cite{curtarolo:art124, curtarolo:art129, curtarolo:art136} and {\\small AFLOW}-online data analysis tools.\nThe link to the advanced search application is highlighted by the orange square, and the application page is shown in Figure~\\ref{fig:aflow_chp:aflow_web_apps}(b).\nThe advanced search application allows users to search for materials that contain (or exclude) specific elements or groups of elements,\nand also to filter and sort the results by properties such as electronic band structure energy gap (under the ``Electronics'' properties filter group)\nand bulk modulus (under the ``Mechanical'' properties filter group).\nThis allows users to identify candidate materials with suitable materials properties for specific applications.\n\nAnother example online application available on the {\\small AFLOW}\\ web portal is the convex hull phase diagram generator.\nThis application can be accessed by clicking on the button highlighted by the orange square in\nFigure~\\ref{fig:aflow_chp:aflow_web_apps}(a), which will bring up a periodic table allowing users to\nselect two or three elements for which they want to generate a convex hull.\nThe application will then access the formation enthalpies and stoichiometries of the materials entries in the\nrelevant alloy systems, and use this data to generate a two or three dimensional convex hull phase diagram\nas depicted in Figure~\\ref{fig:aflow_chp:aflow_web_apps}(c).\nThis application is fully interactive, allowing users to adjust the energy axis scale,\nrotate the diagram to view from different directions, and select specific points to obtain more information on the\ncorresponding entries.\n\n\\subsubsection{Programmatically accessible online repositories of computed materials properties}\n\nIn order to use materials data in machine learning algorithms, it should be stored in a structured online database\nand made programmatically accessible via a \\underline{re}presentational \\underline{s}tate \\underline{t}ransfer {\\small API}\\ ({\\small REST-API}).\nExamples of online repositories of materials data include {\\small AFLOW}~\\cite{aflowlibPAPER, aflowAPI},\nMaterials Project~\\cite{materialsproject.org}, and {\\small OQMD}~\\cite{Saal_JOM_2013}.\nThere are also repositories that aggregate results from multiple sources such as\n{NOMAD}~\\cite{nomad} and Citrine~\\cite{citrine_database}.\n\n{\\small REST-API}{}s facilitate programmatic access to data repositories.\nTypical databases such as {\\small AFLOW}\\ are organized in layers,\nwith the top layer corresponding to a project or catalog (\\nobreak\\mbox{\\it e.g.}, binary alloys),\nthe next layer corresponding to data sets (\\nobreak\\mbox{\\it e.g.}, all of the entries for a particular alloy system),\nand then the bottom layer corresponding to specific materials entries, as illustrated in Figure~\\ref{fig:aflow_chp:aflow_restapi_layers_aurl}(a).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig009}\n\\mycaption[{\\small AFLOW}\\ {\\small REST-API}\\ structure.]\n{\\small (\\textbf{a}) The {\\small AFLOW}\\ database is organized as a multilayered system.\n(\\textbf{b}) Example of an {\\small AURL}\\ which enables direct programmatic access to specific materials entry properties in the {\\small AFLOW}\\ database.}\n\\label{fig:aflow_chp:aflow_restapi_layers_aurl}\n\\end{figure}\n\nIn the case of the {\\small AFLOW}\\ database, there are currently four different ``projects'', namely the\n``ICSD'', ``LIB1'', ``LIB2'' and ``LIB3'' projects; along with three\nmore under construction: ``LIB4'', ``LIB5'' and ``LIB6''.\nThe ``ICSD'' project contains calculated data for previously observed compounds~\\cite{ICSD},\nwhereas the other three projects contain calculated data for single elements, binary alloys,\nand ternary alloys respectively, and are constructed by decorating prototype\nstructures with combinations of different elements.\nWithin ``LIB2'' and ``LIB3'', there are many different data sets, each corresponding to a specific\nbinary or ternary alloy system.\nEach entry in the set corresponds to a specific prototype structure and stoichiometry.\nThe materials properties values for each of these entries are encoded via keywords,\nand the data can be accessed via {\\small URL}{}s constructed from the different layer names and the appropriate keywords.\nIn the case of the {\\small AFLOW}\\ database, the location of each layer and entry is\nidentified by an {\\small \\underline{A}FLOW} \\underline{u}niform \\underline{r}esource \\underline{l}ocator ({\\small AURL})~\\cite{aflowAPI},\nwhich can be converted to a {\\small URL}\\ providing the absolute path to a particular layer, entry or property.\nThe {\\small AURL}\\ takes the form \\url{server:AFLOWDATA\/project\/set\/entry\/?keywords},\nfor example \\url{aflowlib.duke.edu:AFLOWDATA\/LIB2_RAW\/Cu_pvV_sv\/15\/?energy_atom},\nwhere \\url{aflowlib.duke.edu} is the web address of the physical server where the data is located,\n\\url{LIB2_RAW} is the binary alloy project layer, \\url{Cu_pvV_sv} is\nthe set containing the binary alloy system Cu-V, \\url{15} is a specific entry with the composition\nCu$_3$V in a tetragonal lattice, and \\url{energy_atom} is the keyword corresponding\nto the property of energy per atom in units of eV, as shown in Figure~\\ref{fig:aflow_chp:aflow_restapi_layers_aurl}(b).\nEach {\\small AURL}\\ can be converted to a web {\\small URL}\\ by changing the ``\\url{:}'' after the server name to a ``\\url{\/}'',\nso that the {\\small AURL}\\ in Figure~\\ref{fig:aflow_chp:aflow_restapi_layers_aurl}(b) would become the\n{\\small URL}\\ \\url{aflowlib.duke.edu\/AFLOWDATA\/LIB2_RAW\/Cu_pvV_sv\/15\/?energy_atom}.\nThis {\\small URL}, if queried via a web browser or using a UNIX utility such as \\texttt{wget},\nreturns the energy per atom in eV for entry \\url{15} of the Cu-V binary alloy system.\n\nIn addition to the {\\small AURL}, each entry in the {\\small AFLOW}\\ database is also associated with an\n{\\small AUID}~\\cite{aflowAPI},\nwhich is a unique hexadecimal (base 16) number constructed from a checksum of the {\\small AFLOW}\\ output file for that entry.\nSince the {\\small AUID}\\ for a particular entry can always be reconstructed by applying the checksum\nprocedure to the output file, it serves as a permanent, unique specifier for each calculation,\nirrespective of the current physical location of where the data are stored.\nThis enables the retrieval of the results for a particular calculation from different servers,\nallowing for the construction of a truly distributed database that is robust\nagainst the failure or relocation of the physical hardware. Actual database versions can be\nidentified from the version of {\\small AFLOW}\\ used to parse the calculation output files and\npostprocess the results to generate the database entry. This information can be retrieved using the\nkeyword \\url{aflowlib_version}.\n\nThe search and sort functions of the front-end portals can be combined with the programmatic\ndata access functionality of the {\\small REST-API}\\ through the implementation of a Search-{\\small API}.\nThe {\\small AFLUX}\\ Search-{\\small API}\\ uses the {\\small LUX}\\ language to enable the embedding of logical\noperators within {\\small URL}\\ query strings~\\cite{aflux}.\nFor example, the energy per atom of every entry in the {\\small AFLOW}\\ repository containing the element Cu or V,\nbut not the element Ti, with an electronic band gap between 2 and 5~eV, can be retrieved using the command:\n\\url{aflowlib.duke.edu\/search\/API\/species((Cu:V),(!Ti)),Egap(2*,*5),energy_atom}.\nIn this {\\small AFLUX}\\ search query, the comma ``\\verb|,|'' represents the logical {\\small AND} operation, the colon ``\\verb|:|'' the logical {\\small OR} operation,\nthe exclamation mark ``\\verb|!|'' the logical {\\small NOT} operation, and the asterisk ``\\verb|*|'' is the ``loose'' operation that defines a range of values to search within.\nNote that by default {\\small AFLUX}\\ returns only the first 64 entries matching the search query.\nThe number and set of entries can be controlled by appending the \\verb|paging| directive to the end of the search query as follows:\n\\url{aflowlib.duke.edu\/search\/API\/species((Cu:V),(!Ti)),Egap(2*,*5),energy_atom,paging(0)},\nwhere calling the \\verb|paging| directive with the argument ``0'' instructs {\\small AFLUX}\\ to return all of the matching entries\n(note that this could potentially be a large amount of data, depending on the search query).\nThe {\\small AFLUX}\\ Search-{\\small API}\\ allows users to construct and retrieve customized data sets, which they can feed into materials\ninformatics machine learning packages to identify trends and correlations for use in rational materials design.\n\nThe use of {\\small API}{}s to provide programmatic access is being extended beyond materials data retrieval,\nto enable the remote use of pre-trained machine learning algorithms.\nThe {\\small AFLOW-ML}\\ {\\small API}~\\cite{curtarolo:art136} facilitates access to the two machine learning models\nthat are also available online at \\url{aflow.org\/aflow-ml}~\\cite{curtarolo:art124, curtarolo:art129}.\nThe {\\small API}\\ allows users to submit structural data for the material of interest using a utility such as \\verb|cURL|,\nand then returns the results of the model's predictions in {\\small JSON}\\ format.\nThe programmatic access to machine learning predictions enables the incorporation of machine learning into\nmaterials design workflows, allowing for rapid pre-screening to automatically select\npromising candidates for further investigation.\n\n\\clearpage\n\\section{The Structure and Composition Statistics of 6A Binary and Ternary Crystalline Materials}\n\\label{sec:art130}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art130}.\n\n\\subsection{Introduction}\nThe creation of novel materials with optimal properties for diverse applications requires a fundamental\nunderstanding of the factors that govern the formation of crystalline\nsolids from various mixtures of elements.\nCompounds of the non-metallic elements of column 6A, oxygen, sulfur and selenium, are of particular interest.\nThey serve in a large variety of applications\nin diverse fields of technology, \\nobreak\\mbox{\\it e.g.}, chemistry, catalysis, optics,\ngas sensors, electronics, thermoelectrics, piezoelectrics,\ntopological insulators, spintronics and more~\\cite{eranna2004oxide,fortunato2012oxide,tsipis2008electrode,jiang1998new,panda2009review,shi2017,lorenz20162016,ruhle2012all}.\nGiven the very large number of possibilities, many of the alloy systems of these elements have not\nbeen fully investigated, some of them even not at all.\n\nIn recent years, high-throughput computational techniques based on \\nobreak\\mbox{\\it ab-initio}\\ calculations\nhave emerged as a potential route to bridge these experimental gaps and\ngain understanding of the governing principles of compound formation~\\cite{nmatHT}.\nThis led to the creation of large databases of computational materials\ndata~\\cite{aflowPAPER,CMS_Ong2012b}.\nYet, these computational approaches are practically limited by the number and size of structures\nthat can be thoroughly analyzed, and fundamental issues that limit\nthe applicability of standard semi-local {\\small DFT}\\ for non-metallic compounds.\nThe sought-after governing principles are thus still largely unknown.\n\nNevertheless, the considerable body of experimental data that is already available,\nalthough by no means complete, is a useful basis for large-scale data analysis.\nThis experimental data is usually presented in\ncompendiums that lack statistical analysis.\nPresenting this data in a structured manner may be conducive for gaining insights\ninto the essential factors that determine structure formation, and may help to provide\nmaterial scientists with the necessary foundation for rational\nmaterials design.\n\nAnalyses recently carried out for the intermetallic binaries~\\cite{dshemuchadse2014some}\nand ternaries~\\cite{dshemuchadse2015more} have uncovered interesting Bravais lattices distributions and an unexpected large prevalence of unique structure types.\nHere we extend the analysis and discuss trends, as well as special phenomena, across\nbinary and ternary compounds of the 6A non-metals.\nThis analysis reveals the following\ninteresting observations:\n\\begin{itemize}[leftmargin=*]\n\n \\item Considerable overlap exists between the sulfides and selenides:\n about a third of the total number of structure types are shared among\n both compound families.\n In contrast, the overlap between the oxides and the other two families is rather small.\n\n \\item The prevalence of different compound stoichiometries in the sulfide\n\tand selenide families is very similar to each other\n\tbut different from that of the oxides. Some stoichiometries\n\tare abundant in the oxides but are {\\it almost\n absent} in the sulfides or selenides, and vice versa.\n\n \\item The number of ternary oxide stoichiometries, $A_{x}B_{y}$O$_{z}$, decreases when the product of\n binary oxide stoichiometries, of participating elements, increases. This behavior can be explained by general thermodynamic arguments and is discussed in the text.\n\n \\item Overall, oxide compounds tend to have richer oxygen content than the sulfur and selenium content in their corresponding compounds.\n\n \\item Across all three compound families, most structure types are represented\n by only one compound.\n\n \\item High symmetry lattices, \\nobreak\\mbox{\\it e.g.}, the orthorhombic face centered,\n orthorhombic body centered and cubic lattices\n\t are relatively rare among these compounds.\n This reflects the spatial arrangement of the compound forming orbitals of the 6A non-metals,\n whose chemistry does not favor these structures.\n\n\\end{itemize}\n\nIn the analysis presented here, we adopt the ordering of the elements by Mendeleev numbers as\ndefined by Pettifor~\\cite{pettifor:1984,pettifor:1986},\nand complement it by investigating the crystallographic properties of\nthe experimentally reported compounds.\nPettifor maps constructed for these compound families exhibit similar separation between different structure types as the\nclassical Pettifor maps for binary structure types~\\cite{pettifor:1984,pettifor:1986}.\nFor some stoichiometries, the structure types show similar patterns in\nthe maps of the three compound families, suggesting\nthat similar atoms tend to form these stoichiometries with all three elements.\nSuch similarity of patterns is more common between\nthe sulfides and selenides than between either of them and the oxides.\n\nThese findings suggest a few possible guiding principles for directed searches of new compounds.\nElement substitution could be used to examine favorable candidates within the\nimperfect overlaps of the structure distributions, especially between the sulfides and selenides.\nMoreover, the missing stoichiometries and structure symmetries mean that data-driven approaches, \\nobreak\\mbox{\\it e.g.},\nmachine learning, must use training sets not limited to one compound family, even in studies directed at that specific set of compounds.\nThis hurdle may be avoided by augmenting the known structures with those of the other families.\nIn addition, identified gaps in the Mendeleev maps suggest potential new compounds,\nboth within each family or by correlations of similar structure maps across the different families.\n\n\\subsection{Data methodology}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Data extraction numerical summary.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\n & compounds & unique compounds & structure types \\\\\n\\hline\ntotal & 88,373 & 50,294& 13,324 \\\\\n\\hline\nunary & 1752 & 499 & 197\\\\\n\\hline\nbinary & 27,487 & 10,122 & 1,962 \\\\\nbinary oxides & 3,256 & 844 & 538 \\\\\nbinary sulfides & 1,685 & 495 & 270 \\\\\nbinary selenides & 1,050 & 332 & 168 \\\\\n\\hline\nternary & 37,907 & 23,398& 4,409\\\\\nternary oxides & 10,350 & 5,435 & 2,079 \\\\\nternary sulfides & 3,190 & 2,041 & 784 \\\\\nternary selenides & 1,786 & 1,256 & 521\\\\\n\\hline\nquaternary & 15,138 & 11,050 & 3,855 \\\\\n5 atoms & 4,638 & 3,899 & 2,053 \\\\\n6 atoms & 1,219 & 1,101 & 682 \\\\\n7 atoms & 212\t& 201 &\t154 \\\\\n8 atoms & 20 & 20 & 12\\\\\n\\end{tabular}\n\\label{tab:art130:ICSD_DATA}\n\\end{table}\n\nThe {\\small ICSD}~\\cite{ICSD_database} includes approximately 169,800 entries (as of August 2016).\nFor this study we exclude all entries with partial or random occupation and those that do not have full structure data.\nThe remaining set of structures has been filtered using the {\\small AFLOW}\\ software~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI},\nwhich uses an error checking protocol to ensure the integrity of each entry.\n{\\small AFLOW}\\ generates each structure by appropriately propagating the Wyckoff positions of the specified spacegroup.\nThose structures that produce inconsistencies, \\nobreak\\mbox{\\it e.g.}, overlapping atoms or a different stoichiometry\nthan the structure label are ignored.\nIf atoms are detected to be too close ($\\leq 0.6$\\AA), alternative standard ITC\n(International Table of Crystallography)~\\cite{tables_crystallography} settings of the spacegroup are attempted.\nThese settings define different choices for the cell's unique axes, possibly\ncausing atoms to overlap if not reported correctly.\nOverall, these considerations reduce the full set of {\\small ICSD}\\ entries to a\nmuch smaller set of 88,373 ``true'' compounds.\nThese entries are contained in {\\small AFLOW}\\ Database~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nThey include the results of the {\\small AFLOW}\\ generated full symmetry analysis for each structure, \\nobreak\\mbox{\\it i.e.}, Bravais lattice,\nspace group and point group classifications, and Pearson symbol\n(the method and tolerances used for this analysis follow the {\\small AFLOW}\\ standard~\\cite{curtarolo:art104}).\nFor the analysis presented here we identify all the binary and ternary compounds included in this set,\n27,487 binary entries and 37,907 ternary entries.\nFrom these, we extract all the entries that contain oxygen, sulfur or selenium as one of the components.\nOf the binaries, we find 3,256 oxides, 1,685 sulfides and 1,050 selenides.\n10,530 oxides, 3,190 sulfides and 1,786 selenides are found among the ternaries.\nDuplicate entries representing different experimental reports of the same compound,\n\\nobreak\\mbox{\\it i.e.}, the same elements, stoichiometry, space group and Pearson designation, are then eliminated\nto obtain a list in which every reported compound is represented by its most recent corresponding entry in the {\\small ICSD}.\nThis reduces our list of binaries to 844 oxides, 495 sulfides and 332 selenides, and\nthe list of ternaries to 5,435 oxides, 2,041 sulfides and 1,256\nselenides.\nThese results are summarized in Table~\\ref{tab:art130:ICSD_DATA}.\nThroughout the rest of the study, we will refer to these sets of\nbinary and ternary compounds. We choose not to discuss multi-component structures with four or more elements since their\nrelative scarcity in the database most probably indicates incomplete\nexperimental data rather than fundamental issues of their chemistry.\nIt is also instructive to check the effect of element abundance on the number of compounds.\nThe abundance of oxygen in the earth's crust is $\\sim47\\%$ by weight,\naround 1000 times more than that of sulfur ($\\sim697$~ppm) which is around $5,000$\nmore abundant than selenium ($120$~ppb)~\\cite{wedepohl1995composition}.\nComparison with the number of elements (O\/S\/Se) binary compounds, 844\/495\/332,\nor ternary compounds, 5,435\/2,041\/1,256, makes it clear that while a rough correlation\nexists between the elements' abundance and the number of their known compounds,\nit is by no means a simple proportion.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig010}\n\\mycaption[Distributions of the compounds among structure types for binary (inset) and ternary compounds.]\n{Oxides are shown in blue, sulfides in yellow and selenides in green.\nThe binary distributions differ mostly by the length of their single-compound prototypes tails,\nwhile the ternary distribution of the oxides deviates significantly from those of the sulfides and selenides.}\n\\label{fig:art130:prototypes_distribution_curves_log}\n\\end{figure}\n\nIn the next stage, we identify unique structure types.\nStructure types are distinguished by stoichiometry, space group, and Pearson designation, without consideration\nof the specific elemental composition.\nThis implicit definition of structure type is common in the literature~\\cite{Villars2013, PaulingFile},\nand we use it throughout the study as\nproviding a good balance of clarity and simplicity.\nHowever, it should be noted that there are a few rare cases of complex structures where a given\nstructure type under this definition includes a few sub-types (see Figure~\\ref{fig:art130:structure_types_comparison}).\nExamples exist of more complex definitions of structure types, formulated to define similarities\nbetween inorganic crystals structures~\\cite{lima1990nomenclature}.\n\nThe binary structure type lists contain 538 oxides, 270 sulfides and 168 selenides.\nThe ternary lists contain 2,079 oxides, 784 sulfides and 521 selenides.\nThis means that 64\\% of the binary oxides, 55\\% of the sulfides and 51\\% of the selenides are distinct structure types.\nThe corresponding ratios for the ternaries are 38\\% of the oxides, 38\\% of the sulfides and 41\\% of the selenides.\nAll the other entries in the compound lists represent compounds of the same\nstructure types populated by different elements.\nDifferently put, this means that there are on average about 1.6 compounds per structure type in the binary oxides,\n$1.8$ in the binary sulfides and $2$ in the binary selenides.\nAmong the ternaries, the corresponding numbers are 2.6 compounds per structure type in the oxides,\n$2.6$ in the sulfides and $2.4$ in the selenides.\nThese numbers may be compared to the intermetallics, where there are\n20,829 compounds of which 2,166, about 10\\%,\nare unique structure types~\\cite{dshemuchadse2014some}.\nThere are about seven compounds per structure types in the binary intermetallics and about nine in the ternaries.\nThe number for binary intermetallics is considerably larger than for ternary oxides, sulfides or selenides. Together with the higher proportion of unique structure types in the latter, this reflects the limits on\nmaterials chemistry imposed by the presence of one of those 6A elements.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig011}\n\\mycaption[Distributions of structure types among (\\textbf{a}) binary and (\\textbf{b}) ternary stoichiometries.]\n{Oxides are shown in blue, sulfides in yellow and selenides in green.\nThe distributions of the selenides and sulfides are quite similar while those of the\noxides deviate significantly, as detailed in the text.}\n\\label{fig:art130:stoi_hist}\n\\end{figure}\n\nIt should be noted that this structure selection procedure produces lists that partially overlap,\n\\nobreak\\mbox{\\it i.e.}, certain structure types may appear in more than one list,\nsince there might be oxide structure types that are also represented among\nthe sulfide or selenide structures, and vice versa.\n11\\% of the binary oxide structure types also appear in the binary sulfides list and 8\\% are represented\nin the binary selenides list.\n33\\% of the binary sulfide are also represented in the selenides list.\nThe total number of binary oxides, sulfides and selenides structure types is 976, which is reduced by 16\\%,\nto 818 structure types, by removing all overlaps.\nThe corresponding overlap ratios for the ternaries are 10\\% for the oxides and sulfides,\n6\\% for the oxides and selenides and 31\\% for the sulfides and selenides.\nThe total number of entries in the ternary oxides, sulfides, and selenides structure type lists is 3,384,\nwhich is reduced to 2,797 structure types by removing all overlaps, a 17\\% reduction.\nTherefore, the overlaps between these three compound families are similar for the binaries and ternaries.\nIn both, the overlap between the oxides and the other two families is rather small,\nwhereas the overlap between the sulfides and selenides represents about a third of the total number of structure types.\n\nThe sequence of Mendeleev numbers includes 103 elements, from hydrogen to lawrencium\nwith numbers 1-6 assigned to the noble gases, 2-16 to the alkali metals and alkaline earths,\n17-48 to the rare earths and actinides, 49-92 to the metals and metalloids and 93-103 to the non-metals.\nOf these, noble gases are not present in compounds and artificial elements\n(metals heavier than uranium) have very few known compounds.\nWe are thus left with 86 elements, of which the above compounds are composed.\nThat means there are about ten times more binary oxides than\nelement-oxygen combinations, about six times more sulfides than element-sulfur\ncombinations and four times more selenides than element-selenium combinations.\nOxides are much more common than sulfides and selenides.\nThe corresponding numbers for the ternaries are much lower.\nThere are about 1.6 times more ternary oxides than two-element-oxygen ternary possible systems,\nabout 0.6 times less ternary sulfides and about 0.4 times less ternary selenides than the corresponding two-element combinations.\nThe ternaries are relatively quite rare, more so as we progress from oxides to sulfides and then to selenides.\nA similar analysis of the intermetallic binaries in Reference~\\onlinecite{dshemuchadse2014some} shows that of the 20,829 intermetallics,\n277 are unaries (about three times more than possible metal elements), 6,441 are binaries\n(about two times more than possible metal binary systems),\nand 13,026 are ternaries (6.5 times less than possible metal ternary systems).\nThis means that unary metal structures are less common among the metallic\nelements than the oxide, sulfide and binary selenide compounds among their corresponding binary systems.\nThis seems to reflect simply the larger space of stoichiometries available to binaries over unaries.\nHowever, on the contrary, the intermetallic binary compounds are more common among the metallic binary\nsystems than the oxide, sulfide and ternary selenide compounds among their corresponding ternary systems.\nThis discrepancy again reflects either the chemical constraints imposed by the presence of a 6A non-metal on the\nformation of a stable ternary structure, or simply gaps in the\nexperimental data since many ternary systems have not been thoroughly investigated.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.55\\linewidth]{fig012}\n\\mycaption[Composition distributions of binary (\\textbf{a}) oxide, (\\textbf{b}) sulfide, and (\\textbf{c}) selenide stoichiometries.]\n{The count indicates the number of different stoichiometries that include the respective element.\nThe colors go from no stoichiometries (white) to the maximal number of stoichiometries\n(dark blue) which is different for each element, 19\/8\/9 for O\/S\/Se.\nIslands of high prevalence appear for the 4B and 5B transition metals and\nthe heavy alkalis in all three compound families.\nAdditional, smaller islands appear in the sulfides and selenides for the 8 and\n1B transition metals and the 3A and 5A semi-metals.}\n\\label{fig:art130:stoi_periodic_oxide}\n\\end{figure}\n\n\\subsection{Results and discussion}\n\\boldsection{Structure types.}\nThe distribution of the binary and ternary compounds among the corresponding structure types is shown\nin Figure~\\ref{fig:art130:prototypes_distribution_curves_log}.\nDetailed data for the most common structure types is presented in Tables~\\ref{tab:art130:oxide_binary_data}-\\ref{tab:art130:selenide_ternary_data}.\n\nAbout 84\\% of the binary oxide structure types represent a single\ncompound, characterizing the tail end of the binary oxide distribution.\nThey include about 53\\% of the binary oxide compounds.\nThe most common structure type represents 29 compounds,\n3.4\\% of the oxide compounds list.\nAmong the binary sulfides, 76\\% of the structure types represent a single compound.\nThey include 41\\% of the binary sulfide compounds.\nThe most common structure type represents 32 compounds, 6.5\\% of the\nsulfide compounds list.\nAmong the binary selenides, 76\\% of the structure types represent a single compound.\nThey include 39\\% of the binary selenide compounds.\nThe most common structure type represents 31 compounds, 9.3\\% of the selenide compounds list.\n\nIn all three binary lists the most common structure type is rock salt (NaCl).\nThe binary oxide structure type distribution has a much longer tail than the sulfides and selenides,\n\\nobreak\\mbox{\\it i.e.}, more oxide compounds have unique structure types.\nThe most common structure type in these three distributions represents\na similar number of compounds but a smaller proportion of the corresponding compounds in the oxides.\nThe middle regions of the distributions are very similar\n(inset Figure~\\ref{fig:art130:prototypes_distribution_curves_log}).\nThis means that the much larger number of binary oxide compounds, compared to the sulfides and selenides,\nis expressed at the margin of the distribution, in the long tail of unique compounds.\n\nThis discrepancy between the three binary distributions is much less\napparent among the ternary compounds.\n64\\% of the ternary oxide structure types represent a single compound.\nThey include 24\\% of the ternary oxide compounds.\nThe two most common structure types, pyrochlore and perovskite, represent 116 and 115 compounds,\nrespectively, about 2\\% each of the entire compounds list.\nAmong the ternary sulfides, 70\\% of the structure types represent a single compound.\nThey include 34\\% of the ternary sulfide compounds.\nThe most common structure type, delafossite, represents 65 compounds,\n4\\% of the entire compounds list.\nAmong the ternary selenides, 62\\% of the structure types represent a single compound.\nThey include 26\\% of the ternary selenide compounds.\nThe most common structure type, again delafossite, represents 51 compounds, 4\\% of the ternary sulfides.\n\nIn contrast to the binaries, the larger count of ternary oxides, compared to the sulfides and selenides,\nis expressed by a thicker middle region of the structure type distribution,\nwhereas the margins have a similar weight in the distributions of the three compound families.\n\n\\boldsection{Binary stoichiometries.}\nThe structure types stoichiometry distribution for the binary oxide, sulfide and selenide compounds is shown in\nFigure~\\ref{fig:art130:stoi_hist}(a).\nWe define the binaries as $A_xB_y$, where $B$ is O, S or Se, and the number of structure types is shown as a function of $y\/(y+x)$.\nA very clear peak is found for the oxides at the stoichiometry 1:2,\n$A$O$_2$, while both the sulfides and selenides have a major peak at 1:1, $A$S and $A$Se, respectively.\n\nFor $y\/(y+x)<0.5$, there are more gaps in the plot (missing stoichiometries) for the oxides compared\nto the sulfides and selenides, while for $y\/(y+x)>0.6$ there are more gaps in the sulfides and selenides,\nthis behavior is shown in detail in Tables~\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}.\nAn important practical conclusion is that augmenting the binary oxide structure types with\nthose of sulfides and selenides will produce a more extensive coverage of possible stoichiometries.\n\nAnother interesting property is the number of stoichiometries\nfor each of the elements in the periodic table.\nThe prevalence of binary oxide stoichiometries per element is shown in Figure~\\ref{fig:art130:stoi_periodic_oxide}(a).\nA few interesting trends are evident --- the first row of transition metals shows a peak near vanadium (19 stoichiometries)\nand titanium (14 stoichiometries).\nHafnium, which is in the same column of titanium has only a single stoichiometry --- HfO$_{2}$.\nBoth the beginning and end of the $d$-elements exhibit a small amount of stoichiometries --- scandium\nwith only one and zinc with only two.\nThe two most abundant elements, silicon and oxygen, form only a single stoichiometry in the\n{\\small ICSD}\\ --- SiO$_{2}$, with 185 {\\it different} structure types.\nAnother interesting trend is evident for the alkali metals, where rubidium and cesium have more\nstoichiometries --- perhaps related to the participation of $d$-electrons in the chemical bonds.\n\nFigures~\\ref{fig:art130:stoi_periodic_oxide}(b) and (c)\nshow the binary stoichiometries prevalence per element for sulfur and selenium respectively.\nSimilar trends are exhibited --- there are two ``islands'' of large number of stoichiometries\nin the transition metals: one around vanadium and titanium and the other near nickel and copper.\nEvidently, prime candidates for new compounds should be searched among structures in the vicinity of\nthese high density islands, especially for elements that exhibit a considerably higher density in one family.\n\n\\boldsection{Ternary stoichiometries.}\nSimilar to the binaries, the ternary stoichiometries are designated\n$A_xB_yC_z$, where $C$ is O, S or Se.\nThe distributions of the ternaries are, as might be expected, more complex,\nwith maxima at $z\/(x+y+z)=0.6$ for the oxides, $z\/(y+x+z)=0.55$ for the sulfides and $z\/(y+x+z)=0.5$ for the selenides.\nThe major peaks still appear at integer and half integer values, but with more minor peaks at intermediate values.\nThis behavior is shown in Figure~\\ref{fig:art130:stoi_hist}(b).\nThe ternary selenide and sulfides distributions are again nearly identical, and there are\nalmost no compounds with ratios larger than $0.75$ in the oxides or larger than $0.66$ in the sulfides and selenides.\nHowever, there are few sulfide and selenide compounds around 0.8 and 0.85 but no oxides.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Ternary stoichiometry data: $A_xB_yC_z$.]{``$C$-rich'' refers to stoichiometries where $z>x+y$.}\n\\vspace{3mm}\n\\begin{tabular}{ l|r|r|r }\n& oxygen & sulfur & selenium \\\\\n\\hline\nNumber of stoichiometries & 585 & 282 & 206 \\\\\n$C$-rich stoichiometries ratio & 0.85 & 0.67 & 0.66 \\\\\n$C$-rich compound ratio & 0.92 & 0.77 & 0.73 \\\\\n\\end{tabular}\n\\label{tab:art130:ternary_stoi}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig013}\n\\mycaption[Prevalence of stoichiometries among ternary compounds.]\n{Panels include\n(\\textbf{a}) oxide,\n(\\textbf{b}) sulfide,\n(\\textbf{c}) selenide compounds,\nand, for reference,\n(\\textbf{d}) all the possible stoichiometries with up to 12 atoms\nof each component per unit cell.\nIn each figure, the smaller circles are normalized to the biggest one, which denotes the highest prevalence, \\nobreak\\mbox{\\it i.e.},\n718 for oxides, 242 for sulfides, and 145 for selenides,\nin addition a heat map color scheme is used where blue means low prevalence and red means the highest prevalence for each element.\nThe $x$ and $y$ axes denote the atomic fractions in the ternaries $A_xB_yC_z$, where $C$ is O, S or Se, respectively.\n$A$ and $B$ are ordered by Mendeleev number where $M_A>M_B$.}\n\\label{fig:art130:triangles}\n\\end{figure}\n\nAnother perspective of ternary stoichiometries is demonstrated in Figure~\\ref{fig:art130:triangles}\nwhich shows the abundance of the most common stoichiometries.\nThe biggest circle in each diagram denotes the prevalence of the most common stoichiometry\n(number of unique compounds for this stoichiometry),\nwhich is 718 ($x=1$, $y=1$, $z=3$) for oxides, 242 ($x=1$, $y=1$, $z=2$) for sulfides, and 145 ($x=1$, $y=1$, $z=2$) for selenides.\nThe smaller circles in each plot are normalized to the corresponding highest prevalence.\n\nThese diagrams highlight the similarities as well as important differences between the three families of compounds.\nIn all three cases, the most common stoichiometries appear on the symmetry axis of the diagram, \\nobreak\\mbox{\\it i.e.},\nat equal concentrations of the $A$ and $B$ components, or very close to it.\nFor the oxides, they are concentrated near 0.5-0.6 fraction of oxygen, representing the\n$A_1B_1$O$_2$ and $A_1B_1$O$_3$ stoichiometries, respectively,\nand form a very dense cluster with many similar reported stoichiometries of lower prevalence.\nOutside this cluster, the occurrence of reported compositions drops sharply, and other regions\nof the diagram are very sparsely populated, in particular near the vertices of the $B$ and O components.\n\nThe sulfide and selenide diagrams also exhibit prominent clusters on the $AB$ symmetry axes,\nbut they appear at a lower S or Se concentration of about 0.5, \\nobreak\\mbox{\\it i.e.}, $A_1B_1C_2$ stoichiometry.\nThey are considerably more spread out and include a significant contribution at the $ABC$ stoichiometry.\nIn both sulfides and selenides, an additional minor cluster appears closer to the $A$ vertex (Figure~\\ref{fig:art130:triangles}).\nA few members of this cluster are ternary oxides, reflecting the high electronegativity and high Mendeleev number (101) of oxygen.\nThe $B$ and $C$ vertex regions are still sparsely populated, but less so than in the oxides case.\nOverall, the sulfide and selenide diagrams are very similar to each other and different from that of the oxides.\nThey are more spread out, less $AB$ symmetric than the oxide diagram and less tilted towards rich $C$-component concentration.\nThis discrepancy may reflect some uniqueness of oxygen chemistry compared to sulfur and selenium,\nor rather simply reflect the oxygen rich environment in which naturally formed compounds are created in the atmosphere.\nThe number of stoichiometries and the differences in the $C$-component concentration are summarized in Table~\\ref{tab:art130:ternary_stoi}.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Distribution of the oxide, sulfide and selenide compounds and structure types among the 14 Bravais lattices.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\n& \\multicolumn{3}{R{3cm}|}{binary compounds}\n& \\multicolumn{3}{R{3cm}|}{binary structure types}\n& \\multicolumn{3}{R{3cm}|}{binary compounds per structure type}\n& \\multicolumn{3}{R{3cm}|}{ternary compounds}\n& \\multicolumn{3}{R{3cm}|}{ternary structure types}\n& \\multicolumn{3}{R{3cm}}{ternary compounds per structure type}\\\\ \\hline\n& O & S & Se & O & S & Se & O & S & Se & O & S & Se & O & S & Se & O & S & Se \\\\ \\hline\naP & 51 & 13 & 5 & 39 & 12 & 5 & 1.3 & 1.1 & 1 & 378 & 79 & 60 & 219 & 56 & 39 & 1.7 & 1.4 & 1.5 \\\\\nmP & 82 & 54 & 31 & 62 & 36 & 20 & 1.3 & 1.5 & 1.6 & 918 & 318 & 198 & 363 & 166 & 109 & 2.5 & 1.9 & 1.8 \\\\\nmS & 88 & 31 & 22 & 58 & 21 & 15 & 1.5 & 1.5 & 1.5 & 672 & 251 & 170 & 292 & 117 & 77 & 2.3 & 2.1 & 2.2 \\\\\noP & 123 & 82 & 48 & 81 & 37 & 30 & 1.5 & 2.2 & 1.6 & 950 & 481 & 266 & 373 & 139 & 105 & 2.5 & 3.5 & 2.5 \\\\\noS & 39 & 24 & 11 & 36 & 19 & 9 & 1.1 & 1.3 & 1.2 & 334 & 84 & 60 & 133 & 40 & 25 & 2.5 & 2.1 & 2.4 \\\\\noF & 11 & 7 & 11 & 10 & 6 & 4 & 1.1 & 1.2 & 2.8 & 51 & 32 & 23 & 28 & 14 & 8 & 1.8 & 2.3 & 2.9 \\\\\noI & 22 & 5 & 2 & 20 & 4 & 2 & 1.1 & 1.25 & 1 & 89 & 36 & 27 & 39 & 15 & 12 & 2.3 & 2.4 & 2.25 \\\\\ntI & 41 & 20 & 10 & 31 & 17 & 8 & 1.3 & 1.2 & 1.25 & 418 & 80 & 72 & 101 & 34 & 23 & 4.1 & 2.4 & 3.1 \\\\\ntP & 78 & 27 & 28 & 48 & 13 & 16 & 1.6 & 2.1 & 1.75 & 239 & 73 & 52 & 107 & 39 & 26 & 2.2 & 1.9 & 2.0 \\\\\nhP & 94 & 87 & 66 & 62 & 50 & 32 & 1.5 & 1.7 & 2.1 & 435 & 224 & 103 & 198 & 75 & 41 & 2.2 & 3.0 & 2.5 \\\\\nhR & 40 & 44 & 20 & 30 & 33 & 15 & 1.3 & 1.3 & 1.3 & 420 & 230 & 133 & 123 & 49 & 33 & 3.4 & 4.7 & 4.0 \\\\\ncP & 42 & 22 & 20 & 21 & 6 & 4 & 2.0 & 3.7 & 5.0 & 187 & 58 & 43 & 45 & 18 & 13 & 4.2 & 3.2 & 3.3 \\\\\ncF & 75 & 65 & 48 & 19 & 10 & 6 & 3.9 & 6.5 & 8.0 & 251 & 80 & 43 & 27 & 17 & 7 & 9.3 & 4.7 & 3.9 \\\\\ncI & 58 & 14 & 10 & 21 & 6 & 2 & 2.8 & 2.3 & 5.0 & 92 & 15 & 6 & 30 & 5 & 3 & 3.1 & 3.0 & 2.0 \\\\\n\\end{tabular} }\n\\label{table:bravais_lattice_distribution}\n\\end{table}\n\nAnother interesting observation is that while some stoichiometries are abundant in the oxides\nthey are almost absent in the sulfides or the selenides. For example,\nthere are 299 compounds with the $A_2B_2$O$_7$ stoichiometry (ignoring\norder between $M_A$ and $M_B$), but only two $A_2B_2$S$_7$ compounds\nand no $A_2B_2$Se$_7$ compounds. Also, there are 71 $A_1B_3$O$_9$\ncompounds but no $A_1B_3$S$_9$ and $A_1B_3$Se$_9$ compounds. On the\nother hand, there are no $A_4B_{11}X_{22}$ oxides, but 20 sulfides and 8 selenides.\nIf we require that $M_A>M_B$, there are no oxides of the $A_3B_2X_2$\nstoichiometry, but 25 sulfides and 7 selenides.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.55\\linewidth]{fig014}\n\\mycaption[Composition distributions of ternary (\\textbf{a}) oxide, (\\textbf{b}) sulfide, and (\\textbf{c}) selenide stoichiometries.]\n{The count indicates the number of different stoichiometries that include the respective element.\nThe colors go from no stoichiometries (white) to the maximal number of stoichiometries (dark blue)\nwhich is different for each element, 96\/59\/51 for O\/S\/Se.\nHigh prevalence appears for the alkali metals in all three compound families.\nAn additional island in the transition metals is much more pronounced in the oxides.\nThe sulfides and selenides distributions are nearly identical, and show high prevalence of oxygen containing ternaries.}\n\\label{fig:art130:tern_stoi_periodic_oxide}\n\\end{figure}\n\nAgain, an important conclusion is that there are many missing stoichiometries,\nFigure~\\ref{fig:art130:triangles}(d) shows all the possible stoichiometries for $A_xB_yC_z$ for $x,y,z \\le 12$,\nclearly showing rich concentration in the middle, which is not the case for oxides, and also to a\nlesser degree to sulfides and selenides.\n\nWe can repeat the analysis of the binary stoichiometries and ask how many stoichiometries\nper element are there for the ternaries.\nThis is shown in Figure~\\ref{fig:art130:tern_stoi_periodic_oxide}.\nHere, also, the similarity of sulfides and selenides is clear.\nIn addition, while there are similarities between the distributions of binary stoichiometries\nper element to the ternary distributions, there are also obvious differences.\nOne might guess that there should be a correlation between the binary and ternary distributions.\nThis is examined in Figure~\\ref{fig:art130:bintern1a}(a).\n\nIt is evident that the correlation between ternary and binary number of stoichiometries is not strong\nbut the minimal number of ternary stoichiometries tends to grow with the number of binary stoichiometries.\nWe check this further in Figure~\\ref{fig:art130:bintern1a}(b), by comparing the number of ternary stoichiometries of\n$A_xB_y$O$_z$ to the product of stoichiometry numbers of $A_x$O$_y$ and $B_x$O$_y$.\nThe general trend obtained is an inverse correlation, \\nobreak\\mbox{\\it i.e.}, as the product of the numbers of binary\nstoichiometries increases, the number of ternaries decreases. This trend can be explained by the following argument: when the two binaries are rich with stable compounds, the ternaries need to compete with more possibilities of\nbinary phases, which makes the formation of a stable ternary more difficult.\nIn Figure~\\ref{fig:art130:bintern1a}(b), this trend is highlighted for vanadium,\nthe element with the most binary stoichiometries, but this pattern repeats itself for most elements.\nWe analyze this behavior for the sulfides and selenides in Section~\\ref{subsec:art130:prev_stoich_supp}, similar trends are found but\nthey are less pronounced due to a smaller number of known compounds.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig015}\n\\mycaption[Analysis of ternary \\nobreak\\mbox{\\it vs.}\\ binary stoichiometry counts for oxides.]\n{(\\textbf{a}) The number of ternary oxide stoichiometries per element as a function\nof the count of its binary stoichiometries.\nThe dashed line marks perfect similarity $(y=x)$, and the dotted line marks the ratio $y=4x$.\n(\\textbf{b}) The number of ternary oxide stoichiometries as a function of the\nproduct of the numbers of the binary stoichiometries of participating elements.\nThe data for vanadium is shown with red crosses, all the rest is shown with blue circles.}\n\\label{fig:art130:bintern1a}\n\\end{figure}\n\n\\boldsection{Composition and Mendeleev maps.}\nThe occurrence of each element in the binary and ternary compound lists has been\ncounted and tabulated.\nThe results are described in Figure~\\ref{fig:art130:mendeleev_distribution_all_in_one}.\nFor the binary oxides a very prominent peak appears at $M=85$, the\nMendeleev number of silicon.\nIt represents the 185 different silicon oxide\nstructures types reported in the {\\small ICSD}\\ database for just a {\\it single} stoichiometry, SiO$_2$.\nSmaller peaks appear for $M=51$ (titanium, 42 structure types, 14 stoichiometries,\nleading stoichiometry is\nTiO$_2$ with 14 structure types), $M=54$ (vanadium, 42 structure types,\n18 stoichiometries, leading stoichiometry is VO$_2$ with 10 structure types),\n$M=56$ (tungsten, 24 structure types, 9 stoichiometries, leading stoichiometry is WO$_3$ with 13 structure types),\nand $M=45$\n(uranium, 22 structure types, 9 stoichiometries, leading stoichiometries are UO$_2$ and U$_3$O$_8$ with 6 structure types each).\nUnlike the silicon peak which is composed of a single stoichiometry,\nthe other leading peaks evidently include multiple stoichiometries, reflecting the different chemistry of those elements.\nThese differences also carry over into the ternary oxide compounds involving those elements.\nFor example, the stoichiometry distribution of silicon ternary oxides is more tilted towards\nthe silicon poor compounds compared to the corresponding distributions of vanadium and titanium ternary oxides,\nas is shown in Figure~\\ref{fig:art130:specific_triangle_stoichiometries}.\n\nThe distribution of the sulfides is generally much lower than\nthat of the oxides, due to the much smaller total number of known binaries, but is also more uniformly structured.\nIt has one major peak\nfor $M=76$ (zinc, 40 structure types, 2 stoichiometries, leading stoichiometry is ZnS with 39 structure types),\nand quite a few smaller ones such as $M=51$ (titanium, 16 structure types, 5 stoichiometries, leading stoichiometry is TiS$_2$ with 9 structure types),\n$M=61$ (iron, 18 structure types, 5 stoichiometries, leading stoichiometry is FeS with 6 structure types),\n$M=67$ (nickel, 16 structure types, 6 stoichiometries, leading stoichiometry is NiS$_2$ with 8 structure types),\n$M=90$ (phosphorus, 13 structure types, 8 stoichiometries, of which\nP$_2$S$_7$, P$_4$S$_9$, P$_4$S$_6$, P$_4$S$_5$ and P$_4$S$_3$ have 2 structure types each).\nThe $M$~=~8--33 region also exhibits a minor concentration of\nparticipating elements.\nThe selenides distribution is yet smaller than that of the sulfides, and\neven more uniform.\nSeveral peaks appear, $M=51$ (titanium, 13 structure types, 9 stoichiometries, leading stoichiometry is\nTiSe with 3 structure types),\n$M=52$ (niobium, 15 structure types, 8 stoichiometries, leading stoichiometry is\nNbSe$_2$ with 8 structure types),\n$M=53$ (tantalum, 15 structure types, 4 stoichiometries, leading stoichiometry is\nTaSe$_2$ with 10 structure types) and\n$M=79$ (indium, 14 structure types, 5 stoichiometries, leading stoichiometry is In$_2$Se$_3$ with 6 structure types).\nAll distributions cover most of the elements except two obvious gaps, one at $M<9$,\nwhich includes the noble gases and the two heaviest alkali metals, cesium and francium, and another\nat $34\\leq M\\leq 42$ which represents the heavy actinides. Another gap appears in the sulfide and selenide distributions at $91\\leq M\\leq 97$,\nwhich reflects the rarity of polonium and astatine compounds and shows that the elements of the 6A column,\nexcept oxygen, do not coexist, in the known compounds, with each other\nor with the heavier halogen iodine.\n\nThe element occurrence distributions for the ternary oxides, sulfides and\nselenides exhibit greater similarity than the\ncorresponding binary distributions. The most apparent difference, however, is the\nmost common component, which is sulfur, $M=90$, in the\noxides, but oxygen itself, $M=101$, in the sulfides and selenides. The\nsulfide and selenide distributions are almost the same, except for\ngenerally lower numbers in the selenides (due to the smaller total\nnumber of compounds) and an apparent lower participation of the\nlanthanides $M$~=~17--35.\n\nMendeleev maps for the ternaries are shown in\nFigures~\\ref{fig:art130:mendeleev_bigger_x_upper_all_in_one}-\\ref{fig:art130:mendeleev_sulfur_selenium_prototypes}.\nFigure~\\ref{fig:art130:mendeleev_bigger_x_upper_all_in_one} shows the cumulated maps for all\nstoichiometries reported for the respective ternary family.\nThey reflect the same major gaps as the binary distributions.\nThe maps show that most of the reported compositions are represented by one or two compounds\nwith just a few hot-spots that include up to 20 compounds in the oxides and\n10 compounds in the sulfides and selenides.\nThe oxides map is obviously denser, reflecting the much richer, currently known, chemistry of the oxides compared\nto the other two elements.\nThe chemistry becomes more constrained as we proceed down the periodic table column from\noxygen to sulfur and then to selenium.\n\nNext, we examine maps of specific stoichiometries.\nMaps of a few notable oxide stoichiometries and their\nleading structure types are shown in Figure~\\ref{fig:art130:mendeleev_oxide_prototypes}.\nThese maps reflect the dominant features of the\nfull ternary oxides map (Figure~\\ref{fig:art130:mendeleev_bigger_x_upper_all_in_one}),\nbut with significant new additional gaps of absent compounds. These gaps are naturally\nwider for less prevalent stoichiometries, \\nobreak\\mbox{\\it i.e.}, the map of the most\nprevalent stoichiometry, $A_1B_1$O$_3$, is denser than the three\nother maps in Figure~\\ref{fig:art130:mendeleev_oxide_prototypes}.\nDifferent structure types in all stoichiometries tend to accumulate at\nwell defined regions of the map. The separation between them\nis not perfect, but is similar to that exhibited by the classical Pettifor maps for\nbinary structure types~\\cite{pettifor:1984,pettifor:1986}.\nA similar picture is obtained for the sulfide and selenide structure types, although more sparse\n(Figure~\\ref{fig:art130:mendeleev_sulfur_selenium_prototypes}).\nIt is interesting to note that the maps of, \\nobreak\\mbox{\\it e.g.},\n$A_1B_2C_4$ ($C=$ O, S, Se), show similar patterns in the map for oxides\n(Figure~\\ref{fig:art130:mendeleev_oxide_prototypes}) and sulfides\/selenides\n(Figure~\\ref{fig:art130:mendeleev_sulfur_selenium_prototypes}) ---\nsuggesting that similar elements tend to form this stoichiometry.\nIn the same manner, the 2:1:1 stoichiometry shows very similar patterns in oxides, sulfides and selenides (see also Figure~\\ref{fig:art130:mend_211_stoichiometries}).\n\n\\boldsection{Symmetries.}\nThe distribution of the compounds and structure types among the 14 Bravais lattices\nis presented in Table~\\ref{table:bravais_lattice_distribution} and\nFigure~\\ref{fig:art130:bravais_combined}.\nIt is interesting to note that in all six cases (binary and ternary oxides, sulfides and selenides)\nthe distribution is double peaked, with the majority of the compounds belonging to the\nmonoclinic and orthorhombic primitive lattices,\nand a smaller local maximum at the hexagonal and tetragonal lattices.\nAll distributions exhibit a local minimum for the orthorhombic face and body centered lattices.\nThe high symmetry cubic lattices are also relatively rare.\nThis reflects the complex spatial arrangement of the compound forming electrons of oxygen, sulfur and selenium,\nwhich does not favor the high symmetry cubic structures or the\ndensely packed face and body centered orthorhombic structures.\n\nFigure~\\ref{fig:art130:symmetry_distribution_of_structures}\nshows a more detailed distribution of the compounds among the different space groups.\nThe binary compounds show a distinct seesaw structure, with a few\nlocal peaks near the highest symmetry groups of each crystal system.\nThe corresponding ternary distributions have three sharp peaks in the triclinic,\nmonoclinic and orthorhombic systems, and much smaller peaks in the hexagonal and cubic groups.\nIt is interesting to note that the three compound families, exhibit distributions of very similar structure.\nThe oxide distributions are the densest, simply due to the existence of more oxide compounds in the database,\nand become sparser in the sulfide and selenide cases.\nThe compounds of all these families are distributed among a rather limited number of space groups,\nwith most space groups represented by just a single compound or not at all.\n\n\\boldsection{Unit cell size.}\nThe distributions of unit cell sizes (\\nobreak\\mbox{\\it i.e.}, the number of atoms per unit cell) for the six compound families we discuss\nare shown in Figure~\\ref{fig:art130:number_of_atoms_distribution}.\nAll of these distributions have strong dense peaks at small cell sizes and decay sharply at sizes above a few tens of atoms.\nHowever, the details of the distributions differ quite significantly from group to group.\nAmong the binaries, the oxides exhibit the highest and widest peak with\nits maximum of 102 binary oxide compounds located at 12 atoms per cell.\n90\\% of the binary oxides have less than 108 atoms in the unit cell and 50\\% of them have less than 24 atoms.\nThe sulfides distribution has a lower and narrower peak of 70 compounds at 8 atoms.\nThe distribution of the selenides has a still lower peak of 60 compounds at 8 atoms.\nThe fact that oxygen has a peak at 12 atoms in the unit cell and not at 8 as the sulfides and selenides,\nis related to the fact that binary oxides prefer the $A$O$_2$ stoichiometry over $A$O,\nwhere as both sulfides and selenides prefer the 1:1 stoichiometry over 1:2.\nThis is probably related to the different chemistry of oxygen \\nobreak\\mbox{\\it vs.}\\ sulfur and selenium.\nAdditional computational analysis would be required to fully understand the effect of the\ndifferent chemistry on the stoichiometry and number of atoms.\nDetailed data for these dense parts of the distributions is tabulated in Tables~\\ref{tab:art130:Number_of_atoms_in_Binaries_unit_cells}-\\ref{tab:art130:Number_of_atoms_in_Binaries_unit_cells_3}.\nThe oxides distribution exhibits the longest tail of the binaries, with the largest\nbinary oxide unit cell including 576 atoms.\nThe largest binary sulfide and selenide unit cells include 376 and 160 atoms, respectively.\n\nThe distributions of the ternary compounds have higher, wider peaks and longer tails than\ntheir binary counterparts.\nThe relative differences between the oxide, sulfide and selenide distributions\nremain similar to the distributions of the binaries.\nThe ternary oxides exhibit a high and wide peak.\nIts maximum of 465 compounds is located at 24 atoms per cell, and\n90\\% of the compounds have less than 92\natoms in the unit cell and 50\\% of the compounds have less than 32 atoms.\nAs in the binary case, the distribution of the ternary sulfides has a lower and narrower peak than the oxides,\nwhere the maximum of 190 compounds at 28 atoms and 90\\% of the compounds have less than 72 atoms in the unit cell.\nThe distribution of the selenides has a still lower and narrower peak, where the corresponding numbers are\n130 compounds at 28 atoms and 90\\% of the compounds having less than 28 atoms in the unit cell.\nDetailed data for these dense parts of the distributions is shown in Tables~\\ref{tab:art130:Number_of_atoms_in_ternary_unit_cells}-\\ref{tab:art130:Number_of_atoms_in_ternary_unit_cells_3}.\nThe ternary oxides distribution exhibits the longest tail of\nthe three types, with the largest ternary oxide unit cell having 1,080 atoms.\nThe largest ternary sulfide and selenide unit cells have 736 and 756\natoms, respectively.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig016}\n\\mycaption[Distributions of (\\textbf{a}) binary and (\\textbf{b}) ternary compounds among the elements.]\n{The binary oxides exhibit a structures distribution with two prominent peaks. The distributions\nof the binary sulfides and selenides are less structured and more similar to each other.\nThe distributions of the ternary compounds have higher, wider peaks than\ntheir binary counterparts. The relative differences between the oxide, sulfide and selenide\ndistributions remain similar to the distributions of the binaries.}\n\\label{fig:art130:mendeleev_distribution_all_in_one}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.375\\linewidth]{fig017}\n\\mycaption[Mendeleev maps of ternary (\\textbf{a}) oxide $A_xB_y$O$_z$, (\\textbf{b}) sulfide $A_xB_y$S$_z$\nand (\\textbf{c}) selenide $A_xB_y$Se$_z$ compounds.]\n{It is assumed that $x\\geq y$ with the $x$-axis indicating $M_A$\nand the $y$-axis $M_B$.\nIf the stoichiometry is such that $x=y$, the compound is counted as $0.5 A_xB_y$O$_z + 0.5 B_xA_y$O$_z$.\nA color scheme is used to represent the compound count for each composition, blue means the minimal number (one)\nand green means the maximal number which is different for each element.}\n\\label{fig:art130:mendeleev_bigger_x_upper_all_in_one}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig018}\n\\mycaption[Mendeleev maps of the three leading structure types in each of the four leading stoichiometries in ternary oxides.]\n{(\\textbf{a}) $A_1B_1$O$_3$,\n(\\textbf{b}) $A_1B_1$O$_4$,\n(\\textbf{c}) $A_1B_2$O$_4$, and\n(\\textbf{d}) $A_2B_2$O$_7$.\nThe legend box appears at a region with no data points.\nThe number in parenthesis is the number of compounds for this structure type, for ``Other'',\nit refers to the total number of compounds with this stoichiometry.}\n\\label{fig:art130:mendeleev_oxide_prototypes}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig019}\n\\mycaption[Mendeleev maps of the three leading structure types in each of the two leading stoichiometries in sulfur and selenium ternaries.]\n{(\\textbf{a}) $A_1B_2$S$_4$,\n(\\textbf{b}) $A_1B_1$S$_2$,\n(\\textbf{c}) $A_1B_2$Se$_4$, and\n(\\textbf{d}) $A_1B_1$Se$_2$.\nThe number in parenthesis is the number of compounds for this structure type, for ``Other'',\nit refers to the total number of compounds with this stoichiometry.}\n\\label{fig:art130:mendeleev_sulfur_selenium_prototypes}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig020}\n\\mycaption[Number of compounds (\\textbf{a} and \\textbf{b}) and\nstructure types (\\textbf{c} and \\textbf{d}) for each Bravais lattice.]\n{Binaries are on the left (\\textbf{a} and \\textbf{c}) and\nternaries on the right (\\textbf{b} and \\textbf{d}).\nOxides are shown in blue, sulfides in light green and selenides in darker green. All six\ndistributions (binary and ternary oxides, sulfides and selenides) are double peaked with a\nlocal minimum for the orthorhombic face and body centered lattices. The high symmetry\ncubic lattices are also relatively rare. This reflects the complex spatial arrangement of\nthe compound forming electrons of the 6A elements, which does not favor the\nhigh symmetry of these\nstructures.}\n\\label{fig:art130:bravais_combined}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig021}\n\\mycaption[Distributions of compounds (\\textbf{a} and \\textbf{b}) and structure types (\\textbf{c} and \\textbf{d}) among the 230 space groups.]\n{Binaries are on the left (\\textbf{a} and \\textbf{c}) and\nternaries on the right (\\textbf{b} and \\textbf{d}).\nCompounds are depicted on the top (\\textbf{a} and \\textbf{b})\nand structure types on the bottom (\\textbf{c} and \\textbf{d}).}\n\\label{fig:art130:symmetry_distribution_of_structures}\n\\end{figure}\n\nIt should be noted that large unit cells, within the tails of all distributions, tend to have very few representatives,\nwith just one compound with a given unit cell size in most cases.\nNotable exceptions are local peaks near $80$ atoms per unit cell in the binary\ndistributions and near 200 atoms per unit cell in the ternary distributions.\nThe oxide distributions exhibit additional peaks, near 300 atoms per unit cell for the\nbinaries and near 600 atoms per unit cell for the ternaries.\nThese minor peaks may indicate preferable arrangements of cluster-based structures.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.9\\linewidth]{fig022}\n\\mycaption[Unit cell size distributions for oxides, sulfides, and selenides.]\n{Binaries are on the left (\\textbf{a}, \\textbf{c} and \\textbf{e}) and\nternaries on the right (\\textbf{b}, \\textbf{d} and \\textbf{f}).\nOxides are at the top (\\textbf{a} and \\textbf{b}),\nsulfides in the middle (\\textbf{c} and \\textbf{d}) and\nselenides at the bottom (\\textbf{e} and \\textbf{f}).\nThe insets show the compounds with up to 50 atoms per unit cell in each case.\nAll distributions exhibit long tails of rare very large unit cells which extend much further in the oxides.\nThe dense cores of the distributions reflects the higher prevalence of oxides and are very similar for the sulfides and selenides.}\n\\label{fig:art130:number_of_atoms_distribution}\n\\end{figure}\n\n\\subsection{Structure sub-types}\nThe definition of structure type by the combination of stoichiometry,\nPearson symbol and symmetry is common in the literature, but it is not\nnecessarily unique. A given structure, according to this definition,\ncan contain few sub-types.\nAs an example, the structure types\n($A_1B_1$O$_3$:oP20:62),\n($A_1B_1$O$_3$:hR10:167) and ($A_1B_1$O$_3$:cP5:221) contain 115, 36, and\n78 unique compounds, respectively, of mostly perovskites. However, the\noP20 also includes the aragonite structure, the MgSeO$_{3}$\nstructure, and others.\nThe hR10 contains also calcite-like\nstructures. The cP5 group, which has a more strict symmetry, contains\nonly perovskites.\nThese three structure types belong to a common parent class, the high symmetry\ncP5, with two different types of symmetry breaking.\nThe different sub-types within each structure type may be discerned by\nexamining relations between structural descriptors, \\nobreak\\mbox{\\it e.g.}, the volume\nas a function of nearest neighbor distance cubed, as shown in Figure~\\ref{fig:art130:structure_types_comparison}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig023}\n\\mycaption{Comparison of different structure types according to volume per atom \\nobreak\\mbox{\\it vs.}\\\nshortest nearest-neighbor distance cubed $\\left(d^{3}_{\\mathrm{n.n.}}\\right)$.}\n\\label{fig:art130:structure_types_comparison}\n\\end{figure}\n\nIt can be easily seen that the ($A_1B_1$O$_3$:cP5:221) group follows a\nperfect linear relation, as is expected from a uniform structure type.\nHowever, both the ($A_1B_1$O$_3$:oP20:62) and the ($A_1B_1$O$_3$:hR10:167) types\ninclude points that are close to the ($A_1B_1$O$_3$:cP5:221) line but also clusters of\npoints that deviate from it.\n\nThose points represent non-perovskite\nstructures, including those that are close to aragonite and calcite.\n\n\\subsection{Ternary stoichiometry triangles}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig024}\n\\mycaption{Comparison of ternary stoichiometries for (\\textbf{a}) oxygen, (\\textbf{b}) sulfur and (\\textbf{c}) selenium compounds.\nAll stoichiometries of $A_xB_yC_z$, $x,y,z \\leq 12$ are shown in (\\textbf{d}).}\n\\label{fig:triangle_stoichiometries}\n\\end{figure}\n\nFigure~\\ref{fig:art130:stoi_periodic_oxide} shows the prevalence of different ternary\nstoichiometries in a triangle shape.\nThe points inside the triangle are defined by\nthe intersection of lines that connect the vertex points with the\ncorresponding binary stoichiometry on the opposing edge.\nFor example, the stoichiometry $A_{u}B_{v}$O$_{w}$ is represented by the\nintersection of three lines, one from the O vertex to the point\n$u\/(u+v)$ on the $AB$ edge,\nanother from the $A$ vertex to the point $v\/(v+w)$ on the $B$O edge, and\nthe third from the $B$ vertex to the point $u\/(u+w)$ on the $A$O edge.\nThe different stoichiometries in Figure~\\ref{fig:art130:stoi_periodic_oxide} are denoted by\ncircles that vary in size according to the number of compounds for\neach stoichiometry.\nFigure~\\ref{fig:triangle_stoichiometries}(a-c) shows the same data but without reference to prevalence,\nshowing just the stoichiometries locations.\nFigure~\\ref{fig:triangle_stoichiometries}(d) shows, for comparison, the locations of all possible stoichiometries up to\n12 atoms per species ($A_xB_yC_z$ where $x,y,z \\leq 12$). The differences in the distributions of the\nreported compositions of the three compound families are clearly apparent.\n\n\\subsection{Prevalence of structure types among the oxide, sulfide and binary and ternary selenide compounds}\n\nNumerical data for the leading 40 structure types of the oxides,\nsulfides and selenides are shown in Tables\n\\ref{tab:art130:oxide_binary_data} through\n\\ref{tab:art130:selenide_binary_data} for the binaries, and in Tables\n\\ref{tab:art130:oxide_ternary_data} through\n\\ref{tab:art130:selenide_ternary_data} for the ternaries.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the binary oxide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n1:1& cF8 & 225&NaCl&29\\\\\n3:2& cI80 & 206&Bixbyite-Mn$_{2}$O$_{3}$&24\\\\\n2:1& tP6 & 136&KrF$_{2}$&22\\\\\n3:2& hP5 & 164&La$_{2}$O$_{3}$&19\\\\\n2:1& cF12 & 225&Fluorite-CaF$_{2}$&16\\\\\n3:2& mS30 & 12&Sm$_{2}$O$_{3}$&16\\\\\n3:2& cI80 & 199&Sm$_{2}$O$_{3}$ (c180)&15\\\\\n2:1& cP12 & 205&CO$_{2}$ (cP12)&13\\\\\n2:1& mP12 & 14&Baddeleyite-ZrO$_{2}$ (mP12)&8\\\\\n2:1& oP12 & 60&&8\\\\\n2:1& oP12 & 62&&8\\\\\n2:1& oP6 & 58&&8\\\\\n3:2& hR10 & 167&&8\\\\\n1:1& hP4 & 186&&7\\\\\n2:1& oP24 & 61&&7\\\\\n2:1& tI6 & 139&&7\\\\\n1:1& cF8 & 216&&5\\\\\n1:2& cF12 & 225&&5\\\\\n1:2& cP6 & 224&&5\\\\\n3:2& mP20 & 14&&5\\\\\n3:2& oP20 & 60&&5\\\\\n2:1& aP24 & 1&&4\\\\\n2:1& tP12 & 92&&4\\\\\n3:1& mP16 & 14&&4\\\\\n3:2& mS20 & 12&&4\\\\\n3:2& oP20 & 62&&4\\\\\n5:2& mS28 & 15&&4\\\\\n1:1& tP4 & 129&&3\\\\\n2:1& hP9 & 152&&3\\\\\n2:1& mS6 & 12&&3\\\\\n3:1& cP4 & 221&&3\\\\\n3:2& cF80 & 227&&3\\\\\n3:2& hP5 & 150&&3\\\\\n3:2& oS20 & 63&&3\\\\\n7:4& aP22 & 2&&3\\\\\n12:7& hR19 & 148&&3\\\\\n1:1& cP2 & 221&&2\\\\\n1:1& hP4 & 194&&2\\\\\n1:1& mS4 & 12&&2\\\\\n1:1& mS8 & 15&&2\\\\\n\\end{tabular}}\n\\label{tab:art130:oxide_binary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the binary sulfide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n1:1& cF8 & 225&NaCl&32\\\\\n3:2& oP20 & 62&Sb$_{2}$S$_{3}$&20\\\\\n2:1& tP6 & 129&PbClF\/Cu$_{2}$Sb&13\\\\\n2:1& cP12 & 205&Pyrite-Fe$_{2}$S$_{2}$ (cP12)&12\\\\\n2:1& cF24 & 227&Laves(Cub)-Cu$_{2}$Mg&11\\\\\n1:1& hP4 & 194&Nickeline-NiAs&8\\\\\n4:3& cI28 & 220&Th$_{3}$P$_{4}$&8\\\\\n1:1& cF8 & 216&Sphalerite-ZnS (cF8)&7\\\\\n2:1& hP3 & 164&CdI$_{2}$&7\\\\\n2:1& mP12 & 14&CeSe$_{2}$&7\\\\\n1:1& cP2 & 221&&6\\\\\n1:1& hP4 & 186&&6\\\\\n1:2& cF12 & 225&&6\\\\\n2:1& hP6 & 194&&6\\\\\n7:5& mS24 & 12&&6\\\\\n1:1& oP8 & 62&&5\\\\\n2:1& oP6 & 58&&5\\\\\n3:2& hR10 & 167&&5\\\\\n3:2& mP30 & 11&&5\\\\\n1:1& hP2 & 187&&4\\\\\n1:2& hP6 & 194&&4\\\\\n1:2& oP12 & 62&&4\\\\\n2:1& hR3 & 160&&4\\\\\n2:1& oP12 & 62&&4\\\\\n2:1& oP24 & 62&&4\\\\\n3:1& mP8 & 11&&4\\\\\n4:3& cF56 & 227&&4\\\\\n5:2& oP28 & 19&&4\\\\\n1:1& oS8 & 63&&3\\\\\n2:2& hP12 & 189&&3\\\\\n3:2& hP30 & 185&&3\\\\\n3:2& oS20 & 36&&3\\\\\n1:1& hP16 & 186&&2\\\\\n1:1& hP8 & 194&&2\\\\\n1:1& hR2 & 160&&2\\\\\n1:1& hR4 & 166&&2\\\\\n1:1& hR6 & 160&&2\\\\\n1:1& mP8 & 14&&2\\\\\n1:1& mS8 & 5&&2\\\\\n1:1& oS8 & 39&&2\\\\\n\\end{tabular}}\n\\label{tab:art130:sulfide_binary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the binary selenide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n1:1& cF8 & 225&NaCl&31\\\\\n2:1& tP6 & 129&PbClF\/Cu$_{2}$Sb&13\\\\\n2:1& cP12 & 205&Pyrite-FeS$_{2}$ (cP12)&11\\\\\n1:1& hP4 & 186&Wurtzite-ZnS(2H)&9\\\\\n1:1& hP4 & 194&Nickeline-NiAs&9\\\\\n2:1& hP3 & 164&CdI$_{2}$&9\\\\\n3:2& oP20 & 62&Sb$_{2}$S$_{3}$&9\\\\\n4:3& cI28 & 220&Th$_{3}$P$_{4}$&9\\\\\n1:1& cF8 & 216&Sphalerite-ZnS (cF8)&8\\\\\n3:2& oF80 & 70&Sc$_{2}$S$_{3}$&8\\\\\n1:1& cP2 & 221&&7\\\\\n1:2& cF12 & 225&&6\\\\\n4:3& mS14 & 12&&6\\\\\n1:1& oP8 & 62&&4\\\\\n2:1& hP6 & 194&&4\\\\\n2:1& mP12 & 14&&4\\\\\n3:1& mP8 & 11&&4\\\\\n2:1& hP12 & 187&&3\\\\\n2:1& hR3 & 160&&3\\\\\n2:1& oP6 & 58&&3\\\\\n3:2& oS20 & 36&&3\\\\\n4:4& mP32 & 14&&3\\\\\n4:5& tI18 & 87&&3\\\\\n5:2& oP28 & 19&&3\\\\\n1:1& hP8 & 187&&2\\\\\n1:1& hP8 & 194&&2\\\\\n1:1& hR4 & 160&&2\\\\\n1:1& mS8 & 12&&2\\\\\n1:2& oP36 & 58&&2\\\\\n2:1& hP12 & 194&&2\\\\\n2:1& oP12 & 62&&2\\\\\n2:1& oP24 & 62&&2\\\\\n2:2& hP12 & 189&&2\\\\\n2:2& mP16 & 14&&2\\\\\n3:1& mP24 & 11&&2\\\\\n3:2& hR5 & 166&&2\\\\\n3:2& mP10 & 11&&2\\\\\n3:2& mS20 & 9&&2\\\\\n4:3& hP14 & 176&&2\\\\\n8:3& hR11 & 148&&2\\\\\n\\end{tabular}}\n\\label{tab:art130:selenide_binary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the ternary oxide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n7:2:2& cF88 & 227&Pyrochlore&116\\\\\n3:1:1& oP20 & 62&Perovskite-GdFeO$_{3}$ (mostly)&115\\\\\n3:1:1& cP5 & 221&Perovskite-CaTiO$_{3}$&78\\\\\n2:1:1& hR4 & 166&Delafossite-NaCrS$_{2}$&72\\\\\n4:1:1& tI24 & 141&Zircon-ZrSiO$_{4}$&66\\\\\n4:1:2& cF56 & 227&Spinel-Al$_{2}$MgO$_{4}$&66\\\\\n4:1:1& tI24 & 88&Scheelite-CaWO$_{4}$&47\\\\\n4:1:2& oP28 & 62&CaFe$_{2}$O$_{4}$&44\\\\\n4:1:1& mP24 & 14&AgMnO4&43\\\\\n3:1:1& hR10 & 167&Perovskite-NdAlO$_{3}$&36\\\\\n4:1:1& oP24 & 62&Barite-BaSO$_{4}$&34\\\\\n3:1:1& mP20 & 14&&33\\\\\n7:1:3& oS44 & 63&&33\\\\\n2:1:2& hP5 & 164&&32\\\\\n4:1:2& tI14 & 139&&32\\\\\n2:1:2& tI10 & 139&&31\\\\\n4:1:1& oS24 & 63&&31\\\\\n12:3:5& cI160 & 230&&30\\\\\n3:1:1& hR10 & 148&&28\\\\\n2:1:1& hP8 & 194&&27\\\\\n4:1:1& mP12 & 13&&26\\\\\n1:1:1& tP6 & 129&&25\\\\\n5:1:2& mP32 & 14&&25\\\\\n7:2:2& mS22 & 12&&24\\\\\n6:1:2& tP18 & 136&&22\\\\\n11:2:4& mS68 & 15&&20\\\\\n3:1:1& hR10 & 161&&19\\\\\n4:1:2& mP28 & 14&&19\\\\\n7:2:2& mP44 & 14&&19\\\\\n1:1:3& cP5 & 221&&18\\\\\n3:1:1& hR5 & 160&&18\\\\\n1:2:4& tI14 & 139&&17\\\\\n3:1:1& mP40 & 14&&17\\\\\n6:1:2& hP9 & 162&&16\\\\\n7:2:2& aP22 & 2&&16\\\\\n7:2:2& aP44 & 2&&16\\\\\n9:1:3& mP52 & 14&&16\\\\\n1:4:6& hP22 & 186&&15\\\\\n3:1:1& hP30 & 185&&15\\\\\n5:1:2& oP32 & 55&&15\\\\\n\\end{tabular}}\n\\label{tab:art130:oxide_ternary_data}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most common structure types among the ternary sulfide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n2:1:1& hR4 & 166&Delafossite-NaCrS$_{2}$&88\\\\\n4:1:2& oP28 & 62&CaFe$_{2}$O$_{4}$&77\\\\\n4:1:2& cF56 & 227&Spinel-Al$_{2}$MgO$_{4}$&53\\\\\n5:1:2& oP32 & 62&U$_{3}$S$_{5}$&37\\\\\n3:1:1& oP20 & 62&SrZrS$_{3}$&32\\\\\n8:1:6& hR15 & 148&Mo$_{6}$PbS$_{8}$&32\\\\\n1:1:1& mP12 & 14&CeAsS&27\\\\\n2:1:1& hP8 & 194&SnTaS$_{2}$&26\\\\\n6:1:3& mP20 & 11&Tm$_{2}$S$_{3}$&26\\\\\n7:1:4& hP24 & 173&La$_{3}$CuSiS$_{7}$&25\\\\\n1:1:1& tP6 & 129&&23\\\\\n3:1:1& oP20 & 33&&21\\\\\n22:4:11& mS74 & 12&&20\\\\\n1:2:2& hP5 & 164&&19\\\\\n6:1:3& oP40 & 18&&15\\\\\n1:1:1& cP12 & 198&&14\\\\\n2:2:3& hR7 & 166&&14\\\\\n4:1:3& oP32 & 62&&14\\\\\n1:1:1& oP12 & 62&&12\\\\\n4:1:1& tI96 & 142&&12\\\\\n1:1:1& oP24 & 62&&11\\\\\n2:1:1& mP16 & 14&&11\\\\\n2:1:1& oP16 & 62&&11\\\\\n3:1:1& mP40 & 11&&11\\\\\n6:1:3& hP20 & 182&&11\\\\\n12:3:4& hR38 & 161&&11\\\\\n13:4:5& oP44 & 55&&11\\\\\n1:1:4& oP24 & 62&&10\\\\\n2:1:1& hR4 & 160&&10\\\\\n2:1:1& tI16 & 122&&10\\\\\n2:1:2& tI10 & 139&&10\\\\\n3:1:2& oP24 & 62&&10\\\\\n4:1:2& oS28 & 66&&10\\\\\n8:1:5& mS28 & 12&&10\\\\\n3:1:3& oP28 & 62&&9\\\\\n4:1:2& mS14 & 12&&9\\\\\n4:1:2& oF224 & 70&&9\\\\\n4:1:3& cP16 & 223&&9\\\\\n2:1:1& mS64 & 15&&8\\\\\n2:1:2& mP20 & 14&&8\\\\\n\\end{tabular}}\n\\label{tab:art130:sulfide_ternary_data}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of the 40 most structure types among the ternary selenide compounds.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{L{2.5cm}|L{2.5cm}|L{2.5cm}|L{5cm}|R{3.25cm}}\nstoichiometry & Pearson & symmetry & name & compounds count \\\\\n\\hline\n2:1:1& hR4 & 166&Delafossite&51\\\\\n4:1:2& oP28 & 62&CaFe$_{2}$O$_{4}$&48\\\\\n4:1:2& cF56 & 227&Spinel-Al$_{2}$MgO$_{4}$&30\\\\\n8:1:6& hR15 & 148&Mo$_{6}$PbS$_{8}$&25\\\\\n1:1:1& mP12 & 14&CeAsS&22\\\\\n4:1:2& mS14 & 12&CrNb$_{2}$Se$_{4}$-Cr$_{3}$S$_{4}$&18\\\\\n8:1:5& mS28 & 12&Cr$_{5}$CsS$_{8}$&18\\\\\n1:1:1& tP6 & 129&PbClF\/Cu$_{2}$Sb&16\\\\\n1:1:1& cP12 & 198&NiSSb&14\\\\\n3:1:1& oP20 & 62&NH$_{4}$CdCl$_{3}$\/Sn$_{2}$S$_{3}$&14\\\\\n4:1:2& tI14 & 82&&13\\\\\n5:1:2& oP32 & 62&&13\\\\\n4:1:3& oP32 & 62&&12\\\\\n1:2:2& hP5 & 164&&11\\\\\n2:1:1& mP16 & 14&&10\\\\\n2:1:2& tI10 & 139&&10\\\\\n3:1:1& oS20 & 63&&10\\\\\n3:1:3& cP28 & 198&&10\\\\\n6:1:3& hP20 & 182&&10\\\\\n1:1:1& oP12 & 62&&9\\\\\n1:1:3& oP20 & 62&&9\\\\\n2:1:1& tI16 & 122&&9\\\\\n2:1:2& oI20 & 72&&9\\\\\n3:1:3& hP14 & 176&&9\\\\\n4:1:2& oS28 & 66&&9\\\\\n19:2:15& hR72 & 167&&9\\\\\n2:1:1& oP16 & 19&&8\\\\\n6:1:3& oP40 & 58&&8\\\\\n17:1:8& mS52 & 12&&8\\\\\n4:1:6& hP22 & 186&&7\\\\\n6:2:2& mP20 & 14&&7\\\\\n6:2:6& mP28 & 14&&7\\\\\n2:1:1& mS64 & 15&&6\\\\\n2:1:1& tI16 & 140&&6\\\\\n4:1:2& oF224 & 70&&6\\\\\n1:1:4& oS24 & 63&&5\\\\\n2:1:1& hR4 & 160&&5\\\\\n2:1:6& mP18 & 14&&5\\\\\n2:1:12& oF120 & 43&&5\\\\\n2:2:3& hR7 & 166&&5\\\\\n\\end{tabular}}\n\\label{tab:art130:selenide_ternary_data}\n\\end{table}\n\n\\clearpage\n\n\\subsection{Prevalence of stoichiometries}\n\\label{subsec:art130:prev_stoich_supp}\n\nTables~\\ref{tab:art130:Prevalence_of_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Binaries_Stoichiometries_3} list all the binary\nstoichiometries among the three compound families examined in this paper.\nAn interesting finding is that the stoichiometry\n$A_1$O$_2$ has 356 unique compounds, a number that is significantly\nlarger than the number of atoms in the periodic table. This is because\na given chemical composition can have many different structure type\nrealizations.\nThe most prominent example is SiO$_{2}$ which has 185 different\nreported structures, representing the majority of the 356 compounds\nand 244 structure types of this stoichiometry.\nIn contrast, SiS$_{2}$ has only two reported structures, and SiSe$_{2}$\nhas only one.\nChecking other atoms from the same column of Si in the periodic\ntable, we find that GeO$_{2}$ has seven structures and CO$_{2}$ has\nnine.\nThe last observation means that, since CO$_{2}$ is\ngaseous in atmospheric conditions, the {\\small ICSD}\\ compounds of\nCO$_{2}$ are not in atmospheric conditions (either temperature or\npressure or both).\nExamining Tables~\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}\nwe also observe that the $A_xO_y$ set of compounds exhibits\nseveral gaps (missing ratios) along the axis of $y\/(x+y)$. There are\nno reported binary oxides\nfrom 0.51 to (and not including) 0.55, from 0.34 to 0.4,\nfrom 0.26 to 0.3, and from 0.44 to 0.5.\nThose gaps {\\it do not exist} in the sulfides and\nselenides.\nMost of the gaps in the sulfides appear above 0.6, and no\nselenide compounds are reported above 0.65.\nThe maximal ratio for the oxides is 0.84, while the maximal ratio for the sulfides is 0.93.\nTables~\\ref{tab:art130:Elements_stoichiometries}-\\ref{tab:art130:Elements_stoichiometries_3} show the leading stoichiometry for each element\nas well as the number of stoichiometries and unique compounds for this element.\nWhile SiO$_2$ is the {\\it only} stoichiometry of silicon oxide (with 185 structure types),\nvanadium has 18 different stoichiometries and 42 unique compounds,\nVO$_2$ is the stoichiometry with the largest number (10) of structure types.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of binary stoichiometries (1\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:2& 356 & 123 & 79\\\\\n1:1& 99 & 165 & 108\\\\\n2:3& 146 & 58 & 38\\\\\n1:3& 41 & 12 & 10\\\\\n2:1& 27 & 30 & 21\\\\\n2:5& 26 & 8 & 5\\\\\n3:4& 22 & 25 & 21\\\\\n3:1& 9 & 7 & 1\\\\\n6:11& 9 & 0 & 0\\\\\n3:8& 9 & 2 & 2\\\\\n6:1& 4 & 7 & 0\\\\\n3:5& 7 & 2 & 2\\\\\n5:9& 7 & 0 & 0\\\\\n5:7& 0 & 6 & 0\\\\\n4:7& 6 & 1 & 0\\\\\n4:1& 2 & 1 & 5\\\\\n4:3& 2 & 5 & 4\\\\\n6:13& 5 & 0 & 0\\\\\n4:9& 5 & 2 & 1\\\\\n5:4& 0 & 2 & 4\\\\\n12:29& 4 & 0 & 0\\\\\n2:7& 4 & 2 & 0\\\\\n1:4& 4 & 1 & 2\\\\\n8:1& 3 & 1 & 0\\\\\n3:2& 1 & 3 & 2\\\\\n7:8& 0 & 3 & 2\\\\\n4:5& 3 & 3 & 1\\\\\n7:12& 3 & 0 & 0\\\\\n8:15& 3 & 0 & 1\\\\\n4:11& 3 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Binaries_Stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of binary stoichiometries continued (2\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n21:8& 0 & 2 & 0\\\\\n7:3& 2 & 0 & 0\\\\\n5:3& 0 & 0 & 2\\\\\n9:8& 0 & 2 & 1\\\\\n6:7& 0 & 1 & 2\\\\\n9:11& 0 & 0 & 2\\\\\n11:20& 2 & 0 & 0\\\\\n7:13& 2 & 0 & 0\\\\\n9:17& 2 & 0 & 0\\\\\n8:21& 2 & 0 & 0\\\\\n17:47& 2 & 0 & 0\\\\\n5:14& 2 & 0 & 0\\\\\n8:23& 2 & 0 & 0\\\\\n9:26& 2 & 0 & 0\\\\\n2:9& 1 & 0 & 2\\\\\n61:2& 1 & 0 & 0\\\\\n12:1& 0 & 1 & 0\\\\\n7:1& 1 & 0 & 0\\\\\n16:3& 1 & 0 & 0\\\\\n9:2& 1 & 1 & 1\\\\\n15:4& 0 & 1 & 0\\\\\n11:3& 1 & 0 & 0\\\\\n7:2& 0 & 0 & 1\\\\\n34:11& 0 & 0 & 1\\\\\n45:16& 0 & 0 & 1\\\\\n14:5& 0 & 1 & 0\\\\\n11:4& 0 & 0 & 1\\\\\n8:3& 0 & 1 & 1\\\\\n5:2& 0 & 0 & 1\\\\\n16:7& 0 & 1 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Binaries_Stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of binary stoichiometries continued (3\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n31:16& 0 & 1 & 0\\\\\n29:16& 0 & 1 & 0\\\\\n9:5& 0 & 1 & 0\\\\\n7:4& 0 & 1 & 1\\\\\n6:5& 0 & 1 & 1\\\\\n8:7& 0 & 0 & 1\\\\\n17:15& 0 & 1 & 1\\\\\n17:18& 0 & 1 & 0\\\\\n8:9& 0 & 1 & 1\\\\\n5:6& 0 & 1 & 0\\\\\n13:16& 1 & 0 & 0\\\\\n15:19& 0 & 1 & 1\\\\\n8:11& 0 & 1 & 0\\\\\n15:22& 0 & 1 & 0\\\\\n5:8& 1 & 1 & 1\\\\\n9:16& 1 & 0 & 0\\\\\n16:35& 1 & 0 & 0\\\\\n3:7& 1 & 0 & 0\\\\\n5:12& 1 & 0 & 0\\\\\n13:34& 1 & 0 & 0\\\\\n18:49& 1 & 0 & 0\\\\\n25:73& 1 & 0 & 0\\\\\n4:21& 1 & 0 & 0\\\\\n1:8& 0 & 1 & 0\\\\\n1:14& 0 & 1 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Binaries_Stoichiometries_3}\n\\end{table}\n\n\\clearpage\n\nThese differences carry over into the ternary oxide compounds involving those elements,\nwhere the stoichiometry distribution of silicon ternary oxides is much tilted towards the\nsilicon poor compounds than those of vanadium and titanium as shown in Figure~\\ref{fig:art130:specific_triangle_stoichiometries}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig025}\n\\mycaption{Comparison of ternary oxide stoichiometries containing (\\textbf{a}) silicon, (\\textbf{b}) titanium, and (\\textbf{c}) vanadium.}\n\\label{fig:art130:specific_triangle_stoichiometries}\n\\end{figure}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) (1\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:1:3& 718 & 147 & 70\\\\\n1:1:4& 428 & 28 & 8\\\\\n1:2:4& 396 & 242 & 158\\\\\n2:2:7& 304 & 2 & 0\\\\\n1:1:2& 269 & 242 & 145\\\\\n1:2:6& 237 & 8 & 8\\\\\n1:2:5& 149 & 57 & 22\\\\\n1:1:1& 113 & 140 & 90\\\\\n1:2:2& 131 & 64 & 38\\\\\n1:2:3& 100 & 55 & 42\\\\\n2:3:8& 87 & 8 & 4\\\\\n1:3:6& 83 & 63 & 22\\\\\n1:4:4& 78 & 17 & 13\\\\\n1:3:9& 78 & 0 & 0\\\\\n1:3:7& 67 & 1 & 0\\\\\n2:2:5& 64 & 32 & 23\\\\\n2:4:9& 62 & 7 & 3\\\\\n1:3:3& 62 & 50 & 39\\\\\n1:3:4& 58 & 54 & 31\\\\\n1:3:8& 55 & 1 & 0\\\\\n1:2:7& 49 & 3 & 2\\\\\n2:4:11& 46 & 8 & 2\\\\\n3:5:12& 46 & 4 & 0\\\\\n2:3:12& 45 & 0 & 0\\\\\n1:4:1& 15 & 44 & 12\\\\\n1:3:1& 43 & 13 & 30\\\\\n1:2:8& 41 & 2 & 3\\\\\n1:6:8& 4 & 39 & 30\\\\\n1:3:5& 37 & 15 & 6\\\\\n1:1:5& 37 & 2 & 2\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Ternary_Stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) continued (2\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n2:3:9& 34 & 2 & 3\\\\\n1:3:2& 33 & 19 & 11\\\\\n2:3:2& 9 & 32 & 11\\\\\n1:4:7& 32 & 29 & 2\\\\\n2:3:6& 31 & 11 & 11\\\\\n2:2:1& 29 & 30 & 17\\\\\n1:5:8& 29 & 14 & 22\\\\\n2:4:7& 28 & 6 & 0\\\\\n1:2:1& 27 & 12 & 5\\\\\n2:3:4& 17 & 27 & 17\\\\\n2:2:9& 26 & 2 & 2\\\\\n1:5:14& 25 & 0 & 0\\\\\n2:2:3& 25 & 8 & 3\\\\\n2:3:7& 24 & 10 & 3\\\\\n1:4:8& 15 & 24 & 11\\\\\n2:4:1& 24 & 15 & 12\\\\\n2:4:13& 22 & 1 & 1\\\\\n4:6:1& 22 & 5 & 3\\\\\n1:5:4& 21 & 6 & 1\\\\\n4:11:22& 0 & 20 & 3\\\\\n2:6:7& 19 & 5 & 4\\\\\n1:4:6& 18 & 4 & 3\\\\\n3:4:10& 17 & 2 & 0\\\\\n1:4:5& 16 & 0 & 0\\\\\n1:5:2& 3 & 1 & 16\\\\\n1:4:11& 15 & 0 & 0\\\\\n2:4:5& 15 & 4 & 4\\\\\n1:4:3& 14 & 8 & 8\\\\\n3:4:12& 11 & 14 & 0\\\\\n2:4:15& 14 & 0 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) continued (3\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:6:12& 14 & 0 & 0\\\\\n1:5:5& 13 & 3 & 2\\\\\n2:2:11& 13 & 0 & 0\\\\\n4:5:13& 1 & 12 & 0\\\\\n2:12:3& 0 & 12 & 4\\\\\n1:4:9& 12 & 1 & 0\\\\\n1:6:11& 12 & 0 & 0\\\\\n1:4:12& 12 & 1 & 0\\\\\n4:4:11& 11 & 1 & 1\\\\\n3:4:9& 11 & 5 & 3\\\\\n3:5:14& 11 & 0 & 0\\\\\n2:3:10& 11 & 0 & 0\\\\\n1:6:2& 6 & 2 & 11\\\\\n1:6:4& 9 & 10 & 11\\\\\n2:4:3& 5 & 10 & 7\\\\\n1:12:20& 10 & 0 & 0\\\\\n10:14:1& 10 & 1 & 0\\\\\n2:5:13& 10 & 0 & 0\\\\\n2:15:19& 0 & 4 & 10\\\\\n1:8:6& 10 & 6 & 4\\\\\n1:8:14& 10 & 0 & 0\\\\\n1:7:12& 9 & 0 & 0\\\\\n1:5:7& 8 & 1 & 1\\\\\n1:6:6& 8 & 0 & 0\\\\\n2:9:3& 0 & 0 & 8\\\\\n2:6:13& 8 & 1 & 1\\\\\n1:3:12& 8 & 0 & 3\\\\\n1:8:17& 0 & 8 & 8\\\\\n1:12:19& 8 & 0 & 0\\\\\n3:4:8& 8 & 2 & 1\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se; top 120) continued (4\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:5:6& 7 & 1 & 0\\\\\n2:7:2& 0 & 6 & 7\\\\\n1:5:1& 2 & 7 & 5\\\\\n3:4:4& 7 & 2 & 0\\\\\n2:3:1& 2 & 5 & 7\\\\\n2:5:10& 7 & 0 & 0\\\\\n2:5:12& 7 & 0 & 0\\\\\n2:3:11& 4 & 7 & 2\\\\\n4:6:19& 7 & 0 & 0\\\\\n1:1:6& 7 & 3 & 2\\\\\n4:6:13& 5 & 6 & 3\\\\\n1:7:1& 0 & 5 & 6\\\\\n1:10:14& 0 & 6 & 4\\\\\n4:5:15& 6 & 0 & 0\\\\\n3:3:1& 6 & 0 & 0\\\\\n1:12:2& 0 & 1 & 6\\\\\n1:3:10& 6 & 0 & 0\\\\\n2:6:1& 2 & 6 & 4\\\\\n5:9:5& 6 & 0 & 0\\\\\n2:6:15& 5 & 0 & 0\\\\\n2:9:6& 1 & 5 & 0\\\\\n4:4:3& 1 & 1 & 5\\\\\n4:5:12& 5 & 1 & 0\\\\\n2:8:7& 5 & 0 & 0\\\\\n3:6:1& 0 & 5 & 0\\\\\n1:8:2& 0 & 5 & 1\\\\\n1:7:6& 3 & 5 & 4\\\\\n1:8:8& 1 & 5 & 2\\\\\n2:9:2& 1 & 5 & 1\\\\\n3:5:2& 5 & 3 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) (1\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:1:3& 718 & 147 & 70\\\\\n1:1:4& 428 & 28 & 8\\\\\n2:2:7& 304 & 2 & 0\\\\\n1:1:2& 269 & 242 & 145\\\\\n2:1:4& 206 & 101 & 70\\\\\n1:2:4& 190 & 141 & 88\\\\\n1:1:1& 113 & 140 & 90\\\\\n2:1:6& 122 & 7 & 3\\\\\n1:2:6& 115 & 1 & 5\\\\\n1:2:5& 90 & 32 & 14\\\\\n2:1:2& 72 & 37 & 24\\\\\n2:2:5& 64 & 32 & 23\\\\\n3:1:9& 62 & 0 & 0\\\\\n2:3:8& 60 & 6 & 2\\\\\n2:1:5& 59 & 25 & 8\\\\\n1:2:2& 59 & 27 & 14\\\\\n2:1:3& 59 & 23 & 22\\\\\n1:3:6& 34 & 54 & 19\\\\\n3:1:6& 49 & 9 & 3\\\\\n4:1:1& 15 & 43 & 12\\\\\n5:3:12& 43 & 2 & 0\\\\\n3:1:3& 42 & 26 & 23\\\\\n1:2:3& 41 & 32 & 20\\\\\n4:1:4& 40 & 8 & 6\\\\\n4:2:9& 40 & 6 & 1\\\\\n1:4:4& 38 & 9 & 7\\\\\n1:1:5& 37 & 2 & 2\\\\\n2:1:7& 36 & 3 & 1\\\\\n3:1:7& 34 & 1 & 0\\\\\n3:1:4& 32 & 33 & 13\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Ternary_Stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) continued (2\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:3:7& 33 & 0 & 0\\\\\n4:2:11& 31 & 5 & 0\\\\\n2:2:1& 29 & 30 & 17\\\\\n3:1:1& 30 & 12 & 29\\\\\n3:1:8& 30 & 1 & 0\\\\\n6:1:8& 2 & 29 & 21\\\\\n4:1:7& 28 & 3 & 2\\\\\n3:2:2& 1 & 27 & 7\\\\\n3:2:8& 27 & 2 & 2\\\\\n3:2:9& 27 & 2 & 3\\\\\n2:2:9& 26 & 2 & 2\\\\\n1:3:4& 26 & 21 & 18\\\\\n2:1:8& 26 & 2 & 1\\\\\n1:4:7& 4 & 26 & 0\\\\\n5:1:14& 25 & 0 & 0\\\\\n2:2:3& 25 & 8 & 3\\\\\n1:3:8& 25 & 0 & 0\\\\\n2:3:6& 24 & 8 & 8\\\\\n3:1:2& 24 & 12 & 4\\\\\n1:3:3& 20 & 24 & 16\\\\\n1:3:5& 23 & 9 & 5\\\\\n3:2:12& 23 & 0 & 0\\\\\n2:3:12& 22 & 0 & 0\\\\\n2:4:9& 22 & 1 & 2\\\\\n6:4:1& 18 & 1 & 2\\\\\n5:1:4& 17 & 6 & 1\\\\\n2:3:7& 16 & 10 & 2\\\\\n4:11:22& 0 & 16 & 3\\\\\n4:1:5& 16 & 0 & 0\\\\\n1:3:9& 16 & 0 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) continued (3\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n1:2:8& 15 & 0 & 2\\\\\n2:4:11& 15 & 3 & 2\\\\\n1:4:8& 9 & 15 & 9\\\\\n2:4:7& 15 & 4 & 0\\\\\n5:1:8& 15 & 10 & 15\\\\\n3:2:4& 10 & 14 & 9\\\\\n4:2:13& 14 & 1 & 1\\\\\n1:5:8& 14 & 4 & 7\\\\\n3:4:12& 10 & 14 & 0\\\\\n2:1:1& 14 & 1 & 3\\\\\n4:2:1& 14 & 11 & 10\\\\\n3:1:5& 14 & 6 & 1\\\\\n1:2:1& 13 & 11 & 2\\\\\n4:2:7& 13 & 2 & 0\\\\\n4:2:15& 13 & 0 & 0\\\\\n4:1:6& 13 & 0 & 0\\\\\n2:3:4& 7 & 13 & 8\\\\\n1:3:1& 13 & 1 & 1\\\\\n6:1:12& 13 & 0 & 0\\\\\n2:2:11& 13 & 0 & 0\\\\\n1:2:7& 13 & 0 & 1\\\\\n4:1:3& 12 & 5 & 5\\\\\n5:1:5& 12 & 1 & 0\\\\\n5:1:2& 3 & 1 & 12\\\\\n4:1:11& 12 & 0 & 0\\\\\n12:2:3& 0 & 12 & 3\\\\\n2:6:7& 11 & 2 & 0\\\\\n6:1:2& 6 & 2 & 11\\\\\n4:4:11& 11 & 1 & 1\\\\\n5:4:13& 0 & 11 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of ternary stoichiometries ($A_xB_yC_z,~C=$ O, S, Se, with $M_A>M_B$ when $x\\neq y$; top 120) continued (4\/4).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nstoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n4:1:12& 11 & 1 & 0\\\\\n6:1:4& 7 & 10 & 11\\\\\n8:1:6& 10 & 2 & 2\\\\\n3:4:10& 10 & 1 & 0\\\\\n1:6:8& 2 & 10 & 9\\\\\n1:12:20& 10 & 0 & 0\\\\\n14:10:1& 10 & 1 & 0\\\\\n2:4:1& 10 & 4 & 2\\\\\n4:1:9& 9 & 1 & 0\\\\\n3:4:9& 9 & 2 & 1\\\\\n1:3:2& 9 & 7 & 7\\\\\n2:4:5& 9 & 2 & 1\\\\\n4:1:8& 6 & 9 & 2\\\\\n3:2:7& 8 & 0 & 1\\\\\n6:2:7& 8 & 3 & 4\\\\\n9:2:3& 0 & 0 & 8\\\\\n2:4:13& 8 & 0 & 0\\\\\n5:2:13& 8 & 0 & 0\\\\\n12:1:19& 8 & 0 & 0\\\\\n2:3:2& 8 & 5 & 4\\\\\n1:8:17& 0 & 8 & 8\\\\\n8:1:14& 8 & 0 & 0\\\\\n3:2:6& 7 & 3 & 3\\\\\n6:1:11& 7 & 0 & 0\\\\\n2:3:9& 7 & 0 & 0\\\\\n3:4:4& 7 & 1 & 0\\\\\n3:2:10& 7 & 0 & 0\\\\\n6:1:6& 7 & 0 & 0\\\\\n5:1:1& 2 & 7 & 4\\\\\n4:3:10& 7 & 1 & 0\\\\\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of binary stoichiometries (1\/3).]\n{The entries for each element column denote the total number of structure types,\ntotal number of unique compounds and then the leading atom with the total\nnumber of structure types of this stoichiometry in which it appears.\nThe second column shows the stoichiometry $(x:y)$ for $A_xZ_y$, $Z=$ O, S, Se, respectively.}\n\\vspace{3mm}\n\\begin{tabular}{l|l|r|r|r}\nratio $y\/(x+y)$ & stoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n0.032 & (2:61) & 1 1 C(1) & & \\\\\n0.077 & (1:12) & & 1 1 B(1) & \\\\\n0.11 & (1:8) & 3 3 V(1) & 1 1 Ag(1) & \\\\\n0.12 & (1:7) & 1 1 Cs(1) & & \\\\\n0.14 & (1:6) & 4 4 Ti(2) & 7 7 F(5) & \\\\\n0.16 & (3:16) & 1 1 V(1) & & \\\\\n0.18 & (2:9) & 1 1 Rb(1) & 1 1 Zr(1) & 1 1 Ti(1)\\\\\n0.2 & (1:4) & 2 2 Ta(1) & 1 1 Pd(1) & 5 5 Cl(2)\\\\\n0.21 & (4:15) & & 1 1 C(1) & \\\\\n0.21 & (3:11) & 1 1 Cs(1) & & \\\\\n0.22 & (2:7) & & & 1 1 Pd(1)\\\\\n0.24 & (11:34) & & & 1 1 Pd(1)\\\\\n0.25 & (1:3) & 8 9 Zr(3) & 7 7 O(3) & 1 1 O(1)\\\\\n0.26 & (16:45) & & & 1 1 Ti(1)\\\\\n0.26 & (5:14) & & 1 1 Nb(1) & \\\\\n0.27 & (4:11) & & & 1 1 Ti(1)\\\\\n0.27 & (3:8) & & 1 1 Ti(1) & 1 1 Ti(1)\\\\\n0.28 & (8:21) & & 1 2 Zr(1) & \\\\\n0.29 & (2:5) & & & 1 1 O(1)\\\\\n0.3 & (3:7) & 2 2 V(2) & & \\\\\n0.3 & (7:16) & & 1 1 Pd(1) & \\\\\n0.33 & (1:2) & 17 27 H(6) & 18 30 Cu(4) & 15 21 O(4)\\\\\n0.34 & (16:31) & & 1 1 Cu(1) & \\\\\n0.36 & (16:29) & & 1 1 Cu(1) & \\\\\n0.36 & (5:9) & & 1 1 Cu(1) & \\\\\n0.36 & (4:7) & & 1 1 Cu(1) & 1 1 Pd(1)\\\\\n0.38 & (3:5) & & & 2 2 Tl(2)\\\\\n0.4 & (2:3) & 1 1 C(1) & 3 3 Ni(2) & 2 2 Ni(1)\\\\\n0.43 & (3:4) & 2 2 Tl(1) & 4 5 P(2) & 4 4 P(1)\\\\\n0.44 & (4:5) & & 2 2 V(1) & 2 4 V(1)\\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of binary stoichiometries continued (2\/3).]\n{The entries for each element column denote the total number of structure types,\ntotal number of unique compounds and then the leading atom with the total\nnumber of structure types of this stoichiometry in which it appears.\nThe second column shows the stoichiometry $(x:y)$ for $A_xZ_y$, $Z=$ O, S, Se, respectively.}\n\\vspace{3mm}\n\\begin{tabular}{l|l|r|r|r}\nratio $y\/(x+y)$ & stoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n0.45 & (5:6) & & 1 1 N(1) & 1 1 Ni(1)\\\\\n0.47 & (7:8) & & & 1 1 Bi(1)\\\\\n0.47 & (15:17) & & 1 1 Rh(1) & 1 1 Pd(1)\\\\\n0.47 & (8:9) & & 2 2 Ni(1) & 1 1 Co(1)\\\\\n0.5 & (1:1) & 51 99 Mg(12) & 88 165 Zn(39) & 38 108 Ga(5)\\\\\n0.51 & (18:17) & & 1 1 Ni(1) & \\\\\n0.53 & (9:8) & & 1 1 As(1) & 1 1 Bi(1)\\\\\n0.53 & (8:7) & & 3 3 Fe(3) & 2 2 Fe(2)\\\\\n0.54 & (7:6) & & 1 1 In(1) & 2 2 In(2)\\\\\n0.55 & (6:5) & & 1 1 Cr(1) & \\\\\n0.55 & (11:9) & & & 2 2 Mo(2)\\\\\n0.55 & (16:13) & 1 1 V(1) & & \\\\\n0.56 & (5:4) & 3 3 Ti(1) & 2 3 P(2) & 1 1 P(1)\\\\\n0.56 & (19:15) & & 1 1 Mo(1) & 1 1 Mo(1)\\\\\n0.57 & (4:3) & 18 22 Fe(8) & 13 25 Fe(3) & 7 21 Ti(2)\\\\\n0.58 & (11:8) & & 1 1 Tm(1) & \\\\\n0.58 & (7:5) & & 1 6 Y(1) & \\\\\n0.59 & (22:15) & & 1 1 Tm(1) & \\\\\n0.6 & (3:2) & 43 146 Bi(16) & 23 58 Yb(6) & 18 38 In(6)\\\\\n0.62 & (8:5) & 1 1 Mn(1) & 1 1 Cr(1) & 1 1 Cr(1)\\\\\n0.62 & (5:3) & 6 7 V(4) & 2 2 U(2) & 2 2 U(2)\\\\\n0.63 & (12:7) & 1 3 Tb(1) & & \\\\\n0.64 & (7:4) & 4 6 Ti(3) & 1 1 P(1) & \\\\\n0.64 & (16:9) & 1 1 Pr(1) & & \\\\\n0.64 & (9:5) & 6 7 Ti(3) & & \\\\\n0.65 & (20:11) & 1 2 Tb(1) & & \\\\\n0.65 & (11:6) & 7 9 Ti(4) & & \\\\\n0.65 & (13:7) & 1 2 V(1) & & \\\\\n0.65 & (15:8) & 2 3 Ti(2) & & 1 1 Gd(1)\\\\\n0.65 & (17:9) & 1 2 V(1) & & \\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of binary stoichiometries continued (3\/3).]\n{The entries for each element column denote the total number of structure types,\ntotal number of unique compounds and then the leading atom with the total\nnumber of structure types of this stoichiometry in which it appears.\nThe second column shows the stoichiometry $(x:y)$ for $A_xZ_y$, $Z=$ O, S, Se, respectively.}\n\\vspace{3mm}\n\\begin{tabular}{l|l|r|r|r}\nratio $y\/(x+y)$ & stoichiometry & oxides & sulfides & selenides \\\\\n\\hline\n0.67 & (2:1) & 244 356 Si(185) & 50 123 Ti(9) & 34 79 Ta(10)\\\\\n0.68 & (13:6) & 5 5 V(5) & & \\\\\n0.69 & (35:16) & 1 1 U(1) & & \\\\\n0.69 & (9:4) & 5 5 V(2) & 2 2 P(2) & 1 1 Ge(1)\\\\\n0.7 & (7:3) & 1 1 V(1) & & \\\\\n0.71 & (12:5) & 1 1 Cr(1) & & \\\\\n0.71 & (29:12) & 4 4 Nb(4) & & \\\\\n0.71 & (5:2) & 20 26 Nb(6) & 4 8 U(1) & 3 5 Th(1)\\\\\n0.72 & (34:13) & 1 1 U(1) & & \\\\\n0.72 & (21:8) & 2 2 W(1) & & \\\\\n0.73 & (8:3) & 8 9 U(6) & 2 2 Ir(1) & 1 2 Rh(1)\\\\\n0.73 & (49:18) & 1 1 W(1) & & \\\\\n0.73 & (11:4) & 3 3 Mo(3) & & \\\\\n0.73 & (47:17) & 2 2 W(1) & & \\\\\n0.74 & (14:5) & 2 2 W(1) & & \\\\\n0.74 & (23:8) & 2 2 Mo(2) & & \\\\\n0.74 & (26:9) & 2 2 Mo(2) & & \\\\\n0.74 & (73:25) & 1 1 W(1) & & \\\\\n0.75 & (3:1) & 33 41 W(13) & 8 12 Ti(2) & 6 10 Ta(3)\\\\\n0.78 & (7:2) & 4 4 Tc(1) & 2 2 P(2) & \\\\\n0.8 & (4:1) & 3 4 Ru(2) & 1 1 V(1) & 2 2 Nb(1)\\\\\n0.82 & (9:2) & 1 1 P(1) & & 2 2 V(1)\\\\\n0.84 & (21:4) & 1 1 U(1) & & \\\\\n0.89 & (8:1) & & 1 1 O(1) & \\\\\n0.93 & (14:1) & & 1 1 C(1) & \\\\\n\\end{tabular}\n\\label{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of stoichiometries for the elements in the binary compounds, $A_xB_y,~B=$ O, S, Se (1\/3).]\n{The data presented is:\n$y:x(n_1)$, $n_2$, $n_3$, where $y:x$ is the leading stoichiometry, $n_1$ is number of compounds\nfor this stoichiometry, $n_2$ is number of stoichiometries and\n$n_3$ is number of unique compounds for this element.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\natom & oxides & sulfides & selenides \\\\\n\\hline\nAc & 3:2(1),1,1&&\\\\\nAg & 1:1(5),5,11&1:2(3),2,4&1:2(2),2,3\\\\\nAl & 3:2(7),3,9&3:2(3),1,3&3:2(1),1,1\\\\\nAs & 3:2(5),3,8&1:1(4),6,10&1:1(2),3,5\\\\\nAu & 3:2(1),1,1&1:2(1),1,1&1:1(3),1,3\\\\\nB & 3:2(2),3,4&1:12(1),3,3&\\\\\nBa & 1:1(3),2,5&3:1(2),4,5&1:1(2),3,4\\\\\nBe & 1:1(4),1,4&1:1(1),1,1&1:1(2),1,2\\\\\nBi & 3:2(16),4,20&3:2(1),1,1&1:1(4),6,11\\\\\nBr & 1:2(1),2,2&1:1(1),1,1&1:1(2),2,3\\\\\nC & 2:1(9),4,13&4:15(1),5,5&2:1(1),1,1\\\\\nCa & 1:1(2),2,3&1:1(1),1,1&1:1(2),1,2\\\\\nCd & 1:1(2),2,3&1:1(4),2,5&1:1(3),2,4\\\\\nCe & 2:1(5),5,12&2:1(4),4,8&2:1(3),3,6\\\\\nCl & 1:2(1),4,4&1:2(1),2,2&1:4(2),2,3\\\\\nCo & 1:1(5),3,10&8:9(1),4,4&4:3(2),4,6\\\\\nCr & 2:1(5),8,12&1:1(2),5,7&1:1(2),5,7\\\\\nCs & 1:7(1),8,8&1:2(1),5,5&1:2(2),4,5\\\\\nCu & 1:1(3),5,9&1:2(4),7,14&1:2(2),4,7\\\\\nDy & 3:2(4),1,4&2:1(2),5,6&1:1(2),4,5\\\\\nEr & 3:2(4),1,4&3:2(3),4,8&2:1(2),3,4\\\\\nEu & 3:2(4),4,7&1:1(3),3,6&1:1(2),2,3\\\\\nF & &1:6(5),1,5&1:4(1),1,1\\\\\nFe & 4:3(8),4,19&1:1(6),5,18&1:1(4),4,9\\\\\nGa & 3:2(3),1,3&1:1(2),2,3&1:1(5),3,7\\\\\nGd & 3:2(4),3,6&2:1(3),4,6&1:1(1),5,5\\\\\nGe & 2:1(7),1,7&2:1(5),2,7&2:1(5),3,8\\\\\nH & 1:2(6),2,7&1:2(3),1,3&1:2(1),1,1\\\\\nHf & 2:1(5),1,5&1:2(1),3,3&2:1(1),2,2\\\\\nHg & 1:1(5),2,7&1:1(4),1,4&1:1(3),1,3\\\\\n\\end{tabular}\n\\label{tab:art130:Elements_stoichiometries}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of stoichiometries for the elements in the binary compounds, $A_xB_y,~B=$ O, S, Se continued (2\/3).]\n{The data presented is:\n$y:x(n_1)$, $n_2$, $n_3$, where $y:x$ is the leading stoichiometry, $n_1$ is number of compounds\nfor this stoichiometry, $n_2$ is number of stoichiometries and\n$n_3$ is number of unique compounds for this element.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\natom & oxides & sulfides & selenides \\\\\n\\hline\nHo & 3:2(4),1,4&1:1(2),4,7&1:1(1),3,3\\\\\nI & 3:1(2),3,4&&\\\\\nIn & 3:2(5),1,5&1:1(2),5,7&3:2(6),5,14\\\\\nIr & 2:1(1),1,1&2:1(2),3,4&2:1(1),2,2\\\\\nK & 2:1(2),4,5&1:2(2),4,6&1:2(1),4,4\\\\\nLa & 3:2(4),1,4&1:1(5),4,12&1:1(4),3,7\\\\\nLi & 1:2(2),4,6&1:2(3),2,4&1:2(1),1,1\\\\\nLu & 3:2(4),1,4&3:2(3),3,6&1:1(1),3,3\\\\\nMg & 1:1(12),2,13&1:1(2),1,2&1:1(3),2,4\\\\\nMn & 3:2(4),6,14&1:1(4),2,6&1:1(4),2,5\\\\\nMo & 2:1(3),7,15&4:3(2),4,6&11:9(2),4,6\\\\\nN & 2:1(4),5,8&1:1(4),3,6&1:1(2),1,2\\\\\nNa & 2:1(2),4,5&1:1(4),4,9&1:2(1),3,3\\\\\nNb & 5:2(6),6,18&2:1(7),6,14&2:1(7),8,15\\\\\nNd & 3:2(4),2,5&2:1(3),4,6&2:1(2),3,4\\\\\nNi & 1:1(4),2,6&2:1(8),6,16&1:1(2),5,6\\\\\nO & &1:3(3),3,5&1:2(4),3,6\\\\\nOs & 4:1(2),2,3&2:1(1),1,1&2:1(1),1,1\\\\\nP & 5:2(3),5,7&3:4(2),8,13&3:4(1),4,4\\\\\nPa & 2:1(2),2,3&&\\\\\nPb & 1:1(4),5,14&1:1(13),1,13&1:1(3),2,4\\\\\nPd & 1:1(4),3,6&1:4(1),5,5&1:1(2),7,8\\\\\nPm & 3:2(3),1,3&&\\\\\nPr & 3:2(3),7,12&2:1(3),4,6&2:1(2),3,4\\\\\nPt & 2:1(5),3,9&1:1(2),2,3&4:5(1),2,2\\\\\nPu & 3:2(2),3,4&2:1(2),2,3&1:1(1),3,3\\\\\nRb & 3:2(2),7,8&1:2(3),4,7&1:2(1),4,4\\\\\nRe & 3:1(4),3,8&2:1(2),1,2&2:1(2),1,2\\\\\nRh & 3:2(4),2,5&15:17(1),4,4&2:1(2),4,5\\\\\nRu & 2:1(4),2,6&2:1(1),1,1&2:1(1),1,1\\\\\n\\end{tabular}\n\\label{tab:art130:Elements_stoichiometries_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[Prevalence of stoichiometries for the elements in the binary compounds, $A_xB_y,~B=$ O, S, Se continued (3\/3).]\n{The data presented is:\n$y:x(n_1)$, $n_2$, $n_3$, where $y:x$ is the leading stoichiometry, $n_1$ is number of compounds\nfor this stoichiometry, $n_2$ is number of stoichiometries and\n$n_3$ is number of unique compounds for this element.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\natom & oxides & sulfides & selenides \\\\\n\\hline\nS & 3:1(3),3,5&&\\\\\nSb & 3:2(5),3,11&3:2(1),1,1&3:2(1),1,1\\\\\nSc & 3:2(4),1,4&3:2(2),2,3&1:1(1),2,2\\\\\nSe & 2:1(4),3,6&&\\\\\nSi & 2:1(185),1,185&2:1(2),1,2&2:1(1),1,1\\\\\nSm & 3:2(4),2,5&1:1(2),3,4&1:1(2),4,5\\\\\nSn & 2:1(8),3,12&1:1(5),3,8&1:1(3),2,4\\\\\nSr & 1:1(2),2,3&1:1(2),3,4&1:1(1),1,1\\\\\nTa & 5:2(3),6,9&2:1(7),6,13&2:1(10),4,15\\\\\nTb & 3:2(4),5,9&2:1(4),4,7&1:1(2),3,4\\\\\nTc & 2:1(1),2,2&2:1(1),1,1&\\\\\nTe & 2:1(9),4,12&&\\\\\nTh & 2:1(2),1,2&1:1(1),4,4&1:1(2),5,6\\\\\nTi & 2:1(14),14,42&2:1(9),5,16&1:1(3),9,13\\\\\nTl & 1:2(2),3,5&1:1(6),4,9&1:1(3),3,6\\\\\nTm & 3:2(4),1,4&3:2(6),6,12&1:1(1),3,3\\\\\nU & 8:3(6),9,22&5:3(2),7,9&5:3(2),6,8\\\\\nV & 2:1(10),18,42&2:1(3),6,11&4:5(1),5,5\\\\\nW & 3:1(13),9,24&2:1(2),1,2&2:1(1),1,1\\\\\nXe & 3:1(1),1,1&&\\\\\nY & 3:2(5),1,5&3:2(2),4,6&1:1(1),2,2\\\\\nYb & 3:2(2),3,4&3:2(6),4,12&1:1(2),4,5\\\\\nZn & 1:1(4),2,5&1:1(39),2,40&1:1(2),2,3\\\\\nZr & 2:1(7),4,12&1:1(2),7,8&1:2(1),3,3\\\\\n\\end{tabular}\n\\label{tab:art130:Elements_stoichiometries_3}\n\\end{table}\n\n\\clearpage\n\n\\subsection{Correlation between ternary and binary stoichiometries for sulfides and selenides}\nIn this section we analyze the correlation between ternary and binary stoichiometries for sulfides and selenides.\nFigure~\\ref{fig:art130:tern_bin_stoichiometries} shows that, like in the oxides, in both the sulfides and selenides\nwe see a quite scattered pattern. However unlike in the oxides many atoms show points below the line $y=4x$.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig026}\n\\mycaption[The number of ternary (\\textbf{a}) sulfide and (\\textbf{b}) selenide stoichiometries\nper element as a function of the count of its respective binary stoichiometries.]\n{The dashed line marks perfect similarity $y=x$, and the dotted line marks the ratio $y=4x$.}\n\\label{fig:art130:tern_bin_stoichiometries}\n\\end{figure}\n\nWe next analyze in Figure~\\ref{fig:art130:tern_binproduct_stoichiometries} the number of ternary stoichiometries\nas a function of the product of the numbers of the binary stoichiometries of participating atoms.\nAs for the oxides (Figure~\\ref{fig:art130:mendeleev_distribution_all_in_one})\nwe see a trend of inverse correlation, \\nobreak\\mbox{\\it i.e.}, as the product of the numbers of binary\nstoichiometries increases, the number of ternaries decreases.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig027}\n\\mycaption[The number of ternary (\\textbf{a}) sulfide and (\\textbf{b}) selenide stoichiometries\nas a function of the product of the number of the binary stoichiometries of participating elements.]\n{The element with the most binary sulfide\/selenide stoichiometries (P\/Ti) is shown with red ``x'' symbols.\nAll other compounds are shown with blue circles.}\n\\label{fig:art130:tern_binproduct_stoichiometries}\n\\end{figure}\n\n\\subsection{Prevalence of unit cell sizes}\nIn Tables~\\ref{tab:art130:Number_of_atoms_in_Binaries_unit_cells} and\n\\ref{tab:art130:Number_of_atoms_in_ternary_unit_cells}, the number of atoms per unit cell in\nbinary and ternary compounds is shown for systems of up to 100 atoms in the unit cell.\nIn the binary oxides, there is higher prevalence for numbers that are multiples of\n4, 6, and also 12 ---\nfor example --- 12(102), 24(58), 80(47) and 72(20). In addition, 5(24) and a few of its multiples are also common.\nPrime numbers of atoms per unit cell above 10 are very rare ---\n11(2), 19(3), 29(1), 31(2), 67(1) and all the rest do not appear at all.\nIn the ternary oxides, we see a similar behavior: a high prevalence for numbers that are multiple of\n4, 6 and 12 --- for example --- 12 (119), 18 (140), 24 (465), 30(106), 72(102), 80(83), 88(178), 96(51).\nPrime numbers, between 10 to 20 do appear --- 11(15), 13(30), 17(6), 19(15), but those\nabove 20 are very rare.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the binary compounds (1\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n1 & 0 & 0 & 0\\\\\n2 & 7 & 12 & 8\\\\\n3 & 7 & 15 & 13\\\\\n4 & 24 & 20 & 22\\\\\n5 & 24 & 1 & 5\\\\\n6 & 60 & 48 & 24\\\\\n7 & 0 & 3 & 2\\\\\n8 & 63 & 70 & 60\\\\\n9 & 10 & 1 & 1\\\\\n10 & 15 & 9 & 6\\\\\n11 & 2 & 1 & 2\\\\\n12 & 102 & 63 & 50\\\\\n13 & 0 & 1 & 0\\\\\n14 & 19 & 11 & 9\\\\\n15 & 2 & 1 & 0\\\\\n16 & 23 & 22 & 11\\\\\n17 & 0 & 0 & 1\\\\\n18 & 12 & 6 & 5\\\\\n19 & 3 & 0 & 0\\\\\n20 & 37 & 38 & 20\\\\\n21 & 0 & 2 & 0\\\\\n22 & 8 & 2 & 3\\\\\n23 & 0 & 0 & 0\\\\\n24 & 58 & 39 & 14\\\\\n25 & 1 & 0 & 0\\\\\n26 & 2 & 1 & 2\\\\\n27 & 0 & 1 & 0\\\\\n28 & 27 & 24 & 18\\\\\n29 & 1 & 0 & 0\\\\\n30 & 17 & 10 & 2\\\\\n31 & 2 & 0 & 0\\\\\n32 & 18 & 14 & 10\\\\\n33 & 0 & 0 & 0\\\\\n34 & 2 & 1 & 0\\\\\n35 & 0 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_Binaries_unit_cells}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the binary compounds continued (2\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n36 & 25 & 8 & 4\\\\\n37 & 0 & 0 & 0\\\\\n38 & 4 & 0 & 0\\\\\n39 & 0 & 0 & 1\\\\\n40 & 15 & 6 & 2\\\\\n41 & 0 & 0 & 0\\\\\n42 & 0 & 2 & 0\\\\\n43 & 0 & 0 & 0\\\\\n44 & 5 & 5 & 1\\\\\n45 & 0 & 2 & 3\\\\\n46 & 2 & 1 & 0\\\\\n47 & 0 & 0 & 0\\\\\n48 & 27 & 10 & 3\\\\\n49 & 0 & 0 & 0\\\\\n50 & 0 & 0 & 0\\\\\n51 & 0 & 0 & 0\\\\\n52 & 3 & 3 & 2\\\\\n53 & 0 & 0 & 0\\\\\n54 & 2 & 0 & 0\\\\\n55 & 0 & 0 & 0\\\\\n56 & 9 & 9 & 2\\\\\n57 & 0 & 0 & 0\\\\\n58 & 0 & 2 & 0\\\\\n59 & 0 & 0 & 0\\\\\n60 & 5 & 1 & 0\\\\\n61 & 0 & 0 & 0\\\\\n62 & 1 & 0 & 0\\\\\n63 & 0 & 0 & 0\\\\\n64 & 2 & 4 & 1\\\\\n65 & 0 & 0 & 0\\\\\n66 & 0 & 0 & 0\\\\\n67 & 1 & 0 & 0\\\\\n68 & 7 & 3 & 2\\\\\n69 & 0 & 0 & 0\\\\\n70 & 0 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_Binaries_unit_cells_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the binary compounds continued (3\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n71 & 0 & 0 & 0\\\\\n72 & 20 & 2 & 1\\\\\n73 & 0 & 0 & 0\\\\\n74 & 0 & 1 & 0\\\\\n75 & 0 & 0 & 0\\\\\n76 & 3 & 2 & 0\\\\\n77 & 0 & 0 & 0\\\\\n78 & 0 & 0 & 0\\\\\n79 & 0 & 0 & 0\\\\\n80 & 47 & 6 & 12\\\\\n81 & 0 & 0 & 0\\\\\n82 & 2 & 0 & 0\\\\\n83 & 0 & 0 & 0\\\\\n84 & 2 & 0 & 0\\\\\n85 & 0 & 0 & 0\\\\\n86 & 0 & 0 & 0\\\\\n87 & 0 & 0 & 0\\\\\n88 & 1 & 3 & 2\\\\\n89 & 0 & 0 & 0\\\\\n90 & 0 & 0 & 2\\\\\n91 & 0 & 0 & 0\\\\\n92 & 1 & 0 & 0\\\\\n93 & 0 & 0 & 0\\\\\n94 & 1 & 0 & 0\\\\\n95 & 0 & 0 & 0\\\\\n96 & 22 & 1 & 0\\\\\n97 & 0 & 0 & 0\\\\\n98 & 1 & 0 & 0\\\\\n99 & 0 & 0 & 0\\\\\n100 & 0 & 0 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_Binaries_unit_cells_3}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the ternary compounds (1\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n1 & 0 & 0 & 0\\\\\n2 & 0 & 0 & 0\\\\\n3 & 1 & 1 & 2\\\\\n4 & 81 & 112 & 64\\\\\n5 & 173 & 36 & 16\\\\\n6 & 62 & 35 & 23\\\\\n7 & 10 & 29 & 16\\\\\n8 & 64 & 48 & 18\\\\\n9 & 38 & 8 & 4\\\\\n10 & 186 & 33 & 35\\\\\n11 & 15 & 2 & 0\\\\\n12 & 119 & 104 & 76\\\\\n13 & 30 & 5 & 1\\\\\n14 & 116 & 44 & 60\\\\\n15 & 12 & 40 & 30\\\\\n16 & 143 & 106 & 69\\\\\n17 & 6 & 3 & 0\\\\\n18 & 140 & 31 & 26\\\\\n19 & 15 & 0 & 0\\\\\n20 & 363 & 179 & 88\\\\\n21 & 10 & 1 & 0\\\\\n22 & 142 & 25 & 20\\\\\n23 & 1 & 0 & 1\\\\\n24 & 465 & 146 & 57\\\\\n25 & 7 & 0 & 0\\\\\n26 & 65 & 26 & 14\\\\\n27 & 16 & 1 & 0\\\\\n28 & 287 & 190 & 130\\\\\n29 & 1 & 0 & 0\\\\\n30 & 106 & 22 & 12\\\\\n31 & 0 & 0 & 0\\\\\n32 & 181 & 96 & 67\\\\\n33 & 4 & 0 & 0\\\\\n34 & 38 & 16 & 8\\\\\n35 & 0 & 1 & 0\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_ternary_unit_cells}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the ternary compounds continued (2\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n36 & 211 & 67 & 46\\\\\n37 & 5 & 0 & 0\\\\\n38 & 26 & 19 & 3\\\\\n39 & 1 & 2 & 0\\\\\n40 & 216 & 65 & 30\\\\\n41 & 2 & 0 & 0\\\\\n42 & 40 & 4 & 5\\\\\n43 & 3 & 0 & 0\\\\\n44 & 193 & 40 & 22\\\\\n45 & 6 & 2 & 2\\\\\n46 & 24 & 2 & 5\\\\\n47 & 1 & 0 & 0\\\\\n48 & 118 & 28 & 17\\\\\n49 & 6 & 0 & 0\\\\\n50 & 12 & 0 & 0\\\\\n51 & 0 & 0 & 0\\\\\n52 & 114 & 27 & 16\\\\\n53 & 0 & 0 & 0\\\\\n54 & 17 & 8 & 2\\\\\n55 & 1 & 0 & 0\\\\\n56 & 171 & 109 & 56\\\\\n57 & 6 & 0 & 0\\\\\n58 & 14 & 8 & 3\\\\\n59 & 1 & 0 & 0\\\\\n60 & 104 & 31 & 10\\\\\n61 & 1 & 0 & 0\\\\\n62 & 7 & 0 & 3\\\\\n63 & 5 & 0 & 0\\\\\n64 & 86 & 31 & 31\\\\\n65 & 0 & 0 & 0\\\\\n66 & 17 & 0 & 1\\\\\n67 & 0 & 0 & 0\\\\\n68 & 99 & 28 & 14\\\\\n69 & 0 & 0 & 0\\\\\n70 & 6 & 0 & 2\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_ternary_unit_cells_2}\n\\end{table}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption{Prevalence of unit cell sizes among the ternary compounds continued (3\/3).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nnumber of atoms & oxides & sulfides & selenides \\\\\n\\hline\n71 & 0 & 0 & 0\\\\\n72 & 102 & 48 & 39\\\\\n73 & 0 & 0 & 0\\\\\n74 & 2 & 21 & 3\\\\\n75 & 0 & 0 & 0\\\\\n76 & 48 & 6 & 5\\\\\n77 & 0 & 0 & 0\\\\\n78 & 8 & 1 & 0\\\\\n79 & 0 & 0 & 0\\\\\n80 & 83 & 8 & 8\\\\\n81 & 0 & 0 & 0\\\\\n82 & 3 & 1 & 0\\\\\n83 & 0 & 0 & 0\\\\\n84 & 30 & 17 & 7\\\\\n85 & 0 & 0 & 0\\\\\n86 & 7 & 0 & 0\\\\\n87 & 1 & 0 & 0\\\\\n88 & 178 & 12 & 20\\\\\n89 & 0 & 0 & 0\\\\\n90 & 9 & 2 & 1\\\\\n91 & 0 & 0 & 0\\\\\n92 & 20 & 8 & 6\\\\\n93 & 0 & 0 & 0\\\\\n94 & 3 & 0 & 0\\\\\n95 & 0 & 0 & 0\\\\\n96 & 51 & 23 & 6\\\\\n97 & 0 & 0 & 0\\\\\n98 & 1 & 1 & 2\\\\\n99 & 3 & 0 & 0\\\\\n100 & 14 & 3 & 2\\\\\n\\end{tabular}\n\\label{tab:art130:Number_of_atoms_in_ternary_unit_cells_3}\n\\end{table}\n\n\\clearpage\n\n\\subsection{Additional Mendeleev plots}\nThe Mendeleev map for the 1:1:2 stoichiometry are shown in~\\ref{fig:art130:mend_211_stoichiometries}.\nThe maps of the sulfides and selenides cover nearly identical regions, while that of the oxides\nincludes an additional row for hydrogen (Mendeleev number 103).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig028}\n\\mycaption[Comparison of Mendeleev maps for the 211 (\\textbf{a}) oxide, (\\textbf{b}) sulfide and (\\textbf{c}) selenide stoichiometries.]\n{The number in parenthesis is the number of compounds for this structure type, for ``Other'',\nit refers to the total number of compounds with this stoichiometry.}\n\\label{fig:art130:mend_211_stoichiometries}\n\\end{figure}\n\n\\subsection{Summary}\nWe present a comprehensive analysis of the statistics of the binary\nand ternary compounds of oxygen, sulfur and selenium. This analysis and the visualization tools presented here are\nvaluable to finding trends as well as exceptions and peculiar phenomena.\n\nOxygen has a higher electronegativity (3.44) than sulfur\n(2.58) and selenium (2.55), which are similar to each other.\nTherefore, one can expect that oxygen will form compounds with a stronger ionic character.\nOxygen is 1000 times more abundant than sulfur, and more than $10^6$ times than selenium~\\cite{wedepohl1995composition},\nhowever, it has less than two times the number of binary compounds compared to sulfur and $2.5$ that of selenium.\nHence, the abundance of those elements plays a little role in the relative numbers of their known compounds.\nThese important differences are reflected in our analysis by\nthe significantly larger fraction of oxygen rich compounds compared to\nthose that are sulfur or selenium rich.\nStructure type classification also shows that there is little overlap between the oxygen\nstructure types to sulfur or selenium structure types, while\nsulfur and selenium present a much higher overlap. The gaps in these overlaps, especially between\nthe sulfides and selenides, indicate that favorable candidates for new compounds\nmay be obtained by simple element substitution in the corresponding structures.\nIn particular, structures that are significantly more common in one family,\nsuch as KrF$_{2}$ in the oxides, may be good candidates for new compounds in another.\nComparison of these three 6A elements binary and ternary\ncompounds shows significant differences but also some similarities in the symmetry\ndistributions among the various Bravais lattices and their\ncorresponding space groups. In particular, the majority of structure types in all three families have a few or single\ncompound realizations. This prevalence of unique structure types suggests a ripe field for identification of currently unknown compounds,\nby substitution of elements of similar chemical characteristics.\nIn addition, the analysis of the distribution of known compounds among symmetry space groups and,\nin particular, their apparent concentration in specific hot spots of this\nsymmetry space may be serve as a useful insight for searches of potential new compounds.\n\nAn important observation is the existence of different gaps (missing stoichiometries) in the stoichiometry distribution of the oxide\nbinary compounds compared to the sulfides and selenides (Figures~\\ref{fig:triangle_stoichiometries} and \\ref{fig:art130:specific_triangle_stoichiometries},\nand Tables~\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries}-\\ref{tab:art130:Prevalence_of_Elements_in_Binaries_Stoichiometries_3}).\nStoichiometries such as 5:7 appear in the oxides but are missing in the sulfides and selenides.\nMore rare are non-overlapping gaps between the selenides and sulfides, \\nobreak\\mbox{\\it e.g.}, 6:1 and 5:7.\nThese should be prime candidates for new compounds by element substitution between the two families.\nFuture work would be directed at exploiting these discrepancies to search for new compounds within different subsets of those compound families.\n\nSpecific elements tend to present very different stoichiometry distributions, for example, silicon forms only\none oxide stoichiometry (SiO$_2$) while transition metals such as titanium and vanadium present\n14 and 18 different stoichiometries respectively.\nThese differences clearly reflect the different chemistry of those elements, while the large number of reported\nSiO$_2$ structures might reflect research bias into silicon compounds.\n\nAnother important finding is that there is an inverse correlation between the number of ternary stoichiometries\nto the product of binary stoichiometries of participating elements.\nThis can be caused by the fact that there are too many binary phases and hence it becomes\ndifficult to create a stable ternary that competes with all of them.\n\nA Mendeleev analysis of the common structure types of these\nfamilies shows accumulation of different structures at\nwell defined regions of their respective maps, similar to the well-known Pettifor maps of binary structure types.\nFurthermore, at least for some of the stoichiometries, similarity of the maps for a\ngiven stoichiometry is demonstrated across all three elements.\nThese maps should therefore prove useful for predictive purposes regarding the existence\nof yet unknown compounds of the corresponding structure types.\nFuture work will be directed at exploiting identified non-overlapping gaps in the\nMendeleev maps for a directed search of new compounds in these families.\nComplementary properties (\\nobreak\\mbox{\\it e.g.}, partial charges, bond analysis, electronic properties)\nshould be incorporated in the analysis to reveal additional insights of the aforementioned trends among the three elements.\n\\clearpage\n\\section{AFLOW Standard for High-Throughput Materials Science Calculations}\n\\label{sec:art104}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art104},\nwhich was awarded with Comput. Mater. Sci. Editor's Choice.\n\n\\subsection{Introduction} \\label{subsec:art104:intro}\n\nThe emergence of computational materials science over the last two decades has been inextricably linked to the\ndevelopment of complex quantum-mechanical codes that enable accurate evaluation of the electronic and\nthermodynamic properties of a wide range of materials. The continued advancement of this field entails the\nconstruction of large open databases of materials properties that can be easily reproduced and extended.\nOne obstacle to the reproducibility of the data is the unavoidable complexity of the codes used to obtain\nit. Published data usually includes basic information about the underlying calculations that allows rough\nreproduction. However, exact duplication depends on many details, that are seldom reported, and is therefore\ndifficult to achieve.\n\nThese difficulties might limit the utility of the databases currently being created by high-throughput frameworks,\nsuch as {\\small AFLOW}~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI} and the Materials Project~\\cite{APL_Mater_Jain2013,CMS_Ong2012b}.\nFor maximal impact, the data stored in these repositories must be generated and represented in a consistent and robust manner,\nand shared through standardized calculation and communication protocols. Following these guidelines would promote\noptimal use of the results generated by the entire community.\n\nThe {\\small AFLOW}\\ (Automatic FLOW) code is a framework for high-throughput computational materials\ndiscovery~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI, aflowlib.org},\nusing separate {\\small DFT}\\ packages to calculate electronic\nstructure and optimize the atomic geometry. The {\\small AFLOW}\\ framework works with the\n{\\small VASP}~\\cite{vasp_prb1996}\\ {\\small DFT}\\ package,\nand integration with the \\textsc{Quantum {\\small ESPRESSO}}\\ software~\\cite{quantum_espresso_2009}\nis currently in progress.\nThe {\\small AFLOW}\\ framework includes\npreprocessing functions for generating input files for the {\\small DFT}\\ package; obtaining the initial geometric structures\nby extracting the relevant data from crystallographic information files or by generating them using inbuilt prototype\ndatabases, and then transforming them into standard forms which are easiest to calculate. It then runs and monitors\nthe {\\small DFT}\\ calculations automatically, detecting and responding to calculation failures, whether they are due to insufficient\nhardware resources or to runtime errors of the {\\small DFT}\\ calculation itself. Finally, {\\small AFLOW}\\ contains postprocessing\nroutines to extract specific properties from the results of one or more of the {\\small DFT}\\ calculations, such as the band\nstructure or thermal properties~\\cite{curtarolo:art96}.\n\nThe {\\sf \\AFLOW.org}\\ repository~\\cite{aflowlibPAPER, aflowAPI, aflowlib.org} was built according to these principles of consistency and reproducibility,\nand the data it contains can be easily accessed through a representational state transfer application programming\ninterface ({\\small REST-API})~\\cite{aflowAPI}. In this study we present a detailed description of the {\\small AFLOW}\\ standard\nfor high-throughput (HT) materials science calculations by which the data in this repository was created.\n\n\\subsection{AFLOW calculation types} \\label{subsec:art104:AFLOWtypes}\nThe {\\sf \\AFLOW.org}\\ repository~\\cite{aflowlibPAPER} is divided into databases containing calculated\nproperties of over 625,000 materials:\nthe Binary Alloy Project, the Electronic Structure database, the Heusler database, and the Elements database.\nThese are freely accessible online via the {\\sf \\AFLOW.org}~\\cite{aflowlib.org}, as well as through the\n{\\small API}~\\cite{aflowAPI}. The Electronic Structure database consists of entries found in the Inorganic Crystal\nStructures Database, {\\small ICSD}~\\cite{ICSD, ICSD3}, and will thus be referred to as ``{\\small ICSD}'' throughout this publication.\nThe Heusler database consists of ternary compounds, primarily based on the Heusler structure but with other\nstructure types now being added.\n\nThe high-throughput construction of these materials databases relies on a pre-defined set of standard {\\textit{calculation\ntypes}}. These are designed to accommodate the interest in various properties of a given material (\\nobreak\\mbox{\\it e.g.}, the ground\nstate ionic configuration, thermodynamic quantities, electronic and\nmagnetic properties), the program flow of the HT framework that\nenvelopes the {\\small DFT}\\ portions of the calculations, as well as the practical\nneed for computational robustness. The {\\small AFLOW}\\ standard thus deals with the parameters involved in the following\ncalculation types:\n\n\\begin{enumerate}\n \\item {{\\verb!RELAX!}.} Geometry optimizations using algorithms implemented within the {\\small DFT}\\ package. This calculation\n type is concerned with obtaining the ionic configuration and cell\n shape and volume that correspond to a minimum in the\n total energy. It consists of two sequential relaxation steps. The starting point for the first step, {\\verb!RELAX1!},\n can be an entry taken from an external source, such as a library of alloy\n prototypes~\\cite{Massalski, curtarolo:calphad_2005_monster}, the {\\small ICSD}\\ database, or the Pauling\n File~\\cite{PaulingFile}. These initial entries are preprocessed by\n {\\small AFLOW}, and cast into a unit cell that is most convenient\n for calculation, usually the standard primitive cell, in the format appropriate for the {\\small DFT}\\ package in use. The second step, {\\verb!RELAX2!},\n uses the final ionic positions from the first step as its starting point, and serves as a type of annealing step.\n This is used for jumping out of possible local minima resulting from wavefunction artifacts.\n \\item {{\\verb!STATIC!}.} A single-point energy calculation. The starting point is the set of final ionic positions,\n as produced by the {\\verb!RELAX2!} step. The outcome of this calculation is used in the determination of most\n of the thermodynamic and electronic properties included in the various {\\sf \\AFLOW.org}\\ database.\n It therefore applies a more demanding set of parameters than those used on the {\\verb!RELAX!}\n set of runs.\n \\item {{\\verb!BANDS!}.} Electronic band structure generation. The converged {\\verb!STATIC!} charge\n density and ionic positions are used as the starting points, and the wavefunctions are reoptimized along standardized\n high symmetry lines connecting special {\\bf k}-points in the irreducible Brillouin zone (IBZ)~\\cite{aflowBZ}.\n\\end{enumerate}\n\nThese calculation types are performed in the order shown above (\\nobreak\\mbox{\\it i.e.}, {\\verb!RELAX1!} $\\rightarrow$ {\\verb!RELAX2!}\n$\\rightarrow$ {\\verb!STATIC!} $\\rightarrow$ {\\verb!BANDS!}) on all materials found in the Elements,\n{\\small ICSD}, and Heusler databases. Those found in the Binary Alloy database contain data produced only by the two\n{\\verb!RELAX!} calculations.\nSets of these calculation types can be combined to describe more complex\nphenomena than can be obtained from a single calculation. For\nexample, sets of {\\verb!RELAX!} and {\\verb!STATIC!} calculations for different cell\nvolumes and\/or atomic configurations are used to calculate\nthermal and mechanical properties by the {Automatic Gibbs Library}, {\\small AGL}~\\cite{curtarolo:art96},\nand {Automatic Phonon Library}, {\\small APL}~\\cite{aflowPAPER}, methods\nimplemented within the {\\small AFLOW}\\ framework.\nIn the following, we describe the parameter sets used to address the\nparticular challenges of the calculations included in each {\\sf \\AFLOW.org}\\ repository.\n\n\\subsection{The AFLOW Standard parameter set} \\label{subsec:art104:AFLOWstandard}\nThe standard parameters described in this work are classified according to the wide variety of tasks that a typical solid\nstate {\\small DFT}\\ calculation involves: Brillouin zone sampling, Fourier transform meshes, basis sets, potentials,\nself-interaction error (SIE) corrections, electron spin, algorithms guiding SCF convergence and ionic relaxation, and\noutput options.\n\nDue to the intrinsic complexity of the {\\small DFT}\\ codes it is impractical to\nspecify the full set of {\\small DFT}\\ calculation parameters within an HT framework. Therefore, the {\\small AFLOW}\\ standard\nadopts many, but not all, of the internal defaults set by the {\\small DFT}\\ software package. This is most notable in the description of the\nFourier transform meshes, which rely on a discretization scheme that depends on the applied basis and crystal\ngeometry for its specification. Those internal default settings are cast aside when\nerror corrections of failed {\\small DFT}\\ runs, an integral part of {\\small AFLOW}{}'s functionality, take place. The settings\ndescribed in this work are nevertheless prescribed as fully as is practicable, in the interest of providing as\nmuch information as possible to anyone interested in reproducing or building on our results.\n\n\\subsubsection{{\\bf k}-point sampling} \\label{subsubsec:art104:kpointgrid}\nTwo approaches are used when sampling the IBZ: the first consists of uniformly distributing a large number\nof {\\bf k}-points in the IBZ, while the second relies on the construction of paths connecting high symmetry (special)\n{\\bf k}-points in the IBZ. Within {\\small AFLOW}, the second sampling method corresponds to the {\\verb!BANDS!}\ncalculation type, whereas the other calculation types (non-{\\verb!BANDS!}) are performed using the first sampling\nmethod.\n\nSampling in non-{\\verb!BANDS!} calculations is obtained by defining and setting $N_{\\mathrm{KPPRA}}$, the number of\n{\\bf k}-points per atom. This quantity determines the total number of {\\bf k}-points in the IBZ,\ntaking into account the {\\bf k}-points density along each reciprocal lattice vector as well as the number of atoms\nin the simulation cell, via the relation:\n\\begin{equation} \\label{eq:art104:kppra}\n { N_{\\mathrm{KPPRA}} \\leq \\min \\left[ \\prod\\limits_{i=1}^3 N_i \\right] \\times N_{\\mathrm{a} } }\n\\end{equation}\n$N_{\\mathrm{a}}$ is the number of atoms in the cell, and the $N_{i}$ factors correspond to the number\nof sampling points along each reciprocal lattice vector,\n$\\vec{b_{i}}$, respectively. These factors define the grid resolution,\n${\\it \\delta} k{_i} {\\| \\vec{b{_i}} \\|}\/{N{_i} }$, which is made as uniform as possible\nunder the constraint of Equation~\\ref{eq:art104:kppra}. The {\\bf k}-point meshes are then\ngenerated within the Monkhorst-Pack scheme~\\cite{Monkhorst1976}, unless the material belongs to the\n{\\textit{hP}}, or {\\textit{hR}} Bravais lattices, in which case the hexagonal symmetry is preserved by centering the mesh\nat the $\\Gamma$-point.\n\nDefault $N_{\\mathrm{KPPRA}}$ values depend on the calculation type and the\ndatabase. The $N_{\\mathrm{KPPRA}}$ values used for the entries in the Elements\ndatabase are material specific and set manually due to convergence of\nthe total energy calculation. The defaults applied to the\n{\\verb!RELAX!} and {\\verb!STATIC!} calculations are summarized in\nTable~\\ref{tab:art104:kgridnonbands}.\nThese defaults ensure proper convergence of the calculations. They\nmay be too stringent for some cases but enable reliable\napplication within the HT framework, thus presenting a practicable\nbalance between accuracy and calculation cost.\n\n\\begin{table}[tp]\\centering\n\\cprotect\\mycaption{Default $N_{\\mathrm{KPPRA}}$ values used in non-{\\verb!BANDS!} calculations.}\n\\vspace{3mm}\n\\begin{tabular}{l | r r}\n database & {\\verb!STATIC!} & {\\verb!RELAX!} \\\\\n \\hline\n binary alloy & N.A. & 6000 \\\\\n Heusler & 10000 & 6000 \\\\\n {\\small ICSD}\\ & 10000 & 8000 \\\\\n\\end{tabular}\n\\label{tab:art104:kgridnonbands}\n\\end{table}\n\nFor {\\verb!BANDS!} calculations {\\small AFLOW}\\ generates Brillouin zone integration\npaths in the manner described in a previous\npublication~\\cite{aflowBZ}.\nThe {\\bf k}-point sampling density is the {\\textit{line\ndensity}} of {\\bf k}-points along each of the straight-line\nsegments of the path in the IBZ. The default setting\nof {\\small AFLOW}\\ is 128 {\\bf k}-points along each segment connecting high-symmetry {\\bf k}-points in\nthe IBZ for single element structures, and 20 {\\textit\n k}-points for compounds.\n\nThe occupancies at the Fermi edge in all non-{\\verb!RELAX!} type runs are handled via the tetrahedron method with\nBl{\\\"o}chl corrections~\\cite{Bloechl1994a}. This involves the $N_{\\mathrm{KPPRA}}$ parameter, as described above. In\n{\\verb!RELAX!} type calculations, where the determination of accurate forces is important, some type of\nsmearing must be performed. In cases where the material is assumed to be a metal, the\nMethfessel-Paxton approach~\\cite{Methfessel_prb_1989} is adopted, with a smearing width of 0.10~eV.\nGaussian smearing is used in all other types of materials, with a smearing width of 0.05~eV.\n\n\\subsubsection{Potentials and basis set} \\label{subsubsec:art104:pseudopot}\n\nThe interactions involving the valence electron shells are handled with the potentials provided with the {\\small DFT}\\ software\npackage. In {\\small VASP}, these include ultra-soft pseudopotentials (USPP)~\\cite{Vanderbilt, vasp_JPCM_1994} and\nprojector-augmented wavefunction ({\\small PAW}) potentials~\\cite{PAW,kresse_vasp_paw}, which are constructed according to the Local\nDensity Approximation ({\\small LDA})~\\cite{Ceperley_prl_1980, Perdew_prb_1981}, and the Generalized Gradient Approximation\n({\\small GGA}) PW91~\\cite{VASP_PW91_1,VASP_PW91_2} and {\\small PBE}~\\cite{PBE, PBE2} exchange-correlation (XC) functionals.\nThe {\\small ICSD}, Binary Alloy and Heusler databases built according to the {\\small AFLOW}\\ standard use the {\\small PBE}\\ functional combined with\nthe {\\small PAW}\\ potential as the default. The {\\small PBE}\\ functional is among the best studied {\\small GGA}\\ functionals used in crystalline systems, while the {\\small PAW}\\ potentials\nare preferred due to their advantages over the USPP methodology. Nevertheless, defaults have been defined for a number of potential \/ XC functional\ncombinations, and in the case of the Elements database, results are available for {\\small LDA}, {\\small GGA}-PW91 and {\\small GGA}-{\\small PBE}\\ functionals with both USPP and {\\small PAW}\\ potentials.\nAdditionally, there are a small number of entries in the {\\small ICSD}\\ and Binary Alloy databases (less than 1\\% of the total) which have been calculated with the {\\small GGA}-PW91\nfunctional using either the USPP or {\\small PAW}\\ potential. The exact combination of exchange-correlation functional and potential used for a specific entry\nin the {\\sf \\AFLOW.org}\\ database can always be determined by querying the keyword \\verb|dft_type| using the {\\small AFLOW}\\\n{\\small REST-API}~\\cite{aflowAPI}.\n\n{\\small DFT}\\ packages often provide more than one potential of each type per element. The {\\small AFLOW}\\\nstandardized lists of {\\small PAW}\\ and USPP potentials are presented in\nTables~\\ref{tab:art104:tab:pot_paw} and \\ref{tab:art104:pot_uspp}, respectively.\nThe ``Label'' column in these tables corresponds to the naming convention adopted\nby {\\small VASP}. The checksum of each file listed in the tables is included in the accompanying supplement\nfor verification purposes.\n\nEach potential provided with the {\\small VASP}\\ package has two recommended plane-wave kinetic energy cut-off ($E_{\\mathrm{cut}}$)\nvalues, the smaller of which ensures the reliability of a calculation to within a well-defined error. Additionally,\nmaterials with more than one element type will have two or more sets of recommended $E_{\\mathrm{cut}}$ values.\nIn the {\\small AFLOW}\\ standard, the applied $E_{\\mathrm{cut}}$ value is the largest found among the recommendations for all\nspecies involved in the calculation, increased by a factor of 1.4.\n\nIt is possible to evaluate the the non-local parts of the potentials in real space, rather than in the more computationally\nintensive reciprocal space. This approach is prone to aliasing errors, and requires the optimization of real-space\nprojectors if these are to be avoided. The real-space projection scheme is most appropriate for larger systems, \\nobreak\\mbox{\\it e.g.}, surfaces,\nand is therefore not used in the construction of the databases found in the {\\sf \\AFLOW.org}\\ repository.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Projector-Augmented Wavefunction ({\\small PAW}) potentials, parameterized for the {\\small LDA}, PW91, and {\\small PBE}\\\nfunctionals, included in the {\\small AFLOW}\\ standard.]\n{The {\\small PAW}-{\\small PBE}\\ combination is used as the default for {\\small ICSD}\\, Binary Alloy and Heusler databases.\n$\\dagger$: {\\small PBE}\\ potentials only.\n$\\ddagger$: {\\small LDA}\\ and PW91 potentials only.}\n\\vspace{3mm}\n{\\small\n\\begin{tabular}{l r | l r | l r}\n element & label & element & label & element & label \\\\\n \\hline\n H & H & Se & Se & Gd $\\ddagger$ & Gd\\_3 \\\\\n He & He & Br & Br & Tb & Tb\\_3 \\\\\n Li & Li\\_sv & Kr & Kr & Dy & Dy\\_3 \\\\\n Be & Be\\_sv & Rb & Rb\\_sv & Ho & Ho\\_3 \\\\\n B & B\\_h & Sr & Sr\\_sv & Er & Er\\_3 \\\\\n C & C & Y & Y\\_sv & Tm & Tm \\\\\n N & N & Zr & Zr\\_sv & Yb & Yb \\\\\n O & O & Nb & Nb\\_sv & Lu & Lu \\\\\n F & F & Mo & Mo\\_pv & Hf & Hf \\\\\n Ne & Ne & Tc & Tc\\_pv & Ta & Ta\\_pv \\\\\n Na & Na\\_pv & Ru & Ru\\_pv & W & W\\_pv \\\\\n Mg & Mg\\_pv & Rh & Rh\\_pv & Re & Re\\_pv \\\\\n Al & Al & Pd & Pd\\_pv & Os & Os\\_pv \\\\\n Si & Si & Ag & Ag & Ir & Ir \\\\\n P & P & Cd & Cd & Pt & Pt \\\\\n S & S & In & In\\_d & Au & Au \\\\\n Cl & Cl & Sn & Sn & Hg & Hg \\\\\n Ar & Ar & Sb & Sb & Tl & Tl\\_d \\\\\n K & K\\_sv & Te & Te & Pb & Pb\\_d \\\\\n Ca & Ca\\_sv & I & I & Bi & Bi\\_d \\\\\n Sc & Sc\\_sv & Xe & Xe & Po & Po \\\\\n Ti & Ti\\_sv & Cs & Cs\\_sv & At & At \\\\\n V & V\\_sv & Ba & Ba\\_sv & Rn & Rn \\\\\n Cr & Cr\\_pv & La & La & Fr & Fr \\\\\n Mn & Mn\\_pv & Ce & Ce & Ra & Ra \\\\\n Fe & Fe\\_pv & Pr & Pr & Ac & Ac \\\\\n Co & Co & Nd & Nd & Th & Th\\_s \\\\\n Ni & Ni\\_pv & Pm & Pm & Pa & Pa \\\\\n Cu & Cu\\_pv & Sm $\\dagger$ & Sm & U & U \\\\\n Zn & Zn & Sm $\\ddagger$ & Sm\\_3 & Np & Np\\_s \\\\\n Ga & Ga\\_h & Eu & Eu & Pu & Pu\\_s \\\\\n As & As & Gd $\\dagger$ & Gd & & \\\\\n\\end{tabular}}\n\\label{tab:art104:tab:pot_paw}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Ultra-Soft Pseudopotentials (USPP), parameterized for\nthe {\\small LDA}\\ and PW91 functionals, included in the {\\small AFLOW}\\ standard.}\n\\vspace{3mm}\n{\\small\n\\begin{tabular}{l r | l r | l r}\n element & label & element & label & element & label \\\\\n \\hline\n H & H\\_soft & As & As & Tb & Tb\\_3 \\\\\n He & He & Se & Se & Dy & Dy\\_3 \\\\\n Li & Li\\_pv & Br & Br & Ho & Ho\\_3 \\\\\n Be & Be & Kr & Kr & Er & Er\\_3 \\\\\n B & B & Rb & Rb\\_pv & Tm & Tm \\\\\n C & C & Sr & Sr\\_pv & Yb & Yb \\\\\n N & N & Y & Y\\_pv & Lu & Lu \\\\\n O & O & Zr & Zr\\_pv & Hf & Hf \\\\\n F & F & Nb & Nb\\_pv & Ta & Ta \\\\\n Ne & Ne & Mo & Mo\\_pv & W & W \\\\\n Na & Na\\_pv & Tc & Tc & Re & Re \\\\\n Mg & Mg\\_pv & Ru & Ru & Os & Os \\\\\n Al & Al & Rh & Rh & Ir & Ir \\\\\n Si & Si & Pd & Pd & Pt & Pt \\\\\n P & P & Ag & Ag & Au & Au \\\\\n S & S & Cd & Cd & Hg & Hg \\\\\n Cl & Cl & In & In\\_d & Tl & Tl\\_d \\\\\n Ar & Ar & Sn & Sn & Pb & Pb \\\\\n K & K\\_pv & Sb & Sb & Bi & Bi \\\\\n Ca & Ca\\_pv & Te & Te & Po & Po \\\\\n Sc & Sc\\_pv & I & I & At & At \\\\\n Ti & Ti\\_pv & Xe & Xe & Rn & Rn \\\\\n V & V\\_pv & Cs & Cs\\_pv & Fr & Fr \\\\\n Cr & Cr & Ba & Ba\\_pv & Ra & Ra \\\\\n Mn & Mn & La & La & Ac & Ac \\\\\n Fe & Fe & Ce & Ce & Th & Th\\_s \\\\\n Co & Co & Pr & Pr & Pa & Pa \\\\\n Ni & Ni & Nd & Nd & U & U \\\\\n Cu & Cu & Pm & Pm & Np & Np\\_s \\\\\n Zn & Zn & Sm & Sm\\_3 & Pu & Pu\\_s \\\\\n Ga & Ga\\_d & Eu & Eu & & \\\\\n Ge & Ge & Gd & Gd & & \\\\\n\\end{tabular}}\n\\label{tab:art104:pot_uspp}\n\\end{table}\n\n\\subsubsection{Fourier transform meshes} \\label{subsubsec:art104:fftmesh}\n\nAs mentioned previously, it is not practical to describe the precise default settings that are applied by the {\\small AFLOW}\\\nstandard in the specification of the Fourier transform meshes. We\nshall just note that they are defined in terms of the grid\nspacing along each of the reciprocal lattice vectors, $\\vec{b}_i$. These are obtained from the set of real space lattice\nvectors, $\\vec{a}_i$, via $ [\\vec{b}_1 \\vec{b}_2 \\vec{b}_3]^T = 2 \\pi [\\vec{a}_1 \\vec{a}_2 \\vec{a}_3]^{-1} $. A distance\nin reciprocal space is then defined by $d_i={\\|\\vec{b{_i}}\\|} \/n_i$, where the set of $n_i$ are the number\nof grid points along each reciprocal lattice vector, and where the total number of points in the simulation is\n$n_1 \\times n_2 \\times n_3$.\n\nThe {\\small VASP}\\ package relies primarily on the so-called {\\textit{dual grid technique}}, which consists of two overlapping\nmeshes with different coarseness. The least dense of the two is directly dependent on the applied plane-wave basis, $E_{\\mathrm{cut}}$,\nwhile the second is a finer mesh onto which the charge density is mapped. The {\\small AFLOW}\\ standard relies on placing\nsufficient points in the finer mesh such that wrap-around (``aliasing'') errors are avoided. In terms of the quantity $d_i$,\ndefined above, the finer grid is characterized by $d_i \\approx 0.10${\\textit{ \\r{A}}$^{-1}$}, while the coarse grid results\nin $d_i \\approx 0.15${\\textit{ \\r{A}}$^{-1}$}. These two values are approximate, as there is significant dispersion in\nthese quantities across the various databases.\n\n\\subsubsection{DFT$+U$ corrections} \\label{subsubsec:art104:Hubbard}\n\nExtended systems containing {\\textit d} and {\\textit f} block elements are often poorly represented within {\\small DFT}\\ due to\nthe well known self interaction error (SIE)~\\cite{Perdew_prb_1981}. The influence that the SIE has on the energy gap of\ninsulators has long been recognized, and several methods that account for it are available. These include the\n{\\textit{GW}} approximation~\\cite{Hedin_GW_1965}, the rotationally invariant approach introduced by\nDudarev~\\cite{Dudarev_dftu} and Liechtenstein~\\cite{Liechtenstein1995} (denoted here as {\\small DFT}$+U$), as well as the recently\ndeveloped ACBN0 pseudo-hybrid density functional~\\cite{curtarolo:art93}.\n\nThe {\\small DFT}$+U$ approach is currently the best suited for high-throughput investigations, and is therefore included in\nthe {\\small AFLOW}\\ standard for the entire {\\small ICSD}\\ database, and is also used for certain entries in the Heusler\ndatabase containing the elements O, S, Se, and F. It is not used for the Binary Alloy database.\nThis method has a significant dependence on parameters, as each atom is associated with\ntwo numbers, the screened Coulomb parameter, $U$, and the Stoner exchange parameter, $J$. These are usually reported\nas a single factor, combined via $U_{\\mathrm{eff}}=U-J$. The set of $U_{\\mathrm{eff}}$ values associated with the\n{\\textit d} block elements~\\cite{aflowBZ,curtarolo:art68} are presented in Table~\\ref{tab:art104:Ud}, to which the\nelements In and Sn have been added.\n\nA subset of the {\\textit f}-block elements can be found among the systems included in\nthe {\\sf \\AFLOW.org}\\ databases. We are not aware of the existence of a systematic search for the best set\nof $U$ and $J$ parameters for this region of the periodic table, so we have relied on an in-house\nparameterization~\\cite{aflowBZ} in the construction of the databases. The values used are reproduced\nin Table~\\ref{tab:art104:Uf}. Note that by construction the SIE correction must be applied to a pre-selected value of the\n$\\ell$-quantum number, and all elements listed in Table~\\ref{tab:art104:Ud} correspond to $\\ell=2$, while those\nfound in Table~\\ref{tab:art104:Uf} correspond to $\\ell=3$.\n\n\\begin{table}[tp]\\centering\n\\mycaption{$U_{\\mathrm{eff}}$ parameters applied to {\\textit d} orbitals.}\n\\vspace{3mm}\n \\begin{tabular}{l r | l r}\n element & $U_{\\mathrm{eff}}$ & element & $U_{\\mathrm{eff}}$ \\\\\n \\hline\n Sc~\\cite{ScUJ} & 2.9 & W~\\cite{NbUJ} & 2.2 \\\\\n Ti~\\cite{TiUJ} & 4.4 & Tc~\\cite{NbUJ} & 2.7 \\\\\n V~\\cite{VUJ} & 2.7 & Ru~\\cite{NbUJ} & 3.0 \\\\\n Cr~\\cite{CrUJ} & 3.5 & Rh~\\cite{NbUJ} & 3.3 \\\\\n Mn~\\cite{CrUJ} & 4.0 & Pd~\\cite{NbUJ} & 3.6 \\\\\n Fe~\\cite{FeUJ} & 4.6 & Ag~\\cite{AgUJ} & 5.8 \\\\\n Co~\\cite{VUJ} & 5.0 & Cd~\\cite{ZnUJ} & 2.1 \\\\\n Ni~\\cite{VUJ} & 5.1 & In~\\cite{ZnUJ} & 1.9 \\\\\n Cu~\\cite{CrUJ} & 4.0 & Sn~\\cite{SnUJ} & 3.5 \\\\\n Zn~\\cite{ZnUJ} & 7.5 & Ta~\\cite{NbUJ} & 2.0 \\\\\n Ga~\\cite{GaUJ} & 3.9 & Re~\\cite{NbUJ} & 2.4 \\\\\n Sn~\\cite{SnUJ} & 3.5 & Os~\\cite{NbUJ} & 2.6 \\\\\n Nb~\\cite{NbUJ} & 2.1 & Ir~\\cite{NbUJ} & 2.8 \\\\\n Mo~\\cite{NbUJ} & 2.4 & Pt~\\cite{NbUJ} & 3.0 \\\\\n Ta~\\cite{SnUJ} & 2.0 & Au & 4.0 \\\\\n\\end{tabular}\n\\label{tab:art104:Ud}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{$U$ and $J$ parameters applied to selected {\\textit f}-block elements.}\n\\vspace{3mm}\n \\begin{tabular}{l r r | l r r}\n element & {\\textit U} & {\\textit J} & element & {\\textit U} & {\\textit J} \\\\\n \\hline\n La~\\cite{LaUJ} & 8.1 & 0.6 & Dy~\\cite{DyUJ} & 5.6 & 0.0 \\\\\n Ce~\\cite{CeUJ} & 7.0 & 0.7 & Tm~\\cite{TmUJ} & 7.0 & 1.0 \\\\\n Pr~\\cite{PrUJ} & 6.5 & 1.0 & Yb~\\cite{YbUJ} & 7.0 & 0.67 \\\\\n Nd~\\cite{aflowSCINT}& 7.2 & 1.0 & Lu~\\cite{LaUJ} & 4.8 & 0.95 \\\\\n Sm~\\cite{aflowSCINT}& 7.4 & 1.0 & Th~\\cite{ThUJ} & 5.0 & 0.0 \\\\\n Eu~\\cite{aflowSCINT}& 6.4 & 1.0 & U~\\cite{UUJ} & 4.0 & 0.0 \\\\\n Gd~\\cite{GdUJ} & 6.7 & 0.1 & & & \\\\\n\\end{tabular}\n\\label{tab:art104:Uf}\n\\end{table}\n\n\\subsubsection{Spin polarization} \\label{subsubsec:art104:SpinPol}\n\nThe first of the two {\\verb!RELAX!} calculations is always performed in a collinear spin-polarized fashion.\nThe initial magnetic moments in this step are set to the number of atoms in the system, \\nobreak\\mbox{\\it e.g.}, 1.0 $\\mu B\/$atom. If\nthe magnetization resulting from the {\\verb!RELAX1!} step is found to be below 0.025 $\\mu B\/$atom, {\\small AFLOW}\\\neconomizes computational resources by turning spin polarization off in all ensuing calculations. Spin-orbit coupling\nis not used in the current {\\small AFLOW}\\ standard, since it is still\ntoo expensive to include in a HT framework.\n\n\\subsubsection{Calculation methods and convergence criteria} \\label{subsubsec:art104:convergence}\n\nTwo nested loops are involved in the {\\small DFT}\\ calculations used by {\\small AFLOW}\\ in the construction of the databases.\nThe inner loop contains routines that iteratively optimize the electronic degrees of freedom (EDOF), and features\na number of algorithms that are concerned with diagonalizing the Kohn-Sham (KS) Hamiltonian at each iteration.\nThe outer loop performs adjustments to the system geometry (ionic degrees of freedom, IDOF) until the forces acting\non the system are minimized.\n\nThe convergence condition for each loop has been defined in terms of an energy difference, $\\delta E$. If successive\nenergies resulting from the completion of a loop are denoted as $E_{i-1}$ and $E_i$, then\nconvergence is met when the condition $\\delta E \\geqslant E_i - E_{i-1}$ is fulfilled. Note that $E_i$ can either be\nthe electronic energy resulting from the inner loop, or the configurational energy resulting from the outer loop.\nThe electronic convergence criteria will be denoted as $\\delta E_{\\mathrm{elec}}$, and the ionic criteria as $\\delta E_{\\mathrm{ion}}$.\nThe {\\small AFLOW}\\ standard relies on $\\delta E_{\\mathrm{elec}} = 10^{-5}$~eV and $\\delta E_{\\mathrm{ion}} = 10^{-4}$~eV for entries in the\nElements database. All other databases include calculations performed with $\\delta E_{\\mathrm{elec}} = 10^{-3}$~eV and $\\delta E_{\\mathrm{ion}} = 10^{-2}$~eV.\n\nOptimizations of the EDOF depend on sets of parameters that fall under three general themes: initial guesses, diagonalization\nmethods, and charge mixing. The outer loop (optimizations of the IDOF) is concerned with the lattice vectors and the ionic\npositions, and is not as dependent on user input as the inner\nloops. These are described in the following paragraphs.\n\n\\boldsection{Electronic degrees of freedom.}\nThe first step in the process of optimizing the EDOF consists of choosing a trial charge density and a trial\nwavefunction. In the case of the non-{\\verb!BANDS!}-type calculations, the trial wavefunctions are initialized\nusing random numbers, while the trial charge density is obtained from the superposition\nof atomic charge densities. The {\\verb!BANDS!} calculations are not self-consistent, and thus do not feature\na charge density optimization. In these cases the charge density obtained from the previously performed {\\verb!STATIC!}\ncalculation is used in the generation of the starting wavefunctions.\n\nTwo iterative methods are used for diagonalizing the KS Hamiltonian: the Davidson blocked scheme\n(DBS)~\\cite{Liu_rep_1978,Davidson_1983}, and the preconditioned residual minimization method -- direct inversion in\nthe iterative subspace (RMM--DIIS)~\\cite{vasp_prb1996}. Of the two, DBS is known to be the slower and more stable option.\nAdditionally, the subspace rotation matrix is always optimized. These methods are applied in a manner that is dependent\non the calculation type:\n\n\\begin{enumerate}\n \\item {\\verb!RELAX!} calculations. Geometry optimizations contain at least one determination of the system\n forces. The initial determination consists of 5 initial DBS steps,\n followed by as many RMM-DIIS steps as needed to\n fulfill the $\\delta E_{\\mathrm{elec}}$ condition. Later determinations of\n system forces are performed by a similar\n sequence, but only a single DBS step is applied at the outset of the process. Across all\n databases the minimum of number of electronic iterations for {\\verb!RELAX!} calculations is 2. The maximum number is set\n to 120 for entries in the {\\small ICSD}, and 60 for all others.\n \\item non-{\\verb!RELAX!} calculations. In {\\verb!STATIC!} or\n {\\verb!BANDS!} calculations, the diagonalizations are always performed using RMM--DIIS. The minimum number of electronic\n iterations performed during non-{\\verb!RELAX!} calculations is 2, and the maximum is 120.\n\\end{enumerate}\n\nIf the number of iterations in the inner loop somehow exceed the limits listed above, the calculation breaks\nout of this loop, and the system forces and energy are determined. If the $\\delta E_{\\mathrm{ion}}$ convergence condition is\nnot met the calculation re-enters the inner loop, and proceeds normally.\n\nCharge mixing is performed via Pulay's method~\\cite{Pulay_cpl_1980}. The implementation of this charge mixing\napproach in the {\\small VASP}\\ package depends on a series of parameters, of which all but the maximum $\\ell$-quantum number\nhandled by the mixer have been left in their default state. This parameter is modified\nonly in systems included in the {\\small ICSD}\\ database which contain the elements\nlisted in Tables~\\ref{tab:art104:Ud} and \\ref{tab:art104:Uf}. In practical terms, the value applied in these cases is the maximum\n$\\ell$-quantum number found in the {\\small PAW}\\ potential, multiplied by 2.\n\n\\boldsection{Ionic degrees of freedom and lattice vectors.}\nThe {\\verb!RELAX!} calculation type contains determinations of the forces acting on the ions, as well as the full system\nstress tensor. The applied algorithm is the conjugate gradients (CG) approach~\\cite{press1992numerical}, which depends on\nthese quantities for the full optimization of the system geometry, \\nobreak\\mbox{\\it i.e.}, the ionic positions, the lattice vectors, as well\nas modifications of the cell volume. The implementation of CG in {\\small VASP}\\ requires minimal\nuser input, where the only independent parameter is the initial scaling factor which is always left at its\ndefault value. Convergence of the IDOF, as stated above, depends on the value for the $\\delta E_{\\mathrm{ion}}$ parameter,\nas applied across the various databases. The adopted $E_{\\mathrm{cut}}$ (see discussion on ``Potentials and basis set'',\nsection~\\ref{subsubsec:art104:pseudopot}) makes corrections for Pulay stresses unnecessary.\n\nForces acting on the ions and stress tensor are subjected to Harris-Foulkes~\\cite{Harris_prb_1985} corrections.\nMolecular dynamics based relaxations are not performed in the construction of the databases found in the\n{\\sf \\AFLOW.org}\\ repository, so any related settings are not applicable to this work.\n\n\\subsubsection{Output options} \\label{subsubsec:art104:output}\n\nThe reproduction of the results presented on {\\sf \\AFLOW.org}\\ also depends on a select few parameters that\ngovern the output of the {\\small DFT}\\ package. The density of states plots are generated from the {\\verb!STATIC!}\ncalculation. States are plotted with a range of -30~eV to 45~eV, and with a resolution of 5000 points. The band\nstructures are plotted according to the paths of {\\bf k}-points generated for a {\\verb!BANDS!}\ncalculation~\\cite{aflowBZ}. All bands found between -10~eV and 10~eV are included in the plots.\n\n\\subsection{Conclusion} \\label{subsec:art104:conclusion}\n\nThe {\\small AFLOW}\\ standard described here has been applied in the automated creation of the {\\sf \\AFLOW.org}\\ database of\nmaterial properties in a consistent and reproducible manner. The use of standardized parameter sets facilitates\nthe direct comparison of properties between different materials, so that specific trends can be identified to assist\nin the formulation of design rules for accelerated materials development. Following this {\\small AFLOW}\\ standard should\nallow materials science researchers to reproduce the results reported by the {\\small AFLOW}\\ consortium, as well as to\nextend on the database and make meaningful comparisons with their own results.\n\\clearpage\n\\section{Combining the AFLOW GIBBS and Elastic Libraries for Efficiently and Robustly Screening Thermomechanical Properties of Solids}\n\\label{sec:art115}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art115}.\n\n\\subsection{Introduction}\n\nCalculating the thermal and elastic properties of materials is\nimportant for predicting the thermodynamic and mechanical stability of structural\nphases~\\cite{Greaves_Poisson_NMat_2011, Poirier_Earth_Interior_2000,Mouhat_Elastic_PRB_2014, curtarolo:art106}\nand assessing their importance for a variety of applications.\nElastic and mechanical properties such as the shear and bulk moduli are important for predicting the\nhardness of materials~\\cite{Chen_hardness_Intermetallics_2011}, and thus their resistance to\nwear and distortion.\nThermal properties, such as specific heat capacity and lattice thermal conductivity, are important for applications including thermal barrier coatings,\nthermoelectrics~\\cite{zebarjadi_perspectives_2012, aflowKAPPA, Garrity_thermoelectrics_PRB_2016}, and heat sinks~\\cite{Watari_MRS_2001, Yeh_2002}.\n\n\\boldsection{Elasticity.} There are two main methods for calculating the elastic constants,\nbased on the response of either the stress tensor or the total energy to a set of\napplied strains~\\cite{Mehl_TB_Elastic_1996, Mehl_Elastic_1995, Golesorkhtabar_ElaStic_CPC_2013, curtarolo:art100, Silveira_Elastic_CPC_2008, Silveira_Elastic_CPC_2008, Silva_Elastic_PEPI_2007}.\nIn this study, we obtain the elastic constants from the calculated stress tensors for a set of independent deformations of the crystal lattice.\nThis method is implemented within the {\\small AFLOW}\\ framework for\ncomputational materials design\n\\cite{aflowPAPER,curtarolo:art49,monsterPGM}, where it is referred to as the\n\\underline{A}utomatic \\underline{E}lasticity \\underline{L}ibrary ({\\small AEL}).\n{A similar} implementation within the Materials\nProject~\\cite{curtarolo:art100} {allows} extensive\nscreening studies by combining data from these two large\nrepositories of computational materials data.\n\n\\boldsection{Thermal properties.} The determination of the thermal conductivity of materials from first principles requires either calculation of anharmonic\n\\underline{i}nteratomic \\underline{f}orce \\underline{c}onstants (IFCs) for use in the\n\\underline{B}oltzmann \\underline{T}ransport \\underline{E}quation (BTE)~\\cite{Broido2007, Wu_PRB_2012, ward_ab_2009, ward_intrinsic_2010,\nZhang_JACS_2012, Li_PRB_2012, Lindsay_PRL_2013, Lindsay_PRB_2013}, {or molecular dynamics} simulations in combination with\nthe Green-Kubo formula~\\cite{Green_JCP_1954,Kubo_JPSJ_1957}, both of\nwhich are highly demanding computationally even within multiscale approaches~\\cite{curtarolo:art12}.\nThese methods are unsuitable for rapid generation and screening of large databases of materials properties in order to identify trends\nand simple descriptors~\\cite{nmatHT}.\nPreviously, we have implemented the ``{\\small GIBBS}'' quasi-harmonic Debye model\n\\cite{Blanco_CPC_GIBBS_2004, Blanco_jmolstrthch_1996} within both the\n\\underline{A}utomatic \\underline{{\\small G}}{\\small IBBS} \\underline{L}ibrary ({\\small AGL})~\\cite{curtarolo:art96} of the\n{\\small AFLOW}~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI, curtarolo:art104,curtarolo:art110} and\nMaterials Project~\\cite{materialsproject.org,APL_Mater_Jain2013,CMS_Ong2012b} frameworks.\nThis approach does not require large supercell calculations since it\nrelies merely on first-principles calculations of the energy as a function of unit cell volume. It is thus\nmuch more tractable computationally and eminently suited to investigating the thermal properties of\nentire classes of materials in a highly-automated {fashion\nto identify} promising candidates for more in-depth experimental and computational analysis.\n\nThe data set of computed thermal and elastic properties\nproduced for this study is available in the {\\small AFLOW}\\\n\\cite{aflowlibPAPER} online data repository, either using the {\\small AFLOW}\\\n\\underline{RE}presentational \\underline{S}tate \\underline{T}ransfer \\underline{A}pplication \\underline{P}rogramming \\underline{I}nterface\n({\\small REST-API})~\\cite{aflowAPI} or via the {\\sf \\AFLOW.org}\\ web portal~\\cite{aflowlibPAPER,aflowBZ}.\n\n\\subsection{The AEL-AGL methodology}\n\nThe {\\small AEL}-{\\small AGL}\\ methodology combines elastic constants calculations, in\nthe Automatic Elasticity Library ({\\small AEL}), with the calculation of\nthermal properties within the Automatic {\\small GIBBS}\\ Library ({\\small AGL}\\\n\\cite{curtarolo:art96}) - ``{\\small GIBBS}''~\\cite{Blanco_CPC_GIBBS_2004} implementation of the Debye model.\nThis integrated software library includes automatic {error correction} to facilitate high-throughput\ncomputation of thermal and elastic materials properties within the\n{\\small AFLOW}\\ framework~\\cite{aflowPAPER, aflowlibPAPER, aflowAPI, curtarolo:art104,curtarolo:art53,curtarolo:art57,curtarolo:art63,curtarolo:art67,curtarolo:art54}.\nThe principal ingredients of the calculation are described in the following Sections.\n\n\\subsubsection{Elastic properties}\n\\label{subsubsec:art115:aelmethod}\n\nThe elastic constants are evaluated from the stress-strain relations\n\\begin{equation}\n\\left( \\begin{array}{l} s_{11} \\\\ s_{22} \\\\ s_{33} \\\\ s_{23} \\\\ s_{13} \\\\ s_{12} \\end{array} \\right) =\n\\left( \\begin{array}{l l l l l l} c_{11}\\ c_{12}\\ c_{13}\\ c_{14}\\ c_{15}\\ c_{16} \\\\\nc_{12}\\ c_{22}\\ c_{23}\\ c_{24}\\ c_{25}\\ c_{26} \\\\\nc_{13}\\ c_{23}\\ c_{33}\\ c_{34}\\ c_{35}\\ c_{36} \\\\\nc_{14}\\ c_{24}\\ c_{34}\\ c_{44}\\ c_{45}\\ c_{46} \\\\\nc_{15}\\ c_{25}\\ c_{35}\\ c_{45}\\ c_{55}\\ c_{56} \\\\\nc_{16}\\ c_{26}\\ c_{36}\\ c_{46}\\ c_{56}\\ c_{66} \\end{array} \\right)\n\\left( \\begin{array}{c} \\epsilon_{11} \\\\ \\epsilon_{22} \\\\ \\epsilon_{33} \\\\ 2\\epsilon_{23} \\\\ 2\\epsilon_{13} \\\\ 2\\epsilon_{12} \\end{array} \\right)\n\\end{equation}\nwith stress tensor elements $s_{ij}$ calculated\nfor a set of independent normal and shear strains $\\epsilon_{ij}$. The elements of the\nelastic stiffness tensor $c_{ij}$, written in the 6x6 Voigt notation using the mapping~\\cite{Poirier_Earth_Interior_2000}:\n$11 \\mapsto 1$, $22 \\mapsto 2$, $33 \\mapsto 3$, $23 \\mapsto 4$, $13 \\mapsto 5$, $12 \\mapsto 6$;\nare derived from polynomial fits for each independent strain, where the polynomial degree\nis automatically set to be less than the number of strains applied in each independent {direction to} avoid overfitting.\nThe elastic constants are then used to compute the bulk and shear\nmoduli, using either the Voigt approximation\n\\begin{equation}\n\\label{eq:art115:bulkmodvoigt}\nB_{{\\substack{\\scalebox{0.6}{Voigt}}}} = \\frac{1}{9} \\left[ (c_{11} + c_{22} + c_{33}) + 2 (c_{12} + c_{23} + c_{13}) \\right]\n\\end{equation}\nfor the bulk modulus, and\n\\begin{multline}\n\\label{eq:art115:shearmodvoigt}\nG_{{\\substack{\\scalebox{0.6}{Voigt}}}} = \\frac{1}{15} \\left[ (c_{11} + c_{22} + c_{33}) - (c_{12} + c_{23} + c_{13}) \\right]\n+ \\frac{1}{5} (c_{44} + c_{55} + c_{66})\n\\end{multline}\nfor the shear modulus; or the Reuss approximation, which uses the elements of the compliance tensor $s_{ij}$ (the inverse of the stiffness tensor),\nwhere the bulk modulus is given by\n\\begin{equation}\n\\label{eq:art115:bulkmodreuss}\n\\frac{1}{B_{{\\substack{\\scalebox{0.6}{Reuss}}}}} = (s_{11} + s_{22} + s_{33}) + 2 (s_{12} + s_{23} + s_{13})\n\\end{equation}\nand the shear modulus is\n\\begin{multline}\n\\label{eq:art115:shearmodreuss}\n\\frac{15}{G_{{\\substack{\\scalebox{0.6}{Reuss}}}}} = 4(s_{11} + s_{22} + s_{33}) - 4 (s_{12} + s_{23} + s_{13})\n+ 3 (s_{44} + s_{55} + s_{66}).\n\\end{multline}\nFor polycrystalline materials, the Voigt approximation {corresponds to assuming that the strain is uniform and that the stress is supported by the individual grains in parallel, giving} the upper bound on the elastic moduli{;} while the Reuss approximation {assumes that the stress is uniform and that the strain is the sum of the strains of the individual grains in series, giving} the lower bound {on the elastic moduli~\\cite{Poirier_Earth_Interior_2000}}.\nThe two approximations can be combined in the \\underline{V}oigt-\\underline{R}euss-\\underline{H}ill ({\\small VRH})~\\cite{Hill_elastic_average_1952} averages for the bulk modulus\n\\begin{equation}\n\\label{eq:art115:bulkmodvrh}\nB_{{\\substack{\\scalebox{0.6}{VRH}}}} = \\frac{B_{{\\substack{\\scalebox{0.6}{Voigt}}}} + B_{{\\substack{\\scalebox{0.6}{Reuss}}}}}{2};\n\\end{equation}\nand the shear modulus\n\\begin{equation}\n\\label{eq:art115:shearmodvrh}\nG_{{\\substack{\\scalebox{0.6}{VRH}}}} = \\frac{G_{{\\substack{\\scalebox{0.6}{Voigt}}}} + G_{{\\substack{\\scalebox{0.6}{Reuss}}}}}{2}.\n\\end{equation}\nThe Poisson ratio $\\sigma$ is then obtained by:\n\\begin{equation}\n\\label{eq:art115:Poissonratio}\n\\sigma = \\frac{3 B_{{\\substack{\\scalebox{0.6}{VRH}}}} - 2 G_{{\\substack{\\scalebox{0.6}{VRH}}}}}{6 B_{{\\substack{\\scalebox{0.6}{VRH}}}} + 2 G_{{\\substack{\\scalebox{0.6}{VRH}}}}}\n\\end{equation}\n\nThese elastic moduli can also be used to compute the speed of sound for the transverse and longitudinal waves, as well as the\naverage speed of sound in the material~\\cite{Poirier_Earth_Interior_2000}.\nThe speed of sound for the longitudinal waves is\n\\begin{equation}\n\\label{eq:art115:longitudinalsoundspeed}\nv_{\\substack{\\scalebox{0.6}{L}}} = \\left(\\frac{B + \\frac{4}{3}G}{\\rho}\\right)^{\\frac{1}{2}}\\!\\!\\!,\n\\end{equation}\nand for the transverse waves\n\\begin{equation}\n\\label{eq:art115:transversesoundspeed}\nv_{\\substack{\\scalebox{0.6}{T}}} = \\left(\\frac{G}{\\rho}\\right)^{\\frac{1}{2}}\\!\\!\\!,\n\\end{equation}\nwhere $\\rho$ is the mass density of the material. The average speed of\nsound is then evaluated by\n\\begin{equation}\n\\label{eq:art115:speedsound}\n{\\overline v} = \\left[\\frac{1}{3} \\left( \\frac{2}{v_{\\substack{\\scalebox{0.6}{T}}}^3} + \\frac{1}{v_{\\substack{\\scalebox{0.6}{L}}}^3} \\right) \\right]^{-\\frac{1}{3}}\\!\\!\\!.\n\\end{equation}\n\n\\subsubsection{The {\\small AGL}\\ quasi-harmonic Debye-Gr{\\\"u}neisen model}\n\nThe Debye temperature of a solid can be written as~\\cite{Poirier_Earth_Interior_2000}\n\\begin{equation}\n\\label{eq:art115:debyetempv}\n\\theta_{\\substack{\\scalebox{0.6}{D}}} = \\frac{\\hbar}{k_{\\substack{\\scalebox{0.6}{B}}}}\\left[\\frac{6 \\pi^2 n}{V}\\right]^{1\/3} \\!\\! {\\overline v},\n\\end{equation}\nwhere $n$ is the number of atoms in the cell, $V$ is its volume, and\n${\\overline v}$ is the average speed of sound of Equation~\\ref{eq:art115:speedsound}.\nIt can be shown by combining Equations~\\ref{eq:art115:Poissonratio}, \\ref{eq:art115:longitudinalsoundspeed}, \\ref{eq:art115:transversesoundspeed} and \\ref{eq:art115:speedsound}\nthat ${\\overline v}$ is equivalent to~\\cite{Poirier_Earth_Interior_2000}\n\\begin{equation}\n\\label{eq:art115:speedsoundB}\n{\\overline v} = \\sqrt{\\frac{B_{\\substack{\\scalebox{0.6}{S}}}}{\\rho}} f(\\sigma).\n\\end{equation}\nwhere $B_{\\substack{\\scalebox{0.6}{S}}}$ is the adiabatic bulk modulus, $\\rho$ is the density, and $f(\\sigma)$ is a function of the Poisson ratio $\\sigma$:\n\\begin{equation}\n\\label{eq:art115:fpoisson}\nf(\\sigma) = \\left\\{ 3 \\left[ 2 \\left( \\frac{2}{3} \\!\\cdot\\! \\frac{1 + \\sigma}{1 - 2 \\sigma} \\right)^{3\/2} \\!\\!\\!\\!\\!\\!\\!+ \\left( \\frac{1}{3} \\!\\cdot\\! \\frac{1 + \\sigma}{1 - \\sigma} \\right)^{3\/2} \\right]^{-1} \\right\\}^{\\frac{1}{3}}\\!\\!\\!\\!,\n\\end{equation}\nIn an earlier version of {\\small AGL}~\\cite{curtarolo:art96}, the Poisson ratio in Equation~\\ref{eq:art115:fpoisson} was assumed to have the {constant\nvalue $\\sigma = 0.25$ which} is the ratio for a Cauchy solid. This was found to be a reasonable approximation, producing\ngood correlations with experiment.\nThe {\\small AEL}\\ approach, Equation~\\ref{eq:art115:Poissonratio}, directly evaluates $\\sigma$ assuming only that it is independent of temperature and pressure.\nSubstituting Equation~\\ref{eq:art115:speedsoundB} into Equation~\\ref{eq:art115:debyetempv}, the\nDebye temperature is obtained as\n\\begin{equation}\n\\label{eq:art115:debyetemp}\n\\theta_{\\substack{\\scalebox{0.6}{D}}} = \\frac{\\hbar}{k_{\\substack{\\scalebox{0.6}{B}}}}[6 \\pi^2 V^{1\/2} n]^{1\/3} f(\\sigma) \\sqrt{\\frac{B_{\\substack{\\scalebox{0.6}{S}}}}{M}},\n\\end{equation}\nwhere $M$ is the mass of the unit cell.\nThe bulk modulus $B_{\\substack{\\scalebox{0.6}{S}}}$ is obtained from a set of DFT calculations for different volume cells, either by fitting the resulting $E_{\\substack{\\scalebox{0.6}{DFT}}}(V)$\ndata to a phenomenological equation of state or by taking the numerical second derivative of\na polynomial fit\n\\begin{eqnarray}\n\\label{eq:art115:bulkmod}\nB_{\\substack{\\scalebox{0.6}{S}}} (V) &\\approx& B_{\\mathrm{static}} (\\vec{x}) \\approx B_{\\mathrm{static}}(\\vec{x}_{\\substack{\\scalebox{0.6}{opt}}}(V)) \\\\ \\nonumber\n &=&V \\left( \\frac{\\partial^2 E(\\vec{x}_{\\substack{\\scalebox{0.6}{opt}}} (V))}{\\partial V^2} \\right) = V \\left( \\frac{\\partial^2 E(V)}{\\partial V^2} \\right).\n\\end{eqnarray}\nInserting Equation~\\ref{eq:art115:bulkmod} into Equation~\\ref{eq:art115:debyetemp} gives the Debye temperature as a function of volume $\\theta_{\\substack{\\scalebox{0.6}{D}}}(V)$, for each value of\npressure, $p$, and temperature, $T$.\n\nThe equilibrium volume at any particular $(p, T)$ point is obtained by minimizing the Gibbs free energy with\nrespect to volume. First, the vibrational Helmholtz free energy, $F_{\\substack{\\scalebox{0.6}{vib}}}(\\vec{x}; T)$, is calculated in the quasi-harmonic approximation\n\\begin{equation}\nF_{\\substack{\\scalebox{0.6}{vib}}}(\\vec{x}; T) \\!=\\!\\! \\int_0^{\\infty} \\!\\!\\left[\\frac{\\hbar \\omega}{2} \\!+\\! k_{\\substack{\\scalebox{0.6}{B}}} T\\ \\mathrm{log}\\!\\left(1\\!-\\!{\\mathrm e}^{- \\hbar \\omega \/ k_{\\substack{\\scalebox{0.6}{B}}} T}\\right)\\!\\right]\\!g(\\vec{x}; \\omega) d\\omega,\n\\end{equation}\nwhere $g(\\vec{x}; \\omega)$ is the phonon density of states and $\\vec{x}$ describes the geometrical configuration of the system. In the Debye-Gr{\\\"u}neisen model, $F_{\\substack{\\scalebox{0.6}{vib}}}$ can be expressed\nin terms of the Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{D}}}$\n\\begin{equation}\n\\label{eq:art115:helmholtzdebye}\nF_{\\substack{\\scalebox{0.6}{vib}}}(\\theta_{\\substack{\\scalebox{0.6}{D}}}; T) \\!=\\! n k_{\\substack{\\scalebox{0.6}{B}}} T \\!\\left[ \\frac{9}{8} \\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T} \\!+\\! 3\\ \\mathrm{log}\\!\\left(1 \\!-\\! {\\mathrm e}^{- \\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T}\\!\\right) \\!\\!-\\!\\! D\\left(\\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T}\\right)\\!\\!\\right],\n\\end{equation}\nwhere $D(\\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T)$ is the Debye integral\n\\begin{equation}\nD \\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\/T \\right) = 3 \\left( \\frac{T}{\\theta_{\\substack{\\scalebox{0.6}{D}}}} \\right)^3 \\int_0^{\\theta_{\\substack{\\scalebox{0.6}{D}}}\/T} \\frac{x^3}{e^x - 1} dx.\n\\end{equation}\nThe Gibbs free energy is calculated as\n\\begin{equation}\n\\label{eq:art115:gibbsdebye}\n{\\sf G}(V; p, T) = E_{\\substack{\\scalebox{0.6}{DFT}}}(V) + F_{\\substack{\\scalebox{0.6}{vib}}} (\\theta_{\\substack{\\scalebox{0.6}{D}}}(V); T) + pV,\n\\end{equation}\nand fitted by a polynomial of $V$. The equilibrium volume, $V_{\\mathrm{eq}}$, is that which minimizes ${\\sf G}(V; p, T)$.\n\nOnce $V_{\\mathrm{eq}}$ has been determined, $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ can be determined, and then other thermal properties including the Gr{\\\"u}neisen parameter and thermal\nconductivity can be calculated as described in the following Sections.\n\n\\subsubsection{Equations of state}\n\\label{subsubsec:art115:eqnsofstate}\n\nWithin {\\small AGL}\\, the bulk modulus can be determined either numerically from the second derivative of the polynomial fit of $E_{\\substack{\\scalebox{0.6}{DFT}}}(V)$,\nEquation~\\ref{eq:art115:bulkmod}, or by fitting the $(p,V)$ data to a\nphenomenological equation of state ({\\small EOS}). Three different analytic {\\small EOS}\\ have been implemented within\n{\\small AGL}: the Birch-Murnaghan {\\small EOS}~\\cite{Birch_Elastic_JAP_1938, Poirier_Earth_Interior_2000, Blanco_CPC_GIBBS_2004}; the Vinet {\\small EOS}~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004};\nand the Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez spinodal {\\small EOS}~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004}.\n\nThe Birch-Murnaghan {\\small EOS}\\ is\n\\begin{equation}\n\\label{eq:art115:birch}\n\\frac{p}{3 f (1 + 2 f)^\\frac{5}{2}} = \\sum_{i=0}^2 a_i f^i ,\n\\end{equation}\nwhere $p$ is the pressure, $a_i$ are polynomial coefficients, and $f$ is the ``compression'' given by\n\\begin{equation}\n\\label{eq:art115:birchf}\nf = \\frac{1}{2} \\left[\\left(\\frac{V}{V_0} \\right)^{-\\frac{2}{3}}- 1 \\right].\n\\end{equation}\nThe zero pressure bulk modulus is equal to the coefficient $a_0$.\n\nThe Vinet {\\small EOS}\\ is~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:vinet}\n\\log \\left[ \\frac{p x^2}{3 (1 - x)} \\right] = \\log B_0 + a (1 - x),\n\\end{equation}\nwhere $a$ and $\\log B_0$ are fitting parameters and\n\\begin{equation}\n\\label{eq:art115:vinetx}\nx = \\left(\\frac{V}{V_0} \\right)^{\\frac{1}{3}}\\!\\!\\!, \\\na = 3 (B_0' - 1) \/ 2.\n\\end{equation}\nThe isothermal bulk modulus $B_{\\substack{\\scalebox{0.6}{T}}}$ is given by~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:vinetBT}\nB_{\\substack{\\scalebox{0.6}{T}}} = - x^{-2} B_0 e^{a(1-x)} f(x),\n\\end{equation}\nwhere\n\\begin{equation*}\n\\label{eq:art115:vinetfx}\nf(x) = x - 2 - ax (1 - x).\n\\end{equation*}\n\nThe Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez spinodal equation of state has the form~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:bcn}\nV = V_{\\mathrm{sp}} \\exp \\left[ - \\left(\\frac{K^*}{1 - \\beta} \\right) (p - p_{\\mathrm{sp}})^{1 - \\beta} \\right],\n\\end{equation}\nwhere $K^*$, $p_{\\mathrm{sp}}$ and $\\beta$ are the fitting parameters, and $V_{\\mathrm{sp}} $ is given by\n\\begin{equation*}\nV_{\\mathrm{sp}} = V_0 \\exp \\left[ \\frac{\\beta}{\\left(1 - \\beta \\right) B_0'} \\right],\n\\end{equation*}\nwhere $B_0 = [K^*]^{-1} (-p_{\\mathrm{sp}})^{\\beta}$ and $B_0' = (-p_{\\mathrm{sp}})^{-1}\\beta B_0$.\nThe isothermal bulk modulus $B_{\\substack{\\scalebox{0.6}{T}}}$ is then given by~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:bcnBT}\nB_{\\substack{\\scalebox{0.6}{T}}} = \\frac{(p - p_{\\mathrm{sp}})^{\\beta}}{K^*}.\n\\end{equation}\n\n{Note that {\\small AGL}\\ uses $B_{\\substack{\\scalebox{0.6}{T}}}$ instead of $B_{\\substack{\\scalebox{0.6}{S}}}$ in Equation~\\ref{eq:art115:debyetemp} when one of these phenomenological {\\small EOS}\\\nis selected. $B_{\\substack{\\scalebox{0.6}{S}}}$ can then be calculated as\n\\begin{equation}\n\\label{eq:art115:BsBT}\nB_{\\substack{\\scalebox{0.6}{S}}} = B_{\\substack{\\scalebox{0.6}{T}}}(1 + \\alpha \\gamma T),\n\\end{equation}\nwhere $\\gamma$ is the Gr{\\\"u}neisen parameter (described in Section~\\ref{subsubsec:gruneisen} below), and $\\alpha$ is the thermal expansion\n\\begin{equation}\n\\label{eq:art115:thermal_expansion}\n\\alpha = \\frac{\\gamma C_{\\substack{\\scalebox{0.6}{V}}}}{B_{\\substack{\\scalebox{0.6}{T}}} V},\n\\end{equation}\nwhere $C_{\\substack{\\scalebox{0.6}{V}}}$ is the heat capacity at constant volume, given by\n\\begin{equation}\n \\label{eq:art115:heat_capacity}\nC_{\\substack{\\scalebox{0.6}{V}}} = 3 n k_{\\substack{\\scalebox{0.6}{B}}} \\left[4 D\\left(\\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T}\\right) - \\frac{3 \\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T}{\\exp(\\theta_{\\substack{\\scalebox{0.6}{D}}} \/ T) - 1} \\right].\n\\end{equation}\n}\n\n\\subsubsection{The Gr{\\\"u}neisen parameter}\n\\label{subsubsec:gruneisen}\n\nThe Gr{\\\"u}neisen parameter describes the variation of the thermal properties of a material with the unit cell size, and contains\ninformation about higher order phonon scattering which is important\nfor calculating the lattice thermal conductivity\n\\cite{Leibfried_formula_1954, slack, Morelli_Slack_2006, Madsen_PRB_2014, curtarolo:art96},\nand thermal expansion~\\cite{Poirier_Earth_Interior_2000, Blanco_CPC_GIBBS_2004, curtarolo:art114}.\nIt is defined as the phonon frequencies dependence on the unit cell volume\n\\begin{equation}\n\\label{eq:art115:gamma_micro}\n\\gamma_i = - \\frac{V}{\\omega_i} \\frac{\\partial \\omega_i}{\\partial V}.\n\\end{equation}\nDebye's theory assumes that the volume dependence of all mode\nfrequencies is the same as that of the cut-off Debye frequency, so the Gr{\\\"u}neisen parameter can be expressed in terms of $\\theta_{\\substack{\\scalebox{0.6}{D}}}$\n\\begin{equation}\n\\label{eq:art115:gruneisen_theta}\n\\gamma = - \\frac{\\partial \\ \\mathrm{log} (\\theta_{\\substack{\\scalebox{0.6}{D}}}(V))}{\\partial \\ \\mathrm{log} V}.\n\\end{equation}\n\nThis macroscopic definition of the Debye temperature is a weighted\naverage of Equation~\\ref{eq:art115:gamma_micro} with the heat capacities for each branch of the phonon spectrum\n\\begin{equation}\n\\gamma = \\frac{\\sum_i \\gamma_i C_{V, i}} {\\sum_i C_{V,i}}.\n\\end{equation}\n\n{\nWithin {\\small AGL}~\\cite{curtarolo:art96}, the Gr{\\\"u}neisen parameter can\nbe calculated in several different ways, including direct evaluation of Equation~\\ref{eq:art115:gruneisen_theta},\nby using the more stable Mie-Gr{\\\"u}neisen equation~\\cite{Poirier_Earth_Interior_2000},\n\\begin{equation}\n\\label{eq:art115:miegruneisen}\np - p_{T=0} = \\gamma \\frac{U_{\\substack{\\scalebox{0.6}{vib}}}}{V},\n\\end{equation}\nwhere $U_{\\substack{\\scalebox{0.6}{vib}}}$ is the vibrational internal energy~\\cite{Blanco_CPC_GIBBS_2004}\n\\begin{equation}\n\\label{eq:art115:Uvib}\nU_{\\substack{\\scalebox{0.6}{vib}}} = n k_{\\substack{\\scalebox{0.6}{B}}} T\\left[ \\frac{9}{8} \\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T} + 3D \\left( \\frac{\\theta_{\\substack{\\scalebox{0.6}{D}}}}{T} \\right)\\right].\n\\end{equation}\nThe ``Slater gamma'' expression~\\cite{Poirier_Earth_Interior_2000}\n\\begin{equation}\n\\label{eq:art115:slatergamma}\n\\gamma = - \\frac{1}{6} + \\frac{1}{2} \\frac{\\partial B_{\\substack{\\scalebox{0.6}{S}}}}{\\partial p}\n\\end{equation}\nis the default method in the automated workflow used\nfor the {\\small AFLOW}\\ database.\n}\n\n\\subsubsection{Thermal conductivity}\n\nIn the {\\small AGL}\\ framework, the thermal conductivity is calculated using the\nLeibfried-Schl{\\\"o}mann equation~\\cite{Leibfried_formula_1954, slack, Morelli_Slack_2006}\n\\begin{eqnarray}\n\\label{eq:art115:thermal_conductivity}\n\\kappa_{\\mathrm l} (\\theta_{\\mathrm{a}}) &=& \\frac{0.849 \\times 3 \\sqrt[3]{4}}{20 \\pi^3(1 - 0.514\\gamma_{\\mathrm{a}}^{-1} + 0.228\\gamma_{\\mathrm{a}}^{-2})}\n \\left( \\frac{k_{\\substack{\\scalebox{0.6}{B}}} \\theta_{\\mathrm{a}}}{\\hbar} \\right)^2 \\frac{k_{\\substack{\\scalebox{0.6}{B}}} m V^{\\frac{1}{3}}}{\\hbar \\gamma_{\\mathrm{a}}^2}.\n\\end{eqnarray}\nwhere $V$ is the volume of the unit cell and $m$ is the average atomic mass.\nIt should be noted that the Debye temperature and Gr{\\\"u}neisen parameter in this formula, $\\theta_{\\mathrm{a}}$ and $\\gamma_{\\mathrm{a}}$, are slightly\ndifferent {from} the traditional Debye temperature, $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, calculated in Equation~\\ref{eq:art115:debyetemp} and Gr{\\\"u}neisen parameter, $\\gamma$, obtained from\nEquation~\\ref{eq:art115:slatergamma}. Instead, $\\theta_{\\mathrm{a}}$ and $\\gamma_{\\mathrm{a}}$ are obtained by only considering the acoustic modes, based on the assumption that the optical\nphonon modes in crystals do not contribute to heat transport~\\cite{slack}. This $\\theta_{\\mathrm{a}}$ is referred to as the ``acoustic'' Debye temperature\n\\cite{slack, Morelli_Slack_2006}. It can be derived directly from the phonon DOS by integrating only over the acoustic modes~\\cite{slack,\nWee_Fornari_TiNiSn_JEM_2012}. Alternatively, it can be calculated from the traditional Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{D}}}$~\\cite{slack, Morelli_Slack_2006}\n\\begin{equation}\n\\label{eq:art115:acousticdebyetemp}\n\\theta_{\\mathrm{a}} = \\theta_{\\substack{\\scalebox{0.6}{D}}} n^{-\\frac{1}{3}}.\n\\end{equation}\n\n{There is no simple way to extract the ``acoustic'' Gr{\\\"u}neisen parameter from the traditional Gr{\\\"u}neisen parameter.}\nInstead, it must be calculated from Equation~\\ref{eq:art115:gamma_micro} for each phonon branch separately and summed over the acoustic branches~\\cite{curtarolo:art114, curtarolo:art119}.\nThis requires using the quasi-harmonic phonon approximation which involves calculating the full phonon spectrum for different\nvolumes~\\cite{Wee_Fornari_TiNiSn_JEM_2012, curtarolo:art114, curtarolo:art119}, and is therefore too computationally demanding to be used for\nhigh-throughput screening, particularly for large, low symmetry systems. Therefore, we use the approximation\n$\\gamma_{\\mathrm{a}} = \\gamma$ in the {\\small AEL}-{\\small AGL}\\ approach to {calculate} the thermal conductivity. The dependence of the expression in\nEquation~\\ref{eq:art115:thermal_conductivity} on $\\gamma$ is weak~\\cite{curtarolo:art96, Morelli_Slack_2006}, thus\nthe evaluation of $\\kappa_l$ using the traditional Gr{\\\"u}neisen parameter introduces just a small systematic error which is insignificant for\nscreening purposes~\\cite{curtarolo:art119}.\n\nThe thermal conductivity at temperatures other than $\\theta_{\\mathrm{a}}$ is estimated by~\\cite{slack, Morelli_Slack_2006, Madsen_PRB_2014}:\n\\begin{equation}\n\\label{eq:art115:kappa_temperature}\n\\kappa_{\\mathrm l} (T) = \\kappa_{\\mathrm l}(\\theta_{\\mathrm{a}}) \\frac{\\theta_{\\mathrm{a}}}{T}.\n\\end{equation}\n\n\\subsubsection{{\\small DFT}\\ calculations and workflow details}\n\nThe {\\small DFT}\\ calculations to obtain $E(V)$ and the strain tensors were performed using\nthe {\\small VASP}\\ software~\\cite{kresse_vasp} with projector-augmented-wave\npseudopotentials~\\cite{PAW} and the {\\small PBE}\\ parameterization of the\ngeneralized gradient approximation to the exchange-correlation\nfunctional~\\cite{PBE}, using the {parameters described} in the {\\small AFLOW}\\\nStandard~\\cite{curtarolo:art104}. The energies were calculated at zero\ntemperature and pressure, with spin polarization and without zero-point motion or lattice\nvibrations. The initial crystal structures were fully relaxed (cell\nvolume and shape and the basis atom coordinates inside the cell).\n\nFor the {\\small AEL}\\ calculations, 4 strains were applied in each independent lattice direction\n(two compressive and two expansive) with a maximum strain of 1\\% in each direction,\nfor a total of 24 configurations~\\cite{curtarolo:art100}. For cubic systems,\nthe crystal symmetry was used to reduce the number of required strain configurations\nto 8. For each configuration, two ionic positions {\\small AFLOW}\\ Standard {\\verb!RELAX!}~\\cite{curtarolo:art104}\ncalculations at fixed cell volume and shape were followed by a single {\\small AFLOW}\\ Standard {\\verb!STATIC!}~\\cite{curtarolo:art104}\ncalculation.\nThe elastic constants are then calculated by fitting the elements of stress tensor obtained for each independent strain.\nThe stress tensor from the zero-strain configuration\n(\\nobreak\\mbox{\\it i.e.}, the initial unstrained relaxed structure) can also be {included in the set of fitted strains}, although this was found to have negligible effect on the results.\nOnce these calculations are complete, it is verified that the eigenvalues of the stiffness tensor are all positive,\nthat the stiffness tensor obeys the appropriate symmetry rules for the lattice type~\\cite{Mouhat_Elastic_PRB_2014}, and\nthat the applied strain is still within the linear regime, using the method described by de Jong~\\nobreak\\mbox{\\it et al.}~\\cite{curtarolo:art100}.\nIf any of these conditions fail, the calculation is repeated with\nadjusted applied strain.\n\nThe {\\small AGL}\\ calculation of $E(V)$ is fitted to the energy at 28 different\nvolumes of the unit cell obtained by increasing or decreasing the relaxed lattice parameters in fractional\nincrements of 0.01, with a single {\\small AFLOW}\\ Standard\n{\\verb!STATIC!}~\\cite{curtarolo:art104} calculation at each volume.\nThe resulting $E(V)$ data is checked for convexity and to verify that the minimum energy is at the\ninitial volume (\\nobreak\\mbox{\\it i.e.}, at the properly relaxed cell size). If any of these\nconditions fail, the calculation is repeated with adjusted parameters,\n\\nobreak\\mbox{\\it e.g.}, increased k-point grid density.\n\n\\subsubsection{Correlation analysis}\n\nPearson and Spearman correlations {are used to}\nanalyze the results for entire sets of materials. The {Pearson coefficient} $r$ is a measure of the linear\ncorrelation between two variables, $X$ and $Y$. It is calculated by\n\\begin{equation}\n\\label{eq:art115:Pearson}\nr = \\frac{\\sum_{i=1}^{n} \\left(X_i - \\overline{X} \\right) \\left(Y_i - \\overline{Y} \\right) }{ \\sqrt{\\sum_{i=1}^{n} \\left(X_i - \\overline{X} \\right)^2} \\sqrt{\\sum_{i=1}^{n} \\left(Y_i - \\overline{Y} \\right)^2}},\n\\end{equation}\nwhere $\\overline{X}$ and $\\overline{Y}$ are the mean values of $X$ and $Y$.\n\nThe {Spearman coefficient} $\\rho$ is a measure of the monotonicity of the relation between two variables.\nThe raw values of the two variables $X_i$ and $Y_i$ are sorted in ascending order, and are assigned rank values $x_i$ and $y_i$ which\nare equal to their position in the sorted list. If there is more than one variable with the same value, the average of the position values\nare assigned to {all duplicate entries}. The correlation coefficient is then given by\n\\begin{equation}\n\\label{eq:art115:Spearman}\n\\rho = \\frac{\\sum_{i=1}^{n} \\left(x_i - \\overline{x} \\right) \\left(y_i - \\overline{y} \\right) }{ \\sqrt{\\sum_{i=1}^{n} \\left(x_i - \\overline{x} \\right)^2} \\sqrt{\\sum_{i=1}^{n} \\left(y_i - \\overline{y} \\right)^2}}.\n\\end{equation}\nIt is useful for determining how well the ranking order of the values of one variable predict the ranking order of the values of the other variable.\n\nThe discrepancy between the {\\small AEL}-{\\small AGL}\\ predictions and experiment is\nevaluated in terms normalized root-mean-square relative deviation\n\\begin{equation}\n\\label{eq:art115:RMSD}\n{\\mathrm{RMSrD}} = \\sqrt{\\frac{ \\sum_{i=1}^{n} \\left( \\frac{X_i - Y_i}{X_i} \\right)^2 }{N - 1}} ,\n\\end{equation}\n{In contrast} to the correlations described above, lower values of the {\\small RMSrD}\\ indicate better agreement with experiment. This measure is particularly useful for\ncomparing predictions of the same property using different\nmethodologies that may have very similar correlations with, but different\ndeviations from, the experimental results.\n\n\\subsection{Results}\n\nWe used the {\\small AEL}-{\\small AGL}\\ methodology to calculate the mechanical and thermal properties, including the bulk modulus,\nshear modulus, Poisson ratio, Debye temperature, Gr{\\\"u}neisen parameter and thermal conductivity for a set of 74 materials\nwith structures including diamond, zincblende, rocksalt, wurtzite, rhombohedral and body-centered tetragonal.\nThe results have been compared to experimental values (where available), and the correlations between the calculated and\nexperimental values were deduced.\nIn cases where multiple experimental values are present in the literature, we used the most recently reported\nvalue, unless otherwise specified.\n\nIn Section~\\ref{subsubsec:art115:aelmethod}, three different approximations for the bulk and shear moduli are described: Voigt (Equations~\\ref{eq:art115:bulkmodvoigt}, \\ref{eq:art115:shearmodvoigt}),\nReuss (Equations~\\ref{eq:art115:bulkmodreuss}, \\ref{eq:art115:shearmodreuss}), and the Voigt-Reuss-Hill ({\\small VRH}) average (Equations~\\ref{eq:art115:bulkmodvrh}, \\ref{eq:art115:shearmodvrh}).\nThese approximation{s give very similar values for the\nbulk modulus} for the set of materials included in this work, particularly those with cubic symmetry.\nTherefore only {$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$}\nis explicitly cited in the following listed results\n(the values obtained for all three approximations are available in the {\\small AFLOW}\\ database entries for\nthese materials). The values for the shear modulus in these three\napproximations exhibit larger variations, and are therefore all listed and compared to experiment.\nIn several cases, the experimental values of the bulk and shear moduli have been calculated\nfrom the measured elastic constants using Equations~\\ref{eq:art115:bulkmodvoigt} through \\ref{eq:art115:shearmodvrh}, and an experimental Poisson ratio $\\sigma^{\\mathrm{exp}}$\nwas calculated from these values using Equation~\\ref{eq:art115:Poissonratio}.\n\nAs described in Section~\\ref{subsubsec:art115:eqnsofstate}, the bulk modulus in {\\small AGL}\\ can be calculated from a polynomial fit of the $E(V)$ data as shown in Equation~\\ref{eq:art115:bulkmod},\nor by fitting the $E(V)$ data to one of three empirical equations\nof state: Birch-Murnaghan (Equation~\\ref{eq:art115:birch}), Vinet (Equation~\\ref{eq:art115:vinet}), and the Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez\n(Equation~\\ref{eq:art115:bcn}). We compare the results of these four methods, labeled $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$, $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$, and\n$B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$, respectively, with the experimental values $B^{\\mathrm{exp}}$ and those obtained from the\nelastic calculations $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$.\nThe Debye temperatures, Gr{\\\"u}neisen parameters and thermal conductivities depend on the calculated bulk modulus and are\ntherefore also cited below for each of the equations of state.\nAlso included are the Debye temperatures derived from the calculated\nelastic constants and speed of sound as given by Equation~\\ref{eq:art115:speedsound}.\nThe Debye temperatures, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$\n(Equation~\\ref{eq:art115:birch}), $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ (Equation~\\ref{eq:art115:vinet}),\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ (Equation~\\ref{eq:art115:bcn}), calculated using the Poisson ratio $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ obtained from\nEquation~\\ref{eq:art115:Poissonratio}, are compared to $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, obtained from the numerical fit\nof $E(V)$ (Equation~\\ref{eq:art115:bulkmod}) using both $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ and the approximation $\\sigma =\n0.25$ used in Reference~\\onlinecite{curtarolo:art96}, to\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, calculated with the speed of sound obtained\nusing Equation~\\ref{eq:art115:speedsound},\nand to the experimental values $\\theta^{\\mathrm{exp}}$.\nThe values of the acoustic Debye temperature ($\\theta_{\\mathrm{a}}$, Equation~\\ref{eq:art115:acousticdebyetemp})\nare shown, where available, in parentheses below the traditional Debye temperature value.\n\nThe experimental Gr{\\\"u}neisen parameter, $\\gamma^{\\mathrm{exp}}$, is compared to $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ (Equation~\\ref{eq:art115:bulkmod}), obtained using the numerical\npolynomial fit of $E(V)$ and both values of the Poisson ratio\n($\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ and the approximation $\\sigma = 0.25$ from\nReference~\\onlinecite{curtarolo:art96}), and to $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$\n(Equation~\\ref{eq:art115:birch}), $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ (Equation~\\ref{eq:art115:vinet}), and $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ (Equation~\\ref{eq:art115:bcn}), calculated\nusing $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ only. Similarly, the experimental lattice thermal\nconductivity $\\kappa^{\\mathrm{exp}}$ is compared to $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ (Equation~\\ref{eq:art115:bulkmod}),\nobtained using the numerical polynomial fit and both the calculated\nand approximated values of $\\sigma$, and to $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$\n(Equation~\\ref{eq:art115:birch}), $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ (Equation~\\ref{eq:art115:vinet}), and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$\n(Equation~\\ref{eq:art115:bcn}), calculated using only $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$.\n\nThe {\\small AEL}\\ method has been been previously implemented in the Materials Project framework for calculating\nelastic constants~\\cite{curtarolo:art100}. {Data from} the Materials Project database are included\nin the tables below for comparison {for the bulk modulus $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, shear modulus $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, and Poisson ratio $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$.}\n\n\\subsubsection{Zincblende and diamond structure materials}\n\nThe mechanical and thermal properties were calculated for a set of materials with the\nzincblende(spacegroup: $F\\overline{4}3m$,\\ $\\#$216; Pearson symbol: cF8; {\\small AFLOW}\\ prototype: {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_cF8_216_c_a.html}})\nand diamond ($Fd\\overline{3}m$,\\ $\\#$227; cF8; {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A_cF8_227_a.html}}) structures.\nThis {is the same set of materials\nas} in Table I of Reference~\\onlinecite{curtarolo:art96}, which in {turn are from} Table II of\nReference~\\onlinecite{slack} and Table 2.2 of Reference~\\onlinecite{Morelli_Slack_2006}.\n\nThe elastic {properties bulk modulus}, shear modulus and Poisson {ratio calculated} using {\\small AEL}\\ and {\\small AGL}\\ are shown\nin Table~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}, together\nwith experimental values from the literature where available. As can be seen\nfrom the results in Table~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}(a), the $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ values are\ngenerally closest to experiment as shown by the {\\small RMSrD}\\ value of $0.13$, producing an underestimate of the order of 10\\%. The {\\small AGL}\\ values from both the numerical\nfit and the empirical equations of state are generally very similar to each other, while being slightly less than the $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$\nvalues.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of\nzincblende and diamond structure semiconductors.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nC & 442~\\cite{Semiconductors_BasicData_Springer,\n Lam_BulkMod_PRB_1987, Grimsditch_ElasticDiamond_PRB_1975} & 434 & N\/A & 408 & 409 & 403 & 417 & 534~\\cite{Semiconductors_BasicData_Springer, Grimsditch_ElasticDiamond_PRB_1975} & 520 & 516 & 518 & N\/A & 0.069~\\cite{Semiconductors_BasicData_Springer, Grimsditch_ElasticDiamond_PRB_1975} & 0.073 & N\/A \\\\\nSiC & 248~\\cite{Strossner_ElasticSiC_SSC_1987} & 212 & 211 & 203 & 207 & 206 & 206 & 196~\\cite{Fate_ShearSiC_JACeramS_1974} & 195 & 178 & 187 & 187 & 0.145~\\cite{Lam_BulkMod_PRB_1987, Fate_ShearSiC_JACeramS_1974} & 0.160 & 0.16 \\\\\n & 211~\\cite{Semiconductors_BasicData_Springer, Lam_BulkMod_PRB_1987} & & & & & & & 170~\\cite{Semiconductors_BasicData_Springer} & & & & & 0.183~\\cite{Semiconductors_BasicData_Springer} & & \\\\\nSi & 97.8~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 89.1 & 83.0 & 84.2 & 85.9 & 85.0 & 86.1 & 66.5~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 64 & 61 & 62.5 & 61.2 & 0.223~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 0.216 & 0.2 \\\\\n & 98~\\cite{Lam_BulkMod_PRB_1987} & & & & & & & & & & & & & \\\\\nGe & 75.8~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 61.5 & 59.0 & 54.9 & 55.7 & 54.5 & 56.1 & 55.3~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 47.7 & 44.8 & 46.2 & 45.4 & 0.207~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 0.199 & 0.19 \\\\\n & 77.2~\\cite{Lam_BulkMod_PRB_1987} & & & & & & & & & & & & & \\\\\nBN & 367.0~\\cite{Lam_BulkMod_PRB_1987} & 372 & N\/A & 353 & 356 & 348 & 359 & N\/A & 387 & 374 & 380 & N\/A & N\/A & 0.119 & N\/A \\\\\nBP & 165.0~\\cite{Semiconductors_BasicData_Springer, Lam_BulkMod_PRB_1987} & 162 & 161 & 155 & 157 & 156 & 157 & 136~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 164 & 160 & 162 & 162 & 0.186~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 0.125 & 0.12 \\\\\n & 267~\\cite{Semiconductors_BasicData_Springer, Suzuki_ElasticBP_JAP_1983} & & & & & & & & & & & & & \\\\\n & 172~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & & & & & & & & & & & & & \\\\\nAlP & 86.0~\\cite{Lam_BulkMod_PRB_1987} & 82.9 & 85.2 & 78.9 & 80.4 & 79.5 & 80.4 & N\/A & 48.6 & 44.2 & 46.4 & 47.2 & N\/A & 0.264 & 0.27 \\\\\nAlAs & 77.0~\\cite{Lam_BulkMod_PRB_1987} & 67.4 & 69.8 & 63.8 & 65.1 & 64.0 & 65.3 & N\/A & 41.1 & 37.5 & 39.3 & 39.1 & N\/A & 0.256 & 0.26 \\\\\n & 74~\\cite{Greene_ElasticAlAs_PRL_1994} & & & & & & & & & & & & & \\\\\nAlSb & 58.2~\\cite{Lam_BulkMod_PRB_1987, Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 49.4 & 49.2 & 46.5 & 47.8 & 46.9 & 47.8 & 31.9~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 29.7 & 27.4 & 28.5 & 29.6 & 0.268~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 0.258 & 0.25 \\\\\nGaP & 88.7~\\cite{Lam_BulkMod_PRB_1987} & 78.8 & 76.2 & 71.9 & 73.4 & 72.2 & 73.8 & 55.3~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 53.5 & 49.1 & 51.3 & 51.8 & 0.244~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 0.232 & 0.22 \\\\\n & 89.8~\\cite{Boyle_ElasticGaPSb_PRB_1975} & & & & & & & & & & & & & \\\\\nGaAs & 74.8~\\cite{Lam_BulkMod_PRB_1987} & 62.7 & 60.7 & 56.8 & 57.7 & 56.6 & 58.1 & 46.6~\\cite{Bateman_ElasticGaAs_JAP_1975} & 42.6 & 39.1 & 40.8 & 40.9 & 0.244~\\cite{Bateman_ElasticGaAs_JAP_1975} & 0.233 & 0.23 \\\\\n & 75.5~\\cite{Bateman_ElasticGaAs_JAP_1975} & & & & & & & & & & & & & \\\\\nGaSb & 57.0~\\cite{Lam_BulkMod_PRB_1987} & 47.0 & 44.7 & 41.6 & 42.3 & 41.2 & 42.6 & 34.2~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 30.8 & 28.3 & 29.6 & 30.0 & 0.248~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 0.240 & 0.23 \\\\\n & 56.3~\\cite{Boyle_ElasticGaPSb_PRB_1975} & & & & & & & & & & & & & \\\\\nInP & 71.1~\\cite{Lam_BulkMod_PRB_1987, Nichols_ElasticInP_SSC_1980} & 60.4 & N\/A & 56.4 & 57.6 & 56.3 & 57.8 & 34.3~\\cite{Nichols_ElasticInP_SSC_1980} & 33.6 & 29.7 & 31.6 & N\/A & 0.292~\\cite{Nichols_ElasticInP_SSC_1980} & 0.277 & N\/A \\\\\nInAs & 60.0~\\cite{Lam_BulkMod_PRB_1987} & 50.1 & 49.2 & 45.7 & 46.6 & 45.4 & 46.9 & 29.5~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 27.3 & 24.2 & 25.7 & 25.1 & 0.282~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 0.281 & 0.28 \\\\\n & 57.9~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & & & & & & & & & & & & & \\\\\nInSb & 47.3~\\cite{Lam_BulkMod_PRB_1987, DeVaux_ElasticInSb_PR_1956} & 38.1 & N\/A & 34.3 & 35.0 & 34.1 & 35.2 & 22.1~\\cite{DeVaux_ElasticInSb_PR_1956} & 21.3 & 19.0 & 20.1 & N\/A & 0.298~\\cite{DeVaux_ElasticInSb_PR_1956} & 0.275 & N\/A \\\\\n & 48.3~\\cite{Semiconductors_BasicData_Springer, Slutsky_ElasticInSb_PR_1959} & & & & & & & 23.7~\\cite{Semiconductors_BasicData_Springer, Slutsky_ElasticInSb_PR_1959} & & & & & 0.289~\\cite{Semiconductors_BasicData_Springer, Slutsky_ElasticInSb_PR_1959} & \\\\\n & 46.5~\\cite{Vanderborgh_ElasticInSb_PRB_1990} & & & & & & & & & & & & & \\\\\nZnS & 77.1~\\cite{Lam_BulkMod_PRB_1987} & 71.2 & 68.3 & 65.8 & 66.1 & 65.2 & 66.6 & 30.9~\\cite{Semiconductors_BasicData_Springer} & 36.5 & 31.4 & 33.9 & 33.2 & 0.318~\\cite{Semiconductors_BasicData_Springer} & 0.294 & 0.29 \\\\\n & 74.5~\\cite{Semiconductors_BasicData_Springer} & & & & & & & & & & & & & \\\\\nZnSe & 62.4~\\cite{Lam_BulkMod_PRB_1987, Lee_ElasticZnSeTe_JAP_1970} & 58.2 & 58.3 & 53.3 & 53.8 & 52.8 & 54.1 & 29.1~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 29.5 & 25.6 & 27.5 & 27.5 & 0.298~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 0.296 & 0.3\\\\\nZnTe & 51.0~\\cite{Lam_BulkMod_PRB_1987, Lee_ElasticZnSeTe_JAP_1970} & 43.8 & 46.0 & 39.9 & 40.5 & 39.4 & 40.7 & 23.4~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 23.3 & 20.8 & 22.1 & 22.4 & 0.30~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 0.284 & 0.29 \\\\\nCdSe & 53.0~\\cite{Lam_BulkMod_PRB_1987} & 46.7 & 44.8 & 41.5 & 42.1 & 41.1 & 42.3 & N\/A & 16.2 & 13.1 & 14.7 & 15.3 & N\/A & 0.358 & 0.35 \\\\\nCdTe & 42.4~\\cite{Lam_BulkMod_PRB_1987} & 36.4 & 35.3 & 32.2 & 32.7 & 31.9 & 32.8 & N\/A & 14.2 & 11.9 & 13.0 & 13.6 & N\/A & 0.340 & 0.33 \\\\\nHgSe & 50.0~\\cite{Lam_BulkMod_PRB_1987} & 43.8 & 41.2 & 39.0 & 39.7 & 38.5 & 39.9 & 14.8~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 15.6 & 11.9 & 13.7 & 13.3 & 0.361~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 0.358 & 0.35 \\\\\n & 48.5~\\cite{Lehoczky_ElasticHgSe_PR_1969} & & & & & & & & & & & & & \\\\\nHgTe & 42.3~\\cite{Lam_BulkMod_PRB_1987, Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 35.3 & N\/A & 31.0 & 31.6 & 30.8 & 31.9 & 14.7~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 14.4 & 11.6 & 13.0 & N\/A & 0.344~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 0.335 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_elastic}\n\\end{table}\n\nFor the shear modulus, the experimental values $G^{\\mathrm{exp}}$ are compared to the {\\small AEL}\\ values $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$,\n$G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. As can be seen from the values in\nTable~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}(b), the agreement with the experimental values is generally\ngood with a very low {\\small RMSrD}\\ of 0.111 for $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, with the Voigt approximation tending to overestimate and the Reuss approximation tending to underestimate, as would be\nexpected. The experimental values of the Poisson ratio $\\sigma^{\\mathrm{exp}}$ and the {\\small AEL}\\ values $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ (Equation~\\ref{eq:art115:Poissonratio}) are\nalso shown in Table~\\ref{tab:art115:zincblende_elastic} and Figure~\\ref{fig:art115:zincblende_thermal_elastic}(c), and the values are generally in good\nagreement. The Pearson (\\nobreak\\mbox{\\it i.e.}, linear, Equation~\\ref{eq:art115:Pearson}) and Spearman (\\nobreak\\mbox{\\it i.e.}, rank order, Equation~\\ref{eq:art115:Spearman}) correlations between all of\nthe {\\small AEL}-{\\small AGL}\\ elastic property values and experiment are shown in Table~\\ref{tab:art115:zincblende_correlation}, and are generally\nvery high for all of these properties, ranging from 0.977 and 0.982 respectively for $\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, up to 0.999\nand 0.992 for $B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. These very high correlation values demonstrate the validity of using the {\\small AEL}-{\\small AGL}\\\nmethodology to predict the elastic and mechanical properties of\nmaterials.\n\nThe Materials Project values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ for diamond and zincblende structure materials are also shown in\nTable~\\ref{tab:art115:zincblende_elastic}, where available. The Pearson correlations values for the experimental results with the available values of\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ were calculated to be 0.995, 0.987 and 0.952, respectively, while the respective Spearman correlations\nwere 0.963, 0.977 and 0.977, and the {\\small RMSrD}\\ values were 0.149, 0.116 and 0.126. For comparison, the corresponding Pearson correlations for the same\nsubset of materials for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ are 0.997, 0.987, and 0.957 respectively, while the respective Spearman correlations\nwere 0.982, 0.977 and 0.977, and the {\\small RMSrD}\\ values were 0.129, 0.114 and 0.108. These correlation values are very similar, and the general close agreement\n{for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ with $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$}\ndemonstrate that the small differences in the parameters used for the {\\small DFT}\\ calculations make little difference to the results,\nindicating that the parameter set used here is robust for high-throughput calculations.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.98\\linewidth]{fig029}\n\\mycaption[\n({\\bf a}) Bulk modulus,\n({\\bf b}) shear modulus,\n({\\bf c}) Poisson ratio,\n({\\bf d}) lattice thermal conductivity at 300~K,\n({\\bf e}) acoustic Debye temperature and\n({\\bf f}) Gr{\\\"u}neisen parameter of zincblende and\ndiamond structure semiconductors.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.}\n\\label{fig:art115:zincblende_thermal_elastic}\n\\end{figure}\n\nThe thermal {properties Debye} temperature, Gr{\\\"u}neisen parameter and thermal conductivity calculated using {\\small AGL}\\ for this set of materials are\ncompared to the experimental values taken from the literature in Table~\\ref{tab:art115:zincblende_thermal} and are also plotted in Figure~\\ref{fig:art115:zincblende_thermal_elastic}.\nFor the Debye temperature, the experimental values $\\theta^{\\mathrm{exp}}$ are compared {to\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$} in Figure~\\ref{fig:art115:zincblende_thermal_elastic}(e), while {the values} for\nthe empirical equations of state are provided in Table~\\ref{tab:art115:zincblende_thermal_eos}.\nNote that the $\\theta^{\\mathrm{exp}}$ values taken from Reference~\\onlinecite{slack} and\nReference~\\onlinecite{Morelli_Slack_2006} are for $\\theta_{\\mathrm{a}}$, and generally are in good agreement with the $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ values. The\nvalues obtained using the numerical $E(V)$ fit and the three different equations of state are also in good agreement with each other, whereas\nthe values of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated using different $\\sigma$ values differ significantly, indicating that for this property the value\nof $\\sigma$ used is far more important than the equation of state used. The correlation\nbetween $\\theta^{\\mathrm{exp}}$ and the various {\\small AGL}\\ values is also very high,\nof the order of 0.999, and the {\\small RMSrD}\\ is low, of the order of 0.13.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at\n300~K, Debye temperature and Gr{\\\"u}neisen parameter of\nzincblende and diamond structure semiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.\nThe values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 141K for HgTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & \\\\\n & & ($\\sigma = 0.25$)\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)\\cite{curtarolo:art96} & \\\\\n\\hline\nC & 3000~\\cite{Morelli_Slack_2006} & 169.1 & 419.9 & 1450~\\cite{slack, Morelli_Slack_2006} & 1536 & 2094 & 2222 & 0.75~\\cite{Morelli_Slack_2006} & 1.74 & 1.77 \\\\\n & & & & & (1219) & (1662) & & 0.9~\\cite{slack} & & \\\\\nSiC & 360~\\cite{Ioffe_Inst_DB} & 67.19 & 113.0 & 740~\\cite{slack} & 928 & 1106 & 1143 & 0.76~\\cite{slack} & 1.84 & 1.85\t\\\\\n & & & & & (737) & (878) & & & & \\\\\nSi & 166~\\cite{Morelli_Slack_2006} & 20.58 & 26.19 & 395~\\cite{slack, Morelli_Slack_2006} & 568 & 610 & 624 & 1.06~\\cite{Morelli_Slack_2006} & 2.09 & 2.06\t \\\\\n & & & & & (451) & (484) & & 0.56~\\cite{slack} & & \\\\\nGe & 65~\\cite{Morelli_Slack_2006} & 6.44 & 8.74 & 235~\\cite{slack, Morelli_Slack_2006} & 296 & 329 & 342 & 1.06~\\cite{Morelli_Slack_2006} & 2.3 & 2.31 \t \\\\\n & & & & & (235) & (261) & & 0.76~\\cite{slack} & & \\\\\nBN & 760~\\cite{Morelli_Slack_2006} & 138.4 & 281.6 & 1200~\\cite{Morelli_Slack_2006} & 1409 & 1793 & 1887 & 0.7~\\cite{Morelli_Slack_2006} & 1.73 & 1.75\t\\\\\n & & & & & (1118) & (1423) & & & & \\\\\nBP & 350~\\cite{Morelli_Slack_2006} & 52.56 & 105.0 & 670~\\cite{slack, Morelli_Slack_2006} & 811 & 1025 & 1062 & 0.75~\\cite{Morelli_Slack_2006} & 1.78 & 1.79\t\\\\\n & & & & & (644) & (814) & & & & \\\\\nAlP & 90~\\cite{Landolt-Bornstein, Spitzer_JPCS_1970} & 21.16 & 19.34 & 381~\\cite{Morelli_Slack_2006} & 542 & 525 & 531 & 0.75~\\cite{Morelli_Slack_2006} & 1.96 & 1.96\t \\\\\n & & & & & (430) & (417) & & & & \\\\\nAlAs & 98~\\cite{Morelli_Slack_2006} & 12.03 & 11.64 & 270~\\cite{slack, Morelli_Slack_2006} & 378 & 373 & 377 & 0.66~\\cite{slack, Morelli_Slack_2006} & 2.04 & 2.04\t \\\\\n & & & & & (300) & (296) & & & & \\\\\nAlSb & 56~\\cite{Morelli_Slack_2006} & 7.22 & 6.83 & 210~\\cite{slack, Morelli_Slack_2006} & 281 & 276 & 277 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.12 & 2.13 \t \\\\\n & & & & & (223) & (219) & & & & \\\\\nGaP & 100~\\cite{Morelli_Slack_2006} & 11.76 & 13.34 & 275~\\cite{slack, Morelli_Slack_2006} & 396 & 412 & 423 & 0.75~\\cite{Morelli_Slack_2006} & 2.15 & 2.15 \t\\\\\n & & & & & (314) & (327) & & 0.76~\\cite{slack} & & \\\\\nGaAs & 45~\\cite{Morelli_Slack_2006} & 7.2 & 8.0 & 220~\\cite{slack, Morelli_Slack_2006} & 302 & 313\t& 322 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.24 \\\\\n & & & & & (240) & (248) & & & & \\\\\nGaSb & 40~\\cite{Morelli_Slack_2006} & 4.62 & 4.96 & 165~\\cite{slack, Morelli_Slack_2006} & 234 & 240 & 248 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.27 & 2.28 \t \\\\\n & & & & & (186) & (190) & & & & \\\\\nInP & 93~\\cite{Morelli_Slack_2006} & 7.78 & 6.53 & 220~\\cite{slack, Morelli_Slack_2006} & 304 & 286 & 287 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.21 \t \\\\\n & & & & & (241) & (227) & & & & \\\\\nInAs & 30~\\cite{Morelli_Slack_2006} & 5.36 & 4.33 & 165~\\cite{slack, Morelli_Slack_2006} & 246 & 229 & 231 & 0.57~\\cite{slack, Morelli_Slack_2006} & 2.26 & 2.26\t \\\\\n & & & & & (195) & (182) & & & & \\\\\nInSb & 20~\\cite{Morelli_Slack_2006} & 3.64 & 3.02 & 135~\\cite{slack, Morelli_Slack_2006} & 199 & 187 & 190 & 0.56~\\cite{slack, Morelli_Slack_2006} & 2.3 & 2.3 \t \\\\\n & 16.5~\\cite{Snyder_jmatchem_2011} & & & & (158) & (148) & & & & \\\\\nZnS & 27~\\cite{Morelli_Slack_2006} & 11.33 & 8.38 & 230~\\cite{slack, Morelli_Slack_2006} & 379 & 341 & 346 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.01 & 2.00 \t \\\\\n & & & & & (301) & (271) & & & & \\\\\nZnSe & 19~\\cite{Morelli_Slack_2006} & 7.46 & 5.44 & 190~\\cite{slack, Morelli_Slack_2006} & 290 & 260\t& 263 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.07 & 2.06 \t\\\\\n & 33~\\cite{Snyder_jmatchem_2011} & & & & (230) & (206) & & & & \\\\\nZnTe & 18~\\cite{Morelli_Slack_2006} & 4.87 & 3.83 & 155~\\cite{slack, Morelli_Slack_2006} & 228 & 210 & 212 & 0.97~\\cite{slack, Morelli_Slack_2006} & 2.14 & 2.13 \\\\\n & & & & & (181) & (167) & & & & \\\\\nCdSe & 4.4~\\cite{Snyder_jmatchem_2011} & 4.99 & 2.04 & 130~\\cite{Morelli_Slack_2006} & 234 & 173 & 174 & 0.6~\\cite{Morelli_Slack_2006} & 2.19 & 2.18 \\\\\n & & & & & (186) & (137) & & & & \\\\\nCdTe & 7.5~\\cite{Morelli_Slack_2006} & 3.49 & 1.71 & 120~\\cite{slack, Morelli_Slack_2006} & 191 & 150 & 152 & 0.52~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.22\t \\\\\n & & & & & (152) & (119) & & & & \\\\\nHgSe & 3~\\cite{Whitsett_PRB_1973} & 3.22 & 1.32 & 110~\\cite{slack} & 190 & 140\t& 140 & 0.17~\\cite{slack} & 2.4 & 2.38\t \\\\\n & & & & & (151) & (111) & & & & \\\\\nHgTe & 2.5~\\cite{Snyder_jmatchem_2011} & 2.36 & 1.21 & 141~\\cite{Snyder_jmatchem_2011} & 162 & 129 & 130 & 1.9~\\cite{Snyder_jmatchem_2011} & 2.46 & 2.45 \\\\\n & & & & (100)~\\cite{slack} & (129) & (102) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_thermal}\n\\end{table}\n\nThe experimental values $\\gamma^{\\mathrm{exp}}$ of the Gr{\\\"u}neisen parameter are plotted {against\n$\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$, $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$} in Figure~\\ref{fig:art115:zincblende_thermal_elastic}(f), and the values\nare listed in Table~\\ref{tab:art115:zincblende_thermal} and in Table~\\ref{tab:art115:zincblende_thermal_eos}.\nThe very high {\\small RMSrD}\\ values (see Table~\\ref{tab:art115:zincblende_correlation}) show that {\\small AGL}\\ has problems accurately predicting\nthe Gr{\\\"u}neisen parameter for this set of materials, as the calculated value is often 2 to 3 times larger than the experimental one.\nNote also that there are quite large differences between the values obtained for different equations of state, with $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ generally\nhaving the lowest values while $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ has the highest values.\nOn the other hand, in contrast to the case of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, the value of $\\sigma$ used makes little difference to the value\nof $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$. The {correlations} between $\\gamma^{\\mathrm{exp}}$ and the {\\small AGL}\\ values, as shown in Table~\\ref{tab:art115:zincblende_correlation},\nare also quite poor, with no value higher than 0.2 for the Pearson correlations, and {negative Spearman} correlations.\n\nThe experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ is compared in Figure~\\ref{fig:art115:zincblende_thermal_elastic}(d) to the\nthermal conductivities calculated with {\\small AGL}\\ using the Leibfried-Schl{\\\"o}mann equation (Equation~\\ref{eq:art115:thermal_conductivity}): $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$,\nwhile the values are listed in Table~\\ref{tab:art115:zincblende_thermal} and in Table~\\ref{tab:art115:zincblende_thermal_eos}.\nThe absolute agreement between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is quite poor, with {\\small RMSrD}\\ values of the order of 0.8 and discrepancies of tens, or even hundreds, of percent\nquite common. Considerable disagreements also exist between different experimental reports of these properties, in\nalmost all cases where they exist. Unfortunately, the scarcity of experimental data from different sources on the thermal properties of these materials\nprevents reaching definite conclusions regarding the true values of these properties. The available data can thus only be considered as a rough indication\nof their order of magnitude.\n\n{The Pearson} correlations between the {\\small AGL}\\ calculated thermal conductivity values and the experimental\nvalues are high, ranging from $0.871$ to $0.932$, while the Spearman correlations are even higher, ranging from $0.905$\nto $0.954$, as shown in Table~\\ref{tab:art115:zincblende_correlation}. In particular, note that using the $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the {\\small AGL}\\ calculations\nimproves the correlations by about 5\\%, from $0.878$ to $0.927$ and from $0.905$ to $0.954$. For the different equations of state,\n$\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ appear to correlate better with $\\kappa^{\\mathrm{exp}}$ than $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ for this set of\nmaterials.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations and deviations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for zincblende and diamond structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.878 & 0.905 & 0.776 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.927 & 0.95 & 0.796 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.871 & 0.954 & 0.787 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.908 & 0.954 & 0.815 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.932 & 0.954 & 0.771 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.995 & 0.984 & 0.200 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.999 & 0.998 & 0.132 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.999 & 0.998 & 0.132 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.999 & 0.998 & 0.127 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.999 & 0.998 & 0.136 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.137 & -0.187 & 3.51 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.145 & -0.165 & 3.49 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.169 & -0.178 & 3.41 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.171 & -0.234 & 3.63 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.144 & -0.207 & 3.32 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.999 & 0.992 & 0.130 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.999 & 0.986 & 0.201 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.999 & 0.986 & 0.189 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.999 & 0.986 & 0.205 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.999 & 0.986 & 0.185 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.980 & 0.111 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.980 & 0.093 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.980 & 0.152 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.977 & 0.982 & 0.095 \\\\\n\\end{tabular}\n\\label{tab:art115:zincblende_correlation}\n\\end{table}\n\nAs we noted in our previous work on {\\small AGL}~\\cite{curtarolo:art96}, some of the inaccuracy in the thermal conductivity results may be due to the inability of the Leibfried-Schl{\\\"o}mann equation to fully\ndescribe effects such as the suppression of phonon-phonon scattering due to large gaps between the branches of\nthe phonon dispersion~\\cite{Lindsay_PRL_2013}. This can be seen from the thermal conductivity values shown in Table 2.2 of Reference~\\onlinecite{Morelli_Slack_2006}\ncalculated using the experimental values of $\\theta_{\\mathrm{a}}$ and $\\gamma$ in the Leibfried-Schl{\\\"o}mann equation. There are large discrepancies in certain cases such as diamond,\nwhile the Pearson and Spearman correlations of $0.932$ and $0.941$ respectively are very similar to the correlations we calculated using the {\\small AGL}\\ evaluations of\n$\\theta_{\\mathrm{a}}$ and $\\gamma$.\n\nThus, the unsatisfactory quantitative reproduction of these quantities by the Debye quasi-harmonic model\nhas little impact on its effectiveness as a screening tool for identifying high or\nlow thermal conductivity materials. The model can be used when these\nexperimental values are unavailable to help determine the relative values of these quantities and for\nranking {materials conductivity}.\n\n\\subsubsection{Rocksalt structure materials}\n\nThe mechanical and thermal properties were calculated for a set of materials with the rocksalt structure\n(spacegroup: $Fm\\overline{3}m$,\\ $\\#$225; Pearson symbol: cF8;\n{\\small AFLOW}\\ prototype: {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_cF8_225_a_b.html}}).\n This {is the same set of materials\nas} in Table II of Reference~\\onlinecite{curtarolo:art96}, which in turn {are from} the\nsets in Table III of Reference~\\onlinecite{slack} and Table 2.1 of Reference~\\onlinecite{Morelli_Slack_2006}.\n\nThe elastic properties of bulk modulus, shear modulus and Poisson ratio, as calculated using {\\small AEL}\\ and {\\small AGL}\\ are shown\nin Table~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}, together\nwith experimental values from the literature where available. As can be seen\nfrom the results in Table~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(a), for this set of materials the\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ values are closest to experiment, with an {\\small RMSrD}\\ of 0.078. The {\\small AGL}\\ values from both the numerical\nfit and the empirical equations of state are generally very similar to each other, while being slightly less than the $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$\nvalues.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Mechanical properties bulk modulus, shear modulus\nand Poisson ratio of rocksalt structure semiconductors.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nLiH & 33.7~\\cite{Laplaze_ElasticLiH_SSC_1976} & 37.7 & 36.1 & 29.5 & 29.0 & 27.7 & 31.4 & 36.0~\\cite{Laplaze_ElasticLiH_SSC_1976} & 43.4 & 42.3 & 42.8 & 42.9 & 0.106~\\cite{Laplaze_ElasticLiH_SSC_1976} & 0.088 & 0.07 \\\\\nLiF & 69.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 70.4 & 69.9 & 58.6 & 59.9 & 57.5 & 61.2 & 48.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 46.4 & 45.8 & 46.1 & 50.9 & 0.216~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.231 & 0.21 \\\\\nNaF & 48.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 46.9 & 47.6 & 38.7 & 38.6 & 36.8 & 39.3 & 31.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 29.5 & 28.4 & 28.9 & 30.0 & 0.236~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.244 & 0.24 \\\\\nNaCl & 25.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.9 & 22.6 & 20.0 & 20.5 & 19.2 & 20.7 & 14.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 14.0 & 12.9 & 13.5 & 14.3 & 0.255~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.271 & 0.24 \\\\\nNaBr & 20.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 20.5 & 27.1 & 16.3 & 16.9 & 15.7 & 16.9 & 11.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 11.0 & 9.9 & 10.4 & 11.6 & 0.264~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.283 & 0.31 \\\\\nNaI & 15.95~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 16.4 & 15.8 & 12.6 & 13.2 & 12.2 & 13.1 & 8.59~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.35 & 7.31 & 7.83 & 8.47 & 0.272~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.295 & 0.27 \\\\\nKF & 31.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 29.9 & 28.9 & 25.1 & 24.2 & 22.9 & 24.7 & 16.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 16.5 & 15.4 & 15.9 & 16.5 & 0.275~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.274 & 0.26 \\\\\nKCl & 18.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 16.7 & 15.8 & 13.8 & 13.7 & 12.7 & 13.6 & 9.51~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.1 & 8.51 & 9.30 & 9.24 & 0.277~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.265 & 0.26 \\\\\nKBr & 15.4~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 13.8 & 21.6 & 11.1 & 11.4 & 10.5 & 11.2 & 7.85~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.14 & 6.46 & 7.30 & 7.33 & 0.282~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.276 & 0.35 \\\\\nKI & 12.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.9 & 9.52 & 8.54 & 9.03 & 8.28 & 8.84 & 5.96~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.05 & 4.39 & 5.22 & 5.55 & 0.290~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.294 & 0.26 \\\\\nRbCl & 16.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 14.3 & 14.6 & 12.1 & 11.8 & 11.0 & 11.8 & 7.63~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.06 & 6.41 & 7.24 & 7.67 & 0.297~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.284 & 0.28 \\\\\nRbBr & 13.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 12.6 & 13.8 & 10.3 & 9.72 & 9.06 & 9.67 & 6.46~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 7.12 & 5.24 & 6.18 & 6.46 & 0.298~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.289 & 0.3 \\\\\nRbI & 11.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 9.90 & 9.66 & 8.01 & 7.74 & 7.12 & 7.54 & 5.03~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.50 & 3.65 & 4.57 & 4.63 & 0.303~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.300 & 0.29 \\\\\nAgCl & 44.0~\\cite{Hughes_ElasticAgCl_PRB_1996} & 40.6 & N\/A & 33.7 & 34.1 & 33.0 & 34.7 & 8.03~\\cite{Hughes_ElasticAgCl_PRB_1996} & 8.68 & 8.66 & 8.67 & N\/A & 0.414~\\cite{Hughes_ElasticAgCl_PRB_1996} & 0.400 & N\/A \\\\\nMgO & 164~\\cite{Sumino_ElasticMgO_JPE_1976} & 152 & 152 & 142 & 142 & 140 & 144 & 131~\\cite{Sumino_ElasticMgO_JPE_1976} & 119 & 115 & 117 & 119 & 0.185~\\cite{Sumino_ElasticMgO_JPE_1976} & 0.194 & 0.19 \\\\\nCaO & 113~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 105 & 105 & 99.6 & 100 & 98.7 & 101 & 81.0~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 73.7 & 73.7 & 73.7 & 74.2 & 0.210~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.216 & 0.21 \\\\\nSrO & 91.2~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 84.7 & 87.4 &80.0 & 80.2 & 79.1 & 80.8 & 58.7~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 55.1 & 55.0 & 55.1 & 56.0 & 0.235~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.233 & 0.24 \\\\\nBaO & 75.4~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 69.1 & 68.4 & 64.6 & 64.3 & 63.0 & 64.6 & 35.4~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 36.4 & 36.4 & 36.4 & 37.8 & 0.297~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.276 & 0.27 \\\\\nPbS & 52.9~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 53.5 & N\/A & 49.9 & 50.8 & 50.0 & 51.0 & 27.9~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 34.0 & 26.8 & 30.4 & N\/A & 0.276~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 0.261 & N\/A \\\\\nPbSe & 54.1~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 47.7 & N\/A & 43.9 & 44.8 & 43.9 & 44.9 & 26.2~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 31.7 & 23.6 & 27.6 & N\/A & 0.291~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 0.257 & N\/A \\\\\nPbTe & 39.8~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 39.5 & N\/A & 36.4 & 36.6 & 35.8 & 36.8 & 23.1~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 28.7 & 19.8 & 24.3 & N\/A & 0.256~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 0.245 & N\/A \\\\\nSnTe & 37.8~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 40.4 & 39.6 & 38.1 & 38.4 & 37.6 & 38.6 & 20.8~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 31.4 & 22.0 & 26.7 & 27.6 & 0.267~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 0.229 & 0.22 \\\\\n\\end{tabular}}\n\\label{tab:art115:rocksalt_elastic}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.98\\linewidth]{fig030}\n\\mycaption[\n({\\bf a}) Bulk modulus,\n({\\bf b}) shear modulus,\n({\\bf c}) Poisson ratio,\n({\\bf d}) lattice thermal conductivity at 300~K,\n({\\bf e}) Debye temperature and\n({\\bf f}) Gr{\\\"u}neisen parameter of rocksalt structure\nsemiconductors.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\nThe Debye temperatures plotted in ({\\bf b}) are\n$\\theta_{\\mathrm{a}}$, except for SnTe where $\\theta_{\\mathrm D}$ is\nquoted in Reference~\\onlinecite{Snyder_jmatchem_2011}.}\n\\label{fig:art115:rocksalt_thermal_elastic}\n\\end{figure}\n\nFor the shear modulus, the experimental values $G^{\\mathrm{exp}}$ are compared to the {\\small AEL}\\ values $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$,\n$G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. As can be seen from the values in\nTable~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(b), the agreement with the experimental values is generally\ngood with an {\\small RMSrD}\\ of 0.105 for $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, with the Voigt approximation tending to overestimate and the Reuss approximation tending to underestimate, as would be\nexpected. The experimental values of the Poisson ratio $\\sigma^{\\mathrm{exp}}$ and the {\\small AEL}\\ values $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ (Equation~\\ref{eq:art115:Poissonratio}) are\nalso shown in Table~\\ref{tab:art115:rocksalt_elastic} and Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(c), and the values are generally in good\nagreement. The Pearson (\\nobreak\\mbox{\\it i.e.}, linear, Equation~\\ref{eq:art115:Pearson}) and Spearman (\\nobreak\\mbox{\\it i.e.}, rank order, Equation~\\ref{eq:art115:Spearman}) correlations between all of\nthe the {\\small AEL}-{\\small AGL}\\ elastic property values and experiment are shown in Table~\\ref{tab:art115:rocksalt_correlation}, and are generally\nvery high for all of these properties, ranging from 0.959 and 0.827 respectively for $\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, up to 0.998\nand 0.995 for $B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. These very high correlation values demonstrate the validity of using the {\\small AEL}-{\\small AGL}\\\nmethodology to predict the elastic and mechanical properties of materials.\n\n{The values} of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ for rocksalt structure materials are also shown in\nTable~\\ref{tab:art115:rocksalt_elastic}, where available. The Pearson {correlations for} the experimental results with the available values of\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ were calculated to be 0.997, 0.994 and 0.890, respectively, while the respective Spearman correlations\nwere 0.979, 0.998 and 0.817, and the {\\small RMSrD}\\ values were 0.153, 0.105 and 0.126. For comparison, the corresponding Pearson correlations for the same\nsubset of materials for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ are 0.998, 0.995, and 0.951 respectively, while the respective Spearman correlations\nwere 0.996, 1.0 and 0.843, and the {\\small RMSrD}\\ values were 0.079, 0.111 and 0.071. These correlation values are very similar, and the general close agreement\nfor the results for the values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ with those of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$\ndemonstrate that the small differences in the parameters used for the {\\small DFT}\\ calculations make little difference to the results,\nindicating that the parameter set used here is robust for high-throughput calculations.\n\nThe thermal properties of Debye temperature, Gr{\\\"u}neisen parameter and thermal conductivity calculated using {\\small AGL}\\ are\ncompared to the experimental values taken from the literature in Table~\\ref{tab:art115:rocksalt_thermal} and are also plotted in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}.\nFor the Debye temperature, the experimental values $\\theta^{\\mathrm{exp}}$ are compared {to\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$}, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(e), while the actual values for\nthe empirical equations of state are provided in Table~\\ref{tab:art115:rocksalt_thermal_eos}.\nNote that the $\\theta^{\\mathrm{exp}}$ values taken from Reference~\\onlinecite{slack} and\nReference~\\onlinecite{Morelli_Slack_2006} are for $\\theta_{\\mathrm{a}}$, and generally are in good agreement with the $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ values. The\nvalues obtained using the numerical $E(V)$ fit and the three different equations of state are also in good agreement with each other, whereas\nthe values of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated using different $\\sigma$ values differ significantly, indicating that, as in the case of the zincblende\nand diamond structures, the value of $\\sigma$ used is far more important for this property than the equation of state used. The correlation\nbetween $\\theta^{\\mathrm{exp}}$ and the various {\\small AGL}\\ values is also quite high, of the order of 0.98 for the Pearson correlation and 0.92 for the Spearman\ncorrelation.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at 300~K, Debye temperature and Gr{\\\"u}neisen parameter of rocksalt\nstructure semiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\nThe values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 155K for SnTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nLiH & 15~\\cite{Morelli_Slack_2006} & 8.58 & 18.6 & 615~\\cite{slack, Morelli_Slack_2006} & 743 & 962 & 1175 & 1.28~\\cite{slack, Morelli_Slack_2006} & 1.62 & 1.66 \\\\\n & & & & & (590) & (764) & & & & \\\\\nLiF & 17.6~\\cite{Morelli_Slack_2006} & 8.71 & 9.96 & 500~\\cite{slack, Morelli_Slack_2006} & 591 & 617 & 681 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.02 & 2.03 \t \\\\\n & & & & & (469) & (490) & & & & \\\\\nNaF & 18.4~\\cite{Morelli_Slack_2006} & 4.52 & 4.67 & 395~\\cite{slack, Morelli_Slack_2006} & 411 & 416 & 455 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.2 & 2.21 \t \\\\\n & & & & & (326) & (330) & & & & \\\\\nNaCl & 7.1~\\cite{Morelli_Slack_2006} & 2.43 & 2.12 & 220~\\cite{slack, Morelli_Slack_2006} & 284 & 271 & 289 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.23 \t \\\\\n & & & & & (225) & (215) & & & & \\\\\nNaBr & 2.8~\\cite{Morelli_Slack_2006} & 1.66 & 1.33 & 150~\\cite{slack, Morelli_Slack_2006} & 203 & 188 & 198 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.22 \t \\\\\n & & & & & (161) & (149) & & & &\\\\\nNaI & 1.8~\\cite{Morelli_Slack_2006} & 1.17 & 0.851 & 100~\\cite{slack, Morelli_Slack_2006} & 156 & 140 & 147 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.23 \t \\\\\n & & & & & (124) & (111) & & & &\\\\\nKF & N\/A & 2.68 & 2.21 & 235~\\cite{slack, Morelli_Slack_2006} & 305 & 288 & 309\t& 1.52~\\cite{slack, Morelli_Slack_2006} & 2.29 & 2.32 \t\\\\\n & & & & & (242) & (229) & & & &\\\\\nKCl & 7.1~\\cite{Morelli_Slack_2006} & 1.4 & 1.25 & 172~\\cite{slack, Morelli_Slack_2006} & 220 & 213 & 226 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.38 & 2.40 \t \\\\\n & & & & & (175) & (169) & & & &\\\\\nKBr & 3.4~\\cite{Morelli_Slack_2006} & 1.0 & 0.842 & 117~\\cite{slack, Morelli_Slack_2006} & 165 & 156 & 162 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.37 & 2.37 \\\\\n & & & & & (131) & (124) & & & &\\\\\nKI & 2.6~\\cite{Morelli_Slack_2006} & 0.72 & 0.525 & 87~\\cite{slack, Morelli_Slack_2006} & 129 & 116 & 120 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.35 & 2.35 \t \\\\\n & & & & & (102) & (92) & & & &\\\\\nRbCl & 2.8~\\cite{Morelli_Slack_2006} & 1.09 & 0.837 & 124~\\cite{slack, Morelli_Slack_2006} & 168 & 155 & 160 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.34 & 2.37 \t \\\\\n & & & & & (133) & (123) & & & &\\\\\nRbBr & 3.8~\\cite{Morelli_Slack_2006} & 0.76 & 0.558 & 105~\\cite{slack, Morelli_Slack_2006} & 134 & 122 & 129 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.40 & 2.43 \\\\\n & & & & & (106) & (97) & & & &\\\\\nRbI & 2.3~\\cite{Morelli_Slack_2006} & 0.52 & 0.368 & 84~\\cite{slack, Morelli_Slack_2006} & 109 & 97 & 102 & 1.41~\\cite{slack, Morelli_Slack_2006} & 2.47 & 2.47 \t \\\\\n & & & & & (87) & (77) & & & &\\\\\nAgCl & 1.0~\\cite{Landolt-Bornstein, Maqsood_IJT_2003} & 2.58 & 0.613 & 124~\\cite{slack} & 235 & 145 & 148 & 1.9~\\cite{slack} & 2.5 & 2.49 \t \\\\\n & & & & & (187) & (115) & & & &\\\\\nMgO & 60~\\cite{Morelli_Slack_2006} & 31.9 & 44.5 & 600~\\cite{slack, Morelli_Slack_2006} & 758 & 849 & 890\t& 1.44~\\cite{slack, Morelli_Slack_2006} & 1.95 & 1.96 \\\\\n & & & & & (602) & (674) & & & &\\\\\nCaO & 27~\\cite{Morelli_Slack_2006} & 19.5 & 24.3 & 450~\\cite{slack, Morelli_Slack_2006} & 578 & 620 & 638 & 1.57~\\cite{slack, Morelli_Slack_2006} & 2.07 & 2.06 \t \\\\\n & & & & & (459) & (492) & & & &\\\\\nSrO & 12~\\cite{Morelli_Slack_2006} & 12.5 & 13.4 & 270~\\cite{slack, Morelli_Slack_2006} & 399 & 413 & 421 & 1.52~\\cite{slack, Morelli_Slack_2006} & 2.09 & 2.13 \t \\\\\n & & & & & (317) & (328) & & & &\\\\\nBaO & 2.3~\\cite{Morelli_Slack_2006} & 8.88 & 7.10 & 183~\\cite{slack, Morelli_Slack_2006} & 305 & 288 & 292 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.09 & 2.14 \\\\\n & & & & & (242) & (229) & & & &\\\\\nPbS & 2.9~\\cite{Morelli_Slack_2006} & 6.48 & 6.11 & 115~\\cite{slack, Morelli_Slack_2006} & 226 & 220 & 221 & 2.0~\\cite{slack, Morelli_Slack_2006} & 2.02 & 2.00 \t\\\\\n & & & & & (179) & (175) & & & &\\\\\nPbSe & 2.0~\\cite{Morelli_Slack_2006} & 4.88 & 4.81 & 100~\\cite{Morelli_Slack_2006} & 197 & 194 & 196 & 1.5~\\cite{Morelli_Slack_2006} & 2.1 & 2.07 \t \\\\\n & & & & & (156) & (154) & & & &\\\\\nPbTe & 2.5~\\cite{Morelli_Slack_2006} & 4.15 & 4.07 & 105~\\cite{slack, Morelli_Slack_2006} & 170 & 172 & 175 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.04 & 2.09 \t \\\\\n & & & & & (135) & (137) & & & &\\\\\nSnTe & 1.5~\\cite{Snyder_jmatchem_2011} & 4.46 & 5.24 & 155~\\cite{Snyder_jmatchem_2011} & 202 & 210 & 212 & 2.1~\\cite{Snyder_jmatchem_2011} & 2.15 & 2.11 \t \\\\\n & & & & & (160) & (167) & & & &\\\\\n\\end{tabular}}\n\\label{tab:art115:rocksalt_thermal}\n\\end{table}\n\nThe experimental values $\\gamma^{\\mathrm{exp}}$ of the Gr{\\\"u}neisen parameter are plotted {against\n$\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$}, $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$, $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(f), and the values\nare listed in Table~\\ref{tab:art115:rocksalt_thermal} and in Table~\\ref{tab:art115:rocksalt_thermal_eos}.\nThese results show that {\\small AGL}\\ has problems accurately predicting the Gr{\\\"u}neisen parameter for this set of materials as well, as the calculated values\nare often 30\\% to 50\\% larger than the experimental ones and the {\\small RMSrD}\\ values are of the order of 0.5. Note also that there are quite large differences between the values\nobtained for different equations of state, with $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ generally having the lowest values while\n$\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ has the highest values, a similar pattern to that seen above for the zincblende and diamond structure materials. On the other hand, in contrast to the case of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$,\nthe value of $\\sigma$ used makes little difference to the value of $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$. The correlation values between $\\gamma^{\\mathrm{exp}}$\nand the {\\small AGL}\\ values, as shown in Table~\\ref{tab:art115:rocksalt_correlation}, are also quite poor, with values ranging from -0.098 to\n0.118 for the Pearson correlations, and negative values for the Spearman correlations.\n\nThe experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ is compared in Figure~\\ref{fig:art115:rocksalt_thermal_elastic}(d) to the\nthermal conductivities calculated with {\\small AGL}\\ using the Leibfried-Schl{\\\"o}mann equation (Equation~\\ref{eq:art115:thermal_conductivity}): $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$,\nwhile the values are listed in Table~\\ref{tab:art115:rocksalt_thermal} and in Table~\\ref{tab:art115:rocksalt_thermal_eos}.\nThe linear correlation between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is somewhat better than for the zincblende materials set, with a Pearson\ncorrelation as high as $0.94$, although the Spearman correlations are somewhat lower, ranging from $0.445$\nto $0.556$. In particular, note that using the $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the {\\small AGL}\\ calculations improves the correlations by about\n2\\% to 8\\%, from $0.910$ to $0.932$ and from $0.445$ to $0.528$. For the different equations of state, the results for $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$\nappear to correlate best with $\\kappa^{\\mathrm{exp}}$ for this set of materials.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for rocksalt structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.910 & 0.445 & 1.093 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.932 & 0.528 & 1.002 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.940 & 0.556 & 1.038 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.933 & 0.540 & 0.920 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.930 & 0.554 & 1.082 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.985 & 0.948 & 0.253 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.978 & 0.928 & 0.222 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.980 & 0.926 & 0.222 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.979 & 0.925 & 0.218 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.978 & 0.929 & 0.225 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.118 & -0.064 & 0.477 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.036 & -0.110 & 0.486 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & -0.019 & -0.088 & 0.462 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & -0.098 & -0.086 & 0.591 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.023 & -0.110 & 0.443 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 0.995 & 0.078 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.998 & 0.993 & 0.201 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.997 & 0.993 & 0.199 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.997 & 0.990 & 0.239 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.998 & 0.993 & 0.197 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.994 & 0.997 & 0.105 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.991 & 0.990 & 0.157 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.995 & 0.995 & 0.142 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.959 & 0.827 & 0.070 \\\\\n\\end{tabular}\n\\label{tab:art115:rocksalt_correlation}\n\\end{table}\n\nAs in the case of the diamond and zincblende structure materials discussed in the previous Section,\nReference~\\onlinecite{Morelli_Slack_2006} includes values of the thermal conductivity at 300~K for rocksalt structure materials,\ncalculated using the experimental values of $\\theta_{\\mathrm{a}}$ and $\\gamma$ in the Leibfried-Schl{\\\"o}mann equation, in Table 2.1.\nThe correlation values of $0.986$ and $0.761$ with experiment are\nbetter than those obtained for the {\\small AGL}\\ results by a larger margin than for the zincblende materials.\nNevertheless, the Pearson correlation between the calculated and\nexperimental conductivities is high in both calculations, indicating that the {\\small AGL}\\\napproach may be used as a screening tool for high or low conductivity\ncompounds in cases where gaps exist in the experimental data for these\nmaterials.\n\n\\subsubsection{Hexagonal structure materials}\n\nThe experimental data for this set of materials appears in Table III of Reference~\\onlinecite{curtarolo:art96}, taken from Table 2.3 of\nReference~\\onlinecite{Morelli_Slack_2006}. Most of these materials have the wurtzite structure ($P6_3mc$,\\ $\\#$186;\nPearson symbol: hP4; {\\small AFLOW}\\ prototype: {\\sf AB\\_hP4\\_186\\_b\\_b}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_hP4_186_b_b.html}}) except InSe which is $P6_3mmc$,\\ $\\#$194,\nPearson symbol: hP8.\n\nThe calculated elastic properties are shown in Table~\\ref{tab:art115:wurzite_elastic} and Figure~\\ref{fig:art115:wurzite_thermal_elastic}. The bulk moduli\nvalues obtained from a direct calculation of the elastic tensor, $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, are usually slightly higher than those obtained from the\n$E(V)$ curve and are also closer to experiment (Table~\\ref{tab:art115:wurzite_elastic} and Figure~\\ref{fig:art115:wurzite_thermal_elastic}(a)), with the exception of\nInSe where it is noticeably lower.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of hexagonal structure semiconductors.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nSiC & 219~\\cite{Arlt_ELasticSiC_JAAcS_1965} & 213 & 213 & 204 & 208 & 207 & 207 & 198~\\cite{Arlt_ELasticSiC_JAAcS_1965} & 188 & 182 & 185 & 187 & 0.153~\\cite{Arlt_ELasticSiC_JAAcS_1965} & 0.163 & 0.16 \\\\\nAlN & 211~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993} & 195 & 194 & 187 & 190 & 189 & 189 & 135~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993} & 123 & 122 & 122 & 122 & 0.237~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993} & 0.241 & 0.24 \\\\\n & 200~\\cite{Dodd_BulkmodAlN_JMS_2001} & & & & & & & 130~\\cite{Dodd_BulkmodAlN_JMS_2001} & & & & & 0.234~\\cite{Dodd_BulkmodAlN_JMS_2001} &\\\\\nGaN & 195~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 175 & 172 & 166 & 167 & 166 & 168 & 51.6~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 107 & 105 & 106 & 105 & 0.378~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 0.248 & 0.25 \\\\\n & 210~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & & & 123~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & 0.255~\\cite{Polian_ElasticGaN_JAP_1996} & \\\\\nZnO & 143~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 137 & 130 & 128 & 129 & 127 & 129 & 49.4~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 51.7 & 51.0 & 51.4 & 41.2 & 0.345~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 0.334 & 0.36 \\\\\nBeO & 224.4~\\cite{Cline_JAP_1967} & 206 & 208 & 195 & 195 & 192 & 198 & 168~\\cite{Cline_JAP_1967} & 157 & 154 & 156 & 156 & 0.201~\\cite{Cline_JAP_1967} & 0.198 & 0.2 \\\\\nCdS & 60.7~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 55.4 & 53.3 & 49.7 & 50.3 & 49.4 & 50.6 & 18.2~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 17.6 & 17.0 & 17.3 & 17.6 & 0.364~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 0.358 & 0.35 \\\\\nInSe & 37.1~\\cite{Gatulle_ElasticInSe_PSSb_1983} & 19.2 & N\/A & 39.8 & 40.8 & 39.7 & 41.0 & 14.8~\\cite{Gatulle_ElasticInSe_PSSb_1983} & 14.9 & 12.3 & 13.6 & N\/A & 0.324~\\cite{Gatulle_ElasticInSe_PSSb_1983} & 0.214 & N\/A \\\\\nInN & 126~\\cite{Ueno_BulkmodInN_PRB_1994} & 124 & N\/A & 118 & 120 & 119 & 119 & N\/A & 55.4 & 54.4 & 54.9 & N\/A & N\/A & 0.308 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_elastic}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.98\\linewidth]{fig031}\n\\mycaption[\n({\\bf a}) Bulk modulus,\n({\\bf b}) shear modulus,\n({\\bf c}) Poisson ratio,\n({\\bf d}) lattice thermal conductivity,\n({\\bf e}) Debye temperature and\n({\\bf f}) Gr{\\\"u}neisen parameter of hexagonal structure\nsemiconductors.]\n{The Debye temperatures plotted in ({\\bf e}) are\n$\\theta_{\\mathrm{a}}$, except for InSe and InN where $\\theta_{\\mathrm D}$\nvalues are quoted in References~\\onlinecite{Snyder_jmatchem_2011, Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998}.}\n\\label{fig:art115:wurzite_thermal_elastic}\n\\end{figure}\n\nFor the shear modulus, the experimental values $G^{\\mathrm{exp}}$ are compared to the {\\small AEL}\\ values $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$,\n$G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$. As can be seen in\nTable~\\ref{tab:art115:wurzite_elastic} and Figure~\\ref{fig:art115:wurzite_thermal_elastic}(b), the agreement with the experimental values is very\ngood. Similarly good agreement is obtained for the Poisson ratio of most materials (Table~\\ref{tab:art115:wurzite_elastic}\nand Figure~\\ref{fig:art115:wurzite_thermal_elastic}(c)), with\na single exception for InSe where the calculation deviates significantly from the experiment.\nThe Pearson (\\nobreak\\mbox{\\it i.e.}, linear, Equation~\\ref{eq:art115:Pearson}) and Spearman (\\nobreak\\mbox{\\it i.e.}, rank order, Equation~\\ref{eq:art115:Spearman}) correlations between the calculated\nelastic properties and their experimental values are generally\nquite high (Table~\\ref{tab:art115:wurzite_correlation}), ranging from 0.851 and 0.893 respectively for $\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, up to 0.998\nand 1.0 for $G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$.\n\nThe Materials Project values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ for hexagonal structure materials are also shown in\nTable~\\ref{tab:art115:wurzite_elastic}, where available. The Pearson correlations values for the experimental results with the available values of\n$B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ were calculated to be 0.984, 0.998 and 0.993, respectively, while the respective Spearman correlations\nwere 0.943, 1.0 and 0.943, and the {\\small RMSrD}\\ values were 0.117, 0.116 and 0.034. For comparison, the corresponding Pearson correlations for the same\nsubset of materials for $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ are 0.986, 0.998, and 0.998 respectively, while the respective Spearman correlations\nwere 0.943, 1.0 and 1.0, and the {\\small RMSrD}\\ values were 0.100, 0.091 and 0.036. These correlation values are very similar, and the general close agreement\nfor the results for the values of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ with those of $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$, $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ and $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$\ndemonstrate that the small differences in the parameters used for the {\\small DFT}\\ calculations make little difference to the results,\nindicating that the parameter set used here is robust for high-throughput calculations.\n\nThe thermal properties calculated using {\\small AGL}\\ are\ncompared to the experimental values in Table~\\ref{tab:art115:wurzite_thermal} and are also plotted in Figure~\\ref{fig:art115:wurzite_thermal_elastic}.\nFor the Debye temperature, the $\\theta^{\\mathrm{exp}}$ values taken from Reference~\\onlinecite{Morelli_Slack_2006} are for $\\theta_{\\mathrm{a}}$,\nand are mostly in good agreement with the calculated $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ values. As in the case of the other materials sets,\nthe values obtained using the numerical $E(V)$ fit and the three different\nequations of state are very similar to each other, whereas $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated using $\\sigma=0.25$ differs significantly.\nIn fact, the values of $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ calculated with $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ have a lower the correlation with $\\theta^{\\mathrm{exp}}$ than the values calculated with\n$\\sigma = 0.25$ do, although the {\\small RMSrD}\\ values are lower when $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ is used. However, most of this discrepancy appears to be due to the clear\noutlier value for the material InN. When the values for this material are removed from the data set, the Pearson correlation values become very similar\nwhen both the $\\sigma = 0.25$ and $\\sigma = \\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ values are used, increasing to 0.995 and 0.994 respectively.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperature and Gr{\\\"u}neisen parameter\nof hexagonal structure semiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are $\\theta_{\\mathrm{a}}$,\nexcept 190K for InSe~\\cite{Snyder_jmatchem_2011} and 660K for InN~\\cite{Ioffe_Inst_DB,Krukowski_jphyschemsolids_1998}\nwhich are $\\theta_{\\mathrm D}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nSiC & 490~\\cite{Morelli_Slack_2006} & 42.49 & 70.36 & 740~\\cite{Morelli_Slack_2006} & 930 & 1103 & 1138 & 0.75~\\cite{Morelli_Slack_2006} & 1.86 & 1.86 \\\\\n & & & & & (586) & (695) & & & & \\\\\nAlN & 350~\\cite{Morelli_Slack_2006} & 36.73 & 39.0 & 620~\\cite{Morelli_Slack_2006} & 880 & 898 & 917 & 0.7~\\cite{Morelli_Slack_2006} & 1.85 & 1.85 \t \\\\\n & & & & & (554) & (566) & & & & \\\\\nGaN & 210~\\cite{Morelli_Slack_2006} & 18.17 & 18.54 & 390~\\cite{Morelli_Slack_2006} & 592 & 595 & 606 & 0.7~\\cite{Morelli_Slack_2006} & 2.07 & 2.08 \t \\\\\n & & & & & (373) & (375) & & & & \\\\\nZnO & 60~\\cite{Morelli_Slack_2006} & 14.10 & 7.39 & 303~\\cite{Morelli_Slack_2006} & 525 & 422 & 427 & 0.75~\\cite{Morelli_Slack_2006} & 1.97 & 1.94 \t \\\\\n & & & & & (331) & (266) & & & & \\\\\nBeO & 370~\\cite{Morelli_Slack_2006} & 39.26 & 53.36 & 809~\\cite{Morelli_Slack_2006} & 1065 & 1181 & 1235 & 1.38~\\cite{Slack_JAP_1975, Cline_JAP_1967, Morelli_Slack_2006} & 1.76 & 1.76 \t \\\\\n & & & & & (671) & (744) & & & & \\\\\nCdS & 16~\\cite{Morelli_Slack_2006} & 4.40 & 1.76 & 135~\\cite{Morelli_Slack_2006} & 287 & 211 & 213 & 0.75~\\cite{Morelli_Slack_2006} & 2.14 & 2.14 \t \\\\\n & & & & & (181) & (133) & & & & \\\\\nInSe & 6.9~\\cite{Snyder_jmatchem_2011} & 1.84 & 2.34 & 190~\\cite{Snyder_jmatchem_2011} & 230 & 249 & 168 & 1.2~\\cite{Snyder_jmatchem_2011} & 2.24 & 2.24 \t \\\\\n & & & & & (115) & (125) & & & & \\\\\nInN & 45~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 10.44 & 6.82 & 660~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 426 & 369 & 370 & 0.97~\\cite{Krukowski_jphyschemsolids_1998} & 2.17 & 2.18 \t \\\\\n & & & & & (268) & (232) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_thermal}\n\\end{table}\n\nThe experimental and calculated values of the Gr{\\\"u}neisen parameter are listed in Table~\\ref{tab:art115:wurzite_thermal}\nand in Table~\\ref{tab:art115:wurzite_thermal_eos}, and are plotted in Figure~\\ref{fig:art115:wurzite_thermal_elastic}(f).\nAgain, the Debye model does not reproduce the experimental data, as the calculated values\nare often 2 to 3 times too large and the {\\small RMSrD}\\ is larger than 1.5.\nThe corresponding correlation, shown in Table~\\ref{tab:art115:wurzite_correlation}, are also quite poor, with no value higher than 0.160 for\nthe Spearman correlations, and negative values for the Pearson correlations.\n\nThe comparison between the experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ and the calculated values is also quite poor\n(Figure~\\ref{fig:art115:wurzite_thermal_elastic}(d) and Table~\\ref{tab:art115:wurzite_thermal}), with {\\small RMSrD}\\ values of the order of 0.9.\nConsiderable disagreements also exist between different experimental reports for most materials.\nNevertheless, the Pearson correlations between the {\\small AGL}\\ calculated thermal conductivity values and the experimental\nvalues are high, ranging from $0.974$ to $0.980$, while the Spearman correlations are even higher, ranging from $0.976$\nto $1.0$.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for hexagonal structure semiconductors.}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.977 & 1.0 & 0.887 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.980 & 0.976 & 0.911 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.974 & 0.976 & 0.904 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.980 & 0.976 & 0.926 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.980 & 0.976 & 0.895 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.960 & 0.976 & 0.233 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.921 & 0.929 & 0.216 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.921 & 0.929 & 0.217 \\\\\n$\\theta_{\\mathrm{a}}{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.920 & 0.929 & 0.218 \\\\\n$\\theta_{\\mathrm{a}}^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.921 & 0.929 & 0.216 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & -0.039 & 0.160 & 1.566 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & -0.029 & 0.160 & 1.563 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & -0.124 & -0.233 & 1.547 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & -0.043 & 0.012 & 1.677 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ & -0.054 & 0.098 & 1.467 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.990 & 0.976 & 0.201 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.990 & 0.976 & 0.138 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.988 & 0.976 & 0.133 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.988 & 0.976 & 0.139 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.990 & 0.976 & 0.130 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 1.0 & 0.090 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 1.0 & 0.076 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.998 & 1.0 & 0.115 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & 0.851 & 0.893 & 0.143 \\\\\n\\end{tabular}\n\\label{tab:art115:wurzite_correlation}\n\\end{table}\n\nAs for the rocksalt and zincblende material sets, Reference~\\onlinecite{Morelli_Slack_2006} (Table 2.3) includes\nvalues of the thermal conductivity at 300~K for wurtzite structure materials, calculated using the\nexperimental values of the Debye temperature and Gr{\\\"u}neisen parameter in the Leibfried-Schl{\\\"o}mann equation.\nThe Pearson and Spearman correlations are $0.996$ and $1.0$ respectively, which are slightly higher than the correlations obtained using\nthe {\\small AGL}\\ calculated quantities. The difference is insignificant since all of these\ncorrelations are very high and\ncould reliably serve as a screening tool of the thermal conductivity.\nHowever, as we noted in our previous work on {\\small AGL}~\\cite{curtarolo:art96}, the high correlations calculated with the\nexperimental $\\theta_{\\mathrm{a}}$ and $\\gamma$ were obtained using\n$\\gamma=0.75$ for BeO. Table 2.3 of\nReference~\\onlinecite{Morelli_Slack_2006} also cites an alternative value\nof $\\gamma=1.38$ for BeO (Table~\\ref{tab:art115:wurzite_thermal}). Using this outlier\nvalue would severely degrade the results down to $0.7$, for the\nPearson correlation, and $0.829$, for the Spearman correlation.\nThese values are too low for a reliable screening tool. This\ndemonstrates the ability of the\n{\\small AEL}-{\\small AGL}\\ calculations to compensate for anomalies in the\nexperimental data when\nthey exist and still provide a reliable screening method for the\nthermal conductivity.\n\n\\subsubsection{Rhombohedral materials}\n\nThe elastic properties of a few materials with rhombohedral structures\n(spacegroups: $R\\overline{3}mR$,\\ $\\#$166, $R\\overline{3}mH$,\\ $\\#$166; Pearson symbol: hR5; {\\small AFLOW}\\ prototype: {\\sf A2B3\\_hR5\\_166\\_c\\_ac}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A2B3_hR5_166_c_ac.html}};\nand spacegroup: $R\\overline{3}cH$,\\ $\\#$167; Pearson symbol: hR10; {\\small AFLOW}\\ prototype: {\\sf A2B3\\_hR10\\_167\\_c\\_e}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A2B3_hR10_167_c_e.html}})\nare shown in Table~\\ref{tab:art115:rhombo_elastic} (we have left out the material Fe$_2$O$_3$ which was included in\nthe data set in Table IV of Reference~\\onlinecite{curtarolo:art96}, due to convergence issues with some of the\nstrained structures required for the calculation of the elastic tensor).\nThe comparison between experiment and calculation is qualitatively reasonable, but the scarcity of experimental results\ndoes not allow for a proper correlation analysis.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of rhombohedral semiconductors.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nBi$_2$Te$_3$ & 37.0~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972} & 28.8 & 15.0 & 43.7 & 44.4 & 43.3 & 44.5 & 22.4~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972} & 23.5 & 16.3 & 19.9 & 10.9 & 0.248~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972} & 0.219 & 0.21 \\\\\nSb$_2$Te$_3$ & N\/A & 22.9 & N\/A & 45.3 & 46.0 & 45.2 & 46.0 & N\/A & 20.6 & 14.5 & 17.6 & N\/A & N\/A & 0.195 & N\/A \\\\\nAl$_2$O$_3$ & 254~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 231 & 232 & 222 & 225 & 224 & 224 & 163.1~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 149 & 144 & 147 & 147 & 0.235~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 0.238 & 0.24 \\\\\nCr$_2$O$_3$ & 234~\\cite{Alberts_ElasticCr2O3_JMMM_1976} & 203 & 203 & 198 & 202 & 201 & 201 & 129~\\cite{Alberts_ElasticCr2O3_JMMM_1976} & 115 & 112 & 113 & 113 & 0.266~\\cite{Alberts_ElasticCr2O3_JMMM_1976} & 0.265 & 0.27 \\\\\nBi$_2$Se$_3$ & N\/A & 93.9 & N\/A & 57.0 & 57.5 & 56.4 & 57.9 & N\/A & 53.7 & 28.0 & 40.9 & N\/A & N\/A & 0.310 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_elastic}\n\\end{table}\n\nThe thermal properties calculated using {\\small AGL}\\ are\ncompared to the experimental values in Table~\\ref{tab:art115:rhombo_thermal} and the thermal conductivity is also plotted in\nFigure~\\ref{fig:art115:mixed_thermal}(a).\nThe experimental Debye temperatures are $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ for Bi$_2$Te$_3$ and Sb$_2$Te$_3$, and\n$\\theta_{\\mathrm{a}}$ for Al$_2$O$_3$. The values obtained using the numerical $E(V)$ fit and the three different equations of state\n(see Table~\\ref{tab:art115:rhombo_thermal_eos})\nare very similar, but just roughly reproduce the experiments.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperatures and Gr{\\\"u}neisen parameter of rhombohedral\nsemiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$ for\nBi$_2$Te$_3$ and Sb$_2$Te$_3$, and $\\theta_{\\mathrm{a}}$ for Al$_2$O$_3$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nBi$_2$Te$_3$ & 1.6~\\cite{Snyder_jmatchem_2011} & 2.79 & 3.35 & 155~\\cite{Snyder_jmatchem_2011} & 191 & 204 & 161 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.13 & 2.14 \t \\\\\n & & & & & (112) & (119) & & & & \\\\\nSb$_2$Te$_3$ & 2.4~\\cite{Snyder_jmatchem_2011} & 2.90 & 4.46 & 160~\\cite{Snyder_jmatchem_2011} & 217 & 243 & 170 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.2 & 2.11 \t\\\\\n & & & & & (127) & (142) & & & & \\\\\nAl$_2$O$_3$ & 30~\\cite{Slack_PR_1962} & 20.21 & 21.92 & 390~\\cite{slack} & 927 & 952 & 975 & 1.32~\\cite{slack} & 1.91 & 1.91 \t \\\\\n & & & & & (430) & (442) & & & & \\\\\nCr$_2$O$_3$ & 16~\\cite{Landolt-Bornstein, Bruce_PRB_1977} & 10.87 & 12.03 & N\/A & 733 & 717 & 720 & N\/A & 2.26 & 2.10 \t\\\\\n & & & & & (340) & (333) & & & & \\\\\nBi$_2$Se$_3$ & 1.34~\\cite{Landolt-Bornstein} & 3.60 & 2.41 & N\/A & 223 & 199 & 241 & N\/A & 2.08 & 2.12 \t\\\\\n & & & & & (130) & (116) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_thermal}\n\\end{table}\n\nThe calculated Gr{\\\"u}neisen parameters are about 50\\% larger than the experimental ones, and\nthe value of $\\sigma$ used makes a little difference in the calculation.\nThe absolute agreement between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is also quite poor (Figure~\\ref{fig:art115:mixed_thermal}(a)).\nHowever, despite all these discrepancies,\nthe Pearson correlations between the calculated thermal conductivities and the experimental\nvalues are all high, of the order of $0.998$, while the Spearman correlations range from $0.7$ to $1.0$,\nwith all of the different equations of state having very similar correlations with experiment.\nUsing the calculated $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, \\nobreak\\mbox{\\it vs.}\\ the rough Cauchy approximation, improves the Spearman correlation from $0.7$ to $1.0$.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig032}\n\\mycaption{\n({\\bf a}) Lattice thermal conductivity of rhombohedral semiconductors at 300~K.\n({\\bf b}) Lattice thermal conductivity of body-centered tetragonal semiconductors at 300~K.\n}\n\\label{fig:art115:mixed_thermal}\n\\end{figure}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for rhombohedral structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.997 & 0.7 & 0.955 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.998 & 1.0 & 0.821 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.997 & 1.0 & 0.931 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.998 & 1.0 & 0.741 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.997 & 1.0 & 1.002 \\\\\n\\end{tabular}\n\\label{tab:art115:rhombo_correlation}\n\\end{table}\n\n\\subsubsection{Body-centered tetragonal materials}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of body-centered tetragonal semiconductors.]\n{Note that there appears to be an error in Table 1 of Reference~\\onlinecite{Fernandez_ElasticCuInTe_PSSa_1990}\nwhere the bulk modulus values are stated to be in units of $10^{12}$ Pa.\nThis seems unlikely, as that would give a bulk modulus for CuInTe$_2$ an order of magnitude larger than\nthat for diamond.\nAlso, units of $10^{12}$ Pa would be inconsistent with the experimental results listed in\nReference~\\onlinecite{Neumann_ElasticCuInTe_PSSa_1986},\nso therefore it seems that these values are in units of\n$10^{10}$ Pa, which are the values shown here.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ {in} {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nCuGaTe$_2$ & N\/A & 47.0 & N\/A & 42.5 & 43.2 & 42.0 & 43.5 & N\/A & 25.1 & 22.1 & 23.6 & N\/A & N\/A & 0.285 & N\/A \\\\\nZnGeP$_2$ & N\/A & 73.1 & 74.9 & 70.1 & 71.1 & 70.0 & 71.4 & N\/A & 50.5 & 46.2 & 48.4 & 48.9 & N\/A & 0.229 & 0.23 \\\\\nZnSiAs$_2$ & N\/A & 67.4 & 65.9 & 63.4 & 64.3 & 63.1 & 64.6 & N\/A & 44.4 & 40.4 & 42.4 & 42.2 & N\/A & 0.240 & 0.24 \\\\\nCuInTe$_2$ & 36.0~\\cite{Neumann_ElasticCuInTe_PSSa_1986} & 53.9 & N\/A & 38.6 & 39.2 & 38.2 & 39.4 & N\/A & 20.4 & 17.2 & 18.8 & N\/A & 0.313~\\cite{Fernandez_ElasticCuInTe_PSSa_1990} & 0.344 & N\/A \\\\\n & 45.4~\\cite{Fernandez_ElasticCuInTe_PSSa_1990} & & & & & & & & & & & & & \\\\\nAgGaS$_2$ & 67.0~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975} & 70.3 & N\/A & 56.2 & 57.1 & 56.0 & 57.4 & 20.8~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975} & 20.7 & 17.4 & 19.1 & N\/A & 0.359~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975} & 0.375 & N\/A \\\\\nCdGeP$_2$ & N\/A & 65.3 & 65.2 & 60.7 & 61.6 & 60.4 & 61.9 & N\/A & 37.7 & 33.3 & 35.5 & 35.0 & N\/A & 0.270 & 0.27 \\\\\nCdGeAs$_2$ & 69.9~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982} & 52.6 & N\/A & 49.2 & 49.6 & 48.3 & 49.9 & 29.5~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982} & 30.9 & 26.2 & 28.6 & N\/A & 0.315~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982} & 0.270 & N\/A \\\\\nCuGaS$_2$ & 94.0~\\cite{Bettini_ElasticCuGaS_SSC_1975} & 73.3 & N\/A & 69.0 & 69.9 & 68.7 & 70.6 & N\/A & 37.8 & 32.4 & 35.1 & N\/A & N\/A & 0.293 & N\/A \\\\\nCuGaSe$_2$ & N\/A & 69.9 & N\/A & 54.9 & 55.6 & 54.4 & 56.0 & N\/A & 30.3 & 26.0 & 28.1 & N\/A & N\/A & 0.322 & N\/A \\\\\nZnGeAs$_2$ & N\/A & 59.0 & N\/A & 56.2 & 56.7 & 55.5 & 57.1 & N\/A & 39.0 & 35.6 & 37.3 & N\/A & N\/A & 0.239 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:bct_elastic}\n\\end{table}\n\nThe mechanical properties of the body-centered tetragonal materials (spacegroup:\n$I\\overline{4}2d$,\\ $\\#$122; Pearson symbol: tI16; {\\small AFLOW}\\ prototype: {\\sf ABC2\\_tI16\\_122\\_a\\_b\\_d}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/ABC2_tI16_122_a_b_d.html}})\nof Table V of Reference~\\onlinecite{curtarolo:art96} are reported in Table~\\ref{tab:art115:bct_elastic}.\nThe calculated bulk moduli miss considerably the few available experimental results, while the shear moduli\nare well reproduced. Reasonable estimates are also obtained for the Poisson ratio.\n\nThe thermal properties are reported in Table~\\ref{tab:art115:bct_thermal} and Figure~\\ref{fig:art115:mixed_thermal}(b).\nThe $\\theta^{\\mathrm{exp}}$ values are all for $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, and in most cases are in good agreement with the values obtained\nwith the {\\small AEL}\\ calculated $\\sigma$. The\nvalues from the numerical $E(V)$ fit and the three different equations of state are again very similar, but differ significantly\nfrom {$\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$} calculated with $\\sigma=0.25$.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures and Gr{\\\"u}neisen parameter of body-centered tetragonal\nsemiconductors, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nCuGaTe$_2$ & 2.2~\\cite{Snyder_jmatchem_2011} & 1.77 & 1.36 & 226~\\cite{Snyder_jmatchem_2011} & 234 & 215 & 218 & 1.46~\\cite{Snyder_jmatchem_2011} & 2.32 & 2.32 \t \\\\\n & & & & & (117) & (108) & & & & \\\\\nZnGeP$_2$ & 35~\\cite{Landolt-Bornstein, Beasley_AO_1994} & 4.45 & 5.07 & 500~\\cite{Landolt-Bornstein} & 390 & 408 & 411 & N\/A & 2.13 & 2.14 \t \\\\\n & 36~\\cite{Landolt-Bornstein, Beasley_AO_1994} & & & 428~\\cite{Abrahams_JCP_1975} & (195) & (204) & & & & \\\\\n & 18~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & & & & & & & & & \\\\\nZnSiAs$_2$ & 14\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 3.70 & 3.96 & 347~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1981} & 342 & 350 & 354 & N\/A & 2.15 & 2.15 \t \\\\\n & & & & & (171) & (175) & & & & \\\\\nCuInTe$_2$ & 10\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 1.55 & 0.722 & 185~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 215 & 166 & 185 & 0.93~\\cite{Rincon_PSSa_1995} & 2.33 & 2.32 \t \\\\\n & & & & 195~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & (108) & (83) & & & &\\\\\nAgGaS$_2$ & 1.4\\cite{Landolt-Bornstein, Beasley_AO_1994} & 2.97 & 0.993 & 255~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 324 & 224 & 237 & N\/A & 2.20 & 2.20 \t\\\\\n & & & & & (162) & (112) & & & & \\\\\nCdGeP$_2$ & 11~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 3.40 & 2.96 & 340~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 335 & 320 & 324 & N\/A & 2.20 & 2.21 \\\\\n & & & & & (168) & (160) & & & & \\\\\nCdGeAs$_2$ & 42~\\cite{Landolt-Bornstein, Shay_1975} & 2.44 & 2.11 & 241~\\cite{Bohnhammel_PSSa_1981} & 266 & 254 & 255 & N\/A & 2.20 & 2.20 \t\\\\\n & & & & & (133) & (127) & & & &\\\\\nCuGaS$_2$ & 5.09~\\cite{Landolt-Bornstein} & 3.78 & 2.79 & 356~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 387 & 349 & 349 & N\/A & 2.24 & 2.24 \t \\\\\n & & & & & (194) & (175) & & & &\\\\\nCuGaSe$_2$ & 12.9~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 2.54 & 1.46 & 262~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & 294 & 244 & 265 & N\/A & 2.27 & 2.26 \t \\\\\n & & & & & (147) & (122) & & & &\\\\\nZnGeAs$_2$ & 11\\cite{Landolt-Bornstein, Shay_1975} & 2.95 & 3.18 & N\/A & 299 & 307 & 308 & N\/A & 2.16 & 2.17 \t \\\\\n & & & & & (150) & (154) & & & &\\\\\n\\end{tabular}}\n\\label{tab:art115:bct_thermal}\n\\end{table}\n\nThe comparison of the experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ to the calculated values, in Figure~\\ref{fig:art115:mixed_thermal}(b),\nshows poor reproducibility. The available data can thus only be considered a rough indication of their order of magnitude.\nThe Pearson and Spearman correlations are also quite low for all types of calculation,\nbut somewhat better when the calculated $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ is used instead of the Cauchy approximation.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for body-centered tetragonal structure semiconductors.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.265 & 0.201 & 0.812 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.472 & 0.608 & 0.766 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.467 & 0.608 & 0.750 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.464 & 0.608 & 0.778 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.460 & 0.608 & 0.741 \\\\\n\\end{tabular}\n\\label{tab:art115:bct_correlation}\n\\end{table}\n\n\\subsubsection{Miscellaneous materials}\n\nIn this Section we consider materials with various other structures, as in Table VI of Reference~\\onlinecite{curtarolo:art96}:\nCoSb$_3$ and IrSb$_3$\n(spacegroup: $Im\\overline{3}$,\\ $\\#$204; Pearson symbol: cI32; {\\small AFLOW}\\ prototype: {\\sf A3B\\_cI32\\_204\\_g\\_c}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A3B_cI32_204_g_c.html}}),\nZnSb ($Pbca$,\\ $\\#$61; oP16; {\\small AFLOW}\\ prototype: {\\sf AB\\_oP16\\_61\\_c\\_c}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_oP16_61_c_c.html}}),\nSb$_2$O$_3$ ($Pccn$,\\ $\\#$56; oP20), InTe ($Pm\\overline{3}m$,\\ $\\#$221; cP2; {\\small AFLOW}\\ prototype: {\\sf AB\\_cP2\\_221\\_b\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/AB_cP2_221_b_a.html}},\nand $I4\/mcm$,\\ $\\#$140; tI16), Bi$_2$O$_3$ ($P121\/c1,\\ \\#14$; mP20); and SnO$_2$ ($P42\/mnm,\\ \\#136$; tP6; {\\sf A2B\\_tP6\\_136\\_f\\_a}~\\cite{aflowANRL}\\footnote{\\url{http:\/\/aflow.org\/CrystalDatabase\/A2B_tP6_136_f_a.html}}).\nTwo different structures are listed for InTe. In Reference~\\onlinecite{curtarolo:art96}, we\nconsidered its simple cubic structure, but this is a high-pressure phase~\\cite{Chattopadhyay_BulkModInTe_JPCS_1985}, while the ambient\npressure phase is body-centered tetragonal. It appears that the thermal conductivity results should be for the body-centered tetragonal\nphase~\\cite{Spitzer_JPCS_1970}, therefore both sets of results are reported here. The correlation values shown in the tables below\nwere calculated for the body-centered tetragonal structure.\n\nThe elastic properties are shown\nin Table~\\ref{tab:art115:misc_elastic}. Large discrepancies appear between the results of all calculations\nand the few available experimental results.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of materials with various\nstructures.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ {in} {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{MP}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{MP}}}$ \\\\\n\\hline\nCoSb$_3$ & $cI32$ & N\/A & 78.6 & 82.9 & 75.6 & 76.1 & 75.1 & 76.3 & N\/A & 57.2 & 55.1 & 56.2 & 57.0 & N\/A & 0.211 & 0.22 \\\\\nIrSb$_3$ & $cI32$ & N\/A & 97.5 & 98.7 & 94.3 & 94.8 & 93.8 & 95.5 & N\/A & 60.9 & 59.4 & 60.1 & 59.7& N\/A & 0.244 & 0.25 \\\\\nZnSb & $oP16$ & N\/A & 47.7 & 47.8 & 46.7 & 47.0 & 46.0 & 47.7 & N\/A & 29.2 & 27.0 & 28.1 & 28.2 & N\/A & 0.253 & 0.25 \\\\\nSb$_2$O$_3$ & $oP20$ & N\/A & 16.5 & 19.1 & 97.8 & 98.7 & 97.8 & 98.7 & N\/A & 22.8 & 16.4 & 19.6 & 20.4 & N\/A & 0.0749 & 0.11 \\\\\nInTe & $cP2$ & 90.2~\\cite{Chattopadhyay_BulkModInTe_JPCS_1985} & 41.7 & N\/A & 34.9 & 34.4 & 33.6 & 34.7 & N\/A & 8.41 & 8.31 & 8.36 & N\/A& N\/A & 0.406 & N\/A \\\\\nInTe & $tI16$ & 46.5~\\cite{Chattopadhyay_BulkModInTe_JPCS_1985} & 20.9 & N\/A & 32.3 & 33.1 & 32.2 & 33.2 & N\/A & 13.4 & 13.0 & 13.2 & N\/A & N\/A & 0.239 & N\/A \\\\\nBi$_2$O$_3$ & $mP20$ & N\/A & 48.0 & 54.5 & 108 & 110 & 109 & 109 & N\/A & 30.3 & 25.9 & 28.1 & 29.9 & N\/A & 0.255 & 0.27 \\\\\nSnO$_2$ & $tP6$ & 212~\\cite{Chang_ElasticSnO2_JGPR_1975} & 159 & N\/A & 158 & 162 & 161 & 161 & 106~\\cite{Chang_ElasticSnO2_JGPR_1975} & 86.7 & 65.7 & 76.2 & N\/A & 0.285~\\cite{Chang_ElasticSnO2_JGPR_1975} & 0.293 & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:misc_elastic}\n\\end{table}\n\nThe thermal properties are\ncompared to the experimental values in Table~\\ref{tab:art115:misc_thermal}.\nThe experimental Debye temperatures are for $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, except ZnSb for which it is $\\theta_{\\mathrm{a}}$. Good agreement\nis found between calculation and the few available experimental values. Again, the numerical $E(V)$ fit and the three different\nequations of state give similar results.\nFor the Gr{\\\"u}neisen parameter, experiment and calculations again differ considerably, while the changes due to the different\nvalues of $\\sigma$ used in the\ncalculations are negligible.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures\nand Gr{\\\"u}neisen parameter of materials with various structures, comparing the effect of using the\ncalculated value of the Poisson ratio to the previous approximation of $\\sigma = 0.25$.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$,\nexcept ZnSb for which it is $\\theta_{\\mathrm{a}}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ \\\\\n & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & & & & \\\\\n & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & & & & ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & \\\\\n\\hline\nCoSb$_3$ & $cI32$ & 10~\\cite{Snyder_jmatchem_2011} & 1.60 & 2.60 & 307~\\cite{Snyder_jmatchem_2011} & 284 & 310 & 312 & 0.95~\\cite{Snyder_jmatchem_2011} & 2.63 & 2.33 \\\\\n & & & & & & (113) & (123) & & & & \\\\\nIrSb$_3$ & $cI32$ & 16~\\cite{Snyder_jmatchem_2011} & 2.64 & 2.73 & 308~\\cite{Snyder_jmatchem_2011} & 283 & 286 & 286 & 1.42~\\cite{Snyder_jmatchem_2011} & 2.34 & 2.34 \\\\\n & & & & & & (112) & (113) & & & & \\\\\nZnSb & $oP16$ & 3.5~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 1.24 & 1.23 & 92~\\cite{Madsen_PRB_2014} & 244 & 242 & 237 & 0.76~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 2.24 & 2.23 \t \\\\\n & & & & & & (97) & (96) & & & & \\\\\nSb$_2$O$_3$ & $oP20$ & 0.4~\\cite{Landolt-Bornstein} & 3.45 & 8.74 & N\/A & 418 & 572 & 238 & N\/A & 2.13 & 2.12 \t\\\\\n & & & & & & (154) & (211) & & & & \\\\\nInTe & $cP2$ & N\/A & 3.12 & 0.709 & N\/A & 191 & 113 & 116 & N\/A & 2.28 & 2.19 \t\\\\\n & & & & & & (152) & (90) & & & & \\\\\nInTe & $tP16$ & 1.7~\\cite{Snyder_jmatchem_2011, Spitzer_JPCS_1970} & 1.32 & 1.40 & 186~\\cite{Snyder_jmatchem_2011} & 189 & 193 & 150 & 1.0~\\cite{Snyder_jmatchem_2011} & 2.23 & 2.24 \t\\\\\n & & & & & & (95) & (97) & & & & \\\\\nBi$_2$O$_3$ & $mP20$ & 0.8~\\cite{Landolt-Bornstein} & 3.04 & 2.98 & N\/A & 345 & 342 & 223 & N\/A & 2.10 & 2.10 \t \\\\\n & & & & & & (127) & (126) & & & & \\\\\nSnO$_2$ & $tP6$ & 98~\\cite{Turkes_jpcss_1980} & 9.56 & 6.98 & N\/A & 541 & 487 & 480 & N\/A & 2.48 & 2.42 \t \\\\\n & & 55~\\cite{Turkes_jpcss_1980} & & & & (298) & (268) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:misc_thermal}\n\\end{table}\n\nThe experimental thermal conductivity $\\kappa^{\\mathrm{exp}}$ is compared in Table~\\ref{tab:art115:misc_thermal} to the thermal conductivity\ncalculated with {\\small AGL}\\ using the\nLeibfried-Schl{\\\"o}mann equation (Equation~\\ref{eq:art115:thermal_conductivity}) for $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, while the values obtained for $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$\nand $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ are listed in Table~\\ref{tab:art115:misc_thermal_eos}.\nThe absolute agreement between the {\\small AGL}\\ values and $\\kappa^{\\mathrm{exp}}$ is quite poor.\nThe scarcity of\nexperimental data from different sources\non the thermal properties of these materials prevents reaching definite conclusions regarding the true values of these\nproperties. The available data can thus\nonly be considered as a rough indication of their order of magnitude.\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for materials with miscellaneous structures.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.937 & 0.071 & 3.38 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.438 & -0.143 & 8.61 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.498 & -0.143 & 8.81 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.445 & 0.0 & 8.01 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.525 & -0.143 & 9.08 \\\\\n\\end{tabular}\n\\label{tab:art115:misc_correlation}\n\\end{table}\n\nFor these materials, the Pearson correlation between the calculated\nand experimental values of the thermal conductivity ranges from $0.438$ to $0.937$, while the corresponding\nSpearman correlations range from $-0.143$ to $0.071$. In this case, using $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the {\\small AGL}\\\ncalculations does not improve the correlations, instead actually lowering the values somewhat.\nHowever, it should be noted that the Pearson correlation is heavily influenced by the values for SnO$_2$.\nWhen this entry is removed from the list, the Pearson correlation values fall to $-0.471$ and $-0.466$\nwhen the $\\sigma = 0.25$ and $\\sigma = \\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ values are used, respectively.\nThe low correlation values, particularly for the Spearman correlation, for this set of materials demonstrates the\nimportance of the information about the material structure when interpreting results obtained using the {\\small AGL}\\ method\nin order to identify candidate materials for specific thermal applications. This is partly due to the fact that the Gr{\\\"u}neisen\nparameter values tend to be similar for materials with the same\nstructure. Therefore, the effect of the Gr{\\\"u}neisen parameter on the ordinal ranking of\nthe lattice thermal conductivity of materials with the same structure\nis small.\n\n\\subsubsection{Thermomechanical properties from LDA}\n\n{\nThe thermomechanical properties of a randomly-selected subset of the materials investigated in this work were calculated using {\\small LDA}\\\nin order to check the impact of the choice of exchange-correlation functional on the results. For the {\\small LDA}\\ calculations, all structures were\nfirst re-relaxed using the {\\small LDA}\\ exchange-correlation functional with {\\small VASP}\\ using the appropriate parameters and potentials as\ndescribed in the {\\small AFLOW}\\ standard~\\cite{curtarolo:art104}, and then the appropriate strained structures were calculated using {\\small LDA}.\nThese calculations were restricted to a subset of materials to limit the total number of additional first-principles calculations required, and the materials were\nselected randomly from each of the sets in the previous sections so as to cover as wide a range of different structure types as possible, given the available experimental data.\nResults for elastic properties obtained using {\\small LDA}, {\\small GGA}\\ and experimental measurements are shown in Table~\\ref{tab:art115:LDA_elastic}, while the thermal properties are shown in\nTable~\\ref{tab:art115:LDA_thermal}. All thermal properties listed in Table~\\ref{tab:art115:LDA_thermal} were calculated using $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$ in the expression\nfor the Debye temperature.}\n\n{\nIn general, the {\\small LDA}\\ values for elastic and thermal properties are slightly higher than the {\\small GGA}\\ values, as would be generally expected\ndue to their relative tendencies to overbind and underbind, respectively~\\cite{He_GGA_LDA_PRB_2014, Saadaoui_GGA_LDA_EPJB_2015}.\nThe correlations and {\\small RMSrD}\\ of both the {\\small LDA}\\ and {\\small GGA}\\ results with experiment for this set of materials are listed in Table~\\ref{tab:art115:LDA_correlation}.\nThe Pearson and Spearman correlation values for {\\small LDA}\\ and {\\small GGA}\\ are very close to each other for most of the listed properties. The {\\small RMSrD}\\ values show\ngreater differences, although it isn't clear that one of the exchange-correlation functionals consistently gives better predictions than the other.\nTherefore, the choice of exchange-correlation functional will make little difference to the predictive capability of the workflow, so we choose to\nuse {\\small GGA}-{\\small PBE}\\ as it is the functional used for performing the structural relaxation for the entries in the {\\small AFLOW}\\ data repository.\n}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Bulk modulus, shear modulus and Poisson ratio of a subset of the materials investigated in this work,\ncomparing the effect of using different exchange-correlation functionals.]\n{``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $B^{\\mathrm{exp}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $G^{\\mathrm{exp}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $\\sigma^{\\mathrm{exp}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{GGA}}}$ & $\\sigma^{\\substack{\\scalebox{0.6}{LDA}}}$ \\\\\n\\hline\nSi & 97.8~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 89.1& 96.9 & 84.2 & 92.1 & 66.5~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 64 & 65 & 61 & 61.9 & 62.5 & 63.4 & 0.223~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 0.216 & 0.231 \\\\\nBN & 367.0~\\cite{Lam_BulkMod_PRB_1987} & 372 & 402 & 353 & 382 & N\/A & 387 & 411 & 374 & 395 & 380 & 403 & N\/A & 0.119 & 0.124 \\\\\nGaSb & 57.0~\\cite{Lam_BulkMod_PRB_1987} & 47.0 & 58.3 & 41.6 & 52.3 & 34.2~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 30.8 & 35.3 & 28.3 & 32.2 & 29.6 & 33.7 & 0.248~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 0.240 & 0.258 \\\\\nInAs & 60.0~\\cite{Lam_BulkMod_PRB_1987} & 50.1 & 62.3 & 45.7 & 57.4 & 29.5~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 27.3 & 30.1 & 24.2 & 26.4 & 25.7 & 28.2 & 0.282~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 0.281 & 0.303 \\\\\nZnS & 77.1~\\cite{Lam_BulkMod_PRB_1987} & 71.2 & 88.4 & 65.8 & 83.3 & 30.9~\\cite{Semiconductors_BasicData_Springer} & 36.5 & 42.1 & 31.4 & 35.7 & 33.9 & 38.9 & 0.318~\\cite{Semiconductors_BasicData_Springer} & 0.294 & 0.308 \\\\\nNaCl & 25.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.9 & 33.3 & 20.0 & 27.6 & 14.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 14.0 & 19.8 & 12.9 & 16.6 & 13.5 & 18.2 & 0.255~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.271 & 0.269 \\\\\nKI & 12.2~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.9 & 16.3 & 8.54 & 13.3 & 5.96~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.05 & 9.39 & 4.39 & 5.3 & 5.22 & 7.35 & 0.290~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.294 & 0.305 \\\\\nRbI & 11.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 9.90 & 14.8 & 8.01 & 12.1 & 5.03~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.50 & 8.54 & 3.65 & 3.94 & 4.57 & 6.24 & 0.303~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 0.300 & 0.315 \\\\\nMgO & 164~\\cite{Sumino_ElasticMgO_JPE_1976} & 152 & 164 & 142 & 163 & 131~\\cite{Sumino_ElasticMgO_JPE_1976} & 119 & 138 & 115 & 136 & 117 & 137 & 0.185~\\cite{Sumino_ElasticMgO_JPE_1976} & 0.194 & 0.173 \\\\\nCaO & 113~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 105 & 129 & 99.6 & 122 & 81.0~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 73.7 & 87.4 & 73.7 & 86.3 & 73.7 & 86.9 & 0.210~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 0.216 & 0.225 \\\\\nGaN & 195~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 175 & 202 & 166 & 196 & 51.6~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 107 & 116 & 105 & 113 & 106 & 114 & 0.378~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978} & 0.248 & 0.262 \\\\\n & 210~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & 123~\\cite{Polian_ElasticGaN_JAP_1996} & & & & & & & 0.255~\\cite{Polian_ElasticGaN_JAP_1996} & & \\\\\nCdS & 60.7~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 55.4 & 68.2 & 49.7 & 64.1 & 18.2~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 17.6 & 18.4 & 17.0 & 17.8 & 17.3 & 18.1 & 0.364~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980} & 0.358 & 0.378 \\\\\nAl$_2$O$_3$ & 254~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 231 & 259 & 222 & 250 & 163.1~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 149 & 166 & 144 & 163 & 147 & 165 & 0.235~\\cite{Goto_ElasticAl2O3_JGPR_1989} & 0.238 & 0.238 \\\\\nCdGeP$_2$ & N\/A & 65.3 & 78.4 & 60.7 & 74.5 & N\/A & 37.7 & 42.1 & 33.3 & 36.8 & 35.5 & 39.4 & N\/A & 0.270 & 0.285 \\\\\nCuGaSe$_2$ & N\/A & 69.9 & 76.4 & 54.9 & 72.1 & N\/A & 30.3 & 34.7 & 26.0 & 30.0 & 28.1 & 32.3 & N\/A & 0.322 & 0.315 \\\\\nCoSb$_3$ & N\/A & 78.6 & 99.6 & 75.6 & 96.1 & N\/A & 57.2 & 67.1 & 55.1 & 64.2 & 56.2 & 65.7 & N\/A & 0.211 & 0.23 \\\\\n\\end{tabular}}\n\\label{tab:art115:LDA_elastic}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at\n300~K, Debye temperature and Gr{\\\"u}neisen parameter of\na subset of materials, comparing the effect of using different exchange-correlation functionals.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are $\\theta_{\\mathrm{a}}$, except 340~K for CdGeP$_2$~\\cite{Landolt-Bornstein, Abrahams_JCP_1975}, 262K for CuGaSe$_2$\n\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} and 307~K for CoSb$_3$~\\cite{Snyder_jmatchem_2011} which are $\\theta_{\\mathrm D}$.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{GGA}}} $ & $\\kappa^{\\substack{\\scalebox{0.6}{LDA}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{GGA}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{LDA}}}$ \\\\\n & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{GGA}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{LDA}}}$) & & \\\\\n\\hline\nSi & 166~\\cite{Morelli_Slack_2006} & 26.19 & 27.23 & 395~\\cite{slack, Morelli_Slack_2006} & 610 & 614 & 1.06~\\cite{Morelli_Slack_2006} & 2.06 & 2.03\t \\\\\n & & & & & (484) & (487) & 0.56~\\cite{slack} & \\\\\nBN & 760~\\cite{Morelli_Slack_2006} & 281.6 & 312.9 & 1200~\\cite{Morelli_Slack_2006} & 1793 & 1840 & 0.7~\\cite{Morelli_Slack_2006} & 1.75 & 1.72\t\\\\\n & & & & & (1423) & (1460) & & & \\\\\nGaSb & 40~\\cite{Morelli_Slack_2006} & 4.96 & 5.89 & 165~\\cite{slack, Morelli_Slack_2006} & 240 & 254 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.28 & 2.25 \t \\\\\n & & & & & (190) & (202) & & & \\\\\nInAs & 30~\\cite{Morelli_Slack_2006} & 4.33 & 4.92 & 165~\\cite{slack, Morelli_Slack_2006} & 229 & 238 & 0.57~\\cite{slack, Morelli_Slack_2006} & 2.26 & 2.22\t \\\\\n & & & & & (182) & (189) & & & \\\\\nZnS & 27~\\cite{Morelli_Slack_2006} & 8.38 & 9.58 & 230~\\cite{slack, Morelli_Slack_2006} & 341 & 363 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.00 & 2.02 \t \\\\\n & & & & & (271) & (288) & & & \\\\\nNaCl & 7.1~\\cite{Morelli_Slack_2006} & 2.12 & 2.92 & 220~\\cite{slack, Morelli_Slack_2006} & 271 & 312 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.29 \t \\\\\n & & & & & (215) & (248) & & & \\\\\nKI & 2.6~\\cite{Morelli_Slack_2006} & 0.525 & 0.811 & 87~\\cite{slack, Morelli_Slack_2006} & 116 & 137 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.35 & 2.37 \t \\\\\n & & & & & (92) & (109) & & & \\\\\nRbI & 2.3~\\cite{Morelli_Slack_2006} & 0.368 & 0.593 & 84~\\cite{slack, Morelli_Slack_2006} & 97 & 115 & 1.41~\\cite{slack, Morelli_Slack_2006} & 2.47 & 2.45 \t \\\\\n & & & & & (77) & (91) & & & \\\\\nMgO & 60~\\cite{Morelli_Slack_2006} & 44.5 & 58.4 & 600~\\cite{slack, Morelli_Slack_2006} & 849 & 935 & 1.44~\\cite{slack, Morelli_Slack_2006} & 1.96 & 1.95 \\\\\n & & & & & (674) & (742) & & & \\\\\nCaO & 27~\\cite{Morelli_Slack_2006} & 24.3 & 28.5 & 450~\\cite{slack, Morelli_Slack_2006} & 620 & 665 & 1.57~\\cite{slack, Morelli_Slack_2006} & 2.06 & 2.09 \t \\\\\n & & & & & (492) & (528) & & & \\\\\nGaN & 210~\\cite{Morelli_Slack_2006} & 18.54 & 21.34 & 390~\\cite{Morelli_Slack_2006} & 595 & 619 & 0.7~\\cite{Morelli_Slack_2006} & 2.08 & 2.04 \t \\\\\n & & & & & (375) & (390) & & & \\\\\nCdS & 16~\\cite{Morelli_Slack_2006} & 1.76 & 1.84 & 135~\\cite{Morelli_Slack_2006} & 211 & 217 & 0.75~\\cite{Morelli_Slack_2006} & 2.14 & 2.14 \t \\\\\n & & & & & (133) & (137) & & & \\\\\nAl$_2$O$_3$ & 30~\\cite{Slack_PR_1962} & 21.92 & 25.36 & 390~\\cite{slack} & 952 & 1002 & 1.32~\\cite{slack} & 1.91 & 1.91 \t \\\\\n & & & & & (442) & (465) & & & \\\\\nCdGeP$_2$ & 11~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 2.96 & 3.47 & 340~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 320 & 337 & N\/A & 2.21 & 2.18 \\\\\n & & & & & (160) & (169) & & & \\\\\nCuGaSe$_2$ & 12.9~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 1.46 & 2.23 & 262~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & 244 & 281 & N\/A & 2.26 & 2.23 \t \\\\\n & & & & & (122) & (141) & & & \\\\\nCoSb$_3$ & 10~\\cite{Snyder_jmatchem_2011} & 2.60 & 3.25 & 307~\\cite{Snyder_jmatchem_2011} & 310 & 332 & 0.95~\\cite{Snyder_jmatchem_2011} & 2.33 & 2.28 \\\\\n & & & & & (123) & (132) & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:LDA_thermal}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties comparing the {\\small LDA}\\ and {\\small GGA}\\ exchange-correlation functionals\nfor this subset of materials.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.963 & 0.867 & 0.755 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.959 & 0.848 & 0.706 \\\\\n$\\theta^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.996 & 0.996 & 0.119 \\\\\n$\\theta^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\theta^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.996 & 0.996 & 0.174 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.172 & 0.130 & 1.514 \\\\\n$\\gamma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\gamma^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.265 & 0.296 & 1.490 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.995 & 1.0 & 0.111 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.996 & 1.0 & 0.185 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.996 & 1.0 & 0.205 \\\\\n$B^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $B_{\\substack{\\scalebox{0.6}{Static}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.998 & 1.0 & 0.072 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.999 & 0.993 & 0.108 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{VRH}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.997 & 0.986 & 0.153 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.998 & 0.993 & 0.096 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Voigt}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.996 & 0.986 & 0.315 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.999 & 0.993 & 0.163 \\\\\n$G^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $G_{\\substack{\\scalebox{0.6}{Reuss}}}^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.997 & 0.993 & 0.111 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{GGA}}}$ & 0.982 & 0.986 & 0.037 \\\\\n$\\sigma^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\sigma^{\\substack{\\scalebox{0.6}{LDA}}}$ & 0.983 & 0.993 & 0.052 \\\\\n\\end{tabular}\n\\label{tab:art115:LDA_correlation}\n\\end{table}\n\n\\subsubsection{AGL predictions for thermal conductivity}\n\nThe {\\small AEL}-{\\small AGL}\\ methodology has been applied for\nhigh-throughput screening of the elastic and thermal properties of\nover 3000 materials included in the {\\small AFLOW}\\ database~\\cite{aflowAPI}.\nTables~\\ref{tab:art115:highkappa} and \\ref{tab:art115:lowkappa} {list those} found\nto have the highest and lowest thermal conductivities, respectively.\nThe high conductivity list is unsurprisingly dominated by various phases of elemental\ncarbon{, boron nitride, boron carbide and boron carbon nitride,} while {all other}\nhigh-conductivity materials also contain at least one of the elements C, B or N.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Materials from {\\small AFLOW}\\ database with highest thermal conductivities as predicted using\nthe {\\small AEL}-{\\small AGL}\\ methodology.]\n{The {\\small AFLOW}\\ \\underline{u}nique \\underline{id}entifier ({\\small AUID}) is a permanent, server-independent identifier for each entry in the {\\small AFLOW}\\ database~\\cite{aflowAPI}.\nThis identifier allows any of these entries to be retrieved from the repository, and ensures the retrievability and reproducibility of the data\nirrespective of changes in the underlying database structure or hosting location.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r}\ncomp. & Pearson & space group \\# & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & {\\small AUID} \\\\\n\\hline\nC & cF8 & 227 & 420 & 3ab7e139e1c29c9f \\\\\nBN & cF8 & 216 & 282 & fd5539a4f79db51c \\\\\nC & hP4 & 194 & 272 & 440c4eee274b61b6 \\\\\nC & tI8 & 139 & 206 & b2688e84030188b8 \\\\\nBC$_2$N & oP4 & 25 & 188 & c0e7523ff8d34297 \\\\\nBN & hP4 & 186 & 178 & 56d00a95d21b5c3a \\\\\nC & hP8 & 194 & 167 & c42dc8ec018245e5 \\\\\nC & cI16 & 206 & 162 & c969067f8a3bbde9 \\\\\nC & oS16 & 65 & 147 & bdc82cca41c811c6 \\\\\nC & mS16 & 12 & 145 & a59baaad49eb5ab9 \\\\\nBC$_7$ & tP8 & 115 & 145 & 0401731cb29df494 \\\\\nBC$_5$ & oI12 & 44 & 137 & f759c5600121a9e9 \\\\\nBe$_2$C & cF12 & 225 & 129 & 378e092c24555651 \\\\\nCN$_2$ & tI6 & 119 & 127 & 6852d98ddee59417 \\\\\nC & hP12 & 194 & 127 &\tbd79f9fa8154aa95 \\\\\nBC$_7$ & oP8 & 25 & 125 & 4d13f06b9fe563ef \\\\\nB$_2$C$_4$N$_2$ & oP8 & 17 & 120 & 9e325d34d65bd890\\\\\nMnB$_2$ & hP3 & 191 & 117 & 0e5997687be5d3dc \\\\\nC & hP4 & 194 & 117 & 2be120d88682ee01 \\\\\nSiC & cF8 & 216 & 113 & 2cab0c35952c733f \\\\\nTiB$_2$ & hP3 & 191 & 110 & 32d72b1701a0a640 \\\\\nAlN & cF8 & 225 & 107 & 06c4f5b0f1a49096 \\\\\nBP & cF8 & 216 & 105 & 598a7a7328a47d85 \\\\\nC & hP16 & 194 & 105 &\tc9d6a8b917d502f0 \\\\\nVN & hP2 & 187 & 101 &\taa89372868af03a8 \\\\\n\\end{tabular}\n\\label{tab:art115:highkappa}\n\\end{table}\n\nThe low thermal conductivity list tends to contain materials\nwith large unit cells and heavier elements such as Hg, Tl, Pb and Au.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Materials from {\\small AFLOW}\\ database with lowest thermal conductivities as predicted using\nthe {\\small AEL}-{\\small AGL}\\ methodology.]\n{The {\\small AFLOW}\\ \\underline{u}nique \\underline{id}entifier ({\\small AUID}) is a permanent, server-independent identifier for each entry in the {\\small AFLOW}\\ database~\\cite{aflowAPI}.\nThis identifier allows any of these entries to be retrieved from the repository, and ensures the retrievability and reproducibility of the data\nirrespective of changes in the underlying database structure or hosting location.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r}\ncomp. & Pearson & space group \\# & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}} $ & {\\small AUID} \\\\\n\\hline\nHg$_{33}$Rb$_3$ & cP36 & 221 & 0.0113 & 3a84e674e05ac4e6 \\\\\nHg$_{33}$K$_3$ & cP36 & 221 & 0.0116 & ac7610d35123f5c5 \\\\\nCs$_6$Hg$_{40}$ & cP46 & 223 & 0.0136 & 978182b72d30a019 \\\\\nCa$_{16}$Hg$_{36}$ & cP52 & 215 & 0.0751 & fe8eeb1e2af8df90 \\\\\nCrTe & cF8 & 216 & 0.081 & 53c8683bd5648144 \\\\\nHg$_4$K$_2$ & oI12 & 74 & 0.086 & 50b2883feb14cd6e \\\\\nSb$_6$Tl$_{21}$ & cI54 & 229 & 0.089 & f7933008a130dc06 \\\\\nSe & cF24 & 227 & 0.093 & 7d6a2e6c742211e5 \\\\\nCs$_8$I$_{24}$Sn$_4$ & cF36 & 225 & 0.104 & 460691dc51cf5b5d \\\\\nAg$_2$Cr$_4$Te$_8$ & cF56 & 227 & 0.107 & a30bbe2831fa8a18 \\\\\nAsCdLi & cF12 & 216 &\t0.116 & f818510c8952b114 \\\\\nAu$_{36}$In$_{16}$ & cP52 & 215 & 0.117 & bda82cdcf87fa384 \\\\\nCd$_3$In & cP4 & 221 & 0.128 & 3bc3fc68c58fdd1f \\\\\nAuLiSb & cF12 & 216 & 0.130 & bdab7ec2c162ee22 \\\\\nK$_5$Pb$_{24}$ & cI58 & 217 & 0.135 & 58f4471901eff079 \\\\\nK$_8$Sn$_{46}$ & cP54 & 223 & 0.142 & 6b4795df74caacfc \\\\\nAu$_7$Cd$_{16}$Na$_6$ & cF116 & 225 & 0.145 & ec21f32abca24cbd \\\\\nCs & cI2 & 229 & 0.148 & 5acbf212d1783298 \\\\\nCs$_8$Pb$_4$Cl$_{24}$ & cF36 & 225 & 0.157 & 84738cad161f83b3 \\\\\nAu$_{4}$In$_8$Na$_{12}$ & cF96 & 227 & 0.158 & 0393c62d375f5ec6\\\\\nSeTl & cP2 & 221 & 0.164 & 5ebc0f014499d22b \\\\\nCd$_{33}$Na$_6$ & cP39 & 200 & 0.166 & 0e4a5c866567f309 \\\\\nAu$_{18}$In$_{15}$Na$_6$ & cP39 & 200 & 0.168 & f7355e2e7474fb1c \\\\\nCd$_{26}$Cs$_2$ & cF112 & 226 & 0.173 & cfe1448550ccd1d1 \\\\\nAg$_2$I$_2$ & hP4 & 186 & 0.192 & d611e813a85efcb0 \\\\\n\\end{tabular}\n\\label{tab:art115:lowkappa}\n\\end{table}\n\nBy combining the {\\small AFLOW}\\ search for thermal conductivity values with other properties such as chemical, electronic or structural factors,\ncandidate materials for specific engineering applications can be rapidly identified for further in-depth analysis using more accurate\ncomputational methods and for experimental examination. {The full set of thermomechanical properties calculated using\n{\\small AEL}-{\\small AGL}\\ for over 3500 entries can be accessed online at {\\sf \\AFLOW.org}~\\cite{aflowlib.org}, which incorporates search and sort functionality to\ngenerate customized lists of materials.}\n\n\\subsubsection{Results for different equations of state}\n\nThis section includes the results for the thermal conductivity, Debye temperature and the Gr{\\\"u}neisen parameter for the set of 74 materials listed in this work as\ncalculated using the Birch-Murnaghan~\\cite{Birch_Elastic_JAP_1938, Poirier_Earth_Interior_2000, Blanco_CPC_GIBBS_2004}, Vinet~\\cite{Vinet_EoS_JPCM_1989, Blanco_CPC_GIBBS_2004},\nand Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez~\\cite{Baonza_EoS_PRB_1995, Blanco_CPC_GIBBS_2004} equations of state. The experimental values for the lattice thermal conductivity\n$\\kappa^{\\mathrm{exp}}$ are compared to the {\\small AGL}\\ values obtained using the numerical polynomial fit $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, and the three empirical equations of state:\nBirch-Murnaghan, $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$; Vinet, $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$; and Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez, $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$. The experimental values for the Debye temperature\n$\\theta^{\\mathrm{exp}}$ are compared to the {\\small AGL}\\ values obtained using the numerical polynomial fit $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$, and the three empirical equations of state:\nBirch-Murnaghan, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$; Vinet, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$; and Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez, $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$. The {\\small AGL}\\ values listed are for\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$, while the values for $\\theta_{\\mathrm{a}}$ are listed underneath in parentheses. The experimental values for the Gr{\\\"u}neisen parameter\n$\\gamma^{\\mathrm{exp}}$ are compared to the {\\small AGL}\\ values obtained using the numerical polynomial fit $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$, and the three empirical equations of state:\nBirch-Murnaghan, $\\gamma_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$; Vinet, $\\gamma_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$; and Baonza-C{\\'a}ceres-N{\\'u}{\\~n}ez, $\\gamma_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$. The results for the\ndiamond and zincblende structure set of materials are listed in Table~\\ref{tab:art115:zincblende_thermal_eos}, the results for the rocksalt structure set of materials\nare listed in Table~\\ref{tab:art115:rocksalt_thermal_eos}, the results for the hexagonal structure set of materials are listed in Table~\\ref{tab:art115:wurzite_thermal_eos}, the results\nfor the rhombohedral structure set of materials are listed in Table~\\ref{tab:art115:rhombo_thermal_eos}, the results for the body-centered tetragonal structure set of materials\nare listed in Table~\\ref{tab:art115:bct_thermal_eos}, and the results for the miscellaneous structure materials are listed in Table~\\ref{tab:art115:misc_thermal_eos}.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal conductivities, Debye temperatures and Gr{\\\"u}neisen parameters of\nzincblende and diamond structure semiconductors, calculated using the different equations of state.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 141K for HgTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nC & 3000~\\cite{Morelli_Slack_2006} & 419.9 & 298.5 & 307.0 & 466.8 & 1450~\\cite{slack, Morelli_Slack_2006} & 2094 & 2051 & 2056 & 2103 & 0.75~\\cite{Morelli_Slack_2006} & 1.77 & 2.01 &\t1.99 & 1.69 \\\\\n & & & & & & & (1662) & (1628) & (1632) & (1669) & 0.9~\\cite{slack} & & & & \\\\\nSiC & 360~\\cite{Ioffe_Inst_DB} & 113.0 & 120.7 & 101.3 & 125.4 & 740~\\cite{slack} & 1106 & 1108 & 1100 & 1110 & 0.76~\\cite{slack} & 1.85 & 1.80 & 1.93 & 1.78 \t\\\\\n& & & & & & & (878) & (879) & (873) & (881) & & & & & \\\\\nSi & 166~\\cite{Morelli_Slack_2006} & 26.19 & 28.61 & 26.23 & 31.39 & 395~\\cite{slack, Morelli_Slack_2006} & 610 & 611 & 609 & 614 & 1.06~\\cite{Morelli_Slack_2006} & 2.06 & 1.99 & 2.06 & 1.92\t \\\\\n & & & & & & & (484) & (485) & (483) & (487) & 0.56~\\cite{slack} & & & & \\\\\nGe & 65~\\cite{Morelli_Slack_2006} & 8.74 & 9.61 & 8.54 & 10.12 & 235~\\cite{slack, Morelli_Slack_2006} & 329 & 330 & 329 & 331 & 1.06~\\cite{Morelli_Slack_2006} & 2.31 & 2.22 & 2.34 & 2.18\t \\\\\n& & & & & & & (261) & (262) & (261) & (263) & 0.76~\\cite{slack} & & & & \\\\\nBN & 760~\\cite{Morelli_Slack_2006} & 281.6 & 243.1 & 220.5 & 303.4 & 1200~\\cite{Morelli_Slack_2006} & 1793 & 1777 & 1769 & 1798 & 0.7~\\cite{Morelli_Slack_2006} & 1.75 & 1.85 & 1.92 & 1.70\t\\\\\n& & & & & & & (1423) & (1410) & (1404) & (1427) & & & & & \\\\\nBP & 350~\\cite{Morelli_Slack_2006} & 105.0 & 108.8 & 89.95 & 117.8 & 670~\\cite{slack, Morelli_Slack_2006} & 1025 & 1025 & 1016 & 1029 & 0.75~\\cite{Morelli_Slack_2006} & 1.79 & 1.76 & 1.90 & 1.71 \t\\\\\n& & & & & & & (814) & (814) & (806) & (817) & & & & & \\\\\nAlP & 90~\\cite{Landolt-Bornstein, Spitzer_JPCS_1970} & 19.34 & 20.48 & 18.79 & 22.49 & 381~\\cite{Morelli_Slack_2006} & 525 & 526 & 524 & 528 & 0.75~\\cite{Morelli_Slack_2006} & 1.96 & 1.92 & 1.98 & 1.84 \t \\\\\n& & & & & & & (417) & (417) & (416) & (419) & & & & & \\\\\nAlAs & 98~\\cite{Morelli_Slack_2006} & 11.64 & 12.84 & 11.59 & 13.64 & 270~\\cite{slack, Morelli_Slack_2006} & 373 & 374 & 373 & 375 & 0.66~\\cite{slack, Morelli_Slack_2006} & 2.04 & 1.96 & 2.04 & 1.91 \t \\\\\n& & & & & & & (296) & (297) & (296) & (298) & & & & & \\\\\nAlSb & 56~\\cite{Morelli_Slack_2006} & 6.83 & 7.84 & 6.85 & 8.34 & 210~\\cite{slack, Morelli_Slack_2006} & 276 & 277 & 276 & 278 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.13 & 2.01 & 2.12 & 1.96\t \\\\\n& & & & & & & (219) & (220) & (219) & (221) & & & & & \\\\\nGaP & 100~\\cite{Morelli_Slack_2006} & 13.34 & 15.09 & 13.49 & 15.74 & 275~\\cite{slack, Morelli_Slack_2006} & 412 & 414 & 412 & 414 & 0.75~\\cite{Morelli_Slack_2006} & 2.15 & 2.04 & 2.14 & 2.0\t\\\\\n & & & & & & & (327) & (329) & (327) & (329) & 0.76~\\cite{slack} & & & & \\\\\nGaAs & 45~\\cite{Morelli_Slack_2006} & 8.0 & 8.95 & 7.85 & 9.30 & 220~\\cite{slack, Morelli_Slack_2006} & 313 & 315 & 313 & 315 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.24 & 2.15 & 2.26 & 2.11\t \\\\\n& & & & & & & (248) & (250) & (248) & (250) & & & & & \\\\\nGaSb & 40~\\cite{Morelli_Slack_2006} & 4.96 & 5.49 & 4.68 & 5.69 & 165~\\cite{slack, Morelli_Slack_2006} & 240 & 241 & 239 & 241 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.28 & 2.19 & 2.33 & 2.15 \\\\\n& & & & & & & (190) & (191) & (190) & (191) & & & & & \\\\\nInP & 93~\\cite{Morelli_Slack_2006} & 6.53 & 7.40 & 6.57 & 7.71 & 220~\\cite{slack, Morelli_Slack_2006} & 286 & 287 & 286 & 287 & 0.6~\\cite{slack, Morelli_Slack_2006} & 2.21 & 2.1 & 2.2 & 2.06 \t \\\\\n& & & & & & & (227) & (228) & (227) & (228) & & & & & \\\\\nInAs & 30~\\cite{Morelli_Slack_2006} & 4.33 & 4.80 & 4.20 & 4.93 & 165~\\cite{slack, Morelli_Slack_2006} & 229 & 230 & 229 & 230 & 0.57~\\cite{slack, Morelli_Slack_2006} & 2.26 & 2.17 & 2.29 & 2.14\t \\\\\n& & & & & & & (182) & (183) & (182) & (183) & & & & & \\\\\nInSb & 20~\\cite{Morelli_Slack_2006} & 3.02 & 3.33 & 2.76 & 3.44 & 135~\\cite{slack, Morelli_Slack_2006} & 187 & 188 & 186 & 188 & 0.56~\\cite{slack, Morelli_Slack_2006} & 2.3 & 2.22 & 2.38 & 2.18\t \\\\\n & & & & & & & (148) & (149) & (148) & (149) & 16.5~\\cite{Snyder_jmatchem_2011} & & & & \\\\\nZnS & 27~\\cite{Morelli_Slack_2006} & 8.38 & 8.40 & 7.67 & 8.96 & 230~\\cite{slack, Morelli_Slack_2006} & 341 & 341 & 340 & 342 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.0 & 1.99 & 2.07 & 1.94\t \\\\\n& & & & & & & (271) & (271) & (270) & (271) & & & & & \\\\\nZnSe & 19~\\cite{Morelli_Slack_2006} & 5.44 & 5.55 & 4.93 & 5.80 & 190~\\cite{slack, Morelli_Slack_2006} & 260\t& 260 & 259 & 261 & 0.75~\\cite{slack, Morelli_Slack_2006} & 2.06 & 2.04 & 2.14 & 2.01\t\\\\\n & 33~\\cite{Snyder_jmatchem_2011} & & & & & & (206) & (206) & (206) & (207) & & & & & \\\\\nZnTe & 18~\\cite{Morelli_Slack_2006} & 3.83 & 3.95 & 3.44 & 4.10 & 155~\\cite{slack, Morelli_Slack_2006} & 210 & 210 & 209 & 210 & 0.97~\\cite{slack, Morelli_Slack_2006} & 2.13 & 2.1 & 2.23 & 2.07 \\\\\n& & & & & & & (167) & (167) & (166) & (167) & & & & & \\\\\nCdSe & 4.4~\\cite{Snyder_jmatchem_2011} & 2.04 & 2.11 & 1.84 & 2.16 & 130~\\cite{Morelli_Slack_2006} & 173 & 173 & 172 & 173 & 0.6~\\cite{Morelli_Slack_2006} & 2.18 & 2.15 & 2.27 & 2.12 \\\\\n& & & & & & & (137) & (137) & (137) & (137) & & & & & \\\\\nCdTe & 7.5~\\cite{Morelli_Slack_2006} & 1.71 & 1.77 & 1.50 & 1.81 & 120~\\cite{slack, Morelli_Slack_2006} & 150 & 150 & 149 & 150 & 0.52~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.19 & 2.34 & 2.16\t \\\\\n& & & & & & & (119) & (119) & (118) & (119) & & & & & \\\\\nHgSe & 3~\\cite{Whitsett_PRB_1973} & 1.32 & 1.36 & 1.22 & 1.41 & 110~\\cite{slack} & 140\t& 140 & 140 & 140\t& 0.17~\\cite{slack} & 2.38 & 2.35 & 2.47 & 2.31 \\\\\n& & & & & & & (111) & (111) & (111) & (111) & & & & & \\\\\nHgTe & 2.5~\\cite{Snyder_jmatchem_2011} & 1.21 & 1.30 & 1.10 & 1.34 & 141~\\cite{Snyder_jmatchem_2011} & 129\t& 130 & 129 & 130\t& 1.9~\\cite{Snyder_jmatchem_2011} & 2.45 & 2.40 & 2.56 & 2.36 \\\\\n & & & & & & (100)~\\cite{slack} & (102) & (103) & (102) & (103) & 0.46\\cite{slack} & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Thermal properties lattice thermal conductivity at 300~K, Debye temperature and Gr{\\\"u}neisen parameter of rocksalt\nstructure semiconductors, calculated using the different equations of state.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 155K for SnTe which is $\\theta_{\\mathrm D}$~\\cite{Snyder_jmatchem_2011}.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nLiH & 15~\\cite{Morelli_Slack_2006} & 18.6 & 13.54 & 12.37 & 20.98 & 615~\\cite{slack, Morelli_Slack_2006} & 962 & 931 & 927 & 968 & 1.28~\\cite{slack, Morelli_Slack_2006} & 1.66 & 1.84 & 1.90 & 1.58 \\\\\n& & & & & & & (764) & (739) & (734) & (768) & & & & & \\\\\nLiF & 17.6~\\cite{Morelli_Slack_2006} & 9.96 & 10.19 & 8.68 & 11.45 & 500~\\cite{slack, Morelli_Slack_2006} & 617 &\t617 & 610 & 623 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.03 & 2.00 & 2.13 & 1.92\t \\\\\n& & & & & & & (490) & (490) & (485) & (494) & & & & & \\\\\nNaF & 18.4~\\cite{Morelli_Slack_2006} & 4.67 & 4.65 & 3.82 & 4.91 & 395~\\cite{slack, Morelli_Slack_2006} & 416 & 416 & 411 & 417 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.21 & 2.21 & 2.39 & 2.16\t \\\\\n& & & & & & & (330) & (330) & (326) & (331) & & & & & \\\\\nNaCl & 7.1~\\cite{Morelli_Slack_2006} & 2.12 & 2.27 & 1.74 & 2.28 & 220~\\cite{slack, Morelli_Slack_2006} & 271 & 273 & 268 & 272 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.18 & 2.40 & 2.16\t \\\\\n& & & & & & & (215) & (217) & (213) & (216) & & & & & \\\\\nNaBr & 2.8~\\cite{Morelli_Slack_2006} & 1.33 & 1.42 & 1.08 & 1.40 & 150~\\cite{slack, Morelli_Slack_2006} & 188 & 189 & 186 & 188 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.22 & 2.17 & 2.40 & 2.16 \t \\\\\n& & & & & & & (149) & (150) & (148) & (149) & & & & & \\\\\nNaI & 1.8~\\cite{Morelli_Slack_2006} & 0.851 & 0.922 & 0.679 & 0.892 & 100~\\cite{slack, Morelli_Slack_2006} & 140 & 141\t& 138 & 140 & 1.56~\\cite{slack, Morelli_Slack_2006} & 2.23 & 2.17 & 2.43 & 2.18\t \\\\\n& & & & & & & (111) & (112) & (110) & (111) & & & & & \\\\\nKF & N\/A & 2.21 & 2.07 & 1.62 & 2.22 & 235~\\cite{slack, Morelli_Slack_2006} & 288 &\t287\t& 281 & 288 & 1.52~\\cite{slack, Morelli_Slack_2006} & 2.32 & 2.38 & 2.60 & 2.32 \t\\\\\n& & & & & & & (229) & (228) & (224) & (229) & & & & & \\\\\nKCl & 7.1~\\cite{Morelli_Slack_2006} & 1.25 & 1.42 & 1.04 & 1.40 & 172~\\cite{slack, Morelli_Slack_2006} & 213 & 215 & 210 & 214 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.40 & 2.29 & 2.57 & 2.29 \t \\\\\n& & & & & & & (169) & (171) & (167) & (170) & & & & & \\\\\nKBr & 3.4~\\cite{Morelli_Slack_2006} & 0.842 & 0.949 & 0.682 & 0.928 & 117~\\cite{slack, Morelli_Slack_2006} & 156 & 157 & 153 & 156 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.37 & 2.26 & 2.55 & 2.27 \\\\\n& & & & & & & (124) & (125) & (121) & (124) & & & & & \\\\\nKI & 2.6~\\cite{Morelli_Slack_2006} & 0.525 & 0.624 & 0.451 & 0.603 & 87~\\cite{slack, Morelli_Slack_2006} & 116 & 118 & 115 & 117 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.35 & 2.23 & 2.50 & 2.23\t \\\\\n& & & & & & & (92) & (94) & (91) & (93) & & & & & \\\\\nRbCl & 2.8~\\cite{Morelli_Slack_2006} & 0.837 & 0.886 & 0.638 & 0.878 & 124~\\cite{slack, Morelli_Slack_2006} & 155 & 156 & 152 & 155 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.37 & 2.33 & 2.62 & 2.32\t \\\\\n& & & & & & & (123) & (124) & (121) & (123) & & & & & \\\\\nRbBr & 3.8~\\cite{Morelli_Slack_2006} & 0.558 & 0.606 & 0.459 & 0.606 & 105~\\cite{slack, Morelli_Slack_2006} & 122 & 123 & 121 & 123 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.43 & 2.36 & 2.62 & 2.36 \\\\\n& & & & & & & (97) & (98) & (96) & (98) & & & & & \\\\\nRbI & 2.3~\\cite{Morelli_Slack_2006} & 0.368 & 0.434 & 0.320 & 0.415 & 84~\\cite{slack, Morelli_Slack_2006} & 97 & 98 & 96 & 97 & 1.41~\\cite{slack, Morelli_Slack_2006} & 2.47 & 2.32 & 2.60 & 2.34 \t \\\\\n& & & & & & & (77) & (78) & (76) & (77) & & & & & \\\\\nAgCl & 1.0~\\cite{Landolt-Bornstein, Maqsood_IJT_2003} & 0.613 & 0.612 & 0.535 & 0.663 & 124~\\cite{slack} & 145 & 145 & 144 & 146 & 1.9~\\cite{slack} & 2.49 & 2.49 & 2.63 & 2.43 \t \\\\\n& & & & & & & (115) & (115) & (114) & (116) & & & & & \\\\\nMgO & 60~\\cite{Morelli_Slack_2006} & 44.5 & 44.7 & 38.5 & 47.1 & 600~\\cite{slack, Morelli_Slack_2006} & 849 & 848 & 842 & 851\t& 1.44~\\cite{slack, Morelli_Slack_2006} & 1.96 & 1.95 & 2.07 & 1.91 \\\\\n& & & & & & & (674) & (673) & (668) & (675) & & & & & \\\\\nCaO & 27~\\cite{Morelli_Slack_2006} & 24.3 & 24.7 & 22.5 & 25.7 & 450~\\cite{slack, Morelli_Slack_2006} & 620 & 620 & 618 & 621 & 1.57~\\cite{slack, Morelli_Slack_2006} & 2.06 & 2.05 & 2.13 & 2.02\t \\\\\n& & & & & & & (492) & (492) & (491) & (493) & & & & & \\\\\nSrO & 12~\\cite{Morelli_Slack_2006} & 13.4 & 13.3 & 12.2 & 14.0 & 270~\\cite{slack, Morelli_Slack_2006} & 413 & 413 & 412 & 414 & 1.52~\\cite{slack, Morelli_Slack_2006} & 2.13 & 2.13 & 2.21\t& 2.09 \t \\\\\n& & & & & & & (328) & (328) & (327) & (329) & & & & & \\\\\nBaO & 2.3~\\cite{Morelli_Slack_2006} & 7.10 & 6.73 & 6.10 & 6.98 & 183~\\cite{slack, Morelli_Slack_2006} & 288 & 288 & 287 & 288 & 1.5~\\cite{slack, Morelli_Slack_2006} & 2.14 & 2.20 & 2.29 & 2.16 \\\\\n& & & & & & & (229) & (229) & (228) & (229) & & & & & \\\\\nPbS & 2.9~\\cite{Morelli_Slack_2006} & 6.11 & 6.77 & 5.99 & 7.02 & 115~\\cite{slack, Morelli_Slack_2006} & 220 & 221 & 220 & 221 & 2.0~\\cite{slack, Morelli_Slack_2006} & 2.00 & 1.92 & 2.02 & 1.89\t\\\\\n& & & & & & & (175) & (175) & (175) & (175) & & & & & \\\\\nPbSe & 2.0~\\cite{Morelli_Slack_2006} & 4.81 & 5.29 & 4.63 & 5.44 & 100~\\cite{Morelli_Slack_2006} & 194 & 195 & 194 & 195 & 1.5~\\cite{Morelli_Slack_2006} & 2.07 & 2.00 & 2.11 & 1.97\t \\\\\n& & & & & & & (154) & (155) & (154) & (155) & & & & & \\\\\nPbTe & 2.5~\\cite{Morelli_Slack_2006} & 4.07 & 4.11 & 3.50 & 4.32 & 105~\\cite{slack, Morelli_Slack_2006} & 172 & 172 & 171 & 173 & 1.45~\\cite{slack, Morelli_Slack_2006} & 2.09 & 2.08 & 2.22 & 2.05 \t \\\\\n& & & & & & & (137) & (137) & (136) & (137) & & & & & \\\\\nSnTe & 1.5~\\cite{Snyder_jmatchem_2011} & 5.24 & 5.59 & 4.64 & 5.78 & 155~\\cite{Snyder_jmatchem_2011} & 210 & 211 & 209 & 211 & 2.1~\\cite{Snyder_jmatchem_2011} & 2.11 & 2.06 & 2.22 & 2.03 \t \\\\\n& & & & & & & (167) & (167) & (166) & (167) & & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:rocksalt_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperature and Gr{\\\"u}neisen parameter of hexagonal\nstructure semiconductors, calculated using the different equations of state.]\n{The values listed for $\\theta^{\\mathrm{exp}}$ are\n$\\theta_{\\mathrm{a}}$, except 190K for InSe~\\cite{Snyder_jmatchem_2011} and 660K for InN~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998}\nwhich are $\\theta_{\\mathrm D}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nSiC & 490~\\cite{Morelli_Slack_2006} & 70.36 & 75.17 & 62.86 & 77.82 & 740~\\cite{Morelli_Slack_2006} & 1103 & 1105 & 1096 & 1106 & 0.75~\\cite{Morelli_Slack_2006} & 1.86 & 1.80 & 1.94 & 1.78 \\\\\n& & & & & & & (695) & (696) & (690) & (697) & & & & & \\\\\nAlN & 350~\\cite{Morelli_Slack_2006} & 39.0 & 40.53 & 34.49 & 42.3 & 620~\\cite{Morelli_Slack_2006} & 898 & 899 & 893 & 900 & 0.7~\\cite{Morelli_Slack_2006} & 1.85 & 1.82 & 1.95 & 1.79 \t \\\\\n& & & & & & & (566) & (566) & (563) & (567) & & & & & \\\\\nGaN & 210~\\cite{Morelli_Slack_2006} & 18.53 & 16.33 & 16.21 & 20.15 & 390~\\cite{Morelli_Slack_2006} & 595 & 590 & 591 & 596 & 0.7~\\cite{Morelli_Slack_2006} & 2.08 & 2.18 & 2.19 & 2.01\t \\\\\n& & & & & & & (375) & (372) & (372) & (375) & & & & & \\\\\nZnO & 60~\\cite{Morelli_Slack_2006} & 7.39 & 7.72 & 6.80 & 8.06 & 303~\\cite{Morelli_Slack_2006} & 422 & 422 & 420 & 423 & 0.75~\\cite{Morelli_Slack_2006} & 1.94 & 1.91 & 2.01 & 1.87 \t \\\\\n& & & & & & & (266) & (266) & (265) & (266) & & & & & \\\\\nBeO & 370~\\cite{Morelli_Slack_2006} & 53.36 & 54.41 & 46.97 & 56.95 & 809~\\cite{Morelli_Slack_2006} & 1181 & 1182 & 1173 & 1184 & 1.38~\\cite{Slack_JAP_1975, Cline_JAP_1967, Morelli_Slack_2006} & 1.76 & 1.74 & 1.85 & 1.71\t \\\\\n& & & & & & & (744) & (745) & (739) & (746) & & & & & \\\\\nCdS & 16~\\cite{Morelli_Slack_2006} & 1.76 & 1.89 & 1.66 & 1.93 & 135~\\cite{Morelli_Slack_2006} & 211 & 212 & 211 & 212 & 0.75~\\cite{Morelli_Slack_2006} & 2.14 & 2.08 & 2.19 & 2.06\t \\\\\n& & & & & & & (133) & (134) & (133) & (134) & & & & & \\\\\nInSe & 6.9~\\cite{Snyder_jmatchem_2011} & 2.34 & 2.61 & 2.23 & 2.69 & 190~\\cite{Snyder_jmatchem_2011} & 249 & 250 & 248\t& 250 & 1.2~\\cite{Snyder_jmatchem_2011} & 2.24 & 2.14 & 2.28 & 2.11 \t \\\\\n& & & & & & & (125) & (125) & (124) & (125) & & & & & \\\\\nInN & 45~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 6.82 & 6.97 & 5.59 & 7.49 & 660~\\cite{Ioffe_Inst_DB, Krukowski_jphyschemsolids_1998} & 369 & 369 & 365 & 370 & 0.97~\\cite{Krukowski_jphyschemsolids_1998} & 2.18 & 2.15 & 2.35 & 2.09 \t \\\\\n& & & & & & & (232) & (232) & (230) & (233) & & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity, Debye temperatures and Gr{\\\"u}neisen parameter of rhombohedral\nsemiconductors, calculated using the different equations of state.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$ for\nBi$_2$Te$_3$ and Sb$_2$Te$_3$, and $\\theta_{\\mathrm{a}}$ for Al$_2$O$_3$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nBi$_2$Te$_3$ & 1.6~\\cite{Snyder_jmatchem_2011} & 3.35 & 3.63 & 3.17 & 3.73 & 155~\\cite{Snyder_jmatchem_2011} & 204 & 205 & 204 & 205 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.14 & 2.08 & 2.20 & 2.05\t \\\\\n& & & & & & & (119) & (120) & (119) & (120) & & & & & \\\\\nSb$_2$Te$_3$ & 2.4~\\cite{Snyder_jmatchem_2011} & 4.46 & 4.76 & 4.07 & 4.99 & 160~\\cite{Snyder_jmatchem_2011} & 243 & 244 & 242 & 244 & 1.49~\\cite{Snyder_jmatchem_2011} & 2.11 & 2.06 & 2.19 & 2.02\t\\\\\n& & & & & & & (142) & (143) & (142) & (143) & & & & & \\\\\nAl$_2$O$_3$ & 30~\\cite{Slack_PR_1962} & 21.92 & 23.36 & 19.51 & 23.19 & 390~\\cite{slack} & 952 & 954 & 947 & 954 & 1.32~\\cite{slack} & 1.91 & 1.86 & 2.00 & 1.87\t \\\\\n& & & & & & & (442) & (443) & (440) & (443) & & & & & \\\\\nCr$_2$O$_3$ & 16~\\cite{Landolt-Bornstein, Bruce_PRB_1977} & 12.03 & 12.61 & 10.78 & 12.92 & N\/A & 718 & 717 & 713 & 718 & N\/A & 2.10 & 2.05 & 2.19 & 2.04 \t\\\\\n& & & & & & & (333) & (333) & (331) & (333) & & & & & \\\\\nBi$_2$Se$_3$ & 1.34~\\cite{Landolt-Bornstein} & 2.41 & 2.54 & 2.31 & 2.68 & N\/A & 199 & 199 & 199 & 200 & N\/A & 2.12 & 2.07 & 2.16 & 2.03 \t\\\\\n& & & & & & & (116) & (116) & (116) & (117) & & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures and Gr{\\\"u}neisen parameter of body-centered tetragonal\nsemiconductors, calculated using the different equations of state.]\n{``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nCuGaTe$_2$ & 2.2~\\cite{Snyder_jmatchem_2011} & 1.36 & 1.49 & 1.30 & 1.53 & 226~\\cite{Snyder_jmatchem_2011} & 215 & 216 & 215 & 216 & 1.46~\\cite{Snyder_jmatchem_2011} & 2.32 & 2.23 & 2.36 & 2.21 \t \\\\\n& & & & & & & (108) & (108) & (108) & (108) & & & & &\\\\\nZnGeP$_2$ & 35~\\cite{Landolt-Bornstein, Beasley_AO_1994} & 5.07 & 5.54 & 4.95 & 5.73 & 500~\\cite{Landolt-Bornstein} & 408 & 410 & 408 & 410 & N\/A & 2.14 & 2.07 & 2.17 & 2.04 \t \\\\\n& 36~\\cite{Landolt-Bornstein, Beasley_AO_1994} & & & & & & (204) & (205) & (204) & (205) & & & & & \\\\\n& 18~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & & & & & & & & & & & & & & \\\\\nZnSiAs$_2$ & 14\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 3.96 & 4.19 & 3.76 & 4.43 & 347~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1981} & 350 & 350 & 349 & 351 & N\/A & 2.15 & 2.10 & 2.20 & 2.05\t \\\\\n& & & & & & & (175) & (175) & (175) & (176) & & & & &\\\\\nCuInTe$_2$ & 10\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 0.722 & 0.797 & 0.693 & 0.812 & 185~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 166 & 167\t& 166 & 167 & 0.93~\\cite{Rincon_PSSa_1995} & 2.32 & 2.23 & 2.36 & 2.21 \t \\\\\n& & & & & & 195~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & (83) & (84) & (83) & (84) & & & & &\\\\\nAgGaS$_2$ & 1.4\\cite{Landolt-Bornstein, Beasley_AO_1994} & 0.993 & 1.04 & 0.92 & 1.08 & 255~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 224 & 224 & 223 & 224 & N\/A & 2.20 & 2.14 & 2.26 & 2.11\t\\\\\n& & & & & & & (112) & (112) & (112) & (112) & & & & &\\\\\nCdGeP$_2$ & 11~\\cite{Landolt-Bornstein, Shay_1975, Masumoto_JPCS_1966} & 2.96 & 3.18 & 2.85 & 3.31 & 340~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 320 & 321 & 320 & 321 & N\/A & 2.21 & 2.14 & 2.25 & 2.10 \t \\\\\n& & & & & & & (160) & (161) & (160) & (161) & & & & &\\\\\nCdGeAs$_2$ & 42~\\cite{Landolt-Bornstein, Shay_1975} & 2.11 & 2.17 & 1.92 & 2.24 & N\/A & 254 & 254 & 253 & 254 & N\/A & 2.20 & 2.17 & 2.29 & 2.14 \t\\\\\n& & & & & & & (127) & (127) & (127) & (127) & & & & &\\\\\nCuGaS$_2$ & 5.09~\\cite{Landolt-Bornstein} & 2.79 & 2.99 & 2.67 & 3.11 & 356~\\cite{Landolt-Bornstein, Abrahams_JCP_1975} & 349 & 350 & 348 & 350 & N\/A & 2.24 & 2.18 & 2.28 & 2.14 \t \\\\\n& & & & & & & (175) & (175) & (174) & (175) & & & & &\\\\\nCuGaSe$_2$ & 12.9~\\cite{Landolt-Bornstein, Rincon_PSSa_1995} & 1.46 & 1.53 & 1.37 & 1.61 & 262~\\cite{Landolt-Bornstein, Bohnhammel_PSSa_1982} & 244 & 244 & 243 & 245 & N\/A & 2.26 & 2.21 & 2.32 & 2.17\t \\\\\n& & & & & & & (122) & (122) & (122) & (123) & & & & &\\\\\nZnGeAs$_2$ & 11\\cite{Landolt-Bornstein, Shay_1975} & 3.18 & 3.29 & 2.93 & 3.45 & N\/A & 307 & 307 & 306 & 308 & N\/A & 2.17 & 2.13 & 2.24 & 2.10\t \\\\\n& & & & & & & (154) & (154) & (153) & (154) & & & & &\\\\\n\\end{tabular}}\n\\label{tab:art115:bct_thermal_eos}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Lattice thermal conductivity at 300~K, Debye temperatures and Gr{\\\"u}neisen parameter of materials with various\nstructures, calculated using the different equations of state.]\n{The experimental Debye temperatures are $\\theta_{\\mathrm D}$,\nexcept ZnSb for which it is $\\theta_{\\mathrm{a}}$.\n``N\/A'' = Not available for that source.\nUnits: $\\kappa$ in {\\small (W\\,m$^{-1}$K$^{-1}$)}, $\\theta$ in {\\small (K)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $\\kappa^{\\mathrm{exp}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\theta^{\\mathrm{exp}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\theta_{\\substack{\\scalebox{0.6}{D}}}^{\\substack{\\scalebox{0.6}{BCN}}}$ & $\\gamma^{\\mathrm{exp}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{AGL}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BM}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{Vinet}}}$ & $\\gamma^{\\substack{\\scalebox{0.6}{BCN}}}$ \\\\\n& & & & & & & & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{AGL}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BM}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{Vinet}}}$) & ($\\theta_{\\mathrm{a}}^{\\substack{\\scalebox{0.6}{BCN}}}$) & & & & & \\\\\n\\hline\nCoSb$_3$ & cI32 & 10~\\cite{Snyder_jmatchem_2011} & 2.60 & 2.58 & 2.38 & 2.78 & 307~\\cite{Snyder_jmatchem_2011} & 310 & 310 & 309 & 311 & 0.95~\\cite{Snyder_jmatchem_2011} & 2.33 & 2.33 & 2.42 & 2.27 \\\\\n& & & & & & & & (123) & (123) & (123) & (123) & & & & \\\\\nIrSb$_3$ & cI32 & 16~\\cite{Snyder_jmatchem_2011} & 2.73 & 2.89 & 2.67 & 3.01 & 308~\\cite{Snyder_jmatchem_2011} & 286 & 287 & 286 & 287 & 1.42~\\cite{Snyder_jmatchem_2011} & 2.34 & 2.29 & 2.37 & 2.25 \\\\\n& & & & & & & & (113) & (114) & (113) & (114) & & & & \\\\\nZnSb & oP16 & 3.5~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 1.23 & 1.29 & 1.13 & 1.36 & 92~\\cite{Madsen_PRB_2014} & 242 & 242 & 241 & 243 & 0.76~\\cite{Madsen_PRB_2014, Bottger_JEM_2010} & 2.23 & 2.18 & 2.30 & 2.14 \t \\\\\n& & & & & & & & (96) & (96) & (96) & (96) & & & & \\\\\nSb$_2$O$_3$ & oP20 & 0.4~\\cite{Landolt-Bornstein} & 8.74 & 8.93 & 8.18 & 9.20 & N\/A & 572 & 573 & 571 & 573 & N\/A & 2.12 & 2.10 & 2.18 & 2.07\t\\\\\n& & & & & & & & (211) & (211) & (210) & (211) & & & & \\\\\nInTe & cP2 & N\/A & 0.709 & 0.602 & 0.524 & 0.626 & N\/A & 113 & 112 & 111 & 112 & N\/A & 2.19 & 2.33 & 2.45 & 2.29 \\\\\n& & & & & & & & (90) & (89) & (88) & (89) & & & & \\\\\nInTe & tP16 & 1.7~\\cite{Snyder_jmatchem_2011} & 1.40 & 1.53 & 1.27 & 1.55 & 186~\\cite{Snyder_jmatchem_2011} & 193 & 194 & 192 & 194 & 1.0~\\cite{Snyder_jmatchem_2011} & 2.24 & 2.16 & 2.32 & 2.14\t\\\\\n& & & & & & & & (97) & (97) & (96) & (97) & & & & \\\\\nBi$_2$O$_3$ & mP20 & 0.8~\\cite{Landolt-Bornstein} & 2.98 & 3.05 & 2.49 & 3.14 & N\/A & 342 & 342 & 339 & 342 & N\/A & 2.10 & 2.08 & 2.26 & 2.05\t \\\\\n& & & & & & & & (126) & (126) & (125) & (126) & & & & \\\\\nSnO$_2$ & tP6 & 98\\cite{Turkes_jpcss_1980} & 6.98 & 7.76 & 6.52 & 8.31 & N\/A & 487 & 489 & 485 & 490 & N\/A & 2.42 & 2.32 & 2.48 & 2.25\t \\\\\n& & 55~\\cite{Turkes_jpcss_1980} & & & & & & (268) & (269) & (267) & (270) & & & & \\\\\n\\end{tabular}}\n\\label{tab:art115:misc_thermal_eos}\n\\end{table}\n\n\\subsubsection{Elastic constant values}\n\nThe elastic constant values in the 6x6 Voigt notation are shown for zincblende and diamond structure materials in Table~\\ref{tab:art115:zincblende_elastic_supp}, for rocksalt structure materials in Table~\\ref{tab:art115:rocksalt_elastic_supp}, for hexagonal structure materials in Table~\\ref{tab:art115:wurzite_elastic_supp}, for rhombohedral structure materials in Table~\\ref{tab:art115:rhombo_elastic_supp}, for body-centered tetragonal\nternary materials in Table~\\ref{tab:art115:bct_elastic_supp}, for body-centered cubic and simple cubic materials in Table~\\ref{tab:art115:bcc_elastic}, for orthorhombic structures in Table~\\ref{tab:art115:orc_elastic}, and for tetragonal\nstructure materials in Table~\\ref{tab:art115:tet_elastic}.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$ and $c_{44}$ of\nzincblende and diamond structure semiconductors.]\n{The zincblende structure is designated {\\small AFLOW}\\ prototype {\\sf AB\\_cF8\\_216\\_c\\_a}~\\cite{aflowANRL}\nand the diamond structure {\\sf A\\_cF8\\_227\\_a}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\mathrm{exp}}$ & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\mathrm{exp}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\mathrm{exp}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n\\hline\nC & 1076.4~\\cite{Semiconductors_BasicData_Springer} & 1048 & 125.2~\\cite{Semiconductors_BasicData_Springer} & 127 & 577.4~\\cite{Semiconductors_BasicData_Springer} & 560 \\\\\nSiC & 352.3~\\cite{Semiconductors_BasicData_Springer} & 384 & 140.4~\\cite{Semiconductors_BasicData_Springer} & 127 & 232.9~\\cite{Semiconductors_BasicData_Springer} & 240 \\\\\nSi & 165.64~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 153 & 63.94~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 57.1 & 79.51~\\cite{Semiconductors_BasicData_Springer, Hall_ElasticSi_PR_1967} & 74.6 \\\\\nGe & 129.9~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 107 & 48.73~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 38.8 & 68.0~\\cite{Semiconductors_BasicData_Springer, Bruner_ElasticGe_PRL_1961} & 56.7 \\\\\nBN & N\/A & 777 & N\/A & 170 & N\/A & 442 \\\\\nBP & 315.0~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 339 & 100~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 73.3 & 160~\\cite{Semiconductors_BasicData_Springer, Wettling_ElasticBP_SSC_1984} & 185 \\\\\nAlP & N\/A & 125 & N\/A & 61.6 & N\/A & 59.7 \\\\\nAlAs & N\/A & 104 & N\/A & 49.3 & N\/A & 50.4 \\\\\nAlSb & 87.69~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 76.3 & 43.41~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 36.0 & 40.76~\\cite{Semiconductors_BasicData_Springer, Bolef_ElasticAlSb_JAP_1960, Weil_ElasticAlSb_JAP_1972} & 36.0 \\\\\nGaP & 141.4~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 127 & 63.98~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 54.9 & 70.28~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 65.2 \\\\\nGaAs & 188.8~\\cite{Bateman_ElasticGaAs_JAP_1975} & 101 & 53.8~\\cite{Bateman_ElasticGaAs_JAP_1975} & 43.7 & 59.4~\\cite{Bateman_ElasticGaAs_JAP_1975} & 51.9 \\\\\nGaSb & 88.34~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 74.6 & 40.23~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 33.2 & 43.22~\\cite{Boyle_ElasticGaPSb_PRB_1975} & 37.6 \\\\\nInP & 101.1~\\cite{Nichols_ElasticInP_SSC_1980} & 87.7 & 56.1~\\cite{Nichols_ElasticInP_SSC_1980} & 46.7 & 45.6~\\cite{Nichols_ElasticInP_SSC_1980} & 42.3 \\\\\nInAs & 83.29~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 72.4 & 45.26~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 38.9 & 39.59~\\cite{Semiconductors_BasicData_Springer, Gerlich_ElasticAlSb_JAP_1963} & 34.3 \\\\\nInSb & 66.0~\\cite{DeVaux_ElasticInSb_PR_1956} & 55.8 & 38.0~\\cite{DeVaux_ElasticInSb_PR_1956} & 29.3 & 30.0~\\cite{DeVaux_ElasticInSb_PR_1956} & 26.7 \\\\\nZnS & 98.1~\\cite{Semiconductors_BasicData_Springer} & 99.2 & 62.7~\\cite{Semiconductors_BasicData_Springer} & 57.2 & 44.83~\\cite{Semiconductors_BasicData_Springer} & 46.9 \\\\\nZnSe & 85.9~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 81.4 & 50.6~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 46.6 & 40.6~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 37.5 \\\\\nZnTe & 71.1~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 63.2 & 40.7~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 34.1 & 31.3~\\cite{Lee_ElasticZnSeTe_JAP_1970} & 29.2 \\\\\nCdSe & N\/A & 57.7 & N\/A & 41.1 & N\/A & 21.5 \\\\\nCdTe & N\/A & 46.7 & N\/A & 31.2 & N\/A & 18.5 \\\\\nHgSe & 59.5~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 53.3 & 43.07~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 39.0 & 22.015~\\cite{Lehoczky_ElasticHgSe_PR_1969} & 21.2 \\\\\nHgTe & 53.61~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 45.0 & 36.6~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 30.4 & 21.23~\\cite{Semiconductors_BasicData_Springer, Cottam_ElasticHgTe_JPCS_1975} & 19.2 \\\\\n\\end{tabular}}\n\\label{tab:art115:zincblende_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$ and $c_{44}$ of\nrocksalt structure semiconductors.]\n{The rocksalt structure is designated {\\small AFLOW}\\ Prototype {\\sf AB\\_cF8\\_225\\_a\\_b}~\\cite{aflowANRL}.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\mathrm{exp}}$ & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\mathrm{exp}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\mathrm{exp}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n\\hline\nLiH & 67.1~\\cite{Laplaze_ElasticLiH_SSC_1976} & 84.8 & 17.0~\\cite{Laplaze_ElasticLiH_SSC_1976} & 14.2 & 46.0~\\cite{Laplaze_ElasticLiH_SSC_1976} & 48.8 \\\\\nLiF & 113.55~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 124 & 47.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 43.7 & 63.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 50.6 \\\\\nNaF & 97.0~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 96.1 & 24.3~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 22.3 & 28.1~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.6 \\\\\nNaCl & 49.36~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 50.5 & 12.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 12.1 & 12.65~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.6 \\\\\nNaBr & 40.12~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 41.2 & 10.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 10.2 & 9.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 7.97 \\\\\nNaI & 30.25~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 32.7 & 8.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 8.3 & 7.4~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.77 \\\\\nKF & 65.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 59.3 & 14.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 15.3 & 12.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 12.8 \\\\\nKCl & 40.78~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 37.2 & 6.9~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.39 & 6.33~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 6.55 \\\\\nKBr & 34.76~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 31.3 & 5.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.1 & 5.07~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.83 \\\\\nKI & 27.6~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 24.8 & 4.5~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.02 & 3.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 3.17 \\\\\nRbCl & 36.34~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 31.6 & 6.15~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 5.68 & 4.65~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.8 \\\\\nRbBr & 31.57~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 28.7 & 4.95~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 4.5 & 3.8~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 3.8 \\\\\nRbI & 25.83~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 23.1 & 3.7~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 3.3 & 2.78~\\cite{Haussuhl_ElasticRocksalt_ZP_1960} & 2.57 \\\\\nAgCl & 59.6~\\cite{Hughes_ElasticAgCl_PRB_1996} & 52.7 & 36.2~\\cite{Hughes_ElasticAgCl_PRB_1996} & 34.6 & 6.21~\\cite{Hughes_ElasticAgCl_PRB_1996} & 8.4 \\\\\nMgO & 297.8~\\cite{Sumino_ElasticMgO_JPE_1976} & 276 & 97.0~\\cite{Sumino_ElasticMgO_JPE_1976} & 90.7 & 156.3~\\cite{Sumino_ElasticMgO_JPE_1976} & 137 \\\\\nCaO & 221.89~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 202 & 57.81~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 57.0 & 80.32~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 74.6 \\\\\nSrO & 175.47~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 161 & 49.08~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 46.7 & 55.87~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 53.8 \\\\\nBaO & 126.14~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 118 & 50.03~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 44.8 & 33.68~\\cite{Chang_ElasticCaSrBaO_JPCS_1977} & 36.4 \\\\\nPbS & 126.15~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 127 & 16.24~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 16.9 & 17.09~\\cite{Semiconductors_BasicData_Springer, Peresada_ElasticPbS_PSSa_1976} & 20.0 \\\\\nPbSe & 123.7~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 119 & 19.3~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 12.2 & 15.91~\\cite{Semiconductors_BasicData_Springer, Lippmann_ElasticPbSe_PSSa_1971} & 17.2 \\\\\nPbTe & 105.3~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 107 & 7.0~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 5.63 & 13.22~\\cite{Semiconductors_BasicData_Springer, Miller_ElasticPbTe_JPCSS_1981} & 14.1 \\\\\nSnTe & 109.3~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 114 & 2.1~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 3.72 & 9.69~\\cite{Semiconductors_BasicData_Springer, Seddon_ElasticSnTe_SSC_1976} & 15.7 \\\\\n\\end{tabular}\n\\label{tab:art115:rocksalt_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{33}$, $c_{44}$ and $c_{66}$ of hexagonal structure semiconductors.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: $B$ and $G$ in {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nSiC & 494 & 102 & 48.7 & 534 & 151 & 196 \\\\\n& (500~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (92~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (55.8~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (564~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (168~\\cite{Arlt_ELasticSiC_JAAcS_1965}) & (204~\\cite{Arlt_ELasticSiC_JAAcS_1965}) \\\\\nAlN & 377 & 123 & 97.7 & 356 & 113 & 124 \\\\\n& (410.5~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (148.5~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (98.9~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (388.5~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (124.6~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) & (131.0~\\cite{Landolt-Bornstein, McNeil_ElasticAlN_JACerS_1993}) \\\\\nGaN & 329 & 115 & 80.5 & 362 & 90.3 & 109 \\\\\n& (296~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (130.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (158.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (267~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (24.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) & (83.0~\\cite{Semiconductors_BasicData_Springer, Savastenko_ElasticGaN_PSSa_1978}) \\\\\n& (390~\\cite{Polian_ElasticGaN_JAP_1996}) & (145.0~\\cite{Polian_ElasticGaN_JAP_1996}) & (106.0~\\cite{Polian_ElasticGaN_JAP_1996}) & (398~\\cite{Polian_ElasticGaN_JAP_1996}) & (105.0~\\cite{Polian_ElasticGaN_JAP_1996}) & (123.0~\\cite{Polian_ElasticGaN_JAP_1996}) \\\\\nZnO & 210 & 109 & 93.2 & 220 & 46.4 & 51.4 \\\\\n & (207~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (117.7~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (106.1~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (209.5~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (44.8~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (44.6~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) \\\\\nBeO & 427 & 110 & 79.4 & 464 & 138 & 158 \\\\\n & (460.6~\\cite{Cline_JAP_1967}) & (126.5~\\cite{Cline_JAP_1967}) & (88.48~\\cite{Cline_JAP_1967}) & (491.6~\\cite{Cline_JAP_1967}) & (147.7~\\cite{Cline_JAP_1967}) & (167.0~\\cite{Cline_JAP_1967}) \\\\\nCdS & 80.9 & 47.2 & 39.4 & 87.2 & 14.6 & 17.6 \\\\\n & (83.1~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (50.4~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (46.2~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (94.8~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (15.33~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) & (16.3~\\cite{Semiconductors_BasicData_Springer, Kobiakov_ElasticZnOCdS_SSC_1980}) \\\\\nInSe & 58.95 & 18.0 & 7.5 & 19.6 & 9.95 & 20.5 \\\\\n & (73.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (27.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (30.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (36.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (11.7~\\cite{Gatulle_ElasticInSe_PSSb_1983}) & (23.0~\\cite{Gatulle_ElasticInSe_PSSb_1983}) \\\\\nInN & 205 & 94.7 & 77.2 & 213 & 48.1 & 55.4 \\\\\n& N\/A & N\/A & N\/A & N\/A & N\/A & N\/A \\\\\n\\end{tabular}}\n\\label{tab:art115:wurzite_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{14}$, $c_{33}$, $c_{44}$ and $c_{66}$ of rhombohedral\nsemiconductors.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{14}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{14}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nBi$_2$Te$_3$ & 67.6 & 16.6 & 22.05 & 13.9 & 32.7 & 29.25 & 24.6 \\\\\n& (68.47~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (21.77~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (27.04~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (13.25~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (47.68~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (27.38~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) & (23.35~\\cite{Semiconductors_BasicData_Springer, Jenkins_ElasticBi2Te3_PRB_1972}) \\\\\nSb$_2$Te$_3$ & 67.8 & 11.2 & 19.1 & 9.92 & 23.2 & 21.35 & 28.8 \\\\\n & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nAl$_2$O$_3$ & 458 & 133 & 123 & -22.2 & 437 & 138 & 145 \\\\\n& (197.3~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (162.8~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (116.0~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (-21.9~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (500.9~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (146.8~\\cite{Goto_ElasticAl2O3_JGPR_1989}) & (17.25~\\cite{Goto_ElasticAl2O3_JGPR_1989}) \\\\\nCr$_2$O$_3$ & 350 & 145 & 131 & 17.1 & 325 & 128 & 111.5 \\\\\n& (374~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (148~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (175~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (-19~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (362~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (159~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) & (113~\\cite{Alberts_ElasticCr2O3_JMMM_1976}) \\\\\nBi$_2$Se$_3$ & 135 & 85.2 & 69.4 & 43.7 & 145 & 64.7 & 82.9 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\n\\end{tabular}}\n\\label{tab:art115:rhombo_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{33}$, $c_{44}$ and $c_{66}$ of body-centered tetragonal semiconductors.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\ncomp. & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nCuGaTe$_2$ & 67.6 & 36.3 & 37.0 & 66.8 & 32.1 & 31.1 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nZnGeP$_2$ & 118 & 48.95 & 51.7 & 117 & 62.1 & 61.05 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nZnSiAs$_2$ & 108 & 44.7 & 49.3 & 103 & 55.3 & 53.0 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nCuInTe$_2$ & 71.4 & 42.1 & 49.55 & 64.5 & 26.6 & 26.7 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nAgGaS$_2$ & 93.95 & 61.9 & 62.8 & 75.7 & 25.1 & 28.0 \\\\\n& (87.9~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (58.4~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (59.2~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (75.8~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (24.1~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) & (30.8~\\cite{Grimsditch_ElasticAgGaS2_PRB_1975}) \\\\\nCdGeP$_2$ & 102 & 46.25 & 50.6 & 88.2 & 48.0 & 44.9 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nCdGeAs$_2$ & 80.15 & 38.7 & 41.6 & 69.8 & 36.1 & 46.4 \\\\\n& (94.5~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (59.6~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (59.7~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (83.4~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (42.1~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) & (40.8~\\cite{Hailing_ElasticCdGeAs_JPCSS_1982}) \\\\\nCuGaS$_2$ & 102 & 57.0 & 60.5 & 104 & 48.9 & 47.9 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nCuGaSe$_2$ & 93.65 & 57.5 & 58.8 & 92.75 & 39.3 & 37.95 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nZnGeAs$_2$ & 93.7 & 40.65 & 42.6 & 92.6 & 48.2 & 47.1 \\\\\n& (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\n\\end{tabular}}\n\\label{tab:art115:bct_elastic_supp}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$ and $c_{44}$ of materials with BCC and simple\ncubic structures.]\n{``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r|r|r|r}\ncomp. & Pearson & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{11}^{\\mathrm{exp}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\mathrm{exp}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\mathrm{exp}}$ \\\\\n\\hline\nCoSb$_3$ & cI32 & 173 & N\/A & 31.2 & N\/A & 48.0 & N\/A \\\\\nIrSb$_3$ & cI32 & 195 & N\/A & 48.9 & N\/A & 52.85 & N\/A \\\\\nInTe & cP2 & 54.4 & N\/A & 35.3 & N\/A & 7.65 & N\/A \\\\\n\\end{tabular}\n\\label{tab:art115:bcc_elastic}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{23}$, $c_{33}$, $c_{44}$, $c_{55}$ and $c_{66}$ of materials with orthorhombic structures.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{22}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{23}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{55}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& & ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{22}^{\\mathrm{exp}}$) & ($c_{23}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{55}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nZnSb & oP16 & 84.1 & 30.5 & 28.4 & 93.1 & 25.3 & 83.2 & 16.9 & 39.3 & 31.4 \\\\\n& & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nSb$_2$O$_3$ & oP20 & 17.4 & 7.17 & 0.0 & 82.7 & -7.08 & 79.35 & 24.9 & 18.4 & 11.1 \\\\\n& & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\n\\end{tabular}}\n\\label{tab:art115:orc_elastic}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Elastic constants $c_{11}$, $c_{12}$, $c_{13}$, $c_{33}$, $c_{44}$ and $c_{66}$ of materials with tetragonal structures.]\n{Experimental values, where available, are shown in parentheses underneath the calculated values.\n``N\/A'' = Not available for that source.\nUnits: {\\small (GPa)}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\ncomp. & Pearson & $c_{11}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{12}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{13}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{33}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{44}^{\\substack{\\scalebox{0.6}{AEL}}}$ & $c_{66}^{\\substack{\\scalebox{0.6}{AEL}}}$ \\\\\n& & ($c_{11}^{\\mathrm{exp}}$) & ($c_{12}^{\\mathrm{exp}}$) & ($c_{13}^{\\mathrm{exp}}$) & ($c_{33}^{\\mathrm{exp}}$) & ($c_{44}^{\\mathrm{exp}}$) & ($c_{66}^{\\mathrm{exp}}$) \\\\\n\\hline\nInTe & tI16 & 32.4 & 11.55 & 13.4 & 52.8 & 13.4 & 13.45 \\\\\n& & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) & (N\/A) \\\\\nSnO$_2$ & tP6 & 191 & 128 & 123 & 346 & 73.8 & 168 \\\\\n& & (261.7~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (177.2~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (155.5~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (449.6~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (103.07~\\cite{Chang_ElasticSnO2_JGPR_1975}) & (207.4~\\cite{Chang_ElasticSnO2_JGPR_1975}) \\\\\n\\end{tabular}}\n\\label{tab:art115:tet_elastic}\n\\end{table}\n\n\\subsection{Conclusions}\n\nWe have implemented the ``Automatic Elasticity Library'' framework for \\nobreak\\mbox{\\it ab-initio}\\\nelastic constant calculations, and integrated it with the ``Automatic {\\small GIBBS}\\ Library'' implementation of the {\\small GIBBS}\\ quasi-harmonic Debye model within\nthe {\\small AFLOW}\\ and Materials Project ecosystems.\nWe used it\nto automatically calculate the bulk modulus, shear modulus, Poisson ratio, thermal conductivity, Debye temperature and Gr{\\\"u}neisen parameter of materials with\nvarious structures and compared them with available experimental results.\n\nA major aim of high-throughput calculations is to identify useful\nproperty descriptors for screening large datasets of structures~\\cite{nmatHT}.\nHere, we have examined whether the {\\it inexpensive} Debye model, despite its well known deficiencies, can be usefully leveraged for estimating thermal properties of materials by analyzing\ncorrelations between calculated and corresponding experimental quantities.\n\nIt is found that the {\\small AEL}\\ calculation of the elastic moduli\nreproduces the experimental results quite well, within 5\\% to 20\\%,\nparticularly for materials with cubic and\nhexagonal structures. The {\\small AGL}\\ method, using an isotropic approximation\nfor the bulk modulus, tends to provide a slightly worse quantitative\nagreement but still reproduces trends equally well.\nThe correlations are very high, often above $~0.99$.\nUsing different values of the Poisson ratio mainly affects Debye temperatures,\nwhile having very little effect on Gr{\\\"u}neisen parameters.\nSeveral different numerical and empirical equations of state have also been investigated. The differences\nbetween the results obtained from them are\nsmall, but in some cases they are found to introduce an additional\nsource of error compared to a direct evaluation of the bulk modulus\nfrom the elastic tensor or from the $E(V)$ curve.\nUsing the different equations of state has very little effect on Debye temperatures,\nbut has more of an effect on Gr{\\\"u}neisen parameters.\nCurrently, the values for {\\small AGL}\\ properties available in the {\\small AFLOW}\\ repository are those calculated by numerically fitting the $E_{\\substack{\\scalebox{0.6}{DFT}}}(V)$\ndata and calculating the bulk modulus using Equation~\\ref{eq:art115:bulkmod}.\n{The effect of using different exchange-correlation functionals was investigated for a subset of 16 materials. The results showed that\n{\\small LDA}\\ tended to overestimate thermomechanical properties such as bulk modulus or Debye temperature, compared to {\\small GGA}{}'s tendency\nto underestimate. However, neither functional was consistently better than the other at predicting trends. We therefore use {\\small GGA}-{\\small PBE}\\ for\nthe automated {\\small AEL}-{\\small AGL}\\ calculations in order to maintain consistency with the rest of the {\\small AFLOW}\\ data.}\n\nThe {\\small AEL}-{\\small AGL}\\ evaluation of the Debye temperature provides good\nagreement with experiment for this set of materials, whereas the predictions of the Gr{\\\"u}neisen parameter\nare quite poor. However, since the Gr{\\\"u}neisen parameter is slowly varying for materials sharing crystal structures, the {\\small AEL}-{\\small AGL}\\\nmethodology provides a reliable screening tool for identifying materials with very high or very low thermal conductivity.\nThe correlations between the experimental values of the thermal conductivity and those calculated with {\\small AGL}\\ are summarized in\nTable~\\ref{tab:art115:kappa_correlation}. For the entire set of materials examined we find high values of the Pearson correlation\nbetween $\\kappa^{\\mathrm{exp}}$ and $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$, ranging from $0.880$ to $0.933$. It is particularly high, above $0.9$, for materials\nwith high symmetry (cubic, hexagonal or rhombohedral) structures, but significantly lower for anisotropic materials.\nIn our previous work on {\\small AGL}~\\cite{curtarolo:art96}, we used an approximated the value of $\\sigma = 0.25$ in Equation~\\ref{eq:art115:fpoisson}.\nUsing instead the Poisson ratio calculated in {\\small AEL}, $\\sigma^{\\substack{\\scalebox{0.6}{AEL}}}$, the overall correlations are improved\nby about 5\\%, from $0.880$ to $0.928$, in the agreement with previous\nwork on metals~\\cite{Liu_Debye_CMS_2015}. The correlations for\nanisotropic materials, such as the body-centered tetragonal set\nexamined here, improved even more, demonstrating the significance of a\ndirect evaluation of the Poisson ratio.\nThis combined algorithm demonstrates the advantage of an integrated high-throughput materials design framework such as {\\small AFLOW},\nwhich enables the calculation of interdependent properties within a single automated workflow.\n\nA direct {\\small AEL}\\ evaluation of the Poisson ratio, instead of assuming a\nsimple approximation, e.g.\\ a Cauchy solid with $\\sigma = 0.25$,\nconsistently improves the correlations of the {\\small AGL}-Debye temperatures\nwith experiments.\nHowever, it has very little effect on the values obtained for the Gr{\\\"u}neisen parameter.\nSimple approximations lead to more numerically-robust and better system-size scaling calculations,\nas they avoid the complications inherent in obtaining the elastic tensor.\n{Therefore, {\\small AGL}\\ could also be used on its own for initial rapid screening,\nwith {\\small AEL}\\ being performed later for potentially interesting materials to increase the accuracy of the results.}\n\n\\begin{table}[tp]\\centering\n\\mycaption{Correlations between experimental values and {\\small AEL}\\ and {\\small AGL}\\ results for\nelastic and thermal properties for the entire set of materials.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & Pearson & Spearman & {\\small RMSrD}\\ \\\\\n & (linear) & (rank order) \\\\\n\\hline\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ ($\\sigma = 0.25$)~\\cite{curtarolo:art96} & 0.880 & 0.752 & 1.293 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ & 0.928 & 0.720 & 2.614 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BM}}}$ & 0.879 & 0.735 & 2.673 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{Vinet}}}$ & 0.912 & 0.737 & 2.443 \\\\\n$\\kappa^{\\mathrm{exp}}$ \\nobreak\\mbox{\\it vs.}\\ $\\kappa^{\\substack{\\scalebox{0.6}{BCN}}}$ & 0.933 & 0.733 & 2.751 \\\\\n\\end{tabular}\n\\label{tab:art115:kappa_correlation}\n\\end{table}\n\nWith respect to rapid estimation of thermal conductivities,\nthe approximations in the Leibfried-Schl{\\\"o}mann formalism\nmiss some of the details affecting the lattice thermal conductivity, such as the suppression of phonon-phonon scattering due to\nlarge gaps between the branches of the phonon dispersion~\\cite{Lindsay_PRL_2013}.\nNevertheless, the high correlations between $\\kappa^{\\mathrm{exp}}$ and\n$\\kappa^{\\substack{\\scalebox{0.6}{AGL}}}$ found for most of the structure families in this study demonstrate the utility of the {\\small AEL}-{\\small AGL}\\ approach\nas a screening method for large databases of materials where\nexperimental data is lacking or ambiguous.\nDespite its intrinsic limitations, the synergy presented by the {\\small AEL}-{\\small AGL}\\ approach\nprovides the right balance between accuracy and complexity in identifying materials with\npromising properties for further investigation.\n\n\\subsection{AFLOW AEL-AGL REST-API}\n\\label{subsec:art115:restapi_keywords}\n\nThe {\\small AEL}-{\\small AGL}\\ methodology described in this work is being used to calculate the elastic and thermal properties of materials in a high-throughput\nfashion by the {\\small AFLOW}\\ consortium. The results are now available on the {\\small AFLOW}\\ database~\\cite{aflowlib.org, aflowlibPAPER}\nvia the {\\small AFLOW}\\ {\\small REST-API}~\\cite{aflowAPI}. The following optional materials keywords have now been added to the {\\small AFLOW}\\ {\\small REST-API}\\\nto facilitate accessing this data.\n\n\\def\\item {{\\it Description:}\\ }{\\item {{\\it Description.}\\ }}\n\\def\\item {{\\it Type:}\\ }{\\item {{\\it Type.}\\ }}\n\\def\\item {{\\it Example.}\\ }{\\item {{\\it Example.}\\ }}\n\\def\\item {{\\it Units:}\\ }{\\item {{\\it Units.}\\ }}\n\\def\\item {{\\it Request syntax.}\\ }{\\item {{\\it Request syntax.}\\ }}\n\n\\begin{itemize}\n\n\\item\n\\verb|ael_bulk_modulus_reuss|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ bulk modulus as calculated using the Reuss average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_bulk_modulus_reuss=105.315|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_bulk_modulus_reuss|.\n\\end{itemize}\n\n\\item\n\\verb|ael_bulk_modulus_voigt|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ bulk modulus as calculated using the Voigt average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_bulk_modulus_voigt=105.315|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_bulk_modulus_voigt|.\n\\end{itemize}\n\n\\item\n\\verb|ael_bulk_modulus_vrh|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ bulk modulus as calculated using the\nVoigt-Reuss-Hill ({\\small VRH}) average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_bulk_modulus_vrh=105.315|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_bulk_modulus_vrh|.\n\\end{itemize}\n\n\\item\n\\verb|ael_elastic_anisotropy|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ elastic anisotropy.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } dimensionless.\n\\item {{\\it Example.}\\ } \\verb|ael_elastic_anistropy=0.000816153|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_elastic_anisotropy|.\n\\end{itemize}\n\n\\item\n\\verb|ael_poisson_ratio|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ Poisson ratio.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } dimensionless.\n\\item {{\\it Example.}\\ } \\verb|ael_poisson_ratio=0.21599|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_poisson_ratio|.\n\\end{itemize}\n\n\\item\n\\verb|ael_shear_modulus_reuss|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ shear modulus as calculated using the Reuss average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_shear_modulus_reuss=73.7868|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_shear_modulus_reuss|.\n\\end{itemize}\n\n\\item\n\\verb|ael_shear_modulus_voigt|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ shear modulus as calculated using the Voigt average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_shear_modulus_voigt=73.7989|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_shear_modulus_voigt|.\n\\end{itemize}\n\n\\item\n\\verb|ael_shear_modulus_vrh|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ shear modulus as calculated using the\nVoigt-Reuss-Hill ({\\small VRH}) average.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|ael_shear_modulus_vrh=73.7929|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_shear_modulus_vrh|.\n\\end{itemize}\n\n\\item\n\\verb|ael_speed_of_sound_average|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ average speed of sound calculated from the transverse and longitudinal speeds of sound.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } m\/s.\n\\item {{\\it Example.}\\ } \\verb|ael_speed_of_sound_average=500.0|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_speed_of_sound_average|.\n\\end{itemize}\n\n\\item\n\\verb|ael_speed_of_sound_longitudinal|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ speed of sound in the longitudinal direction.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } m\/s.\n\\item {{\\it Example.}\\ } \\verb|ael_speed_of_sound_longitudinal=500.0|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_speed_of_sound_longitudinal|.\n\\end{itemize}\n\n\\item\n\\verb|ael_speed_of_sound_transverse|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AEL}\\ speed of sound in the transverse direction.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } m\/s.\n\\item {{\\it Example.}\\ } \\verb|ael_speed_of_sound_transverse=500.0|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?ael_speed_of_sound_transverse|.\n\\end{itemize}\n\n\\end{itemize}\n\n\\begin{itemize}\n\n\\item\n\\verb|agl_acoustic_debye|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ acoustic Debye temperature.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } K.\n\\item {{\\it Example.}\\ } \\verb|agl_acoustic_debye=492|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_acoustic_debye|.\n\\end{itemize}\n\n\\item\n\\verb|agl_bulk_modulus_isothermal_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ isothermal bulk modulus at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|agl_bulk_modulus_isothermal_300K=96.6|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_bulk_modulus_isothermal_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_bulk_modulus_static_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ static bulk modulus at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } GPa.\n\\item {{\\it Example.}\\ } \\verb|agl_bulk_modulus_static_300K=99.59|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_bulk_modulus_static_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_debye|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ Debye temperature.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } K.\n\\item {{\\it Example.}\\ } \\verb|agl_debye=620|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_debye|.\n\\end{itemize}\n\n\\item\n\\verb|agl_gruneisen|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ Gr{\\\"u}neisen parameter.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } dimensionless.\n\\item {{\\it Example.}\\ } \\verb|agl_gruneisen=2.06|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_gruneisen|.\n\\end{itemize}\n\n\\item\n\\verb|agl_heat_capacity_Cv_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ heat capacity at constant volume (C$_V$) at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } k$_\\mathrm{B}$\/cell.\n\\item {{\\it Example.}\\ } \\verb|agl_heat_capacity_Cv_300K=4.901|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_heat_capacity_Cv_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_heat_capacity_Cp_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ heat capacity at constant pressure (C$_p$) at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } k$_\\mathrm{B}$\/cell.\n\\item {{\\it Example.}\\ } \\verb|agl_heat_capacity_Cp_300K=5.502|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_heat_capacity_Cp_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_poisson_ratio_source|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns source of Poisson ratio used to calculate Debye temperature in {\\small AGL}. Possible sources include \\verb|ael_poisson_ratio_|, in\nwhich case the Poisson ratio was calculated from first principles using {\\small AEL}; \\verb|empirical_ratio_|, in which case the value was taken\nfrom the literature; and \\verb|Cauchy_ratio_0.25|, in which case the default value of 0.25 of the Poisson ratio of a Cauchy solid\nwas used.\n\\item {{\\it Type:}\\ } \\verb|string|.\n\\item {{\\it Example.}\\ } \\verb|agl_poisson_ratio_source=ael_poisson_ratio_0.193802|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_poisson_ratio_source|.\n\\end{itemize}\n\n\\item\n\\verb|agl_thermal_conductivity_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ thermal conductivity at 300~K.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } W\/m*K.\n\\item {{\\it Example.}\\ } \\verb|agl_thermal_conductivity_300K=24.41|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_thermal_conductivity_300K|.\n\\end{itemize}\n\n\\item\n\\verb|agl_thermal_expansion_300K|\n\\begin{itemize}\n\\item {{\\it Description:}\\ } Returns {\\small AGL}\\ thermal expansion at 300~K and zero pressure.\n\\item {{\\it Type:}\\ } \\verb|number|.\n\\item {{\\it Units:}\\ } 1\/K.\n\\item {{\\it Example.}\\ } \\verb|agl_thermal_expansion_300K=4.997e-05|.\n\\item {{\\it Request syntax.}\\ } \\verb|$aurl\/?agl_thermal_expansion_300K|.\n\\end{itemize}\n\n\\end{itemize}\n\\clearpage\n\\chapter{Data-driven Approaches}\n\\section{AFLOW-CHULL: Cloud-Oriented Platform for Autonomous Phase Stability Analysis}\n\\label{sec:art146}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art146}.\n\n\\subsection{Introduction}\nAccelerating the discovery of new functional materials demands an efficient determination of synthesizability.\nIn general, materials synthesis is a multifaceted problem, spanning\n\\textbf{i.} technical challenges, such as experimental apparatus design and growth conditions~\\cite{Jansen_AngChemInt_2002,Potyrailo_ACSCombSci_2011},\nas well as\n\\textbf{ii.} economic and environmental obstacles, including accessibility and handling of necessary components~\\cite{Kuzmin_JPCM_2014,curtarolo:art109}.\nPhase stability is a limiting factor.\nOften, it accounts for the gap between\nmaterials prediction and experimental realization.\nAddressing stability requires an understanding of how phases compete thermodynamically.\nDespite the wealth of available experimental phase diagrams~\\cite{ASMAlloyInternational},\nthe number of systems explored represents a negligible fraction of\nall hypothetical structures~\\cite{Walsh_NChem_2015,curtarolo:art124}.\nLarge materials databases~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux,nomad,APL_Mater_Jain2013,Saal_JOM_2013,cmr_repository,Pizzi_AiiDA_2016}\nenable the construction of calculated phase diagrams,\nwhere aggregate structural and energetic materials data is employed.\nThe analysis delivers many fundamental thermodynamic descriptors,\nincluding stable\/unstable classification,\nphase coexistence, measures of robust stability, and determination of\ndecomposition reactions~\\cite{curtarolo:art109,curtarolo:art113,Bechtel_PRM_2018,Li_CMS_2018,Balachandran_PRM_2018}.\n\nAs with all informatics-based approaches, \\nobreak\\mbox{\\it ab-initio}\\ phase diagrams require an abundance of data:\nwell-converged enthalpies from a variety of different phases.\nMany thermodynamic descriptors computed\nfrom the {\\sf \\AFLOW.org}\\ repository\nhave already demonstrated predictive power in characterizing phase\nstability~\\cite{curtarolo:art49,curtarolo:art51,curtarolo:art53,curtarolo:art57,curtarolo:art63,curtarolo:art67,curtarolo:art70,curtarolo:art74,monsterPGM,curtarolo:art106,curtarolo:art109,curtarolo:art112,curtarolo:art113,curtarolo:art117,curtarolo:art126,curtarolo:art130},\nincluding one investigation that resulted in the synthesis of\ntwo new magnets --- the first ever discovered by computational approaches~\\cite{curtarolo:art109}.\nAs exploration embraces more complex systems, such analyses are expected to\nbecome increasingly critical in confining the search space.\nIn fact, prospects for stable ordered phases diminish with every new component (dimension), despite the growing number of combinations.\nThis is due to increased competition with\n\\textbf{i.} phases of lower dimensionality, \\nobreak\\mbox{\\it e.g.}, ternary phases competing with stable binary phases~\\cite{curtarolo:art130}, and\n\\textbf{ii.} disordered (higher entropy) phases~\\cite{curtarolo:art99,curtarolo:art122,curtarolo:art139}.\n\nTo address the challenge, a new module has been implemented in the autonomous, open-source~\\cite{gnu_license}\n{\\small AFLOW}\\ (\\underline{A}utomatic \\underline{Flow}) framework for \\nobreak\\mbox{\\it ab-initio}\\ calculations~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}.\n{\\small \\AFLOWHULLtitle}\\ ({\\small AFLOW}\\ \\underline{c}onvex \\underline{hull}) offers a thermodynamic characterization that can be employed\nlocally from any {\\small UNIX}-like machine, including those running Linux and macOS.\nBuilt-in data curation and validation schemes ensure results are well-converged:\nadhering to proper hull statistics, performing outlier detection, and determining structural equivalence.\n{\\small \\AFLOWHULLtitle}\\ is powered by the {\\small AFLUX}\\ Search-{\\small API}\\ (\\underline{a}pplication \\underline{p}rogramming \\underline{i}nterface)~\\cite{aflux},\nwhich enables access to more than 2 million compounds from the {\\sf \\AFLOW.org}\\ repository.\nWith {\\small AFLUX}\\ integration, data-bindings are flexible enough to serve any materials database,\nincluding large heterogeneous repositories such as {NOMAD}~\\cite{nomad}.\n\nSeveral analysis output types have been created for integration\ninto a variety of design workflows, including plain text and\n{\\small JSON}\\ (\\underline{J}ava\\underline{S}cript \\underline{O}bject \\underline{N}otation) file types.\nA small set of example scripts have been included demonstrating\nhow to employ {\\small \\AFLOWHULLtitle}\\ from within a Python environment, much in the spirit of {\\small AFLOW-SYM}~\\cite{curtarolo:art135}.\nThe {\\small JSON}\\ output also powers an interactive, online web application offering enhanced presentation of thermodynamic descriptors and\nvisualization of 2-\/3-dimensional hulls.\nThe application can be accessed through the {\\sf \\AFLOW.org}\\ portal under ``Apps and Docs'' or directly at {\\sf aflow.org\/aflow-chull}.\n\nAs a test-bed, the module is applied to all 2 million compounds available in the {\\sf \\AFLOW.org}\\ repository.\nAfter enforcing stringent hull convergence criteria, the module resolves a thermodynamic characterization\nfor more than 1,300 binary and ternary systems.\nStable phases are screened for previously explored systems and ranked by their\nrelative stability criterion, a dimensionless quantity capturing the\neffect of the phase on the minimum energy surface~\\cite{curtarolo:art109}.\nSeveral promising candidates are identified, including\n17\\ $C15_{b}$-type structures $\\left(F\\overline{4}3m~\\#216\\right)$ and two half-Heuslers.\nHence, screening criteria based on these thermodynamic descriptors can accelerate the\ndiscovery of new stable phases.\nMore broadly, the design of more challenging materials, including ceramics~\\cite{curtarolo:art80} and metallic glasses~\\cite{curtarolo:art112},\nbenefit from autonomous, integrated platforms such as {\\small \\AFLOWHULLtitle}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig033}\n\\mycaption[Example hull illustrations in 2-\/3-dimensions as generated by {\\small \\AFLOWHULLtitle}.]\n{(\\textbf{a}) Co-Ti and (\\textbf{b}) Mn-Pd-Pt.}\n\\label{fig:art146:hull_examples}\n\\end{figure}\n\n\\subsection{Methods}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig034}\n\\mycaption[Illustration of the convex hull construction for a binary system with {\\small \\AFLOWHULLtitle}.]\n{The approach is inspired by the {\\small Qhull}\\ algorithm~\\cite{qhull}.\nThe points on the plot represent structures from the {\\sf \\AFLOW.org}\\ database~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\n(\\textbf{a}) and (\\textbf{g}) denote the beginning and end of the algorithm, respectively.\n(\\textbf{c}-\\textbf{f}) denote the iterative loop that continues until the\ncondition denoted by (\\textbf{b}) is no longer satisfied.\nPoints are marked with crosses if, by that step in the algorithm, they have been determined to be inside the hull,\nand otherwise are marked with circles.\nThe furthest point from the facet in (\\textbf{d}) is marked with a triangle.\nPoints and facets of interest are highlighted in \\textcolor{pranab_red}{{\\bf red}} and \\textcolor{pranab_green}{{\\bf green}}, respectively.}\n\\label{fig:art146:hull_workflow}\n\\end{figure}\n\n\\boldsection{Defining thermodynamic stability.}\nFor a multicomponent system at a fixed temperature ($T$) and pressure ($p$),\nthe minimum Gibbs free energy $G$ (per atom) defines the thermodynamic equilibrium:\n\\begin{equation}\nG(T,p,\\{x_{i}\\})=H-TS\n\\label{eq:art146:gibbs_free_energy}\n\\end{equation}\nwhere $x_{i}$ is the atomic concentration of the $i$-species,\n$H$ is the enthalpy, and $S$ is the entropy.\nA binary phase $A_{x_{A}}B_{x_{B}}$ is stable at equilibrium with respect to its components\n$A$ and $B$ if the corresponding formation reaction releases energy:\n\\begin{equation}\nx_{A} A + x_{B} B \\xrightarrow[]{\\Delta G<0} A_{x_{A}}B_{x_{B}},\n\\label{eq:art146:formation_reaction}\n\\end{equation}\nwhere $\\Delta G$ is the energy difference between the mixed phase\nand the sum of its components.\nConversely, a positive $\\Delta G$ suggests the decomposition of $A_{x_{A}}B_{x_{B}}$ is preferred, and\nis thus unstable.\nIn general, the magnitude of $\\Delta G$ quantifies the propensity for the reaction,\nand the sign determines the direction.\n\nRelative stability can be visualized on a free-energy-concentration diagram\n--- $G$ \\nobreak\\mbox{\\it vs.}\\ $\\left\\{ x_i \\right\\}$ ---\nwhere $\\Delta G$ is depicted as the energetic vertical-distance between $A_{x_{A}}B_{x_{B}}$ and the\ntie-line connecting $A$ and $B$ end-members (elemental phases).\nEnd-members constitute only a single pathway to formation\/decomposition, and\nall feasible reactions should be considered for system-wide stability.\n{Identification of equilibrium phases} is mathematically equivalent to the construction\nof the convex hull --- the set of the most extreme or ``outside'' points (Figure~\\ref{fig:art146:hull_examples}(a)).\n{The convex hull characterizes the phase stability of the system at equilibrium\nand does not include kinetic considerations for synthesis.\nGrowth conditions affect the final outcome leading to formation of polymorphs and\/or metastable phases,\nwhich could differ from the equilibrium phases.\nThis is a formidable task for high-throughput characterization.\nTo help identify kinetic pathways for synthesis, {\\small \\AFLOWHULLtitle}\\\nincludes (more in future releases) potential kinetic descriptors,\n\\nobreak\\mbox{\\it e.g.}, chemical decompositions, distance from stability, entropic temperature~\\cite{curtarolo:art98},\nglass formation ability~\\cite{curtarolo:art112}, and spectral entropy analysis for high-entropy systems.}\n\nIn the zero temperature limit (as is the case for ground-state density functional theory),\nthe entropic term of Equation~\\ref{eq:art146:gibbs_free_energy} vanishes,\nleaving the formation enthalpy term (per atom) as the driving force:\n\\begin{equation}\n H_\\mathrm{f}=H_{A_{x_{A}}B_{x_{B}}}-\\left(x_{A} H_{A} + x_{B} H_{B} \\right).\n\\end{equation}\nBy construction, formation enthalpies of stable elemental phases are zero, restricting\nthe convex hull to the lower hemisphere.\n{Zero-point energies are not yet included in the {\\sf \\AFLOW.org}\\ repository and thus are neglected from the enthalpy calculations.\nEfforts to incorporate vibrational characterizations are underway~\\cite{curtarolo:art96,Nath_QHA_2016}.\nThis contribution could have a large impact on compounds containing light-elements, such as\nhydrogen~\\cite{Majzoub_PRB_2005}, which comprise a small minority (less than 1\\%) of the overall repository.}\n\nBy offsetting the enthalpy with that of the elemental phases,\n$H_\\mathrm{f}$ quantifies the energy gain from forming new bonds between\nunlike components,\\footnote{The formation enthalpy is not to be confused with the cohesive energy, which quantifies\nthe energy difference between the phase and its fully gaseous (single atoms) counterpart, \\nobreak\\mbox{\\it i.e.},\nthe energy in all bonds.} \\nobreak\\mbox{\\it e.g.}, $A-B$.\n{Currently, the {\\small \\AFLOWHULLtitle}\\ framework does not allow the renormalization of chemical potentials to\nimprove the calculation of formation enthalpies when gas phases are involved.\nA new first-principles approach is being developed and tested in {\\small AFLOW},\nand will be implemented in future versions of the {\\small \\AFLOWHULLtitle}\\\nsoftware together with the available approaches}~\\cite{CrUJ,Lany_Zunger_FERE_2012}.\n\nThe tie-lines connecting stable phases in Figure~\\ref{fig:art146:hull_examples}(a)\ndefine regions of phase separation where the two phases coexist at equilibrium.\nThe chemical potentials are equal for each component among coexisting phases,\nimplying the common tangent tie-line construction~\\cite{Ganguly_thermo_2008,Darken_pchemmetals_1953}.\n{Under thermodynamic equilibrium,} phases above a tie-line will decompose into a linear combination of the stable phases that\ndefine the tie-line (Figure~\\ref{fig:art146:hull_analyses}(d)).\nThe Gibbs phase rule~\\cite{McQuarrie} dictates the shape of tie-lines for $N$-ary systems,\nwhich generalizes to $\\left(N-1\\right)$-dimensional triangles (simplexes) and correspond to facets of the convex hull,\n\\nobreak\\mbox{\\it e.g.}, lines in two dimensions (Figure~\\ref{fig:art146:hull_examples}(a)),\ntriangles in three dimensions (Figure~\\ref{fig:art146:hull_examples}(b)),\nand tetrahedra in four.\nThe set of equilibrium facets define the $N$-dimensional minimum energy surface.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig035}\n\\mycaption[Illustration of the {\\small \\AFLOWHULLtitle}\\ iterative hull scheme.]\n{The convex hull and associated properties are first calculated for the binary\nhulls, and then propagated to the ternary hull.\nThis is generalized for $N$-dimensions.}\n\\label{fig:art146:dimensions}\n\\end{figure}\n\n\\boldsection{Hull construction.}\n{\\small \\AFLOWHULLtitle}\\ calculates the $N$-dimensional convex hull corresponding to an $N$-ary system\nwith an algorithm partially inspired by {\\small Qhull}~\\cite{qhull}.\nThe algorithm\nis efficient in identifying the most important points for construction of facets,\nwhich are treated as hyperplanes instead of boundary-defining inequalities.\n{\\small \\AFLOWHULLtitle}\\ uniquely accommodates thermodynamic hulls,\n\\nobreak\\mbox{\\it i.e.}, data occupying the lower half hemisphere and\ndefined by stoichiometric coordinates $\\left(0 \\leq x_{i} \\leq 1 \\right)$.\nPoints corresponding to individual phases are characterized by their stoichiometric and energetic coordinates:\n\\begin{equation}\n\\mathbf{p}=\\left[x_{1}, x_{2}, \\ldots, x_{N-1}, H_\\mathrm{f}\\right] = \\left[\\mathbf{x}, H_\\mathrm{f}\\right],\n\\label{eq:art146:point}\n\\end{equation}\nwhere $x_{N}$ is implicit $\\left(\\sum_{i}x_i=1\\right)$.\nData preparation includes the\n\\textbf{i.} elimination of phases unstable with respect to end-members (points above the zero $H_{\\mathrm{f}}$ tie-line)\nand \\textbf{ii.} organization of phases by stoichiometry and sorted by energy.\nThrough this stoichiometry group structure, all but the minimum energy phases are eliminated from\nthe convex hull calculation.\n\nThe workflow is illustrated in Figure~\\ref{fig:art146:hull_workflow}.\n{\\small \\AFLOWHULLtitle}\\ operates by partitioning space, iteratively defining\n``inside'' \\nobreak\\mbox{\\it vs.}\\ ``outside'' half-spaces until all points are either on the hull or inside of it.\nFirst, a simplex is initialized (Figure~\\ref{fig:art146:hull_workflow}(a)) with the most extreme points:\nstable end-members and the globally stable mixed phase (lowest energy).\nA facet is described as:\n\\begin{equation}\n\\mathbf{n} \\cdot \\mathbf{r} + D = 0,\n\\label{eq:art146:plane_eq}\n\\end{equation}\nwhere $\\mathbf{n}$ is the characteristic normal vector, $\\mathbf{r}$ is the position vector,\nand $D$ is the offset.\nA general hyperplane is defined by $N$ points and $k=\\left(N-1\\right)$ corresponding edges\n$\\mathbf{v}_{k}=\\mathbf{p}_{k}-\\mathbf{p}_{\\mathrm{origin}}$.\nTo construct $\\mathbf{n}$, {\\small \\AFLOWHULLtitle}\\ employs a generalized cross product approach~\\cite{Massey_AMM_1983},\nwhere $n_{i \\in \\{1,\\ldots,N\\}}$ (unnormalized) is the $i$-row cofactor\n$\\left(C_{i,j=0}\\right)$ of the matrix $\\mathbf{V}$ containing $\\mathbf{v}_k$ in its columns:\n\\begin{equation}\n n_{i} = \\left(-1\\right)^{i+1}M_{i,j=0}\\left(\n\\begin{bmatrix}\n | & & | \\\\\n \\mathbf{v}_{1} & \\ldots & \\mathbf{v}_{k} \\\\\n | & & | \\\\\n\\end{bmatrix}\n\\right)\n\\label{eq:art146:hyperplane_normal}\n\\end{equation}\nHere, $M_{i,j=0}\\left(\\mathbf{V}\\right)$ denotes the\n$i$-row minor of $\\mathbf{V}$,\n\\nobreak\\mbox{\\it i.e.}, the determinant of the submatrix formed by removing the $i$-row.\n\nThe algorithm then enters a loop over the facets of the convex hull until no points are declared ``outside'',\ndefined in the hyperplane description by the signed point-plane distance (Figure~\\ref{fig:art146:hull_workflow}(b)).\nEach point outside of the hull is singularly assigned to the outside set of a facet (\\textcolor{pranab_red}{{\\bf red}}\nin Figure~\\ref{fig:art146:hull_workflow}(c)).\nThe furthest point from each facet --- by standard point-plane distance --- is selected from the outside set\n(marked with a triangle in Figure~\\ref{fig:art146:hull_workflow}(d)).\nEach neighboring facet is visited to determine whether the furthest point is also outside of it, defining\nthe set of visible planes (\\textcolor{pranab_green}{{\\bf green}}) and its boundary,\nthe horizon ridges (\\textcolor{pranab_red}{{\\bf red}}) (Figure~\\ref{fig:art146:hull_workflow}(d)).\nThe furthest point is combined with each ridge of the horizon to form new facets (Figure~\\ref{fig:art146:hull_workflow}(e)).\nThe visible planes --- the dotted line in Figure~\\ref{fig:art146:hull_workflow}(e) --- are then removed from the\nconvex hull (Figure~\\ref{fig:art146:hull_workflow}(f)).\nThe fully constructed convex hull --- with all points on the hull or inside of it --- is\nsummarized in Figure~\\ref{fig:art146:hull_workflow}(g).\n\nA challenge arises with lower dimensional data in higher dimensional convex hull constructions.\nFor example, binary phases composed of the same species all exist on the same (vertical) plane in three dimensions.\nA half-space partitioning scheme can make no ``inside'' \\nobreak\\mbox{\\it vs.}\\ ``outside'' differentiation between such points.\nThese ambiguously-defined facets\\nocite{qhull}\\footnote{Ambiguously-defined facets occur when a set of $d+1$ points (or more) define a $(d-1)$-flat~\\cite{qhull}.}\nconstitute a hull outside the scope of the {\\small Qhull}\\ algorithm~\\cite{qhull}.\nIn the case of three dimensions, the creation of ill-defined facets with collinear edges can result.\nHyper-collinearity --- planes defined with collinear edges, tetrahedra defined with coplanar faces, \\nobreak\\mbox{\\it etc.}\\ ---\nis prescribed by the content (hyper-volume) of the facet.\nThe quantity resolves the length of the line ($1$-simplex), the area of a triangle ($2$-simplex),\nthe volume of a tetrahedron ($3$-simplex), \\nobreak\\mbox{\\it etc.},\nand is calculated for a simplex of $N$-dimensions via the Cayley-Menger determinant~\\cite{sommerville_1929_n_dimensional_geometry}.\nBoth vertical and content-less facets are problematic for thermodynamic characterizations,\nparticularly when calculating hull distances, which require facets within finite energetic distances\nand well-defined normals.\n\nA dimensionally-iterative scheme is implemented in {\\small \\AFLOWHULLtitle}\\ to solve the issue.\nIt calculates the convex hull for each dimension consecutively\n(Figure~\\ref{fig:art146:dimensions}).\nIn the case of a ternary hull, the three binary hulls are calculated first, and the relevant\nthermodynamic data is extracted and then propagated forward.\nThough vertical and content-less facets are still created in higher dimensions, no thermodynamic\ndescriptors are extracted from them.\nTo optimize the calculation, only stable binary structures are propagated forward to\nthe ternary hull calculation, and this approach is generalized for $N$-dimensions.\nThe scheme is the default for thermodynamic hulls, resorting back to\nthe general convex hull algorithm otherwise.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig036}\n\\mycaption[Illustrations of various automated convex hull analyses in {\\small \\AFLOWHULLtitle}.]\n{(\\textbf{a}) A plot showing an egregious outlier in the Al-Co convex hull.\n(\\textbf{b}) The corrected Al-Co convex hull with the outlier removed.\n(\\textbf{c}) The Te-Zr convex hull with the traditional compound labels replaced\nwith the corresponding {\\small ICSD}\\ number designations as determined by a structure\ncomparison analysis.\nIf multiple {\\small ICSD}\\ entries are found for the same stoichiometry, the lowest number\n{\\small ICSD}\\ entry is chosen (chronologically reported, usually).\n(\\textbf{d}) The Pd-Pt convex hull. The decomposition energy of Pd$_{2}$Pt$_{3}$ is plotted in\n\\textcolor{pranab_red}{{\\bf red}}, and highlighted in \\textcolor{pranab_green}{{\\bf green}} is\nthe equilibrium facet directly below it.\nThe facet is defined by ground-state phases PdPt$_{3}$ and PdPt.\n(\\textbf{e}) The Pd-Pt convex hull. The stability criterion $\\delta_{\\mathrm{sc}}$ of PdPt is plotted\nin \\textcolor{pranab_green}{{\\bf green}}, with the pseudo-hull plotted with dashed lines.\n(\\textbf{f}) The B-Sm convex hull plotted with the\nideal ``{\\it iso-max-latent-heat}'' lines of the grand-canonical ensemble~\\cite{monsterPGM,curtarolo:art98}\nfor the ground-state structures.}\n\\label{fig:art146:hull_analyses}\n\\end{figure}\n\n\\boldsection{Thermodynamic data.}\nStructural and energetic data employed to construct the convex hull\nis retrieved from the {\\sf \\AFLOW.org}~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}{} repository, which contains more than 2 million compounds and\n200 million calculated properties.\nThe database is generated by the autonomous, \\nobreak\\mbox{\\it ab-initio}\\ framework {\\small AFLOW}~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}{}\nfollowing the {\\small AFLOW}\\ Standard for high-throughput materials science\ncalculations~\\cite{curtarolo:art104}.\nIn particular, calculations are performed\nwith {\\small VASP}\\ (\\underline{V}ienna \\textit{\\underline{A}b initio} \\underline{S}imulation \\underline{P}ackage)~\\cite{vasp_prb1996}.\nWavefunctions are represented by a large basis set, including\nall terms with kinetic energy up to a threshold 1.4 times larger than the recommended defaults.\n{\\small AFLOW}\\ also leverages a large $\\mathbf{k}$-point mesh --- as standardized by\na $\\mathbf{k}$-points-per-reciprocal-atom scheme~\\cite{curtarolo:art104} ---\nwhich is critical for convergence and reliability of calculated properties.\nInvestigations show that the {\\small AFLOW}\\ Standard of at least $6,000$ $\\mathbf{k}$-points-per-reciprocal-atom\nfor structural relaxations and $10,000$ for the static calculations ensures\nrobust convergence of the energies to within one meV\/atom in more than 95\\% of systems\n(including metals which suffer from the discontinuity in the occupancy function at zero temperature),\nand within three meV\/atom otherwise~\\cite{Wisesa_Kgrids_PRB_2016}.\n\nSpecial consideration is taken for the calculation of $H_{\\mathrm{f}}$.\nThe reference energies for the elemental phases are calculated and stored in the\n{\\small LIB1}\\ catalog for unary phases in the {\\sf \\AFLOW.org}\\ repository, and include variations for different\nfunctionals and pseudopotentials.\nFor consistency, {\\small \\AFLOWHULLtitle}\\ only employs data calculated with the \\underline{P}erdew-\\underline{B}urke-\\underline{E}rnzerhof\nGeneralized Gradient Approximation functional~\\cite{PBE}\nand pseudopotentials calculated with the\n\\underline{p}rojector \\underline{a}ugmented \\underline{w}ave method~\\cite{PAW} ({\\small PAW}-{\\small PBE}).\n{Calculations employing {\\small DFT}$+U$ corrections to rectify self-interaction errors and energy-gap issues for\nelectronic properties~\\cite{curtarolo:art104} are neglected.\nIn general, these corrections are parameterized\nand material-specific~\\cite{curtarolo:art93}.\nThey artificially augment the energy of the system affecting the reliability of thermodynamic properties.}\nIt is possible to encounter stable (lowest energy) elemental phases with energy differences from the reference\nof order meV\/atom, which is the result of duplicate entries (by relaxation or otherwise)\nas well as reruns with new parameters, \\nobreak\\mbox{\\it e.g.}, a denser $\\mathbf{k}$-point mesh.\nTo avoid any issues with the convex hull calculation, the algorithm fixes\nthe half-space plane at zero.\nHowever, a ``warning'' is prompted in the event that the stable elemental phase differs from\nthe reference energy by more than 15 meV\/atom, yielding a ``skewed'' hull.\n\nData is retrieved via the {\\small AFLUX}\\ Search-{\\small API}~\\cite{aflux}, designed for accessing\nproperty-specific datasets efficiently.\nThe following is an example of a relevant request:\n\\begin{center}\n\\noindent{\\sf http:\/\/aflowlib.duke.edu\/search\/API\/?species(Mn,Pd),nspecies(2),*,paging(0)}\n\\end{center}\nwhere {\\sf http:\/\/aflowlib.duke.edu\/search\/API\/} is the {\\small URL}\\ for the {\\small AFLUX}\\ server and\n{\\sf species(Mn,Pd),nspecies(2),*,paging(0)} is the query.\n{\\sf species(Mn,Pd)} queries for any entry containing the elements\nMn or Pd, {\\sf nspecies(2)} limits the search to binaries only, {\\sf *} returns the data\nfor all available fields, and {\\sf paging(0)} amalgamates all data into a single response\nwithout paginating (warning, this can be a large quantity of data).\nSuch queries are constructed combinatorially for each dimension, \\nobreak\\mbox{\\it e.g.},\na general ternary hull $ABC$ constructs the following seven queries:\n{\\sf species($A$)},\n{\\sf species($B$)}, and\n{\\sf species($C$)} with {\\sf nspecies(1)},\n{\\sf species($A$,$B$)},\n{\\sf species($A$,$C$)}, and\n{\\sf species($B$,$C$)} with {\\sf nspecies(2)}, and\n{\\sf species($A$,$B$,$C$)} with {\\sf nspecies(3)}.\n\n\\boldsection{Validation schemes.}\nVarious statistical analyses and data curation procedures are employed\nby {\\small \\AFLOWHULLtitle}\\ to maximize fidelity.\nAt a minimum, each binary hull must contain 200 structures to ensure\na sufficient sampling size for inference.\nThere is never any guarantee that all stable structures have been identified~\\cite{curtarolo:art54,monsterPGM},\nbut convergence is approached with larger datasets.\nWith continued growth of {\\small LIB3}\\ (ternary phases) and beyond, higher dimensional parameters will be incorporated,\nthough it is expected that the parameters are best defined along tie-lines (\\nobreak\\mbox{\\it vs.}\\ tie-surfaces).\nA comprehensive list of available alloys and structure counts are included in the\nSupporting Information of Reference~\\cite{curtarolo:art146}.\n\n\\boldsection{Outlier detection.} In addition to having been calculated with a standard set of parameters~\\cite{curtarolo:art104},\ndatabase entries\nshould also be well-converged.\nPrior to the injection of new entries into the {\\sf \\AFLOW.org}\\ database,\nvarious verification tests are employed to ensure convergence, including an analysis of the\nrelaxed structure's stress tensor~\\cite{aflux}.\nIssues stemming from poor convergence and failures in the functional parameterization~\\cite{curtarolo:art54,curtarolo:art113}\ncan change the topology of the convex hull,\nresulting in contradictions with experiments.\nHence, an outlier detection algorithm is applied before the hull is constructed:\nstructures are classified as outliers and discarded if\nthey have energies that fall well below the first\nquartile by a multiple of the interquartile range (conservatively set to 3.25 by default)~\\cite{Miller_QJEPSA_1991}.\nOnly points existing in the lower half-space (phases stable against end-members)\nare considered for the outlier analysis, and hence systems need to show\nsome miscibility, \\nobreak\\mbox{\\it i.e.}, at least four points for a proper interquartile range determination.\nDespite its simplicity, the interquartile range is the preferred estimate of scale\nover other measures such as the standard deviation or the median absolute deviation,\nwhich require knowledge of the underlying distribution (normal or otherwise)~\\cite{Leys_JESP_2013}.\nAn example hull (Al-Co) showing an outlier is plotted in Figure~\\ref{fig:art146:hull_analyses}(a)\nand the corrected hull with the outlier removed is presented in Figure~\\ref{fig:art146:hull_analyses}(b).\n\n\\boldsection{Duplicate detection.} A procedure for identifying duplicate entries is also employed.\nBy database construction, near-exact duplicates of elemental phases exist in {\\small LIB2},\nwhich is created spanning the full range of compositions for each alloy system (including elemental phases).\nThese degenerate entries are detected and removed by comparing composition, prototype,\nand formation enthalpy.\nOther structures may have been created distinctly, but converge to duplicates\nvia structural relaxation.\nThese equivalent structures are detected via {\\small AFLOW-XTAL-MATCH}\\\n({\\small AFLOW}\\ crys\\underline{tal} \\underline{match})~\\cite{aflow_compare_2018},\nwhich determines structural\/material uniqueness via the Burzlaff criteria~\\cite{Burzlaff_ActaCrystA_1997}.\nTo compare two crystals, a commensurate representation between structures is resolved by\n\\textbf{i.} identifying common unit cells,\n\\textbf{ii.} exploring cell orientations and origin choices,\nand \\textbf{iii.} matching atomic positions.\nFor each description, the structural similarity is measured by\na composite misfit quantity based on the lattice deviations and mismatch of the mapped atomic positions,\nwith a match occurring for sufficiently small misfit values ($<0.1$).\nDepending on the size of the structures, the procedure can be quite expensive,\nand only applied to find duplicate stable structures.\nCandidates are first screened by composition, space group, and\nformation enthalpies (must be within 15~meV\/atom of the relevant stable configuration).\n{By identifying duplicate stable phases, {\\small \\AFLOWHULLtitle}\\ can\ncross-reference the {\\sf \\AFLOW.org}\\ {\\small ICSD}\\ (\\underline{I}norganic \\underline{C}rystal \\underline{S}tructure \\underline{D}atabase)\ncatalog~\\cite{ICSD,ICSD3}{} to reveal whether the structure has already been observed.}\nThe analysis is depicted in Figure~\\ref{fig:art146:hull_analyses}(c), where\nthe Te-Zr convex hull is plotted with the \\verb|compound| labels replaced with the\ncorresponding {\\small ICSD}\\ number designation.\n\n\\boldsection{Thermodynamic descriptors.}\nA wealth of properties can be extracted from the convex hull construction beyond\na simple determination of stable\/unstable phases.\nFor unstable structures, the energetic vertical-distance to the hull $\\Delta H_{\\mathrm{f}}$,\ndepicted in Figure~\\ref{fig:art146:hull_analyses}(d), serves as a useful metric for quasi-stability.\n$\\Delta H_{\\mathrm{f}}$ is the magnitude of the energy driving the decomposition reaction.\nWithout the temperature and pressure contributions to the energy,\nnear-stable structures should also be considered \\text{(meta-)stable} candidates,\n\\nobreak\\mbox{\\it e.g.}, those within $k_{\\mathrm{B}}T=25$~meV (room temperature) of the hull.\nHighly disordered systems can be realized with even larger distances~\\cite{Sato_Science_2006,curtarolo:art113}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig037}\n\\mycaption[Distance to the hull algorithm.]\n{(\\textbf{a}) The correct distance (shown in \\textcolor{pranab_green}{{\\bf green}}) for $d_1$ is the minimum distance of structure $S_1$\nto all hyperplanes defining the convex hull.\nIn case of structure $S_2$, the minimum distance is not $d_2$ (\\textcolor{pranab_green}{{\\bf green}}) line), an artifact of the hyperplane\ndescription for hull facets.\n(\\textbf{b}) Projecting the points to the zero energy line guarantees that all points will lie within the hull,\nthus enabling the use of minimization algorithm to calculate the correct distance.\nThe distance to the hull $d$ is given as the difference of the projected distance $d_2$ from the distance to the zero energy line $d_1$.\nThe image is adapted from Figure A10 in Reference~\\cite{curtarolo:art113}.}\n\\label{fig:art146:hyperplane_confusion}\n\\end{figure}\n\nTo calculate $\\Delta H_{\\mathrm{f}}$ of phase $\\mathbf{p}$ (Equation~\\ref{eq:art146:point}),\n{\\small \\AFLOWHULLtitle}\\ first resolves the energy of the hull $H_{\\mathrm{hull}}$ at\nstoichiometric coordinates $\\mathbf{x}$, and then\nsubtracts it from\nthe phase's formation enthalpy $H_{\\mathrm{f}}$:\n\\begin{equation}\n\\Delta H_{\\mathrm{f}}[\\mathbf{p}]=\\left|H_{\\mathrm{f}}-H_{\\mathrm{hull}}[\\mathbf{x}]\\right|.\n\\label{eq:art146:dist2hull}\n\\end{equation}\nThe procedure is depicted in Figure~\\ref{fig:art146:hull_analyses}(d), which involves\nidentifying the facet (highlighted in \\textcolor{pranab_green}{{\\bf green}}) that encloses $\\mathbf{x}$ and thus defines\n$H_{\\mathrm{hull}}(\\mathbf{x})$.\nHere, the hyperplane description can be misleading (Equations~\\ref{eq:art146:plane_eq}~and~\\ref{eq:art146:hyperplane_normal}) as\nit lacks information about facet boundaries (Figure~\\ref{fig:art146:hyperplane_confusion}).\nThe enclosing facet is identified as that which\nminimizes the distance to the zero $H_{\\mathrm{f}}$ tie-line at $\\mathbf{x}$:\n\\begin{equation}\nH_{\\mathrm{hull}}[\\mathbf{x}]=-\\min_{\\mathrm{facets}\\in \\mathrm{hull}}\\left|n_N^{-1} \\left(D + \\sum_{i=1}^{N-1} n_i x_i\\right)\\right|.\n\\label{eq:art146:energy_hull}\n\\end{equation}\nVertical facets and those showing hyper-collinearity (having no content) are excluded from the calculation.\n\nWith the appropriate facet identified, the $l$ coefficients of the balanced decomposition reaction\nare derived to yield the full equation.\nThe decomposition of an $N$-ary phase into $l-1$ stable phases\ndefines an $\\left(l \\times N\\right)$-dimensional chemical composition matrix $\\mathbf{C}$,\nwhere $C_{j,i}$ is the atomic concentration of the $i$-species\nof the $j$-phase (the first of which is the unstable mixed phase).\nTake, for example, the decomposition of $\\mathrm{Pd}_{2}\\mathrm{Pt}_{3}$\nto $\\mathrm{PdPt}$ and $\\mathrm{PdPt}_{3}$ as presented in Figure~\\ref{fig:art146:hull_analyses}(d):\n\\begin{equation}\nN_{1}~\\mathrm{Pd}_{0.4}\\mathrm{Pt}_{0.6} \\to N_{2}~\\mathrm{Pd}_{0.5}\\mathrm{Pt}_{0.5} + N_{3}~\\mathrm{Pd}_{0.25}\\mathrm{Pt}_{0.75},\n\\label{eq:art146:decomp_reaction}\n\\end{equation}\nwhere $N_{j}$ is the balanced chemical coefficient for the $j$-phase.\nIn this case, $\\mathbf{C}$ is defined as:\n\\begin{equation}\n\\begin{bmatrix}\nx_{\\mathrm{Pd}} \\in \\mathrm{Pd}_{2}\\mathrm{Pt}_{3} & x_{\\mathrm{Pt}} \\in \\mathrm{Pd}_{2}\\mathrm{Pt}_{3} \\\\\n-x_{\\mathrm{Pd}} \\in \\mathrm{PdPt} & -x_{\\mathrm{Pt}} \\in \\mathrm{PdPt} \\\\\n-x_{\\mathrm{Pd}} \\in \\mathrm{PdPt}_{3} & -x_{\\mathrm{Pt}} \\in \\mathrm{PdPt}_{3} \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.4 & 0.6 \\\\\n-0.5 & -0.5 \\\\\n-0.25 & -0.75 \\\\\n\\end{bmatrix},\n\\end{equation}\nwhere a negative sign differentiates the right hand side of the equation from the left.\nReference~\\onlinecite{Thorne_ARXIV_2011} shows that $N_{j}$ can be extracted from the null space of $\\mathbf{C}$.\n{\\small \\AFLOWHULLtitle}\\ accesses the null space via a full $\\mathbf{QR}$ decomposition of $\\mathbf{C}$, specifically employing a general\nHouseholder algorithm~\\cite{trefethen1997numerical}.\nThe last column of the $\\left(l \\times l\\right)$-dimensional $\\mathbf{Q}$ orthogonal matrix spans the null space $\\mathbf{N}$:\n\\begin{equation}\n \\mathbf{Q} =\n\\begin{bmatrix}\n | & | & 0.8111 \\\\\n \\mathbf{q}_{1} & \\mathbf{q}_{2} & 0.4867 \\\\\n | & | & 0.3244 \\\\\n\\end{bmatrix}.\n\\end{equation}\nBy normalizing $\\mathbf{N}$ such that the first element $N_{1}=1$, the approach yields $N_{2}=0.6$ and $N_{3}=0.4$,\nwhich indeed balances Equation~\\ref{eq:art146:decomp_reaction}.\nThese coefficients can be used to verify\nthe decomposition energy\nobserved in Figure~\\ref{fig:art146:hull_analyses}(d).\nThe formation enthalpies of Pd$_{2}$Pt$_{3}$, PdPt, and PdPt$_{3}$ are\n\\mbox{-286~meV\/(10~atoms)}, \\mbox{-72~meV\/(2~atoms)}, and \\mbox{-104~meV\/(4~atoms)}, respectively.\nThe decomposition energy is calculated as:\n\\begin{equation}\n0.6 H_{\\mathrm{f}}\\left[\\mathrm{PdPt}\\right] + 0.4 H_{\\mathrm{f}}\\left[\\mathrm{PdPt}_{3}\\right] - H_{\\mathrm{f}}\\left[\\mathrm{Pd}_{2}\\mathrm{Pt}_{3}\\right]\n= -3~\\mathrm{meV\/atom},\n\\end{equation}\n\nFor a given stable structure, {\\small \\AFLOWHULLtitle}\\ determines the phases with which it is in equilibrium.\nFor instance, PdPt is in two-phase equilibria with Pd$_{3}$Pt as well as\nwith PdPt$_{3}$ (Figure~\\ref{fig:art146:hull_analyses}(d)).\nPhase coexistence plays a key role in defining a descriptor for precipitate-hardened superalloys.\nCandidates are chosen if a relevant composition is in two-phase equilibrium with the host matrix,\nsuggesting that the formation of coherent precipitates in the matrix is feasible~\\cite{Kirklin_ActaMat_2016,curtarolo:art113}.\n\nAn analysis similar to that quantifying instability $\\left(\\Delta H_{\\mathrm{f}}\\right)$\ndetermines the robustness of stable structures.\nThe stability criterion $\\delta_{\\mathrm{sc}}$ is defined as the distance of a stable\nstructure to the pseudo-hull constructed without it\n(Figure~\\ref{fig:art146:hull_analyses}(e)).\nIts calculation is identical to that of $\\Delta H_{\\mathrm{f}}$ for the pseudo-hull (Equations~\\ref{eq:art146:dist2hull}~and~\\ref{eq:art146:energy_hull}).\nThis descriptor quantifies the effect of the structure on the minimum energy surface, as\nwell as the structure's susceptibility to destabilization by a new phase that has yet to be explored.\nAs with the decomposition analysis, $\\delta_{\\mathrm{sc}}$ also serves to anticipate\nthe effects of temperature and pressure on the minimum energy surface.\nThe descriptor played a pivotal role in screening Heusler structures for new magnetic systems~\\cite{curtarolo:art109}.\n$\\delta_{\\mathrm{sc}}$ calls for the recalculation of facets local to the structure\nand all relevant duplicates as well, thus employing the results of the structure comparison\nprotocol.\n\n{\\small \\AFLOWHULLtitle}\\ can also plot the entropic temperature envelopes characterizing nucleation\nin hyper-thermal synthesis methods for binary systems~\\cite{curtarolo:art98}.\nThe entropic temperature is the ratio of the formation enthalpy to the mixing entropy for an ideal solution ---\na simple quantification for the resilience against disorder~\\cite{monsterPGM}.\nThe ideal ``{\\it iso-max-latent-heat}'' lines\nshown in Figure~\\ref{fig:art146:hull_analyses}(f)\ntry to reproduce the phase's capability to absorb latent heat, which can\npromote its nucleation over more stable phases when starting from large\nQ reservoirs\/feedstock.\nThe descriptor successfully predicts the synthesis of SmB$_{6}$ over SmB$_{4}$\nwith hyper-thermal plasma co-sputtering~\\cite{monsterPGM,curtarolo:art98}.\n\n\\subsection{Results}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig038}\n\\mycaption[Excerpt from the Ag-Au-Cd thermodynamic analysis report.]\n{The document is generated by {\\small \\AFLOWHULLtitle}\\ and showcases\nentry-specific data from the {\\sf \\AFLOW.org}\\ database as well as calculated thermodynamic descriptors.\nStructures highlighted in \\textcolor{pranab_green}{{\\bf green}} are structurally equivalent stable structures,\nand those in \\textcolor{orange}{{\\bf orange}} are structurally similar (same relaxed space group).\nThe working document includes a variety of links,\nincluding hyperlinks to the entry page of each phase (see prototypes)\nand links to relevant parts of the report (see decomposition reaction and\n$N$-phase equilibria).}\n\\label{fig:art146:report}\n\\end{figure}\n\n\\boldsection{Analysis output.}\nFollowing the calculation of the convex hull and relevant thermodynamic descriptors,\n{\\small \\AFLOWHULLtitle}\\ generates a {\\small PDF}\\ file summarizing the results.\nIncluded in the {\\small PDF}\\ are \\textbf{i.} an illustration of the convex hull as shown in\nFigure~\\ref{fig:art146:hull_examples} (for binary and ternary systems)~\\cite{pgfplots_manual} and\n\\textbf{ii.} a report with the aforementioned calculated\nthermodynamic descriptors --- an excerpt is shown in Figure~\\ref{fig:art146:report}.\n\nIn the illustrations, color is used to differentiate points with different enthalpies\nand indicate depth of the facets (3-dimensions).\nThe report includes entry-specific data from the {\\sf \\AFLOW.org}\\ database (prototype, {\\small AUID},\noriginal and relaxed space groups, spin, formation enthalpy $H_{\\mathrm{f}}$, and entropic temperature $T_{\\mathrm{S}}$)\nas well as calculated thermodynamic data (distance to the hull $\\Delta H_{\\mathrm{f}}$,\nthe balanced decomposition reaction for unstable phases, the\nstability criterion $\\delta_{\\mathrm{sc}}$ for stable phases, and\nphases in coexistence).\nStable phases (and those that are structurally equivalent) are highlighted in \\textcolor{pranab_green}{{\\bf green}},\nand similar phases (comparing relaxed space groups) are highlighted in \\textcolor{orange}{{\\bf orange}}.\nLinks are also incorporated in the report, including external\nhyperlinks to entry pages on {\\sf \\AFLOW.org}\\ (see prototypes) and internal\nlinks to relevant parts of the report (see decomposition reaction and $N$-phase equilibria).\nInternal links are also included on the convex hull illustration (see Supporting Information of Reference~\\cite{curtarolo:art146}).\nThe information is provided in the form of plain text and {\\small JSON}\\ files.\nKeys and format are explained in the Supporting Information.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.95\\linewidth]{fig039}\n\\mycaption[The convex hull web application powered by {\\small \\AFLOWHULLtitle}.]\n{(\\textbf{a}) An example 2-dimensional convex hull illustration (Mo-Ti).\n(\\textbf{b}) An example 3-dimensional convex hull illustration (Fe-Rh-Zr).\n(\\textbf{c}) The information component of the hull application.\nPertinent thermodynamic data for selected points is displayed within the grid of cards.\nEach card includes a link to the {\\sf \\AFLOW.org}\\ entry page and the option to remove a point.\nAs points are selected within the visualization, more cards will be added to the grid.\n(\\textbf{d}) The comparison component of the hull application.\nEach hull visualization is displayed as part of a grid of cards.\nFrom this page, new hulls can be added to the store by typing a query in the search box (sidebar).}\n\\label{fig:art146:hull_app}\n\\end{figure}\n\n\\boldsection{Web application.}\nA modern web application has been developed to provide an enhanced, command-line-free platform for {\\small \\AFLOWHULLtitle}.\nThe project includes a rich feature set consisting of binary and ternary convex\nhull visualizations, {\\sf \\AFLOW.org}\\ entry data retrieval, and a convex hull comparison interface.\nThe application is divided into four components: the periodic table, the visualization viewport,\nthe selected entries list, and the comparison page.\n\nThe periodic table component is initially displayed.\nHulls can be queried by selecting\/typing in the elemental combination.\nAs elements are added to the search, the periodic table reacts to the query depending on the\nreliability of the hull:\n\\textcolor{pranab_green}{{\\bf green}} (fully reliable, $N_{\\mathrm{entries}} \\geq 200$),\n\\textcolor{orange}{{\\bf orange}} (potentially reliable, $100 \\leq N_{\\mathrm{entries}} < 200$),\n\\textcolor{pranab_red}{{\\bf red}} (unreliable, $N_{\\mathrm{entries}} < 100$), and\n\\textcolor{gray}{\\bf gray} (unavailable, $N_{\\mathrm{entries}} =0$).\nEach new hull request triggers a fresh data download and analysis,\noffering the most up-to-date results given that new calculations are injected into the\n{\\sf \\AFLOW.org}\\ repository daily.\nOnce the analysis is performed and results are retrieved,\nthe application loads the visualization viewport\nprompting a redirect to the {\\small URL}\\ endpoint of the selected hull, \\nobreak\\mbox{\\it e.g.}, {\\sf \/hull\/AlHfNi}.\nThe {\\small URL}\\ is ubiquitous and can be shared\/cited.\n\nWhen a binary convex hull is selected, the viewport reveals a traditional\n2-dimensional plot (Figure~\\ref{fig:art146:hull_app}(a)),\nwhile a ternary hull yields a 3-dimensional visualization (Figure~\\ref{fig:art146:hull_app}(b)).\nThe scales of both are tunable, and the 3-dimensional visualization offers\nmouse-enabled pan and zoom.\n\nCommon to both types is the ability to select and highlight points.\nWhen a point is selected, its name will appear within the sidebar.\nThe information component is populated with a grid of cards containing properties of each\nselected point (entry), including a link to the {\\sf \\AFLOW.org}\\ entry page (Figure~\\ref{fig:art146:hull_app}(c)).\n\nThe application environment stores all previously selected hulls,\nwhich are retrievable via the hull comparison component (Figure~\\ref{fig:art146:hull_app}(d)).\nOn this page each hull visualization is displayed as a card on a grid.\nThis grid serves as both a history and a means to compare hulls.\n\n\\begin{table}[tp]\\centering\n\\mycaption[The 25 binary phases predicted to be most stable by {\\small \\AFLOWHULLtitle}.]\n{Phases with equivalent structures in the {\\small AFLOW}\\ {\\small ICSD}\\ catalog are excluded.\nThe list is sorted by the absolute value ratio between the stability criterion $\\left(\\delta_{\\mathrm{sc}}\\right)$\nand the formation enthalpy $\\left(H_{\\mathrm{f}}\\right)$ (shown as a percentage).\n${}^{\\dagger}$ indicates no binary phase diagram is available on the\n{\\small ASM}\\ Alloy Phase Diagram database~\\cite{ASMAlloyInternational}.\n{\\small POCC}\\ denotes a \\underline{p}artially-\\underline{occ}upied (disordered) structure~\\cite{curtarolo:art110}.\nComparisons with the {\\small ASM}\\ database include phases that are observed at high temperatures and pressures.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|l|r|r|r|R{4.75in}}\ncompound & {\\small AUID} & relaxed space group & $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$ & Figure & comparison with {\\small ASM}\\ Alloy Phase Diagrams~\\cite{ASMAlloyInternational} \\\\\n\\hline\n\\href{http:\/\/aflow.org\/material.php?id=aflow:38ecc639e4504b9d}{Hf$_{5}$Pb}$^{\\dagger}$ & \\texttt{aflow:38ecc639e4504b9d} & $P4\/mmm~\\#123$ & 78\\% & \\ref{fig:art146:HfPb_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:11ba11a3ee157f2e}{AgIn$_{3}$} & \\texttt{aflow:11ba11a3ee157f2e} & $P6_{3}\/mmc~\\#194$ & 54\\% & \\ref{fig:art146:AgIn_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:b60c1f9a1528ba5b}{AgIn$_{2}$} (space group $I4\/mcm$, $\\Delta H_{\\mathrm{f}}$ = 53 meV\/atom) and \\href{http:\/\/aflow.org\/material.php?id=aflow:d30bd203dd3b4049}{In} (space group $I4\/mmm$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1da75eb5f31b6dd5}{Hf$_{3}$In$_{4}$}$^{\\dagger}$ & \\texttt{aflow:1da75eb5f31b6dd5} & $P4\/mbm~\\#127$ & 45\\% & \\ref{fig:art146:HfIn_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:66dda41a34fe3ad6}{AsTc$_{2}$}$^{\\dagger}$ & \\texttt{aflow:66dda41a34fe3ad6} & $C2\/m~\\#12$ & 41\\% & \\ref{fig:art146:AsTc_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:57e1a1246f813f27}{MoPd$_{8}$} & \\texttt{aflow:57e1a1246f813f27} & $I4\/mmm~\\#139$ & 40\\% & \\ref{fig:art146:MoPd_binary_hull_supp} & composition not found, nearest are Mo$_{0.257}$Pd$_{0.743}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) and \\href{http:\/\/aflow.org\/material.php?id=aflow:53b1a8ec286d7fe5}{Pd} (space group $Fm\\overline{3}m$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:32051219452f8e0f}{Ga$_{4}$Tc}$^{\\dagger}$ & \\texttt{aflow:32051219452f8e0f} & $Im\\overline{3}m~\\#229$ & 39\\% & \\ref{fig:art146:GaTc_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:7bd140d7b4c65bc1}{Pd$_{8}$V} & \\texttt{aflow:7bd140d7b4c65bc1} & $I4\/mmm~\\#139$ & 36\\% & \\ref{fig:art146:PdV_binary_hull_supp} & composition not found, nearest are V$_{0.1}$Pd$_{0.9}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) and \\href{http:\/\/aflow.org\/material.php?id=aflow:4c0207df2fbbd51e}{VPd$_{3}$} (space group $I4\/mmm$, $\\Delta H_{\\mathrm{f}}$ = 5 meV\/atom) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:e7ed70c4711eb718}{InSr$_{3}$} & \\texttt{aflow:e7ed70c4711eb718} & $P4\/mmm~\\#123$ & 35\\% & \\ref{fig:art146:InSr_binary_hull_supp} & composition not found, nearest are Sr$_{28}$In$_{11}$ (space group $Imm2$) and \\href{http:\/\/aflow.org\/material.php?id=aflow:cb9aeb10d6379029}{Sr} (space group $Fm\\overline{3}m$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:f5cc5eaf65e692a9}{CoNb$_{2}$} & \\texttt{aflow:f5cc5eaf65e692a9} & $I4\/mcm~\\#140$ & 35\\% & \\ref{fig:art146:CoNb_binary_hull_supp} & composition not found, nearest are Nb$_{6.7}$Co$_{6.3}$ (space group $R\\overline{3}m$, {\\small POCC}\\ structure) and Nb$_{0.77}$Co$_{0.23}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:6ee057decaf093d0}{Ag$_{3}$In$_{2}$} & \\texttt{aflow:6ee057decaf093d0} & $Fdd2~\\#43$ & 34\\% & \\ref{fig:art146:AgIn_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:89453842555b9d95}{Ag$_{9}$In$_{4}$} (space group $P\\overline{4}3m$, $\\Delta H_{\\mathrm{f}}$ = 21 meV\/atom) and \\href{http:\/\/aflow.org\/material.php?id=aflow:b60c1f9a1528ba5b}{AgIn$_{2}$} (space group $I4\/mcm$, $\\Delta H_{\\mathrm{f}}$ = 53 meV\/atom) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:360240dae753fec6}{AgPt} & \\texttt{aflow:360240dae753fec6} & $P\\overline{6}m2~\\#187$ & 34\\% & \\ref{fig:art146:AgPt_binary_hull_supp} & polymorph found (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:bd3056780447faf0}{OsY$_{3}$} & \\texttt{aflow:bd3056780447faf0} & $Pnma~\\#62$ & 34\\% & \\ref{fig:art146:OsY_binary_hull_supp} & composition found, one-to-one match \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:96142e32718a5ee0}{RuZn$_{6}$} & \\texttt{aflow:96142e32718a5ee0} & $P4_{1}32~\\#213$ & 33\\% & \\ref{fig:art146:RuZn_binary_hull_supp} & composition found, one-to-one match \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1ba6b4b5c0ed9788}{Ag$_{2}$Zn} & \\texttt{aflow:1ba6b4b5c0ed9788} & $P\\overline{6}2m~\\#189$ & 33\\% & \\ref{fig:art146:AgZn_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:46dec61deb1ed379}{Ag} (space group $Fm\\overline{3}m$) and Ag$_{4.5}$Zn$_{4.5}$ (space group $P\\overline{3}$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:87d6637b32224f7b}{MnRh} & \\texttt{aflow:87d6637b32224f7b} & $Pm\\overline{3}m~\\#221$ & 32\\% & \\ref{fig:art146:MnRh_binary_hull_supp} & \\href{http:\/\/aflow.org\/material.php?id=aflow:19c39238f5d3feb5}{polymorph} found (space group $P4\/mmm$, $\\Delta H_{\\mathrm{f}}$ = 156 meV\/atom) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:f08f2f61de18aa61}{AgNa$_{2}$} & \\texttt{aflow:f08f2f61de18aa61} & $I4\/mcm~\\#140$ & 32\\% & \\ref{fig:art146:AgNa_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:a174f130a5b9b61f}{NaAg$_{2}$} (space group $Fd\\overline{3}m$, $\\Delta H_{\\mathrm{f}}$ = 208 meV\/atom) and \\href{http:\/\/aflow.org\/material.php?id=aflow:95da3ef7fcc58eea}{Na} (space group $R\\overline{3}m$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:7ce4fcc3660c16cf}{BeRe$_{2}$} & \\texttt{aflow:7ce4fcc3660c16cf} & $I4\/mcm~\\#140$ & 31\\% & \\ref{fig:art146:BeRe_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:2bb092148157834d}{Be$_{2}$Re} (space group $P6_{3}\/mmc$) and \\href{http:\/\/aflow.org\/material.php?id=aflow:47d6720be60b12f3}{Re} (space group $P6_{3}\/mmc$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:e94ab366799a008c}{As$_{2}$Tc}$^{\\dagger}$ & \\texttt{aflow:e94ab366799a008c} & $C2\/m~\\#12$ & 30\\% & \\ref{fig:art146:AsTc_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:eec0d7b6b0d1dfa0}{Be$_{2}$Mn}$^{\\dagger}$ & \\texttt{aflow:eec0d7b6b0d1dfa0} & $P6_{3}\/mmc~\\#194$ & 30\\% & \\ref{fig:art146:BeMn_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:6f3f5b696f5aa391}{AgAu} & \\texttt{aflow:6f3f5b696f5aa391} & $P4\/mmm~\\#123$ & 29\\% & \\ref{fig:art146:AgAu_binary_hull_supp} & polymorph found (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:ca051dbe25c55b92}{Nb$_{5}$Re$_{24}$} & \\texttt{aflow:ca051dbe25c55b92} & $I\\overline{4}3m~\\#217$ & 29\\% & \\ref{fig:art146:NbRe_binary_hull_supp} & composition not found, nearest are Nb$_{0.25}$Re$_{0.75}$ (space group $I\\overline{4}3m$, {\\small POCC}\\ structure) and Nb$_{0.01}$Re$_{0.99}$ (space group $P6_{3}\/mmc$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:a9daa69940d3a59a}{La$_{3}$Os}$^{\\dagger}$ & \\texttt{aflow:a9daa69940d3a59a} & $Pnma~\\#62$ & 28\\% & \\ref{fig:art146:LaOs_binary_hull_supp} & no diagram \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8ce84acfd6f9ea44}{Be$_{5}$Pt} & \\texttt{aflow:8ce84acfd6f9ea44} & $F\\overline{4}3m~\\#216$ & 28\\% & \\ref{fig:art146:BePt_binary_hull_supp} & composition found, one-to-one match \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:487f7cf6c3fb13f0}{Ir$_{8}$Ru} & \\texttt{aflow:487f7cf6c3fb13f0} & $I4\/mmm~\\#139$ & 27\\% & \\ref{fig:art146:IrRu_binary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:1513b1faeafa2d61}{Ir} (space group $Fm\\overline{3}m$) and Ru$_{0.3}$Ir$_{0.7}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:66af8171e22dc212}{InK} & \\texttt{aflow:66af8171e22dc212} & $R\\overline{3}m~\\#166$ & 27\\% & \\ref{fig:art146:InK_binary_hull_supp} & composition not found, nearest are K$_{8}$In$_{11}$ (space group $R\\overline{3}c$) and \\href{http:\/\/aflow.org\/material.php?id=aflow:a9c9107790b0344c}{K} (space group $Im\\overline{3}m$) \\\\\n\\end{tabular}}\n\\label{tab:art146:stable_binaries}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[The 25 ternary phases predicted to be most stable by {\\small \\AFLOWHULLtitle}.]\n{Phases with equivalent structures in the {\\small AFLOW}\\ {\\small ICSD}\\ catalog are excluded.\nThe list is sorted by the absolute value ratio between the stability criterion $\\left(\\delta_{\\mathrm{sc}}\\right)$\nand the formation enthalpy $\\left(H_{\\mathrm{f}}\\right)$ (shown as a percentage).\n${}^{\\dagger}$ indicates no ternary phase diagram is available on the\n{\\small ASM}\\ Alloy Phase Diagram database~\\cite{ASMAlloyInternational},\nwhile ${}^{\\ddagger}$ indicates all three relevant binaries are available.\n{\\small POCC}\\ denotes a \\underline{p}artially-\\underline{occ}upied (disordered) structure~\\cite{curtarolo:art110}.\nComparisons with the {\\small ASM}\\ database include phases that are observed at high temperatures and pressures.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|l|r|r|r|R{4.75in}}\ncompound & {\\small AUID} & relaxed space group & $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$ & Figure & comparison with {\\small ASM}\\ Alloy Phase Diagrams~\\cite{ASMAlloyInternational} \\\\\n\\hline\n\\href{http:\/\/aflow.org\/material.php?id=aflow:df0cdf0f1ad3110d}{MgSe$_{2}$Zn$_{2}$}$^{\\dagger}$ & \\texttt{aflow:df0cdf0f1ad3110d} & $Fmmm~\\#69$ & 58\\% & \\ref{fig:art146:MgSeZn_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Mg-Se) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8c51c7ab71f25d11}{Be$_{4}$OsTi}$^{\\dagger}$ & \\texttt{aflow:8c51c7ab71f25d11} & $F\\overline{4}3m~\\#216$ & 38\\% & \\ref{fig:art146:BeOsTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:4e5711451dc4b601}{Be$_{4}$OsV}$^{\\dagger}$ & \\texttt{aflow:4e5711451dc4b601} & $F\\overline{4}3m~\\#216$ & 38\\% & \\ref{fig:art146:BeOsV_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr} & \\texttt{aflow:1684c02e75b0d950} & $Fm\\overline{3}m~\\#225$ & 35\\% & \\ref{fig:art146:AgInZr_ternary_hull_supp} & composition not found, nearest are Ag$_{0.8}$In$_{0.2}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure), Zr$_{0.5}$In$_{0.5}$ (space group $Fm\\overline{3}m$, {\\small POCC}\\ structure), and AgZr$_{5}$In$_{3}$ (space group $P6_{3}\/mcm$) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:b85addbb42c47ae9}{Be$_{4}$RuTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:b85addbb42c47ae9} & $F\\overline{4}3m~\\#216$ & 32\\% & \\ref{fig:art146:BeRuTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:cabd6decf5b6c991}{Be$_{4}$FeTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:cabd6decf5b6c991} & $F\\overline{4}3m~\\#216$ & 29\\% & \\ref{fig:art146:BeFeTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:7010472778d429f7}{Be$_{4}$ReV}$^{\\dagger \\ddagger}$ & \\texttt{aflow:7010472778d429f7} & $F\\overline{4}3m~\\#216$ & 29\\% & \\ref{fig:art146:BeReV_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:e4cc9eea02d9d303}{Ba$_{2}$RhZn}$^{\\dagger}$ & \\texttt{aflow:e4cc9eea02d9d303} & $Cm~\\#8$ & 29\\% & \\ref{fig:art146:BaRhZn_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Ba-Rh) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:2ace5c5383f8ea10}{Be$_{4}$HfOs}$^{\\dagger}$ & \\texttt{aflow:2ace5c5383f8ea10} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeHfOs_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:de79192a0c4e751f}{Be$_{4}$ReTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:de79192a0c4e751f} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeReTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:d484b95ba623f9f7}{Be$_{4}$TcV}$^{\\dagger}$ & \\texttt{aflow:d484b95ba623f9f7} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeTcV_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Tc) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:c13660b990eb9570}{Be$_{4}$TcTi}$^{\\dagger}$ & \\texttt{aflow:c13660b990eb9570} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeTcTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Tc) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:07840d9e13694f7e}{Be$_{4}$RuV}$^{\\dagger \\ddagger}$ & \\texttt{aflow:07840d9e13694f7e} & $F\\overline{4}3m~\\#216$ & 27\\% & \\ref{fig:art146:BeRuV_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:5778f3b725d5f850}{AsCoTi}$^{\\dagger \\ddagger}$ & \\texttt{aflow:5778f3b725d5f850} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:AsCoTi_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:9a10dd8a8224e158}{Be$_{4}$MnTi}$^{\\dagger}$ & \\texttt{aflow:9a10dd8a8224e158} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:BeMnTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Mn) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:de412213bdefbd14}{Be$_{4}$OsZr}$^{\\dagger}$ & \\texttt{aflow:de412213bdefbd14} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:BeOsZr_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Os) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:07bcc161f57da109}{Be$_{4}$IrTi}$^{\\dagger}$ & \\texttt{aflow:07bcc161f57da109} & $F\\overline{4}3m~\\#216$ & 26\\% & \\ref{fig:art146:BeIrTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Ir) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:90b98cdcd6eea146}{Mg$_{2}$ScTl}$^{\\dagger}$ & \\texttt{aflow:90b98cdcd6eea146} & $P4\/mmm~\\#123$ & 25\\% & \\ref{fig:art146:MgScTl_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Sc-Tl) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:086b4a89f8d62804}{Be$_{4}$MnV}$^{\\dagger}$ & \\texttt{aflow:086b4a89f8d62804} & $F\\overline{4}3m~\\#216$ & 25\\% & \\ref{fig:art146:BeMnV_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Mn) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:0595e3d45678a85c}{AuBe$_{4}$Cu}$^{\\dagger \\ddagger}$ & \\texttt{aflow:0595e3d45678a85c} & $F\\overline{4}3m~\\#216$ & 25\\% & \\ref{fig:art146:AuBeCu_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:d7fed8d4996290f4}{BiRhZr}$^{\\dagger \\ddagger}$ & \\texttt{aflow:d7fed8d4996290f4} & $F\\overline{4}3m~\\#216$ & 24\\% & \\ref{fig:art146:BiRhZr_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:80bf8ad33a5bb33b}{LiMg$_{2}$Zn} & \\texttt{aflow:80bf8ad33a5bb33b} & $Fm\\overline{3}m~\\#225$ & 21\\% & \\ref{fig:art146:LiMgZn_ternary_hull_supp} & composition not found, nearest are \\href{http:\/\/aflow.org\/material.php?id=aflow:a66c0917c0faf13f}{Li} (space group $Im\\overline{3}m$, $\\Delta H_{\\mathrm{f}}$ = 2 meV\/atom), \\href{http:\/\/aflow.org\/material.php?id=aflow:b83b8ffef10abaa0}{Mg} (space group $P6_{3}\/mmc$), and Li$_{0.77}$MgZn$_{1.23}$ (space group $Fd\\overline{3}m$, {\\small POCC}\\ structure) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:faa814b1222e8aea}{Be$_{4}$RhTi}$^{\\dagger}$ & \\texttt{aflow:faa814b1222e8aea} & $F\\overline{4}3m~\\#216$ & 21\\% & \\ref{fig:art146:BeRhTi_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Be-Rh) \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:26cc4fc55644b0d8}{AuCu$_{4}$Hf}$^{\\dagger \\ddagger}$ & \\texttt{aflow:26cc4fc55644b0d8} & $F\\overline{4}3m~\\#216$ & 21\\% & \\ref{fig:art146:AuCuHf_ternary_hull_supp} & no diagram, all three binary phase diagrams found \\\\\n\\href{http:\/\/aflow.org\/material.php?id=aflow:ab57b1ae74f4c6d4}{Mg$_{2}$SeZn$_{2}$}$^{\\dagger}$ & \\texttt{aflow:ab57b1ae74f4c6d4} & $Fmmm~\\#69$ & 21\\% & \\ref{fig:art146:MgSeZn_ternary_hull_supp} & no diagram, two of three binary phase diagrams found (no Mg-Se) \\\\\n\\end{tabular}}\n\\label{tab:art146:stable_ternaries}\n\\end{table}\n\n\\boldsection{Candidates for synthesis.}\nTo demonstrate the capability of {\\small \\AFLOWHULLtitle}, all binary and ternary systems in the {\\sf \\AFLOW.org}\\ repository\nare explored for ones yielding well-converged thermodynamic properties.\nSince reliability constraints are built-in, {no pre-filtering is required and}\nall potential elemental combinations {are attempted.}\nAcross all catalogs present in the database, there exist materials composed of\n86 elements, including:\nH, He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl,\nAr, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge, As, Se, Br, Kr, Rb,\nSr, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, In, Sn, Sb, Te, I, Xe, Cs, Ba, La,\nCe, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Ta, W, Re, Os, Ir,\nPt, Au, Hg, Tl, Pb, Bi, Ac, Th, and Pa.\nHulls are eliminated if systems\n\\textbf{i.} are unreliable based on count (fewer than 200 entries among binary combinations), and\n\\textbf{ii.} show significant immiscibility (fewer than 50 points below the zero $H_{\\mathrm{f}}$ tie-line).\nTernary systems are further screened for those containing ternary ground-state structures.\nThe analysis resulted in the full thermodynamic characterization of 493\\ binary and 873\\ ternary systems.\nThe results are provided in the Supporting Information of Reference~\\cite{curtarolo:art146}.\n\nLeveraging the {\\small JSON}\\ outputs, reliable hulls are further explored for new stable phases.\nPhases are first screened (eliminated) if an equivalent structure exists in the {\\sf \\AFLOW.org}\\ {\\small ICSD}\\\ncatalog, and candidates are sorted by their relative stability criterion,\n\\nobreak\\mbox{\\it i.e.}, $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$.\nThis dimensionless quantity captures the effect of the phase on the minimum energy\nsurface relative to its depth, enabling comparisons across hulls.\n{An example Python script that performs this analysis is provided in the Supporting Information.}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig040}\n\\mycaption[Illustration of the most prevalent stable ternary structures.]\n{(\\textbf{a}) The conventional cubic cell of the ``quaternary-Heusler'' structure, LiMgPdSn~\\cite{Eberz_ZfNaturfB_35_1341_1980,anrl_pt2_2018}.\nEach species occupies a Wyckoff site of space group $F\\overline{4}3m~\\#216$:\nSn (purple) (4a),\nMg (yellow) (4b),\nPd (gray) (4c), and\nLi (blue) (4d).\n(\\textbf{b}) The conventional cubic cell of the Heusler structure, here represented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}.\nEach species occupies a Wyckoff site of space group $Fm\\overline{3}m~\\#225$:\nIn (pink) (4a), Zr (green) (4b), Ag (light gray) (8c).\n(\\textbf{c}) The conventional cubic cell of the half-Heusler $C1_{b}$ structure, here represented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:5778f3b725d5f850}{AsCoTi}.\nEach species occupies a Wyckoff site of space group $F\\overline{4}3m~\\#216$:\nTi (light blue) (4a), As (purple) (4b), Co (dark blue) (4c).\nThe (4d) site is empty.\n(\\textbf{d}) The conventional cubic cell of the $C15_{b}$-type crystal, here represented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8c51c7ab71f25d11}{Be$_{4}$OsTi}.\nEach species occupies a Wyckoff site of space group $F\\overline{4}3m~\\#216$:\nTi (light blue) (4a),\nOs (brown) (4c), and\nBe (light green) (8e).\nThe (4d) site is empty, and the Be atoms form a tetrahedron centered around the (4b) site of (\\textbf{a}).}\n\\label{fig:art146:heuslers}\n\\end{figure}\n\nThe top 25 most stable binary and ternary phases are presented in Tables~\\ref{tab:art146:stable_binaries}\nand \\ref{tab:art146:stable_ternaries}, respectively, for which extended analysis is performed\nbased on information stored in the {\\small ASM}\\ (\\underline{A}merican \\underline{S}ociety for \\underline{M}etals)\nAlloy Phase Diagram database~\\cite{ASMAlloyInternational}.\nThe {\\small ASM}\\ database is the largest of its kind, aggregating a wealth of experimental phase diagram information:\n40,300 binary and ternary alloy phase diagrams from over 9,000 systems.\nUpon searching the {\\small ASM}\\ website, many binary systems from Table~\\ref{tab:art146:stable_binaries}\nare unavailable and denoted by the symbol ${}^{\\dagger}$.\nAmong those that are available, some stable phases have already been observed,\nincluding\n\\href{http:\/\/aflow.org\/material.php?id=aflow:bd3056780447faf0}{OsY$_{3}$},\n\\href{http:\/\/aflow.org\/material.php?id=aflow:96142e32718a5ee0}{RuZn$_{6}$},\nand\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8ce84acfd6f9ea44}{Be$_{5}$Pt}.\nFor\n\\href{http:\/\/aflow.org\/material.php?id=aflow:360240dae753fec6}{AgPt},\n\\href{http:\/\/aflow.org\/material.php?id=aflow:87d6637b32224f7b}{MnRh},\nand\n\\href{http:\/\/aflow.org\/material.php?id=aflow:6f3f5b696f5aa391}{AgAu},\nthe composition is successfully predicted,\nbut polymorphs (structurally distinct phases) are observed instead.\nFor all other phases on the list, the composition has not been observed.\nThe discrepancy may be isolated to the phase, or indicative of a more extreme\ncontradiction in the topology of the hull, and thus, nearby phases are also analyzed.\nFor the Be-Re system, though \\href{http:\/\/aflow.org\/material.php?id=aflow:7ce4fcc3660c16cf}{BeRe$_{2}$}\nhas not been observed,\nboth \\href{http:\/\/aflow.org\/material.php?id=aflow:2bb092148157834d}{Be$_{2}$Re}\nand \\href{http:\/\/aflow.org\/material.php?id=aflow:47d6720be60b12f3}{Re}\nare successfully identified.\nMost of the remaining phases show the nearest phase to be a disordered (partially\noccupied) structure, which are excluded from the {\\sf \\AFLOW.org}\\ repository.\nAddressing disorder is a particularly challenging task in \\nobreak\\mbox{\\it ab-initio}\\ studies.\nHowever, recent high-throughput techniques~\\cite{curtarolo:art110} show promise for future investigations\nand will be integrated in future releases of the code.\n\nAmong the most stable ternary phases, only two systems have available\nphase diagrams in the {\\small ASM}\\ database, Ag-In-Zr and Li-Mg-Zn.\nFor the Ag-In-Zr system, the composition of\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}\nis not observed and the nearest stable phases include disordered structures and\nAgZr$_{5}$In$_{3}$, which has not yet been included the {\\sf \\AFLOW.org}\\ repository.\nFor Li-Mg-Zn, the composition of\n\\href{http:\/\/aflow.org\/material.php?id=aflow:80bf8ad33a5bb33b}{LiMg$_{2}$Zn}\nis also not observed and the nearest stable phases include unaries\n\\href{http:\/\/aflow.org\/material.php?id=aflow:a66c0917c0faf13f}{Li},\n\\href{http:\/\/aflow.org\/material.php?id=aflow:b83b8ffef10abaa0}{Mg},\nand a disordered structure.\nAll other ternary systems are entirely unexplored.\nTernary phases with all three binary phase diagrams available\nare denoted with the symbol ${}^{\\ddagger}$, suggesting experimental feasibility.\n\nA striking feature of Table~\\ref{tab:art146:stable_ternaries}\nis that most of the stable structures are\nfound to be in space group $F\\overline{4}3m~\\#216$.\nThis structure has a face-centered cubic lattice with symmetry operations that\ninclude a four-fold rotation about the ${<}001{>}$ axes, a three-fold rotation\nabout the ${<}111{>}$ axes, and no inversion.\nFurther study reveals that these phases, as well as $Fm\\overline{3}m~\\#225$\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}\nand\n\\href{http:\/\/aflow.org\/material.php?id=aflow:80bf8ad33a5bb33b}{LiMg$_{2}$Zn},\ncan be obtained from the ``quaternary-Heusler'' structure,\nLiMgPdSn~\\cite{Eberz_ZfNaturfB_35_1341_1980,anrl_pt2_2018} (Figure~\\ref{fig:art146:heuslers}(a)).\nThe prototype can be considered a $2\\times2\\times2$ supercell of the body-centered cubic structure.\nThe Sn, Mg, Au and Li atoms all occupy different Wyckoff positions of space group\n$F\\overline{4}3m$ and each atom has two sets of nearest neighbors, each four-fold coordinated.\nVarious decorations of these Wyckoff positions generate the other structures:\n\\begin{itemize}\n\\item By decorating two second-neighbor atom sites identically, a Heusler alloy forms\n({\\em Strukturbericht} symbol $L2_{1}$)~\\cite{Bradley_PRSL_A144_340_1934,aflowANRL}.\nFor example, the following substitutions generate\n\\href{http:\/\/aflow.org\/material.php?id=aflow:1684c02e75b0d950}{Ag$_{2}$InZr}\n(Figure~\\ref{fig:art146:heuslers}(b)):\nPd $\\rightarrow$ Ag, Li $\\rightarrow$ Ag,\nSn $\\rightarrow$ In, and Mg $\\rightarrow$ Zr.\nSince the crystal now has an inversion center, the space group becomes\n$Fm\\overline{3}m~\\#225$.\nAs in LiMgPdSn, each atom has two sets of four-fold coordinated nearest neighbors,\neach arranged as a tetrahedron.\nNow, however, one species (Ag) has second-neighbors of the same type.\n\\item By removing the Li atom completely, a half-Heusler forms\n($C1_{b}$)~\\cite{Nowotny_Z_f_Metallk_33_391_1941,aflowANRL}.\nThere are two half-Heusler systems in Table~\\ref{tab:art146:stable_ternaries}:\n\\href{http:\/\/aflow.org\/material.php?id=aflow:5778f3b725d5f850}{AsCoTi}\n(Figure~\\ref{fig:art146:heuslers}(c)) and\n\\href{http:\/\/aflow.org\/material.php?id=aflow:d7fed8d4996290f4}{BiRhZr}.\nThe structure does differ from that of LiMgPdSn and $L2_{1}$,\nas the Ag and Ti atoms are four-fold coordinated, with only Co having the\ncoordination seen in the previous structures.\n\\item The majority of structures in Table~\\ref{tab:art146:stable_ternaries}\nare type $C15_{b}$, prototype AuBe$_{5}$~\\cite{Batchelder_Acta_Crist_11_122_1958,aflowANRL}\n({\\small AFLOW}\\ prototype: \\verb|AB5_cF24_216_a_ce|~\\cite{AB5_cF24_216_a_ce}),\nrepresented by\n\\href{http:\/\/aflow.org\/material.php?id=aflow:8c51c7ab71f25d11}{Be$_{4}$OsTi}\nshown in Figure~\\ref{fig:art146:heuslers}(d).\nCompared to the $C1_{b}$, $C15_{b}$ contains an (8e) Wyckoff position\nforming a tetrahedra centered around the (4b) Wyckoff position.\nReplacing the tetrahedra with a single atom returns the $C1_{b}$ structure.\n\\end{itemize}\nHence, of the 25 most stable ternary structures, 21 are of related structure.\n\nSampling bias likely plays a role in the high prominence of space group $F\\overline{4}3m~\\#216$\nstructures in Table~\\ref{tab:art146:stable_ternaries}, but cannot fully account for the anomaly.\nSpace group $F\\overline{4}3m~\\#216$ constitutes about 17\\% of the {\\small LIB3}\\ catalog,\ncontaining the bulk of the {\\sf \\AFLOW.org}\\ repository (at over 1.5 million ternary systems)\ngenerated largely by small structure prototypes.\nFor context, space group $F\\overline{4}3m~\\#216$ is ranked about twentieth of the\nmost common space groups in the {\\small ICSD}~\\cite{Urusov_JSC_2009},\nappearing in about 1\\% of all entries.\nFurther exploration of larger structure ternary prototypes covering the full range\nof space groups is needed to fully elucidate the nature of this structure's stability.\n\nThe \\mbox{regular-}, inverse-, and half-Heusler prototypes were added to {\\small LIB3}\\\nfor the exploration of new magnets, of which two were discovered~\\cite{curtarolo:art109}.\nThe Heusler set includes more 236,000 structures, most of which remains unexplored.\nThe fully sorted lists of stable binary and ternary phases are presented in the\nSupporting Information of Reference~\\cite{curtarolo:art146}.\n\n\\clearpage\n\n\\subsection{Convex hulls of most stable candidates}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig041}\n\\mycaption{Hf-Pb binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:HfPb_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig042}\n\\mycaption{Ag-In binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgIn_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig043}\n\\mycaption{Hf-In binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:HfIn_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig044}\n\\mycaption{As-Tc binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AsTc_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig045}\n\\mycaption{Mo-Pd binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MoPd_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig046}\n\\mycaption{Ga-Tc binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:GaTc_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig047}\n\\mycaption{Pd-V binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:PdV_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig048}\n\\mycaption{In-Sr binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:InSr_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig049}\n\\mycaption{Co-Nb binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:CoNb_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig050}\n\\mycaption{Ag-Pt binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgPt_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig051}\n\\mycaption{Os-Y binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:OsY_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig052}\n\\mycaption{Ru-Zn binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:RuZn_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig053}\n\\mycaption{Ag-Zn binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgZn_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig054}\n\\mycaption{Mn-Rh binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MnRh_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig055}\n\\mycaption{Ag-Na binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgNa_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig056}\n\\mycaption{Be-Re binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRe_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig057}\n\\mycaption{Be-Mn binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeMn_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig058}\n\\mycaption{Ag-Au binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgAu_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig059}\n\\mycaption{Nb-Re binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:NbRe_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig060}\n\\mycaption{La-Os binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:LaOs_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig061}\n\\mycaption{Be-Pt binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BePt_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig062}\n\\mycaption{Ir-Ru binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:IrRu_binary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig063}\n\\mycaption{In-K binary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:InK_binary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig064}\n\\mycaption{Mg-Se-Zn ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MgSeZn_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig065}\n\\mycaption{Be-Os-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeOsTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig066}\n\\mycaption{Be-Os-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeOsV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig067}\n\\mycaption{Ag-In-Zr ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AgInZr_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig068}\n\\mycaption{Be-Ru-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRuTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig069}\n\\mycaption{Be-Fe-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeFeTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig070}\n\\mycaption{Be-Re-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeReV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig071}\n\\mycaption{Ba-Rh-Zn ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BaRhZn_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig072}\n\\mycaption{Be-Hf-Os ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeHfOs_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig073}\n\\mycaption{Be-Re-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeReTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig074}\n\\mycaption{Be-Tc-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeTcV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig075}\n\\mycaption{Be-Tc-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeTcTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig076}\n\\mycaption{Be-Ru-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRuV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig077}\n\\mycaption{As-Co-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AsCoTi_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig078}\n\\mycaption{Be-Mn-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeMnTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig079}\n\\mycaption{Be-Os-Zr ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeOsZr_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig080}\n\\mycaption{Be-Ir-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeIrTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig081}\n\\mycaption{Mg-Sc-Tl ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:MgScTl_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig082}\n\\mycaption{Be-Mn-V ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeMnV_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig083}\n\\mycaption{Au-Be-Cu ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AuBeCu_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig084}\n\\mycaption{Bi-Rh-Zr ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BiRhZr_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig085}\n\\mycaption{Li-Mg-Zn ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:LiMgZn_ternary_hull_supp}\n\\end{figure}\n\n\\vspace{\\fill}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig086}\n\\mycaption{Be-Rh-Ti ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:BeRhTi_ternary_hull_supp}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.75\\linewidth]{fig087}\n\\mycaption{Au-Cu-Hf ternary convex hull as plotted by {\\small \\AFLOWHULLtitle}.}\n\\label{fig:art146:AuCuHf_ternary_hull_supp}\n\\end{figure}\n\n\\def\\item {{\\it Description:}\\ }{\\item {{\\it Description:}\\ }}\n\\def\\item {{\\it Type:}\\ }{\\item {{\\it Type:}\\ }}\n\\def\\item {{\\it Units:}\\ }{\\item {{\\it Units:}\\ }}\n\\def\\item {{\\it Similar to:}\\ }{\\item {{\\it Similar to:}\\ }}\n\\def{\\color{blue}{\\item {{\\it Description:}\\ }}}{{\\color{blue}{\\item {{\\it Description:}\\ }}}}\n\\def{\\color{blue}{\\item {{\\it Type:}\\ }}}{{\\color{blue}{\\item {{\\it Type:}\\ }}}}\n\\def{\\color{blue}{\\item {{\\it Similar to:}\\ }}}{{\\color{blue}{\\item {{\\it Similar to:}\\ }}}}\n\n\\subsection{AFLOW-CHULL manual}\n\n\\boldsection{Command-line options.}\n{\\small \\AFLOWHULLtitle}\\ is an integrated module of the {\\small AFLOW}\\ \\nobreak\\mbox{\\it ab-initio}\\ framework\nwhich runs on any {\\small UNIX}-like computer, including those running macOS.\nThe most up-to-date binary can be downloaded from {\\sf aflow.org\/src\/aflow}:\ncurrent version 3.1.207.\n{\\small \\AFLOWHULLtitle}\\ depends on the compiled binary executable and an internet connection,\nas all data is retrieved and analyzed \\textit{in-situ}.\nThe default output option also requires the \\LaTeX\\ package.\nThe results (graphics and {\\small PDF}\\ reports) presented herein are\ncompiled using pdf\\TeX, Version 3.14159265-2.6-1.40.18 (\\TeX\\ Live 2017).\n\n\\vspace{0.5cm}\n\n\\noindent Primary commands:\n\\begin{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY!}\n \\begin{itemize}\n \\item{Calculates and returns the convex hull for system In-Ni-Y.}\n \\end{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY! \\\\ \\verb!--distance_to_hull=aflow:375066afdfb5a93f!}\n \\begin{itemize}\n \\item{Calculates and returns the distance to the hull $\\Delta H_{\\mathrm{f}}$ for \\href{http:\/\/aflow.org\/material.php?id=aflow:375066afdfb5a93f}{InNiY$_{4}$}.}\n \\end{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY! \\\\ \\verb!--stability_criterion=aflow:60a36639191c0af8!}\n \\begin{itemize}\n \\item{Calculates and returns the stability criterion $\\delta_{\\mathrm{sc}}$ for \\href{http:\/\/aflow.org\/material.php?id=aflow:60a36639191c0af8}{InNi$_{4}$Y}.\n The structure and relevant duplicates (if any) are removed to create the pseudo-hull.}\n \\end{itemize}\n \\item{\\verb!aflow --chull --alloy=InNiY --hull_formation_enthalpy=0.25,0.25!}\n \\begin{itemize}\n \\item{Calculates and returns the formation enthalpy of the minimum energy surface at In$_{0.25}$Ni$_{0.25}$Y$_{0.5}$.\n The input composition is specified by implicit coordinates (refer to Equation~\\ref{eq:art146:point}), where the last coordinate\n offers an optional energetic shift.\n }\n \\end{itemize}\n \\item{\\verb!aflow --chull --usage!}\n \\begin{itemize}\n \\item{Prints full set of commands to the screen.}\n \\end{itemize}\n \\item{\\verb!aflow --readme=chull!}\n \\begin{itemize}\n \\item{Prints a verbose manual (commands and descriptions) to the screen.}\n \\end{itemize}\n\\end{itemize}\n\n\\vspace{0.5cm}\n\n\\noindent General options:\n\\begin{myitemize}\n \\item{\\verb!--output=pdf!}\n \\begin{myitemize}\n \\item{Selects the output format. Options include: \\verb|pdf|, \\verb|png|, \\verb|json|, \\verb|txt|, and \\verb|full|. For multiple output, provide a comma-separated value list. A file with the corresponding extension is created, \\nobreak\\mbox{\\it e.g.}, {\\sf aflow\\_InNiY\\_hull.pdf}.}\n \\end{myitemize}\n\\item{\\verb!--destination=$HOME\/!}\n \\begin{myitemize}\n \\item{Sets the output path to {\\sf \\${\\small HOME}}. All output will be redirected to this destination.}\n \\end{myitemize}\n\\item \\verb!--keep=log!\n \\begin{myitemize}\n \\item{Creates a log file with verbose output of the calculation, \\nobreak\\mbox{\\it e.g.}, {\\sf aflow\\_InNiY\\_hull.log}.}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\noindent Loading options:\n\\begin{myitemize}\n\\item \\verb!--load_library=icsd!\n \\begin{myitemize}\n \\item{Limits the catalogs from which entries are loaded. Options include: \\verb!icsd!, \\verb!lib1!, \\verb!lib2!, and \\verb!lib3!. For multiple catalogs, provide a comma-separated value list.}\n \\end{myitemize}\n\\item \\verb!--load_entries_entry_output!\n \\begin{myitemize}\n \\item{Prints verbose output of the entries loaded. This output is included in the log file by default.}\n \\end{myitemize}\n\\item \\verb!--neglect=aflow:60a36639191c0af8,aflow:3f24d2be765237f1!\n \\begin{myitemize}\n \\item{Excludes individual points from the convex hull calculation.}\n \\end{myitemize}\n\\item \\verb!--see_neglect!\n \\begin{myitemize}\n \\item{Prints verbose output of the entries neglected from the calculation, including ill-calculated entries, duplicates, outliers, and those requested via \\verb!--neglect!.}\n \\end{myitemize}\n\\item \\verb!--remove_extreme_points=-1000!\n \\begin{myitemize}\n \\item{Excludes all points with formation enthalpies below -1000 meV\/atom.}\n \\end{myitemize}\n\\item \\verb!--include_paw_gga!\n \\begin{myitemize}\n \\item{Includes all entries calculated with {\\small PAW}-{\\small GGA}\\ (in addition to those calculated with {\\small PAW}-{\\small PBE}).\n {\\small PAW}-{\\small GGA}\\ refers to the \\underline{G}eneralized \\underline{G}radient \\underline{A}pproximation functional~\\cite{PBE}\n with pseudopotentials calculated with the\n \\underline{p}rojector \\underline{a}ugmented \\underline{w}ave method~\\cite{PAW}.\n This flag is needed to generate Figure~\\ref{fig:art146:hull_analyses}(f).}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\noindent Analysis options:\n\\begin{myitemize}\n\\item \\verb!--skip_structure_comparison!\n \\begin{myitemize}\n \\item{Avoids determination of structures equivalent to stable phases (speed).}\n \\end{myitemize}\n\\item \\verb!--skip_stability_criterion_analysis!\n \\begin{myitemize}\n \\item{Avoids determination of the stability criterion of stable phases (speed).}\n \\end{myitemize}\n\\item \\verb!--include_skewed_hulls!\n \\begin{myitemize}\n \\item{Proceeds to calculate the hull in the event that it is determined ``skewed'', \\nobreak\\mbox{\\it i.e.},\n the stable elemental phase differs from the reference energy by more than 15~meV\/atom.\n This flag is needed to generate Figure~\\ref{fig:art146:hull_analyses}(f).}\n \\end{myitemize}\n\\item \\verb!--include_unreliable_hulls!\n \\begin{myitemize}\n \\item{Proceeds to calculate the hull in the event that it is determined unreliable (fewer than 200 entries along the binary hulls).}\n \\end{myitemize}\n\\item \\verb!--include_outliers!\n \\begin{myitemize}\n \\item{Includes outliers in the calculation.}\n \\end{myitemize}\n\\item \\verb!--force!\n \\begin{myitemize}\n \\item{Forces an output, ignoring all warnings.}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\noindent {\\small PDF}\/\\LaTeX\\ options:\n\\begin{myitemize}\n\\item \\verb!--image_only!\n \\begin{myitemize}\n \\item{Creates a {\\small PDF}\\ with the hull illustration only.}\n \\end{myitemize}\n\\item \\verb!--document_only!\n \\begin{myitemize}\n \\item{Creates a {\\small PDF}\\ with the thermodynamic report only. Default for dimensions $N>3$.}\n \\end{myitemize}\n\\item \\verb!--keep=tex!\n \\begin{myitemize}\n \\item{Saves the \\LaTeX\\ input file (deleted by default), allowing for customization of the resulting {\\small PDF}, \\nobreak\\mbox{\\it e.g.}, {\\sf aflow\\_InNiY\\_hull.tex}.}\n \\end{myitemize}\n\\item \\verb!--latex_interactive!\n \\begin{myitemize}\n \\item{Displays the \\LaTeX\\ compilation output and enables interaction with the program.}\n \\end{myitemize}\n\\item \\verb!--plot_iso_max_latent_heat!\n \\begin{myitemize}\n \\item{Plots the entropic temperature envelopes shown in Figure~\\ref{fig:art146:hull_analyses}(f). Limited to binary systems only.}\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{{{\\small AFLOW}}rc options.}\nThe {\\sf .aflow.rc} file is a new protocol for specifying {\\small AFLOW}\\ default options.\nThe file emulates the {\\sf .bashrc} script that runs when initializing an interactive environment in\nBash (\\underline{B}ourne \\underline{a}gain \\underline{sh}ell).\nA fresh {\\sf .aflow.rc} file is created in {\\sf \\${\\small HOME}} if one is not already\npresent.\n\n\\noindent Relevant {\\small \\AFLOWHULLtitle}\\ options include:\n\\begin{myitemize}\n\\item \\verb!DEFAULT_CHULL_ALLOWED_DFT_TYPES=\"PAW_PBE\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the allowed entries based on \\underline{d}ensity \\underline{f}unctional \\underline{t}heory ({\\small DFT}) calculation type (comma-separated value).\n Options include: \\verb|US|, \\verb|GGA|, \\verb|PAW_LDA|, \\verb|PAW_GGA|, \\verb|PAW_PBE|, \\verb|GW|, and \\verb|HSE06|~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_ALLOW_ALL_FORMATION_ENERGIES=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Allows all entries independent of {\\small DFT}\\ calculation type~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_COUNT_THRESHOLD_BINARIES=200!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the minimum number of entries for a reliable binary hull.\n \\item {{\\it Type:}\\ } \\verb|integer|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_PERFORM_OUTLIER_ANALYSIS=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Enables determination of outliers.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_OUTLIER_ANALYSIS_COUNT_THRESHOLD_BINARIES=50!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the minimum number of entries for a reliable outlier analysis.\n Only phases stable with respect to their end-members are considered for the outlier analysis (below the zero $H_{\\mathrm{f}}$ tie-line).\n \\item {{\\it Type:}\\ } \\verb|integer|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_OUTLIER_MULTIPLIER=3.25!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the bounds beyond the interquartile range for which points are considered outliers~\\cite{Miller_QJEPSA_1991}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_IGNORE_KNOWN_ILL_CONVERGED=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AFLOW}\\ maintains a list of (older) prototypes known to have converged poorly.\n These entries are likely outliers, \\nobreak\\mbox{\\it e.g.}, see prototype $549$ in Figure~\\ref{fig:art146:hull_analyses}(a).\n If this flag is on (\\verb|1|), then the entries are removed before the analysis.\n Turning this flag off (\\verb|0|) is not recommended.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_PLOT_UNARIES=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Incorporates the end-members in the convex hull illustration.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_PLOT_OFF_HULL=-1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Incorporates off-hull phases in the convex hull illustration, but excludes phases unstable with respect to their end-members (above the zero $H_{\\mathrm{f}}$ tie-line).\n Only three values are accepted: \\verb|-1| (default: true for 2-dimensional systems, false for 3-dimensional systems), \\verb|0| (false), \\verb|1| (true).\n \\item {{\\it Type:}\\ } \\verb|-1 (default), 0 (false), or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_PLOT_UNSTABLE=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Incorporates all unstable phases in the convex hull illustration.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_FILTER_SCHEME=\"energy-axis\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the exclusion scheme for the convex hull illustration.\n In contrast to \\verb!--neglect!, this scheme is limited to the illustration, and points are still included in the analysis\/report.\n The following strings are accepted: \\verb!energy-axis!, \\verb!distance!, and an empty string.\n \\verb!energy-axis! refers to a scheme that eliminates structures from the illustration based on their formation enthalpies.\n On the other hand, \\verb!distance! refers to a scheme that eliminates structures from the illustration based on their distances to the hull.\n An empty string signifies no exclusion scheme.\n The criteria (value) for elimination is defined by \\\\ \\verb!DEFAULT_CHULL_LATEX_FILTER_VALUE!.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_FILTER_VALUE=50!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the value beyond which points are excluded per the scheme defined with \\verb!DEFAULT_CHULL_LATEX_FILTER_SCHEME!.\n In this case, \\\\ {\\small \\AFLOWHULLtitle}\\ would filter points with formation enthalpies greater than 50 meV.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_COLOR_BAR=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines whether to show the color bar graphic (3-dimensional illustration only). Colors can still be incorporated without the color bar graphic.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_HEAT_MAP=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines whether to color facets with heat maps illustrating their depth (3-dimensional illustration only).\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_COLOR_GRADIENT=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines whether to incorporate a color scheme at all in the illustration.\n Turning this flag off will also turn off \\verb!DEFAULT_CHULL_LATEX_COLOR_BAR! and \\verb!DEFAULT_CHULL_LATEX_HEAT_MAP!.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_COLOR_MAP=\"\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the color map, options are presented in Reference~\\onlinecite{pgfplots_manual}.\n Default is \\\\ \\verb!rgb(0pt)=(0.035,0.270,0.809); rgb(63pt)=(1,0.644,0)!.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_LINKS=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the links scheme. True\/false, \\nobreak\\mbox{\\it i.e.}, \\verb|0|\/\\verb|1|, will toggle all links on\/off.\n \\verb|2| enables external hyperlinks only (no links to other sections of the {\\small PDF}).\n \\verb|3| enables internal links only (no links to external pages).\n \\item {{\\it Type:}\\ } \\verb|0 (false), 1 (true), 2 (external-only), or 3 (internal-only)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_LABEL_NAME=\"\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the labeling scheme for phases shown on the convex hull.\n By default, the \\verb|compound| label is shown, while the \\verb|prototype| label can also be specified.\n \\verb|icsd| shows the {\\small ICSD}\\ entry number designation\n (lowest for multiple equivalent ground-state structures reflecting \\verb|icsd_canonical_auid|) if appropriate,\n as shown in Figure~\\ref{fig:art146:hull_analyses}(c).\n Also acceptable: \\verb|both| (\\verb|compound| and \\verb|prototype|) and \\verb|none|.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_META_LABELS=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Enables verbose labels, including \\verb|compound|, \\verb|prototype|, $H_{\\mathrm{f}}$, $T_{\\mathrm{S}}$,\n and $\\Delta H_{\\mathrm{f}}$. Warning, significant overlap of labels should be expected.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_LABELS_OFF_HULL=0!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Enables labels for off-hull points.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_HELVETICA_FONT=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Switches the font scheme from Computer Modern (default) to Helvetica.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_FONT_SIZE=\"\"!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Defines the font size of the labels on the convex hull illustration. Warning,\n other settings may override this default. Options include: \\verb|tiny|, \\verb|scriptsize|,\n \\verb|footnotesize|, \\verb|small|, \\verb|normalsize|, \\verb|large| (default), \\verb|Large|,\n \\verb|LARGE|, \\verb|huge|, and \\verb|Huge|.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_ROTATE_LABELS=1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Toggles whether labels are rotated.\n \\item {{\\it Type:}\\ } \\verb|0 (false) or 1 (true)|\n \\end{myitemize}\n\\item \\verb!DEFAULT_CHULL_LATEX_BOLD_LABELS=-1!\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Toggles whether labels are bolded.\n Three values are accepted: \\verb|-1| (default: false unless phase is a ternary), \\verb|0| (false), \\verb|1| (true).\n \\item {{\\it Type:}\\ } \\verb|-1 (default), 0 (false), or 1 (true)|\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{Image generation.}\nInstructions for generating the images herein are provided below.\nMany of these images can be generated automatically with {\\small \\AFLOWHULLtitle}. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_examples}(a): run \\verb|aflow --chull --alloy=CoTi --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_examples}(b): run \\verb|aflow --chull --alloy=MnPdPt --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_workflow}: \\textbf{i.} the Pd-Pt hull was first generated by running \\\\ \\verb|aflow --chull --alloy=PdPt --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input file ({\\sf aflow\\_PdPt\\_hull.tex}) was modified by hand and compiled to get the various hull illustrations,\n\\textbf{iii.} the overall flowchart was constructed with Microsoft PowerPoint. \\\\\n\\noindent Figure~\\ref{fig:art146:dimensions}: \\textbf{i.} the Al-Ni, Al-Ti, and Ni-Ti binary hulls were first generated by running \\\\ \\verb|aflow --chull --alloy=AlNi,AlTi,NiTi --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input files ({\\sf aflow\\_AlNi\\_hull.tex}, {\\sf aflow\\_AlTi\\_hull.tex}, and {\\sf aflow\\_NiTi\\_hull.tex}) were modified by hand and compiled to get the binary hull images,\n\\textbf{iii.} a snapshot of the Al-Ni-Ti ternary hull was taken from the web application at {\\sf aflow.org\/aflow-chull},\n\\textbf{iv.} the overall illustration was constructed with Adobe Illustrator. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(a): set \\verb|DEFAULT_CHULL_IGNORE_KNOWN_ILL_CONVERGED=0| in the {\\sf .aflow.rc} and run \\verb|aflow --chull --alloy=AlCo --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(b): set \\verb|DEFAULT_CHULL_IGNORE_KNOWN_ILL_CONVERGED=1| in the {\\sf .aflow.rc} and run \\verb|aflow --chull --alloy=AlCo --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(c): set \\verb|DEFAULT_CHULL_LATEX_LABEL_NAME=``icsd''| in the {\\sf .aflow.rc} and run \\verb|aflow --chull --alloy=TeZr --image_only|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(d): \\textbf{i.} the Pd-Pt hull was first generated by running \\\\ \\verb|aflow --chull --alloy=PdPt --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input file ({\\sf aflow\\_PdPt\\_hull.tex}) was modified by hand and compiled to get the hull illustration.\n$\\Delta H_{\\mathrm{f}}[\\text{aflow:71bc1b15525ffa35}]$ can be calculated individually by running \\verb|aflow --chull --alloy=PdPt --distance_to_hull=aflow:71bc1b15525ffa35|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(e): \\textbf{i.} the Pd-Pt hull was first generated by running \\\\ \\verb|aflow --chull --alloy=PdPt --image_only --keep=tex|,\n\\textbf{ii.} the resulting \\LaTeX\\ input file ({\\sf aflow\\_PdPt\\_hull.tex}) was modified by hand and compiled to get the hull illustration.\n$\\delta_{\\mathrm{sc}}[\\text{aflow:f31b0e27897cd162}]$ can be calculated individually by running \\verb|aflow --chull --alloy=PdPt| \\\\ \\verb|--stability_criterion=aflow:f31b0e27897cd162|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_analyses}(f): run \\verb|aflow --chull --alloy=BSm --image_only| \\\\ \\verb|--plot_iso_max_latent_heat --include_paw_gga --include_skewed_hulls|. \\\\\n\\noindent Figure~\\ref{fig:art146:report}: run \\verb|aflow --chull --alloy=AgAuCd|. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(a): navigate to {\\sf aflow.org\/aflow-chull} and select the Mo-Ti hull. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(b): navigate to {\\sf aflow.org\/aflow-chull} and select the Fe-Rh-Zr hull. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(c): navigate to {\\sf aflow.org\/aflow-chull}, select the Au-Cu-Zr hull, click on several points in the 3-dimensional illustration to populate the ``Select Points'' table on the left side of the screen, then click on one of the points in the table. \\\\\n\\noindent Figure~\\ref{fig:art146:hull_app}(d): navigate to {\\sf aflow.org\/aflow-chull}, select the Au-Cu-Zr, Au-Cu, and AuZr hulls by clicking ``Periodic Table'' from the navigation bar on the top right corner of the screen between selections, and click ``Hull History'' from the navigation bar on the top right corner of the screen. \\\\\n\\noindent Figures~\\ref{fig:art146:heuslers}(a-d): the structures were visualized with the CrystalMaker X software.\n\n\\vspace{0.5cm}\n\n\\boldsection{Python environment.}\nA module has been created that employs {\\small \\AFLOWHULLtitle}\\\nwithin a Python environment.\nThe module and its description closely follow that of the {\\small AFLOW-SYM}\\ Python module~\\cite{curtarolo:art135}.\nIt connects to a local {\\small AFLOW}\\ installation and imports the {\\small \\AFLOWHULLtitle}\\ results into a\n\\verb|CHull| class.\nA \\verb|CHull| object is initialized with:\n\n\\begin{python}\nfrom aflow_hull import CHull\nfrom pprint import pprint\n\nchull = CHull(aflow_executable = '.\/aflow')\nalloy = 'AlCuZr'\noutput = chull.get_hull(alloy)\npprint(output)\n\\end{python}\n\n\\noindent By default, the \\verb|CHull| object searches for an {\\small AFLOW}\\ executable in\nthe {\\sf \\${\\small PATH}}.\nHowever, the location of an {\\small AFLOW}\\ executable can be specified as\nfollows:\n\n\\noindent \\verb|CHull(aflow_executable=$HOME\/bin\/aflow)|.\n\n\\noindent The \\verb|CHull| object contains built-in methods corresponding to the command line calls mentioned previously:\n\n\\begin{myitemize}\n\\item \\verb|get_hull(`InNiY', options = `--keep=log')|\n\\item \\verb|get_distance_to_hull(`InNiY', `aflow:375066afdfb5a93f',| \\\\ \\verb|options = `--keep=log')|\n\\item \\verb|get_stability_criterion(`InNiY', `aflow:60a36639191c0af8',| \\\\ \\verb|options = `--keep=log')|\n\\item \\verb|get_hull_energy(`InNiY', [0.25,0.25], options = `--keep=log')|\n\\end{myitemize}\nEach method requires an input alloy string and allows an additional parameters\/flags string to be passed via \\verb|options|.\n\\verb|get_distance_to_hull| and \\\\ \\verb|get_stability_criterion| require an additional string input for the\n{\\small AUID}, while \\verb|get_hull_energy| takes an array of doubles as its input\nfor the composition.\n\n\\vspace{0.5cm}\n\n\\boldsection{Python module.}\nThe module to run the aforementioned {\\small \\AFLOWHULLtitle}\\ commands\nis provided below.\nThis module can be modified to incorporate additional\/customized options.\n\\begin{python}\nimport json\nimport subprocess\nimport os\n\nclass CHull:\n\n def __init__(self, aflow_executable='aflow'):\n self.aflow_executable = aflow_executable\n\n def aflow_command(self, cmd):\n try:\n return subprocess.check_output(\n self.aflow_executable + cmd,\n shell=True\n )\n except subprocess.CalledProcessError:\n print('Error aflow executable not found at: ' + self.aflow_executable)\n\n def get_hull(self, alloy, options = None):\n command = ' --chull'\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\n def get_distance_to_hull(self, alloy, off_hull_point, options = None):\n command = ' --chull --distance_to_hull=' + off_hull_point\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\n def get_stability_criterion(self, alloy, hull_point, options = None):\n command = ' --chull --stability_criterion=' + hull_point\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\n def get_hull_energy(self, alloy, composition, options = None):\n command = ' --chull --hull_energy=' + ','.join([ str(comp) for comp in composition ])\n if options:\n command += ' ' + options\n\n output = ''\n output = self.aflow_command(\n command + ' --print=json --screen_only --alloy=' + alloy\n )\n res_json = json.loads(output)\n return res_json\n\\end{python}\n\n\\vspace{0.5cm}\n\n\\noindent{\\textbf{Stability analysis.}\nA Python script is provided below demonstrating\nhow to perform the stability analysis presented in the Results\nsection.\nThe script gathers the most stable binary compounds generated\nfrom 2-element combinations of \\texttt{elements}.\nCompounds are filtered for binary ground-state structures not in the {\\small ICSD}.\nOnly unique compositions are saved.\nThe script writes the results to the {\\small JSON}\\ file {\\sf most\\_stable\\_binaries.json}\nand prints them to screen.\nThe script can be adapted to incorporate the full set of elements and\nfor the calculation of ternary systems.\nConsidering the number of combinations, it is recommended that the script be adapted\nto generate the hulls in parallel.\n}\n\\begin{python}\nfrom aflow_hull import CHull\nimport json\nfrom pprint import pprint\n\nelements = ['Mn', 'Pd', 'Pt'] #extend as needed\nelements.sort()\n\nmost_stable_binaries = [] #final list\nsaved_points_rc = [] #easy way to avoid adding duplicate compositions\n\nchull = CHull(aflow_executable = '.\/aflow') #initialize hull object\nfor i in range(len(elements)): #generate binary alloy combinations\n for j in range(i + 1, len(elements)): #generate binary alloy combinations\n alloy = elements[i]+elements[j] #generate binary alloy combinations\n output = chull.get_hull(alloy) #get hull data\n points_data = output['points_data'] #grab points data\n for point in points_data:\n #filter for binary ground-state structures not in the {\\small ICSD}\\\n if point['ground_state'] and not point['icsd_ground_state'] and point['nspecies'] == 2:\n #easy way to avoid adding duplicate compositions\n if point['reduced_compound'] not in saved_points_rc:\n saved_points_rc.append(point['reduced_compound'])\n #save only what is necessary\n abridged_entry = {}\n abridged_entry['compound'] = point['compound']\n abridged_entry['prototype'] = point['prototype']\n abridged_entry['auid'] = point['auid']\n abridged_entry['aurl'] = point['aurl']\n abridged_entry['relative_stability_criterion'] = point['relative_stability_criterion']\n most_stable_binaries.append(abridged_entry)\n\nmost_stable_binaries = sorted(most_stable_binaries, key=lambda point: -point['relative_stability_criterion']) #sort in descending order\n\n#save data to JSON file\nwith open('most_stable_binaries.json', 'w') as fout:\n json.dump(most_stable_binaries, fout)\n\n#also print output to screen\npprint(most_stable_binaries)\n\\end{python}\n\n\\vspace{0.5cm}\n\n\\boldsection{Output list.}\nThis section details the output fields for the thermodynamic analysis.\nThe lists describe the keywords as they appear in the {\\small JSON}\\ format.\nSimilar keywords are used for the standard text output.\n\n\\boldsection{Points data} (\\verb|points_data|).\n\\begin{myitemize}\n\\item \\verb|auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small \\underline{A}FLOW} \\underline{u}nique \\underline{ID} ({\\small AUID})~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|aurl|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small \\underline{A}FLOW} \\underline{u}niform \\underline{r}esource \\underline{l}ocator ({\\small AURL})~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Compound name~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|enthalpy_formation_atom|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Formation enthalpy per atom $\\left(H_{\\mathrm{f}}\\right)$~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } meV\/atom\n \\end{myitemize}\n\\item \\verb|enthalpy_formation_atom_difference|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The energetic vertical-distance to the hull $\\left(\\Delta H_{\\mathrm{f}}\\right)$, \\nobreak\\mbox{\\it i.e.}, the magnitude of the energy driving the decomposition reaction.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } meV\/atom\n \\end{myitemize}\n\\item \\verb|entropic_temperature|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The ratio of the formation enthalpy and the ideal mixing entropy $\\left(T_{\\mathrm{S}}\\right)$~\\cite{monsterPGM}.\n This term defines the ideal ``{\\it iso-max-latent-heat}'' lines of the grand-canonical ensemble~\\cite{monsterPGM,curtarolo:art98}. Refer to Figure~\\ref{fig:art146:hull_analyses}(f).\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } Kelvin\n \\end{myitemize}\n\\item \\verb|equivalent_structures_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of structurally equivalent entries. This analysis is limited to stable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|ground_state|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True for stable phases, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|icsd_canonical_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of an equivalent {\\small ICSD}\\ entry. If there are multiple equivalent {\\small ICSD}\\ entries, the one with the lowest number designation is chosen (original usually). This analysis is limited to stable phases only.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|icsd_ground_state|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True for stable phases with an equivalent {\\small ICSD}\\ entry, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|nspecies|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The number of species in the system (\\nobreak\\mbox{\\it e.g.}, binary = 2 and ternary = 3).\n \\item {{\\it Type:}\\ } \\verb|integer|\n \\end{myitemize}\n\\item \\verb|phases_decomposition_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of the products of the decomposition reaction (stable phases). This analysis is limited to unstable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|phases_decomposition_coefficient|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Coefficients of the decomposition reaction normalized to reactant, \\nobreak\\mbox{\\it i.e.}, $\\textbf{N}$ from Equation~\\ref{eq:art146:decomp_reaction}. Hence, the first entry is always 1. This analysis is limited to unstable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of doubles|\n \\end{myitemize}\n\\item \\verb|phases_decomposition_compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } \\verb|compound| of the products of the decomposition reaction (stable phases). This analysis is limited to unstable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|phases_equilibrium_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of phases in coexistence. This analysis is limited stable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|phases_equilibrium_compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } \\verb|compound| of phases in coexistence. This analysis is limited stable phases only.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|prototype|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AFLOW}\\ prototype designation~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|relative_stability_criterion|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } A dimensionless quantity capturing the effect of the phase on the minimum energy surface relative to its depth, \\nobreak\\mbox{\\it i.e.}, $\\left|\\delta_{\\mathrm{sc}}\/H_{\\mathrm{f}}\\right|$.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\end{myitemize}\n\\item \\verb|space_group_orig|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The space group (symbol and number) of the structure pre-relaxation as determined by {\\small AFLOW-SYM}~\\cite{curtarolo:art135}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|space_group_relax|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The space group (symbol and number) of the structure post-relaxation as determined by {\\small AFLOW-SYM}~\\cite{curtarolo:art135}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\item \\verb|spin_atom|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The magnetization per atom for spin polarized calculations~\\cite{aflowAPI}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } $\\mu_{\\mathrm{B}}$\/atom.\n \\end{myitemize}\n\\item \\verb|stability_criterion|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } A metric for robustness of a stable phase $\\left(\\delta_{\\mathrm{sc}}\\right)$, \\nobreak\\mbox{\\it i.e.},\n the distance of a stable phase from the pseudo-hull constructed without it.\n This analysis is limited to stable phases only.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } meV\/atom\n \\end{myitemize}\n\\item \\verb|url_entry_page|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The {\\small URL}\\ to the entry page: \\\\ {\\sf http:\/\/aflow.org\/material.php?id=aflow:60a36639191c0af8}.\n \\item {{\\it Type:}\\ } \\verb|string|\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{Facets data} (\\verb|facets_data|).\n\\begin{myitemize}\n\\item \\verb|artificial|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True if the facet is artificial, \\nobreak\\mbox{\\it i.e.}, defined solely by artificial end-points, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|centroid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The centroid of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of doubles|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|content|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The content (hyper-volume) of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of doubles|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|hypercollinearity|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True if the facet has no content, \\nobreak\\mbox{\\it i.e.}, exhibits hyper-collinearity, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|normal|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The normal vector characterizing the facet, \\nobreak\\mbox{\\it i.e.}, $\\mathbf{n}$ in Equation~\\ref{eq:art146:plane_eq}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|offset|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } The offset characterizing the facet, \\nobreak\\mbox{\\it i.e.}, $D$ in Equation~\\ref{eq:art146:plane_eq}.\n \\item {{\\it Type:}\\ } \\verb|double|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\item \\verb|vertical|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } True if the facet is vertical along the energetic axis, and false otherwise.\n \\item {{\\it Type:}\\ } \\verb|boolean|\n \\end{myitemize}\n\\item \\verb|vertices_auid|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } {\\small AUID}\\ of the phases that define the vertices of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|vertices_compound|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } \\verb|compound| of the phases that define the vertices of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of strings|\n \\end{myitemize}\n\\item \\verb|vertices_position|\n \\begin{myitemize}\n \\item {{\\it Description:}\\ } Coordinates that define the vertices of the facet.\n \\item {{\\it Type:}\\ } \\verb|array of arrays of doubles|\n \\item {{\\it Units:}\\ } Stoichiometric-energetic coordinates as defined by Equation~\\ref{eq:art146:point}.\n \\end{myitemize}\n\\end{myitemize}\n\n\\vspace{0.5cm}\n\n\\boldsection{{\\small AFLOW}\\ forum.}\nUpdates about {\\small \\AFLOWHULLtitle}\\ are discussed in the {\\small AFLOW}\\ forum ({\\sf aflow.org\/forum}):\n``Thermodynamic analysis''.\n\n\\subsection{Conclusions}\nThermodynamic analysis is a critical step for any effective materials design workflow.\nBeing a collective characterization, thermodynamics requires comparisons between many configurations of the system.\nThe availability of large databases \\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux,nomad,APL_Mater_Jain2013,Saal_JOM_2013,cmr_repository}\nallows the construction of computationally-based phase diagrams.\n{\\small \\AFLOWHULLtitle}\\ presents a complete software infrastructure, including\nflexible protocols for data retrieval, analysis, and verification~\\cite{aflowPI,nomad}.\nThe module is exhaustively applied to the {\\sf \\AFLOW.org}\\ repository and identified\nseveral new candidate phases: 17\\ promising $C15_{b}$-type structures and two half-Heuslers.\nThe extension of {\\small \\AFLOWHULLtitle}\\ to repositories beyond {\\sf \\AFLOW.org}\\ {can}\nbe performed {by adapting} the open-source \\texttt{C++} code and\/or Python module.\nComputational platforms such as {\\small \\AFLOWHULLtitle}\\ are valuable tools for guiding synthesis, including high-throughput and\neven autonomous approaches~\\cite{Xiang06231995,Takeuchi:2003fe,koinuma_nmat_review2004,nmatHT}.\n\n\\clearpage\n\\section{Modeling Off-Stoichiometry Materials with a High Throughput \\nobreak\\mbox{\\it Ab-initio}\\ Approach}\n\\label{sec:art110}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art110}.\nAuthor contributions are as follows:\nStefano Curtarolo designed the study.\nKesong Yang and Corey Oses implemented the {\\small AFLOW-POCC}\\ framework and performed proof of concept studies.\nAll authors discussed the results and their implications and contributed to the paper.\n\n\\subsection{Introduction}\n\nCrystals are characterized by their regular, repeating structures.\nSuch a description allows us to reduce our focus from the macroscopic material to a microscopic subset of\nunique atoms and positions.\nA full depiction of material properties, including mechanical, electronic, and magnetic features,\nfollows from an analysis of the primitive lattice.\nFirst principles quantum mechanical calculations have been largely successful in reproducing ground state properties of\nperfectly ordered crystals~\\cite{DFT,Hohenberg_PR_1964,nmatHT}.\nHowever, such perfection does not exist in nature.\nInstead, crystals display a degree of randomness, or disorder, in their lattices.\nThere are several types of disorder; including topological, spin, substitutional, and vibrational~\\cite{Elliott_PoAM_1990}.\nThis work focuses on substitutional disorder, in which equivalent sites of a crystal are not uniquely or fully occupied.\nRather, each site is characterized by a statistical, or partial, occupation.\nSuch disorder is intrinsic in many technologically significant systems, including those used in fuel cells~\\cite{Xie_ACatB_2015},\nsolar cells~\\cite{Kurian_JPCC_2013}, high-temperature superconductors~\\cite{Bednorz_ZPBCM_1986,Maeno_Nature_1994}, low thermal conductivity\nthermoelectrics~\\cite{Winter_JACerS_2007}, imaging and communications devices~\\cite{Patra_JAP_2012}, as well as promising\nrare-earth free materials for use in sensors, actuators, energy-harvesters, and spintronic devices~\\cite{Wang_SR_2013}.\nHence, a comprehensive computational study of substitutionally disordered materials at the atomic scale is of paramount importance for\noptimizing key physical properties of materials in technological applications.\n\nUnfortunately, structural parameters with partial occupancy cannot be used directly in first principles\ncalculations --- a significant hindrance for computational studies of disordered systems.\nTherefore, additional efforts must be made to model disorder or aperiodic systems~\\cite{curtarolo:art25,Mihalkovic_PHM_2006,\nNordheim_1931_AP_VCA,Vanderbilt_2000_PRB_VCA,Soven_PhysRev_1967,Korringa1947392,\nKohn_1954_PhysRev,Stocks_PRL_1978,zunger_sqs,Shan_PRL_1999,Popescu_PRL_2010,Faulkner_PMS_1982}.\nA rigorous statistical treatment of substitutional disorder at the atomic scale requires utility of large ordered supercells\ncontaining a composition consistent with the compound's stoichiometry~\\cite{sod,Habgood_PCCP_2011,Haverkort_ArXiv_2011}.\nHowever, the computational cost of such large supercell calculations has traditionally inhibited their use.\nFortunately, the emergence of high-throughput (HT) computational techniques \\cite{nmatHT}\ncoupled with the exponential growth of computational power is\nnow allowing the study of disordered systems from first principles~\\cite{MGI}.\n\nHerein, we present an approach to perform such a treatment working within the HT computational framework\n{\\small AFLOW}~\\cite{curtarolo:art104,aflowPAPER}.\nWe highlight three novel and attractive features central to this method: complete implementation into an automatic high throughput framework (optimizing speed without\nmitigating accuracy), utility of a novel occupancy optimization algorithm, and use of the Universal Force Field method \\cite{Rappe_1992_JCAS_UFF}\nto reduce the number of {\\small DFT}\\ calculations needed per system.\nTo illustrate the effectiveness of the approach, {\\small AFLOW-POCC}\\ is applied to three disordered systems,\na zinc chalcogenide (ZnS$_{1-x}$Se$_x$), a wide-gap oxide semiconductor (Mg$_{x}$Zn$_{1-x}$O), and an iron alloy (Fe$_{1-x}$Cu$_{x}$).\nExperimental observations are successfully reproduced and new phenomena are predicted:\n\\begin{itemize}\n\\item ZnS$_{1-x}$Se$_x$ shows a small, yet smooth optical bowing over the complete compositional space.\nAdditionally, the stoichiometrically-evolving ensemble average DOS demonstrates that\nthis system is of the amalgamation type and not of the persistence type.\n\\item Mg$_{x}$Zn$_{1-x}$O exhibits an abrupt transition in optical bowing consistent with a phase transition\nover its compositional range.\n\\item The ferromagnetic behavior of Fe$_{1-x}$Cu$_{x}$ is predicted to be smoothly stifled as more\ncopper is introduced into the structure, even through a phase transition.\n\\end{itemize}\nOverall, these systems exhibit highly-tunable properties already exploited in many technologies.\nThrough the approach, these features are not only recovered, but additional insight into the underlying physical mechanisms is\nalso revealed.\n\n\\subsection{Methodology}\n\nThis section details the technicalities of representing a partially occupied disordered system as a series of unique supercells.\nHere is an outline of the approach:\n\\begin{enumerate}\n\\item For a given disordered material, optimize its partial occupancy values and determine the size of the derivative superlattice $n$.\n\\item\n\\begin{enumerate}\n\\item Use the superlattice size $n$ to generate a set of unique derivative superlattices and corresponding sets of\nunique supercells with the required stoichiometry.\n\\item Import these non-equivalent supercells into the automatic computational framework {\\small AFLOW}\\ for HT\nfirst principles electronic structure calculations.\n\\end{enumerate}\n\\item Obtain and use the relative formation enthalpy to calculate the equilibrium probability of each\nsupercell as a function of temperature $T$ according to the Boltzmann distribution.\n\\item Determine the disordered system's material properties through ensemble averages of those calculated for each supercell.\nSpecifically, the following properties are resolved: the density of states (DOS), band gap energy $E_{\\mathrm{gap}}$, and magnetic moment $M$.\n\\end{enumerate}\n\nIn the following sections, a model disordered system, Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$, is presented\nto illustrate the technical procedures mentioned above.\nThis disordered system has two partially occupied sites: one shared between silver and zirconium, and another shared between\ncadmium and a vacancy.\nWorking within the {\\small AFLOW}\\ framework~\\cite{aflowBZ}, a simple structure file has been designed for partially occupied systems.\nAdapted from {\\small VASP}{}'s {\\small POSCAR}~\\cite{vasp_cms1996,vasp_prb1996}, the {\\small PARTCAR}\\ contains within it a description of lattice parameters and\nsite coordinates\/occupants, along\nwith a concentration tolerance (explained in the next section), and (partial) occupancy values for each site.\nTo see more details about this structure or its {\\small PARTCAR}, see Section~\\ref{subsec:art110:PARCAR}.\n\n\\subsubsection{Determining superlattice size}\nIn order to fully account for the partial occupancy of the disordered system, the set of superlattices of\na size corresponding to the lowest common denominator of the fractional partial occupancy values should be generated.\nWith partial occupancy values of 0.733 (733\/1000) and 0.267 (267\/1000) in the disordered system Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$,\nsuperlattices of size 1000 would need to be constructed.\nNot only would this require working with correspondingly large supercells (16,000 atoms per supercell in this example),\nbut the number of unique supercells in the set would be substantial.\nThis extends well beyond the capabilities of first principles calculations, and thus, is not practical.\nIt is therefore necessary to optimize the partial occupancy values to produce an appropriate superlattice size.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Evolution of the algorithm used to optimize the partial occupancy values and superlattice size for the disordered system\nAg$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$.]\n{$f_i$ indicates the iteration's choice fraction for each partially occupied site, ($i$ = 1, 2, 3, \\ldots);\n$e_i$ indicates the error between the iteration's choice fraction and the actual partial occupancy value.\n$e_{\\mathrm{max}}$ is the maximum error of the system.}\n\\vspace{3mm}\n\\begin{tabular}{llrlrlrrr}\n\\multirow{2}{*}{$n^{\\prime}$} & \\multicolumn{2}{c}{occup. 1 (Ag)} & \\multicolumn{2}{c}{occup. 2 (Zr)} & \\multicolumn{2}{c}{occup. 3 (Cd)} & \\multirow{2}{*}{\\textit{$e_{\\mathrm{max}}$} } & \\multirow{2}{*}{$n$} \\\\\n\\cline{2-7}\n & \\textit{$f_{1}$} & \\textit{$e_{1}$} & \\textit{$f_{2}$} & \\textit{$e_{2}$} & \\textit{$f_{3}$} & \\textit{$e_{3}$} & & \\\\\n\\hline\n1\t& 1\/1 & 0.267 & 0\/1 & 0.267 & 1\/1 & 0.2 & 0.267 & 1 \\\\\n\\hline\n2\t& 1\/2 & 0.233 & 1\/2 & 0.233 & 2\/2 & 0.2 & 0.233 & 2 \\\\\n\\hline\n3\t& 2\/3 & 0.067 & 1\/3 & 0.067 & 2\/3 & 0.133 & 0.133 & 3 \\\\\n\\hline\n4\t& 3\/4 & 0.017 & 1\/4 & 0.017 & 3\/4 & 0.05 & 0.05 & 4 \\\\\n\\hline\n5\t& 4\/5 & 0.067 & 1\/5 & 0.067 & 4\/5 & 0 & 0.067 & 5 \\\\\n\\hline\n6\t& 4\/6 & 0.067 & 2\/6 & 0.067 & 5\/6 & 0.033 & 0.067 & 6 \\\\\n\\hline\n7\t& 5\/7 & 0.019 & 2\/7 & 0.019 & 6\/7 & 0.057 & 0.057 & 7 \\\\\n\\hline\n8\t& 6\/8 & 0.017 & 2\/8 & 0.017 & 6\/8 & 0.05 & 0.05 & 4 \\\\\n\\hline\n9\t& 7\/9 & 0.044 & 2\/9 & 0.044 & 7\/9 & 0.022 & 0.044 & 9 \\\\\n\\hline\n10\t& 7\/10 & 0.033 & 3\/10 & 0.033 & 8\/10 & 0 & 0.033 & 10 \\\\\n\\hline\n11\t& 8\/11 & 0.006 & 3\/11 & 0.006 & 9\/11 & 0.018 & 0.018 & 11 \\\\\n\\hline\n12\t& 9\/12 & 0.017 & 3\/12 & 0.017 & 10\/12 & 0.033 & 0.033 & 12 \\\\\n\\hline\n13\t& 10\/13 & 0.036 & 3\/13 & 0.036 & 10\/13 & 0.031 & 0.036 & 13 \\\\\n\\hline\n14\t& 10\/14 & 0.019 & 4\/14 & 0.019 & 11\/14 & 0.014 & 0.019 & 14 \\\\\n\\hline\n15\t& 11\/15 & 0.00003 & 4\/15 & 0.00003 & 12\/15 & 0 & 0.00003 & 15 \\\\\n\\end{tabular}\n\\label{tab:art110:pocc_algo}\n\\end{table}\n\nAn efficient algorithm is presented to calculate the optimized partial occupancy values and corresponding superlattice size\nwith the example disordered system Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$ in Table~\\ref{tab:art110:pocc_algo}.\nFor convenience, the algorithm's iteration step is referred as $n^{\\prime}$,\nthe superlattice index, and $n$ as the superlattice size.\nQuite simply, the algorithm iterates, increasing the superlattice index from 1 to $n^{\\prime}$ until the optimized partial occupancy values reach the required accuracy.\nAt each iteration, a fraction is generated for each partially occupied site, all of which have the common denominator $n^{\\prime}$.\nThe numerator is determined to be the integer that reduces the overall fraction's error relative to the actual site's fractional partial occupancy value.\nThe superlattice size corresponds to the lowest common denominator of the irreducible fractions (\\nobreak\\mbox{\\it e.g.}, see iteration step 8).\nThe maximum error among all of the sites is chosen to be the accuracy metric for the system.\n\nFor the disordered system Ag$_{8.733}$Cd$_{3.8}$Zr$_{3.267}$, given a tolerance of 0.01, the calculated superlattice size is 15\n(240 atoms per supercell).\nBy choosing a superlattice with a nearly equivalent stoichiometry as the disordered system, the supercell size has been\nreduced by over a factor of 60 and entered the realm of feasibility with this calculation.\nNotice that the errors in partial occupancy values calculated for\nsilver and zirconium are the same, as they share the same site.\nThe same holds true for cadmium and its vacant counterpart (not shown).\nTherefore, the algorithm only needs to determine one choice fraction per site, instead of per occupant (as shown).\nSuch an approach reduces computational costs by guaranteeing that only the smallest supercells (both in number and size)\nwith the lowest tolerable error in composition are funneled into the HT first principles calculation framework.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig088}\n\\mycaption[Structure enumeration for off-stoichiometric materials modeling.]\n{For the off-stoichiometric material ZnS$_{0.25}$Se$_{0.75}$, a superlattice of size $n=4$ accommodates the stoichiometry exactly.\nBy considering all possibilities of decorated supercells and eliminating duplicates by UFF energies, seven structures are identified as unique.\nThese representative structures are fully characterized by {\\small AFLOW}\\ and {\\small VASP}, and are ensemble-averaged to resolve the system-wide properties.}\n\\label{fig:art110:pocc}\n\\end{figure}\n\n\\subsubsection{Unique supercells generation}\nWith the optimal superlattice size $n$, the unique derivative superlattices of the disordered system can be generated using\nHermite Normal Form (HNF) matrices~\\cite{enum1} as depicted in Figure~\\ref{fig:art110:pocc}.\nEach HNF matrix generates a superlattice of a size corresponding to its determinant, $n$.\nThere exists many HNF matrices with the same determinant, each creating a variant superlattice.\nFor each unique superlattice, a complete set of possible supercells is generated with the required stoichiometry by exploring all\npossible occupations of partially occupied sites.\nHowever, not all of these combinations are unique --- nominally warranting an involved structure comparison analysis that becomes\nextremely time consuming for large supercells~\\cite{enum1}.\nInstead, duplicates are identified by estimating the total energy of each supercell in a HT manner based on the Universal Force Field (UFF)\nmethod~\\cite{Rappe_1992_JCAS_UFF}.\nThis classical molecular mechanics force field approximates the energy of a structure by considering its composition,\nconnectivity, and geometry, for which parameters have been tabulated.\nOnly supercells with the same total energy are structurally compared and potentially treated as duplicate structures to be discarded, if necessary.\nThe count of duplicate structures determines the degeneracy of the structure.\nOnly non-equivalent supercells are imported into the automatic computational framework {\\small AFLOW}\\ for HT\nquantum mechanics.\n\n\\subsubsection{Supercell equilibrium probability calculation}\nThe unique supercells representing a partially occupied disordered material are labeled as {$S_1$, $S_2$, $S_3$, \\ldots,\n$S_n$}.\nTheir formation enthalpies (per atom) are labeled as {$H_{\\mathrm{f},1}$, $H_{\\mathrm{f},2}$, $H_{\\mathrm{f},3}$, \\ldots, $H_{\\mathrm{f},n}$}, respectively.\nThe formation enthalpy of each supercell is automatically calculated from HT first principles calculations using the {\\small AFLOW}\\\nframework~\\cite{curtarolo:art104,aflowPAPER}.\nThe supercell with the lowest formation enthalpy is selected as a reference (ground state structure), and its formation enthalpy is denoted as $H_{\\mathrm{f},0}$.\nThe relative formation enthalpy of the \\emph{i}th supercell is calculated as $\\Delta {H_{\\mathrm{f},i}} = {H_{\\mathrm{f},i}} - {H_{\\mathrm{f},0}}$\nand characterizes its disorder relative to the ground state.\nThe probability $P_i$ of the \\emph{i}th supercell is determined by the Boltzmann factor:\n\\begin{equation}\n{P_i} = \\frac{{{g_ie^{ - \\Delta {H_{\\mathrm{f},i}}\/{k_{\\mathrm{B}}}T}}}}{{\\sum\\limits_{i = 1}^n {{g_ie^{ - \\Delta {H_{\\mathrm{f},i}}\/{k_{\\mathrm{B}}}T}}} }},\n\\end{equation}\nwhere $g_i$ is the degeneracy of the \\emph{i}th supercell,\n$\\Delta {H_{\\mathrm{f},i}}$ is the relative formation enthalpy of the \\emph{i}th supercell,\n$k_{\\mathrm{B}}$ is the Boltzmann constant,\nand $T$ is a virtual ``roughness'' temperature.\n$T$ is not a true temperature \\textit{per se}, but instead a parameter describing how much disorder has been\nstatistically explored during synthesis.\nTo elaborate further, consider two extremes in the ensemble average (ignoring structural degeneracy):\n\\begin{enumerate}\n\\item $k_{\\mathrm{B}} T \\lesssim \\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)$\nneglecting highly disordered structures $(\\Delta {H_{\\mathrm{f},i}} \\ggg 0)$\nas $T\\to 0$, and\n\\item $k_{\\mathrm{B}} T\\ggg\\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)$\nrepresenting the annealed limit ($T\\to \\infty$) in which all structures are equiprobable.\n\\end{enumerate}\nThe probability $P_i$ describes the weight of the \\emph{i}th supercell among the thermodynamically equivalent states of the disordered material\nat equilibrium.\n\n\\subsubsection{Ensemble average density of states, band gap energy, and magnetic moment}\nWith the calculated material properties of each supercell and its equilibrium probability in hand, the overall system properties\ncan be determined by ensemble averages of those calculated for each supercell.\nThis work focuses on the calculation of the ensemble average density of states (DOS), band gap energy $E_{\\mathrm{gap}}$, and magnetic moment $M$.\nThe DOS of the \\emph{i}th supercell is labeled as $N_i(E)$ and indicates the number of electronic states per energy interval.\nThe ensemble average DOS of the system is then determined by the following formula:\n\\begin{equation}\nN(E) = \\sum\\limits_{i = 1}^n {{P_i} \\times {N_i}(E)}.\n\\end{equation}\nAdditionally, a band gap $E_{\\mathrm{gap},i}$ can be extracted from the DOS of each supercell.\nIn this fashion, an ensemble average band gap $E_{\\mathrm{gap}}$ can be calculated for the system.\nIt is important to note that standard density functional theory ({\\small DFT}) calculations are limited to a description of the ground\nstate~\\cite{DFT,Hohenberg_PR_1964,nmatHT}.\nAs such, calculated excited state properties may contain substantial errors.\nIn particular, {\\small DFT}\\ tends to underestimate the band gap~\\cite{Perdew_IJQC_1985}.\nDespite these known hindrances in the theory, the framework is capable of predicting significant trends\nspecific to the disordered systems.\nAs a bonus, the calculation of these results are performed in a high-throughput fashion.\nIt is expected that a more accurate, fine-grained description of the electronic structure in such systems will be obtained through a combination of\nthis software framework and more advanced first principles approaches~\\cite{GW,Hedin_GW_1965,Heyd2003,Liechtenstein1995,curtarolo:art86,curtarolo:art93,curtarolo:art103}.\n\nIn the same spirit as the $N(E)$ and $E_{\\mathrm{gap}}$, {\\small AFLOW-POCC}\\ calculates the ensemble average magnetic moment $M$ of the system.\nThe magnetic moment of the \\emph{i}th supercell is labeled as $M_i$.\nIf the ground state of the \\emph{i}th structure is non-spin-polarized, then its magnetic moment is set to zero, \\nobreak\\mbox{\\it i.e.}, $M_i=0$.\nTaking into account the impact of signed spins on the ensemble average, this approach is limited only to ferromagnetic solutions.\nAdditionally, as an initialization for the self-consistent run, the same ferromagnetic alignment is assumed among all of the spins in the system\n(an {\\small AFLOW}\\ calculation standard)~\\cite{curtarolo:art104}.\nFinally, the ensemble average magnetic moment of the system is calculated with the following formula:\n\\begin{equation}\nM = \\sum\\limits_{i = 1}^n {{P_i} \\times } |{M_i}|.\n\\end{equation}\n\n\\subsection{Example applications}\nThree disordered systems of technological importance are analyzed using {\\small AFLOW-POCC}:\na zinc chalcogenide, a wide-gap oxide semiconductor, and an iron alloy.\nUnless otherwise stated, the supercells used in these calculations were generated with the lowest superlattice size $n_{\\mathrm{xct}}$ needed to represent\nthe composition exactly.\n\n\\subsubsection{Zinc chalcogenides}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig089}\n\\mycaption[Disordered ZnS$_{1-x}$Se$_x$.]\n{(\\textbf{a}) A comparison of the experimental \\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985} \\nobreak\\mbox{\\it vs.}\\\ncalculated compositional dependence of the band gap energy $E_{\\mathrm{gap}}$ at room temperature.\nA rigid shift in the $E_{\\mathrm{gap}}$ axis relative to the experimental results of ZnSe (second ordinate axis) accounts for the expected systematic\ndeviation in {\\small DFT}\\ calculations~\\cite{Perdew_IJQC_1985}.\nOnly the lowest empirical $E_{\\mathrm{gap}}$ trends are shown.\nError bars indicate the weighted standard deviation of the ensemble average $E_{\\mathrm{gap}}$.\n(\\textbf{b}) Calculated density of states plots for various compositions:\n$x_{\\mathrm{Se}}=0.00$ ($n=1$),\n$0.33$ ($n=3$),\n$0.67$ ($n=3$), and\n$1.00$ ($n=1$).\nThe straight black line indicates the position of the valence band maximum,\nwhile the straight magenta and cyan lines indicate the positions of the valence band minimum at $x_{\\mathrm{Se}}=0.33$\nand the conduction band minimum at $x_{\\mathrm{Se}}=0.00$, respectively. }\n\\label{fig:art110:ZnSSe}\n\\end{figure}\n\nOver the years, zinc chalcogenides have garnered interest for a dynamic range of applications --- beginning with the creation\nof the first blue-light emitting laser diodes~\\cite{Haase_APL_1991}, and recently have been studied\nas inorganic graphene analogues (IGAs) with potential applications in flexible and transparent nanodevices~\\cite{Sun_NComm_2012}.\nThese wide-gap II-VI semiconductors have demonstrated a smoothly tunable band gap energy $E_{\\mathrm{gap}}$ with respect to\ncomposition~\\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985}.\nBoth linear and quadratic dependencies have been observed, with the latter phenomenon referred to as\n\\textit{optical bowing}~\\cite{Bernard_PRB_1986}.\nSpecifically, given the pseudo-ternary system $A_{x}B_{1-x}C$,\n\\begin{equation}\nE_{\\mathrm{gap}}(x)=\\left[x \\epsilon_{AC}+(1-x)\\epsilon_{BC}\\right] - b x(1-x),\n\\end{equation}\nwith $b$ characterizing the bowing.\nWhile Larach \\nobreak\\mbox{\\it et al.}\\ reported a linear dependence ($b=0$)~\\cite{Larach_PR_1957},\nEbina \\nobreak\\mbox{\\it et al.}\\ \\cite{Ebina_PRB_1974} and\nEl-Shazly \\nobreak\\mbox{\\it et al.}\\ \\cite{El-Shazly_APA_1985} reported similar bowing parameters of\n$b=0.613\\pm0.027$~eV and $b=0.457\\pm0.044$~eV, respectively, averaged over the two observed direct transitions.\n\nAs a proof of concept, {\\small AFLOW-POCC}\\ is employed to calculate the compositional dependence of the $E_{\\mathrm{gap}}$ and\nDOS for ZnS$_{1-x}$Se$_x$ at room temperature (annealed limit).\nOverall, this system shows relatively low disorder ($\\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)\\sim 0.005$~eV),\nexhibiting negligible variations in the ensemble average properties at higher temperatures.\nThese results are compared to experimental measurements~\\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985} in Figure~\\ref{fig:art110:ZnSSe}.\nCommon among all three trends (Figure~\\ref{fig:art110:ZnSSe}(a)) is the $E_{\\mathrm{gap}}$ shrinkage with increasing $x_{\\mathrm{Se}}$,\nas well as a near 1~eV tunable $E_{\\mathrm{gap}}$ range.\nThe calculated trend demonstrates a non-zero bowing similar to that observed by both Ebina \\nobreak\\mbox{\\it et al.}~\\cite{Ebina_PRB_1974} and\nEl-Shazly \\nobreak\\mbox{\\it et al.}~\\cite{El-Shazly_APA_1985}.\nA fit shows a bowing parameter of $b=0.585\\pm0.078$~eV, lying in the range between the two experimental bowing parameters.\n\nThe ensemble average DOS plots at room temperature are illustrated in Figure~\\ref{fig:art110:ZnSSe}(b) for $x_{\\mathrm{Se}}=0.00$ ($n=1$), $0.33$ ($n=3$), $0.67$ ($n=3$),\nand $1.00$ ($n=1$).\nThe plots echo the negatively correlated band gap relationship illustrated in Figure~\\ref{fig:art110:ZnSSe}(a), highlighting\nthat the replacement of sulfur with selenium atoms reduces the band gap.\nSpecifically, two phenomena are observed as the concentration of selenium increases: (\\textcolor{red}{\\bf red arrows})\nthe reduction of the valence band width\n(with the exception of $x_{\\mathrm{Se}} = 0.00$ (ZnS) concentration), and (\\textcolor{blue}{\\bf blue arrows})\na shift of the conduction band peak back towards the Fermi energy.\nThe valence band of ZnS more closely resembles that of its extreme concentration counterpart at $x_{\\mathrm{Se}} = 1.00$\n(ZnSe) than the others.\nThe extreme concentration conduction peaks appear more defined than their intermediate concentration counterparts, which is likely an artifact\nof the ensemble averaging calculation.\n\nFinally, a partial-DOS analysis is performed in both species and orbitals (not shown).\nIn the valence band, sulfur and selenium account for the majority of the states, in agreement with their relative concentrations.\nMeanwhile, zinc accounts for the majority of the states in the conduction band at all concentrations.\nCorrespondingly, at all concentrations, the $p$-orbitals make up the majority of the valence band,\nwhereas the conduction band consists primarily of $s$- and $p$-orbitals.\nThese observations are consistent with conclusions drawn from previous optical reflectivity measurements that optical transitions\nare possible from sulfur or selenium valence bands to zinc conduction bands~\\cite{Kirschfeld_PRL_1972}.\n\nOverall, the concentration-evolving $E_{\\mathrm{gap}}$ trend and DOS plots support a continuing line of\nwork~\\cite{Larach_PR_1957,Ebina_PRB_1974,El-Shazly_APA_1985} corroborating that this system\nis of the amalgamation type~\\cite{Onodera_JPSJ_1968}\nand not of the persistence type~\\cite{Kirschfeld_PRL_1972}.\nNotably, however, reflectivity spectra shows that the peak position in the $E_{\\mathrm{gap}}$ for ZnS rich alloys may remain\nstationary~\\cite{Ebina_PRB_1974},\nwhich may have manifested itself in the aforementioned anomaly observed in this structure's valence band width.\n\n\\subsubsection{Wide-gap oxide semiconductor alloys}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig090}\n\\mycaption[Disordered Mg$_{x}$Zn$_{1-x}$O.]\n{(\\textbf{a}) A comparison of the experimental~\\cite{Ohtomo_SST_2005,Takeuchi_JAP_2003,\nChen_JAPCM_2003,Takagi_JJAP_2003,Choopun_APL_2002,Minemoto_TSF_2000,Sharma_APL_1999,Ohtomo_APL_1998} \\nobreak\\mbox{\\it vs.}\\ calculated compositional\ndependence of the band gap energy $E_{\\mathrm{gap}}$ at room temperature.\nA rigid shift in the $E_{\\mathrm{gap}}$ axis relative to the experimental results of MgO (second ordinate axis) accounts for the expected\nsystematic deviation in {\\small DFT}\\ calculations~\\cite{Perdew_IJQC_1985}.\nThe \\textcolor{blue}{\\bf wurtzite} and \\textcolor{red}{\\bf rocksalt} structures are highlighted in blue and red, respectively,\nwhile the mixed phase structures are shown in black.\nError bars indicate the weighted standard deviation of the ensemble average $E_{\\mathrm{gap}}$.\n(\\textbf{b}) Calculated density of states plots for various compositions:\n$x_{\\mathrm{Mg}}=0.00$ ($n=1$),\n$0.33$ ($n=3$),\n$0.67$ ($n=3$), and\n$1.00$ ($n=1$).\nThe straight black line indicates the position of the valence band maximum,\nwhile the straight cyan line indicates the position of the conduction band minimum at $x_{\\mathrm{Mg}}=0.00$.}\n\\label{fig:art110:MgZnO}\n\\end{figure}\n\nZinc oxide (ZnO) has proven to be a pervasive material, with far reaching applications such as paints, catalysts,\npharmaceuticals (sun creams), and optoelectronics~\\cite{Takeuchi_MgZnO_Patent}.\nIt has long been investigated for its electronic properties, and falls into the class of transparent conducting\noxides~\\cite{Ellmer_ZnO_2007}.\nJust as with the previous zinc chalcogenide example, ZnO is a wide-gap II-VI semiconductor that has demonstrated\na tunable band gap energy $E_{\\mathrm{gap}}$ with composition.\nIn particular, ZnO has been engineered to have an $E_{\\mathrm{gap}}$ range as large as 5~eV by synthesizing it with magnesium.\nThis pairing has been intensively studied because of the likeness in ionic radius between zinc and magnesium\nwhich results in mitigated misfit strain in the heterostructure~\\cite{Yoo_TSF_2015}.\nWhile the solubility of MgO and ZnO is small, synthesis has been made possible throughout the full compositional\nspectrum~\\cite{Ohtomo_SST_2005,Takeuchi_JAP_2003,Chen_JAPCM_2003,Takagi_JJAP_2003,Choopun_APL_2002,\nMinemoto_TSF_2000,Sharma_APL_1999,Ohtomo_APL_1998}.\n\nAs another proof of concept, the compositional dependence of the $E_{\\mathrm{gap}}$ and DOS for Mg$_{x}$Zn$_{1-x}$O\nare modeled at room temperature (annealed limit).\nIn particular, this disordered system is chosen to illustrate the breath of materials which this framework can model.\nSimilar to ZnS$_{1-x}$Se$_x$, this system shows relatively low disorder ($\\max\\left(\\Delta {H_{\\mathrm{f},i}}\\right)\\sim 0.007$~eV),\nexhibiting negligible variations in the ensemble average properties at higher temperatures.\nThe results are compared to that observed empirically~\\cite{Ohtomo_SST_2005,Takeuchi_JAP_2003,Chen_JAPCM_2003,Takagi_JJAP_2003,Choopun_APL_2002,\nMinemoto_TSF_2000,Sharma_APL_1999,Ohtomo_APL_1998} in Figure~\\ref{fig:art110:MgZnO}.\nAs illustrated in Figure~\\ref{fig:art110:MgZnO}(a), Ohtomo \\nobreak\\mbox{\\it et al.}\\ observed a composition dependent phase transition\nfrom a wurtzite to a rocksalt structure with increasing $x_{\\mathrm{Mg}}$; the transition occurring around the mid concentrations.\nThis transition is enforced in the calculations.\nEmpirically, the overall trend in the wurtzite phase shows a negligible bowing in the $E_{\\mathrm{gap}}$ trend,\ncontrasting the significant bowing observed in the rocksalt phase.\nThe wurtzite phase $E_{\\mathrm{gap}}$ trend shows a slope of $2.160\\pm0.080$~eV, while the rocksalt phase shows a bowing\nparameter of $3.591\\pm0.856$~eV.\nCalculated trends are shown in Figure~\\ref{fig:art110:MgZnO}(a).\nQualitatively, linear and non-linear $E_{\\mathrm{gap}}$ trends are also observed in the wurtzite and rocksalt phases, respectively.\nThe fits are as follows: a slope of $2.147\\pm0.030$~eV in the wurtzite phase and a bowing parameter of\n$5.971\\pm1.835$~eV in the rocksalt phase.\nThese trends match experiment well within the margins of error.\nA larger margin of error is detected in the rocksalt phase, particular in the phase separated region\n($0.4\\lesssim x_{\\mathrm{Mg}} \\lesssim 0.6$).\nThis may be indicative of the significant shear strain and complex nucleation behavior characterizing the region~\\cite{Takeuchi_JAP_2003}.\n\nThe ensemble average DOS plots at room temperature are illustrated in Figure~\\ref{fig:art110:MgZnO}(b) for $x_{\\mathrm{Mg}}=0.00$ ($n=1$), $0.33$ ($n=3$), $0.67$ ($n=3$),\nand $1.00$ ($n=1$)\nThe plots not only echo the positively correlated band gap relationship illustrated in Figure~\\ref{fig:art110:MgZnO}(a),\nbut also exhibit the aforementioned change from a linear to non-linear trend.\nThis is most easily seen by observing the shift in the conduction band away from the Fermi energy,\nhighlighted by the \\textcolor{blue}{\\bf blue arrows}.\nContrasting ZnS$_{1-x}$Se$_x$, a significant change in width of the valence band is not observed over the range of the stoichiometry.\n\nFinally, a partial-DOS analysis is performed in both species and orbitals (not shown).\nOverall, the constant oxygen backbone plays a major role in defining the shape of both the valence and conduction bands,\nparticularly as $x_{\\mathrm{Mg}}$ increases.\nThis resonates with the strong $p$-orbital presence in both bands throughout all concentrations.\nZinc and its $d$-orbitals play a particularly dominant role in the valence band in magnesium-poor structures.\n\n\\subsubsection{Iron alloys}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig091}\n\\mycaption[Disordered Fe$_{1-x}$Cu$_{x}$.]\n{(\\textbf{a}) A comparison of the experimental~\\cite{Sumiyama_JPSJ_1984} \\nobreak\\mbox{\\it vs.}\\ calculated compositional\ndependence of the magnetic moment $M$.\nThe calculations mimic the following phases observed at 4.2~K:\n$x_{\\mathrm{Cu}}\\leq0.42$ \\textcolor{blue}{\\bf bcc} phase shown in blue,\n$0.42\\!0.95\\right)$ features,\nthe final feature vector captures 2,494 total descriptors.\n\nDescriptor construction is inspired by the topological charge indices~\\cite{Galvez_JCICS_1995}\nand the Kier-Hall\nelectro-topological state indices~\\cite{Kier_Electrotopological_1999}.\nLet $\\mathbf{M}$ be the matrix obtained by multiplying the adjacency\nmatrix $\\mathbf{A}$ by the reciprocal square distance matrix $\\mathbf{D}$ $\\left(D_{ij}=1\/r_{i,j}^{2}\\right)$:\n\\begin{equation}\n\\mathbf{M}=\\mathbf{A} \\cdot \\mathbf{D}.\n\\end{equation}\nThe matrix $\\mathbf{M}$, called the Galvez matrix, is a square $n \\times n$ matrix,\nwhere $n$ is the number of atoms in the unit cell.\nFrom $\\mathbf{M}$, descriptors of reference property $\\mathbf{q}$ are calculated as\n\\begin{equation}\nT^{\\mathrm{E}}=\\sum_{i=1}^{n-1}\\sum_{j=i+1}^{n}\\left|q_{i}-q_{j}\\right|M_{ij}\n\\end{equation}\nand\n\\begin{equation}\nT_{{\\substack{\\scalebox{0.6}{bond}}}}^{\\mathrm{E}}=\\sum_{\\{i,j\\}\\in\\mathrm{bonds}}\\left|q_{i}-q_{j}\\right|M_{ij},\n\\end{equation}\nwhere the first set of indices count over all pairs of atoms and the second\nis restricted to all pairs $i,j$ of bonded atoms.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig095}\n\\mycaption[Outline of the modeling work-flow.]\n{{\\small ML}\\ models are represented by orange diamonds. Target properties predicted by these models are highlighted in green.}\n\\label{fig:art124:figure2}\n\\end{figure}\n\n\\boldsection{Quantitative materials structure-property relationship modeling.}\nIn training the models, the same {\\small ML}\\ method and descriptors are employed without any hand tuning or variable selection.\nSpecifically, models are constructed using gradient boosting decision tree\n({\\small GBDT}) technique~\\cite{Friedman_AnnStat_2001}.\nAll models were validated through $y$-randomization (label scrambling).\nFive-fold cross validation is used to assess how well each model will generalize to an independent dataset.\nHyperparameters are determined with grid searches on the training set and 10-fold cross validation.\n\nThe gradient boosting decision trees ({\\small GBDT}) method~\\cite{Friedman_AnnStat_2001}\nevolved from the application of boosting\nmethods~\\cite{gbm} to regression trees~\\cite{Loh_ISR_2014}.\nThe boosting method is based on the observation that finding many weakly accurate\nprediction rules can be a lot easier than finding a single, highly accurate rule~\\cite{Schapire_ML_1990}.\nThe boosting algorithm calls this ``weak'' learner repeatedly, at each stage feeding it\na different subset of the training examples.\nEach time it is called, the weak learner generates a new weak prediction rule.\nAfter many iterations, the boosting algorithm combines these weak rules into\na single prediction rule aiming to be much more accurate than any single weak rule.\n\nThe {\\small GBDT}\\ approach is an additive model of the following form:\n\\begin{equation}\nF(\\mathbf{x};\\{\\gamma_{m},\\mathbf{a}\\}_{1}^{M})=\\sum_{m=1}^{M}\\gamma_{m} h_{m}(\\mathbf{x};\\mathbf{a}_{m}),\n\\end{equation}\nwhere $h_{m}(\\mathbf{x};\\mathbf{a}_{m})$ are the weak learners (decision trees in this case)\ncharacterized by parameters\n$\\mathbf{a}_{m}$, and $M$ is the total\ncount of decision trees obtained through boosting.\n\nIt builds the additive model in a forward stage-wise fashion:\n\\begin{equation}\nF_m(\\mathbf{x})=F_{m-1}(\\mathbf{x})+\\gamma_{m} h_{m}(\\mathbf{x};\\mathbf{a}_{m}).\n\\end{equation}\nAt each stage $\\left(m=1,2,\\ldots,M\\right)$, $\\gamma_{m}$ and $\\mathbf{a}_{m}$ are chosen to minimize the loss function\n$f_L$ given the current model $F_{m-1}(x_{i})$ for all data points (count $N$),\n\\begin{equation}\n\\left(\\gamma_{m},\\mathbf{a}_{m}\\right)=\\argmin_{\\gamma,\\mathbf{a}} \\sum_{i=1}^{N}\nf_{L} \\left[y_{i},F_{m-1}\\left(\\mathbf{x_{i}}\\right)+\\gamma h\\left(\\mathbf{x}_{i};\\mathbf{a}\\right)\\right].\n\\end{equation}\nGradient boosting attempts to solve this minimization problem numerically via steepest descent.\nThe steepest descent direction is the negative gradient of the loss function\nevaluated at the current model $F_{m-1}$, where the step length is chosen using line search.\n\nAn important practical task is to quantify variable importance.\nFeature selection in decision tree ensembles cannot differentiate between primary\neffects and effects caused by interactions between variables.\nTherefore, unlike regression coefficients, a direct comparison of captured effects is prohibited.\nFor this purpose, variable influence is quantified in the\nfollowing way~\\cite{Friedman_AnnStat_2001}.\nLet us define the influence of variable $j$ in a single tree $h$.\nConsider that the tree has $l$ splits and therefore $l-1$ levels.\nThis gives rise to the definition of the variable influence,\n\\begin{equation}\nK_{j}^2(h)=\\sum_{i=1}^{l-1} I_{i}^{2} \\mathbbm{1}\\left(x_{i}=j\\right),\n\\end{equation}\nwhere $I_{i}^{2}$ is the empirical squared improvement resulting from this split,\nand $\\mathbbm{1}$ is the indicator function.\nHere, $\\mathbbm{1}$ has a value of one if the split at node $x_{i}$ is on variable $j$, and\nzero otherwise,\n\\nobreak\\mbox{\\it i.e.}, it measures the number of times a variable $j$ is selected for splitting.\nTo obtain the overall influence of variable $j$ in the ensemble of decision trees (count $M$),\nit is averaged over all trees,\n\\begin{equation}\nK_{j}^{2}={M}^{-1} \\sum_{m=1}^{M} K_{j}^{2} (h_{m}).\n\\end{equation}\nThe influences $K_{j}^{2}$ are normalized so that they add to one.\nInfluences capture the importance of the variable, but\nnot the direction of the response (positive or negative).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig096}\n\\mycaption[Five-fold cross validation plots for the eight {\\small ML}\\ models predicting electronic and thermomechanical properties.]\n{\\textbf{(a)} Receiver operating characteristic ({\\small ROC}) curve for the classification {\\small ML}\\ model.\n\\textbf{(b)}-\\textbf{(h)} Predicted \\nobreak\\mbox{\\it vs.}\\ calculated values\nfor the regression {\\small ML}\\ models:\n\\textbf{(b)} band gap energy $\\left(E_{\\scriptstyle \\mathrm{BG}}\\right)$,\n\\textbf{(c)} bulk modulus $\\left(B_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(d)} shear modulus $\\left(G_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(e)} Debye temperature $\\left(\\theta_{\\scriptstyle \\mathrm{D}}\\right)$,\n\\textbf{(f)} heat capacity at constant pressure $\\left(C_{\\scriptstyle \\mathrm{P}}\\right)$,\n\\textbf{(g)} heat capacity at constant volume $\\left(C_{\\scriptstyle \\mathrm{V}}\\right)$, and\n\\textbf{(h)} thermal expansion coefficient $\\left(\\alpha_{\\scriptstyle \\mathrm{V}}\\right)$.\n}\n\\label{fig:art124:figure3}\n\\end{figure}\n\n\\boldsection{Integrated modeling work-flow.}\nEight predictive models are developed in this work, including:\na binary classification model that predicts if a material is a metal or an insulator\nand seven regression models that predict:\nthe band gap energy $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\right)$ for insulators,\nbulk modulus $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$,\nshear modulus $\\left(G_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$,\nDebye temperature $\\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\\right)$,\nheat capacity at constant pressure $\\left(C_{\\substack{\\scalebox{0.6}{p}}}\\right)$,\nheat capacity at constant volume $\\left(C_{\\substack{\\scalebox{0.6}{V}}}\\right)$, and\nthermal expansion coefficient $\\left(\\alpha_{\\substack{\\scalebox{0.6}{V}}}\\right)$.\n\nFigure~\\ref{fig:art124:figure2} shows the overall application work-flow.\nA novel candidate material is first classified as a metal or an insulator.\nIf the material is classified as an insulator, $E_{\\substack{\\scalebox{0.6}{BG}}}$ is predicted,\nwhile classification as a metal implies that the material has no $E_{\\substack{\\scalebox{0.6}{BG}}}$.\nThe six thermomechanical properties are then predicted independent of the material's metal\/insulator classification.\nThe integrated modeling work-flow has been implemented as a web application at\n\\href{http:\/\/aflow.org\/aflow-ml}{aflow.org\/aflow-ml},\nrequiring only the atomic species and positions as input for predictions.\n\nWhile all three models were trained independently, the accuracy of the\n$E_{\\substack{\\scalebox{0.6}{BG}}}$ regression model is inherently dependent on the accuracy of the metal\/insulator classification model\nin this work-flow.\nHowever, the high accuracy of the metal\/insulator classification model suggests this not to be a practical concern.\n\n\\boldsection{Model generalizability.}\nOne technique for assessing model quality is five-fold cross validation, which gauges how well\nthe model is expected to generalize to an independent dataset.\nFor each model, the scheme involves randomly partitioning the set into five groups and predicting the value of\neach material in one subset while training the model on the other four subsets.\nHence, each subset has the opportunity to play the role of the ``test set''.\nFurthermore, any observed deviations in the predictions are addressed.\nFor further analysis, all predicted and calculated results are available in\nSupplementary Note 2 of Reference~\\cite{curtarolo:art124}.\n\nThe accuracy of the metal\/insulator classifier is reported as the\narea under the curve ({\\small AUC})\nof the receiver operating characteristic ({\\small ROC}) plot (Figure~\\ref{fig:art124:figure3}(a)).\nThe {\\small ROC}\\ curve illustrates the model's ability to differentiate between metallic and insulating input materials.\nIt plots the prediction rate for insulators (correctly \\nobreak\\mbox{\\it vs.}\\ incorrectly predicted) throughout the\nfull spectrum of possible prediction thresholds.\nAn area of 1.0 represents a perfect test, while an area of 0.5 characterizes a random guess (the dashed line).\nThe model shows excellent external predictive power with the {\\small AUC}\\ at 0.98,\nan insulator-prediction success rate (sensitivity) of 0.95,\na metal-prediction success rate (specificity) of 0.92,\nand an overall classification rate ({\\small CCR}) of 0.93.\nFor the complete set of \\PLMFelectronicTotal\\ materials, this corresponds to\n2,103 misclassified materials, including 1,359 misclassified metals and 744 misclassified insulators.\nEvidently, the model exhibits positive bias toward predicting insulators, where bias refers to whether a\n{\\small ML}\\ model tends to over- or under-estimate the predicted property.\nThis low false-metal rate is fortunate as the model is unlikely to\nmisclassify a novel, potentially interesting semiconductor as a metal.\nOverall, the metal classification model is robust enough to handle the full complexity of the periodic table.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical summary of the five-fold cross-validated predictions for the seven regression models.]\n{The summary corresponds with Figure~\\ref{fig:art124:figure3}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & {\\small RMSE}\\ & {\\small MAE}\\ & $r^{2}$ \\\\\n\\hline\n$E_{\\substack{\\scalebox{0.6}{BG}}}$ & 0.51~eV & 0.35~eV & 0.90 \\\\\n$B_{\\substack{\\scalebox{0.6}{VRH}}}$ & 14.25~GPa & 8.68~GPa & 0.97 \\\\\n$G_{\\substack{\\scalebox{0.6}{VRH}}}$ & 18.43~GPa & 10.62~GPa & 0.88 \\\\\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$ & 56.97~K & 35.86~K & 0.95 \\\\\n$C_{\\substack{\\scalebox{0.6}{p}}}$ & 0.09~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.05~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.95 \\\\\n$C_{\\substack{\\scalebox{0.6}{V}}}$ & 0.07~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.04~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.95 \\\\\n$\\alpha_{\\substack{\\scalebox{0.6}{V}}}$ & $1.47 \\times 10^{-5}$~K$^{-1}$ & $5.69 \\times 10^{-6}$~K$^{-1}$ & 0.91 \\\\\n\\end{tabular}\n\\label{tab:art124:table1}\n\\end{table}\n\nThe results of the five-fold cross validation analysis for the band gap energy $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\right)$ regression model\nare plotted in Figure~\\ref{fig:art124:figure3}(b).\nAdditionally, a statistical profile of these predictions, along with that of the six thermomechanical regression models,\nis provided in Table~\\ref{tab:art124:table1}, which includes metrics such as\nthe root-mean-square error ({\\small RMSE}),\nmean absolute error ({\\small MAE}), and coefficient of determination $\\left(r^2\\right)$.\nSimilar to the classification model, the $E_{\\substack{\\scalebox{0.6}{BG}}}$ model exhibits a positive predictive bias.\nThe biggest errors come from materials with narrow band gaps,\n\\nobreak\\mbox{\\it i.e.}, the scatter in the lower left corner in Figure~\\ref{fig:art124:figure3}(b).\nThese materials predominantly include complex fluorides and nitrides.\nN$_{2}$H$_{6}$Cl$_{2}$ ({\\small ICSD}\\ \\#23145)\nexhibits the worst prediction accuracy with signed error SE = 3.78 eV~\\cite{Donohue_JCP_1947}.\nThe most underestimated materials are HCN ({\\small ICSD}\\ \\#76419) and, respectively\nN$_{2}$H$_{6}$Cl$_{2}$ ({\\small ICSD}\\ \\#240903) with SE = -2.67 and -3.19 eV~\\cite{Dulmage_ActaCrist_1951,Kruszynski_ActaCristE_2007}, respectively.\nThis is not surprising considering that all three are molecular crystals.\nSuch systems are anomalies in the {\\small ICSD}, and fit better in other databases, such as\nthe Cambridge Structural Database~\\cite{Groom_CSD_2016}.\nOverall, 10,762 materials are predicted within 25\\% accuracy of calculated values,\nwhereas 824 systems have errors over 1 eV.\n\nFigures~\\ref{fig:art124:figure3}(c-h) and Table~\\ref{tab:art124:table1} showcase the results of the five-fold cross validation analysis\nfor the six thermomechanical regression models.\nFor both bulk $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$ and shear $\\left(G_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$ moduli,\nover 85\\% of materials are predicted within 20~GPa of their calculated values.\nThe remaining models also demonstrate high accuracy, with\nat least 90\\% of the full training set $\\left(>2,546~\\mathrm{systems}\\right)$\npredicted to within 25\\% of the calculated values.\nSignificant outliers in predictions of the bulk modulus include\ngraphite ({\\small ICSD}\\ \\#187640, SE = 100 GPa, likely\ndue to extreme anisotropy) and two theoretical high-pressure boron nitrides ({\\small ICSD}\\ \\#162873 and \\#162874,\nunder-predicted by over 110 GPa)~\\cite{Lian_JCP_2013,Doll_PRB_2008}.\nOther theoretical systems are ill-predicted throughout the six properties, including\nZN ({\\small ICSD}\\ \\#161885), CN$_{2}$ ({\\small ICSD}\\ \\#247676), C$_{3}$N$_{4}$ ({\\small ICSD}\\ \\#151782),\nand CH ({\\small ICSD}\\ \\#187642)~\\cite{EscorciaSalas_MJ_2008,Li_PCCP_2012,Marques_PRB_2004,Lian_JCP_2013}.\nPredictions for the $G_{\\substack{\\scalebox{0.6}{VRH}}}$, Debye temperature $\\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\\right)$, and thermal expansion coefficient\n$\\left(\\alpha_{\\substack{\\scalebox{0.6}{V}}}\\right)$\ntend to be slightly underestimated, particularly for higher calculated values.\nAdditionally, mild scattering can be seen for $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ and\n$\\alpha_{\\substack{\\scalebox{0.6}{V}}}$, but not enough to have a significant\nimpact on the error or correlation metrics.\n\nDespite minimal deviations, both {\\small RMSE}\\ and {\\small MAE}\\ are within 4\\% of the ranges covered for each property,\nand the predictions demonstrate excellent correlation with the calculated properties.\nNote the tight clustering of points just below 3 $k_{\\substack{\\scalebox{0.6}{B}}}$\/atom for the heat\ncapacity at constant volume $\\left(C_{\\substack{\\scalebox{0.6}{V}}}\\right)$.\nThis is due to $C_{\\substack{\\scalebox{0.6}{V}}}$ saturation in accordance with the Dulong-Petit law occurring at or below\n300 K for many compounds.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig097}\n\\mycaption[Semi-log scatter plot of the full dataset (\\PLMFelectronicTotal\\ unique materials) in a dual-descriptor space.]\n{$\\avg\\left(\\Delta H_{\\scriptstyle \\mathrm{fusion}}\\lambda^{-1}\\right)$ \\nobreak\\mbox{\\it vs.}\\\n$\\avg\\left(V_{\\scriptstyle \\mathrm{molar}}r_{\\scriptstyle \\mathrm{cov}}^{-1}\\right)$.\nInsulators and metals are colored in red and blue, respectively.}\n\\label{fig:art124:figure4}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig098}\n\\mycaption[Partial dependence plots of the $E_{\\scriptstyle \\mathrm{BG}}$, $B_{\\scriptstyle \\mathrm{VRH}}$, and\n$\\theta_{\\scriptstyle \\mathrm{D}}$ models.]\n{\\textbf{(a)} Partial dependence of $E_{\\scriptstyle \\mathrm{BG}}$ on the $\\avg\\left(\\Delta IP_{\\mathrm{bond}}\\right)$\ndescriptor.\nFor $E_{\\scriptstyle \\mathrm{BG}}$,\nthe 2D interaction between $\\std\\left(\\Delta IP_{\\mathrm{bond}}\\right)$ and $\\avg\\left(\\Delta IP_{\\mathrm{bond}}\\right)$\nand between $\\rho$ (density) and $\\avg\\left(\\Delta IP_{\\mathrm{bond}}\\right)$ are illustrated in panels\n\\textbf{(b)} and \\textbf{(c)}, respectively.\n\\textbf{(d)} Partial dependence of the $B_{\\scriptstyle \\mathrm{VRH}}$ on the crystal volume per atom descriptor.\nFor $\\theta_{\\scriptstyle \\mathrm{D}}$,\nthe 2D interaction between\n$\\avg\\left(\\Delta EA_{\\scriptstyle \\mathrm{bond}}\\right)$ and\n$\\std\\left(\\Delta H_{\\scriptstyle \\mathrm{vapor}} \\Delta H_{\\scriptstyle \\mathrm{atom}}^{-1}\\right)$\nand between\ncrystal lattice parameters $b$ and $c$ are illustrated\nin panels \\textbf{(e)} and \\textbf{(f)}, respectively.}\n\\label{fig:art124:figure5}\n\\end{figure}\n\n\\boldsection{Model interpretation.}\nModel interpretation is of paramount importance in any {\\small ML}\\ study.\nThe significance of each descriptor is determined in order to gain insight into\nstructural features that impact molecular properties of interest.\nInterpretability is a strong advantage of decision tree methods, particularly with the {\\small GBDT}\\ approach.\nOne can quantify the predictive power of a specific descriptor by analyzing the reduction\nof the {\\small RMSE}\\ at each node of the tree.\n\nPartial dependence plots offer yet another opportunity for {\\small GBDT}\\ model interpretation.\nSimilar to the descriptor significance analysis, partial dependence resolves the\neffect of a variable (descriptor) on a property, but only after marginalizing over all other\nexplanatory variables~\\cite{Hastie_StatLearn_2001}.\nThe effect is quantified by the change of that property as relevant descriptors are varied.\nThe plots themselves highlight the most important interactions among relevant descriptors\nas well as between properties and their corresponding descriptors.\nWhile only the most important descriptors are highlighted and discussed,\nan exhaustive list of relevant descriptors and their relative contributions\ncan be found in\nSupplementary Note 1 of Reference~\\cite{curtarolo:art124}.\n\nFor the metal\/insulator classification model, the descriptor significance analysis\nshows that two descriptors have the highest importance (equally), namely\n$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)$ and\n$\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)$.\n$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)$ is the ratio between the\nfusion enthalpy $\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\right)$\nand the thermal conductivity $\\left(\\lambda\\right)$ averaged over all atoms in the material, and\n$\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)$ is the ratio between the\nmolar volume $\\left(V_{\\substack{\\scalebox{0.6}{molar}}}\\right)$\nand the covalent radius $\\left(r_{\\substack{\\scalebox{0.6}{cov}}}\\right)$ averaged over all atoms in the material.\nBoth descriptors are simple node-specific features.\nThe presence of these two prominent descriptors accounts for the high accuracy of the classification model.\n\nFigure~\\ref{fig:art124:figure4} shows the projection of the full dataset onto the dual-descriptor space of\n$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)$ and $\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)$.\nIn this 2D space, metals and insulators are substantially partitioned.\nTo further resolve this separation, the plot is split into four quadrants\n(see dashed lines) with an origin approximately at\n$\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)=11$,~$\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)=2$.\nInsulators are predominately located in quadrant I.\nThere are several clusters (one large and several small) parallel to the $x$-axis.\nMetals occupy a compact square block in quadrant III within intervals\n$5<\\avg\\left(V_{\\substack{\\scalebox{0.6}{molar}}} r_{\\substack{\\scalebox{0.6}{cov}}}^{-1}\\right)<12$ and $0.02<\\avg\\left(\\Delta H_{\\substack{\\scalebox{0.6}{fusion}}}\\lambda^{-1}\\right)<2$.\nQuadrant II is mostly empty with a few materials scattered about the origin.\nIn the remaining quadrant (IV), materials have mixed character.\n\nAnalysis of the projection shown in Figure~\\ref{fig:art124:figure4} suggests a simple heuristic rule:\nall materials within quadrant I are classified as insulators $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\!>\\!0\\right)$,\nand all materials outside of this quadrant are metals.\nRemarkably, this unsupervised projection approach achieves a very high\nclassification accuracy of 86\\% for the entire dataset of \\PLMFelectronicTotal\\ materials.\nThe model misclassifies only 3,621 materials:\n2,414 are incorrectly predicted as insulators and 1,207 are incorrectly predicted as metals.\nThis example illustrates how careful model analysis of the most significant descriptors\ncan yield simple heuristic rules for materials design.\n\nThe regression model for the band gap energy $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\right)$ is more complex.\nThere are a number of descriptors in the model with comparable contributions,\nand thus, all individual contributions are small.\nThis is expected as a number of conditions can affect $E_{\\substack{\\scalebox{0.6}{BG}}}$.\nThe most important are $\\avg\\left(\\chi Z_{\\mathrm{eff}}^{-1}\\right)$ and $\\avg\\left(C \\lambda^{-1}\\right)$ with\nsignificance scores of 0.075 and 0.071,\nrespectively, where $\\chi$ is the electronegativity, $Z_{\\mathrm{eff}}$ is the effective nuclear charge,\n$C$ is the specific heat capacity, and $\\lambda$ is the thermal conductivity of each atom.\n\nFigure~\\ref{fig:art124:figure5} shows partial dependence plots focusing on $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ as an example.\nIt is derived from edge fragments of bonded atoms $\\left(l=1\\right)$ and defined as an absolute difference in\nionization potentials averaged over the material.\nIn other words, it is a measure of bond polarity, similar to electronegativity.\nFigure~\\ref{fig:art124:figure5}(a) shows a steady monotonic increase in $\\Delta E_{\\substack{\\scalebox{0.6}{BG}}}$ for larger\nvalues of $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$.\nThe effect is small, but captures an expected physical principle:\npolar inorganic materials (\\nobreak\\mbox{\\it e.g.}, oxides, fluorides) tend to have larger $E_{\\substack{\\scalebox{0.6}{BG}}}$.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig099}\n\\mycaption[Model performance evaluation for the six {\\small ML}\\ models predicting thermomechanical properties\nof \\PLMFthermoTestTotal\\ newly characterized materials.]\n{Predicted \\nobreak\\mbox{\\it vs.}\\ calculated values for the regression {\\small ML}\\ models:\n\\textbf{(a)} bulk modulus $\\left(B_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(b)} shear modulus $\\left(G_{\\scriptstyle \\mathrm{VRH}}\\right)$,\n\\textbf{(c)} Debye temperature $\\left(\\theta_{\\scriptstyle \\mathrm{D}}\\right)$,\n\\textbf{(d)} heat capacity at constant pressure $\\left(C_{\\scriptstyle \\mathrm{P}}\\right)$,\n\\textbf{(e)} heat capacity at constant volume $\\left(C_{\\scriptstyle \\mathrm{V}}\\right)$, and\n\\textbf{(f)} thermal expansion coefficient $\\left(\\alpha_{\\scriptstyle \\mathrm{V}}\\right)$.}\n\\label{fig:art124:figure6}\n\\end{figure}\n\nGiven the number of significant interactions involved with this phenomenon,\ntailoring $E_{\\substack{\\scalebox{0.6}{BG}}}$ involves the\noptimization of a highly non-convex, multidimensional object.\nFigure~\\ref{fig:art124:figure5}(b) illustrates a 2D slice of this object as\n$\\std\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ and $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ vary simultaneously.\nLike $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$,\n$\\std\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ is the standard deviation of the set of absolute differences in $IP$ among\nall bonded atoms.\nIn the context of these two variables, $E_{\\substack{\\scalebox{0.6}{BG}}}$ responds to deviations in $\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}$\namong the set of bonded atoms, but remains constant across shifts in $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$.\nThis suggests an opportunity to tune $E_{\\substack{\\scalebox{0.6}{BG}}}$ by considering another composition that varies the deviations among bond polarities.\nAlternatively, a desired $E_{\\substack{\\scalebox{0.6}{BG}}}$ can be maintained\nby considering another composition that preserves the deviations among bond polarities, even as the overall average\nshifts.\nSimilarly, Figure~\\ref{fig:art124:figure5}(c) shows the partial dependence on both\nthe density $\\left(\\rho\\right)$ and $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$.\nContrary to the previous trend, larger $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$\nvalues correlate with smaller $E_{\\substack{\\scalebox{0.6}{BG}}}$, particularly for low density structures.\nMaterials with higher density and lower $\\avg\\left(\\Delta IP_{{\\substack{\\scalebox{0.6}{bond}}}}\\right)$ tend to have higher $E_{\\substack{\\scalebox{0.6}{BG}}}$.\nConsidering the elevated response (compared to Figure~\\ref{fig:art124:figure5}(b)), the inverse correlation of $E_{\\substack{\\scalebox{0.6}{BG}}}$ with the average\nbond polarity in the context of density suggests an even more effective means of tuning $E_{\\substack{\\scalebox{0.6}{BG}}}$.\n\nA descriptor analysis of the thermomechanical property models reveals the importance of\none descriptor in particular, the volume per atom of the crystal.\nThis conclusion certainly resonates with the nature of these properties, as they generally correlate\nwith bond strength~\\cite{curtarolo:art115}.\nFigure~\\ref{fig:art124:figure5}(d) exemplifies such a relationship, which shows\nthe partial dependence plot of the bulk modulus $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$ on the volume per atom.\nTightly bound atoms are generally indicative of stronger bonds.\nAs the interatomic distance increases, properties like $B_{\\substack{\\scalebox{0.6}{VRH}}}$ generally reduce.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical summary of the new predictions for the six thermomechanical regression models.]\n{The summary corresponds with Figure~\\ref{fig:art124:figure6}.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nproperty & {\\small RMSE}\\ & {\\small MAE}\\ & $r^{2}$ \\\\\n\\hline\n$B_{\\substack{\\scalebox{0.6}{VRH}}}$ & 21.13~GPa & 12.00~GPa & 0.93 \\\\\n$G_{\\substack{\\scalebox{0.6}{VRH}}}$ & 18.94~GPa & 13.31~GPa & 0.90 \\\\\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$ & 64.04~K & 42.92~K & 0.93 \\\\\n$C_{\\substack{\\scalebox{0.6}{p}}}$ & 0.10~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.06~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.92 \\\\\n$C_{\\substack{\\scalebox{0.6}{V}}}$ & 0.07~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.05~$k_{\\substack{\\scalebox{0.6}{B}}}$\/atom & 0.95 \\\\\n$\\alpha_{\\substack{\\scalebox{0.6}{V}}}$ & $1.95 \\times 10^{-5}$~K$^{-1}$ & $5.77 \\times 10^{-6}$~K$^{-1}$ & 0.76 \\\\\n\\end{tabular}\n\\label{tab:art124:table2}\n\\end{table}\n\nTwo of the more interesting dependence plots are also shown in Figure~\\ref{fig:art124:figure5}(e-f),\nboth of which offer opportunities for tuning the Debye temperature ($\\theta_{\\substack{\\scalebox{0.6}{D}}}$).\nFigure~\\ref{fig:art124:figure5}(e) illustrates the interactions among two descriptors,\nthe absolute difference in electron affinities among bonded atoms\naveraged over the material\n$\\left(\\avg\\left(\\Delta EA_{\\substack{\\scalebox{0.6}{bond}}}\\right)\\right)$, and\nthe standard deviation of the set of ratios of the enthalpies of vaporization $\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}}\\right)$\nand atomization $\\left(\\Delta H_{\\substack{\\scalebox{0.6}{atom}}}\\right)$ for all atoms in the material\n$\\left(\\std\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}} \\Delta H_{\\substack{\\scalebox{0.6}{atom}}}^{-1}\\right)\\right)$.\nWithin these dimensions, two distinct regions emerge of increasing\/decreasing $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ separated by a\nsharp division\nat about $\\avg\\left(\\Delta EA_{\\substack{\\scalebox{0.6}{atom}}}\\right) = 3$.\nWithin these partitions, there are clusters of maximum gradient in $\\theta_{\\substack{\\scalebox{0.6}{D}}}$---peaks within the left\npartition and troughs within the right.\nThe peaks and troughs alternate with varying $\\std\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}} \\Delta H_{\\substack{\\scalebox{0.6}{atom}}}^{-1}\\right)$.\nAlthough $\\std\\left(\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}} \\Delta H_{\\substack{\\scalebox{0.6}{atom}}}^{-1}\\right)$\nis not an immediately intuitive descriptor, the alternating clusters may be a manifestation\nof the periodic nature of $\\Delta H_{\\substack{\\scalebox{0.6}{vapor}}}$ and $\\Delta H_{\\substack{\\scalebox{0.6}{atom}}}$~\\cite{webelements_periodicity}.\nAs for the partitions themselves,\nthe extremes of $\\avg\\left(\\Delta EA_{\\substack{\\scalebox{0.6}{atom}}}\\right)$ characterize covalent and ionic materials, as\nbonded atoms with similar $EA$ are likely to share electrons, while those\nwith varying $EA$ prefer to donate\/accept electrons.\nConsidering that $EA$ is also periodic, various opportunities for carefully tuning $\\theta_{\\substack{\\scalebox{0.6}{D}}}$\nshould be available.\n\nFinally, Figure~\\ref{fig:art124:figure5}(f) shows the partial dependence of $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ on the lattice parameters $b$ and $c$.\nIt resolves two notable correlations:\n\\textit{(i)} uniformly increasing the cell size of the system decreases $\\theta_{\\substack{\\scalebox{0.6}{D}}}$, but\n\\textit{(ii)} elongating the cell ($c\/b \\gg 1$) increases it.\nAgain, \\textit{(i)} can be attributed to the\ninverse relationship between volume per atom and bond strength,\nbut does little to address \\textit{(ii)}.\nNevertheless, the connection between elongated, or layered, systems and the Debye temperature is certainly not\nsurprising---anisotropy can be leveraged to enhance phonon-related interactions associated with\nthermal conductivity~\\cite{Minnich_PRB_2015}\nand superconductivity~\\cite{Shimahara_PRB_2002,Jha_PT_1989,Klein_SSC_1980}.\nWhile the domain of interest is quite narrow,\nthe impact is substantial, particularly in comparison to that shown in Figure~\\ref{fig:art124:figure5}(e).\n\n\\boldsection{Model validation.}\nWhile the expected performances of the {\\small ML}\\ models can be projected through five-fold cross validation,\nthere is no substitute for validation against an independent dataset.\nThe {\\small ML}\\ models for the thermomechanical properties are leveraged to make predictions\nfor materials previously uncharacterized, and subsequently validated\nthese predictions via the {\\small AEL}-{\\small AGL}\\ integrated framework~\\cite{curtarolo:art96, curtarolo:art115}.\nFigure~\\ref{fig:art124:figure6} illustrates the models' performance on the set of \\PLMFthermoTestTotal\\ additional materials,\nwith relevant statistics displayed in Table~\\ref{tab:art124:table2}.\nFor further analysis, all predicted and calculated results are available in\nSupplementary Note 3 of Reference~\\cite{curtarolo:art124}.\n\nComparing with the results of the generalizability analysis shown in Figure~\\ref{fig:art124:figure3} and Table~\\ref{tab:art124:table1},\nthe overall errors are consistent with five-fold cross validation.\nFive out of six models have $r^2$ of 0.9 or higher.\nHowever, the $r^2$ value for the thermal expansion coefficient\n$\\left(\\alpha_{\\substack{\\scalebox{0.6}{V}}}\\right)$ is lower than forecasted.\nThe presence of scattering suggests the need for a larger training set---as new,\nmuch more diverse materials were likely introduced in the test set.\nThis is not surprising considering the number of variables that can affect thermal expansion~\\cite{Figge_APL_2009}.\nOtherwise, the accuracy of these predictions confirm the effectiveness of the {\\small PLMF}\\ representation,\nwhich is particularly compelling considering:\n\\textit{(i)} the limited diversity training dataset (only about 11\\% as large as that available for\npredicting the electronic properties), and\n\\textit{(ii)} the relative size of the test set (over a quarter the size of the training set).\n\nIn the case of the bulk modulus $\\left(B_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$, 665 systems (86\\% of test set) are predicted within 25\\%\nof calculated values.\nOnly the predictions of four materials, Bi ({\\small ICSD}\\ \\#51674), PrN ({\\small ICSD}\\ \\#168643),\nMg$_{3}$Sm ({\\small ICSD}\\ \\#104868), and ZrN ({\\small ICSD}\\ \\#161885), deviate beyond 100~GPa from calculated values.\nBi is a high-pressure phase (Bi-III) with a caged, zeolite-like structure~\\cite{McMahon_BiIII_2001}.\nThe structures of zirconium nitride (wurtzite phase) and praseodymium nitride (B3 phase) were hypothesized and\ninvestigated via {\\small DFT}\\ calculations~\\cite{EscorciaSalas_MJ_2008,Kocak_PSCB_2010} and have yet to be observed\nexperimentally.\n\nFor the shear modulus $\\left(G_{\\substack{\\scalebox{0.6}{VRH}}}\\right)$, 482 materials (63\\% of the test set) are predicted within 25\\%\nof calculated values.\nJust one system, C$_{3}$N$_{4}$ ({\\small ICSD}\\ \\#151781), deviates beyond 100~GPa from its calculated value.\nThe Debye temperature $\\left(\\theta_{\\substack{\\scalebox{0.6}{D}}}\\right)$ is predicted to within 50 K accuracy for 540 systems (70\\% of the test set).\nBeF$_{2}$ ({\\small ICSD}\\ \\#173557), yet another cage (sodalite) structure~\\cite{Zwijnenburg_JACS_2008}, has among the largest errors\nin three models including $\\theta_{\\substack{\\scalebox{0.6}{D}}}$ (SE = -423 K) and both heat capacities\n($C_{\\substack{\\scalebox{0.6}{p}}}$: SE = 0.65 $k_{\\substack{\\scalebox{0.6}{B}}}$\/atom; $C_{\\substack{\\scalebox{0.6}{V}}}$: SE = 0.61 $k_{\\substack{\\scalebox{0.6}{B}}}$\/atom).\nSimilar to other ill-predicted structures, this polymorph is theoretical, and has yet to be synthesized.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig100}\n\\mycaption[Comparison of the {\\small AEL}-{\\small AGL}\\ calculations and {\\small ML}\\ predictions with experimental values for three thermomechanical properties.]\n{\\textbf{(a)} bulk modulus $\\left(B\\right)$,\n\\textbf{(b)} shear modulus $\\left(G\\right)$,\nand\n\\textbf{(c)} Debye temperature $\\left(\\theta_{\\scriptstyle \\mathrm{D}}\\right)$.\n}\n\\label{fig:art124:figure7}\n\\end{figure}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical summary of the {\\small AEL}-{\\small AGL}\\ calculations and\n{\\small ML}\\ predictions \\nobreak\\mbox{\\it vs.}\\ experimental values for three thermomechanical properties.]\n{The summary corresponds with Figure~\\ref{fig:art124:figure7}.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r}\n\\multirow{2}{*}{property} & \\multicolumn{2}{c|}{{\\small RMSE}} & \\multicolumn{2}{c|}{{\\small MAE}} & \\multicolumn{2}{c}{$r^{2}$} \\\\\n\\cline{2-7}\n & exp. \\nobreak\\mbox{\\it vs.}\\ calc. & exp. \\nobreak\\mbox{\\it vs.}\\ pred. & exp. \\nobreak\\mbox{\\it vs.}\\ calc. & exp. \\nobreak\\mbox{\\it vs.}\\ pred. & exp. \\nobreak\\mbox{\\it vs.}\\ calc. & exp. \\nobreak\\mbox{\\it vs.}\\ pred. \\\\\n\\hline\n$B$ & 8.90~GPa & 10.77~GPa & 6.36~GPa & 8.12~GPa & 0.99 & 0.99 \\\\\n$G$ & 7.29~GPa & 9.15~GPa & 4.76~GPa & 6.09~GPa & 0.99 & 0.99 \\\\\n$\\theta_{\\substack{\\scalebox{0.6}{D}}}$ & 76.13~K & 65.38~K & 49.63~K & 42.92~K & 0.97 & 0.97 \\\\\n\\end{tabular}}\n\\label{tab:art124:table3}\n\\end{table}\n\n\\boldsection{Comparison with experiments.}\nA comparison between calculated, predicted, and experimental results is presented in\nFigure~\\ref{fig:art124:figure7}, with relevant statistics summarized in Table~\\ref{tab:art124:table3}.\nData is considered for the bulk modulus $B$, shear modulus $G$, and (acoustic) Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{a}}}$\nfor 45 well-characterized materials with\ndiamond (SG\\# 227, {\\small AFLOW}\\ prototype \\texttt{A\\_cF8\\_227\\_a}),\nzincblende (SG\\# 216, \\texttt{AB\\_cF8\\_216\\_c\\_a}),\nrocksalt (SG\\# 225, \\texttt{AB\\_cF8\\_225\\_a\\_b}),\nand wurtzite (SG\\# 186, \\texttt{AB\\_hP4\\_186\\_b\\_b})\nstructures~\\cite{Morelli_Slack_2006,Semiconductors_BasicData_Springer}.\nExperimental $B$ and $G$ are compared to the $B_{\\substack{\\scalebox{0.6}{VRH}}}$ and $G_{\\substack{\\scalebox{0.6}{VRH}}}$ values predicted here, and\n$\\theta_{\\substack{\\scalebox{0.6}{a}}}$ is converted to the traditional Debye temperature $\\theta_{\\substack{\\scalebox{0.6}{D}}}=\\theta_{\\substack{\\scalebox{0.6}{a}}} n^{1\/3}$,\nwhere $n$ is the number of atoms in the unit cell.\nAll relevant values are listed in\nSupplementary Note 4 of Reference~\\cite{curtarolo:art124}.\n\nExcellent agreement is found between experimental and calculated values,\nbut more importantly, between experimental and predicted results.\nWith error metrics close to or under expected tolerances from the generalizability analysis,\nthe comparison highlights effective experimental confidence in the approach.\nThe experiments\/prediction validation is clearly the ultimate objective of the research presented here.\n\n\\subsection{Discussion}\nTraditional trial-and-error approaches have proven ineffective in discovering practical materials.\nComputational models developed with {\\small ML}\\ techniques may provide\na truly rational approach to materials design.\nTypical high-throughput {\\small DFT}\\ screenings involve exhaustive\ncalculations of all materials in the database, often without\nconsideration of previously calculated results.\nEven at high-throughput rates, an average {\\small DFT}\\ calculation of a medium\nsize structure (about 50 atoms per unit cell) takes about 1,170 CPU-hours of\ncalculations or about 37 hours on a 32-CPU cores node.\nHowever, in many cases, the desired range of values for the target property is known.\nFor instance,\nthe optimal band gap energy and thermal conductivity for optoelectronic applications\nwill depend on the power and voltage conditions of the device~\\cite{Figge_APL_2009,Zhou_JACerS_2016}.\nSuch cases offer an opportunity to leverage previous results and savvy {\\small ML}\\ models,\nsuch as those developed in this work, for rapid pre-screening of potential materials.\nResearchers can quickly narrow the list of candidate materials and avoid many extraneous\n{\\small DFT}\\ calculations---saving money, time, and computational resources.\nThis approach takes full advantage of previously calculated results,\ncontinuously accelerating materials discovery.\nWith prediction rates of about 0.1 seconds per material, the same 32-CPU cores node can screen\nover 28 million material candidates per day with this framework.\n\nFurthermore, interaction diagrams as depicted in Figure~\\ref{fig:art124:figure5} offer a pathway to design\nmaterials that meet certain constraints and requirements.\nFor example, substantial differences in thermal expansion coefficients among the materials used\nin high-power, high-frequency optoelectronic applications leads to bending and cracking of the structure\nduring the growth process~\\cite{Figge_APL_2009,Zhou_JACerS_2016}.\nNot only would this work-flow facilitate the search for semiconductors with large band gap energies,\nhigh Debye temperatures (thermal conductivity),\nbut also materials with similar thermal expansion coefficients.\n\nWhile the models themselves demonstrate excellent predictive power with minor deviations, outlier analysis reveals\ntheoretical structures to be among the worst offenders.\nThis is not surprising, as the true stability conditions (\\nobreak\\mbox{\\it e.g.}, high-pressure\/high-temperature) have yet\nto be determined, if they exist at all.\nThe {\\small ICSD}\\ estimates that structures for over 7,000 materials (or roughly 4\\%) come\nfrom calculations rather than actual experiment.\nSuch discoveries exemplify yet another application for {\\small ML}\\ modeling, rapid\/robust curation of large datasets.\n\nTo improve large-scale high-throughput computational screening for the identification\nof materials with desired properties, fast and accurate data mining approaches\nshould be incorporated into the standard work-flow.\nIn this work, we developed a universal {\\small QMSPR}\\ framework for predicting electronic\nproperties of inorganic materials.\nIts effectiveness is validated through the prediction of eight key materials properties\nfor stoichiometric inorganic crystalline materials, including\nthe metal\/insulator classification,\nband gap energy, bulk and shear moduli, Debye temperature, heat capacity (at constant\npressure and volume), and thermal expansion coefficient.\nIts applicability extends to all 230 space groups and the vast majority of\nelements in the periodic table.\nAll models are freely available at \\href{http:\/\/aflow.org\/aflow-ml}{aflow.org\/aflow-ml}.\n\n\\subsection{Methods}\n\\boldsection{Data preparation.}\nTwo independent datasets were prepared for the creation and validation of the {\\small ML}\\ models.\nThe training set includes\nelectronic~\\cite{aflowlibPAPER,aflowPAPER,aflowBZ,curtarolo:art67,monsterPGM,curtarolo:art49}\nand thermomechanical properties~\\cite{curtarolo:art96, curtarolo:art115} for a broad diversity of\ncompounds already characterized in the {\\small AFLOW}\\ database.\nThis set is used to build and analyze the {\\small ML}\\ models, one model per property.\nThe constructed thermomechanical models are then employed to make predictions of previously uncharacterized compounds in the {\\small AFLOW}\\ database.\nBased on these predictions and consideration of computational cost, several compounds are selected to validate the models' predictive\npower.\nThese compounds and their newly computed properties define the test set.\nThe compounds used in both datasets are specified in\nSupplementary Notes 2 and 3 of Reference~\\cite{curtarolo:art124}, respectively.\n\n\\boldsection{Training set.}\n{\\bf I.}\nBand gap energy data for 49,934 materials were extracted from the {\\small AFLOW}\\\nrepository~\\cite{aflowlibPAPER,aflowPAPER,aflowBZ,curtarolo:art67,monsterPGM,curtarolo:art49}, representing approximately\n60\\% of the known stoichiometric inorganic crystalline materials listed in the\nInorganic Crystal Structure Database ({\\small ICSD})~\\cite{ICSD,ICSD3}.\nWhile these band gap energies are generally underestimated with respect to experimental\nvalues~\\cite{Perdew_IJQC_1985}, {\\small DFT}+$U$ is robust enough to\ndifferentiate between metallic (no $E_{\\substack{\\scalebox{0.6}{BG}}}$) and insulating $\\left(E_{\\substack{\\scalebox{0.6}{BG}}}\\!>\\!0\\right)$ systems~\\cite{curtarolo:art104}.\nAdditionally, errors in band gap energy prediction are typically systematic.\nTherefore, the band gap energy values can be corrected \\textit{ad-hoc} with fitting\nschemes~\\cite{Yazyev_PRB_2012,Zheng_PRL_2011}.\nPrior to model development, both {\\small ICSD}\\ and {\\small AFLOW}\\ data were curated:\nduplicate entries, erroneous structures, and ill-converged calculations were corrected or removed.\nNoble gases crystals are not considered.\nThe final dataset consists of \\PLMFelectronicTotal\\ unique materials (\\PLMFmetalTotal\\ with no $E_{\\substack{\\scalebox{0.6}{BG}}}$\nand \\PLMFinsulatorTotal\\ with $E_{\\substack{\\scalebox{0.6}{BG}}}\\!>\\!0$),\ncovering the seven lattice systems, 230 space groups, and 83 elements\n(H-Pu, excluding noble gases, Fr, Ra, Np, At, and Po).\nAll referenced {\\small DFT}\\ calculations were performed with the Generalized Gradient Approximation\n({\\small GGA}) {\\small PBE}~\\cite{PBE}\nexchange-correlation functional and projector-augmented wavefunction ({\\small PAW})\npotentials~\\cite{PAW,kresse_vasp_paw} according to the\n{\\small AFLOW}\\ Standard for High-Throughput (HT) Computing~\\cite{curtarolo:art104}.\nThe Standard ensures reproducibility of the data, and provides visibility\/reasoning for any parameters\nset in the calculation, such as accuracy thresholds, calculation\npathways, and mesh dimensions.\n{\\bf II.}\nThermomechanical properties data for just over 3,000 materials were extracted from the {\\small AFLOW}\\\nrepository~\\cite{curtarolo:art115}.\nThese properties include the bulk modulus, shear modulus, Debye temperature, heat capacity at constant pressure,\nheat capacity at constant volume, and thermal expansion coefficient, and were\ncalculated using the {\\small AEL}-{\\small AGL}\\ integrated framework~\\cite{curtarolo:art96, curtarolo:art115}.\nThe {\\small AEL}\\ ({\\small AFLOW}\\ Elasticity Library)\nmethod~\\cite{curtarolo:art115} applies a set of independent normal and shear strains to the structure, and then fits the calculated stress\ntensors to obtain the elastic constants~\\cite{curtarolo:art100}.\nThese can then be used to calculate the elastic moduli in\nthe Voigt and Reuss approximations, as well as the Voigt-Reuss-Hill ({\\small VRH}) averages which are the values of the bulk and\nshear moduli modeled in this work.\nThe {\\small AGL}\\ ({\\small AFLOW}\\ {\\small GIBBS}\\ Library) method~\\cite{curtarolo:art96}\nfits the energies from a set of isotropically\ncompressed and expanded volumes of a structure to a quasiharmonic Debye-Gr{\\\"u}neisen model~\\cite{Blanco_CPC_GIBBS_2004}\nto obtain thermomechanical\nproperties, including the bulk modulus, Debye temperature, heat capacity, and thermal expansion coefficient.\n{\\small AGL}\\ has been\ncombined with {\\small AEL}\\ in a single workflow, so that it can utilize the Poisson ratios obtained from {\\small AEL}\\ to improve the\naccuracy of the thermal properties predictions~\\cite{curtarolo:art115}.\nAfter a similar curation of ill-converged calculations, the final dataset consists of\n\\PLMFthermoTrainingTotal\\ materials.\nIt covers the seven lattice systems, includes unary, binary, and ternary compounds, and\nspans broad ranges of each thermomechanical property, including\nhigh thermal conductivity systems such as C ({\\small ICSD}\\ \\#182729), BN ({\\small ICSD}\\ \\#162874), BC$_{5}$ ({\\small ICSD}\\ \\#166554),\nCN$_{2}$ ({\\small ICSD}\\ \\#247678), MnB$_{2}$ ({\\small ICSD}\\ \\#187733), and SiC ({\\small ICSD}\\ \\#164973), as well as\nlow thermal conductivity systems such as Hg$_{33}$(Rb,K)$_{3}$ ({\\small ICSD}\\ \\#410567 and \\#410566),\nCs$_{6}$Hg$_{40}$ ({\\small ICSD}\\ \\#240038), Ca$_{16}$Hg$_{36}$ ({\\small ICSD}\\ \\#107690), CrTe ({\\small ICSD}\\ \\#181056),\nand Cs ({\\small ICSD}\\ \\#426937).\nMany of these systems additionally exhibit extreme values of the bulk and shear moduli,\nsuch as C (high bulk and shear moduli) and Cs (low bulk and shear moduli).\nInteresting systems such as\nRuC ({\\small ICSD}\\ \\#183169) and NbC ({\\small ICSD}\\ \\#189090)\nwith a high bulk modulus ($B_{\\substack{\\scalebox{0.6}{VRH}}}$ = 317.92 GPa, 263.75 GPa) but\nlow shear modulus ($G_{\\substack{\\scalebox{0.6}{VRH}}}$ = 16.11 GPa, 31.86 GPa)\nalso populate the set.\n\n\\boldsection{Test set.}\nWhile nearly all {\\small ICSD}\\ compounds are characterized electronically within the {\\small AFLOW}\\ database,\nmost have not been characterized thermomechanically due to the added computational cost.\nThis presented an opportunity to validate the {\\small ML}\\ models.\nOf the remaining compounds, several were prioritized for immediate characterization via\nthe {\\small AEL}-{\\small AGL}\\ integrated framework~\\cite{curtarolo:art96, curtarolo:art115}.\nIn particular, focus was placed on systems predicted to have a large bulk modulus, as this property\nis expected to scale well with the other aforementioned thermomechanical\nproperties~\\cite{curtarolo:art96, curtarolo:art115}.\nThe set also includes various other small cell, high symmetry systems expected to span the full\napplicability domains of the models.\nThis effort resulted in the characterization of \\PLMFthermoTestTotal\\ additional compounds.\n\n\\boldsection{Data availability.}\nAll the \\nobreak\\mbox{\\it ab-initio}\\ data are freely available to the public as\npart of the {\\small AFLOW}\\ online repository and can be accessed through {\\sf \\AFLOW.org}\\\nfollowing the {\\small REST-API}\\ interface~\\cite{aflowPAPER}.\n\\clearpage\n\\chapter{Applications}\n\\section{Materials Cartography: Representing and Mining Materials Space Using Structural and Electronic Fingerprints}\n\\label{sec:art094}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art94},\nwhich was awarded with ACS Editors' Choice.\nAuthor contributions are as follows:\nStefano Curtarolo and Alexander Tropsha designed the study.\nOlexandr Isayev and Denis Fourches developed the fingerprinting and cartography methods.\nEugene N. Muratov adapted the SiRMS method for materials.\nCorey Oses and Kevin M. Rasch prepared the data and worked with the {\\sf \\AFLOW.org}\\ database.\nAll authors discussed the results and their implications and contributed to the paper.\n\n\\subsection{Introduction}\nDesigning materials with desired physical and chemical properties is recognized as an\noutstanding challenge in materials research~\\cite{Rajan_materialstoday_2005,nmatHT,Potyrailo_ACSCombSci_2011}.\nMaterial properties directly depend on a large number of key variables, often making the property prediction complex.\nThese variables include constitutive elements, crystal forms, and geometrical and electronic characteristics; among others.\nThe rapid growth of materials research has led to the accumulation of vast amounts of data.\nFor example, the Inorganic Crystal Structure Database ({\\small ICSD}) includes more than 170,000 entries~\\cite{ICSD}.\nExperimental data are also included in other databases, such as MatWeb~\\cite{MatWeb} and MatBase~\\cite{Matbase}.\nIn addition, there are several large databases such as the {\\sf \\AFLOW.org}\\ repository~\\cite{aflowBZ,aflowSCINT},\nthe Materials Project~\\cite{APL_Mater_Jain2013},\nand the Harvard Clean Energy Project~\\cite{Hachmann_JPCL_2011,Hachmann_EES_2014}\nthat contain thousands of unique materials and their theoretically calculated properties.\nThese properties include electronic structure profiles estimated with quantum mechanical methods.\nThe latter databases have great potential to serve as a source of novel functional materials.\nPromising candidates from these databases may in turn be selected for experimental\nconfirmation using rational design approaches~\\cite{MGI}.\n\nThe rapidly growing compendium of experimental and theoretical materials data offers\na unique opportunity for scientific discovery.\nSpecialized data mining and data visualization methods are being developed within\nthe nascent field of materials\ninformatics~\\cite{Rajan_materialstoday_2005,Suh_MST_2009,Olivares-Amaya_EES_2011,Potyrailo_ACSCombSci_2011,nmatHT,Schuett_PRB_2014,Seko_PRB_2014}.\nSimilar approaches have been used extensively in cheminformatics with resounding success.\nFor example, in many cases, these approaches have served to help identify and design\nsmall organic molecules with desired biological activity and acceptable\nenvironmental\/human-health safety profiles~\\cite{Laggner_NCB_2012,Besnard_Nature_2012,Cherkasov_JMC_2013,Lusci_JCIM_2013}.\nApplication of cheminformatics approaches to materials science would allow researchers to\n{\\bf i.} define, visualize, and navigate through materials space,\n{\\bf ii.} analyze and model structural and electronic characteristics of materials\nwith regard to a particular physical or chemical property, and\n{\\bf iii.} employ predictive materials informatics models to forecast the experimental properties of\n{\\it de novo} designed or untested materials.\nSuch rational design approaches in materials science constitute a rapidly growing\nfield~\\cite{Olivares-Amaya_EES_2011,Balachandran_PRSA_2011,Kong_JCIM_2012,Balachandran_ActaCristB_2012,Srinivasan_MAT_2013,Schuett_PRB_2014,Seko_PRB_2014,Broderick_APL_2014,Dey_CMS_2014}.\n\nHerein, we introduce a novel materials fingerprinting approach.\nWe combine this with graph theory, similarity searches, and machine learning algorithms.\nThis enables the unique characterization, comparison, visualization, and design of materials.\nWe introduce the concept and describe the development of materials fingerprints that encode\nmaterials' band structures, density of states ({\\small DOS}), crystallographic, and constitutional information.\nWe employ materials fingerprints to visualize this territory via advancing the new concept of ``{\\it materials cartography}''.\nWe show this technology identifies clusters of materials with similar properties.\nFinally, we develop Quantitative Materials Structure-Property Relationship ({\\small QMSPR}) models\nthat rely on these materials fingerprints.\nWe then employ these models to discover novel materials with desired properties that\nlurk within the materials databases.\n\n\\subsection{Methods}\n\\label{subsec:art094:methods}\n\n\\subsubsection{{\\sf \\AFLOW.org}\\ repository and data}\nThe {\\sf \\AFLOW.org}\\ repository of density functional theory ({\\small DFT}) calculations is managed\nby the software package {\\small AFLOW}~\\cite{aflowPAPER,aflowlibPAPER}.\nAt the time of the study, the {\\sf \\AFLOW.org}\\ database included the results of calculations\ncharacterizing over 20,000 crystals, but has since grown to include 50,000 entries ---\nrepresenting about a third of the contents of the {\\small ICSD}~\\cite{ICSD}.\nOf the characterized systems, roughly half are metallic and half are insulating.\n{\\small AFLOW}\\ leverages the {\\small VASP}\\ Package~\\cite{vasp_cms1996} to calculate the total energy\nof a given crystal structure with {\\small PAW}\\ pseudopotentials~\\cite{PAW} and the {\\small PBE}~\\cite{PBE} exchange-correlation functional.\nThe entries of the repositories have been described previously~\\cite{aflowBZ,aflowlibPAPER,aflowAPI}.\n\n\\subsubsection{Data set of superconducting materials}\nWe have compiled experimental data for superconductivity critical temperatures,\n$T_{\\mathrm{c}}$, for more than 700 records from the Handbook of Superconductivity~\\cite{Poole_Superconductivity_2000} and the\nCRC Handbook of Chemistry and Physics~\\cite{Lide_CRC_2004}, as well as the SuperCon Database~\\cite{SuperCon}.\nAs we have shown recently~\\cite{Fourches_JCIM_2010}, data curation is a necessary\nstep for any Quantitative Structure-Property Relationship ({\\small QSAR}) modeling.\nIn the compiled data set, several $T_{\\mathrm{c}}$ values have been measured under strained conditions,\nsuch as different pressures and magnetic fields.\nWe have only kept records taken under standard pressure and with no external magnetic fields.\nFor materials with variations in reported $T_{\\mathrm{c}}$ values in excess of 4~K,\noriginal references were revisited and records have been discarded when no reliable information was available.\n$T_{\\mathrm{c}}$ values with a variation of less than 3~K have been averaged.\nOf the remaining 465 materials ($T_{\\mathrm{c}}$ range of 0.1-133~K), most records show\na variability in $T_{\\mathrm{c}}$ of $\\pm$1~K between different sources.\nSuch a level of variability would be extremely influential in materials with\nlow $T_{\\mathrm{c}}$ ($T_{\\mathrm{c}}\\!<\\!1$~K) because we have used the decimal\nlogarithm of the experimentally measured critical temperature ($\\log(T_{\\mathrm{c}})$) as our target property.\n\nTo appropriately capture information inherent to materials over the full range of\n$T_{\\mathrm{c}}$, we have constructed two data sets for the development of three models.\nThe {\\bf continuous model} serves to predict $T_{\\mathrm{c}}$ and utilizes\nrecords excluding materials with $T_{\\mathrm{c}}$ values less than 2~K.\nThis data set consists of 295 unique materials with a $\\log(T_{\\mathrm{c}})$ range of 0.30-2.12.\nThe {\\bf classification model} serves to predict the position of $T_{\\mathrm{c}}$\n(above\/below) with respect to the threshold $T_{\\mathrm{thr}}$\n(unbiasedly set to 20~K as observed in Figure~\\ref{fig:art094:bands}(e), see the \\nameref{subsec:art094:results} section).\nIt utilizes records incorporating the aforementioned excluded materials,\nas well as lanthanum cuprate (La$_2$CuO$_4$, {\\small ICSD}\\ \\#19003).\nLanthanum cuprate had been previously discarded for high variability\n($T_{\\mathrm{c}}$ = 21-39~K), but now satisfies the classification criteria.\nThis data set consists of 464 materials (29 with $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\nand 435 with $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$).\nFinally, the {\\bf structural model} serves to identify geometrical components that most\ninfluence $T_{\\mathrm{c}}$. It utilizes the same data set as the continuous model.\n\n\\subsubsection{Materials fingerprints}\nFollowing the central paradigms of structure-property relationships, we assume that\n{\\bf i.} properties of materials are a direct function of their structure and\n{\\bf ii.} materials with similar structures (as determined by constitutional,\ntopological, spatial, and electronic characteristics) are likely to have similar physical and chemical properties.\n\nThus, encoding material characteristics in the form of numerical arrays,\nnamely descriptors~\\cite{nmatHT,Schuett_PRB_2014} or\n``{\\it fingerprints}''~\\cite{Valle_ActaCristA_2010}, enables the use of classical cheminformatics and\nmachine-learning approaches to mine, visualize, and model any set of materials.\nWe have encoded the electronic structure diagram for each material as two distinct types of arrays\n(Figure~\\ref{fig:art094:fingerprints_construction}):\na {\\it symmetry-dependent fingerprint} (band structure based ``B-fingerprint'') and a\n{\\it symmetry-independent fingerprint} ({\\small DOS}\\ based ``D-fingerprint'').\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig101}\n\\mycaption[Construction of materials fingerprints from the band structure and {\\small DOS}.]\n{For simplicity, we illustrate the idea of B-fingerprints with only 8 bins.}\n\\label{fig:art094:fingerprints_construction}\n\\end{figure}\n\n\\boldsection{B-fingerprint.} Along every special high-symmetry point of the Brillouin zone ({\\small BZ}),\nthe energy diagram has been discretized into 32 bins to serve as our fingerprint array.\nEach {\\small BZ}\\ has a unique set of high-symmetry points~\\cite{aflowBZ}.\nThe comparison set of high-symmetry points belonging to a single {\\small BZ}\\ type is considered symmetry-dependent.\nTo name a few examples, the Brillouin zone path of a cubic lattice\n($\\Gamma\u2013X\u2013M\u2013\\Gamma\u2013R\u2013X\\!\\!\\mid\\!\\! M\u2013R$) is encoded with just four points ($\\Gamma, M, R, X$),\ngiving rise to a fingerprint array of length 128.\nThe body-centered orthorhombic lattice is more complex~\\cite{aflowBZ,aflowSCINT}\n($\\Gamma\u2013X\u2013L\u2013T\u2013W\u2013R\u2013X_1\u2013Z\u2013\\Gamma\u2013Y\u2013S\u2013W\\!\\mid\\!L_1\u2013Y\\!\\mid\\!Y_1\u2013Z$)\nand is represented by 13 points ($\\Gamma, L, L_1, L_2, R, S, T, W, X, X_1, Y, Y_1, Z)$,\ngiving a fingerprint array of length 416.\nConversely, the comparison of identical {\\bf k}-points not specifically belonging to any {\\small BZ}\\\nis always possible when only restricted to $\\Gamma$.\nConsequently, we limit our models to the $\\Gamma$ point B-fingerprint in the present work.\n\n\\boldsection{D-fingerprint.} A similar approach can be taken for the {\\small DOS}\\ diagrams,\nwhich are sampled in 256 bins (from min to max) and the magnitude of each bin is discretized in 32 bits.\nTherefore, the D-fingerprint is a total of 1024 bytes.\nOwing to the complexity and limitations of the symmetry-dependent B-fingerprints,\nwe have only generated symmetry-independent D-fingerprints.\nThe length of these fingerprints is tunable depending on the objects, applications, and other factors.\nWe have carefully designed the domain space and length of these fingerprints to avoid\nthe issues of enhancing boundary effects or discarding important features.\n\n\\boldsection{SiRMS descriptors for materials.}\nTo characterize the structure of materials from several different perspectives,\nwe have developed descriptors similar to those used for small organic molecules\nthat can reflect their compositional, topological, and spatial (stereochemical) characteristics.\nClassical cheminformatics tools can only handle small organic molecules.\nTherefore, we have modified the Simplex (SiRMS) approach~\\cite{Kuzmin_JCAMD_2008}\nbased on our experience with mixtures~\\cite{Muratov_SC_2013,Muratov_MI_2012}\nin order to make this method suitable for computing descriptors for materials.\n\nThe SiRMS approach~\\cite{Kuzmin_JCAMD_2008} characterizes small organic molecules\nby splitting them into multiple molecular fragments called simplexes.\nSimplexes are tetratomic fragments of fixed composition (1D), topology (2D), and chirality and symmetry (3D).\nThe occurrences of each of these fragments in a given compound are then counted.\nAs a result, each molecule of a given data set can be characterized by its SiRMS fragment profiles.\nThese profiles take into account atom types, connectivity, \\nobreak\\mbox{\\it etc.}~\\cite{Kuzmin_JCAMD_2008}.\nHere, we have adapted the SiRMS approach to describe materials with their fragmental compositions.\n\nEvery material is represented according to the structure of its crystal unit cell (Figure~\\ref{fig:art094:sirms_generation}).\nComputing SiRMS descriptors for materials is equivalent to the computation of\nSiRMS fragments for nonbonded molecular mixtures.\nBounded simplexes describe only a single component of the mixture.\nUnbounded simplexes could either belong to a single component, or could span up to four components of the unit cell.\nA special label is used during descriptor generation to distinguish ``mixture''\nsimplexes (belonging to different molecular moieties) from those incorporating elements from a single compound~\\cite{Muratov_MI_2012}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig102}\n\\mycaption{Generation of SiRMS descriptors for materials.}\n\\label{fig:art094:sirms_generation}\n\\end{figure}\n\nThus, the structure of every material is characterized by both bounded and unbounded\nSiRMS descriptors as illustrated in Figure~\\ref{fig:art094:sirms_generation}.\nThe descriptor value of a given simplex fragment is equal to the number of its occurrences in the system.\nIn the case of materials, this value has been summed throughout all the constituents of a system;\ntaking into account their stoichiometric ratios and crystal lattices (see Figure~\\ref{fig:art094:sirms_generation}).\n``Mixture'' descriptors are weighted according to the smallest stoichiometric\nratio of constituents within this mixture, and added throughout all the mixtures in a system.\nAtoms in simplexes are differentiated according to their type (element) and partial charge.\nFor the latter, atoms are divided into six groups corresponding to their partial charge:\n$A\\!\\leq\\!-2\\!<\\!B\\!\\leq\\!-1\\!<\\!C\\!\\leq\\!0\\!<\\!D\\!\\leq\\!1\\!<\\!E\\!\\leq\\!2\\!<\\!F$.\nIn addition, we have developed a special differentiation of atoms in simplexes to account for their groups on the periodic table.\nThat is, all elements belonging to the same group are encoded by the same symbol.\n\n\\subsubsection{Network representation (materials cartograms)}\nTo represent the library of materials as a network, we considered each material, encoded by its fingerprints, as a node.\nEdges exist between nodes with similarities greater than or equal to certain thresholds.\nIn this study, we use fingerprint-based Tanimoto similarity and a threshold $S=0.7$.\nThis network representation of materials is defined as the graph $G(V,E)$, where $V=\\left\\{\\nu_1|\\nu_2\\in L\\right\\}$ and\n$E\\!=\\!\\left\\{(\\nu_1,\\nu_2)\\mid\\mathrm{sim}(\\nu_1,\\nu_2)\\geq T\\right\\}$.\nHere, $L$ denotes a materials library, $\\mathrm{sim}(\\nu_1,\\nu_2)$\ndenotes a similarity between materials $\\nu_1$ and $\\nu_2$, and $T$ denotes a similarity threshold.\n\nTo examine if the materials networks are scale-free, we analyzed the degree distributions of the networks.\nNetworks are considered scale-free if the distribution of vertex degrees of the nodes follows the power law:\n$p(x)=kx^{-\\alpha}$ where $k$ is the normalization constant, and $\\alpha$ is the exponent.\nThe materials networks have been visualized using the Gephi package~\\cite{Bastian_ICWSM_2009}.\nThe ForceAtlas 2 algorithm~\\cite{Jacomy_PLoS_2014}, a type of force-directed layout\nalgorithm, has been used for the graph layout.\nA force-directed layout algorithm considers a force between any two nodes,\nand minimizes the ``energy'' of the system by moving the nodes and changing the forces between them.\nThe algorithm guarantees that the topological similarity among nodes determines their vicinity, leading to accurate and\nvisually-informative representations of materials space.\n\n\\subsection{Results and discussion}\n\\label{subsec:art094:results}\n\n\\subsubsection{Similarity search in materials space}\nIn the first phase of this study, the optimized geometries, symmetries,\nband structures, and {\\small DOS}{}s available in the {\\sf \\AFLOW.org}\\ repository were converted\ninto fingerprints, or arrays of numbers.\n\nWe encoded the electronic structure diagram for each material as two distinct types of\nfingerprints (Figure~\\ref{fig:art094:fingerprints_construction}):\nband structure symmetry-dependent fingerprints (B-fingerprints) and\n{\\small DOS}\\ symmetry-independent fingerprints (D-fingerprints).\nThe B-fingerprint is defined as a collated digitalized histogram of energy eigenvalues\nsampled at the high-symmetry reciprocal points with 32 bins.\nThe D-fingerprint is a string containing 256 4-byte real numbers,\neach characterizing the strength of the {\\small DOS}\\ in one of the 256 bins dividing the [-10, 10]~eV interval.\nMore details are in the \\nameref{subsec:art094:methods} section.\n\nThis unique, condensed representation of materials enabled the use of cheminformatics methods,\nsuch as similarity searches, to retrieve materials with similar properties but different compositions from the {\\sf \\AFLOW.org}\\ database.\nAs an added benefit, our similarity search can also quickly find duplicate records.\nFor example, we have identified several barium titanate (BaTiO$_3$) records with identical fingerprints\n({\\small ICSD}\\ \\#15453, \\#27970, \\#6102, and \\#27965 in the {\\sf \\AFLOW.org}\\ database).\nThus, fingerprint representation afforded rapid identification of duplicates,\nwhich is the standard first step in our cheminformatics data curation workflow~\\cite{Fourches_JCIM_2010}.\nIt is well known that standard {\\small DFT}\\ has severe limitations in the description of excited states, and needs to be substituted\nwith more advanced approaches to characterize semiconductors and\ninsulators~\\cite{Hedin_GW_1965,GW,Heyd2003,Liechtenstein1995,Cococcioni_reviewLDAU_2014}.\nHowever, there is a general trend of {\\small DFT}\\ errors being comparable in similar classes of systems.\nThese errors may thus be considered ``systematic'', and are irrelevant when one seeks only similarities between materials.\n\nThe first test case is gallium arsenide, GaAs ({\\small ICSD}\\ \\#41674),\na very important material for electronics~\\cite{INSPEC_PGA_1986} in the {\\sf \\AFLOW.org}\\ database.\nGaAs is taken as the reference material, and the remaining 20,000+ materials from the\n{\\sf \\AFLOW.org}\\ database are taken as the virtual screening library.\nThe pairwise similarity between GaAs and any of the materials represented by our D-fingerprints\nis computed using the Tanimoto similarity coefficient ($S$)~\\cite{Maggiora_JMC_2014}.\nThe top five materials (GaP, Si, SnP, GeAs, InTe) retrieved show very high similarity ($S\\!>\\!0.8$)\nto GaAs, and all five are known to be semiconductor materials~\\cite{Lide_CRC_2004,Littlewood_CRSSMS_1983,Madelung_Semiconductors_2004}.\n\nIn addition, we have searched the {\\sf \\AFLOW.org}\\ database for materials similar to BaTiO$_3$\nwith the perovskite structure ({\\small ICSD}\\ \\#15453) using B-fingerprints.\nBaTiO$_3$ is widely used as a ferroelectric ceramic or piezoelectric~\\cite{Bhalla_MRI_2000}.\nOut of the six most similar materials with $S>0.8$, five (BiOBr, SrZrO$_3$, BaZrO$_3$, KTaO$_3$ and KNbO$_3$)\nare well known for their optical properties~\\cite{Rabe_Ferroelectrics_2010}.\nThe remaining material, cubic YbSe ({\\small ICSD}\\ \\#33675), is largely unexplored.\nOne can therefore formulate a testable hypothesis suggesting that this material may be ferroelectric or piezoelectric.\n\nWe also investigated the challenging case of topological insulators.\nThey form a rare group of insulating materials with conducting surface-segregated states (or interfaces)~\\cite{nmatTI}\narising from a combination of spin-orbit coupling and time-reversal symmetry~\\cite{RevModPhys.82.3045}.\nAlthough {\\small DFT}\\ calculations conducted for materials in the {\\sf \\AFLOW.org}\\ repository do not\nincorporate spin-orbit coupling for the most part~\\cite{nmatTI}, various topological insulators show exceptionally\nhigh band-structure similarities --- validating the B-fingerprints scheme.\nThe two materials most similar to Sb$_2$Te$_3$~\\cite{RevModPhys.82.3045} (based on B-fingerprints)\nwith $S\\!>\\!0.9$ are Bi$_2$Te$_3$~\\cite{Chen09science,zhang_PRL_2009} and Sb$_2$Te$_2$Se~\\cite{Xu2010arxiv1007}.\nFive out of six materials most similar to Bi$_2$Te$_2$Se~\\cite{Xu2010arxiv1007,Arakane2010NC}\nare also known topological insulators: Bi$_2$Te$_2$S, Bi$_2$Te$_3$, Sb$_2$Te$_2$Se,\nGeBi$_2$Te$_4$~\\cite{Xu2010arxiv1007}, and Sb$_2$Se$_2$Te~\\cite{nmatTI,Zhang_Nat.Phys._2009}.\n\nThese examples demonstrate proof of concept and illustrate the power of simple yet uncommon\nfingerprint-based similarity searches for rapid and effective identification of\nmaterials with similar properties in large databases.\nThey also illuminate the intricate link between structures and properties of materials by demonstrating\nthat similar materials (as defined by their fingerprint similarity) have similar\nproperties (such as being ferroelectric or insulating).\nThis observation sets the stage for building and exploring {\\small QMSPR}\\ models; as discussed in the following sections.\n\n\\subsubsection{Visualizing and exploring materials space}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig103}\n\\mycaption[Materials cartograms with D- (top) and B-fingerprint network representations (bottom).]\n{({\\bf a}) D-fingerprint network representation of materials. Materials are color-coded\naccording to the number of atoms per unit cell.\nRegions corresponding to pure elements, binary, ternary and quaternary compounds are outlined.\n({\\bf b}) Distribution of connectivity within the network.\n({\\bf c}) Mapping band gaps of materials. Points colored in deep blue are metals;\ninsulators are color-coded according to the band gap value. Four large communities are outlined.\n({\\bf d}) Mapping the superconductivity critical temperature, $T_{\\mathrm{c}}$, with relevant regions outlined.}\n\\label{fig:art094:cartograms}\n\\end{figure}\n\nThe use of fingerprint representation and similarity concepts led us to develop the materials network.\nCompounds are mapped as nodes.\nWe use the ``{\\it force directed graph drawing}'' algorithm~\\cite{Herman_IEEEtvcg_2000}\nin which positions of the compounds are initially taken randomly.\nThere is a force between the nodes: a repulsive Coulomb component and an optional\nattractive contribution with a spring constant equal to the Tanimoto coefficient\nbetween D-fingerprints (effective when $S\\ge0.7$).\nTwo nodes are connected only when the coefficient is greater than or equal to the threshold.\nThe model is equilibrated through a series of heating and quenching steps.\nFigure~\\ref{fig:art094:cartograms}(a) shows the result in which we add\nBezier-curved lines depicting regions of accumulation.\nWe shall refer to this approach to visualizing and analyzing materials and their properties as ``{\\it materials cartography}''.\n\nThe network shown in Figure~\\ref{fig:art094:cartograms}(a) is color-coded according to overall complexity.\nPure systems, 79\\% of the total 246 unary nodes, are confined in a small, enclosed region.\nBinary nodes cover more configurational space, with 82\\% of the 3700+ binaries lying in a compact region.\nTernaries are scattered. They mostly populate the center of the space (91\\% of the 5300+ ternaries).\nQuaternaries and beyond are located at the top part of the network (92\\% of the 1080 nodes).\nThis region is the most distant from that of the unary nodes, which tends to be disconnected from the others.\nIndeed, overlap between binaries and ternaries is substantial.\nThe diversification of electronic properties and thickness of the compact envelope grows with structural complexity.\nOrphans are defined as nodes with a very low degree of connectivity: only the vertices (materials)\nconnected by edges are shown ($\\sim$39\\% of the database).\nInterestingly, of the 200 materials with connectivity smaller than 12,\nmost are La-based (36 bimetallic and 126 polymetallic) or Ce-based (10 nodes).\n\n\\begin{table}[tp]\\centering\n\\mycaption[Topological properties for constructed materials cartograms.]\n{In network theory, a ``component'' is a group of nodes that are all connected to each other.\nA ``giant component'' is a connected component of a given random graph that contains a constant\nfraction of the entire graph's vertices~\\cite{Chung_Complex_2006}.\nFigures in parenthesis are calculated by fitting only the asymptotic portion of the curve in Figure~\\ref{fig:art094:cartograms}(b).\n}\n\\vspace{3mm}\n\\begin{tabular}{l | r r}\n & D-fingerprints network & B-fingerprints network \\\\\n\\hline\ntotal number of cases & 17420 & 17420 \\\\\ngiant component & 10521 (60.4\\%) & 15535 (89.2\\%)\\\\\nedges & 466,000 & 564,000 \\\\\naverage degree & 88.60 & 72.59 \\\\\nnetwork diameter (edges) & 27 & 23 \\\\\npower law $\\gamma$ & 2.745 & 0.916 (2.04) \\\\\n\\end{tabular}\n\\label{tab:art094:cartograms}\n\\end{table}\n\nThe degree of connectivity is illustrated in Figure~\\ref{fig:art094:cartograms}(b).\nThe panel indicates the log-log distribution of connectivity across the sample set.\nThe red and blue points measure the D-fingerprints (Figure~\\ref{fig:art094:cartograms}(a))\nand B-fingerprints connectivity (Figure~\\ref{fig:art094:cartograms}(c)), respectively.\nTable~\\ref{tab:art094:cartograms} contains relevant statistical information about the cartograms.\nAlthough the power law distribution of Figure~\\ref{fig:art094:cartograms}(b) is typical of\nscale-free networks and similar to many networks examined in cheminformatics and\nbioinformatics~\\cite{Girvan_PNAS_2002,Newman_SiRev_2003,Yildirim_NB_2007}, in our case, connectivity differs.\nIn previous examples~\\cite{Girvan_PNAS_2002,Newman_SiRev_2003,Yildirim_NB_2007},\nmost of the nodes have only a few connections; with a small minority being highly\nconnected to a small set of ``hubs''~\\cite{Jeong_Nature_2000,Barabasi_Science_1999}.\nIn contrast, the {\\sf \\AFLOW.org}\\ database is highly heterogeneous:\nmost of the hubs' materials are concentrated along the long, narrow belt along the middle of the network.\nThe top 200 nodes (ranked by connectivity) are represented by 83 polymetallics\n(CoCrSi, Al$_2$Fe$_3$Si$_3$, Al$_8$Cr$_4$Y, \\nobreak\\mbox{\\it etc.}),\n102 bimetallics (Al$_3$Mo, As$_3$W$_2$, FeZn$_{13}$, \\nobreak\\mbox{\\it etc.}),\n14 common binary compounds (GeS, AsIn, \\nobreak\\mbox{\\it etc.}), and boron ({\\small ICSD}\\ \\#165132).\nThis is not entirely surprising, since these materials are well studied\nand represent the lion's share of the {\\small ICSD}\\ database.\nAl$_3$FeSi$_2$ ({\\small ICSD}\\ \\#79710), an uncommonly used material, has the highest connectivity of 946.\nMeanwhile, complex ceramics and exotic materials are relatively disconnected.\n\nA second network, built with B-fingerprints, is illustrated in Figure~\\ref{fig:art094:cartograms}(c).\nWhile this network preserves most of the topological features described\nin the D-fingerprint case (Figure~\\ref{fig:art094:cartograms}(a)), critical distinctions appear.\nThe B-fingerprint network separates metals from insulators.\nClustering and subsequent community analyses show four large groups of materials.\nGroup-A ($\\sim$3000 materials) consists predominately of insulating compounds (63\\%) and semiconductors (10\\%).\nGroup-B distinctly consists of compounds with polymetallic character (70\\% of $\\sim$2500 materials).\nIn contrast, Group-C includes $\\sim$500 zero band gap materials with nonmetal atoms,\nincluding halogenides, carbides, silicides, \\nobreak\\mbox{\\it etc.}\\\nLastly, Group-D has a mixed character with $\\sim$300 small band gap materials (below 1.5~eV);\nand $\\sim$500 semimetals and semiconductors.\n\nLithium scandium diphosphate, LiScP$_2$O$_7$ ({\\small ICSD}\\ \\#91496), has the highest connectivity\nof 746 in the B-fingerprint network.\nVery highly connected materials are nearly evenly distributed between Groups-A and -B,\nforming dense clusters within their centers.\nAs in the case of the D-fingerprint network, the connectivity distribution follows a power law\n(Figure~\\ref{fig:art094:cartograms}(b), see Table~\\ref{tab:art094:cartograms} for\nadditional statistics); indicating that this is a scale-free network.\n\nTo illustrate one possible application of the materials networks, we chose superconductivity ---\none of the most elusive challenges in solid-state physics.\nWe have compiled experimental data for 295 stoichiometric superconductors that\nare also available in the {\\sf \\AFLOW.org}\\ repository.\nAll materials in the data set are characterized with the fingerprints specified in\nthe \\nameref{subsec:art094:methods} section.\nThe data set includes both prominently high temperature superconducting materials\nsuch as layered cuprates, ferropnictides, iron arsenides-122, MgB$_2$; as well as more\nconventional compounds such as A15, ternary pnictides, \\nobreak\\mbox{\\it etc.}\\\nOur model does not consider the effect of phonons, which play a\ndominant role in many superconductors~\\cite{tinkham_superconductivity}.\nHigh-throughput parameterization of phonon spectra is still in its infancy~\\cite{curtarolo:Ru},\nand only recently have vibrational descriptors been\nadapted to large databases~\\cite{curtarolo:art96}.\nWe envision that future development of vibrational fingerprints\nfollowing these guidelines will capture similarities between\nknown, predicted, and verified superconductors (\\nobreak\\mbox{\\it i.e.},\nMgB$_2$ \\nobreak\\mbox{\\it vs.}\\ LiB$_2$~\\cite{curtarolo:art21,curtarolo:art26} and MgB$_2$ \\nobreak\\mbox{\\it vs.}\\\nFe-B compounds~\\cite{Kolmogorov_FeB_PRL2010,Gou_PRL_2013_FeB_superconductor}).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig104}\n\\mycaption[Comparison high-low $T_{\\mathrm{c}}$ aligned band structures and $T_{\\mathrm{c}}$ predictions.]\n{({\\bf a}) Band structure of Ba$_2$Ca$_2$Cu$_3$HgO$_8$ ($T_{\\mathrm{c}}=$133~K).\n({\\bf b}) Band structure of SrCuO$_2$ ({\\small ICSD}\\ \\#16217, $T_{\\mathrm{c}}=$91~K~\\cite{Takahashi_PSCC_1994_SrCuO2_superconductor}).\n({\\bf c}) Aligned B-fingerprints for the 15 materials with the highest and lowest $T_{\\mathrm{c}}$.\n({\\bf d}) Band structure of Nb$_2$Se$_3$ ({\\small ICSD}\\ \\#42981, $T_{\\mathrm{c}}=$0.4~K).\n({\\bf e}) Plot of the predicted \\nobreak\\mbox{\\it vs.}\\ experimental critical temperatures for the continuous model.\nMaterials are color-coded according to the classification model: solid\/open green (red) circles indicate\ncorrect\/incorrect predictions in $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\n($T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$), respectively.}\n\\label{fig:art094:bands}\n\\end{figure}\n\nAll materials are identified and marked on the B-fingerprint network, and are\ncolor-coded according to their critical temperature, $T_{\\mathrm{c}}$ (Figure~\\ref{fig:art094:cartograms}(d)).\nAll high-$T_{\\mathrm{c}}$ superconductors are localized in a relatively compact region.\nThe distribution is centered on a tight group of Ba$_2$Cu$_3X$O$_7$ compounds\n(the so-called Y123, where $X$= lanthanides).\nThe materials with the two highest $T_{\\mathrm{c}}$ values in our set are\nBa$_2$Ca$_2$Cu$_3$HgO$_8$ ({\\small ICSD}\\ \\#75730, $T_{\\mathrm{c}}=$133~K) and\nBa$_2$CaCu$_2$HgO$_6$ ({\\small ICSD}\\ \\#75725, $T_{\\mathrm{c}}=$125~K).\nTheir close grouping manifests a significant superconductivity hot-spot of materials with similar fingerprints.\nWe aligned the B-fingerprints for the 15 superconductors with the highest $T_{\\mathrm{c}}$\nvalues in Figure~\\ref{fig:art094:bands}(c).\n\nAll the top 15 high $T_{\\mathrm{c}}$ superconductors are layered cuprates,\nwhich have dominated high $T_{\\mathrm{c}}$ superconductor research since 1986~\\cite{Bednorz_ZPBCM_1986}.\nThese compounds are categorized as Charge-Transfer Mott Insulators (CTMI)~\\cite{Zaanen_PRL_1985}.\nThere are three distinct bands that are conserved for these structures around -6, -1, and 4~eV\nrelative to the Fermi energy at $\\Gamma$ (within the simple {\\small DFT}+$U$ description available in the {\\sf \\AFLOW.org}\\ repository,\nFigure~\\ref{fig:art094:bands}(c)).\nThese features are consistent with the three-band Hubbard-like picture characteristic of\nCTMIs~\\cite{Manske_Superconductors_2004,Emery_PRL_1987}.\n\nMeanwhile, the fingerprint distribution for the 15 materials with the lowest\n$T_{\\mathrm{c}}$ is random (Figure~\\ref{fig:art094:bands}(c)).\nThe importance of band structure features in superconductivity has long\nbeen recognized~\\cite{Zaanen_NPhys_2006,Micnas_RMP_1990,Orenstein_Science_2000}.\nThus, materials cartography based on the B-fingerprint network allows us to visualize this phenomenon concisely.\n\n\\subsubsection{Predictive QMSPR modeling}\nWe developed {\\small QMSPR}\\ models (continuous~\\cite{Bramer_PDM_2007}, classification, and structural)\nto compute superconducting properties of materials from their structural characteristics.\nTo achieve this objective, we compiled two superconductivity data sets consisting of\n{\\bf i.} 295 materials with continuous $T_{\\mathrm{c}}$ values ranging from 2~K to 133~K; and\n{\\bf ii.} 464 materials with binary $T_{\\mathrm{c}}$ values.\nThe models were generated with Random Forest (RF)~\\cite{Breiman_ML_2001} and\nPartial Least Squares (PLS)~\\cite{Wold_CILS_2001} techniques.\nThese used both B- and D-fingerprints, as well as Simplex (SiRMS)~\\cite{Kuzmin_JCAMD_2008} descriptors.\nThese fingerprints were adapted for materials modeling for the first time in this study\n(see the \\nameref{subsec:art094:methods} section).\nAdditionally, we incorporated atomic descriptors that differentiate\nby element, charge, and group within the periodic table.\nStatistical characteristics for all 464 materials used for the {\\small QMSPR}\\ analysis\nare reported in Tables~\\ref{tab:art094:continuous}-\\ref{tab:art094:fragments}.\n\nAttempts to develop {\\small QMSPR}\\ models using B- and D-fingerprints for both data sets were not satisfactory,\nindicating that our fingerprints, while effective in qualitative clustering,\ndo not contain enough information for quantitatively predicting target properties\n({\\small QMSPR}\\ model acceptance criteria has been discussed previously~\\cite{Tropsha_MI_2010}).\nThus, we employed more sophisticated chemical fragment descriptors,\nsuch as SiRMS~\\cite{Kuzmin_JCAMD_2008}, and adapted them for\nmaterials modeling (see the \\nameref{subsec:art094:methods} section).\n\n\\boldsection{Continuous model.}\nWe constructed a continuous model which serves to predict the value of\n$T_{\\mathrm{c}}$ with a consensus RF- and PLS-SiRMS approach.\nIt has a cross-validation determination coefficient of $Q^2=0.66$ (five-fold external CV;\nsee Table~\\ref{tab:art094:continuous}).\nFigure~\\ref{fig:art094:bands}(e) shows predicted \\nobreak\\mbox{\\it vs.}\\ experimental $T_{\\mathrm{c}}$\nvalues for the continuous model: all materials having\n$\\log(T_{\\mathrm{c}})$$\\leq$1.3 are scattered, but within the correct range.\nInterestingly, we notice that systems with\n$\\log(T_{\\mathrm{c}})$$\\geq$1.3 received higher accuracy, with the exceptions of\nMgB$_{2}$ ({\\small ICSD}\\ \\#26675), Nb$_{3}$Ge ({\\small ICSD}\\ \\#26573), Cu$_{1}$Nd$_{2}$O$_{4}$ ({\\small ICSD}\\ \\#4203),\nAs$_{2}$Fe$_{2}$Sr ({\\small ICSD}\\ \\#163208), Ba$_{2}$CuHgO$_{4}$ ({\\small ICSD}\\ \\#75720), and\nClHfN ({\\small ICSD}\\ \\#87795) (all highly underestimated).\nNot surprisingly MgB$_2$~\\cite{Buzea_SST_2001_MgB2} is an outlier in our statistics.\nThis is in agreement with the fact that to date\nno superconductor with an electronic structure similar to MgB$_2$ has been found.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical characteristics of the continuous {\\small QMSPR}\\ models for superconductivity.]\n{$Q^{2}$(ext) refers to the leave-one-out five-fold external cross-validation coefficient,\nRMSE refers to root-mean-square error,\nMAE refers to the mean absolute error,\nRF-SiRMS refers to the application of the Random Forest technique with Simplex descriptors,\nPLS-SiRMS refers to the application of the Partial Least Squares regression technique with Simplex descriptors,\nand consensus refers to the average of the RF-SiRMS and PLS-SiRMS results.}\n\\vspace{3mm}\n\\begin{tabular}{l | r r r r}\nmodel & $N$ & $Q^{2}$(ext) & RMSE & MAE \\\\\n\\hline\nRF-SiRMS & 295 & 0.64 & 0.24 & 0.18\\% \\\\\nPLS-SiRMS & 295 & 0.61 & 0.25 & 0.20\\% \\\\\nconsensus & 295 & 0.66 & 0.23 & 0.18\\% \\\\\n\\end{tabular}\n\\label{tab:art094:continuous}\n\\end{table}\n\n\\boldsection{Classification model.}\nBy observing the existence of the threshold $T_{\\mathrm{thr}}$=20~K ($\\log(T_{\\mathrm{thr}})$=1.3),\nwe developed a classification model.\nIt is based on the same RF-SiRMS technique, but it is strictly used to predict the position of\n$T_{\\mathrm{c}}$ with respect to the threshold, above or below.\nThe classification model has a balanced accuracy of 0.97 with five-fold external CV analysis.\nThe type of points in Figure~\\ref{fig:art094:bands}(e) illustrates the classification model outcome:\nsolid\/open green (red) circles for correct\/incorrect\npredictions in $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ ($T_{\\mathrm{c}}\\leq T_{\\mathrm{thr}}$), respectively.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Statistical characteristics of the classification {\\small QMSPR}\\ models for superconductivity.]\n{AD refers to applicability domain~\\cite{Tropsha_CPD_2007}.\nAccuracy is determined by the ratio of correct predictions to the total number of predictions,\nsensitivity is determined by the ratio of correctly predicted $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\nto the number of empirical $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$,\nspecificity is determined by the ratio of correctly predicted $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$\nto the number of empirical $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$,\nCCR (correct classification rate) is the average of the sensitivity and the specificity,\nand coverage is determined by the ratio of the total number of predictions to the total number of cases.}\n\\vspace{3mm}\n\\begin{tabular}{l | r r}\n & no AD & with AD \\\\\n\\hline\ntotal number of cases & 464 & 464\\\\\ntotal number of predictions & 464 & 451\\\\\nnumber of correct predictions & 452 & 446\\\\\nnumber of wrong predictions & 12 & 5\\\\\nnumber of empirical $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ & 29 & 22\\\\\nnumber of empirical $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ & 435 & 429\\\\\nnumber of correctly predicted $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ & 19 & 17\\\\\nnumber of correctly predicted $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ & 433 & 429\\\\\nnumber of incorrectly predicted $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ & 2 & 0\\\\\nnumber of incorrectly predicted $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ & 10 & 5\\\\\n$T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ prediction value & 0.90 & 1.00\\\\\n$T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ prediction value & 0.98 & 0.99\\\\\naccuracy & 0.97 & 0.99\\\\\nsensitivity & 0.66 & 0.77\\\\\nspecificity & 1.00 & 1.00\\\\\nCCR & 0.83 & 0.89\\\\\ncoverage & 1.00 & 0.97\\\\\n\\end{tabular}\n\\label{tab:art094:classification}\n\\end{table}\n\nFor $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$ and $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$,\naccuracies of prediction are 98\\% and 90\\% (cumulative 94\\%).\n(Figure~\\ref{fig:art094:bands}(e), see Table~\\ref{tab:art094:classification} for additional statistics).\nAmong the 464 materials, ten systems with experimental $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ are predicted to have\n$T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$)\n[FeLaAsO ({\\small ICSD}\\ \\#163496), AsFeO$_{3}$Sr$_{2}$V ({\\small ICSD}\\ \\#165984), As$_{2}$EuFe$_{2}$ ({\\small ICSD}\\ \\#163210),\nAs$_{2}$Fe$_{2}$Sr, CuNd$_{2}$O$_{4}$ ({\\small ICSD}\\ \\#86754), As$_{2}$BaFe$_{2}$\n({\\small ICSD}\\ \\#166018), MgB$_{2}$, ClHfN, La$_{2}$CuO$_{4}$, and Nb$_{3}$Ge].\nOnly two with experimental $T_{\\mathrm{c}}\\!\\leq\\!T_{\\mathrm{thr}}$\nare predicted with $T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$\n(AsFeLi ({\\small ICSD}\\ \\#168206), As$_{2}$CaFe$_{2}$ ({\\small ICSD}\\ \\#166016)).\nOwing to the spread around the threshold, additional information about\nborates and Fe-As compounds is required for proper training of the learning algorithm.\n\nIn the past, it has been shown that {\\small QSAR}\\ approaches can be used for the\ndetection of mis-annotated chemical compounds, a critical step in data curation~\\cite{Fourches_JCIM_2010}.\nWe have employed a similar approach here.\nIn our models, three materials, ReB$_2$ ({\\small ICSD}\\ \\#23871),\nLi$_2$Pd$_3$B ({\\small ICSD}\\ \\#84931), and La$_2$CuO$_4$, were significantly mis-predicted.\nMore careful examination of the data revealed that the $T_{\\mathrm{c}}$'s of\nReB$_2$ and Li$_2$Pd$_3$B were incorrectly extracted from literature.\nWe also found that La$_2$CuO$_4$ has the largest variation of reported values within the data set.\nTherefore, it was excluded from the regression.\nThis approach illustrates that {\\small QMSPR}\\ modeling should be automatically implemented to reduce and correct erroneous entries.\n\n\\boldsection{Structural model.}\nWe also developed a structural model meant to capture the geometrical\nfeatures that most influence $T_{\\mathrm{c}}$.\nIt employs SiRMS descriptors, PLS approaches, and five-fold external cross-validation.\nThe predictive performance of this model ($Q^2=0.61$) is comparable to that\nof the SiRMS-based RF model (see Table~\\ref{tab:art094:continuous} for additional statistics).\nThe top 10 statistically significant geometrical fragments and their contributions to\n$T_{\\mathrm{c}}$ variations are shown in Table~\\ref{tab:art094:classification}.\nAll descriptor contributions were converted to atomic contributions\n(details discussed previously~\\cite{Muratov_FMC_2010}) and related to material structures.\nExamples of unit cell structures for pairs of similar materials\nwith different $T_{\\mathrm{c}}$ values were color-coded according to\natomic contributions to $T_{\\mathrm{c}}$, and are shown in Figure~\\ref{fig:art094:structure_fragments}\n(green for $T_{\\mathrm{c}}\\!\\uparrow$, red for $T_{\\mathrm{c}}\\!\\downarrow$, and gray for neutral).\n\n\\begin{table}[tp]\\centering\n\\mycaption[Top statistically significant fragments and their contributions to $T_{\\mathrm{c}}$ variation.]\n{``-'' demonstrates that the collection is bonded, while ``and'' demonstrates that the collection is not bonded.}\n\\vspace{3mm}\n\\begin{tabular}{l | r}\nfragment name & contribution to $log(T_{\\mathrm{c}})$ score\\\\\n\\hline\nO-Cu-O & 18\\%\\\\\nperiodic groups IB-IIB-IVA & 14\\%\\\\\nperiodic groups IIA and IB & 12\\%\\\\\nAs, As, Fe fragment count & 5\\%\\\\\nperiodic groups IIB-IVA & 5\\%\\\\\nperiodic groups IIA and IVA & 5\\%\\\\\ncharges~\\cite{bader_atoms_1994} (-1.5)(-1.5)(+2.5) & 3\\%\\\\\nO element count & 2\\%\\\\\nCu element count & 2\\%\\\\\nO, O, O fragment count & 2\\%\\\\\ncharge~\\cite{bader_atoms_1994} (+2.5) & 2\\%\\\\\nNb element count & 2\\%\\\\\ncharge~\\cite{bader_atoms_1994} (-1.5) & 2\\%\\\\\n\\end{tabular}\n\\label{tab:art094:fragments}\n\\end{table}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.00\\linewidth]{fig105}\n\\mycaption[Materials color-coded according to atom contributions to $\\log(T_{\\mathrm{c}})$.]\n{Atoms and structural fragments that decrease superconductivity critical temperatures\nare colored in red and those enhancing $T_{\\mathrm{c}}$ are shown in green.\nNon-influential fragments are in gray.\n({\\bf a}) Ba$_2$Ca$_2$Cu$_3$HgO$_8$,\n({\\bf b}) As$_2$Ni$_2$O$_6$Sc$_2$Sr$_4$,\n({\\bf c}) Mo$_6$PbS$_8$,\n({\\bf d}) Mo$_6$NdS$_8$,\n({\\bf e}) Li$_2$Pd$_3$B,\n({\\bf f}) Li$_2$Pt$_3$B,\n({\\bf g}) FeLaAsO, and\n({\\bf h}) FeLaPO. }\n\\label{fig:art094:structure_fragments}\n\\end{figure}\n\nExamples of fragments for materials having\n$T_{\\mathrm{c}}\\!>\\!T_{\\mathrm{thr}}$ [Ba$_2$Ca$_2$Cu$_3$HgO$_8$, {\\small ICSD}\\ \\#75730,\n$\\log(T_{\\mathrm{c}})$=2.12]\nand $T_{\\mathrm{c}}\\!\\!\\leq T_{\\mathrm{thr}}$\n[As$_2$Ni$_2$O$_6$Sc$_2$Sr$_4$, {\\small ICSD}\\ \\#180270, $\\log(T_{\\mathrm{c}})$=0.44]\nare shown in Figures~\\ref{fig:art094:structure_fragments}(a) and~\\ref{fig:art094:structure_fragments}(b), respectively.\nThey indicate that individual atom contributions are nonlocal as they strongly\ndepend upon the atomic environment (Figures~\\ref{fig:art094:structure_fragments}(c)-\\ref{fig:art094:structure_fragments}(h)).\nFor example, Mo$_6$PbS$_8$ [{\\small ICSD}\\ \\#644102, $\\log(T_{\\mathrm{c}})$=1.13] and\nMo$_6$NdS$_8$ [{\\small ICSD}\\ \\#603458, $\\log(T_{\\mathrm{c}})$=0.54]\ndiffer by a substitution --- yet the difference in $T_{\\mathrm{c}}$ is substantial.\nFurthermore, substitution of Nd for Pb affects contributions to the\ntarget property from all the remaining atoms in the unit cell\n(Figure~\\ref{fig:art094:structure_fragments}(c) and~\\ref{fig:art094:structure_fragments}(d)).\nThe same observation holds for Li$_2$Pd$_3$B [{\\small ICSD}\\ \\#84931,\n$\\log(T_{\\mathrm{c}})$=0.89] and Li$_2$Pt$_3$B [{\\small ICSD}\\ \\#84932,\n$\\log(T_{\\mathrm{c}})$=0.49] Figure~\\ref{fig:art094:structure_fragments}(e) and\n~\\ref{fig:art094:structure_fragments}(f); as well as FeLaAsO [{\\small ICSD}\\ \\#163496, $\\log(T_{\\mathrm{c}})$=1.32]\nand FeLaPO [{\\small ICSD}\\ \\#162724, $\\log(T_{\\mathrm{c}})$=0.82]\nFigure~\\ref{fig:art094:structure_fragments}(g) and~\\ref{fig:art094:structure_fragments}(h).\n\n\\subsection{Conclusion}\nWith high-throughput approaches in materials science\nincreasing the data-driven content of the field,\nthe gap between accumulated-information and derived knowledge widens.\nThe issue can be overcome by adapting the data-analysis approaches\ndeveloped during the past decade for chem- and bioinformatics.\n\nOur study gives an example of this.\nWe introduce novel materials fingerprint descriptors that lead to the generation of\nnetworks called ``{\\it materials cartograms}'': nodes represent compounds;\nconnections represent similarities.\nThe representation can identify regions with distinct physical and chemical properties,\nthe key step in searching for interesting, yet unknown compounds.\n\nStarting from atomic-compositions, bond-topologies, structure-geometries,\nand electronic properties of materials publicly available in the {\\sf \\AFLOW.org}\\ repository,\nwe have introduced cheminformatics models leveraging novel materials fingerprints.\nWithin our formalism, simple band-structure and {\\small DOS}\\ fingerprints are adequate to\nlocate metals, semiconductors, topological insulators, piezoelectrics, and superconductors.\nMore complex {\\small QMSPR}\\ modeling~\\cite{Kuzmin_JCAMD_2008} are used to tackle\nqualitative and quantitative values of superconducting critical temperature\nand geometrical features helping\/hindering criticality~\\cite{Kuzmin_JCAMD_2008}.\n\nIn summary, the fingerprinting cartography introduced in this work\nhas demonstrated its utility in an initial set of problems.\nThis shows the possibility of designing new materials and gaining\ninsight into the relationship between the structure\nand physical properties of materials.\nFurther advances in the analysis and exploration of databases may become the\nfoundation for rationally designing novel compounds with desired properties.\n\\clearpage\n\\section{Machine Learning Modeling of Superconducting Critical Temperature}\n\\label{sec:art137}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art137}.\nAuthor contributions are as follows:\nValentin Stanev, Ichiro Takeuchi and Aaron Gilad Kusne designed the research.\nValentin Stanev worked on the model.\nCorey Oses and Stefano Curtarolo performed the {\\small AFLOW}\\ calculations.\nValentin Stanev, Ichiro Takeuchi, Efrain Rodriguez and Johnpierre Paglione analyzed the results.\nValentin Stanev, Corey Oses, Ichiro Takeuchi and Efrain Rodriguez wrote the text of the manuscript.\nAll authors discussed the results and commented on the manuscript.\n\n\\subsection{Introduction}\n\nSuperconductivity, despite being the subject of intense physics,\nchemistry and materials science research for more than a century,\nremains among one of the most puzzling scientific topics~\\cite{SCSpecial_PSCC_2015}.\nIt is an intrinsically quantum phenomenon caused by a finite attraction between paired electrons,\nwith unique properties including zero DC resistivity, Meissner and Josephson effects, and\nwith an ever-growing list of current and potential applications.\nThere is even a profound connection between phenomena in the\nsuperconducting state and the Higgs mechanism in particle physics~\\cite{PWAnderson_PR_1963}.\nHowever, understanding the relationship between superconductivity and materials'\nchemistry and structure presents significant theoretical and experimental challenges.\nIn particular, despite focused research efforts in the last 30 years,\nthe mechanisms responsible for high-temperature superconductivity in\ncuprate and iron-based families remain elusive~\\cite{Chu_PSCC_2015,Paglione_NatPhys_2010}.\n\nRecent developments, however, allow a different approach to investigate what ultimately determines the superconducting\ncritical temperatures $\\left(T_{\\mathrm{c}}\\right)$ of materials.\nExtensive databases covering various measured and calculated materials properties have been created over\nthe years~\\cite{ICSD,aflowPAPER,cmr_repository,Saal_JOM_2013,APL_Mater_Jain2013}.\nThe shear quantity of accessible information also makes possible,\nand even necessary,\nthe use of data-driven approaches, \\nobreak\\mbox{\\it e.g.}, statistical and machine learning (ML)\nmethods~\\cite{Agrawal_APLM_2016,Lookman_MatInf_2016,Jain_JMR_2016,Mueller_MLMS_2016}.\nSuch algorithms can be\ndeveloped\/trained on the variables collected in these databases,\nand employed to predict macroscopic properties\nsuch as the melting temperatures of binary compounds~\\cite{Seko_PRB_2014},\nthe likely crystal structure at a given composition~\\cite{Balachandran_SR_2015},\nband gap energies~\\cite{Pilania_SR_2016,curtarolo:art124} and\ndensity of states~\\cite{Pilania_SR_2016} of certain classes of materials.\n\nTaking advantage of this immense increase of readily accessible and potentially relevant information, we develop several\nML methods modeling $T_{\\mathrm{c}}$ from\nthe complete list of reported (inorganic) superconductors~\\cite{SuperCon}.\nIn their simplest form, these methods take as input a number of predictors\ngenerated from the elemental composition of each material.\nModels developed with these basic features are surprisingly accurate, despite\nlacking information of relevant properties, such as space group, electronic structure,\nand phonon energies.\nTo further improve the predictive power of the models,\nas well as the ability to extract useful information out of them,\nanother set of features are constructed based on crystallographic and electronic information\ntaken from the {\\small AFLOW}\\ Online Repositories~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\n\nApplication of statistical methods in the context of superconductivity began in the early\neighties with simple clustering methods~\\cite{Villars_PRB_1988,Rabe_PRB_1992}.\nIn particular, three ``golden'' descriptors confine the sixty known (at the time) superconductors with\n$T_{\\mathrm{c}} > 10$~K to three small islands in space:\nthe averaged valence-electron numbers, orbital radii differences, and metallic\nelectronegativity differences.\nConversely, about $600$ other superconductors with $T_{\\mathrm{c}} < 10$~K appear randomly dispersed\nin the same space.\nThese descriptors were selected heuristically due to their\nsuccess in classifying binary\/ternary structures and predicting stable\/metastable ternary quasicrystals.\nRecently, an investigation stumbled on this clustering problem again by observing a\nthreshold $T_{\\mathrm{c}}$ closer to $\\log\\left(T_{\\mathrm{c}}^{\\mathrm{thres}}\\right)\\approx1.3$\n$\\left(T_{\\mathrm{c}}^{\\mathrm{thres}}=20~\\text{K}\\right)$~\\cite{curtarolo:art94}.\nInstead of a heuristic approach, random forests and simplex fragments were\nleveraged on the structural\/electronic properties\ndata from the {\\small AFLOW}\\ Online Repositories to find the optimum clustering descriptors.\nA classification model was developed showing good performance.\nSeparately, a sequential learning framework was evaluated on superconducting materials,\nexposing the limitations of relying on random-guess (trial-and-error) approaches for\nbreakthrough discoveries~\\cite{Ling_IMMI_2017}.\nSubsequently, this study also highlights the impact machine learning can have\non this particular field.\nIn another early work,\nstatistical methods were used to find correlations between normal state properties and $T_{\\mathrm{c}}$\nof the metallic elements in the first six rows of the periodic table~\\cite{Hirsch_PRB_1997}.\nOther contemporary works hone in on specific materials~\\cite{Owolabi_JSNM_2015,Ziatdinov_NanoTech_2016}\nand families of superconductors~\\cite{Klintenberg_CMS_2013,Owolabi_APTA_2014} (see also Reference~\\cite{Norman_RPP_2016}).\n\nWhereas previous investigations explored several hundred compounds at most,\nthis work considers more than $16,000$ different compositions.\nThese are extracted from the SuperCon database, which contains an exhaustive\nlist of superconductors, including many closely-related materials varying only by small changes in stoichiometry (doping plays a significant role in optimizing $T_{\\mathrm{c}}$).\nThe order-of-magnitude increase in training data\n\\textbf{i.} presents crucial subtleties in chemical composition among related compounds,\n\\textbf{ii.} affords family-specific modeling exposing different superconducting mechanisms, and\n\\textbf{iii.} enhances model performance overall.\nIt also enables the optimization of several model construction procedures.\nLarge sets of independent variables can be constructed and rigorously filtered\nby predictive power (rather than selecting them by intuition alone).\nThese advances are crucial to uncovering insights into the\nemergence\/suppression of superconductivity with composition.\n\nAs a demonstration of the potential of ML methods in looking for novel superconductors, we combined and\napplied several models to search for candidates among the roughly\n$110,000$ different compositions contained in the Inorganic Crystallographic Structure Database ({\\small ICSD}),\na large fraction of which have not been tested for superconductivity.\nThe framework highlights 35 compounds with predicted $T_{\\mathrm{c}}$'s\nabove 20~K for experimental validation.\nOf these, some exhibit interesting chemical and structural similarities to cuprate superconductors, demonstrating the ability of the ML models to identify meaningful patterns in the data.\nIn addition, most materials from the list share a peculiar feature in their electronic band structure:\none (or more) flat\/nearly-flat bands just below the energy of the highest occupied electronic state.\nThe associated large peak in the density of states (infinitely large in the limit of truly flat bands)\ncan lead to strong electronic instability, and has been discussed recently as one possible way to\nhigh-temperature superconductivity~\\cite{Kopnin_PRB_2011,Peotta_NComm_2015}.\n\n\\subsection{Results}\n\\boldsection{Data and predictors.}\nThe success of any ML method ultimately depends on access to reliable and plentiful data.\nSuperconductivity data used in this work is extracted from the SuperCon database~\\cite{SuperCon},\ncreated and maintained by the Japanese National Institute for Materials Science.\nIt houses information such as the $T_{\\mathrm{c}}$\nand reporting journal publication for superconducting materials known from experiment.\nAssembled within it is a uniquely exhaustive list of all reported superconductors,\nas well as related non-superconducting compounds.\nAs such, SuperCon is the largest database of its kind, and has never before been employed\n{\\it en masse} for machine learning modeling.\n\nFrom SuperCon, we have extracted a list of approximately $16,400$ compounds,\nof which $4,000$ have no $T_{\\mathrm{c}}$ reported (see Methods for details).\nOf these, roughly $5,700$ compounds are cuprates and $1,500$ are iron-based\n(about 35\\% and 9\\%, respectively), reflecting the significant research efforts invested in these two families.\nThe remaining set of about $8,000$ is a mix of various materials, including conventional phonon-driven superconductors\n(\\nobreak\\mbox{\\it e.g.}, elemental superconductors, A15 compounds), known unconventional superconductors like the\nlayered nitrides and heavy fermions, and many materials for which the mechanism of superconductivity\nis still under debate (such as bismuthates and borocarbides).\nThe distribution of materials by $T_{\\mathrm{c}}$ for the three groups is shown in Figure~\\ref{fig:art137:Class_score}(a).\n\nUse of this data for the purpose of creating ML models can be problematic.\nML models have an intrinsic applicability domain, \\nobreak\\mbox{\\it i.e.}, predictions\nare limited to the patterns\/trends encountered in the training set.\nAs such, training a model only on superconductors can lead to significant selection bias\nthat may render it ineffective when applied to new\nmaterials\\footnote{\\textit{N.B.}, a model suffering from selection bias\ncan still provide valuable statistical information about known superconductors.}.\nEven if the model learns to correctly recognize factors promoting superconductivity,\nit may miss effects that strongly inhibit it.\nTo mitigate the effect, we incorporate about $300$ materials found by H.\nHosono's group not to display superconductivity~\\cite{Hosono_STAM_2015}.\nHowever, the presence of non-superconducting materials,\nalong with those without $T_{\\mathrm{c}}$ reported in SuperCon, leads to a conceptual problem.\nSome of these compounds emerge as non-superconducting ``end-members'' from\ndoping\/pressure studies, indicating no superconducting transition was observed despite some efforts to find one.\nHowever, the transition may still exist,\nalbeit at experimentally difficult to reach or altogether inaccessible temperatures\n(for most practical purposes below $10$~mK)\\footnote{There are theoretical arguments for this --- according\nto the Kohn-Luttinger theorem, a\nsuperconducting instability should be present as $T \\rightarrow 0$ in any fermionic metallic system\nwith Coulomb interactions~\\cite{Kohn_PRL_1965}.}.~\\nocite{Kohn_PRL_1965}\nThis presents a conundrum:\nignoring compounds with no reported $T_{\\mathrm{c}}$ disregards a potentially important\npart of the dataset, while assuming $T_{\\mathrm{c}} = 0$~K prescribes an inadequate description\nfor (at least some of) these compounds.\nTo circumvent the problem,\nmaterials are first partitioned in two groups by their $T_{\\mathrm{c}}$,\nabove and below a threshold temperature $\\left(T_{\\mathrm{sep}}\\right)$,\nfor the creation of a classification model.\nCompounds with no reported critical temperature can be classified in the ``below-$T_{\\mathrm{sep}}$'' group\nwithout the need to specify a $T_{\\mathrm{c}}$ value (or assume it is zero).\nThe ``above-$T_{\\mathrm{sep}}$'' bin also enables the development of a regression model\nfor $\\ln{(T_{\\mathrm{c}})}$, without problems arising in the $T_{\\mathrm{c}}\\to0$ limit.\n\nFor most materials, the SuperCon database provides\nonly the chemical composition and $T_{\\mathrm{c}}$.\nTo convert this information into meaningful features\/predictors (used interchangeably),\nwe employ the Materials Agnostic Platform for Informatics and Exploration (Magpie)~\\cite{Ward_ML_GFA_NPGCompMat_2016}.\nMagpie computes a set of attributes for each material, including elemental property\nstatistics like the mean and the standard deviation of 22 different elemental properties\n(\\nobreak\\mbox{\\it e.g.}, period\/group on the periodic table,\natomic number, atomic radii, melting temperature), as well as electronic structure attributes, such as the average\nfraction of electrons from the $s$, $p$, $d$ and $f$ valence shells among all\nelements present.\n\nThe application of Magpie predictors, though appearing to lack \\textit{a priori} justification,\nexpands upon past clustering approaches by Villars and Rabe~\\cite{Villars_PRB_1988,Rabe_PRB_1992}.\nThey show that, in the space of a few judiciously chosen\nheuristic predictors, materials separate and cluster according to their\ncrystal structure and even complex properties such as high-temperature\nferroelectricity and superconductivity.\nSimilar to these features, Magpie predictors capture significant chemical information, which\nplays a decisive role in determining\nstructural and physical properties of materials.\n\nDespite the success of Magpie predictors in modeling materials properties~\\cite{Ward_ML_GFA_NPGCompMat_2016},\ninterpreting their connection to superconductivity presents a serious challenge.\nThey do not encode (at least directly) many important properties, particularly those\npertinent to superconductivity.\nIncorporating features\nlike lattice type and density of states would undoubtedly lead to significantly more powerful and interpretable models.\nSince such information is not generally available in SuperCon,\nwe employ data from the {\\small AFLOW}\\ Online Repositories~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nThe materials database houses more than 200 million properties calculated with\nthe software package {\\small AFLOW}~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}.\nIt contains information for the vast majority of compounds in the {\\small ICSD}~\\cite{ICSD}.\nAlthough the {\\small AFLOW}\\ Online Repositories contain calculated properties,\nthe density functional theory ({\\small DFT}) results have been extensively validated with\n{\\small ICSD}\\ records~\\cite{curtarolo:art94,curtarolo:art96,curtarolo:art112,curtarolo:art115,curtarolo:art120,curtarolo:art124}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig106}\n\\mycaption[Schematic of the random forest ML approach.]\n{Example of a single decision tree used to classify materials depending on whether\n$T_{\\mathrm{c}}$ is above or below $10$~K.\nA tree can have many levels, but only the three top are shown.\nThe decision rules leading to each subset are written inside individual rectangles.\nThe subset population percentage\nis given by ``samples'', and the node color\/shade\nrepresents the degree of separation,\n\\nobreak\\mbox{\\it i.e.}, dark blue\/orange illustrates a high proportion of\n$T_{\\mathrm{c}} >10$~K\/$T_{\\mathrm{c}} < 10$~K materials\n(the exact value is given by ``proportion'').\nA random forest consists of a large number --- could be hundreds or thousands --- of such individual trees.}\n\\label{fig:art137:tree_example}\n\\end{figure}\n\nUnfortunately, only a small subset of materials in SuperCon overlaps with those in the {\\small ICSD}:\nabout $800$ with finite $T_{\\mathrm{c}}$ and less than $600$ are contained within {\\small AFLOW}.\nFor these, a set of 26 predictors are incorporated\nfrom the {\\small AFLOW}\\ Online Repositories, including structural\/chemical information like the lattice type, space group,\nvolume of the unit cell, density, ratios of the lattice parameters,\nBader charges and volumes, and formation energy (see Methods for details).\nIn addition, electronic properties are considered, including the\ndensity of states near the Fermi level as calculated by {\\small AFLOW}.\nPrevious investigations exposed limitations in applying ML methods to a similar dataset\nin isolation~\\cite{curtarolo:art94}.\nInstead, a framework is presented here for combining models built on Magpie descriptors\n(large sampling, but features limited to compositional data) and {\\small AFLOW}\\ features\n(small sampling, but diverse and pertinent features).\n\nOnce we have a list of relevant predictors, various ML models can be applied to the\ndata~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.\nAll ML algorithms in this work are\nvariants of the random forest method~\\cite{randomforests}.\nFundamentally, this approach combines many individual decision trees, where\neach tree is a non-parametric supervised learning method used for\nmodeling either categorical or numerical variables (\\nobreak\\mbox{\\it i.e.},\nclassification or regression modeling).\nA tree predicts the value of a target variable by learning simple decision rules\ninferred from the available features (see Figure~\\ref{fig:art137:tree_example} for an example).\n\nRandom forest is one of the most powerful, versatile, and widely-used ML methods~\\cite{Caruana_2006}.\nThere are several advantages that make it especially suitable for this problem.\nFirst, it can learn complicated non-linear dependencies from the data.\nUnlike many other methods (\\nobreak\\mbox{\\it e.g.}, linear regression),\nit does not make assumptions about the functional form of the relationship between the predictors\nand the target variable (\\nobreak\\mbox{\\it e.g.}, linear, exponential or some other {\\it a priori} fixed function).\nSecond, random forests are quite tolerant to heterogeneity in the training data.\nIt can handle both numerical and categorical data which, furthermore, does not\nneed extensive and potentially dangerous preprocessing, such as scaling or normalization.\nEven the presence of strongly correlated predictors is not a problem for model\nconstruction (unlike many other ML algorithms).\nAnother significant advantage of this method is that, by combining information from\nindividual trees, it can estimate the importance of each predictor, thus making the model more interpretable.\nHowever, unlike model construction, determination of predictor importance is complicated by the presence of\ncorrelated features.\nTo avoid this, standard feature selection procedures are employed along with\na rigorous predictor elimination scheme (based on their strength and correlation with others).\nOverall, these methods\nreduce the complexity of the models and improve our\nability to interpret them.\n\n\\boldsection{Classification models.}\nAs a first step in applying ML methods to the dataset, a sequence of classification models\nare created, each designed to separate materials into two distinct groups depending on whether\n$T_{\\mathrm{c}}$ is above or below some predetermined value.\nThe temperature that separates the two groups ($T_{\\mathrm{sep}}$)\nis treated as an adjustable parameter of the model, though some physical\nconsiderations should guide its choice as well.\nClassification ultimately allows compounds with no reported $T_{\\mathrm{c}}$ to be used\nin the training set by including them in the below-$T_{\\mathrm{sep}}$ bin.\nAlthough discretizing continuous variables is not generally recommended, in this case\nthe benefits of including compounds without $T_{\\mathrm{c}}$ outweigh\nthe potential information loss.\n\nIn order to choose the optimal value of $T_{\\mathrm{sep}}$, a series of random forest models\nare trained with different threshold temperatures separating the two classes.\nSince setting $T_{\\mathrm{sep}}$ too low or too high creates strongly imbalanced classes\n(with many more instances in one group), it is important to compare the models using several different metrics.\nFocusing only on the accuracy (count of correctly-classified instances)\ncan lead to deceptive results.\nHypothetically, if $95\\%$ of the observations in the dataset are in the below-$T_{\\mathrm{sep}}$ group,\nsimply classifying all materials as such would\nyield a high accuracy ($95\\%$), while being trivial in any other sense\\footnote{There are more sophisticated techniques to deal with severely imbalanced datasets, like undersampling the majority class or generating synthetic data points for the minority class (see, for example, Reference~\\cite{SMOTE}.}~\\nocite{SMOTE}.\nTo avoid this potential pitfall, three other standard metrics\nfor classification are considered: precision, recall, and $F_{\\mathrm{1}}$ score.\nThey are defined using the values $tp$, $tn$, $fp$, and $fn$ for\nthe count of true\/false positive\/negative\npredictions of the model:\n\\begin{eqnarray}\n\\text{accuracy} \\equiv \\frac{tp+tn}{tp+tn+fp+fn},\n\\label{eq:art137:accur}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\text{precision}\\equiv\\frac{tp}{tp+fp},\n\\label{eq:art137:precision}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\text{recall} \\equiv\\frac{tp}{tp+fn},\n\\label{eq:art137:recall}\n\\end{eqnarray}\n\\begin{eqnarray}\nF_{\\mathrm{1}}\\equiv2*\\frac{\\text{precision}*\\text{recall}}{\\text{precision}+\\text{recall}},\n\\label{eq:art137:f1}\n\\end{eqnarray}\nwhere positive\/negative refers to above-$T_{\\mathrm{sep}}$\/below-$T_{\\mathrm{sep}}$.\nThe accuracy of a classifier is the total proportion of correctly-classified materials,\nwhile precision measures the proportion of correctly-classified\nabove-$T_{\\mathrm{sep}}$ superconductors out of all predicted above-$T_{\\mathrm{sep}}$.\nThe recall is the proportion of correctly-classified above-$T_{\\mathrm{sep}}$\nmaterials out of all truly above-$T_{\\mathrm{sep}}$ compounds.\nWhile the precision measures the probability that a\nmaterial selected by the model actually has $T_{\\mathrm{c}} > T_{\\mathrm{sep}}$,\nthe recall reports how sensitive the model is to above-$T_{\\mathrm{sep}}$ materials.\nMaximizing the precision or recall would require some compromise with\nthe other, \\nobreak\\mbox{\\it i.e.}, a model that labels all materials as above-$T_{\\mathrm{sep}}$ would have perfect recall but dismal precision.\nTo quantify the trade-off between recall and precision, their harmonic mean ($F_{\\mathrm{1}}$ score) is\nwidely used to measure the performance of a classification model.\nWith the exception of accuracy, these metrics are not symmetric with respect to the exchange of positive and negative labels.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.90\\linewidth]{fig107}\n\\mycaption[SuperCon dataset and classification model performance.]\n{(\\textbf{a}) Histogram of materials categorized by\n$T_{\\mathrm{c}}$ (bin size is $2$~K, only those with finite $T_{\\mathrm{c}}$ are counted).\nBlue, green, and red denote low-$T_{\\mathrm{c}}$, iron-based, and cuprate superconductors, respectively.\nIn the inset: histogram of materials categorized by $\\ln{(T_{\\mathrm{c}})}$\nrestricted to those with $T_{\\mathrm{c}} >10$~K.\n(\\textbf{b}) Performance of different classification models as a function of the threshold temperature\n$\\left(T_{\\mathrm{sep}}\\right)$ that separates materials in two classes by $T_{\\mathrm{c}}$.\nPerformance is measured by accuracy (gray), precision (red), recall (blue), and $F_{\\mathrm{1}}$ score (purple).\nThe scores are calculated from predictions on an independent test set, \\nobreak\\mbox{\\it i.e.}, one separate\nfrom the dataset used to train the model.\nIn the inset: the dashed red curve gives the proportion of materials in the above-$T_{\\mathrm{sep}}$ set.\n(\\textbf{c}) Accuracy, precision, recall, and $F_{\\mathrm{1}}$\nscore as a function of the size of the training set with a fixed test set.\n(\\textbf{d}) Accuracy, precision, recall, and $F_{\\mathrm{1}}$ as a function of the number of\npredictors.}\n\\label{fig:art137:Class_score}\n\\end{figure}\n\nFor a realistic estimate of the performance of each model,\nthe dataset is randomly split ($85\\%\/15\\%$) into training and test subsets.\nThe training set is employed to fit the model, which is then applied to the test set for subsequent benchmarking.\nThe aforementioned metrics (Equations~\\ref{eq:art137:accur}-\\ref{eq:art137:f1}) calculated on the test set provide\nan unbiased estimate of how well the model is expected to generalize to a new (but similar) dataset.\nWith the random forest method, similar estimates can be obtained intrinsically at the training stage.\nSince each tree is trained only on a bootstrapped subset of the data,\nthe remaining subset can be used as an internal test set.\nThese two methods for quantifying model performance usually yield very similar results.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig108}\n\\mycaption[Scatter plots of $3,000$ superconductors in the space of the four most important classification predictors.]\n{Blue\/red represent below-$T_{\\mathrm{sep}}$\/above-$T_{\\mathrm{sep}}$ materials, where $T_{\\mathrm{sep}} = 10$~K.\n(\\textbf{a}) Feature space of the first and second most important predictors:\nstandard deviations of the column numbers and electronegativities (calculated over the values for the constituent elements in each compound).\n(\\textbf{b}) Feature space of the third and fourth most important predictors:\nstandard deviation of the elemental melting temperatures and average of the\natomic weights.}\n\\label{fig:art137:Class_features}\n\\end{figure}\n\nWith the procedure in place, the models' metrics are evaluated for a range of $T_{\\mathrm{sep}}$ and illustrated\nin Figure~\\ref{fig:art137:Class_score}(b).\nThe accuracy increases as $T_{\\mathrm{sep}}$ goes from $1$~K to $40$~K,\nand the proportion of above-$T_{\\mathrm{sep}}$ compounds drops from above $70\\%$ to about $15\\%$,\nwhile the recall and $F_{\\mathrm{1}}$ score generally decrease.\nThe region between $5-15$~K is especially appealing in (nearly) maximizing\nall benchmarking metrics while balancing the sizes of the bins.\nIn fact, setting $T_{\\mathrm{sep}}=10$~K is a particularly convenient choice.\nIt is also the temperature used in References~\\cite{Villars_PRB_1988,Rabe_PRB_1992}\nto separate the two classes, as it is just above the\nhighest $T_{\\mathrm{c}}$ of all elements and pseudoelemental\nmaterials (solid solution whose range of composition includes a pure element).\nHere, the proportion of above-$T_{\\mathrm{sep}}$ materials is approximately $38\\%$ and\nthe accuracy is about $92\\%$, \\nobreak\\mbox{\\it i.e.}, the model can\ncorrectly classify nine out of ten materials --- much better than random guessing.\nThe recall --- quantifying how well all above-$T_{\\mathrm{sep}}$ compounds are labeled and,\nthus, the most important metric when searching for new superconducting materials --- is even higher.\n(Note that the models' metrics also depend on random factors such as the composition of the\ntraining and test sets, and their exact values can vary.)\n\n\\begin{table}[tp]\\centering\n\\mycaption[The most relevant predictors and their importances for the classification and general regression models.]\n{$\\avg(x)$ and $\\std(x)$ denote the composition-weighted average and\nstandard deviation, respectively,\ncalculated over the vector of elemental values for each compound~\\cite{Ward_ML_GFA_NPGCompMat_2016}.\nFor the classification model, all predictor importances are quite close.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l | l r| l r}\npredictor & \t\\multicolumn{4}{c}{model}\n\\\\\n\\cline{2-5}\nrank& \\multicolumn{2}{l|}{classification} & \\multicolumn{2}{l}{regression (general; $T_{\\mathrm{c}}>10$ K)} \\\\\n\\hline\n1 & $\\std ($column number$)$ & \\setlength{\\tabcolsep}{4pt} 0.26 & $\\avg ($number of unfilled orbitals$)$ & \\setlength{\\tabcolsep}{4pt} 0.26\\\\\n2 & $\\std ($electronegativity$)$ & \\setlength{\\tabcolsep}{4pt} 0.26 & $\\std ($ground state volume$)$ & \\setlength{\\tabcolsep}{4pt} 0.18\\\\\n3 & $\\std ($melting temperature$)$ & \\setlength{\\tabcolsep}{4pt} 0.23 & $\\std ($space group number$)$ & \\setlength{\\tabcolsep}{4pt} 0.17\\\\\n4 & $\\avg ($atomic weight$)$ & \\setlength{\\tabcolsep}{4pt} 0.24 & $\\avg ($number of $d$ unfilled orbitals$)$ & \\setlength{\\tabcolsep}{4pt} 0.17\\\\\n5 & & \\setlength{\\tabcolsep}{4pt} - & $\\std ($number of $d$ valence electrons$)$ & \\setlength{\\tabcolsep}{4pt} 0.12\\\\\n6 & & \\setlength{\\tabcolsep}{4pt} - & $\\avg ($melting temperature$)$ & \\setlength{\\tabcolsep}{4pt} 0.1\\\\\n\\end{tabular}}\n\\label{tab:art137:Table1}\n\\end{table}\n\nThe most important factors that determine the model's performance are the size of the available\ndataset and the number of meaningful predictors.\nAs can be seen in Figure~\\ref{fig:art137:Class_score}(c), all\nmetrics improve significantly with the increase of the training set size. The effect is most dramatic for sizes between several hundred and few thousands instances, but there is no obvious saturation even for the largest available datasets. This validates efforts herein to incorporate as much relevant data as possible into model training.\nThe number of predictors is another very important model parameter.\nIn Figure~\\ref{fig:art137:Class_score}(d),\nthe accuracy is calculated at each step of the backward feature elimination process.\nIt quickly saturates when the number of predictors reaches $10$.\nIn fact, a model using only the five most informative predictors, selected out of the full list of 145 ones,\nachieves almost $90\\%$ accuracy.\n\nFor an understanding of what the model has learned, an analysis of the chosen predictors is needed.\nIn the random forest method, features can be ordered by their importance quantified via\nthe so-called Gini importance or\n``mean decrease in impurity''~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.\nFor a given feature, it is the sum of the Gini impurity\\footnote{Gini impurity is calculated as\n\\unexpanded{$\\sum_i p_i \\left(1-p_i\\right)$},\nwhere \\unexpanded{$p_i$} is the probability of randomly chosen data point\nfrom a given decision tree leaf to be in class\n\\unexpanded{$i$}~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.}~\\nocite{Bishop_ML_2006,Hastie_StatLearn_2001}\nover the number of splits that include the feature, weighted by the number of samples\nit splits, and averaged over the entire forest.\nDue to the nature of the algorithm, the closer to the top of the tree a predictor is used,\nthe greater number of predictions it impacts.\n\nAlthough correlations between predictors do not affect the model's ability to learn, it can distort importance estimates.\nFor example, a material property with a strong effect on $T_{\\mathrm{c}}$ can be shared\namong several correlated predictors.\nSince the model can access the same information through any of these variables,\ntheir relative importances are diluted across the group.\nTo reduce the effect and limit the list of predictors to a manageable size,\nthe backward feature elimination method is employed.\nThe process begins with a model constructed with the full list of predictors,\nand iteratively removes the least significant one, rebuilding the model and recalculating importances\nwith every iteration.\n(This iterative procedure is necessary since the ordering of the predictors by importance can change at each step.)\nPredictors are removed until the overall accuracy of the model drops by $2\\%$, at which point there are only five left.\nFurthermore, two of these predictors are strongly correlated with each other, and we remove the less important one. This\nhas a negligible impact on the model performance,\nyielding four predictors total (see Table~\\ref{tab:art137:Table1})\nwith an above $90\\%$ accuracy score --- only slightly worse than the full model.\nScatter plots of the pairs of the most important predictors are shown in Figure~\\ref{fig:art137:Class_features}, where\nblue\/red denotes whether the material is in the below-$T_{\\mathrm{sep}}$\/above-$T_{\\mathrm{sep}}$ class.\nFigure~\\ref{fig:art137:Class_features}(a) shows a scatter plot of $3,000$ compounds\nin the space spanned by the standard deviations of the column numbers and electronegativities\ncalculated over the elemental values.\nSuperconductors with $T_{\\mathrm{c}} > 10$~K tend to\ncluster in the upper-right corner of the plot and in a relatively thin elongated region extending to the left of it.\nIn fact, the points in the upper-right corner represent mostly cuprate materials,\nwhich with their complicated compositions and large number of elements are likely\nto have high standard deviations in these variables.\nFigure~\\ref{fig:art137:Class_features}(b) shows\nthe same compounds projected in the space of the standard deviations of\nthe melting temperatures and the averages of the atomic weights of the elements forming each compound.\nThe above-$T_{\\mathrm{sep}}$ materials tend to cluster in areas with lower mean atomic weights --- not\na surprising result given the role of phonons in conventional superconductivity.\n\nFor comparison, we create another classifier based on the average number of valence electrons,\nmetallic electronegativity differences, and orbital radii differences, \\nobreak\\mbox{\\it i.e.}, the predictors used\nin References~\\cite{Villars_PRB_1988,Rabe_PRB_1992} to cluster materials with $T_{\\mathrm{c}} > 10$ K.\nA classifier built only with these three predictors\nis less accurate than both the full and the truncated models presented herein,\nbut comes quite close: the full model has about $3\\%$ higher\naccuracy and $F_{\\mathrm{1}}$ score, while the truncated model with four predictors is less that $2\\%$ more accurate.\nThe rather small (albeit not insignificant) differences demonstrates that even on the scale of the\nentire SuperCon dataset, the predictors used by Villars and Rabe~\\cite{Villars_PRB_1988,Rabe_PRB_1992}\ncapture much of the relevant chemical information for superconductivity.\n\n\\begin{table}[tp]\\centering\n\\mycaption[The most significant predictors and their importances for the three material-specific regression models.]\n{$\\avg(x)$, $\\std(x)$, $\\max(x)$ and $\\fraction(x)$ denote the composition-weighted average,\nstandard deviation, maximum, and fraction, respectively,\ntaken over the elemental values for each compound.\n$l^2$-norm of a composition is calculated by $||x||_{2} = \\sqrt{\\sum_i x_i^2}$, where $x_i$ is the proportion of each element $i$ in the compound.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l | l r| l r |l r}\npred. & \t\\multicolumn{6}{c}{model}\n\\\\\n\\cline{2-7}\nrank & \\multicolumn{2}{l|}{regression (low-$T_{\\mathrm{c}}$)} & \\multicolumn{2}{l}{regression (cuprates)}& \\multicolumn{2}{|l} {regression (Fe-based)} \\\\\n\\hline\n1 & $\\fraction (d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.18 & $\\avg ($number of unfilled orbitals$)$ &\\setlength{\\tabcolsep}{4pt} 0.22 & $\\std ($column number$)$ &\\setlength{\\tabcolsep}{4pt} 0.17\\\\\n2 & $\\avg ($number of $d$ unfilled orbitals$)$ &\\setlength{\\tabcolsep}{4pt} 0.14 & $\\std ($number of $d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\avg ($ionic character$)$ &\\setlength{\\tabcolsep}{4pt} 0.15\\\\\n3 & $\\avg ($number of valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\fraction (d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\std ($Mendeleev number$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n4 & $\\fraction (s$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.11 & $\\std ($ground state volume$)$ &\\setlength{\\tabcolsep}{4pt} 0.13 & $\\std ($covalent radius$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n5 & $\\avg ($number of $d$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.09 & $\\std ($number of valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.1 & $\\max ($melting temperature$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n6 & $\\avg ($covalent radius$)$ &\\setlength{\\tabcolsep}{4pt} 0.09 & $\\std ($row number$)$ &\\setlength{\\tabcolsep}{4pt} 0.08 & $\\avg ($Mendeleev number$)$ &\\setlength{\\tabcolsep}{4pt} 0.14\\\\\n7 & $\\avg ($atomic weight$)$ &\\setlength{\\tabcolsep}{4pt} 0.08 & $||$composition$||_{2}$ &\\setlength{\\tabcolsep}{4pt} 0.07 & $||$composition$||_{2}$ &\\setlength{\\tabcolsep}{4pt} 0.11\\\\\n8 & $\\avg ($Mendeleev number$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & $\\std ($number of $s$ valence electrons$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & &\\setlength{\\tabcolsep}{4pt} -\\\\\n9 & $\\avg ($space group number$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & $\\std ($melting temperature$)$ &\\setlength{\\tabcolsep}{4pt} 0.07 & &\\setlength{\\tabcolsep}{4pt} -\\\\\n10 & $\\avg ($number of unfilled orbitals$)$ &\\setlength{\\tabcolsep}{4pt} 0.06 & &\\setlength{\\tabcolsep}{4pt} - & &\\setlength{\\tabcolsep}{4pt} -\\\\\n\\end{tabular}}\n\\label{tab:art137:Table2}\n\\end{table}\n\n\\boldsection{Regression models.}\nAfter constructing a successful classification model, we now move to the more difficult challenge of predicting $T_{\\mathrm{c}}$.\nCreating a regression model may enable better understanding of the factors controlling\n$T_{\\mathrm{c}}$ of known superconductors,\nwhile also serving as an organic part of a system for identifying potential new ones.\nLeveraging the same set of elemental predictors as the classification model, several regression models are presented\nfocusing on materials with $T_{\\mathrm{c}} > 10$~K.\nThis approach avoids the problem of materials with no reported $T_{\\mathrm{c}}$ with the assumption that,\nif they were to exhibit superconductivity at all, their critical temperature would be below $10$~K.\nIt also enables the substitution of $T_{\\mathrm{c}}$ with $\\ln{(T_{\\mathrm{c}})}$ as the target variable\n(which is problematic as $T_{\\mathrm{c}}\\to0$), and thus addresses the problem of the uneven distribution\nof materials along the $T_{\\mathrm{c}}$ axis (see Figure~\\ref{fig:art137:Class_score}(a)).\nUsing $\\ln{(T_{\\mathrm{c}})}$ creates a more uniform distribution (Figure~\\ref{fig:art137:Class_score}(a) inset),\nand is also considered a best practice when the range of a target variable covers more than one\norder of magnitude (as in the case of $T_{\\mathrm{c}}$).\nFollowing this transformation, the dataset is parsed randomly ($85\\%$\/$15\\%$) into training\nand test subsets (similarly performed for the classification model).\n\nPresent within the dataset are distinct families of superconductors with different driving\nmechanisms for superconductivity, including cuprate and iron-based high-temperature superconductors,\nwith all others denoted ``low-$T_{\\mathrm{c}}$'' for brevity (no specific mechanism in this group).\nSurprisingly, a single regression model does reasonably well among the\ndifferent families -- benchmarked on the test set,\nthe model achieves $R^2 \\approx 0.88$ (Figure~\\ref{fig:art137:Rerg_r2}(a)).\nIt suggests that the random forest algorithm is flexible and powerful enough\nto automatically separate the compounds into groups\nand create group-specific branches with distinct predictors (no explicit group labels were used during training and testing).\nAs validation, three separate models are constructed, each trained only on a specific family, namely the\nlow-$T_{\\mathrm{c}}$, cuprate, and iron-based superconductors, respectively.\nBenchmarking on mixed-family test sets, the models performed well on compounds belonging\nto their training set family while demonstrating no predictive power on the others.\nFigures~\\ref{fig:art137:Rerg_r2}(b)-(d) illustrate a cross-section of this comparison.\nSpecifically, the model trained on low-$T_{\\mathrm{c}}$ compounds dramatically underestimates\nthe $T_{\\mathrm{c}}$ of both high-temperature superconducting families (Figures~\\ref{fig:art137:Rerg_r2}(b) and (c)),\neven though this test set only contains compounds with $T_{\\mathrm{c}} < 40$~K.\nConversely, the model trained on the cuprates tends to overestimate the $T_{\\mathrm{c}}$\nof low-$T_{\\mathrm{c}}$ (Figure~\\ref{fig:art137:Rerg_r2}(d)) and iron-based (Figure~\\ref{fig:art137:Rerg_r2}(e)) superconductors.\nThis is a clear indication that superconductors from these groups have different factors determining their $T_{\\mathrm{c}}$.\nInterestingly, the family-specific models do not perform better than the general regression containing\nall the data points: $R^2$ for the low-$T_{\\mathrm{c}}$ materials is about $0.85$, for cuprates is just below $0.8$,\nand for iron-based compounds is about $0.74$.\nIn fact, it is a purely geometric effect that\nthe combined model has the highest $R^2$.\nEach group of superconductors contributes mostly to a distinct $T_{\\mathrm{c}}$ range, and, as a result, the combined regression is better determined over longer temperature interval.\n\nIn order to reduce the number of predictors and increase the interpretability of these models without\nsignificant detriment to their performance, a backward feature elimination process is again employed.\nThe procedure is very similar to the one described previously for the classification model,\nwith the only difference being that the reduction is guided by $R^2$ of the model, rather than the accuracy\n(the procedure stops when $R^2$ drops by $3\\%$).\n\nThe most important predictors for the four models (one general and three family-specific) together with\ntheir importances are shown in Tables~\\ref{tab:art137:Table1} and \\ref{tab:art137:Table2}.\nDifferences in important predictors across the family-specific models reflect the fact that\ndistinct mechanisms are responsible for driving superconductivity among these groups.\nThe list is longest for the low-$T_{\\mathrm{c}}$ superconductors, reflecting the eclectic nature of\nthis group.\nSimilar to the general regression model,\ndifferent branches are likely created for distinct sub-groups.\nNevertheless, some important predictors have straightforward interpretation.\nAs illustrated in Figure~\\ref{fig:art137:Tc_atomWeigth}(a),\nlow average atomic weight is a necessary (albeit not sufficient) condition for\nachieving high $T_{\\mathrm{c}}$ among the low-$T_{\\mathrm{c}}$ group.\nIn fact, the maximum $T_{\\mathrm{c}}$ for a given weight roughly follows $1\/\\sqrt{m_A}$.\nMass plays a significant role in conventional superconductors\nthrough the Debye frequency of phonons, leading to the well-known formula $T_{\\mathrm{c}} \\sim 1\/\\sqrt{m}$,\nwhere $m$ is the ionic mass\\footnote{See, for example, References~\\cite{Maxwell_PR_1950,Reynolds_PR_1950,Reynolds_PR_1951}.}~\\nocite{Maxwell_PR_1950,Reynolds_PR_1950,Reynolds_PR_1951}.\nOther factors like density of states are also important,\nwhich explains the spread in $T_{\\mathrm{c}}$ for a given $m_A$.\nOutlier materials clearly lying above the $\\sim 1\/\\sqrt{m_A}$ line include\nbismuthates and chloronitrates, suggesting the conventional electron-phonon mechanism is not driving\nsuperconductivity in these materials.\nIndeed, chloronitrates exhibit a very weak isotope effect~\\cite{Kasahara_PSCC_2015}, though\nsome unconventional electron-phonon coupling could still be relevant for superconductivity~\\cite{Yin_PRX_2013}.\nAnother important feature for low-$T_{\\mathrm{c}}$ materials\nis the average number of valence electrons.\nThis recovers the empirical relation first discovered by Matthias more than sixty years ago~\\cite{Matthias_PR_1955}.\nSuch findings validate the ability of ML approaches\nto discover meaningful patterns that encode true physical phenomena.\n\nSimilar $T_{\\mathrm{c}}$-\\nobreak\\mbox{\\it vs.}-predictor plots reveal more interesting and subtle features.\nA narrow cluster of materials with $T_{\\mathrm{c}} > 20$~K emerges in the context of the mean covalent radii of compounds\n(Figure ~\\ref{fig:art137:Tc_atomWeigth}(b)) --- another\nimportant predictor for low-$T_{\\mathrm{c}}$ superconductors.\nThe cluster includes (left-to-right) alkali-doped C$_{60}$, MgB$_2$-related compounds, and bismuthates.\nThe sector likely characterizes a region of strong covalent bonding and corresponding high-frequency phonon modes\nthat enhance $T_{\\mathrm{c}}$ (however, frequencies that are too high become irrelevant for superconductivity).\nAnother interesting relation appears in the context of the average number of $d$ valence electrons.\nFigure~\\ref{fig:art137:Tc_atomWeigth}(c) illustrates a fundamental bound on\n$T_{\\mathrm{c}}$ of all non-cuprate and non-iron-based superconductors.\n\nA similar limit exists for cuprates based on the average number of unfilled orbitals (Figure ~\\ref{fig:art137:Tc_atomWeigth}(d)).\nIt appears to be quite rigid --- several data points found above it on inspection are actually\nincorrectly recorded entries in the database and were subsequently removed.\nThe connection between $T_{\\mathrm{c}}$ and the average number of unfilled orbitals\\footnote{The\nnumber of unfilled orbitals refers to the\nelectron configuration of the substituent elements before combining to form oxides.\nFor example, Cu has one unfilled orbital ([Ar]$4s^23d^9$) and Bi has\nthree ([Xe]$4f^{14}6s^25d^{10}6p^3$).\nThese values are averaged per formula unit.}\nmay offer new insight into the mechanism for superconductivity in this family.\nKnown trends include higher $T_{\\mathrm{c}}$'s for structures that\n\\textbf{i.} stabilize more than one superconducting Cu-O plane per unit cell\nand \\textbf{ii.} add more polarizable cations such as Tl$^{3+}$ and Hg$^{2+}$ between these planes.\nThe connection reflects these observations,\nsince more copper and oxygen per formula unit\nleads to lower average number of unfilled orbitals (one for copper, two for oxygen).\nFurther, the lower-$T_{\\mathrm{c}}$ cuprates typically consist of Cu$^{2-}$\/Cu$^{3-}$-containing\nlayers stabilized by the addition\/substitution of hard cations,\nsuch as Ba$^{2+}$ and La$^{3+}$, respectively.\nThese cations have a large number of unfilled orbitals, thus increasing the compound's average.\nTherefore, the ability of between-sheet cations\nto contribute charge to the Cu-O planes may be indeed quite important.\nThe more polarizable the $A$ cation, the more electron density it can contribute\nto the already strongly covalent Cu$^{2+}$--O bond.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig109}\n\\mycaption[Benchmarking of regression models predicting $\\ln(T_{\\mathrm{c}})$.]\n{(\\textbf{a}) Predicted\n\\nobreak\\mbox{\\it vs.}\\ measured $\\ln(T_{\\mathrm{c}})$ for the general regression model.\nThe test set comprises of a mix of low-$T_{\\mathrm{c}}$, iron-based, and cuprate superconductors\nwith $T_{\\mathrm{c}}>10$~K.\nWith an $R^2$ of about $0.88$, this one model can accurately predict\n$T_{\\mathrm{c}}$ for materials in different superconducting groups.\n(\\textbf{b} and \\textbf{c}) Predictions of the regression model\ntrained solely on low-$T_{\\mathrm{c}}$ compounds\nfor test sets containing cuprate and iron-based materials.\n(\\textbf{d} and \\textbf{e}) Predictions of the regression model\ntrained solely on cuprates for test sets containing low-$T_{\\mathrm{c}}$ and iron-based superconductors.\nModels trained on a single group have no predictive power for materials from other groups.}\n\\label{fig:art137:Rerg_r2}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig110}\n\\mycaption[Scatter plots of $T_{\\mathrm{c}}$ for superconducting materials in the space of significant,\nfamily-specific regression predictors.]\n{For $4,000$ ``low-$T_{\\mathrm{c}}$'' superconductors (\\nobreak\\mbox{\\it i.e.}, non-cuprate and non-iron-based),\n$T_{\\mathrm{c}}$ is plotted\n\\nobreak\\mbox{\\it vs.}\\ the\n(\\textbf{a}) average atomic weight,\n(\\textbf{b}) average covalent radius, and\n(\\textbf{c}) average number of $d$ valence electrons.\nThe dashed red line in (\\textbf{a}) is $\\sim 1\/\\sqrt{m_A}$.\nHaving low average atomic weight and low average number of $d$ valence\nelectrons are necessary (but not sufficient) conditions for achieving high $T_{\\mathrm{c}}$\nin this group.\n(\\textbf{d}) Scatter plot of $T_{\\mathrm{c}}$ for all known superconducting cuprates \\nobreak\\mbox{\\it vs.}\\ the mean number of unfilled orbitals.\n(\\textbf{c} and \\textbf{d}) suggest that the values of these predictors lead to\nhard limits on the maximum achievable $T_{\\mathrm{c}}$.}\n\\label{fig:art137:Tc_atomWeigth}\n\\end{figure}\n\n\\boldsection{Including AFLOW.}\nThe models described previously demonstrate\nsurprising accuracy and predictive power, especially considering the difference between the\nrelevant energy scales of most Magpie predictors (typically in the range of eV) and superconductivity (meV scale).\nThis disparity, however, hinders the interpretability of the models,\n\\nobreak\\mbox{\\it i.e.}, the ability to extract meaningful physical correlations.\nThus, it is highly desirable to create accurate ML models with features based on\nmeasurable macroscopic properties of the actual compounds\n(\\nobreak\\mbox{\\it e.g.}, crystallographic and electronic properties)\nrather than composite elemental predictors.\nUnfortunately, only a small subset of materials in SuperCon\nis also included in the {\\small ICSD}:\nabout $1,500$ compounds in total, only about $800$ with finite $T_{\\mathrm{c}}$,\nand even fewer are characterized with \\nobreak\\mbox{\\it ab initio}\\ calculations\\footnote{Most of the superconductors in\n\\protect{\\small ICSD}\\ but not in \\protect{\\small AFLOW}\\ are non-stoichiometric\/doped compounds, and thus not amenable to conventional {\\small DFT}\\ methods.\nFor the others, \\protect{\\small AFLOW}\\ calculations were attempted but did not converge to a reasonable solution.}.\nIn fact, a good portion of known superconductors are disordered (off-stoichiometric) materials and\nnotoriously challenging to address with {\\small DFT}\\ calculations.\nCurrently, much faster and efficient methods are becoming available~\\cite{curtarolo:art110}\nfor future applications.\n\nTo extract suitable features, data is incorporated from\nthe {\\small AFLOW}\\ Online Repositories --- a database of\n{\\small DFT}\\ calculations managed by the software package {\\small AFLOW}.\nIt contains information for the vast majority of compounds\nin the {\\small ICSD}\\ and about 550 superconducting materials.\nIn Reference~\\onlinecite{curtarolo:art94}, several\nML models using a similar set of materials are presented.\nThough a classifier shows good accuracy, attempts to create a\nregression model for $T_{\\mathrm{c}}$ led to disappointing results.\nWe verify that using Magpie predictors for the superconducting compounds in the {\\small ICSD}\\\nalso yields an unsatisfactory regression model.\nThe issue is not the lack of compounds \\textit{per se}, as\nmodels created with randomly drawn subsets from SuperCon with\nsimilar counts of compounds perform much better.\nIn fact, the\nproblem is the chemical sparsity of superconductors in the {\\small ICSD}, \\nobreak\\mbox{\\it i.e.},\nthe dearth of closely-related compounds (usually created by chemical substitution).\nThis translates to compound scatter in predictor space --- a challenging learning environment for the model.\n\nThe chemical sparsity in {\\small ICSD}\\ superconductors is a significant hurdle, even when both sets of predictors\n(\\nobreak\\mbox{\\it i.e.}, Magpie and {\\small AFLOW}\\ features) are combined via feature fusion.\nAdditionally, this approach alone neglects the majority of the $16,000$ compounds available via SuperCon.\nInstead, we constructed separate models employing\nMagpie and {\\small AFLOW}\\ features, and then judiciously combined the results\nto improve model metrics --- known as late or decision-level fusion.\nSpecifically, two independent classification models are developed,\none using the full SuperCon dataset and Magpie predictors, and another based on\nsuperconductors in the {\\small ICSD}\\ and {\\small AFLOW}\\ predictors.\nSuch an approach can improve the recall, for example, in the case where we classify ``high-$T_{\\mathrm{c}}$''\nsuperconductors as those predicted by either model to be above-$T_{\\mathrm{sep}}$.\nIndeed, this is the case here where, separately, the models obtain a recall of $40\\%$ and $ 66\\%$, respectively, and\ntogether achieve a recall of about $76\\%$\\footnote{These numbers are based on (a relatively small) test set benchmarking and their uncertainty is roughly $3\\%$.}.\nIn this way, the models' predictions complement each other in a constructive way such that\nabove-$T_{\\mathrm{sep}}$ materials missed by one model (but not the other) are now accurately classified.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig111}\n\\mycaption[DOS of four compounds identified by the ML algorithm as potential materials with $T_{\\mathrm{c}} > 20$~K.]\n{The partial DOS contributions from $s$, $p$ and $d$ electrons and total DOS are shown in blue, green, red, and black, respectively.\nThe large peak just below $E_F$ is a direct consequence of the flat band(s) present in all these materials.\nThese images were generated automatically via {\\small AFLOW}~\\cite{curtarolo:art53}.\nIn the case of substantial overlap among \\textbf{k}-point labels, the right-most label is offset below.}\n\\label{fig:art137:flat_bands}\n\\end{figure}\n\n\\boldsection{Searching for new superconductors in the ICSD.}\nAs a final proof of concept demonstration,\nthe classification and regression models\ndescribed previously are integrated in one pipeline\nand employed to screen the entire {\\small ICSD}\\ database for candidate ``high-$T_{\\mathrm{c}}$'' superconductors.\n(Note that ``high-$T_{\\mathrm{c}}$'' is a simple label,\nthe precise meaning of which can be adjusted.)\nSimilar tools power high-throughput screening workflows for materials with desired\nthermal conductivity and magnetocaloric properties~\\cite{curtarolo:art120,Bocarsly_ChemMat_2017}.\nAs a first step, the full set of Magpie predictors are generated for all\ncompounds in SuperCon.\nA classification model similar to the one presented above is constructed,\nbut trained only on materials in SuperCon and not in the {\\small ICSD}\\ (used\nas an independent test set).\nThe model is then applied on the {\\small ICSD}\\ set\nto create a list of materials with predicted $T_{\\mathrm{c}}$ above $10$~K.\nOpportunities for model benchmarking are limited to those\nmaterials both in the SuperCon and {\\small ICSD}\\ datasets, though this test\nset is shown to be problematic.\nThe set includes about 1,500 compounds, with $T_{\\mathrm{c}}$ reported for only about half of them.\nThe model achieves an impressive accuracy of $0.98$, which is overshadowed by the fact that\n$96.6\\%$ of these compounds belong to the $T_{\\mathrm{c}} < 10$~K class.\nThe precision, recall, and $F_{\\mathrm{1}}$ scores are about $0.74$,\n$0.66$, and $0.70$, respectively.\nThese metrics are lower than the estimates\ncalculated for the general classification model,\nwhich is expected given that this set cannot\nbe considered randomly selected.\nNevertheless, the performance suggests a good opportunity to identify new candidate superconductors.\n\n\\begin{table}[tp]\\centering\n\\mycaption[List of potential superconductors identified by the pipeline.]\n{Also shown are their {\\small ICSD}\\ numbers and symmetries.\nNote that for some compounds there are several entries.\nAll of the materials contain oxygen.}\n\\vspace{3mm}\n{\\small\n\\begin{tabular}{l|r|r}\ncompound & {\\small ICSD}\\ & SYM \\\\\n\\hline\nCsBe(AsO$_4$) & 074027 & orthorhombic \\\\\nRbAsO$_2$ & 413150 & orthorhombic \\\\\nKSbO$_2$ & 411214 & monoclinic \\\\\nRbSbO$_2$ & 411216 & monoclinic \\\\\nCsSbO$_2$ & 059329 & monoclinic \\\\\n\\hline\nAgCrO$_2$ & 004149\/025624 & hexagonal \\\\\nK$_{0.8}$(Li$_{0.2}$Sn$_{0.76}$)O$_2$ & 262638 & hexagonal \\\\\n\\hline\nCs(MoZn)(O$_3$F$_3$)& 018082 & cubic \\\\\n\\hline\nNa$_3$Cd$_2$(IrO$_6$) & 404507 & monoclinic \\\\\nSr$_3$Cd(PtO$_6$) & 280518 & hexagonal \\\\\nSr$_3$Zn(PtO$_6$) & 280519 & hexagonal \\\\\n\\hline\n(Ba$_5$Br$_2)$Ru$_2$O$_9$ & 245668 & hexagonal \\\\\n\\hline\nBa$_4$(AgO$_2$)(AuO$_4)$ & 072329 & orthorhombic \\\\\nSr$_5$(AuO$_4$)$_2$ & 071965 & orthorhombic \\\\\n\\hline\nRbSeO$_2$F & 078399 & cubic \\\\\nCsSeO$_2$F & 078400 & cubic \\\\\nKTeO$_2$F & 411068 & monoclinic \\\\\n\\hline\nNa$_2$K$_4$(Tl$_2$O$_6$) & 074956 & monoclinic \\\\\n\\hline\nNa$_3$Ni$_2$BiO$_6$ & 237391 & monoclinic \\\\\nNa$_3$Ca$_2$BiO$_6$ & 240975 & orthorhombic\\\\\n\\hline\n\nCsCd(BO$_3$) & 189199 & cubic \\\\\n\\hline\nK$_2$Cd(SiO$_4)$ & 083229\/086917 & orthorhombic \\\\\nRb$_2$Cd(SiO$_4$) & 093879 & orthorhombic \\\\\nK$_2$Zn(SiO$_4$) & 083227 & orthorhombic \\\\\nK$_2$Zn(Si$_2$O$_6$) & 079705 & orthorhombic \\\\\n\\hline\n\nK$_2$Zn(GeO$_4$) & 069018\/085006\/085007 & orthorhombic \\\\\n(K$_{0.6}$Na$_{1.4})$Zn(GeO$_4)$ & 069166 & orthorhombic \\\\\nK$_2$Zn(Ge$_2$O$_6$) & 065740 & orthorhombic \\\\\nNa$_6$Ca$_3$(Ge$_2$O$_6$)$_3$ & 067315 & hexagonal \\\\\nCs$_3$(AlGe$_2$O$_7$) & 412140 & monoclinic \\\\\nK$_4$Ba(Ge$_3$O$_9$) & 100203 & monoclinic \\\\\nK$_{16}$Sr$_4$(Ge$_3$O$_9$)$_{4}$ & 100202 & cubic \\\\\nK$_3$Tb[Ge$_3$O$_8$(OH)$_2$] & 193585 & orthorhombic \\\\\nK$_3$Eu[Ge$_3$O$_8$(OH)$_2$] & 262677 & orthorhombic \\\\\n\\hline\nKBa$_6$Zn$_4$(Ga$_7$O$_{21}$) & 040856 & trigonal \\\\\n\\end{tabular}}\n\\label{tab:art137:Table3}\n\\end{table}\n\nNext in the pipeline, the list is fed into a random forest regression\nmodel (trained on the entire SuperCon database)\nto predict $T_{\\mathrm{c}}$.\nFiltering on the materials with $T_{\\mathrm{c}} > 20$~K,\nthe list is further reduced to about 2,000 compounds.\nThis count may appear daunting, but should\nbe compared with the total number of compounds in the database --- about 110,000.\nThus, the method selects less than two percent of all materials,\nwhich in the context of the training set (containing more than $20\\%$ with ``high-$T_{\\mathrm{c}}$''),\nsuggests that the model is not overly biased toward predicting high critical temperatures.\n\nThe vast majority of the compounds identified as\ncandidate superconductors are cuprates,\nor at least compounds that contain copper and oxygen.\nThere are also some materials clearly related to the iron-based superconductors.\nThe remaining set has 35 members, and is composed of materials that are not obviously\nconnected to any high-temperature superconducting families (see Table~\\ref{tab:art137:Table3})\\footnote{For at least one compound\nfrom the list --- Na$_3$Ni$_2$BiO$_6$ --- low-temperature measurements have been performed and no signs\nof superconductivity were observed~\\cite{Seibel_InChem_2013}.}~\\nocite{Seibel_InChem_2013}.\nNone of them is predicted to have\n$T_{\\mathrm{c}}$ in excess of $40$~K, which is not surprising, given that no such instances exist in the training dataset. All contain oxygen --- also not a surprising result, since the group of\nknown superconductors with $T_{\\mathrm{c}} > 20$~K is dominated by oxides.\n\nThe list comprises several distinct groups.\nMost of the materials are insulators, similar to stoichiometric (and underdoped) cuprates that\ngenerally require charge doping and\/or pressure to drive these materials into a superconducting state.\nEspecially interesting are the compounds containing heavy metals (such as Au, Ir, Ru), metalloids (Se, Te),\nand heavier post-transition metals (Bi, Tl), which are or could be pushed into interesting\/unstable oxidation states.\nThe most surprising and non-intuitive of the compounds in the list are the silicates and the germanates.\nThese materials form corner-sharing SiO$_4$ or GeO$_4$ polyhedra, similar to quartz glass,\nand also have counter cations with full or empty shells such as Cd$_2$$^+$ or K$^+$.\nConverting these insulators to metals (and possibly superconductors) likely requires\nsignificant charge doping. However, the similarity between these compounds and cuprates is meaningful.\nIn compounds like K$_2$CdSiO$_4$ or K$_2$ZnSiO$_4$, K$_2$Cd (or K$_2$Zn) unit carries\na 4+ charge that offsets the (SiO$_4$)$^{4-}$ (or (GeO$_4$)$^{4-}$) charges.\nThis is reminiscent of the way Sr$_2$ balances the (CuO$_4$)$^{4-}$ unit in Sr$_2$CuO$_4$.\nSuch chemical similarities based on charge balancing and stoichiometry were likely identified and exploited by the ML algorithms.\n\nThe electronic properties calculated by {\\small AFLOW}\\ offer additional insight into the results of the search, and suggest a possible connection among these candidate.\nPlotting the electronic structure of the potential superconductors exposes a rather unusual feature shared\nby almost all --- one or several (nearly) flat bands just below the energy of the highest occupied electronic state\\footnote{The\nflat band attribute is unusual for a superconducting material: the average DOS of the known superconductors in the \\protect{\\small ICSD}\\\n(at least those available in the \\protect{\\small AFLOW}\\ Online Repositories) has no distinct features, demonstrating roughly uniform distribution of electronic states.\nIn contrast, the average DOS of the potential superconductors in Table~\\ref{tab:art137:Table3} shows a sharp peak just below $E_{\\mathrm{F}}$.\nAlso, most of the flat bands in the potential superconductors we discuss have a notable contribution from the oxygen $p$-orbitals.\nAccessing\/exploiting the potential strong instability this electronic structure feature creates can require significant charge doping.}.\nSuch bands lead to a large peak in the DOS (see Figure~\\ref{fig:art137:flat_bands}) and\ncan cause a significant enhancement in $T_{\\mathrm{c}}$.\nPeaks in the DOS elicited by van Hove singularities can enhance $T_{\\mathrm{c}}$\nif sufficiently close to $E_{\\mathrm{F}}$~\\cite{Labbe_PRL_1967,Hirsch_PRL_1986,Dzyaloshinskii_JETPLett_1987}.\nHowever, note that unlike typical van Hove points, a true flat band creates divergence\nin the DOS (as opposed to its derivatives), which in turn leads to a critical temperature\ndependence that is linear in the pairing interaction strength, rather than the usual exponential relationship\nyielding lower $T_{\\mathrm{c}}$~\\cite{Kopnin_PRB_2011}.\nAdditionally, there is significant similarity\nwith the band structure and DOS of layered\nBiS$_2$-based superconductors~\\cite{Yazici_PSCC_2015}.\n\nThis band structure feature came as the surprising\nresult of applying the ML model.\nIt was not sought for, and, moreover,\nno explicit information about the electronic band structure has been\nincluded in these predictors.\nThis is in contrast to the algorithm presented in Reference~\\onlinecite{Klintenberg_CMS_2013},\nwhich was specifically designed to filter {\\small ICSD}\\ compounds based on several preselected electronic structure features.\n\nWhile at the moment it is not clear if some (or indeed any) of these compounds are really superconducting,\nlet alone with $T_{\\mathrm{c}}$'s above 20~K,\nthe presence of this highly unusual electronic structure feature is encouraging.\nAttempts to synthesize several of these compounds are already underway.\n\n\\subsection{Discussion}\nHerein, several machine learning tools are developed to study the critical temperature of superconductors.\nBased on information from the SuperCon database, initial coarse-grained\nchemical features are generated using the Magpie software.\nAs a first application of ML methods, materials are divided into two classes depending on\nwhether $T_{\\mathrm{c}}$ is above or below $10$~K.\nA non-parametric random forest classification model is constructed\nto predict the class of superconductors.\nThe classifier shows excellent performance, with out-of-sample accuracy and $F_{\\mathrm{1}}$\nscore of about $92\\%$.\nNext,\nseveral successful random forest regression models are created to predict the value of $T_{\\mathrm{c}}$,\nincluding separate models for three material sub-groups, \\nobreak\\mbox{\\it i.e.},\ncuprate, iron-based, and low-$T_{\\mathrm{c}}$ compounds.\nBy studying the importance of predictors for each family of superconductors,\ninsights are obtained about the\nphysical mechanisms driving superconductivity among the different groups.\nWith the incorporation of crystallographic-\/electronic-based features\nfrom the {\\small AFLOW}\\ Online Repositories, the ML models are further improved.\nFinally, we combined these models into one integrated pipeline, which is employed to search the entire\n{\\small ICSD}\\ database for new inorganic superconductors.\nThe model identified 35 oxides as candidate materials.\nSome of these are chemically and structurally similar to cuprates (even though no explicit structural information was provided during training of the model). Another feature that unites almost all of these materials is the presence of flat or nearly-flat bands just below the energy of the highest occupied electronic state.\n\nIn conclusion, this work demonstrates the important role\nML models can play in superconductivity research.\nRecords collected over several decades in SuperCon and other relevant databases can be consumed by ML models,\ngenerating insights and promoting better understanding of the connection\nbetween materials' chemistry\/structure and superconductivity.\nApplication of sophisticated ML algorithms has the potential to dramatically accelerate\nthe search for candidate high-temperature superconductors.\n\n\\subsection{Methods}\n\n\\boldsection{Superconductivity data.}\nThe SuperCon database consists of two separate subsets: ``Oxide \\& Metallic''\n(inorganic materials containing metals, alloys, cuprate high-temperature superconductors, \\nobreak\\mbox{\\it etc.})\nand ``Organic'' (organic superconductors).\nDownloading the entire inorganic materials dataset and removing compounds with\nincompletely-specified chemical compositions leaves about $22,000$ entries.\nIf a single $T_{\\mathrm{c}}$ record exists for a given material, it is taken to accurately reflect the critical temperature of this material.\nIn the case of multiple records for the same compound,\nthe reported material's $T_{\\mathrm{c}}$'s are averaged, but only if\ntheir standard deviation is less than $5$~K, and discarded otherwise.\nThis brings the total down to about $16,400$ compounds,\nof which around $4,000$ have no critical temperature reported. Each entry in the set contains fields for the chemical composition,\n$T_{\\mathrm{c}}$, structure, and a journal reference to the information source.\nHere, structural information is ignored as it is not always available.\n\nThere are occasional problems with the validity and consistency of some of the data.\nFor example, the database includes some reports based on tenuous experimental evidence and\nonly indirect signatures of superconductivity, as well as reports of inhomogeneous (surface, interfacial)\nand nonequilibrium phases.\nEven in cases of \\textit{bona fide} bulk superconducting phases, important relevant variables\nlike pressure are not recorded.\nThough some of the obviously erroneous records were removed from the data,\nthese issues were largely ignored\nassuming their effect on the entire dataset to be relatively modest. The data cleaning and processing is carried out using the Python Pandas package for data analysis~\\cite{Mckinney_Pandas_2012}.\n\n\\boldsection{Chemical and structural features.}\nThe predictors are calculated using the Magpie software \\cite{magpie_software}.\nIt computes a set of 145 attributes\nfor each material, including:\n\\textbf{i.} stoichiometric features (depends only on the ratio of elements and\nnot the specific species);\n\\textbf{ii.} elemental property statistics: the mean, mean absolute deviation, range, minimum,\nmaximum, and mode of 22 different elemental properties\n(\\nobreak\\mbox{\\it e.g.}, period\/group on the periodic table,\natomic number, atomic radii, melting temperature);\n\\textbf{iii.} electronic structure attributes: the average\nfraction of electrons from the $s$, $p$, $d$ and $f$ valence shells among all\nelements present; and\n\\textbf{iv.} ionic compound features that include whether it is possible to form an ionic\ncompound assuming all elements exhibit a single oxidation state.\n\nML models are also constructed with the\nsuperconducting materials in the {\\small AFLOW}\\ Online Repositories.\n{\\small AFLOW}\\ is a high-throughput \\nobreak\\mbox{\\it ab initio}\\ framework that manages density functional theory ({\\small DFT})\ncalculations in accordance with the {\\small AFLOW}\\ Standard~\\cite{curtarolo:art104}.\nThe Standard ensures that the calculations and derived properties are empirical (reproducible), reasonably\nwell-converged, and above all, consistent (fixed set of parameters), a particularly attractive feature for ML modeling.\nMany materials properties important for superconductivity have been calculated within the {\\small AFLOW}\\ framework,\nand are easily accessible through the {\\small AFLOW}\\ Online Repositories.\nThe features are built with the following properties:\nnumber of atoms, space group, density, volume, energy per atom, electronic entropy per atom, valence of the cell,\nscintillation attenuation length, the ratios of the unit cell's dimensions, and Bader charges and volumes.\nFor the Bader charges and volumes (vectors), the following statistics\nare calculated and incorporated:\nthe maximum, minimum, average, standard deviation, and range.\n\n\\boldsection{Machine learning algorithms.}\nOnce we have a list of relevant predictors, various ML models can be applied to the\ndata~\\cite{Bishop_ML_2006,Hastie_StatLearn_2001}.\nAll ML algorithms in this work are\nvariants of the random forest method~\\cite{randomforests}.\nIt is based on creating a set of individual decision trees (hence the ``forest''),\neach built to solve the same classification\/regression problem.\nThe model then combines their results, either by voting or averaging depending on the problem.\nThe deeper individual tree are, the more complex the relationships the model can learn,\nbut also the greater the danger of overfitting, \\nobreak\\mbox{\\it i.e.}, learning\nsome irrelevant information or just ``noise''.\nTo make the forest more robust to overfitting, individual trees in the ensemble are\nbuilt from samples drawn with replacement (a bootstrap sample) from the training set.\nIn addition, when splitting a node during the construction of a tree, the model chooses the best split\nof the data only considering a random subset of the features.\n\nThe random forest models above are developed using scikit-learn --- a powerful and efficient machine\nlearning Python library \\cite{Pedregosa_JMLR_2011}.\nHyperparameters of these models include the number of trees in the forest,\nthe maximum depth of each tree, the minimum number of samples required to split an internal node,\nand the number of features to consider when looking for the best split.\nTo optimize the classifier and the combined\/family-specific regressors, the\nGridSearch function in scikit-learn is employed, which generates and compares candidate models from a grid of parameter values.\nTo reduce computational expense, models are not optimized at each step of the backward feature selection process.\n\nTo test the influence of using log-transformed target variable $\\ln(T_{\\mathrm{c}})$,\na general regression model is trained and tested on raw $T_{\\mathrm{c}}$ data.\nThis model is very similar to the one described in section ``Results'', and its $R^2$ value is fairly similar as well\n(although comparing $R^2$ scores of models built using different target data can be misleading).\nHowever, note the relative sparsity of data points in some $T_{\\mathrm{c}}$ ranges, which makes the model susceptible to outliers.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig112}\n\\mycaption[Regression model predictions of $T_{\\mathrm{c}}$.]\n{Predicted\n\\nobreak\\mbox{\\it vs.}\\ measured $T_{\\mathrm{c}}$ for general regression model.\n$R^2$ score is comparable to the one obtained testing regression modeling $\\ln(T_{\\mathrm{c}})$.}\n\\label{fig:art137:Regr_non_loc}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig113}\n\\mycaption[Histograms of $\\Delta\\ln(T_{\\mathrm{c}}) * \\ln(T_{\\mathrm{c}})^{-1}$ for the four regression models.]\n{$\\Delta\\ln(T_{\\mathrm{c}}) \\equiv (\\ln(T^{\\mathrm{meas}}_{\\mathrm{c}}) - \\ln(T^{\\mathrm{pred}}_{\\mathrm{c}}))$\nand $\\ln(T_{\\mathrm{c}}) \\equiv \\ln(T^{\\mathrm{meas}}_{\\mathrm{c}})$.}\n\\label{fig:art137:Regr_err}\n\\end{figure}\n\n\\boldsection{Prediction errors of the regression models.}\nPreviously, several regression models were described,\neach one designed to predict the critical temperatures of materials from different superconducting groups.\nThese models achieved an impressive $R^{2}$ score, demonstrating\ngood predictive power for each group.\nHowever, it is also important to consider the accuracy of the predictions\nfor individual compounds (rather than on the aggregate set),\nespecially in the context of searching for new materials.\nTo do this, we calculate the prediction errors for about 300 materials from a test set.\nSpecifically, we consider the difference between the logarithm of the predicted and measured\ncritical temperature $[\\ln(T^{\\mathrm{meas}}_{\\mathrm{c}})- \\ln(T^{\\mathrm{pred}}_{\\mathrm{c}})]$\nnormalized by the value of $\\ln(T^{\\mathrm{meas}}_{\\mathrm{c}})$\n(normalization compensates the different $T_{\\mathrm{c}}$ ranges of different groups).\nThe models show comparable spread of errors.\nThe histograms of errors for the four models\n(combined and three group-specific) are shown in Fig.~\\ref{fig:art137:Regr_err}.\nThe errors approximately follow a normal distribution,\ncentered not at zero but at a small negative value.\nThis suggests the models are marginally biased, and on average tend to slightly underestimate $T_{\\mathrm{c}}$.\nThe variance is comparable for all models, but largest for the model trained\nand tested on iron-based materials, which also shows the smallest $R^2$.\nPerformance of this model is expected to benefit from a larger training set.\n\n\\boldsection{Data availability.} The superconductivity data used to generate the results\nin this work can be downloaded from \\url{https:\/\/github.com\/vstanev1\/Supercon}.\n\\clearpage\n\\section{High Throughput Thermal Conductivity of High Temperature Solid Phases: The Case of Oxide and Fluoride Perovskites}\n\\label{sec:art120}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art120}.\n\n\\subsection{Introduction}\n\\label{subsec:art120:Introduction}\n\nHigh throughput \\nobreak\\mbox{\\it ab-initio}\\ screening of materials is a new and rapidly\ngrowing discipline~\\cite{nmatHT}. Amongst the basic properties\nof materials, thermal conductivity is a particularly relevant one.\nThermal management is a crucial factor to a vast range of technologies,\nincluding power electronics, CMOS interconnects, thermoelectric energy\nconversion, phase change memories, turbine thermal coatings and many\nothers~\\cite{Cahill_APR_2014}. Thus, rapid determination\nof thermal conductivity for large pools of compounds is a desirable\ngoal in itself, which may enable the identification of suitable compounds\nfor targeted applications. A few recent works have investigated thermal\nconductivity in a high throughput fashion~\\cite{aflowKAPPA,Seko_PRL_2015}.\nA drawback of these studies is that they were restricted to use the\nzero Kelvin phonon dispersions. This is often fine when the room temperature\nphase is mechanically stable at 0~K. It however poses a problem\nfor materials whose room or high temperature phase is not the 0~K\nstructure: when dealing with structures exhibiting displacive distortions,\nincluding temperature effects in the phonon spectrum is a crucial\nnecessity.\n\nSuch a phenomenon often happens for perovskites. Indeed, the perovskite\nstructure can exhibit several distortions from the ideal cubic lattice,\nwhich is often responsible for rich phase diagrams. When the structure\nis not stable at low temperatures, a simple computation of the phonon\nspectrum using forces obtained from density functional theory and\nthe finite displacement method yields imaginary eigenvalues. This\nprevents us from assessing the mechanical stability of those compounds\nat high temperatures or calculating their thermal conductivity. Moreover,\ntaking into account finite-temperature effects in phonon calculations\nis currently a very demanding task, especially for a high-throughput\ninvestigation.\n\nIn this study, we are interested in the \\textit{high-temperature}\nproperties of perovskites, notably for thermoelectric applications.\nFor this reason, we focus on perovskites with the highest symmetry\ncubic structure, which are most likely to exist at high temperatures\n\\cite{Landau_CTP5_SP_1969,Howard_ActaCrisA_2005,Thomas_PRL_1968,Cochran_PSSB_1968,Angel_PRL_2005}.\nWe include the effects of anharmonicity in our \\nobreak\\mbox{\\it ab-initio}\\ calculations\nof mechanical and thermal properties.\n\n\\subsection{Finite-temperature calculations of mechanical stability and thermal properties}\n\\label{subsec:art120:Finite-T_calculations}\n\nRecently, several methods have been developed to deal with anharmonic\neffects at finite temperatures in solids~\\cite{Souvatzis_PRL_2008,Hellman_PRB_2011,Hellman_PRB_2013,Errea_PRL_2013,Tadano_PRB_2015,VanRoekeghem_ARXIV_2016}.\nIn this study, we use the method presented in Reference~\\cite{VanRoekeghem_ARXIV_2016}\nto compute the temperature-dependent interatomic force constants,\nwhich uses a regression analysis of forces from density functional\ntheory coupled with a harmonic model of the quantum canonical ensemble.\nThis is done in an iterative way to achieve self-consistency of the\nphonon spectrum.\nThe workflow is summarized in Figure~\\ref{fig:art120:finite-T-phonon}.\nIn the following (in particular Section~\\ref{subsec:art120:PCA-regression}),\nit will be referred as ``SCFCS'' -- standing for self-consistent\nforce constants. As a trade-off between accuracy and throughput, we\nchoose a 3x3x3 supercell and a cutoff of 5~\\AA\\ for the third order\nforce constants. Special attention is paid to the computation of the\nthermal displacement matrix~\\cite{VanRoekeghem_ARXIV_2016}, due to the imaginary\nfrequencies that can appear during the convergence process, as well\nas the size of the supercell that normally prevents us from sampling\nthe usual soft modes at the corners of the Brillouin zone (see Supplementary Material of Reference~\\cite{curtarolo:art120}).\nThis allows us to assess the stability at 1000~K of the\n391 hypothetical compounds mentioned in Section~\\ref{subsec:art120:Introduction}.\nAmong this set, we identify 92 mechanically stable compounds, for\nwhich we also check the stability at 300~K. The phonon spectra of\nthe stable compounds are provided in the Supplementary Material of Reference~\\cite{curtarolo:art120}.\nFurthermore, we compute the thermal conductivity using the finite temperature force\nconstants and the full solution of the Boltzmann transport equation\nas implemented in the ShengBTE code~\\cite{Li_ShengBTE_CPC_2014}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig114}\n\\mycaption{Workflow of the method used to calculate the phonon spectrum and thermal\nconductivity including finite-temperature anharmonic effects.}\n\\label{fig:art120:finite-T-phonon}\n\\end{figure}\n\nWe list the stable compounds and their thermal conductivities in Table\n\\ref{tab:art120:List-of-perovskites}.\nRemarkably, this list contains 37\nperovskites that have been reported experimentally in the ideal cubic\nstructure (see References in Table~\\ref{tab:art120:List-of-perovskites}),\nwhich lends support to our screening method.\nOn the other hand, we\nalso find that 11 compounds are reported only in a non-perovskite\nform. This is not necessarily indicative of mechanical instability,\nbut instead suggests thermodynamical stability may be an issue for\nthese compounds, at least near this temperature and pressure. 36 compounds\nremain unreported experimentally in the literature to our knowledge.\nThus, by screening only for mechanical stability at high-temperatures,\nwe reduce the number of potential new perovskites by a factor of 10.\nFurthermore, we find that 50 of them are mechanically stable in the\ncubic form close to room temperature.\n\nOf the full list of perovskites, only a few measurements of thermal\nconductivity are available in the literature. They are displayed in\nparentheses in Table~\\ref{tab:art120:List-of-perovskites} along with their\ncalculated values. Our method tends to slightly underestimate the\nvalue of the thermal conductivity, due to the compromises we made\nto limit the computational cost of the study (see Supplementary Material of Reference~\\cite{curtarolo:art120}).\nThis discrepancy could also be partially related to the electronic\nthermal conductivity, which was not subtracted in the measurements.\nStill, we expect the order of magnitude of the thermal conductivity\nand the relative classification of different materials to be consistent.\nMore importantly, this large dataset allows us to analyze the global\ntrends driving thermal conductivity. These trends are discussed in\nSection~\\ref{subsec:art120:Descriptors}.\n\n\\newcommand{\\fluoridestabfootone}{\nAuMgF$_{3}$ was mentioned theoretically in Reference~\\cite{Uetsuji_TJSME_2006}.}\n\\newcommand{\\fluoridestabfoottwo}{\nThe thermal diffusivity of BaLiF$_{3}$ was measured at 300~K in\nReference~\\cite{Duarte_MSEB_1994} as $\\alpha$=0.037~cm$^{2}$s$^{-1}$.}\n\n\\clearpage\n\n\\begin{table}[tp]\\centering\n\\mycaption[List of cubic perovskites found to be mechanically stable at 1000~K\nand their corresponding computed lattice thermal conductivity (in\nW\/m\/K).]\n{We also report the computed lattice thermal conductivity at\n300~K (in W\/m\/K) when we obtain stability at that temperature. We\nhighlight in blue the compounds that are experimentally reported in\nthe ideal cubic perovskite structure, and in red those that are reported\nonly in non-perovskite structures (references provided in the table).\nWhen no reference is provided, no mention of the compound in this\nstoichiometry has been found in the experimental literature. Experimental\nmeasurements of the thermal conductivity are reported in parentheses,\nand in italics when the structure is not cubic.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|rcl|r|r|r|r|rcl|r|r|r|r|r}\n & $\\kappa_{1000}$ & & $\\kappa_{300}$ & & References & & & $\\kappa_{1000}$ & & $\\kappa_{300}$ & & References & & & $\\kappa_{1000}$ & & $\\kappa_{300}$ & & References\\tabularnewline\n\\cline{1-6} \\cline{8-13} \\cline{15-20}\n\\textcolor{blue}{CaSiO$_{3}$} & 4.89 & & & & \\cite{Komabayashi_EPSL_2007} & & CdYF$_{3}$ & 1.29 & & 3.51 & & & & TlOsF$_{3}$ & 0.62 & & 0.95 & & \\tabularnewline\n\\textcolor{blue}{RbTaO$_{3}$} & 3.61 & & & & \\cite{Lebedev_PhysSolStat_2015} & & \\textcolor{blue}{RbCaF$_{3}$} & 1.15 & & 2.46 & (3.2) & \\cite{Ludekens_ActaCrist_1952,Ridou_Ferroelectrics_1976,Martin_Phonons_1976} & & InZnF$_{3}$ & 0.61 & & 1.86 & & \\tabularnewline\n\\textcolor{blue}{NaTaO$_{3}$} & 3.45 & & & & \\cite{Kennedy_JPCM_1999} & & HgInF$_{3}$ & 1.15 & & 3.85 & & & & \\textcolor{blue}{CsCdF$_{3}$} & 0.59 & & 1.73 & & \\cite{Rousseau_PRB_1975}\\tabularnewline\n\\textcolor{red}{CuCF$_{3}$} & 3.32 & & 8.79 & & \\cite{Zanardi_JACS_2011} & & AlFeF$_{3}$ & 1.14 & & & & & & AlMgF$_{3}$ & 0.56 & & & & \\tabularnewline\n\\textcolor{blue}{SrSiO$_{3}$} & 3.23 & & 10.10 & & \\cite{Xiao_AM_2013} & & \\textcolor{blue}{PbHfO$_{3}$} & 1.12 & & & & \\cite{Kwapulinski_JPCM_1994} & & AuZnF$_{3}$ & 0.53 & & & & \\tabularnewline\n\\textcolor{blue}{NaNbO$_{3}$} & 3.05 & & & (\\textit{1.5}) & \\cite{Shirane_PR_1954,Mishra_PRB_2011,Tachibana_APL_2008} & & \\textcolor{blue}{AgMgF$_{3}$} & 1.11 & & & & \\cite{Portier_CRASC_1970} & & InOsF$_{3}$ & 0.52 & & & & \\tabularnewline\n\\textcolor{blue}{BaHfO$_{3}$} & 3.04 & (4.5) & 8.26 & (10.4) & \\cite{Maekawa_BaHfO3_SrHfO3_JAC_2006} & & ZnScF$_{3}$ & 1.10 & & 3.66 & & & & \\textcolor{blue}{RbSrF$_{3}$} & 0.51 & & & & \\cite{Pies_Landolt_Bornstein_1973}\\tabularnewline\n\\textcolor{blue}{KNbO$_{3}$} & 2.94 & & & (\\textit{10}) & \\cite{Shirane_PR_1954,Tachibana_APL_2008} & & \\textcolor{blue}{RbFeF$_{3}$} & 1.09 & & 4.62 & & \\cite{Kestigian_IC_1966} & & \\textcolor{blue}{CsSrF$_{3}$} & 0.50 & & 1.13 & & \\cite{Pies_Landolt_Bornstein_1973}\\tabularnewline\n\\textcolor{red}{TlTaO$_{3}$} & 2.86 & & & & \\cite{Ramadass_SSC_1975} & & \\textcolor{black}{TlMgF$_{3}$} & 1.06 & & 3.42 & & \\cite{Arakawa_JPCM_2006} & & BeYF$_{3}$ & 0.48 & & 2.34 & & \\tabularnewline\n\\textcolor{blue}{AgTaO$_{3}$} & 2.77 & & & & \\cite{Kania_PT_1981,Pawelczyk_PT_1987} & & \\textcolor{blue}{KCaF$_{3}$} & 1.06 & & & & \\cite{Demetriou_SSI_2005} & & BeScF$_{3}$ & 0.48 & & 1.59 & & \\tabularnewline\n\\textcolor{blue}{KMgF$_{3}$} & 2.74 & & 8.25 & (10) & \\cite{Wood_JACryst_2002,Martin_Phonons_1976} & & HgScF$_{3}$ & 1.01 & & 5.42 & & & & \\textcolor{blue}{TlCdF$_{3}$} & 0.44 & & & & \\cite{Rousseau_PRB_1975}\\tabularnewline\n\\textcolor{red}{GaTaO$_{3}$} & 2.63 & & & & \\cite{Xu_Thesis_2000,Armiento_PRB_2011,Castelli_EES_2012} & & \\textcolor{blue}{CsCaF$_{3}$} & 0.98 & & 3.03 & & \\cite{Rousseau_SSC_1981} & & \\textcolor{blue}{RbHgF$_{3}$} & 0.43 & & & & \\cite{Hoppe_ZAAC_1969}\\tabularnewline\n\\textcolor{blue}{BaTiO$_{3}$} & 2.51 & & 4.99 & (\\textit{4-5}) & \\cite{Tachibana_APL_2008,Strukov_JPCM_2003} & & AuMgF$_{3}$ & 0.96 & & & & \\tablefootnote{\\fluoridestabfootone} & & PdYF$_{3}$ & 0.43 & & 0.99 & & \\tabularnewline\n\\textcolor{blue}{PbTiO$_{3}$} & 2.42 & & & (\\textit{5}) & \\cite{Tachibana_APL_2008} & & InMgF$_{3}$ & 0.96 & & 3.53 & & & & AlZnF$_{3}$ & 0.39 & & & & \\tabularnewline\n\\textcolor{blue}{SrTiO$_{3}$} & 2.36 & (4) & 6.44 & (10.5) & \\cite{Muta_JAC_2005,Popuri_RSCA_2014,Yamanaka_JSSC_2004} & & \\textcolor{blue}{RbZnF$_{3}$} & 0.91 & & 2.64 & & \\cite{Daniel_PRB_1995} & & \\textcolor{black}{KHgF$_{3}$} & 0.37 & & & & \\cite{Hoppe_ZAAC_1969}\\tabularnewline\n\\textcolor{blue}{SrHfO$_{3}$} & 2.20 & (\\textit{2.7}) & & (\\textit{5.2}) & \\cite{Kennedy_PRB_1999,Yamanaka_JSSC_2004} & & ZnInF$_{3}$ & 0.88 & & 1.89 & & & & \\textcolor{red}{RbSnF$_{3}$} & 0.37 & & 0.82 & & \\cite{Tran_JSSC_2014}\\tabularnewline\n\\textcolor{blue}{BaZrO$_{3}$} & 2.13 & (2.9) & 5.61 & (5.2) & \\cite{Yamanaka_JAC_2003} & & \\textcolor{black}{BaSiO$_{3}$} & 0.87 & & & & \\cite{Yusa_AM_2007} & & ZnBiF$_{3}$ & 0.37 & & 1.29 & & \\tabularnewline\nXeScF$_{3}$ & 1.87 & & 4.40 & & & & TlCaF$_{3}$ & 0.86 & & & & & & \\textcolor{blue}{CsHgF$_{3}$} & 0.37 & & 1.00 & & \\cite{Hoppe_ZAAC_1969}\\tabularnewline\nHgYF$_{3}$ & 1.84 & & 5.37 & & & & CdScF$_{3}$ & 0.85 & & 2.37 & & & & \\textcolor{red}{KSnF$_{3}$} & 0.35 & & & & \\cite{Tran_JSSC_2014}\\tabularnewline\n\\textcolor{blue}{AgNbO$_{3}$} & 1.79 & & & & \\cite{Lukaszewski_PT_1983,Sciau_JPCM_2004} & & XeBiF$_{3}$ & 0.82 & & 2.13 & & & & CdBiF$_{3}$ & 0.33 & & 0.98 & & \\tabularnewline\n\\textcolor{red}{TlNbO$_{3}$} & 1.75 & & & & \\cite{Ramadass_SSC_1975} & & \\textcolor{blue}{AgZnF$_{3}$} & 0.80 & & & & \\cite{Portier_CRASC_1970} & & \\textcolor{black}{RbPbF$_{3}$} & 0.32 & & & & \\cite{Yamane_SSI_2008}\\tabularnewline\n\\textcolor{blue}{KFeF$_{3}$} & 1.72 & & 6.37 & (3.0) & \\cite{Okazaki_JPSJ_1961,Suemune_kappa_JPSJ_1964} & & PdScF$_{3}$ & 0.79 & & 1.63 & & & & BeAlF$_{3}$ & 0.30 & & 1.70 & & \\tabularnewline\nSnSiO$_{3}$ & 1.66 & & 4.22 & & \\cite{Clark_IC_2001,Armiento_PRB_2014} & & \\textcolor{blue}{KCdF$_{3}$} & 0.75 & & & & \\cite{Hidaka_SSC_1977,Hidaka_PT_1990} & & \\textcolor{red}{KPbF$_{3}$} & 0.30 & & & & \\cite{Hull_JPCM_1999}\\tabularnewline\n\\textcolor{red}{PbSiO$_{3}$} & 1.66 & & 3.69 & & \\cite{Mackay_MM_1952,Xiao_AM_2012} & & \\textcolor{blue}{BaLiF$_{3}$} & 0.73 & & 2.21 & \\tablefootnote{\\fluoridestabfoottwo} & \\cite{Mortier_SSC_1994,Duarte_MSEB_1994} & & CsBaF$_{3}$ & 0.29 & & & & \\tabularnewline\n\\textcolor{black}{AuNbO$_{3}$} & 1.56 & & & & \\cite{Wu_AngChemInt_2013} & & HgBiF$_{3}$ & 0.72 & & 2.37 & & & & InCdF$_{3}$ & 0.29 & & & & \\tabularnewline\n\\textcolor{red}{CaSeO$_{3}$} & 1.42 & & & & \\cite{Wildner_NJMA_2007} & & ZnAlF$_{3}$ & 0.72 & & 1.92 & & & & BaCuF$_{3}$ & 0.28 & & & & \\tabularnewline\n\\textcolor{red}{NaBeF$_{3}$} & 1.40 & & 2.53 & & \\cite{ODaniel_NJMMAA_1945,Roy_JACerS_1953} & & GaZnF$_{3}$ & 0.69 & & & & & & \\textcolor{red}{TlSnF$_{3}$} & 0.27 & & 0.63 & & \\cite{Foulon_EJSSIC_1993}\\tabularnewline\n\\textcolor{blue}{RbMgF$_{3}$} & 1.37 & & 4.54 & & \\cite{Shafer_JoPACoS_1969} & & \\textcolor{blue}{RbCdF$_{3}$} & 0.68 & & 1.46 & & \\cite{Rousseau_PRB_1975} & & \\textcolor{blue}{TlHgF$_{3}$} & 0.26 & & & & \\cite{Hebecker_Naturwissenschaften_1973}\\tabularnewline\nGaMgF$_{3}$ & 1.34 & & 2.11 & & & & GaRuF$_{3}$ & 0.67 & & & & & & CdSbF$_{3}$ & 0.26 & & & & \\tabularnewline\n\\textcolor{blue}{KZnF$_{3}$} & 1.33 & & 4.15 & (5.5) & \\cite{Suemune_JPSJ_1964,Martin_Phonons_1976} & & \\textcolor{black}{CsZnF$_{3}$} & 0.67 & & 1.12 & & \\cite{Longo_JSSC_1969} & & \\textcolor{blue}{TlPbF$_{3}$} & 0.22 & & & & \\cite{Buchinskaya_RCR_2004}\\tabularnewline\nZnYF$_{3}$ & 1.32 & & 3.72 & & & & \\textcolor{black}{TlZnF$_{3}$} & 0.64 & & 1.96 & & \\cite{Babel_TlZnF3_1967} & & & & & & & \\tabularnewline\n\\end{tabular}}\n\\label{tab:art120:List-of-perovskites}\n\\end{table}\n\n\\clearpage\n\nWe also investigate the (potentially) negative thermal expansion of\nthese compounds. Indeed, the sign of the coefficient of thermal expansion\n$\\alpha_{\\text{V}}$ is the same as the sign of the weighted Gr\\\"{u}neisen\nparameter $\\gamma$, following $\\alpha_{\\text{V}}=\\frac{\\gamma c_{\\text{V}}\\rho}{K_{\\text{T}}}$,\nwhere $K_{\\text{T}}$ is the isothermal bulk modulus, $c_{\\text{V}}$ is the isochoric\nheat capacity and $\\rho$ is the density~\\cite{Gruneisen_AnnPhys_1912,ashcroft_mermin}.\nThe weighted Gr\\\"{u}neisen parameter is obtained by summing the contributions\nof the mode-dependent Gr\\\"{u}neisen parameters: $\\gamma=\\sum\\gamma_{i}c_{Vi}\/\\sum c_{Vi}$.\nFinally the mode-dependent parameters are related to the volume variation\nof the mode frequency $\\omega_{i}$ via $\\gamma_{i}=-(V\/\\omega_{i})(\\partial\\omega_{i}\/\\partial V)$.\nIn our case, we calculate those parameters directly using the second\nand third order force constants at a given temperature~\\cite{Fabian_PRL_1997,Broido_PRB_2005,Hellman_PRB_2013}:\n\\begin{equation}\n\\gamma_{m}=-\\frac{1}{6\\omega_{m}^{2}}\\sum_{ijk\\alpha\\beta\\gamma}\\frac{\\epsilon_{mi\\alpha}^{*}\\epsilon_{mj\\beta}}{\\sqrt{M_{i}M_{j}}}r_{k}^{\\gamma}\\Psi_{ijk}^{\\alpha\\beta\\gamma}e^{i\\mathbf{q}\\cdot\\mathbf{r}_{j}}\n\\end{equation}\n\nThis approach has been very successful in predicting the thermal expansion\nbehavior in the empty perovskite ScF$_{3}$~\\cite{VanRoekeghem_ARXIV_2016},\nwhich switches from negative to positive around 1100~K~\\cite{Greve_JACS_2010}.\nIn our list of filled perovskites, we have found only two candidates\nwith negative thermal expansion around room temperature: TlOsF$_{3}$\nand BeYF$_{3}$, and none at 1000~K.\nThis shows that filling the perovskite structure is probably detrimental to the negative thermal\nexpansion.\n\nWe also examine the evolution of the thermal conductivity as a function\nof temperature, for the compounds that are mechanically stable at\n300~K and 1000~K. There is substantial evidence that the thermal\nconductivity in cubic perovskites generally decreases more slowly\nthan the model $\\kappa\\propto T^{-1}$ behavior~\\cite{Peierls_AnnPhys_1929,Roufosse_JGPR_1974}\nat high temperatures, in contrast to the thermal conductivity of \\nobreak\\mbox{\\it e.g.}\\\nSi or Ge that decreases faster than $\\kappa\\propto T^{-1}$~\\cite{Glassbrenner_PR_1964}.\nThis happens for instance in SrTiO$_{3}$~\\cite{Muta_JAC_2005,Popuri_RSCA_2014},\nKZnF$_{3}$~\\cite{Suemune_JPSJ_1964,Martin_Phonons_1976},\nKMgF$_{3}$~\\cite{Martin_Phonons_1976}, KFeF$_{3}$~\\cite{Suemune_kappa_JPSJ_1964},\nRbCaF$_{3}$~\\cite{Martin_Phonons_1976}, BaHfO$_{3}$\n\\cite{Maekawa_BaHfO3_SrHfO3_JAC_2006}, BaSnO$_{3}$~\\cite{Maekawa_BaSnO3_JAC_2006}\nand BaZrO$_{3}$~\\cite{Yamanaka_JAC_2003}.\nWe also predicted an anomalous behavior in ScF$_{3}$ using \\nobreak\\mbox{\\it ab-initio}\\ calculations,\ntracing its origin to the important anharmonicity of the soft modes\n\\cite{VanRoekeghem_ARXIV_2016}.\nFigure~\\ref{fig:art120:Kappa} displays several experimentally\nmeasured thermal conductivities from the literature on a logarithmic\nscale, along with the results of our high-throughput calculations.\nAs discussed above, the absolute values of the calculated thermal\nconductivities are generally underestimated, but their relative magnitude\nand the overall temperature dependence are generally consistent. Although\nthe behavior of the thermal conductivity $\\kappa(T)$ is in general\nmore complex than a simple power-law behavior, we model the deviation\nto the $\\kappa\\propto T^{-1}$ law by using a parameter $\\alpha$\nthat describes approximately the temperature-dependence of $\\kappa$\nbetween 300~K and 1000~K as $\\kappa\\propto T^{-\\alpha}$. For\ninstance, in Figure~\\ref{fig:art120:Kappa}, KMgF$_{3}$ appears to have\nthe fastest decreasing thermal conductivity with $\\alpha=0.9$ both\nfrom experiment and calculations, while SrTiO$_{3}$ is closer to\n$\\alpha=0.6$. At present, there are too few experimental measurements\nof the thermal conductivities in cubic perovskites to state that the\n$\\kappa\\propto T^{-\\alpha}$ behavior with $\\alpha<1$ is the general\nrule in this family. However, the large number of theoretical predictions\nprovides a way to assess this trend. Of the 50 compounds that we found\nto be mechanically stable at room temperature, we find a mean $\\alpha\\simeq0.85$,\nsuggesting that this behavior is likely general and correlated to\nstructural characteristics of the perovskites.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=8.5cm]{fig115}\n\\mycaption{A comparison between total thermal conductivities from References~\\cite{Suemune_JPSJ_1964,Martin_Phonons_1976,Muta_JAC_2005,Popuri_RSCA_2014,Maekawa_BaHfO3_SrHfO3_JAC_2006,Yamanaka_JAC_2003},\nhigh-throughput calculations of the lattice thermal conductivity at\n300~K and 1000~K, and model behaviors in $\\kappa\\propto T^{-1}$\nand $\\kappa\\propto T^{-0.7}$.}\n\\label{fig:art120:Kappa}\n\\end{figure}\n\n\\subsection{Accelerating the discovery of stable compounds at high temperature}\n\\label{subsec:art120:PCA-regression}\n\nThrough brute-force calculations of the initial list of 391 compounds,\nwe extracted 92 that are mechanically stable at 1000~K. However,\nthis type of calculation is computationally expensive. Thus, it is\ndesirable for future high-throughput searches of other material classes\nto define a strategy for exploring specific parts of the full combinatorial\nspace. In this section, we propose and test such a strategy based\non an iterative machine-learning scheme using principal component\nanalysis and regression.\n\nWe begin by calculating the second order force constants $\\Phi_{0\\text{~K}}$\nof all compounds using the finite displacement method, which is more\nthan an order of magnitude faster than finite-temperature calculations.\nThis gives us a list of 29 perovskites that are mechanically stable\nin the cubic phase at 0~K. Since this is the highest symmetry phase,\nthey are likely also mechanically stable at high-temperatures\\footnote{However, we note that transitions to other structures can take place,\nin particular with one of hexagonal symmetry, such as in BaTiO$_{3}$\n\\cite{Glaister_PPS_1960}, RbZnF$_{3}$\\cite{Daniel_PRB_1995}\nor RbMgF$_{3}$~\\cite{Shafer_JoPACoS_1969}.\nThis phase transition is of\nfirst order, in contrast to displacive transitions that are of second\norder.}.\nWe calculate their self-consistent finite-temperature force constants\n$\\Phi{}_{1000\\text{~K}}^{\\text{SCFCS}}$ as described in Section~\\ref{subsec:art120:Finite-T_calculations}.\nThis initial set allows us to perform principal component analysis\nof the 0~K force constants so that we obtain a transformation that\nretains the 10 most important components. In a second step, we use\nregression analysis to find a relation between the principal components\nat 0~K and at 1000~K. This finally gives us a model that extracts\nthe principal components of the force constants at 0~K, interpolate\ntheir values at 1000~K, and reconstruct the full force constants\nmatrix at 1000~K: $\\Phi_{1000\\text{~K}}^{\\text{model}}$. We say that this\nmodel has been ``trained'' on the particular set of compounds described\nabove. Applying it to the previously calculated $\\Phi_{0\\text{~K}}$\nfor all compounds, we can efficiently span the full combinatorial\nspace to search for new perovskites with a phonon spectrum that is\nunstable at 0~K but stable at 1000~K. For materials determined\nmechanically stable with $\\Phi_{1000\\text{~K}}^{\\text{model}}$, we calculate\n$\\Phi_{1000\\text{~K}}^{\\text{SCFCS}}$. If the mechanical stability is confirmed,\nwe add the new compound to the initial set and subsequently train\nthe model again with the enlarged set. When no new compounds with\nconfirmed mechanical stability at high temperatures are found, we\nstop the search. This process is summarized in Figure~\\ref{fig:art120:PCA-regression}.\nFollowing this strategy, we find 79 perovskites that are stable according\nto the model, 68 of which are confirmed to be stable by the full calculation.\nThis means that we have reduced the total number of finite-temperature\ncalculations by a factor of 5, and that we have retrieved mechanically\nstable compounds with a precision of 86\\% and a recall of 74\\% \\footnote{Precision is defined as the fraction of true positives in all positives\nreported by the model and recall as the fraction of true positives\nfound using the model with respect to all true positives.}. It allows us to obtain approximate phonon spectra for unstable compounds,\nwhich is not possible with our finite-temperature calculations scheme\n(see Supplementary Material of Reference~\\cite{curtarolo:art120}). It also allows us to find compounds\nthat had not been identified as mechanically stable by the first exhaustive\nsearch due to failures in the workflow. Considering the generality\nof the approach, we expect this method to be applicable to other families\nof compounds as well. Most importantly, it reduces the computational\nrequirements, particularly if the total combinatorial space is much\nlarger than the space of interest.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.6\\linewidth]{fig116}\n\\mycaption{Depiction of strategy for exploring the relevant combinatorial space\nof compounds that are mechanically stable at high temperature.}\n\\label{fig:art120:PCA-regression}\n\\end{figure}\n\n\\subsection{Simple descriptors of the thermal conductivity}\n\\label{subsec:art120:Descriptors}\n\nWe now focus on the analysis of the thermal conductivity data provided\nin Table~\\ref{tab:art120:List-of-perovskites}. We note that this set contains\nabout two times more fluorides than oxides. This was already the case\nafter the first screening in which we kept only the semiconductors,\nand it can be explained by the strong electronegativity of fluorine,\nwhich generally forms ionic solids with the alkali and alkaline earth\nmetals easily, as well as with elements from groups 12, 13 and 14.\nThis is shown on Figure~\\ref{fig:art120:Columns}, in which we display histograms\nof the columns of elements at sites \\textit{A} and \\textit{B} of the\nperovskite in our initial list of paramagnetic semiconductors and\nafter screening for mechanical stability.\n\nWe can also see that the oxides tend to display a higher thermal conductivity\nthan the fluorides, as shown on the density plot of Figure~\\ref{fig:art120:Fluorides_vs_oxides}.\nThis is once again due to the charge of the fluorine ion, which is\nhalf that of the oxygen ion. In a model of a purely ionic solid, this\nwould cause the interatomic forces created by electrostatic interactions\nto be divided by two in fluorides as compared to oxides. This is roughly\nwhat we observe in our calculations of the second order force constants.\nIt translates into smaller phonon frequencies and mean group velocities\nin fluorides as compared to oxides. Fluorides also have smaller heat\ncapacities, due to their larger lattice parameters (see Supplementary material of Reference~\\cite{curtarolo:art120}).\nThose two factors mainly drive the important discrepancy\nof the thermal conductivity between fluorides and oxides. Following\nthe same reasoning, it means that halide perovskites in general should\nhave a very low thermal conductivity.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=8.5cm]{fig117}\n\\mycaption[Column number of the element at site (\\textbf{a}) \\textit{A} and (\\textbf{b}) \\textit{B}\nof the perovskite \\textit{ABX}$_{3}$.]\n{Counts in the initial list of fluorides (red) and oxides (blue) paramagnetic semiconductors and\nafter screening for mechanical stability are shown in violet and cyan, respectively.}\n\\label{fig:art120:Columns}\n\\end{figure}\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=8.5cm]{fig118}\n\\mycaption[Distribution of compounds as a function of the lattice thermal conductivity\nat 1000~K.]\n{The red curve corresponds to the distribution for all\nmechanically stable compounds. The blue curve corresponds to the distribution\nfor fluorides only. The green curve corresponds to the distribution\nfor oxides only.}\n\\label{fig:art120:Fluorides_vs_oxides}\n\\end{figure}\n\nFinally, we analyze the correlations between the thermal conductivity\nand different simple structural descriptors. Figure~\\ref{fig:art120:Correlograms}\ndisplays the correlograms for fluorides and oxides between the following\nvariables: the thermal conductivity $\\kappa$, the thermal conductivity\nin the small grain limit $\\kappa_{\\text{sg}}$~\\cite{curtarolo:art85,aflowKAPPA},\nthe mean phonon group velocity v$_{\\text{g}}$, the heat capacity\nc$_{\\text{V}}$, the root mean square Gr\\\"{u}neisen parameter $\\gamma_{\\text{rms}}$\n\\cite{Madsen_PRB_2014,Madsen_PSSA_2016}, the masses of atoms\nat sites \\textit{A} and \\textit{B} of the perovskite \\textit{ABX}$_{3}$,\ntheir electronegativity, their Pettifor number~\\cite{pettifor:1984},\ntheir ionic radius, the lattice parameter of the compound and its\nelectronic gap. Remarkably, sites \\textit{A} and \\textit{B} play very\ndifferent roles in fluorides and oxides. In particular, the thermal\nconductivity of fluorides is mostly influenced by substitutions of\nthe atom inside the fluorine octahedron (site \\textit{B}), while the\ninterstitial atom at site \\textit{A} has a negligible impact. The\nopposite is true for the oxides. This means that when searching for\nnew compounds with a low lattice thermal conductivity, substitutions\nat the \\textit{A} site of fluorides can be performed to optimize cost\nor other considerations without impacting thermal transport. It is\nalso interesting to note that the gap is largely correlated with the\nelectronegativity of atom \\textit{B}, suggesting the first electronic\nexcitations likely involve electron transfer from the anion to the\n\\textit{B} atom.\n\nCommon to both fluorides and oxides, the lattice parameter is mostly\ncorrelated with the ionic radius of atom \\textit{B} rather than atom\n\\textit{A}. Interestingly, the lattice parameter is larger for fluorides,\nalthough the ionic radius of fluorine is smaller than for oxygen.\nThis is presumably due to partially covalent bonding in oxides (see\n\\nobreak\\mbox{\\it e.g.}, Reference~\\onlinecite{Kolezynski_Ferroelectrics_2005}). In contrast, fluorides\nare more ionic: the mean degree of ionicity of the \\textit{X-B} bond\ncalculated from Pauling's electronegativities~\\cite{Pauling_JACS_1932}\n$e_{X}$ and $e_{B}$ as $I{}_{XB}=100\\left(1-e^{\\left(e_{X}-e_{B}\\right)\/4}\\right)$\nyields a value of 56\\% for oxides \\nobreak\\mbox{\\it vs.}\\ 74\\% for fluorides. Ionicity\nis also reflected by the band structure, as can be seen from the weak\ndispersion and hybridization of the F-2$p$ bands \\footnote{See for instance the band structure of SrTiO$_{3}$~\\cite{vanBenthem_JAP_2001}\ncompared to the one of KCaF$_{3}$~\\cite{Ghebouli_SSS_2015}. In those\ntwo compounds, the degree of ionicity of the \\textit{X}-\\textit{B}\nbond calculated from Pauling's electronegativity is 59\\% and 89\\%,\nrespectively.}. This may explain why the role of atoms at site \\textit{A} and \\textit{B}\nis so different between the two types of perovskites. We think that\nthe more ionic character combined to the small nominal charge in fluorides\nmakes the octahedron cage enclosing the atom \\textit{B} less rigid,\nsuch that the influence of the atom \\textit{B} on the thermal conductivity\nbecomes more significant.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig119}\n\\mycaption[Correlograms among properties of\nmechanically stable (\\textbf{a}) fluorides and\n(\\textbf{b}) oxides at 1000~K.]\n{Properties compared include the thermal conductivity $\\kappa$, the thermal\nconductivity in the small grain limit $\\kappa_{\\text{sg}}$, the mean\nphonon group velocity v$_{\\text{g}}$, the heat capacity c$_{\\text{V}}$,\nthe root mean square Gr\\\"{u}neisen parameter $\\gamma_{\\text{rms}}$,\nthe masses m$_{A}$ and m$_{B}$\nof atoms at sites \\textit{A} and \\textit{B} of the perovskite \\textit{ABX}$_{3}$,\ntheir electronegativity e$_{A}$, e$_{B}$,\ntheir Pettifor scale $\\chi_{A}$, $\\chi_{B}$,\ntheir ionic radius\nr$_{A}$, r$_{B}$,\nthe lattice parameter of the compound a$_{\\text{latt}}$\nand its electronic gap.}\n\\label{fig:art120:Correlograms}\n\\end{figure}\n\n\\subsection{Conclusion}\n\nEmploying finite-temperature \\nobreak\\mbox{\\it ab-initio}\\ calculations of force\nconstants in combination with machine learning techniques, we have\nassessed the mechanical stability and thermal conductivity of hundreds\nof oxides and fluorides with cubic perovskite structures at high temperatures.\nWe have shown that the thermal conductivities of fluorides are generally\nmuch smaller than those of oxides, and we found new potentially stable\nperovskite compounds. We have also shown that the thermal conductivity\nof cubic perovskites generally decreases more slowly than the inverse\nof temperature. Finally, we provide simple ways of tuning the thermal\nproperties of oxides and fluorides by contrasting the effects of substitutions\nat the \\textit{A} and \\textit{B} sites. We hope that this work will\ntrigger further interest in halide perovskites for applications that\nrequire a low thermal conductivity.\n\\clearpage\n\\section{Accelerated Discovery of New Magnets in the Heusler Alloy Family}\n\\label{sec:art109}\n\nThis study follows from a collaborative effort described in Reference~\\cite{curtarolo:art109}.\nAuthor contributions are as follows:\nThe initial idea for the project was developed by Stefano Sanvito and Stefano Curtarolo.\nJunkai Xue and Thomas Archer constructed the Heusler database.\nCorey Oses and Mario \\v{Z}ic performed additional {\\small DFT}\\ calculations for tetragonally distorted Heusler alloys.\nAnurag Tiwari performed the regression analysis for the $T_\\mathrm{C}$.\nCorey Oses also performed the convex hull calculations.\nCrystal growth and experimental characterization has been performed by Pelin Tozman under the supervision of\nMunuswamy Venkatesan and J. Michael D. Coey.\nThe project was supervised by Stefano Sanvito, Stefano Curtarolo and J. Michael D. Coey, who also produced the manuscript.\n\n\\subsection{Introduction}\nVery few types of macroscopic order in condensed matter are as sensitive to details as magnetism. The magnetic\ninteraction is usually based on the $m$-$J$ paradigm, where localized magnetic moments, $m$, are magnetically\ncoupled through the exchange interaction, $J$. Only a few elements in the periodic table can provide localized moments\nin the solid state, namely 3$d$ transition metals, 4$f$ rare earths and some 4$d$ ions. Lighter 2$p$\nelements are prone to form close shells, while in heavier ones the Hund's coupling is not strong enough to\nsustain a high-spin configuration~\\cite{Janak_PRB_1977}. The magnetic coupling then depends on how the wave-functions\nof the magnetic ions overlap with each other, either directly, through other ions or via delocalized electrons.\nThis generates a multitude of mechanisms for magnetic coupling, operating at both sides of the\nmetal\/insulator transition boundary, and specific to the details of the chemical environment. In general $J$ is sensitive\nto the bond length, the bond angle, the magnetic ion valence. It is then not surprising\nthat among the $\\sim$100,000 unique inorganic compounds known to mankind~\\cite{ICSD4}, only about 2,000\nshow magnetic order of any kind~\\cite{CoeyBook}.\n\nWhen one focuses on the magnets that are useful for consumer applications, then the choice becomes even more restricted\nwith no more than two dozen compounds taking practically the entire global market. A useful magnet, regardless of the particular\ntechnology, should operate in the -50$^\\circ$C to +120$^\\circ$C range, imposing the ordering temperature, $T_\\mathrm{C}$,\nto be at least 300$^\\circ$C. Specific technologies then impose additional constraints. Permanent magnets should display a\nlarge magnetization and hysteresis~\\cite{CoeyBook}. Magnetic electrodes in high-performance magnetic tunnel junctions should\ngrow epitaxially on a convenient insulator and have a band-structure suitable for spin-filtering~\\cite{Handbook_Spin_Mag_2011}. If the same tunnel\njunction is used as spin-transfer torque magnetic random access memory element, the magnet should also have a low Gilbert\ndamping coefficient and a high Fermi-level spin polarization~\\cite{Handbook_Spin_Mag_2011}. Indeed, there are not many magnets matching all the\ncriteria, hence the design of a new one suitable for a target application is a complex and multifaceted task.\n\nThe search for a new magnet usually proceeds by trial and error, but the path may hide surprises. For instance,\nchemical intuition suggests that SrTcO$_3$ should be a poor magnet, since all Sr$X$O$_3$ perovskites with $X$ in\nthe chemical neighborhood of Tc are either low-temperature magnetic ($X$ = Ru, Cr, Mn, Fe) or do not present any\nmagnetic order ($X$ = Mo). Yet, SrTcO$_3$ is a G-type antiferromagnet~\\cite{Rodriguez_STO2} with a remarkably\nhigh N\\'{e}el temperature, 750$^\\circ$C, originating from a subtle interplay between $p$-$d$ hybridization and Jahn-Teller\ndistortion~\\cite{Franchini_STOus}. This illustrates that often a high-performance magnet may represent a singularity\nin physical\/chemical trends and that its search can defy intuition. For this reason we take a completely different\napproach to the discovery process and demonstrate that a combination of advanced electronic structure theory and\nmassive database creation and search, the high-throughput computational materials design approach~\\cite{nmatHT},\ncan provide a formidable tool for finding new magnetic materials.\n\nOur computational strategy consists of three main steps. Firstly, we construct an extensive database containing the\ncomputed electronic structures of potential novel magnetic materials. Here we consider Heusler alloys (HAs), a prototypical\nfamily of ternary compounds populated with several high-performance magnets~\\cite{Graf_PSSC_2011}. A rough stability analysis,\nbased on evaluating the enthalpy of formation against reference single-phase compounds provides a first\nscreening of the database. This, however, is not a precise measure of the thermodynamic stability of a material, since\nit does not consider decomposition into competing phases (single-element, binary, and ternary compounds). Such analysis\nrequires the computation of the electronic structure of all possible decomposition members associated with the given Heusler compounds.\nThis is our second step and it is carried out here only for intermetallic HAs, for which an extensive binary database is\navailable~\\cite{aflowlibPAPER}. Finally, we analyze the magnetic order of the predicted stable magnetic intermetallic HAs and, via\na regression trained on available magnetic data, estimate their $T_\\mathrm{C}$. The theoretical screening is then validated by\nexperimental synthesis of a few of the predicted compounds.\n\n\\subsection{Construction of the database}\n\nThe prototypical HA, $X_2YZ$ (Cu$_2$MnAl-type), crystallizes in the {\\it Fm$\\overline{3}$m} cubic\nspace group, with the $X$ atoms occupying the 8$c$ Wyckoff position (1\/4, 1\/4, 1\/4) and the $Y$ and $Z$ atoms\nbeing respectively at 4$a$ (0, 0, 0) and 4$b$ (1\/2, 1\/2, 1\/2). The crystal can be described as four interpenetrating\n{\\it fcc} lattices with $Y$ and $Z$ forming an octahedral-coordinated rock-salt structure, while the $X$ atoms occupy the\ntetrahedral voids [see Figure~\\ref{fig:art109:Fig1}(a)]. Two alternative structures also exist. In the inverse Heusler $(XY)XZ$\n(Hg$_2$CuTi-type), now $X$ and $Z$ form the rock-salt lattice, while the remaining $X$ and the $Y$ atoms fill the\ntetrahedral sites [Figure~\\ref{fig:art109:Fig1}(b)], so that one $X$ atom presents sixfold octahedral coordination, while the other\nfourfold tetrahedral coordination. The second structure, the half-Heusler $XYZ$ (MgCuSb-type), is obtained by removing one of\nthe $X$ atoms, thus leaving a vacancy at one of the tetrahedral site [Figure~\\ref{fig:art109:Fig1}(c)].\nThe minimal unit cell describing all three types can be constructed as a tetrahedral {\\it F$\\overline{4}$3m} cell, containing\n4 (3 for the case of the half Heusler) atoms [Figure~\\ref{fig:art109:Fig1}(d)]. Such a cell allows for a ferromagnetic spin configuration\nand for a limited number of antiferromagnetic ones.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig120}\n\\mycaption[Heusler structures.]\n{(\\textbf{a}) regular Heusler,\n(\\textbf{b}) inverse Heusler, and\n(\\textbf{c}) half Heusler.\n(\\textbf{d}) the tetrahedral {\\it F$\\overline{4}$3m} cell used to construct the electronic structure database.\n(\\textbf{e}) Ternary convex hull diagram for Al-Mn-Ni.\nNote the presence of the stable HA, Ni$_2$MnAl.}\n\\label{fig:art109:Fig1}\n\\end{figure}\n\n\\subsection{Results}\n\nWe construct the HAs database by considering all possible three-element combinations made of atoms from\nthe 3$d$, 4$d$ and 5$d$ periods and some elements from group III, IV, V and VI. In particular we use Ag, Al, As, Au,\nB, Ba, Be, Bi, Br, Ca, Cd, Cl, Co, Cr, Cu, Fe, Ga, Ge, Hf, Hg, In, Ir, K, La, Li, Mg, Mn, Mo, Na, Nb, Ni, Os, P, Pb, Pd,\nPt, Re, Rh, Ru, Sb, Sc, Se, Si, Sn, Sr, Ta, Tc, Te, Ti, Tl, V, W, Y, Zn and Zr. Note that we have deliberately excluded\nrare earths, responding to the global need to design new magnets with a reduced rare earth content.\nFurthermore, we have not imposed constraints on the total number of valence electrons~\\cite{zhang_sorting_2012,Yan_NComm_2015},\nsince magnetism is found for a broad range of electron counts.\nFor each combination of three elements ($X$, $Y$, $Z$) all the possible regular, inverse and half HAs are constructed.\nThese total to 236,115 decorations. The electronic structure of all the structures is computed by density functional theory\n({\\small DFT}) in the generalized gradient approximation ({\\small GGA}) of the exchange correlation functional as parameterized by\nPerdew-Burke-Ernzerhof~\\cite{PBE}. Our {\\small DFT}\\ platform is the {\\small VASP}\\ code~\\cite{vasp_cms1996} and each structure is fully relaxed.\nThe typical convergence tolerance is 1~meV\/atom and this is usually achieved by sampling the Brillouin zone\nover a dense grid of 3000-4000 $k$-points per reciprocal atom. A much denser grid of 10,000 $k$-points is employed\nfor the static run to obtain accurate charge densities and density of states. The large volume of data is managed by the\nAFLOW code~\\cite{aflowPAPER}, which creates the appropriate entries for the AFLOW database~\\cite{aflowlibPAPER}.\nMore details about the computational method are in Reference~\\cite{curtarolo:art104}.\n\nLet us begin our analysis by providing a broad overview of the database. Among the 236,115 decorations only 104,940\nare unique, meaning that only a single structure is likely to form for a given stoichiometry. Strictly speaking, this is not true\nsince there are many examples of HAs presenting various degrees of site occupation disorder, and the\nestimate gives an initial idea on how many compounds one may expect. Then a minimal criterion of stability is that the\nenthalpy of formation of the $X_2YZ$ structure, $H_{X_2YZ}$, is lower than the sum of the enthalpies of formation\nof its elementary constituents, namely $H_{\\mathrm{f}}=H_{X_2YZ}-(2H_{X}+H_{Y}+H_{Z})<0$.\nSuch criterion returns us 35,602 compounds, with 6,778 presenting a magnetic moment. Note that this number can be\nslightly underestimated as our unit cell can describe only a handful of possible anti-ferromagnetic configurations, meaning that\ncompounds where the magnetic cell is larger than the unit cell may then converge to a diamagnetic solution.\nIn any case, such a number is certainly significantly larger than the actual number of stable magnetic HAs. This can\nonly be established by computing the entire phase diagram of each ternary compound, \\nobreak\\mbox{\\it i.e.}, by assessing the stability of\nany given $X_2YZ$ structure against decomposition over all the possible alternative binary and ternary prototypes (for example\n$X_2YZ$ can decompose into $XY$+$XZ$, $X_2Y$+$Z$, $XYZ$+$X$, \\nobreak\\mbox{\\it etc.}). Such a calculation is extremely intensive. An\ninformative phase diagram for a binary alloy needs to be constructed over approximately 10,000 prototypes~\\cite{monsterPGM}, which\nmeans that at least 30,000 calculations are needed for every ternary. As a consequence mapping the stability of every calculated HA\nwill require the calculation of approximately 15,000,000 prototypes, quite a challenging task.\n\nWhen the electronic structure and the enthalpy of formation of the relevant binaries are available, then one can\nconstruct the convex hull diagram for the associated ternary compounds~\\cite{Lukas_CALPHAD_2007}. An example of such convex\nhull diagram for Al-Mn-Ni is presented in Figure~\\ref{fig:art109:Fig1}(e). The figure shows that there is a stable phase, Ni$_2$MnAl,\nwith a formation energy of -404~meV\/atom. In this case, there are also three other unstable ternary structures with\n$H_{\\mathrm{f}}<0$, namely Mn$_2$NiAl, NiMnAl and Al$_2$MnNi. The enthalpy of formation of Mn$_2$NiAl is\n$H_{\\mathrm{f}}=-209$~meV\/atom and it is 121~meV\/atom higher than the tie-plane, that of NiMnAl is\n-39~meV\/atom (400~meV\/atom above the tie plane), and that of Al$_2$MnNi is\n-379~meV\/atom (100~meV above the tie plane). This illustrates that $H_{\\mathrm{f}}<0$\nalone is not a stringent criterion for stability and that a full analysis needs to be performed before making the call on\na given ternary. Notably, Ni$_2$MnAl has been synthesized in a mixture of B2 and L2$_1$\nphases~\\cite{Ziebeck_JPFMP_1975} and it is a well-established magnetic shape memory alloy.\n\nGiven the enormous computational effort of mapping the stability of the entire database we have limited\nfurther analysis to intermetallic HAs made only with elements of the 3$d$, 4$d$ and 5$d$\nperiods. These define 36,540 structures, for which the corresponding binaries are available in the {\\sf \\AFLOW.org}\\\ndatabase~\\cite{aflowlibPAPER}. Our convex hull analysis then returns 248 thermodynamically stable compounds (full\nlist provided in Tables~\\ref{fig:art109:magnetic_heusler_1}-\\ref{fig:art109:magnetic_heusler_8}), of which only 22 possess a magnetic ground state in the tetrahedral\n{\\it F$\\overline{4}$3m} unit cell. The details of their electronic structure are presented in Table~\\ref{tab:art109:BLtab}.\nNote that in the last column of the table we include an estimate of the robustness of a particular compound against\ndecomposition, $\\delta_{\\mathrm{sc}}^{30}$. A material is deemed as decomposable (`Y' in the table) if its enthalpy of formation\nis negative but less than 30~meV\/atom lower than the most stable balanced decomposition. In contrast a material is\ndeemed robust (`N' in the table) when $H_{\\mathrm{f}}$ is more than 30~meV\/atom away from that of the closest balanced\ndecomposition. When such a criterion is applied we find that 14 of the predicted HAs can potentially decompose,\nwhile the other 8 are robust.\n\nWe have further checked whether such magnetic ground states are stable against tetragonal distortion, which may\noccur in HAs in particular with the Mn$_2YZ$ composition. Indeed we find that the ground state of five structures, namely\nCo$_2$NbZn, Co$_2$TaZn, Pd$_2$MnAu, Pd$_2$MnZn and Pt$_2$MnZn, is tetragonally distorted. Furthermore for\ntwo of them, Co$_2$NbZn and Co$_2$TaZn, the tetragonal distortion suppresses the magnetic order indicating that\nthe competition between the Stoner and band Jahn-Teller instability~\\cite{Labbe_JPFrance_1966} favors a distorted non-magnetic ground\nstate. The analysis so far tells us that the incidence of stable magnetic HAs among the possible intermetallics is\nabout 0.057\\%. When this is extrapolated to the entire database we can forecast a total of about 140 stable magnetic\nalloys, of which about 60 are already known. In the same way we can estimate approximately 1,450 stable non-magnetic\nHAs, although this is just a crude forecast, since regions of strong chemical stability may be present in the complete\ndatabase and absent in the intermetallic subset.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Electronic structure parameters of the 22 magnetic HAs found among all possible\nintermetallics.]\n{The table lists the unit cell volume of the {\\it F$\\overline{4}$3m} cell, the $c\/a$ ratio for tetragonal\ncells, $a$, the Mn-Mn distance for Mn-containing alloys, $d_\\mathrm{Mn-Mn}$, the magnetic moment per formula\nunit, $m$, the spin polarization at the Fermi level, $P_\\mathrm{F}$, the enthalpy of formation $H_{\\mathrm{f}}$, the entropic\ntemperature, $T_\\mathrm{S}$, and the magnetic ordering temperature, $T_\\mathrm{C}$. Compounds labeled with\n$*$ are not stable against tetragonal distortion (Co$_2$NbZn and Co$_2$TaZn become diamagnetic after distortion).\nNote that $T_\\mathrm{C}$ is evaluated only for Co$_2YZ$ and $X_2$Mn$Z$ compounds for which a sufficiently large\nnumber of experimental data are available for other chemical compositions. In the case of Mn$_2YZ$ compounds we\nreport the magnetic moment of the ground state and in brackets that of the ferromagnetic solution. The last column\nprovides a more stringent criterion of stability. $\\delta_{\\mathrm{sc}}^{30}=$~Y if the given compound has an enthalpy within 30\nmeV\/atom from that of its most favorable balanced decomposition (potentially decomposable), and $\\delta_{\\mathrm{sc}}^{30}=$~N if\nsuch enthalpy is more than 30~meV\/atom lower (robust).}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r|r|r}\nalloy & volume (\\AA$^3$) & $c\/a$ & $a$ (\\AA) & $d_\\mathrm{Mn-Mn}$ (\\AA) & $m$ ($\\mu_\\mathrm{B}$\/f.u.) & $P_\\mathrm{F}$\n& $H_{\\mathrm{f}}$ (eV\/atom) & $T_\\mathrm{S}$~{\\small (K)}\\ & $T_\\mathrm{C}$~{\\small (K)}\\ & $\\delta_{\\mathrm{sc}}^{30}$\\\\\n\\hline\nMn$_2$PtRh & 58.56 & & 6.16 & 3.08 & 0.00 (9.05) & 0.00 (0.86) & -0.29 & 3247 & -- & N \\\\\nMn$_2$PtCo & 54.28 && 6.00 & 3.00 & 1.13 (9.04) & 0.00 (0.86) & -0.17 & 1918 & -- & Y \\\\\nMn$_2$PtPd & 60.75 && 6.24 & 3.12 & 0.00 (8.86) & 0.00 (0.38) & -0.29 & 3218 & -- & N \\\\\nMn$_2$PtV & 55.73 && 6.06 & 3.03 & 4.87 (4.87) & 0.67 & -0.30 & 3353 & -- & Y \\\\\nMn$_2$CoCr & 47.19 && 5.73 & 2.87 & 4.84 (4.84) & 0.016 & -0.05 & 529 & -- & N \\\\\nCo$_2$MnTi & 49.68 && 5.84 & & 4.92 & 0.58 & -0.28 & 3122 & 940 & N \\\\\nCo$_2$VZn & 46.87 && 5.73 & & 1.01 & 0.93 & -0.15 & 1653 & 228 & Y \\\\\nCo$_2$NbZn$^*$ & 51.87 &1.0& 5.9 &&1.00 & 0.95 & -0.18 & 2034 & 212 & Y \\\\\nCo$_2$NbZn & 51.52 & 1.15 & 5.63 && 0.0 & 0.0 & -0.20& 2034 & 0 & Y \\\\\nCo$_2$TaZn$^*$ & 51.80&1.0& 5.92 && 0.98& 0.63 & -0.22 & 2502 & 125 & N \\\\\nCo$_2$TaZn & 51.55 & 1.12 & 5.70 && 0.0 & 0.0 & -0.23& 2502 & 0 & N \\\\\nRh$_2$MnTi & 58.08 && 6.15 & 4.35 & 4.80 & 0.51 & -0.58 & 6500 & 417 & Y\\\\\nRh$_2$MnZr & 64.50 && 6.37 &4.50&4.75 & 0.34 & -0.58 & 6518 & 338 & Y \\\\\nRh$_2$MnHf & 63.22 && 6.32 & 4.47&4.74 & 0.34 & -0.67 & 7474 & 364 & Y \\\\\nRh$_2$MnSc & 61.62 && 6.27& 4.43&4.31 & 0.77 & -0.63 & 7031 & 429 & N \\\\\nRh$_2$MnZn & 54.95 && 6.03&4.27&3.37 & 0.63 & -0.31 & 3444 & 372 & Y \\\\\nPd$_2$MnAu$^*$ & 64.21 &1.0& 6.36& 4.49&4.60 & 0.06 & -0.20 & 2203 & 853 & Y \\\\\nPd$_2$MnAu & 63.50 & 1.35 & 5.75&4.07 & 4.28 & 0.28 & -0.33 & 2203 & 331 & Y \\\\\nPd$_2$MnCu & 57.63 && 6.13&4.34&4.53 & 0.06 & -0.22 & 2492 & 415 & Y \\\\\nPd$_2$MnZn$^*$ & 58.88 & 1.0 &6.17&4.37& 4.33 & 0.38 & -0.39& 4399 & 894 & Y \\\\\nPd$_2$MnZn & 58.74 & 1.18 &5.84&4.13& 4.22 & 0.16 & -0.47& 4399 & 402 & Y \\\\\nPt$_2$MnZn$^*$ & 59.23 &1.0&6.19&4.37&4.34 & 0.34 & -0.45 & 5035 & 694 & Y \\\\\nPt$_2$MnZn & 58.95 & 1.22 &5.79&4.10& 4.13 & 0.017 & -0.65& 5035 & 381 & Y \\\\\nRu$_2$MnNb & 59.64 &&6.20&4.39& 4.07 & 0.85 & -0.19 & 2068 & 276 & Y \\\\\nRu$_2$MnTa & 59.72 &&6.20&4.39& 4.06 & 0.86 & -0.26 & 2912 & 305 & N \\\\\nRu$_2$MnV & 54.38 &&6.01&4.25& 4.00 & 0.707 & -0.16 & 1832 & 342 & Y \\\\\nRh$_2$FeZn & 54.60 &&6.02&& 4.24 & 0.49 & -0.28 & 3150 & -- & N\\\\\n\\end{tabular}}\n\\label{tab:art109:BLtab}\n\\end{table}\n\nIn Table~\\ref{tab:art109:BLtab}, together with structural details, the magnetic moment per formula unit, $m$, and the enthalpy\nof formation we report a few additional quantities that help us in understanding the potential of a given alloy as\nhigh-performance magnet. The spin polarization of the density of states at the Fermi level, $n_\\mathrm{F}^\\sigma$\n($\\sigma=\\uparrow, \\downarrow$) is calculated as~\\cite{Mazin_PRL1999}\n\\begin{equation}\nP_\\mathrm{F}=\\frac{n_\\mathrm{F}^\\uparrow-n_\\mathrm{F}^\\downarrow}{n_\\mathrm{F}^\\uparrow+n_\\mathrm{F}^\\downarrow}\\:,\n\\end{equation}\nand expresses the ability of a metal to sustain spin-polarized currents~\\cite{Coey_JPDAP_2004}. We find a broad distribution of\n$P_\\mathrm{F}$s with values ranging from 0.93 (Co$_2$VZn) to 0.06 (Pd$_2$MnCu). None of the HAs display\nhalf-metallicity, and in general their spin-polarization is similar to those of the elementary 3$d$ magnets\n(Fe, Co and Ni).\n\nWe then calculate the entropic temperature~\\cite{nmatHT,monsterPGM,curtarolo:art98}, $T_\\mathrm{S}$. For simplicity we give\nthe definition for a $XY$ binary alloy, although all our calculations are performed for its ternary equivalent,\n\\begin{equation}\nT_\\mathrm{S}=\\max_i\\left[\\frac{H_{\\mathrm{f}}(X_{x_i}Y_{1-x_i})}{k_\\mathrm{B}[x_i\\log x_i+(1-x_i)\\log(1-x_i)]}\n\\right]\\:,\n\\end{equation}\nwhere $k_\\mathrm{B}$ is the Boltzmann constant and $i$ counts all the stable compounds in the $XY$ binary system.\nEffectively $T_\\mathrm{S}$ is a concentration-maximized formation enthalpy weighted by the inverse of its ideal entropic\ncontribution (random alloy). It measures the ability of an ordered phase to resist deterioration into a temperature-driven,\nentropically-promoted, disordered mixture. The sign of $T_\\mathrm{S}$ is chosen such that a positive temperature\nis needed for competing against the compound stability (note that $T_\\mathrm{S}<0$ if $H_{\\mathrm{f}}>0$), and one expects\n$T_\\mathrm{S}\\rightarrow0$ for a compound spontaneously decomposing into a disordered mixture.\nIf we analyze the $T_\\mathrm{S}$ distribution for all the intermetallic HAs with $H_{\\mathrm{f}}<0$\n(8776 compounds) we find the behavior to closely follow that of a two-parameter Weibull distribution with a shape of\n1.13 and a scale of 2585.63 (see histogram in Figure~\\ref{fig:art109:histogram_full}). The same distribution for the 248 stable intermetallic HAs is\nrather uniform in the range 1,000-10,000~K and presents a maximum at around 3,500~K. A similar trend is observed\nfor the 20 stable magnetic HAs, suggesting that several of them may be highly disordered.\n\nFinally, Table~\\ref{tab:art109:BLtab} includes an estimate of the magnetic ordering temperatures, $T_\\mathrm{C}$. These\nhave been calculated based on available experimental data. Namely we have collected the experimental\n$T_\\mathrm{C}$'s of approximately 40 known magnetic Heusler compounds (see Section~\\ref{subsec:art109:tc_known_heuslers}) and performed a linear\nregression correlating the experimental $T_\\mathrm{C}$'s with a range of calculated electronic structure properties,\nnamely equilibrium volume, magnetic moment per formula unit, spin-decomposition and number of valence\nelectrons. The regression is possible only for those compounds for which the set of available experimental data\nis large enough, namely for Co$_2YZ$ and $X_2$Mn$Z$ HAs. We have trained the regression over the existing\ndata and found that for the two classes Co$_2YZ$ and $X_2$Mn$Z$ the typical error in the $T_\\mathrm{C}$ estimate\nis in the range of 50~K, which is taken as our uncertainty.\n\n\\subsection{Discussion}\n\nWe have found three different classes of stable magnetic HAs, namely Co$_2YZ$, $X_2$Mn$Z$ and Mn$_2YZ$. In\naddition we have predicted also Rh$_2$FeZn to be stable. This is rather unique since there are no other HAs with Fe\nin octahedral coordination and no magnetic ions at the tetrahedral positions.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig121}\n\\mycaption[Slater-Pauling curve for magnetic HAs of the form Co$_2YZ$.]\n{The magnetic moment per formula unit, $m$,\nis plotted against the number of valence electron, $N_\\mathrm{V}$, in the left panel, while $T_\\mathrm{C}$ is displayed\non the right. Red symbols corresponds to predicted HAs, while the black ones to existing materials. For the sake of clarity\nseveral compounds have been named collectively on the picture. Co$_2AB$ 1: Co$_2$FeGa, Co$_2$FeAl, Co$_2$MnSi,\nCo$_2$MnGe, Co$_2$MnSn; Co$_2AB$ 2: Co$_2$TaAl, Co$_2$ZrAl, Co$_2$HfGa, Co$_2$HfAl, Co$_2$TaGa;\nCo$_2AB$ 3: Co$_2$ZrAl, Co$_2$HfAl, Co$_2$HfGa, Co$_2$TaGa.}\n\\label{fig:art109:Co2XY}\n\\end{figure}\n\nThe first class is Co$_2YZ$, a class which is already populated by about 25 known compounds all lying on the\nSlater-Pauling curve~\\cite{Graf_PSSC_2011}. Our analysis reveals four new stable alloys, three of them with the low valence\nelectron counts of 25 (Co$_2$VZn, Co$_2$NbZn, Co$_2$TaZn) and one, Co$_2$MnTi, presenting the large count\nof 29. The regression correctly places these four on the Slater-Pauling curve (see Figure~\\ref{fig:art109:Co2XY}) and predicts\nfor Co$_2$MnTi the remarkably high $T_\\mathrm{C}$ of 940~K. This is a rather interesting since only about\ntwo dozen magnets are known to have a $T_\\mathrm{C}$ in that range~\\cite{CoeyBook}. Therefore, the discovery\nof Co$_2$MnTi has to be considered as exceptional. The other three new compounds in this class are all predicted to\nhave a $T_\\mathrm{C}$ around 200~K, but two of them become non-magnetic upon tetragonal distortion leaving\nonly Co$_2$VZn magnetic ($T_\\mathrm{C}\\sim228$~K).\n\nThe second class is $X_2$Mn$Z$ in which we find 13 new stable magnets, most of them including a 4$d$ ion (Ru, Rh\nand Pd) in the tetrahedral $X$ position. In general, these compounds have a magnetic moment per formula unit ranging between\n4~$\\mu_\\mathrm{B}$ and 5~$\\mu_\\mathrm{B}$, consistent with the nominal 2+ valence of Mn in octahedral coordination.\nThe regression, run against 18 existing compounds of which 13 are with $X$ = Ru, Rh or Pd, establishes a correlation\nbetween the Mn-Mn nearest neighbors distance, $d_\\mathrm{Mn-Mn}$, and $T_\\mathrm{C}$ as shown in\nFigure~\\ref{fig:art109:X2MnY}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig122}\n\\mycaption[Magnetic data for $X_2$Mn$Z$ magnets.]\n{$T_\\mathrm{C}$ (left) and magnetic moment per formula unit (right)\nas a function of the Mn-Mn distance, $d_\\mathrm{Mn-Mn}$.\nNote that the $T_\\mathrm{C}$ is limited to about 550~K and\npeaks at a volume of about 60~\\AA$^3$. In contrast the magnetic moment is approximately constant with values in between\n4~$\\mu_\\mathrm{B}$ and 5~$\\mu_\\mathrm{B}$. Close circles (with associated chemical compositions) correspond to the\npredicted compounds, while the other symbols correspond to experimental data. Different colors correspond to different\nnumber of valence electrons, $N_\\mathrm{V}$. Blue chemical formulas correspond to compound displaying tetragonal\ndistortion. The two red lines are Castelliz-Konamata curves, while the black one is to guide the eye.}\n\\label{fig:art109:X2MnY}\n\\end{figure}\n\nWe find that $T_\\mathrm{C}$ is a non-monotonic function of $d_\\mathrm{Mn-Mn}$ with a single maximum at\n$d_0$$\\sim$4.4~\\AA\\ corresponding to a temperature of 550~K (the maximum coincides approximately with\nCu$_2$MnSn). The only apparent exception to such trend is the prototypical Cu$_2$MnAl, which displays a large\n$T_\\mathrm{C}$ and relatively small $d_\\mathrm{Mn-Mn}$~\\cite{Oxley1963}. A strong sensitivity of the $T_\\mathrm{C}$\nof Mn-containing compounds to $d_\\mathrm{Mn-Mn}$ was observed long time ago and rationalized in an empirical\n$T_\\mathrm{C}$-$d_\\mathrm{Mn-Mn}$ curve by Castelliz~\\cite{Castelliz_ZM_1955}. This predicts that $T_\\mathrm{C}$ is not\nmonotonically dependent on $d_\\mathrm{Mn-Mn}$ and has a maximum at around $d_\\mathrm{Mn-Mn}=3.6$. The curve\nhas been validated for a number of HAs and it has been used to explain the positive pressure coefficient of\n$T_\\mathrm{C}$, $(1\/T_\\mathrm{C})(\\mathrm{d}T_\\mathrm{C}\/\\mathrm{d}P)$, found, for instance, in\nRh$_2$MnSn~\\cite{Adachi200437}. Refinements of the Castelliz curve predict that the rate of change of $T_\\mathrm{C}$\nwith $d_\\mathrm{Mn-Mn}$ in HAs is related to the valence count~\\cite{Kanomata_JMMM_1987}, although the position\nof the maximum is not. In general the results of Figure~\\ref{fig:art109:X2MnY}, including several experimental data, seems to contradict\nthe picture since a monotonically decreasing $T_\\mathrm{C}$ is expected for any $d_\\mathrm{Mn-Mn}>3.6$~\\AA, \\nobreak\\mbox{\\it i.e.},\npractically for any HAs of the form $X_2$Mn$Z$. There are a few possible reasons for such disagreement. Firstly, the\nCastelliz curve assumes that only Mn presents a magnetic moment, which is unlikely since many of the $X_2$Mn$Z$\ncompounds of Figure~\\ref{fig:art109:X2MnY} have Rh or Pd in the X position, two highly spin-polarizable ions. Secondly,\nmany HAs in Figure~\\ref{fig:art109:X2MnY} present various levels of disorder, meaning that Mn-Mn pairs separated\nby less than the nominal $d_\\mathrm{Mn-Mn}$ are likely to be present in actual samples. We then propose that the\ntrend of Figure~\\ref{fig:art109:X2MnY} (see dashed black lines) represents a new empirical curve, valid for $X_2$Mn$Z$ HAs,\nand taking into account such effects.\n\nThe last class of predicted magnetic HAs is populated by Mn$_2YZ$ compounds. These have recently\nreceived significant attention because of their high $T_\\mathrm{C}$ and the possibility of displaying tetragonal\ndistortion and hence large magneto-crystalline anisotropy~\\cite{Kreiner2014}. Experimentally when the 4$c$ position\nis occupied by an element from group III, IV or V one finds the regular Heusler structure if the atomic number of\nthe $Y$ ion is smaller than that of Mn, $Z$($Y$)$<$$Z$(Mn), and the inverse one for $Z$($Y$)$>$$Z$(Mn). To date only\nMn$_2$VAl and Mn$_2$VGa have been grown with a $Y$ element lighter than Mn, so that except those two all other\nMn$_2YZ$ HAs crystallize with the inverse structure (see Figure~\\ref{fig:art109:Mn2YZ}). In the case of the two regular\nHAs, Mn$_2$VAl and Mn$_2$VGa, the magnetic order is ferrimagnetic with the two Mn ions at the tetrahedral\nsites being anti-ferromagnetically coupled to V~\\cite{Nakamichi1983,Itoh1983,Kumar2008}. In contrast for\nthe inverse Mn$_2$-based HAs the antiferromagnetic alignment is between the two Mn ions and the magnetic\nground state then depends on whether there are other magnetic ions in the compound. In general, however, site disorder is\nnot uncommon (see Section~\\ref{subsec:art109:tet_disorder_Mn2PtPd}) and so is tetragonal distortion, so that the picture becomes more complicated. There are\nalso some complex cases, such as that of Mn$_3$Ga, presenting a ground state with a non-collinear arrangement of\nboth the spin and angular momentum~\\cite{Rode_PRB_2013}.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=0.65\\linewidth]{fig123}\n\\mycaption[Enthalpy of formation difference between the regular and inverse Heusler structure, $\\Delta H_\\mathrm{RI}$,\nfor Mn$_2$-containing compounds as a function of the cell volume.]\n{The solid red squares (with chemical formulas) are\nthe predicted stable intermetallic materials, while the open red squares are existing compounds. For completeness we\nalso include data for Co$_2$-based HAs, again with open symbols for existing compounds and solid one for predicted.\nIn brackets beside the chemical formulas we report the value for the entropic temperature, $T_\\mathrm{S}$, in {\\small (K)}.}\n\\label{fig:art109:Mn2YZ}\n\\end{figure}\n\nIf we now turn our attention to the predicted compounds we find five stable compositions of which three match the\n$\\delta_{\\mathrm{sc}}^{30}$ robustness criterion. Most intriguingly the regular {\\it Fm$\\overline{3}$m} structure\nappears to be the ground state for all the compounds, regardless of their chemical composition. This sets Mn$_2$-based\nintermetallic compounds aside from those with elements from the main groups. In Figure~\\ref{fig:art109:Mn2YZ} we present the\nenthalpy of formation difference between the regular and the inverse structure, $\\Delta H_\\mathrm{RI}=H_\\mathrm{f,R}-H_\\mathrm{f,I}$,\nfor the computed and the experimentally known Mn$_2$-based HAs, together with their $T_\\mathrm{S}$ and reference\ndata for Co$_2$-based alloys. In general we find that $\\Delta H_\\mathrm{RI}$ for the Mn$_2YZ$ class is significantly smaller\nthan for the Co$_2YZ$ one. In fact there are cases, \\nobreak\\mbox{\\it e.g.}, Mn$_2$PtGa and Mn$_2$PtIn, in which the two phases are almost\ndegenerate and different magnetic configurations can favor one over the other. Overall, one then expects such compounds\nto be highly disordered. Finally, we take a look at the magnetic ground state. In all cases the compounds present some\ndegree of antiferromagnetic coupling, which results in either a zero-moment ground state when Mn is the only magnetic ion,\nand in a ferrimagnetic configuration when other magnetic ions are present.\n\nThe last step in our approach consists in validating the theoretical predictions by experiments. We have attempted the\nsynthesis of four HAs, namely Co$_2$MnTi, Mn$_2$PtPd, Mn$_2$PtCo and Mn$_2$PtV. Co$_2$MnTi\nis chosen because of its high Curie temperature, while among the Mn$_2$-based alloys we have selected two\npresenting ferrimagnetic ground state (Mn$_2$PtCo and Mn$_2$PtV) and one meeting the stringent\n$\\delta_{\\mathrm{sc}}^{30}$ robustness criterion (Mn$_2$PtPd). The alloys have been prepared by arc melting in high-purity Ar, with\nthe ingots being remelted four times to ensure homogeneity. An excess of 3 \\% wt. Mn is added in order to\ncompensate for Mn losses during arc melting (see Section~\\ref{subsec:art109:exp_data_Mn2_based} for details). Structural characterization has been carried\nout by powder X-ray diffraction (XRD), while magnetic measurements were made using a superconducting magnetometer\nin a field of up to 5~T. Furthermore, the microstructure has been analyzed by scanning electron microscopy\nof the polished bulk samples, while the compositions are determined by Energy Dispersive X-ray (EDX)\nspectroscopy.\n\nTwo of the four HAs have been successfully synthesized, Co$_2$MnTi and Mn$_2$PtPd, while the other two,\nMn$_2$PtCo and Mn$_2$PtV, decompose into binary compounds (see Section~\\ref{subsec:art109:exp_data_Mn2_based} for details).\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig124}\n\\mycaption[Experimental magnetic characterization of Co$_2$MnTi.]\n{(\\textbf{a}) magnetization curve at 4~K and 300~K (inset: zero-field cooled magnetization\ncurve as a function of temperature in magnetic field of 1~T);\n(\\textbf{b}) XRD spectrum (inset: EDX chemical composition\nanalysis).\nCo$_2$MnTi crystallizes in a single {\\it Fm$\\overline{3}$m} phase corresponding to a regular Heusler.\nThe $T_\\mathrm{C}$ extrapolated from the magnetization curve is around 900~K.}\n\\label{fig:art109:Co2MnTi}\n\\end{figure}\n\nIn Figure~\\ref{fig:art109:Co2MnTi} we present the structural and magnetic characterization of Co$_2$MnTi. It crystallizes in\nthe regular {\\it Fm$\\overline{3}$m} Heusler structure with no evidence of secondary phases and a lattice parameter of\n$a=5.89$~\\AA\\, in close agreement with theory, $a=5.84$~\\AA. The magnetization curve displays little temperature\ndependence and a saturation moment of 4.29~$\\mu_\\mathrm{B}$\/f.u. at 4~K, fully consistent with the calculated\nferromagnetic ground state (see Table~\\ref{tab:art109:BLtab}). Most notably, the $T_\\mathrm{C}$ extrapolated from the\nzero-field cooled magnetization curve in a field of 1~T is found to be 938~K, essentially identical that predicted\nby our regression, 940~K.\nThis is a remarkable result, since it is the first time that a new high-temperature ferromagnet\nhas been discovered by HT means.\n\n\\begin{figure}[tp]\\centering\n\\includegraphics[width=\\linewidth]{fig125}\n\\mycaption[Experimental magnetic characterization of Mn$_2$PtPd.]\n{(\\textbf{a}) field cooled and zero-field cooled magnetization curve as a function of temperature in\na magnetic field of 0.1~T (inset: magnetization curve at 4~K and 300~K);\n(\\textbf{b}) XRD spectrum (inset: EDX chemical\ncomposition analysis). Mn$_2$PtPd crystallizes in a single {\\it I}4{\\it\/mmm} (TiAl$_3$-type) phase corresponding\nto a regular tetragonal distorted Heusler.\nSEM images confirm that the bulk sample is mainly of Mn$_2$PtPd\ncomposition (gray color) with a small amount of a secondary Mn-O inclusions, which have spherical shape of\ndiameter 400-900~nm and do not appear in the XRD spectrum.}\n\\label{fig:art109:Mn2PtPd}\n\\end{figure}\n\nAlso in the case of Mn$_2$PtPd a single phase is found without evidence of decomposition. The XRD pattern\n[Figure~\\ref{fig:art109:Mn2PtPd}(b)] corresponds to a tetragonally-distorted regular Heusler with space group {\\it I}4{\\it\/mmm}\n(TiAl$_3$-type) and lattice parameters $a=4.03$~\\AA\\ and $c=7.24$~\\AA. Our magnetic data show a magnetic\ntransition at $\\sim$320~K, which shifts to a slightly higher temperature upon field cooling [Figure~\\ref{fig:art109:Mn2PtPd}(a)].\nMagnetization curves at room temperature and 4~K show no hysteresis or spontaneous magnetization indicating\nthat the compound is antiferromagnetic at low temperature.\nFrom Table~\\ref{tab:art109:BLtab} it will appear that the only difference between the calculated and experimental\ndata for Mn$_2$PtPd concerns the tetragonal distortion. However, the search for tetragonal distortion reported in the\ntable was performed only for the ferromagnetic state. Further analysis for the antiferromagnetic ground state (see Section~\\ref{subsec:art109:tet_disorder_Mn2PtPd})\nreveals that indeed Mn$_2$PtPd is antiferromagnetic and tetragonal distorted with a $c\/a$ ratio of around 1.3, in good\nagreement with experiments.\n\n\\subsection{Table of \\texorpdfstring{$T_\\mathrm{C}$}{Tc} of known Heusler alloys} \\label{subsec:art109:tc_known_heuslers}\n\nHere we present experimental data, collected from the literature, for known magnetic Heusler alloys.\nThese data have been used to perform the regression used to extract the $T_\\mathrm{C}$ of the\nnew predicted compounds.\n\n\\begin{table}[tp]\\centering\n\\mycaption[Summary Table for the magnetic Heusler alloys of the type Co$_2XY$.]\n{Here are reported the compound,\nthe magnetic moment per formula unit, $m$, and the experimental $T_\\mathrm{C}$, together with the\nappropriate reference. The quantities labeled with a `*' have been used to run the\nregression.}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r|r|r|r}\nmaterial & $m$\/f.u. ($\\mu_\\mathrm{B}$) & $m^*$\/f.u. ($\\mu_\\mathrm{B}$) & $T_\\mathrm{C}$~{\\small (K)}\\ & $T_\\mathrm{C}^*$~{\\small (K)}\\ & source & reference \\\\\n\\hline\nCo$_2$TiAl & 0.74 & 0.74 & 134 & 134 & Exp. & \\onlinecitesq{PhysRevB.76.024414} \\\\\nCo$_2$TiGa & 0.82 & 0.82 & 128 & 128 & Exp. & \\onlinecitesq{LandBorn1,Sasaki2001406} \\\\\nCo$_2$TiSi & 1.96 & 1.96 & 380 & 380 & Exp. & \\onlinecitesq{LandBorn1,Barth28092011} \\\\\nCo$_2$TiGe & 1.94 & 1.94 & 380 & 380 & Exp. & \\onlinecitesq{LandBorn1,Barth28092011} \\\\\nCo$_2$TiSn & 1.97 & 1.97 & 355 & 355 & Exp. & \\onlinecitesq{LandBorn1,Barth28092011} \\\\\nCo$_2$ZrSn & 1.56 & 1.56 & 448 & 448 & Exp. & \\onlinecitesq{Zhang2006255} \\\\\nCo$_2$VGa & 2.04 & 2.04 & 357 & 357 & Exp. & \\onlinecitesq{PhysRevB.76.024414,PhysRevB.82.144415} \\\\\nCo$_2$VSn & 1.21 & 1.21 & 95 & 95 & Exp. & \\onlinecitesq{PhysRevB.76.024414,0022-3727-40-6-S01} \\\\\nCo$_2$VAl & 1.86 & 1.86 & 342 & 342 & Exp. & \\onlinecitesq{LandBorn1,PhysRevB.82.144415} \\\\\nCo$_2$ZrAl & 0.74 & 0.74 & 185 & 185 & Exp. & \\onlinecitesq{LandBorn1,Kanomata200526} \\\\\nCo$_2$ZrSn & 1.51 & 1.51 & 444 & 444 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$NbAl & 1.35 & 1.35 & 383 & 383 & Exp. & \\onlinecitesq{LandBorn1,0022-3727-40-6-S01} \\\\\nCo$_2$NbSn & 0.52 & 0.52 & 119 & 119 & Exp. & \\onlinecitesq{LandBorn1,PhysRevB.66.174428} \\\\\nCo$_2$HfAl & 0.81 & 0.81 & 193 & 193 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$HfGa & 0.54 & 0.54 & 186 & 186 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$HfSn & 1.55 & 1.55 & 394 & 394 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$CrGa & 3.01 & 3.01 & 495 & 495 & Exp. & \\onlinecitesq{PhysRevB.76.024414} \\\\\nCo$_2$CrAl & 1.55 & 1.55 & 334 & 334 & Exp. & \\onlinecitesq{PhysRevB.76.024414,Hakimi20103443,JEPT2013Svyazhin} \\\\\nCo$_2$MnAl & 4.01-4.04 & 4.04 & 693-697 & 697 & Exp. & \\onlinecitesq{LandBorn1,PhysRevB.76.024414} \\\\\nCo$_2$MnGa & 4.05 & 4.05 & 694 & 694 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$MnGe & 5.11 & 5.11 & 905 & 905 & Exp. & \\onlinecitesq{LandBorn1} \\\\\nCo$_2$MnSi & 4.90 & 4.90 & 985 & 985 & Exp. & \\onlinecitesq{PhysRevB.76.024414,LandBorn1} \\\\\nCo$_2$MnSn & 5.08 & 5.08 & 829 & 829 & Exp. & \\onlinecitesq{PhysRevB.76.024414,LandBorn1} \\\\\nCo$_2$FeSi & 6.00 & 6.00 & 1100 & 1100 & Exp. & \\onlinecitesq{PhysRevB.76.024414} \\\\\nCo$_2$FeAl & 4.96 & 4.96 & 1000 & 1000 & Exp. & \\onlinecitesq{Trudel:2013p323} \\\\\nCo$_2$FeGa & 5.15 & 5.15 & $>$1100 & 1100 & Exp. & \\onlinecitesq{Trudel:2013p323} \\\\\nCo$_2$TaAl & 0.75 & 0.75 & 260 & 260 & Exp. & \\onlinecitesq{Carbonari1996} \\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[tp]\\centering\n\\mycaption[Summary Table magnetic Heuslers of the type $X_2$Mn$Y$.]\n{Here are reported the compound,\nthe magnetic moment per formula unit, $m$, the experimental $T_\\mathrm{C}$, the volume of\nthe $F\\overline{4}3m$ cell, and the number of valence electrons per formula unit, $N_\\mathrm{V}$,\ntogether with the appropriate reference. The quantity labeled with a `*' are those, which have\nbeen used to run the regression.}\n\\vspace{3mm}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|r|r|r|r|r|r|r|r}\nmaterial & $m$\/f.u. ($\\mu_\\mathrm{B}$) & $m^*$\/f.u. ($\\mu_\\mathrm{B}$) & $T_\\mathrm{C}$~{\\small (K)}\\ & $T_\\mathrm{C}^*$~{\\small (K)}\\ & volume (\\AA$^3$) & $N_\\mathrm{V}$ & order & reference \\\\\n\\hline\nRh$_2$MnGe & 4.17-4.62 & 4.62 & 400-470 & 450 & 56.46 & 29 & FM & \\onlinecitesq{LandBorn1,Klaer2009,Suits1976,Adachi200437,Hames1971} \\\\\nRh$_2$MnSn & 3.10-3.93 & 3.10 & 412-431 & 412 & 62.22 & 29 & FM & \\onlinecitesq{LandBorn1,Suits1976,Adachi200437} \\\\\nRh$_2$MnPb & 4.12 & 4.12 & 338 & 338 & 65.58 & 29 & FM & \\onlinecitesq{LandBorn1,Suits1976} \\\\\nRh$_2$MnAl & 4.1 & 4.1 & 85-105 & 95 & 54.96 & 28 & FM & \\onlinecitesq{Wijn1,Suits1976} \\\\\nCu$_2$MnSn & 4.11 & 4.11 & 530 & 530 & 60.36 & 33 & FM & \\onlinecitesq{LandBorn1,Wijn1,Oxley1963} \\\\\nCu$_2$MnAl & 3.73-4.12 & 4.12 & 603 & 603 & 51.93 & 32 & FM & \\onlinecitesq{LandBorn1,Wijn1,Oxley1963} \\\\\nCu$_2$MnIn & 3.95 & 3.95 & 510 & 510 & 59.45 & 32& FM & \\onlinecitesq{Oxley1963,Coles1949} \\\\\nPd$_2$MnAl & 4.4 & 4.4 & 240 & 240 & 58.89 & 30 & AFM & \\onlinecitesq{Wijn1,Webster1968} \\\\\nPd$_2$MnSn & 4.23 & 4.23 & 189 & 189 & 66.00 & 31 & FM & \\onlinecitesq{LandBorn1,Wijn1,Campbell1977,Webster1967} \\\\\nPd$_2$MnSb & 4.40 & 4.40 & 247 & 247 & 67.58 & 32 & FM & \\onlinecitesq{LandBorn1,Wijn1,Webster1967} \\\\\nPd$_2$MnGe & 3.2 & 3.2 & 170 & 170 & 60.49 & 31 & FM & \\onlinecitesq{Wijn1} \\\\\nPd$_2$MnIn & 4.3 & 4.3 & 142 & 142 & 65.88 & 30 & AFM & \\onlinecitesq{Wijn1,Webster1967} \\\\\nAu$_2$MnAl & 4.2 & 4.2 & 233 & 233 & 65.37 & 32 & FM & \\onlinecitesq{Wijn1,Bacon1967} \\\\\nAu$_2$MnZn & 4.6 & 4.6 & 253 & 253 & 65.32 & 31 & FM & \\onlinecitesq{Wijn1,Bacon1973} \\\\\nRu$_2$MnGe & 3.2-3.8 & 3.8 & 316 & 316 & 54.33 & 27 & AFMII & \\onlinecitesq{Kanomata2006,Gotoh1995} \\\\\nRu$_2$MnSi & 2.8 & 2.8 & 313 & 313 & 51.82 & 27 & AFMII & \\onlinecitesq{Kanomata2006} \\\\\nRu$_2$MnSb & 3.9-4.4 & 4.4 & 195 & 195 & 58.98 & 28 & AFMII & \\onlinecitesq{Kanomata2006,Gotoh1995} \\\\\nRu$_2$MnSn & 2.8 & 2.8 & 296 & 296 & 58.92 & 27 & AFMII & \\onlinecitesq{Kanomata2006} \\\\\n\\end{tabular}}\n\\end{table}\n\n\\newcommand{\\heuslerstabfootone}{\nNote that the tetragonal phase is obtained when annealing at $400^\\circ$C.\nA higher annealing temperature of $800^\\circ$C results in a disorder pseudo-cubic phase.\nNo magnetic data are available for this second phase.}\n\\newcommand{\\heuslerstabfoottwo}{\nNote that Mn$_2$NiGa is a shape memory alloy, displaying a martensitic transformation at a critical temperature\n$T_{\\mathrm{m}}=270$~K.\nThe structure is cubic for $T>T_{\\mathrm{m}}$ and tetragonal for $T0$).]\n{The continuous red line is our best fit to a two-parameter Weibull distribution with a shape of 1.13\nand a scale of 2585.63.}\n\\label{fig:art109:histogram_full}\n\\end{figure}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=1.0\\linewidth]{fig134}\n\\mycaption[Histogram of the entropic temperature, $T_\\mathrm{S}$, for all the 248 intermetallic Heuslers estimated stable\nafter the construction of the convex hull diagrams for the ternary phase.]\n{The red lines indicate three compounds present in the {\\small ICSD}\\ database.}\n\\end{figure}\n\n\\clearpage\n\n\\subsection{Tetragonal distortion for \\texorpdfstring{Mn$_2$}{Mn2}PtPd} \\label{subsec:art109:tet_disorder_Mn2PtPd}\n\nThe total energy of Mn$_2$PtPd is calculated for different $c\/a$ ratio (and constant volume) for both the\nferromagnetic and antiferromagnetic state. Note that, while in the ferromagnetic configuration the energy\nminimum is found for the cubic solution, in the antiferromagnetic case (lower in energy) this is found for\n$c\/a=1.3$, in agreement with the experimental data.\n\n\\vspace{1cm}\n\n\\begin{figure}[htp!]\\centering\n\\includegraphics[width=0.5\\linewidth]{fig135}\n\\mycaption{Total energy as a function of the $c\/a$ ratio for Mn$_2$PtPd calculated with {\\small GGA}-{\\small DFT}.}\n\\end{figure}\n\n\\clearpage\n\n\\subsection{List of all stable intermetallic Heuslers}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers (1\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nZn$_2$AgAu &\t64.64\t&\t-0.15\t&\t1723\t\\\\\nPd$_2$AgCd &\t68.13\t&\t-0.27\t&\t2958\t\\\\\nAg$_2$CdSc\t&\t77.7808\t&\t-0.248922\t&\t2778.26\t\\\\\nAg$_2$CdY\t&\t85.8372\t&\t-0.301022\t&\t3359.76\t\\\\\nAg$_2$CdZr\t&\t77.9156\t&\t-0.098514\t&\t1099.53\t\\\\\nHg$_2$AgLa\t&\t99.102\t&\t-0.392808\t&\t4384.2\t\\\\\nPd$_2$AgHg\t&\t69.3436\t&\t-0.146835\t&\t1638.85\t\\\\\nHg$_2$AgSc\t&\t81.9652\t&\t-0.256216\t&\t2859.68\t\\\\\nSc$_2$AgHg\t&\t84.1676\t&\t-0.364257\t&\t4065.54\t\\\\\nHg$_2$AgY\t&\t89.648\t&\t-0.349733\t&\t3903.43\t\\\\\nSc$_2$AgOs\t&\t72.1864\t&\t-0.376927\t&\t4206.95\t\\\\\nSc$_2$AgRu\t&\t72.4904\t&\t-0.44129\t&\t4925.31\t\\\\\nY$_2$AgRu\t&\t87.0664\t&\t-0.346082\t&\t3862.68\t\\\\\nAu$_2$CdLa\t&\t94.5472\t&\t-0.66943\t&\t7471.63\t\\\\\nPd$_2$AuCd\t&\t68.5296\t&\t-0.301286\t&\t3362.7\t\\\\\nAu$_2$CdY\t&\t85.436\t&\t-0.674423\t&\t7527.36\t\\\\\nAu$_2$CdZr\t&\t78.2676\t&\t-0.457602\t&\t5107.38\t\\\\\nCu$_2$AuPd\t&\t57.166\t&\t-0.115899\t&\t1293.57\t\\\\\nAu$_2$CuZn\t&\t62.4144\t&\t-0.142872\t&\t1594.62\t\\\\\nAu$_2$HfZn\t&\t71.7456\t&\t-0.438785\t&\t4897.35\t\\\\\nAu$_2$HgLa\t&\t94.9036\t&\t-0.627046\t&\t6998.57\t\\\\\nPd$_2$AuHg\t&\t69.7384\t&\t-0.162896\t&\t1818.12\t\\\\\nZn$_2$AuRh\t&\t59.324\t&\t-0.312353\t&\t3486.23\t\\\\\nSc$_2$AuRu\t&\t71.926\t&\t-0.675774\t&\t7542.43\t\\\\\nAu$_2$TiZn\t&\t66.8216\t&\t-0.352571\t&\t3935.11\t\\\\\nAu$_2$ZnZr\t&\t73.2548\t&\t-0.467891\t&\t5222.21\t\\\\\nCu$_2$CdZr\t&\t65.8668\t&\t-0.155451\t&\t1735.01\t\\\\\nRh$_2$CdHf\t&\t67.1976\t&\t-0.68254\t&\t7617.94\t\\\\\nHg$_2$CdLa\t&\t103.328\t&\t-0.460008\t&\t5134.23\t\\\\\nHg$_2$CdSc\t&\t86.474\t&\t-0.265346\t&\t2961.58\t\\\\\nHg$_2$CdY\t&\t94.0524\t&\t-0.381128\t&\t4253.84\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_1}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (2\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nPd$_2$CdSc\t&\t70.7684\t&\t-0.725422\t&\t8096.56\t\\\\\nPd$_2$CdY\t&\t78.308\t&\t-0.731543\t&\t8164.88\t\\\\\nPd$_2$CdZr\t&\t72.2028\t&\t-0.58712\t&\t6552.94\t\\\\\nRh$_2$CdSc\t&\t67.0008\t&\t-0.622274\t&\t6945.31\t\\\\\nRh$_2$CdZr\t&\t68.4788\t&\t-0.627501\t&\t7003.65\t\\\\\nHf$_2$CoRe\t&\t66.8792\t&\t-0.412526\t&\t4604.27\t\\\\\nCo$_2$HfSc\t&\t61.3956\t&\t-0.38894\t&\t4341.02\t\\\\\nHf$_2$CoTc\t&\t66.1292\t&\t-0.493898\t&\t5512.48\t\\\\\nCo$_2$HfZn\t&\t53.9212\t&\t-0.326005\t&\t3638.6\t\\\\\nSc$_2$CoIr\t&\t64.4424\t&\t-0.71918\t&\t8026.89\t\\\\\nTi$_2$CoIr\t&\t56.8924\t&\t-0.622184\t&\t6944.3\t\\\\\nTi$_2$CoMn\t&\t52.0108\t&\t-0.382265\t&\t4266.53\t\\\\\nTi$_2$CoRe\t&\t56.7704\t&\t-0.444075\t&\t4956.4\t\\\\\nSc$_2$CoRu\t&\t63.6324\t&\t-0.467309\t&\t5215.72\t\\\\\nTi$_2$CoTc\t&\t56.0352\t&\t-0.510928\t&\t5702.56\t\\\\\nZr$_2$CoTc\t&\t68.3008\t&\t-0.359379\t&\t4011.09\t\\\\\nCo$_2$TiZn\t&\t48.8244\t&\t-0.350328\t&\t3910.07\t\\\\\nCo$_2$ZnZr\t&\t55.166\t&\t-0.268346\t&\t2995.06\t\\\\\nV$_2$CrFe\t&\t47.8092\t&\t-0.167619\t&\t1870.82\t\\\\\nTi$_2$CrIr\t&\t57.4292\t&\t-0.551684\t&\t6157.44\t\\\\\nV$_2$CrMn\t&\t48.2312\t&\t-0.193973\t&\t2164.97\t\\\\\nNb$_2$CrOs\t&\t62.176\t&\t-0.200243\t&\t2234.95\t\\\\\nTa$_2$CrOs\t&\t62.2812\t&\t-0.311877\t&\t3480.92\t\\\\\nV$_2$CrOs\t&\t52.3748\t&\t-0.302942\t&\t3381.19\t\\\\\nV$_2$CrRe\t&\t53.0104\t&\t-0.258046\t&\t2880.1\t\\\\\nTa$_2$CrRu\t&\t61.7168\t&\t-0.280556\t&\t3131.34\t\\\\\nV$_2$CrRu\t&\t51.9164\t&\t-0.25086\t&\t2799.9\t\\\\\nHf$_2$CuRe\t&\t69.632\t&\t-0.296279\t&\t3306.82\t\\\\\nHf$_2$CuTc\t&\t69.12\t&\t-0.339081\t&\t3784.54\t\\\\\nCu$_2$HfZn\t&\t58.7964\t&\t-0.19888\t&\t2219.74\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_2}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (3\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nSc$_2$CuIr\t&\t68.0976\t&\t-0.699208\t&\t7803.98\t\\\\\nSc$_2$CuOs\t&\t67.3716\t&\t-0.408716\t&\t4561.75\t\\\\\nZr$_2$CuOs\t&\t71.076\t&\t-0.345336\t&\t3854.35\t\\\\\nPd$_2$CuZn\t&\t56.2556\t&\t-0.403379\t&\t4502.19\t\\\\\nSc$_2$CuPt\t&\t70.4452\t&\t-0.801364\t&\t8944.16\t\\\\\nRh$_2$CuTa\t&\t58.3456\t&\t-0.455697\t&\t5086.11\t\\\\\nSc$_2$CuRu\t&\t67.33\t&\t-0.46985\t&\t5244.08\t\\\\\nY$_2$CuRu\t&\t81.6212\t&\t-0.318052\t&\t3549.83\t\\\\\nZr$_2$CuTc\t&\t71.3516\t&\t-0.26889\t&\t3001.13\t\\\\\nCu$_2$TiZn\t&\t53.8756\t&\t-0.169069\t&\t1887\t\\\\\nCu$_2$ZnZr\t&\t60.172\t&\t-0.223658\t&\t2496.28\t\\\\\nHf$_2$FeOs\t&\t65.9256\t&\t-0.524889\t&\t5858.38\t\\\\\nTi$_2$FeMn\t&\t51.8856\t&\t-0.336061\t&\t3750.83\t\\\\\nTi$_2$FeOs\t&\t55.8712\t&\t-0.568209\t&\t6341.88\t\\\\\nHf$_2$IrMn\t&\t66.5552\t&\t-0.641543\t&\t7160.37\t\\\\\nHf$_2$IrMo\t&\t70.62\t&\t-0.605585\t&\t6759.04\t\\\\\nHf$_2$IrRe\t&\t69.8952\t&\t-0.743454\t&\t8297.82\t\\\\\nHf$_2$IrTc\t&\t69.3832\t&\t-0.854328\t&\t9535.3\t\\\\\nIr$_2$HfZn\t&\t63.0396\t&\t-0.732469\t&\t8175.21\t\\\\\nHf$_2$MoRh\t&\t70.6316\t&\t-0.529099\t&\t5905.36\t\\\\\nTc$_2$HfMo\t&\t64.7628\t&\t-0.293247\t&\t3272.98\t\\\\\nTc$_2$HfNb\t&\t67.0276\t&\t-0.447695\t&\t4996.8\t\\\\\nNi$_2$HfZn\t&\t55.6964\t&\t-0.431443\t&\t4815.41\t\\\\\nHf$_2$OsRu\t&\t68.636\t&\t-0.769146\t&\t8584.58\t\\\\\nOs$_2$HfSc\t&\t67.5148\t&\t-0.560224\t&\t6252.76\t\\\\\nHf$_2$OsTc\t&\t69.0408\t&\t-0.626591\t&\t6993.49\t\\\\\nHf$_2$PdRe\t&\t71.6136\t&\t-0.559388\t&\t6243.42\t\\\\\nHf$_2$PdTc\t&\t71.1996\t&\t-0.620052\t&\t6920.51\t\\\\\nPd$_2$HfZn\t&\t65.598\t&\t-0.675223\t&\t7536.29\t\\\\\nHf$_2$ReRh\t&\t69.9132\t&\t-0.700075\t&\t7813.66\t\\\\\nHf$_2$ReZn\t&\t71.8108\t&\t-0.299023\t&\t3337.45\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_3}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (4\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nHf$_2$RhTc\t&\t69.3144\t&\t-0.78918\t&\t8808.18\t\\\\\nRh$_2$HfZn\t&\t61.8484\t&\t-0.857463\t&\t9570.3\t\\\\\nRu$_2$HfSc\t&\t66.8988\t&\t-0.728346\t&\t8129.19\t\\\\\nHf$_2$RuTc\t&\t68.6544\t&\t-0.669049\t&\t7467.38\t\\\\\nTc$_2$HfTa\t&\t66.898\t&\t-0.509943\t&\t5691.57\t\\\\\nTc$_2$HfW\t&\t64.88\t&\t-0.346078\t&\t3862.64\t\\\\\nTi$_2$IrMn\t&\t56.4032\t&\t-0.694021\t&\t7746.08\t\\\\\nTi$_2$IrMo\t&\t61.0552\t&\t-0.626827\t&\t6996.13\t\\\\\nSc$_2$IrNi\t&\t65.8112\t&\t-0.801053\t&\t8940.69\t\\\\\nSc$_2$IrPd\t&\t69.9072\t&\t-0.991899\t&\t11070.8\t\\\\\nY$_2$IrPd\t&\t83.818\t&\t-0.881856\t&\t9842.55\t\\\\\nTi$_2$IrRe\t&\t60.1416\t&\t-0.756525\t&\t8443.71\t\\\\\nSc$_2$IrRh\t&\t67.5008\t&\t-1.04502\t&\t11663.7\t\\\\\nY$_2$IrRh\t&\t81.2688\t&\t-0.841385\t&\t9390.84\t\\\\\nSc$_2$IrRu\t&\t66.5136\t&\t-0.830031\t&\t9264.12\t\\\\\nSc$_2$IrZn\t&\t70.6392\t&\t-0.724073\t&\t8081.51\t\\\\\nTi$_2$IrTc\t&\t59.6952\t&\t-0.840269\t&\t9378.39\t\\\\\nZr$_2$IrTc\t&\t71.522\t&\t-0.693835\t&\t7744.01\t\\\\\nIr$_2$TiZn\t&\t57.6528\t&\t-0.695974\t&\t7767.89\t\\\\\nIr$_2$ZnZr\t&\t64.3208\t&\t-0.634848\t&\t7085.65\t\\\\\nMn$_2$NbTi\t&\t54.8624\t&\t-0.227403\t&\t2538.09\t\\\\\nTi$_2$MnNi\t&\t53.3032\t&\t-0.342964\t&\t3827.88\t\\\\\nTi$_2$MnOs\t&\t56.2816\t&\t-0.502285\t&\t5606.09\t\\\\\nTi$_2$MnRh\t&\t56.0512\t&\t-0.577568\t&\t6446.34\t\\\\\nMn$_2$TaTi\t&\t54.9728\t&\t-0.27885\t&\t3112.3\t\\\\\nMn$_2$TiV\t&\t49.6572\t&\t-0.274813\t&\t3067.23\t\\\\\nMn$_2$TiW\t&\t52.8752\t&\t-0.237692\t&\t2652.92\t\\\\\nNb$_2$MoOs\t&\t65.8108\t&\t-0.281237\t&\t3138.94\t\\\\\nNb$_2$MoRe\t&\t66.5\t&\t-0.250455\t&\t2795.37\t\\\\\nNb$_2$MoRu\t&\t65.5172\t&\t-0.256633\t&\t2864.32\t\\\\\nMo$_2$NbTa\t&\t67.5352\t&\t-0.166161\t&\t1854.55\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_4}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (5\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nNb$_2$MoTc\t&\t66.0904\t&\t-0.252979\t&\t2823.54\t\\\\\nMo$_2$NbW\t&\t65.644\t&\t-0.113515\t&\t1266.96\t\\\\\nTi$_2$MoNi\t&\t58.4756\t&\t-0.291743\t&\t3256.19\t\\\\\nTa$_2$MoOs\t&\t65.8048\t&\t-0.393033\t&\t4386.71\t\\\\\nV$_2$MoOs\t&\t56.3172\t&\t-0.312104\t&\t3483.45\t\\\\\nTi$_2$MoPd\t&\t62.484\t&\t-0.393388\t&\t4390.67\t\\\\\nTi$_2$MoPt\t&\t62.1896\t&\t-0.647475\t&\t7226.58\t\\\\\nTa$_2$MoRe\t&\t66.5052\t&\t-0.341358\t&\t3809.96\t\\\\\nRe$_2$MoTi\t&\t61.2988\t&\t-0.294221\t&\t3283.85\t\\\\\nV$_2$MoRe\t&\t56.986\t&\t-0.280895\t&\t3135.12\t\\\\\nTi$_2$MoRh\t&\t60.9012\t&\t-0.515086\t&\t5748.97\t\\\\\nTa$_2$MoRu\t&\t65.4864\t&\t-0.37169\t&\t4148.5\t\\\\\nV$_2$MoRu\t&\t56.0544\t&\t-0.266878\t&\t2978.68\t\\\\\nTa$_2$MoTc\t&\t66.1236\t&\t-0.348888\t&\t3894\t\\\\\nMo$_2$TaW\t&\t65.598\t&\t-0.138309\t&\t1543.69\t\\\\\nTc$_2$MoTi\t&\t60.4636\t&\t-0.348792\t&\t3892.93\t\\\\\nMo$_2$TiW\t&\t63.4916\t&\t-0.13852\t&\t1546.05\t\\\\\nMo$_2$VW\t&\t61.3504\t&\t-0.111196\t&\t1241.07\t\\\\\nOs$_2$NbSc\t&\t65.272\t&\t-0.455798\t&\t5087.24\t\\\\\nTa$_2$NbOs\t&\t67.6176\t&\t-0.32288\t&\t3603.72\t\\\\\nNb$_2$OsW\t&\t66.2136\t&\t-0.199311\t&\t2224.54\t\\\\\nRe$_2$NbTa\t&\t65.8404\t&\t-0.370123\t&\t4131.01\t\\\\\nNb$_2$ReTc\t&\t65.302\t&\t-0.339447\t&\t3788.63\t\\\\\nRe$_2$NbTi\t&\t63.3304\t&\t-0.398599\t&\t4448.84\t\\\\\nRh$_2$NbZn\t&\t60.0404\t&\t-0.492704\t&\t5499.15\t\\\\\nRu$_2$NbSc\t&\t64.652\t&\t-0.549806\t&\t6136.48\t\\\\\nTa$_2$NbRu\t&\t67.4116\t&\t-0.270899\t&\t3023.55\t\\\\\nRu$_2$NbZn\t&\t59.4144\t&\t-0.275852\t&\t3078.83\t\\\\\nTc$_2$NbTa\t&\t64.8712\t&\t-0.435369\t&\t4859.23\t\\\\\nTc$_2$NbTi\t&\t62.4432\t&\t-0.468812\t&\t5232.5\t\\\\\nTc$_2$NbZr\t&\t67.9524\t&\t-0.372639\t&\t4159.09\t\\\\\nSc$_2$NiOs\t&\t65.1596\t&\t-0.499591\t&\t5576.03\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_5}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (6\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nSc$_2$NiPt\t&\t68.0772\t&\t-0.888355\t&\t9915.08\t\\\\\nTi$_2$NiRe\t&\t57.7528\t&\t-0.434031\t&\t4844.29\t\\\\\nZn$_2$NiRh\t&\t52.1512\t&\t-0.344857\t&\t3849.01\t\\\\\nSc$_2$NiRu\t&\t64.7916\t&\t-0.579752\t&\t6470.71\t\\\\\nTi$_2$NiTc\t&\t57.1568\t&\t-0.486273\t&\t5427.38\t\\\\\nNi$_2$TiZn\t&\t50.568\t&\t-0.405025\t&\t4520.55\t\\\\\nSc$_2$OsPd\t&\t68.9692\t&\t-0.693907\t&\t7744.82\t\\\\\nSc$_2$OsPt\t&\t68.3896\t&\t-0.8455\t&\t9436.77\t\\\\\nTa$_2$OsRe\t&\t65.1812\t&\t-0.351313\t&\t3921.07\t\\\\\nTi$_2$OsRu\t&\t59.0072\t&\t-0.744346\t&\t8307.77\t\\\\\nZr$_2$OsRu\t&\t70.7628\t&\t-0.593618\t&\t6625.47\t\\\\\nOs$_2$ScTa\t&\t65.0652\t&\t-0.533786\t&\t5957.68\t\\\\\nSc$_2$OsZn\t&\t69.8296\t&\t-0.439858\t&\t4909.33\t\\\\\nOs$_2$ScZr\t&\t68.6728\t&\t-0.476955\t&\t5323.38\t\\\\\nTa$_2$OsTc\t&\t64.6524\t&\t-0.405699\t&\t4528.07\t\\\\\nOs$_2$TaTi\t&\t62.2512\t&\t-0.496833\t&\t5545.25\t\\\\\nTa$_2$OsW\t&\t66.2384\t&\t-0.299962\t&\t3347.93\t\\\\\nTi$_2$OsTc\t&\t59.5392\t&\t-0.632543\t&\t7059.92\t\\\\\nV$_2$OsTc\t&\t55.1728\t&\t-0.345037\t&\t3851.02\t\\\\\nZr$_2$OsTc\t&\t71.0824\t&\t-0.476841\t&\t5322.1\t\\\\\nSc$_2$PdPt\t&\t72.2524\t&\t-1.08971\t&\t12162.5\t\\\\\nZn$_2$PdRh\t&\t56.2356\t&\t-0.51684\t&\t5768.54\t\\\\\nSc$_2$PdRu\t&\t68.874\t&\t-0.78368\t&\t8746.79\t\\\\\nPd$_2$ScZn\t&\t64.8108\t&\t-0.783946\t&\t8749.75\t\\\\\nTi$_2$PdTc\t&\t61.282\t&\t-0.542858\t&\t6058.93\t\\\\\nZr$_2$PdTc\t&\t73.412\t&\t-0.522887\t&\t5836.03\t\\\\\nPd$_2$TiZn\t&\t60.4704\t&\t-0.57928\t&\t6465.44\t\\\\\nPd$_2$ZnZr\t&\t67.0388\t&\t-0.641322\t&\t7157.91\t\\\\\nZn$_2$PtRh\t&\t56.5432\t&\t-0.518725\t&\t5789.58\t\\\\\nSc$_2$PtRu\t&\t68.1764\t&\t-0.962407\t&\t10741.6\t\\\\\nPt$_2$ScZn\t&\t65.1192\t&\t-0.926966\t&\t10346\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_6}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (7\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nSc$_2$PtZn\t&\t73.0916\t&\t-0.836692\t&\t9338.47\t\\\\\nZn$_2$PtSc\t&\t62.8012\t&\t-0.667899\t&\t7454.54\t\\\\\nTi$_2$PtTc\t&\t61.1772\t&\t-0.743086\t&\t8293.71\t\\\\\nTi$_2$ReRh\t&\t60.04\t&\t-0.667826\t&\t7453.72\t\\\\\nTa$_2$ReRu\t&\t64.9088\t&\t-0.376608\t&\t4203.39\t\\\\\nTa$_2$ReTc\t&\t65.2984\t&\t-0.465143\t&\t5191.55\t\\\\\nRe$_2$TaTi\t&\t63.47\t&\t-0.457933\t&\t5111.07\t\\\\\nTa$_2$ReW\t&\t66.8768\t&\t-0.278984\t&\t3113.79\t\\\\\nRe$_2$TiV\t&\t58.2544\t&\t-0.402985\t&\t4497.78\t\\\\\nRe$_2$TiW\t&\t61.6096\t&\t-0.350499\t&\t3911.97\t\\\\\nTi$_2$ReZn\t&\t61.3876\t&\t-0.316406\t&\t3531.47\t\\\\\nSc$_2$RhRu\t&\t66.6712\t&\t-0.818639\t&\t9136.97\t\\\\\nRh$_2$ScZn\t&\t60.8824\t&\t-0.779193\t&\t8696.71\t\\\\\nRh$_2$TaZn\t&\t59.9676\t&\t-0.548351\t&\t6120.24\t\\\\\nTi$_2$RhTc\t&\t59.5088\t&\t-0.741616\t&\t8277.31\t\\\\\nZr$_2$RhTc\t&\t71.5328\t&\t-0.64339\t&\t7180.99\t\\\\\nRh$_2$TiZn\t&\t56.792\t&\t-0.783097\t&\t8740.29\t\\\\\nRh$_2$VZn\t&\t55.0032\t&\t-0.416055\t&\t4643.66\t\\\\\nRh$_2$ZnZr\t&\t63.148\t&\t-0.778072\t&\t8684.2\t\\\\\nRu$_2$ScTa\t&\t64.4184\t&\t-0.625766\t&\t6984.29\t\\\\\nRu$_2$ScTi\t&\t62.122\t&\t-0.656051\t&\t7322.31\t\\\\\nRu$_2$ScV\t&\t59.5772\t&\t-0.460194\t&\t5136.31\t\\\\\nSc$_2$RuZn\t&\t70.06\t&\t-0.491623\t&\t5487.1\t\\\\\nRu$_2$ScZr\t&\t68.1104\t&\t-0.649445\t&\t7248.57\t\\\\\nTa$_2$RuTc\t&\t64.312\t&\t-0.412004\t&\t4598.45\t\\\\\nRu$_2$TaTi\t&\t61.542\t&\t-0.554291\t&\t6186.54\t\\\\\nTa$_2$RuW\t&\t66.0004\t&\t-0.285381\t&\t3185.19\t\\\\\nRu$_2$TaY&\t70.2656\t&\t-0.340037\t&\t3795.22\t\\\\\nRu$_2$TaZn\t&\t59.4956\t&\t-0.344438\t&\t3844.33\t\\\\\nTi$_2$RuTc\t&\t59.1864\t&\t-0.643868\t&\t7186.32\t\\\\\nV$_2$RuTc\t&\t54.8572\t&\t-0.320533\t&\t3577.52\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_7}\n\\end{table}\n\n\\begin{table}[htp!]\\centering\n\\mycaption{Summary table of magnetic Heuslers continued (8\/8).}\n\\vspace{3mm}\n\\begin{tabular}{l|r|r|r}\nmaterial & volume (\\AA$^3$) & $H_{\\mathrm{f}}$ (eV) & $T_\\mathrm{S}$~{\\small (K)}\\ \\\\\n\\hline\nZr$_2$RuTc\t&\t70.7428\t&\t-0.524685\t&\t5856.11\t\\\\\nRu$_2$VZn\t&\t54.4836\t&\t-0.218631\t&\t2440.18\t\\\\\nRu$_2$WZn\t&\t57.7132\t&\t-0.126584\t&\t1412.82\t\\\\\nTc$_2$TaTi\t&\t62.5436\t&\t-0.530531\t&\t5921.35\t\\\\\nTc$_2$TaZr\t&\t67.8028\t&\t-0.431941\t&\t4820.97\t\\\\\nTc$_2$TiV\t&\t57.4784\t&\t-0.450416\t&\t5027.17\t\\\\\nTc$_2$TiW\t&\t60.7084\t&\t-0.403323\t&\t4501.56\t\\\\\nTi$_2$TcZn\t&\t61.0412\t&\t-0.346197\t&\t3863.97\t\\\\\nTc$_2$WZr\t&\t65.7664\t&\t-0.258379\t&\t2883.81\t\\\\\n\\end{tabular}\n\\label{fig:art109:magnetic_heusler_8}\n\\end{table}\n\n\\subsection{Conclusion}\n\nIn conclusion we have demonstrated a new systematic pathway to the discovery of novel magnetic materials. We have\ncreated an extensive library of Heusler compounds including about 250,000 structures. For the sub-class of intermetallic\nalloys we have been able to establish the materials stability against decomposition of 20 novel magnetic HAs,\nbelonging to Co$_2YZ$, Mn$_2YZ$ and $X_2$Mn$Z$ classes. A simple machine learning method, correlating calculated\nmicroscopic electronic structure quantities with macroscopic measured properties, has been used to predict the magnetic\n$T_\\mathrm{C}$ of such compounds. The method has been put to the test with the experimental synthesis of four\ncompounds and validated by the growth of two. In particular we have discovered a new high-performance ferromagnet,\nCo$_2$MnTi and a tetragonally distorted antiferromagnet, Mn$_2$PtPd. Our method offers a new high-throughput tool\nfor the discovery of new magnets, which can now be applied to other structural families, opening new possibilities for\ndesigning materials for energy, data storage and spintronics applications.\n\\clearpage\n\\chapter{Conclusion}\nModeling approaches promise a direct and systematic path to materials discovery.\nTo justify their application, these methods need to bridge several gaps:\n\\textbf{i.} prediction of synthesizability (as property prediction\/optimization becomes irrelevant if the material cannot form),\n\\textbf{ii.} treatment of more ``real-world'' phenomena (\\nobreak\\mbox{\\it vs.}\\ the ideal systems modeled \\nobreak\\mbox{\\it ab initio}),\nand\n\\textbf{iii.} identification of structure-property relationships (harnessing the information for practical design rules).\nRecent progress has been driven by data-centric approaches~\\cite{curtarolo:art13}\nfacilitated by large, programmatically-accessible materials databases.\n\nFrameworks like {\\small AFLOW}~\\cite{curtarolo:art49,curtarolo:art53,curtarolo:art57,aflowBZ,curtarolo:art63,aflowPAPER,curtarolo:art85,curtarolo:art110,monsterPGM,aflowANRL,aflowPI}\\ have characterized millions of compounds\nwithout the need for laborious human intervention~\\cite{aflowlibPAPER,aflowAPI,curtarolo:art104,aflux}.\nCombinatorial exploration of various structure prototypes offers a means for sampling\ncandidate stable structures~\\cite{curtarolo:art130,aflowANRL}.\nThe gamut of extractable features derives from electronic, magnetic, chemical, crystallographic, thermomechanical,\nand thermodynamic characterizations --- each warranting\nrobust algorithms\nthat scale with the panoply of structures in the database.\nFor example, convenient definitions for the primitive cell representation~\\cite{aflowPAPER} and\nhigh-symmetry Brillouin Zone path~\\cite{aflowBZ} have not only standardized electronic structure calculations,\nbut also optimized their computation.\nMoreover, careful treatment of spatial tolerance and proper validation schemes have finally\nenabled accurate and autonomous determination of the\ncomplete symmetry profile of crystals~\\cite{curtarolo:art134}.\nElasticity~\\cite{curtarolo:art115} and phonon~\\cite{aflowPAPER,curtarolo:art114,curtarolo:art119,curtarolo:art125}\ncalculations are incredibly sensitive to the quality of the symmetry analysis.\nThe scheme resolves experimentally-validated space groups and\naccommodates even the most skewed unit cells, meeting the demand for high-throughput thermomechanical characterizations.\n\nThe development of the {\\sf \\AFLOW.org}\\ repository has motivated both\nbroad-scale thermodynamic formability modeling and adoption of {\\small ML}\\ algorithms.\nEnsembles of ordered phases are successfully employed to\n\\textbf{i.} construct phase diagrams forecasting stability~\\cite{curtarolo:art146}\nand\n\\textbf{ii.} formulate descriptors and models to predict the formation\/properties of disordered materials~\\cite{curtarolo:art110}.\nThese methods go beyond standard modeling approaches, leveraging\nseveral \\nobreak\\mbox{\\it ab-initio}\\ calculations in each analysis\nand encouraging\nthe continued expansion of these large materials databases.\n\nAs the proliferation of high-throughput approaches\nincreases the wealth of data in the field, the gap between accumulated-information and derived-knowledge widens.\nThe divergence must be addressed autonomously, reciprocating the pace of data generation.\n{\\small ML}\\ models\nare constructed for rapid predictions and exposing subtle\/hidden trends\nthat would have otherwise evaded human detection\/understanding.\nUseful examples include models\npredicting electronic and thermomechanical properties\nfrom basic features of the structure and composition, \\nobreak\\mbox{\\it i.e.}, not requiring additional calculations\nor experiments, affording easy integration into virtually any materials design workflow~\\cite{curtarolo:art124}.\n\n{\\small ML}\\ models are also employed to identify meaningful correlations among materials\/properties,\nleading to enhanced understanding of fundamental physical mechanisms.\nFor many phenomena, the connection between the arrangement of elements into solid compounds\nand the observed macroscopic behavior is still largely unknown, as with\nhigh-temperature superconductors.\nThese materials are particularly difficult to address within automated \\nobreak\\mbox{\\it ab-initio}\\ frameworks because\nthe underlying {\\small DFT}\\ theory fails to capture the strong interactions and correlations\nresponsible for the effect~\\cite{DFT}.\nHowever, as demonstrated by the materials cartography approach~\\cite{curtarolo:art94},\nother similarities between materials, such as the electronic density of states and band structure,\ncan be exploited to reveal interesting candidates.\nAlternatively, {\\small DFT}\\ data can be avoided altogether.\nInstead, models have been constructed leveraging empirical\ninformation retrieved from the SuperCon database~\\cite{SuperCon} for more than 12,000 materials~\\cite{curtarolo:art137}.\nDistinct driving mechanisms are resolved by comparing important features\nof a general model, trained on all data,\nwith that of family-specific models, trained on\nlow-$T_{\\mathrm{c}}$, cuprate, and iron-based superconductors, respectively.\n\nStructure-property relationships have also been resolved\nin perovskites ($ABX_{3}$ where $X$ = F and O) for high-temperature thermoelectric applications~\\cite{curtarolo:art120}.\nThe thermal conductivity of fluorides is strongly influenced by substitutions of the $B$ site,\nwhile in oxides the same is true for the $A$ site --- presenting a useful engineering opportunity.\nFor example, to mitigate costs in device production, substitutions in the less influential site\ncan be expected not to affect the thermoelectric performance.\n\nFinally, thermodynamic descriptors and regression analyses among classes of ground-state compounds\ncontributed to the screening of 36,540 Heusler compounds for new magnetic systems~\\cite{curtarolo:art109}.\nAn attempt to synthesize four candidates yielded two novel materials.\nOf these, Co$_{2}$MnTi promises to be a high-performance ferromagnet with $T_{\\mathrm{C}}=938$~K,\nas predicted by the Slater-Pauling curve --- illustrating the predictive power of data-driven approaches.\nThese methods will accelerate the path to synthesis and, ultimately,\ntransform the practice of traditional materials discovery to one of rational and autonomous materials design.\n\\clearpage\n\n\\newcommand{Ozoli\\c{n}\\v{s}}{Ozoli\\c{n}\\v{s}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOver the last decades, multi-agent systems have been considered in a variety of applications such as connectivity and formation control \\cite{magnus}\nor coverage \\cite{coverage}. The complexity of these applications has motivated the need of an expressive language, capable of describing complex task specifications for planning and control synthesis. \n\nRecently, extensive interest has been shown in planning under high-level task specifications expressed by Linear Temporal Logic (LTL) \\cite{tl1,tl2}. In these methods the temporal formula, the environment and the agent dynamics are abstracted into finite-transition systems. Then, graph-based methods are employed to find a discrete path satisfying the LTL specifications which is finally followed using continuous control laws. An important limitation of the aforementioned methods is the increasing computational complexity as the number of the agents in the team becomes larger. Towards minimizing the computational costs, large effort has been devoted to the decomposition of a global LTL formula into local LTL tasks whose satisfaction depends on subsets of agents. Existing methods, applied to heterogeneous agents \\cite{dimos2,ltl_services}, most often employ exhausting automata-based approaches \\cite{dimos1,dimos2,ltl_services} or more recently, cross-entropy optimization methods limited, though, to homogeneous agents \\cite{ltldec_entropy}. \n\nAll methods presented so far consider the satisfaction of LTL tasks without explicit time constraints. On the other hand, Signal Temporal Logic (STL) \\cite{stl} can express complex tasks under strict deadlines. An advantage of STL over LTL, is the robust semantics \\cite{stlrob, fainekos} it offers that allow the evaluation of the satisfaction of the task over a continuous-time signal, rendering the abstractions of the agents' dynamics obsolete. \n\nExisting methods for planning under STL specifications consider a global STL formula and find plans as solutions to computationally prohibitive MILPs \\cite{mpc_raman,mpc_sadra} or to scalable convex programs \\cite{lars2,kunal}. Other approaches compose \\cite{decentralized} or assume the existence \\cite{lars_linear} of local STL tasks whose satisfaction involves only a small subset of agents. This facilitates the design of decentralized frameworks that are inherently more robust to agents' failures and often cheaper in terms of communication. Towards decentralized control under global task specifications, a satisfiability modulo theories (SMT) approach has been proposed in \\cite{stldec2} for tasks described in caSTL, in which both the global formula and the team of agents is decomposed. Here, the decomposition is based on a set of services required for each task and a set of utility functions specifying the capabilities of the agents. Nevertheless, the decomposition of a global STL formula in continuous space and time remains an open problem.\n\nIn this paper we propose a novel framework for the decomposition of a global STL formula imposed on a multi-agent system into a set of local tasks when the team of agents is a-priori divided into disjoint sub-teams. The goal of the decomposition is to make the satisfaction of every local task dependent only to a subset of agents that belong to the same sub-team. Initially, the predicate functions corresponding to STL formulas forming the local tasks are parameterized as functions of the infinity norm of the agents' states while their parameters are found as part of the solution to a convex program that aims at maximizing the volume of their zero level-set. Although the choice of the parametric family of the predicate functions is not restrictive, our current choice allows us to draw conclusions on the volume of a continuous state-space set by incorporating a finite, but possibly large, number of constraints in the convex program. The number of these constraints differs per global STL task but depends solely on the number of the agents' states involved in its satisfaction. Two definitions of the local tasks that differ on the definition of the STL tasks originating from eventually formulas are introduced. Finally, for both definitions the satisfaction of the global STL formula is proven when the conjunction of the local tasks is satisfied.\n\nThe remainder of the paper is as follows: Section II includes the preliminaries and problem formulation. Section III introduces the proposed method for STL decomposition. Simulations are shown in Section IV and conclusions are summarized in Section V.\n\n\n\\section{Preliminaries and Problem Formulation}\nThe set of real and non-negative real numbers are denoted by $\\mathbb{R}$ and $\\mathbb{R}_{\\geq 0}$ respectively. True and false are denoted by $\\top, \\bot$ respectively. Scalars and vectors are denoted by non-bold and bold letters respectively. The infinity norm of a vector $\\mathbf{x} \\in \\mathbb{R}^n$ is defined as $\\Vert \\mathbf{x} \\Vert_{\\infty}=\\max_i\\vert \\mathbf{x}_i \\vert$, where $\\mathbf{x}=\\begin{bmatrix} \\mathbf{x}_1 & \\ldots & \\mathbf{x}_n \\end{bmatrix}^T$.\nGiven a finite set $ \\mathcal{V}$, $\\prod_{k\\in \\mathcal{V}}\\mathbb{X}_k$ denotes the Cartesian product of the sets $\\mathbb{X}_k, k\\in \\mathcal{V}$. Given a rectangular matrix $A\\in M_{n\\times m}(\\mathbb{R})$ we define the set $A\\mathbb{X}$ as $A\\mathbb{X}=\\{A\\mathbf{x}: \\mathbf{x} \\in \\mathbb{X}\\}$. A square matrix $P\\in M_n(\\{0,1\\})$ is called a \\textit{permutation matrix} \\cite[Ch. 0.9.5]{horn} if exactly\none entry in each row and column is equal to 1 and all other entries are 0. Consider the vectors $\\mathbf{x} \\in \\mathbb{R}^n, \\mathbf{y}\\in \\mathbb{R}^m$ with $n\\leq m$ satisfying $\\mathbf{x}=B\\mathbf{y}$. The matrix $B=[b_{ij}]$ is called a \\textit{selection matrix} if it has the following properties: 1) $b_{ij}\\in \\{0,1\\}$, 2) $\\sum_{j=1}^m b_{ij}=1, \\forall i=1,\\ldots,n$ and 3) $\\sum_{i=1}^n b_{ij}=1, \\forall j=1,\\ldots,m$.\n\n\\subsection{Signal Temporal Logic (STL)}\nSignal Temporal Logic (STL) determines whether a predicate $\\mu$ is true or false. The validity of each predicate $\\mu$ is evaluated based on a continuously differentiable function $h:\\mathbb{R}^n \\rightarrow \\mathbb{R}$ as follows:\n\\begin{equation*}\n \\mu=\\begin{cases} \\top, &h(\\mathbf{x}) \\geq 0 \\\\ \\bot, & h(\\mathbf{x})< 0 \\end{cases}\n\\end{equation*}\nfor $\\mathbf{x} \\in \\mathbb{R}^n$. The basic STL formulas are given by the grammar:\n$$ \\phi:= \\top \\; | \\; \\psi \\;| \\;\\neg \\phi \\; | \\; \\phi_1 \\land \\phi_2 \\; |\\; \\mathcal{G}_{[a,b]} \\phi \\;| \\; \\mathcal{F}_{[a,b]} \\phi \\;| \\; \\phi_1 \\; \\mathcal{U}_{[a,b]} \\; \\phi_2 $$\nwhere $\\phi_1, \\phi_2 $ are STL formulas and $\\mathcal{G}_{[a,b]},\\; \\mathcal{F}_{[a,b]}, \\; \\mathcal{U}_{[a,b]}$ is the always, eventually and until operator defined over the interval $[a,b]$ with $0 \\leq a \\leq b$. Let $ \\mathbf{x} \\models \\phi$ denote the satisfaction of the formula $\\phi$ by a signal $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$. The formula $\\phi$ is satisfiable if $\\exists \\; \\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$ such that $\\mathbf{x} \\models \\phi$. The STL semantics for a signal $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$ are recursively given and can be found, e.g., in \\cite{lars2}. STL is equipped with robustness metrics determining how robustly an STL formula $\\phi$ is satisfied at time $t$ by a signal $\\mathbf{x}$. These semantics are defined as follows \\cite{stlrob, fainekos}: $\\rho^{\\mu}(\\mathbf{x},t)=h(\\mathbf{x}(t))$, $\\rho^{\\neg \\phi}(\\mathbf{x},t)=-\\rho^{\\phi}(\\mathbf{x},t)$, $ \\rho^{\\phi_1 \\wedge \\phi_2}(\\mathbf{x},t)=\\min(\\rho^{\\phi_1}(\\mathbf{x},t),\\rho^{\\phi_2}(\\mathbf{x},t))$, $\\rho^{\\phi_1 \\; \\mathcal{U}_{[a,b]} \\; \\phi_2}(\\mathbf{x},t)=\\max_{t_1 \\in [t+a,t+b]} \\min(\\rho^{\\phi_2}(\\mathbf{x},t_1), \\min_{t_2\\in [t,t_1]} \\rho^{\\phi_1}(\\mathbf{x},t_2)) $, $\\rho^{\\mathcal{F}_{[a,b]} \\phi}(\\mathbf{x},t)=\\max_{t_1\\in [t+a,t+b]} \\rho^{\\phi}(\\mathbf{x},t_1) $, $\\rho^{\\mathcal{G}_{[a,b]} \\phi}(\\mathbf{x},t)=\\min_{t_1\\in [t+a,t+b]} \\rho^{\\phi}(\\mathbf{x},t_1)$. Finally, it should be noted that $\\mathbf{x} \\models \\phi$ if $\\rho^{\\phi}(\\mathbf{x},0)>0$.\n\n\n\\subsection{Problem Formulation}\n\nIn this work we consider the following STL fragment:\n\\begin{subequations}\n\\begin{align}\n \\psi &:= \\; \\mu \\;| \\;\\neg \\mu \\label{eq:f1} \\\\\n \\varphi &:= \\mathcal{G}_{[a,b]} \\psi \\;| \\; \\mathcal{F}_{[a,b]} \\psi \\label{eq:f2}\\\\\n \\phi&:=\\bigwedge_{i=1}^{p} \\varphi_i \\label{eq:f3}\n\\end{align}\n\\end{subequations}\nwhere $0\\leq a \\leq b < \\infty$ and $p\\geq 1$.\n\\begin{remark}\nThe STL fragment defined by \\eqref{eq:f1}-\\eqref{eq:f3} is expressive enough to accommodate until STL formulas of the form $\\varphi=\\psi_1 \\mathcal{U}_{[a,b]} \\psi_2$ where $\\psi_i, i=1,2$ are defined by \\eqref{eq:f1}. By definition, for any $t^* \\in [a,b]$ the until formula $\\varphi=\\psi_1 \\mathcal{U}_{[a,b]} \\psi_2$ can be written as $\\varphi=\\mathcal{G}_{[a,t^*]} \\psi_1 \\wedge \\mathcal{F}_{[t^*,t^*]} \\psi_2$. Hence, if for a given time instant $t^* \\in [a,b]$ there exists a signal $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{R}^n$ such that $\\mathbf{x} \\models \\big(\\mathcal{G}_{[a,t^*]} \\psi_1 \\wedge \\mathcal{F}_{[t^*,t^*]} \\psi_2\\big)$ then $\\mathbf{x} \\models \\varphi$.\n\\end{remark}\n\nConsider a team of $R$ agents with each agent identified by its index $k \\in \\mathcal{V}=\\{1, \\ldots, R\\}$. For every agent $k$ let $ \\mathbf{x}_k \\in \\mathbb{X}_k$ denote its state vector, where $\\mathbb{X}_k\\subseteq \\mathbb{R}^{\\bar{n}_k} $ is a known, bounded, convex set for every $k\\in \\mathcal{V}$. Let $n=\\sum_{k\\in \\mathcal{V}} \\bar{n}_k$ and $\\mathbf{x}=\\begin{bmatrix} \\mathbf{x}_1^T & \\ldots & \\mathbf{x}_R^T \\end{bmatrix}^T \\in \\mathbb{X}$ where $\\mathbb{X}=\\prod_{k\\in \\mathcal{V}} \\mathbb{X}_k$ is convex as the Cartesian product of convex sets. Assume that the agents are decomposed in $v$ smaller teams $\\{\\mathcal{V}_1, \\ldots, \\mathcal{V}_v\\}$, $\\mathcal{V}_l \\subseteq \\mathcal{V}, l=1,\\ldots,v$ that are disjoint, i.e., for any $l_1, l_2\\in \\{1,\\ldots, v\\}$ with $l_1\\neq l_2$ it holds that $\\mathcal{V}_{l_1}\\cap \\mathcal{V}_{l_2}=\\emptyset$ and satisfy $\\bigcup_{l=1}^v \\mathcal{V}_l=\\mathcal{V}$.\n\nConsider a global STL formula $\\phi$ of the form \\eqref{eq:f3} with $\\mathcal{I}= \\{1,\\ldots, p\\}$ and sub-formulas $\\varphi_i, \\; i\\in \\mathcal{I}$ satisfying \\eqref{eq:f1}-\\eqref{eq:f2}. Let $[a_i,b_i]$ be the interval of satisfaction associated with the temporal operator of $\\varphi_i, i\\in \\mathcal{I}$ and define the sets of always and eventually formulas of $\\phi$ as $\\mathcal{I}_{\\mathcal{G}}=\\big\\{i\\in \\mathcal{I}: \\varphi_i=\\mathcal{G}_{[a_i,b_i]} \\psi_i\\big\\}$ and $\\mathcal{I}_{\\mathcal{F}}=\\big\\{i\\in \\mathcal{I}: \\varphi_i=\\mathcal{F}_{[a_i,b_i]} \\psi_i\\big\\}$ respectively. Observe that by definition of the STL fragment in \\eqref{eq:f1}-\\eqref{eq:f3} it holds that $\\mathcal{I}=\\mathcal{I}_{\\mathcal{G}} \\cup \\mathcal{I}_{\\mathcal{F}}$. Assume without loss of generality that the satisfaction of each $\\varphi_i, i\\in \\mathcal{I}$ depends on multiple agents of different teams $\\mathcal{V}_l$ and let $V_i\\subseteq \\{1,\\ldots,v\\}, i\\in \\mathcal{I}$ denote the set of indices of the agents' groups that have at least one member contributing to the satisfaction of $\\varphi_i$. Since $\\phi$ is a global task its satisfaction requires agents to be fully aware of the actions of their peers. However, in real-time scenarios communication between all agents may often be hard to establish, especially when the working environment of the agents is large. Addressing this problem, in this paper we propose decomposing the initial task $\\phi$ into local tasks the satisfaction of which depends only on the agents in the same team $\\mathcal{V}_l$. This problem is formally introduced as:\n\n\\begin{problem}\nGiven a global STL formula $\\phi$ defined by \\eqref{eq:f3} and the disjoint sets of agents $\\mathcal{V}_l, l=1,\\ldots,v$ satisfying $\\bigcup_{l=1}^v \\mathcal{V}_l=\\mathcal{V}$ find STL formulas $\\phi_1, \\ldots, \\phi_v$ such that: 1) each STL formula $\\phi_l$ depends on the agents in $\\mathcal{V}_l$ and 2) $\\mathbf{x} \\models \\big(\\phi_1\\wedge \\ldots\\wedge \\phi_v\\big) \\Rightarrow \\mathbf{x} \\models \\phi$ if such $\\mathbf{x}:\\mathbb{R}_{\\geq 0}\\rightarrow \\mathbb{X}$ exists.\n\\end{problem}\n\n\\section{Decomposition of STL Formulas}\n\nIn this Section we design a number of STL tasks the satisfaction of which depends on a known subset of agents. Consider the formula $\\phi$ defined by \\eqref{eq:f3}. Let the predicate function $h_i:\\mathbb{X} \\rightarrow \\mathbb{R}$ associated with the formula $\\varphi_i,i\\in \\mathcal{I}$. Then, the zero level-set of $h_i(\\mathbf{x})$ is defined as follows:\n\\begin{equation}\n \\mathcal{S}_i=\\{\\mathbf{x} \\in \\mathbb{X}: h_i(\\mathbf{x})\\geq 0\\} \\label{eq:levelset}\n\\end{equation}\nHere, we assume that $h_i(\\mathbf{x}),i\\in \\mathcal{I}$ is a function whose value may depend on the states of all agents in $\\mathcal{V}$. As a result guaranteeing the satisfaction of $\\varphi_i, i\\in \\mathcal{I}$ may require the knowledge of all agents' actions and thus global communication. In real-time scenarios communication among all agents can become costly or hard due to packet losses or communications delays. On the other hand, decentralized approaches allow agents to communicate with a subset of their peers and optimize their actions with respect to a limited number of agents thus improving the computational complexity of the problem. \n\nIn the context of STL control synthesis a decentralized approach involves the requirement of assigning to agents tasks whose satisfaction depends only to a subset of agents with established communication links , i.e., to $\\mathcal{V}_l, l=1,\\ldots,v$ while guaranteeing the satisfaction of the global task $\\phi$. To that end, in this paper we propose a set of STL tasks $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l,\\; l=1,\\ldots,v$ whose satisfaction depends on the corresponding set of agents $\\mathcal{V}_l$. Here, $\\bar{\\varphi}_{q_i}^l$ denotes the $q_i^l$-th formula of $\\phi_l$ that is considered to be the result of the decomposition of the sub-formula $\\varphi_i$ of \\eqref{eq:f3}. If it is clear from context, we may omit the subscript of the index $q_i\\in \\{1,\\ldots,p_l\\}$. \n\nLet $\\mathbf{z}_l \\in \\mathcal{Z}_l \\subset \\mathbb{R}^{n_l}$ be the states of the agents in $\\mathcal{V}_l$ where $n_l=\\sum_{k\\in \\mathcal{V}_l} \\bar{n}_k$ and $\\mathcal{Z}_l=\\prod_{k\\in \\mathcal{V}_l} \\mathbb{X}_k$. The vector $\\mathbf{z}_l, l=1,\\ldots,v$ can be obtained from $\\mathbf{x}$ using the following equation:\n\\begin{equation}\n \\mathbf{z}_l=E_l \\mathbf{x} \\label{eq:z2x}\n\\end{equation}\nwhere $E_l \\in M_{n_l\\times n}(\\{0,1\\})$ is a selection matrix. Additionally, the vector $\\mathbf{x}$ can be written with respect to the vectors $\\mathbf{z}_l, l=1,\\ldots,v$ as:\n\\begin{equation}\n \\mathbf{x}=A \\mathbf{z} \\label{eq:permutation}\n\\end{equation}\nwhere $\\mathbf{z}=\\begin{bmatrix} \\mathbf{z}_1^T & \\ldots & \\mathbf{z}_v^T \\end{bmatrix}^T$ and $A\\in M_n(\\{0,1\\})$ is an appropriately chosen permutation matrix. Let $[a_{q}^l,b_{q}^l]$ and $h_{q}^l:\\mathcal{Z}_l\\rightarrow \\mathbb{R}, q=1,\\ldots,p_l, \\; l=1,\\ldots v$ denote the interval of satisfaction and predicate function corresponding to $\\bar{\\varphi}_{q}^l$ respectively. Here, for every $l=1,\\ldots,v$ we assume that $h_{q_i}^l(\\mathbf{z}_l)=h_{q_i}^l(\\mathbf{z}_l;\\bm{\\theta}_i^l), q_i=1,\\ldots,p_l$ belongs to a known family of functions and its value depends on a set of parameters $\\bm{\\theta}_i^l\\in \\Theta_i^l \\subseteq \\mathbb{R}^{m_i^l}$ to be tuned towards maximizing the volume of the zero level-set of $h_{q_i}^l(\\mathbf{z}_l)$ defined as:\n\\begin{equation}\n S_{q_i}^l=\\big\\{\\mathbf{z}_l \\in \\mathcal{Z}_l: h_{q_i}^l(\\mathbf{z}_l)\\geq 0 \\big\\} \\label{eq:set}\n\\end{equation}\n\nBased on the above we propose the following method for designing $\\phi_l,l=1,\\ldots,v$:\n\\begin{theorem}\nConsider the global STL formula $\\phi$ defined by \\eqref{eq:f1}-\\eqref{eq:f3} and the predicate function $h_i(\\mathbf{x})$ associated to $\\varphi_i, i\\in \\mathcal{I}$. Assume that $\\mathcal{S}_i\\neq \\emptyset$, where $\\mathcal{S}_i, i\\in \\mathcal{I}$ is defined in \\eqref{eq:levelset}. For every $i\\in \\mathcal{I}$ derive the functions $h_{q_i}^l(\\mathbf{z}_l)$ as solutions to the following optimization problem:\n\\begin{subequations}\\label{eq:dec}\n\\begin{align}\n \\max_{\\bm{\\theta}_i^l\\in \\Theta_i^l, l\\in V_i} \\sum_{l\\in V_i}\\textit{vol}(S_{q_i}^l) \\tag{\\ref{eq:dec}}\n \\end{align}\nsubject to:\n\\begin{align}\n \\mathbf{z}_l&\\in S_{q_i}^l, \\quad l\\in V_i\\\\\n \\mathbf{x}&\\in \\mathcal{S}_i \\label{eq:basiceq}\\\\\n \\mathbf{z}_l&=E_l \\mathbf{x}, \\quad l\\in V_i\n\\end{align}\n\\end{subequations}\nwhere $\\textit{vol}(S_{q_i}^l)$ denotes the volume of the set $S_{q_i}^l$ defined in \\eqref{eq:set}. For every $l=1,\\ldots,v$ define the formulas $\\bar{\\varphi}_{q_i}^l$ as follows:\n\\begin{equation}\n \\bar{\\varphi}_{q_i}^l=\\begin{cases} \\mathcal{F}_{[a_{q_i}^l,b_{q_i}^l]} \\bar{\\mu}_{q_i}^l, \\quad i \\in \\mathcal{I}_{\\mathcal{F}}\\\\\\mathcal{G}_{[a_{q_i}^l,b_{q_i}^l]} \\bar{\\mu}_{q_i}^l, \\quad i\\in \\mathcal{I}_{\\mathcal{G}}\n \\end{cases} \\label{eq:newformula}\n\\end{equation}\nwith \n\\begin{subequations}\n\\begin{align}\n [a_{q_i}^l,b_{q_i}^l]&=\\begin{cases}[t_i,t_i], \\quad i \\in \\mathcal{I}_{\\mathcal{F}}\\\\ [a_i,b_i],\\quad i\\in \\mathcal{I}_{\\mathcal{G}} \\end{cases} \\label{eq:interval}\\\\\n \\bar{\\mu}_{q_i}^l&=\\begin{cases} \\top, & h_{q_i}^l(\\mathbf{z}_l)\\geq 0 \\\\ \\bot, & h_{q_i}^l(\\mathbf{z}_l)< 0 \\end{cases} \\label{eq:predicate}\n\\end{align}\n\\end{subequations}\nwhere $\\mathcal{I}=\\mathcal{I}_{\\mathcal{G}} \\cup \\mathcal{I}_{\\mathcal{F}}$, $t_i\\in [a_i,b_i]$ and $[a_i,b_i]$ is the interval of satisfaction associated with each $\\varphi_i$ of the global formula $\\phi$. Let $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l$, $l=1,\\ldots,v$. If there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$, then $\\rho^\\phi(\\mathbf{x},0)>0$.\n\\end{theorem}\n\n\\begin{proof}\nFor every $i\\in \\mathcal{I}$, \\eqref{eq:dec} aims at maximizing the volume of the $S_{q_i}^l,l\\in V_i$ which underapproximates the projection set of $\\mathcal{S}_i$ onto $\\mathcal{Z}_l$. Since $\\mathcal{S}_i\\neq \\emptyset$ for every $i\\in \\mathcal{I}$, \\eqref{eq:dec} is always feasible.\nBy definition of the robust semantics and the definition of the min operator it holds that:\n\\begin{equation*}\n \\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)\\leq \\rho^{\\phi_l}(\\mathbf{x},0), \\; l=1,\\ldots,v\n\\end{equation*}\nAs a result if there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$ then $\\rho^{\\phi_l}(\\mathbf{x},0)> 0$ for every $l=1,\\ldots,v$. By design, the satisfaction of $\\phi_l$ depends on a subset of agents, thus $\\rho^{\\phi_l}(\\mathbf{x},0)=\\rho^{\\phi_l}(\\mathbf{z}_l,0)> 0$ where $\\mathbf{z}_l$ satisfies \\eqref{eq:z2x}. Then, by the definition of the robust semantics for every $l=1,\\ldots,v$ and $ q_i=1,\\ldots,p_l$ it holds that:\n\\begin{equation*}\n 0< \\rho^{\\phi_l}(\\mathbf{z}_l,0)=\\min_{q_i=1,\\ldots,p_l} \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)\\leq \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)\n\\end{equation*}\nIf $i\\in \\mathcal{I}_{\\mathcal{G}}$, then $\\varphi_{q_i}^l$ is an always formula. Hence due to \\eqref{eq:newformula}, $ \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)>0$ implies $h_{q_i}^l(\\mathbf{z}_l(t))> 0$ for every $t\\in [a_i,b_i]$, $l\\in V_i$. Since $h_{q_i}^l(\\mathbf{z}_l), l\\in V_i$ is a feasible solution of \\eqref{eq:dec}, it holds that $h_i(\\mathbf{x}(t))>0, \\forall t\\in [a_i,b_i]$ where $\\mathbf{x}(t)=A\\mathbf{z}(t)$ and $\\mathbf{z}(t)=\\begin{bmatrix}\\mathbf{z}_1^T(t) &\\ldots & \\mathbf{z}_v^T(t) \\end{bmatrix}^T$. Hence, $\\rho^{\\varphi_i}(\\mathbf{x},0)>0$. If $i\\in \\mathcal{I}_{\\mathcal{F}}$, then due to \\eqref{eq:newformula} and \\eqref{eq:interval}-\\eqref{eq:predicate}, for every $l\\in V_i$ it holds that: $ \\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)=h_{q_i}^l(\\mathbf{z}_l(t_i))>0$. Following a similar argument as before, we can conclude that $h_i(\\mathbf{x}(t_i))>0$ where $\\mathbf{x}(t_i)=A\\mathbf{z}(t_i)$. This implies that $\\max_{t\\in [a_i,b_i]}h_i(\\mathbf{x}(t))\\geq h_i(\\mathbf{x}(t_i))>0$ leading to $\\rho^{\\varphi_i}(\\mathbf{x},0)>0$. Then, the result follows by the fact that $\\rho^{\\phi}(\\mathbf{x},0)=\\min\\big( \\min_{i\\in \\mathcal{I}_{\\mathcal{G}}} \\rho^{\\varphi_i}(\\mathbf{x},0), \\min_{i\\in \\mathcal{I}_{\\mathcal{F}}} \\rho^{\\varphi_i}(\\mathbf{x},0)\\big)$.\n\\end{proof}\n\nIn problem \\eqref{eq:dec} the goal is to maximize the volume of set $S_{q_i}^l$ by exhaustively evaluating $h_{q_i}^l(\\mathbf{z}_l)$ over the continuous set $\\mathcal{Z}_l$, which is in practice intractable. Another limitation of the proposed problem is often the lack of a known formula for computing the volume of a set, unless $h_{q_i}^l(\\mathbf{z}_l)$ belongs to a specific class of functions such as the class of ellipsoids.\n\nAiming at reducing the computational complexity of the STL decomposition problem described above, we propose a convex formulation for designing the predicate functions corresponding to \\eqref{eq:newformula} for every $l=1,\\ldots,v$. The computational benefits of the proposed approach are related to the number of points in $\\mathcal{Z}_l, l\\in V_i$ that are considered for evaluation of the satisfaction of \\eqref{eq:basiceq}. More specifically, contrary to \\eqref{eq:dec}, in this approach only a finite number of points is evaluated that depends on the number of states $\\mathbf{x}_k$ of the agents $k\\in \\mathcal{V}_l, l\\in V_i$ involved in the satisfaction of $h_i(\\mathbf{x})$. Let $d_i^l\\geq 1$ be the number of states in $\\mathbf{z}_l, l\\in V_i$ contributing to $h_i(\\mathbf{x})$. Since the global formula is a-priori given, the elements of $\\mathbf{z}_l , l\\in V_i$ on which the predicate function $h_i(\\mathbf{x})$ depends are known. Hence, we may write $h_i(\\mathbf{x}),i \\in \\mathcal{I}$ as:\n\\begin{subequations}\n\\begin{align}\n h_i(\\mathbf{x})&=h_i(\\mathbf{y}_{\\alpha(1)},\\ldots,\\mathbf{y}_{\\alpha(\\vert V_i \\vert)}) \\label{eq:dependency2}\\\\\n\\mathbf{y}_{\\alpha(c)}&=B_i^{\\alpha(c)} \\mathbf{z}_{\\alpha(c)}, \\quad c=1,\\ldots, \\vert V_i \\vert \\label{eq:dependency}\n\\end{align}\n\\end{subequations}\nwhere $\\alpha: \\{1,\\ldots,\\vert V_i \\vert\\} \\rightarrow V_i$ is an injective function defined as $\\alpha(c)=l$ and where $B_i^{\\alpha(c)}\\in M_{d_i^l\\times n_l}(\\{0,1\\})$ is an appropriate selection matrix and $\\mathbf{z}_{\\alpha(c)} \\in \\mathcal{Z}_{\\alpha(c)}$.\n\nBased on the above, we can consider a special class of concave functions of the following form:\n\\begin{equation}\n h_{q_i}^l(\\mathbf{z}_l)=r_{q_i}^l-\\Vert B_i^l(\\mathbf{z}_l-\\mathbf{c}_{q_i}^l) \\Vert_{\\infty}, \\quad q_i=1,\\ldots,p_l \\label{eq:prinf}\n\\end{equation}\nwhere $r_{q_i}^l\\in \\mathbb{R}_{\\geq 0}$, $\\mathbf{c}_{q_i}^l\\in \\mathcal{Z}_l$ and $B_i^l\\in M_{d_i^l\\times n_l}(\\{0,1\\})$ is the same selection matrix considered in \\eqref{eq:dependency} with $\\alpha(c)=l$.\nLet $J_{q_i}^l\\subseteq \\{1,\\ldots,n_l\\}$ denote the set of indices of the columns of $B_i^l$ with non-zero entries. Given the predicate functions defined by \\eqref{eq:prinf}, it follows that:\n\\begin{equation}\n h_{q_i}^l(\\mathbf{z}_l)\\geq 0 \\Leftrightarrow \\quad \\mathbf{z}_l(\\eta) \\in [-r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta),r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta)]\n\\end{equation}\nfor every $\\eta \\in J_{q_i}^l$ where $\\mathbf{z}_l(\\eta),\\mathbf{c}_{q_i}^l(\\eta)$ denote the $\\eta$-th element of the vectors $\\mathbf{z}_l,\\mathbf{c}_{q_i}^l$ respectively. For every $i\\in\\mathcal{I}$ and $l\\in\\{1,\\ldots,v\\}$ consider the following set of vectors:\n\\begin{equation}\n\\begin{split}\n \\mathcal{P}_i^l=\\big\\{\\bm{\\xi}\\in \\mathcal{Z}_l: \\bm{\\xi}(\\eta)&=-r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta) \\; \\text{or} \\\\ \\bm{\\xi}(\\eta)&=r_{q_i}^l+\\mathbf{c}_{q_i}^l(\\eta), \\eta \\in J_{q_i}^l \\big\\} \\label{eq:vertexset}\n \\end{split}\n\\end{equation}\nwhere $\\bm{\\xi}(\\eta)$ denotes the $\\eta$-th element of $\\bm{\\xi}$. If $r_{q_i}^l\\geq 0$, the set $B_i^l\\mathcal{P}_i^l$ consists of the vertices of a hypercube in $\\mathbb{R}^{d_i^l}$ of edge length $r_{q_i}^l$ and center $\\mathbf{c}_{q_i}^l$.\nHence, its cardinality will be equal to $2^{d_i^l}$. To guarantee the convexity of the proposed problem we pose the following assumption:\n\n\\begin{assumption}\nFor every $i\\in \\mathcal{I}$ the predicate function $h_i(\\mathbf{x})$ is concave in $\\mathbb{X}$.\n\\end{assumption}\n\n\\begin{theorem}\nConsider the global STL formula $\\phi$ defined by \\eqref{eq:f1}-\\eqref{eq:f3} and the predicate functions $h_i(\\mathbf{x}), i\\in \\mathcal{I}$ associated to $\\varphi_i$. Let Assumption 1 hold. For every $i\\in \\mathcal{I}$ assume that $\\mathcal{S}_i\\neq \\emptyset$, where $\\mathcal{S}_i$ is defined in \\eqref{eq:levelset}. Consider the functions $ h_{q_i}^l(\\mathbf{z}_l), \\; q_i=1,\\ldots,p_l, \\; l=1,\\ldots,v$ defined by \\eqref{eq:prinf} where $\\mathbf{c}_{q_i}^l,r_{q_i}^l$ are parameters found as the solution to the following optimization problem:\n\\begin{subequations}\\label{eq:convex}\n\\begin{align}\n \\max_{\\mathbf{c}_{q_i}^l,r_{q_i}^l} \\sum_{l\\in V_i} r_{q_i}^l \\tag{\\ref{eq:convex}}\n \\end{align}\nsubject to:\n\\begin{align}\n h_i(\\mathbf{y}_{\\alpha(1)},\\ldots,\\mathbf{y}_{\\alpha(\\vert V_i \\vert)}) &\\geq 0 \\\\\n \\mathbf{y}_{\\alpha(c)}&\\in B_i^{\\alpha(c)}\\mathcal{P}_i^{\\alpha(c)}, \\quad c=1,\\ldots,\\vert V_i \\vert\n\\end{align}\n\\end{subequations}\nwhere $\\alpha: \\{1,\\ldots,\\vert V_i \\vert\\} \\rightarrow V_i$ and $B_i^{\\alpha(c)}\\in M_{d_i^l\\times n_l}(\\{0,1\\})$ is the injective function and selection matrix respectively considered in \\eqref{eq:dependency2}-\\eqref{eq:dependency} and $\\mathcal{P}_i^{\\alpha(c)}$ is the set defined by \\eqref{eq:vertexset} for every $\\alpha(c)=l\\in V_i$.\nFor every $l=1,\\ldots,v$ define the formulas $\\bar{\\varphi}_{q_i}^l$ based on \\eqref{eq:newformula} and \\eqref{eq:interval}-\\eqref{eq:predicate}\nand consider the decomposed STL formulas $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l$, $l=1,\\ldots,v$. If there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$, then $\\rho^\\phi(\\mathbf{x},0)>0$.\n\\end{theorem}\n\\begin{proof}\nFor every $i\\in \\mathcal{I}$, \\eqref{eq:convex} finds the maximum volume sets $B_i^lS_{q_i}^l,l\\in V_i$ which are underapproximations of the projection sets of $\\mathcal{S}_i$ onto $B_i^l\\mathcal{Z}_l$. Since $\\mathcal{S}_i\\neq \\emptyset$ for every $i\\in \\mathcal{I}$, \\eqref{eq:convex} is always feasible. To simplify notation let the sets $W=\\big\\{\\mathbf{y}: \\mathbf{y}_{l}=B_i^{l}\\mathbf{z}_{l}, \\; \\mathbf{z}_l \\in S_{q_i}^l, \\; l\\in V_i\\big\\}$ and $W^\\prime=\\big\\{\\mathbf{y}: \\mathbf{y}_{l}=B_i^{l}\\bm{\\xi}_l, \\; \\bm{\\xi}_l\\in \\mathcal{P}_i^l, \\; l\\in V_i\\big\\}$ where $\\mathbf{y}=\\begin{bmatrix} \\mathbf{y}_{\\alpha(1)}^T & \\ldots & \\mathbf{y}_{\\alpha(\\vert V_i \\vert)}^T \\end{bmatrix}^T$. The sets $B_i^lS_{q_i}^l$ are convex since they are projection sets of the zero-level sets of the concave function $h_{q_i}^l(\\mathbf{z}_l)$ defined by \\eqref{eq:prinf}. Hence, $W$ is convex as the Castesian product of convex sets. By Caratheodory's theorem \\cite[Th. 17.1]{rockafellar} every point $\\mathbf{y}\\in W$ can be written as a convex combination of $d_i+1$ points where $d_i=dim(W)=\\sum_{l\\in V_i} d_i^l$. Observe that $W^\\prime \\subset W$ with $\\vert W^\\prime \\vert= 2^{d_i}> d_i$. Applying Caratheodory's theorem, we write any point $\\mathbf{y}\\in W$ as a convex combination of the form: $\\mathbf{y}=\\sum_{j=1}^{d_i+1} \\lambda_j \\mathbf{y}_j^\\prime$ where $\\mathbf{y}_j^\\prime \\in W^\\prime, \\; \\lambda_j\\geq 0 $ and $ \\sum_{j=1}^{d_i+1}\\lambda_j=1$. By feasibility of \\eqref{eq:convex} and due to Assumption 1 we can conclude that $h_i(\\mathbf{y}_{\\alpha(1)},\\ldots,\\mathbf{y}_{\\alpha(\\vert V_i \\vert)})\\geq 0$ for any $\\mathbf{y}\\in W$ with $\\mathbf{y}=\\begin{bmatrix} \\mathbf{y}_{\\alpha(1)}^T & \\ldots & \\mathbf{y}_{\\alpha(\\vert V_i \\vert)}^T \\end{bmatrix}^T$. The rest of the proof is similar to that of Theorem 1.\n\\end{proof}\n\nFor $i\\in \\mathcal{I}_{\\mathcal{F}}$ the new STL tasks, defined by \\eqref{eq:newformula}, are expected to be satisfied at a specific time instant $t_i\\in [a_i,b_i]$ which is considered a designer's choice. However, in many cases pre-determining the time instant of satisfaction of a formula may lead to conservatism and reduced performance. An alternative would be to allow satisfaction of the local formulas over time intervals $[a_q^l,b_q^l]\\subseteq [a_i,b_i]$. Then, in order to guarantee the satisfaction of the global formula we can define the local tasks corresponding to $\\varphi_i, i\\in \\mathcal{I}_{\\mathcal{F}}$ as STL tasks of the form $\\mathcal{G}_{[a_{q_i}^l,b_{q_i}^l]} \\mu$.\nThis is depicted in the following Proposition:\n\\begin{figure*}[!t]\n \\centering\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{ s11}\n \\caption{Trajectories of agents 1,4,5}\n \\label{fig:ev1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s12}\n \\caption{Trajectories of agents 2,3}\n \\label{fig:ev2}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s13}\n \\caption{Barrier Function Evolution}\n \\label{fig:ev3}\n \\end{subfigure}\n \\caption{Agents' Trajectories under the local STL tasks defined based on \\eqref{eq:newformula}, \\eqref{eq:interval}-\\eqref{eq:predicate} and Barrier Function Evolution}\n \\label{fig:evall}\n\\end{figure*}\n\\begin{proposition}\nConsider the global STL formula $\\phi$ defined by \\eqref{eq:f1}-\\eqref{eq:f3}. Let Assumption 1 hold. For every $i\\in \\mathcal{I}$ assume that $\\mathcal{S}_i\\neq \\emptyset$, where $\\mathcal{S}_i$ is defined by \\eqref{eq:levelset}. For every $i\\in \\mathcal{I}$ consider the functions $h_{q_i}^l(\\mathbf{z}_l), l\\in V_i$ defined by \\eqref{eq:prinf} with their parameters found as solutions to \\eqref{eq:convex}. Let the STL formula $\\bar{\\varphi}_{q_i}^l$ be defined as: \n\\begin{equation}\n \\bar{\\varphi}_{q_i}^l= \\mathcal{G}_{[a_{q_i}^l,b_{q_i}^l]} \\bar{\\mu}_{q_i}^l \\label{eq:always}\n\\end{equation}\nwhere\n\\begin{equation}\n [a_{q_i}^l,b_{q_i}^l]\\begin{cases}\\subseteq[a_i,b_i], \\quad i \\in \\mathcal{I}_{\\mathcal{F}}\\\\= [a_i,b_i],\\quad i\\in \\mathcal{I}_{\\mathcal{G}} \\end{cases} \\label{eq:alwaysint}\n\\end{equation}\nand $\\bar{\\mu}_{q_i}^l$ is a predicate defined by \\eqref{eq:predicate}. Let $\\phi_l=\\bigwedge_{q_i=1}^{p_l} \\bar{\\varphi}_{q_i}^l$, $l=1,\\ldots,v$. If there exists $\\mathbf{x}:\\mathbb{R}_{\\geq 0} \\rightarrow \\mathbb{X}$ such that $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$, then $\\rho^\\phi(\\mathbf{x},0)>0$.\n\\end{proposition}\n\n\\begin{proof}\nFor $i\\in \\mathcal{I}_{\\mathcal{G}}$ the proof follows similar arguments to Theorem 1. For $i\\in \\mathcal{I}_{\\mathcal{F}}$, if $\\rho^{\\phi_1\\wedge \\ldots\\wedge \\phi_v}(\\mathbf{x},0)>0$ then, by \\eqref{eq:always}-\\eqref{eq:alwaysint} and the definition of the robust semantics, $\\rho^{\\bar{\\varphi}_{q_i}^l}(\\mathbf{z}_l,0)>0$ implies $ h_{q_i}^l(\\mathbf{z}_l(t))>0$ for every $ t\\in [a_{q_i}^l,b_{q_i}^l]$ and $l\\in V_i$. Since $ h_{q_i}^l(\\mathbf{z}_l), l \\in V_i$ are feasible solutions of \\eqref{eq:convex} we may conclude that $h_i(\\mathbf{x}(t))>0$, where $\\mathbf{x}(t)=A\\mathbf{z}(t)$ and $\\mathbf{z}(t)=\\begin{bmatrix}\\mathbf{z}_1^T(t) &\\ldots & \\mathbf{z}_v^T(t) \\end{bmatrix}^T$, for every $t\\in [a_{q_i}^l,b_{q_i}^l]\\subseteq [a_i,b_i]$. Hence, $\\rho^{\\varphi_i}(\\mathbf{x},0)=\\max_{t\\in [a_i,b_i]} h_i(\\mathbf{x}(t))\\geq \\max_{t\\in [a_{q_i}^l,b_{q_i}^l]} h_i(\\mathbf{x}(t))>0$. The rest of the proof is similar to that of Theorem 1.\n\\end{proof}\n\n\\begin{figure*}[!t]\n \\centering\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{ s21}\n \\caption{Trajectories of agents 1,4,5}\n \\label{fig:alw1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s22}\n \\caption{Trajectories of agents 2,3}\n \\label{fig:alw2}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{s23}\n \\caption{Barrier Function Evolution}\n \\label{fig:alw3}\n \\end{subfigure}\n \\caption{Agents' Trajectories under the local STL tasks defined based on \\eqref{eq:predicate}, \\eqref{eq:always} and \\eqref{eq:alwaysint} and Barrier Function Evolution}\n \\label{fig:alwall}\n\\end{figure*}\n\\section{Simulations}\nConsider a team of $R=5$ agents. Without loss of generality the team is decomposed in 5 sub-teams: $\\mathcal{V}_k=\\{k\\}, k\\in \\mathcal{V}$. The agents' states $\\mathbf{x}_k, k\\in \\mathcal{V}$ evolve over time based on the following equation:\n\\begin{equation*}\n \\dot{\\mathbf{x}}_k=A_k\\mathbf{x}_k+\\mathbf{u}_k , \\; k=1,\\ldots,5\n\\end{equation*}\nwhere $A_k=\\begin{bmatrix}-0.5 &0\\\\1 & -1 \\end{bmatrix}$ for every $ k\\in \\{1,2,5\\}$ and $A_k=\\begin{bmatrix}-1 &-1\\\\0 & -3 \\end{bmatrix}$ for $k\\in \\{3,4\\}$. The states and inputs of the agents are subject to constraints, i.e., $\\mathbf{x}_k \\in \\mathbb{X}$, $\\mathbf{u}_k \\in \\mathbb{U}$ where $\\mathbb{X}=\\{\\mathbf{x}\\in \\mathbb{R}^2: \\Vert \\mathbf{x} \\Vert_2 \\leq d_x\\}$, $\\mathbb{U}=\\{\\mathbf{u}\\in \\mathbb{R}^2: \\Vert \\mathbf{u} \\Vert_2 \\leq d_u\\}$, $d_x=1$ and $d_u=5$. Consider the global STL formula $\\phi=\\bigwedge_{i=1}^4 \\varphi_i$ where $\\varphi_i,i\\in \\mathcal{I}$ are defined as: $ \\varphi_1=\\mathcal{G}_{[0,2.1]}( \\Vert \\mathbf{x}_1-\\mathbf{x}_2-p_x \\Vert_2^2 \\leq 0.1)$, $\\varphi_2= \\mathcal{G}_{[2,4]}( \\Vert \\mathbf{x}_3-\\mathbf{x}_4 \\Vert_{2}^2 \\leq 0.2)$, $ \\varphi_3= \\mathcal{F}_{[3,7]}(\\Vert \\mathbf{x}_5-\\mathbf{x}_4\\Vert_{P_1}^2 \\leq 0.2)$ and $\\varphi_4= \\mathcal{F}_{[8,10]}(\\Vert \\mathbf{x}_5-\\mathbf{x}_2\\Vert_{P_2}^2 \\leq 0.25)$, where $p_x=\\begin{bmatrix} 0.3 & 0.5\\end{bmatrix}^T$ and $P_1=\\text{diag}(4,1), \\; P_2=\\text{diag}(0.1,0.4)$ are positive definite weight matrices. Since the predicate functions corresponding to $\\varphi_i, i\\in \\mathcal{I}$ are quadratic, the proposed problem \\eqref{eq:convex} becomes a Quadratically Constrained Quadratic Program (QCQP) and is efficiently solved using \nthe \\textit{Opti Toolbox} \\cite{opti}. The average computational time of the QCQPs is 0.052sec on an Intel Core i7-8665U with 16GB RAM using MATLAB. \n\nTo verify the validity of Theorem 2 and Proposition 1 we design agents' trajectories using the MPC scheme proposed in \\cite{ecc} with a sampling frequency of 10 Hz and optimization horizon length $N=1$. Each agent $k$ solves a local MPC problem without communicating with its peers since the satisfaction of the assigned tasks depends only on its own behavior. Here, a single, time-varying barrier $b_k(\\mathbf{x}_k,t), k\\in \\mathcal{V}$ is considered and designed offline encoding the local STL task specifications $\\phi_k$ corresponding to $\\mathcal{V}_k$. For every subtask of $\\phi_k$ a temporal behavior is designed for agent $k$ such that the satisfaction of $\\phi_k$ with a robustness value $r=0.005$ is guaranteed when $b_k(\\mathbf{x}_k,t)\\geq 0$ is true for every $t\\in [0,10]$. For details on the design of the barrier function $b_k(\\mathbf{x}_k,t)$ see \\cite{lars_linear,ecc}. The local STL task $\\phi_k$ assigned to each agent $k$ is defined by \\eqref{eq:newformula}, \\eqref{eq:interval}-\\eqref{eq:predicate} as follows:\n\\begin{align*}\n \\phi_1&=\\mathcal{G}_{[0,2.1]}\\;\\bar{\\mu}_1^1=\\bar{\\varphi}_1^1\\\\\n \\phi_2&= (\\mathcal{G}_{[0,2.1]}\\;\\bar{\\mu}_1^2)\\wedge(\\mathcal{F}_{[9,9]}\\; \\bar{\\mu}_4^2)=\\bar{\\varphi}_1^2 \\wedge \\bar{\\varphi}_4^2\\\\\n \\phi_3&= \\mathcal{G}_{[2,4]}\\;\\bar{\\mu}_2^3=\\bar{\\varphi}_2^3 \\\\\n \\phi_4&= (\\mathcal{G}_{[2,4]}\\;\\bar{\\mu}_2^4)\\wedge(\\mathcal{F}_{[7,7]}\\; \\bar{\\mu}_3^4)=\\bar{\\varphi}_2^4 \\wedge \\bar{\\varphi}_3^4\\\\\n \\phi_5&= (\\mathcal{F}_{[7,7]}\\;\\bar{\\mu}_3^5)\\wedge(\\mathcal{F}_{[9,9]}\\; \\bar{\\mu}_4^5)=\\bar{\\varphi}_3^5 \\wedge \\bar{\\varphi}_4^5\n\\end{align*}\nIn Figure \\eqref{fig:ev1}, \\eqref{fig:ev2} the agents' trajectories and the zero level sets $S_q^k$ of the predicate functions $h_q^k(\\mathbf{x}_k)$ are shown when the parameters $c_q^k,r_q^k$ are found as solutions to \\eqref{eq:convex}. Since the agents move on $\\mathbb{R}^2$ and $r_q^k\\neq 0$ for every $q$ and $k$ the zero level sets define square areas with edge length $r_q^k$. In Figure \\eqref{fig:ev3} the evolution of the local barrier functions $b_k(\\mathbf{x}_k,t)$ is shown. Since $\\min_k \\inf_{t\\in [0,10]} b_k(\\mathbf{x}_k,t)\\geq 5.38\\cdot 10^{-4}$ is true, we can conclude that $\\rho^{\\phi_k}(\\mathbf{x}_k,0)\\geq 0.005$ for every $k\\in \\mathcal{V}$. To validate Theorem 1 and given the trajectories of the agents found by the local MPC controllers we aim at designing a barrier function $b_c(\\mathbf{x},t)$ encoding the global specifications described by $\\phi$ and evaluating its value over the interval $[0,10]$. If $b_c(\\mathbf{x},t)\\geq 0$ is true for every $t\\in [0,10]$, then the global formula $\\phi$ is satisfied. From Figure \\eqref{fig:ev3} we have that $\\inf_{t\\in [0,10]} b_c(\\mathbf{x},t)\\geq 0.0234$. Hence, $\\mathbf{x} \\models \\phi$.\n\nNext, we consider the alternative definition of the local tasks as described in Proposition 1. Observe that the local tasks $\\phi_1, \\phi_3$ remain the same. The new local tasks $\\phi_2, \\phi_4, \\phi_5$ are defined as: $\\phi_2=\\bar{\\varphi}_1^2 \\wedge\\bar{\\varphi}_4^2$, $\\phi_4=\\bar{\\varphi}_2^4 \\wedge \\bar{\\varphi}_3^4$ and $\\phi_5=\\bar{\\varphi}_3^5 \\wedge \\bar{\\varphi}_4^5 $, where $\\bar{\\varphi}_1^2=\\mathcal{G}_{[0,2.1]}\\;\\bar{\\mu}_1^2$, $\\bar{\\varphi}_4^2=\\mathcal{G}_{[9,10]}\\; \\bar{\\mu}_4^2$, $\\bar{\\varphi}_2^4=\\mathcal{G}_{[2,4]}\\;\\bar{\\mu}_2^4$, $\\bar{\\varphi}_3^4=\\mathcal{G}_{[5,7]}\\; \\bar{\\mu}_3^4$, $\\bar{\\varphi}_3^5=\\mathcal{G}_{[5,7]}\\;\\bar{\\mu}_3^5$ and $\\bar{\\varphi}_4^5=\\mathcal{G}_{[9,10]}\\; \\bar{\\mu}_4^5$. In Figure \\eqref{fig:alw1} and \\eqref{fig:alw2} the agents' trajectories are shown. Following a similar procedure as before, we design a set of local barrier functions $b_k(\\mathbf{x}_k,t)$ and a function $b_c(\\mathbf{x},t)$ with robustness $r=0.005$. Based on Figure \\eqref{fig:alw3}, $\\min_k \\inf_{t\\in [0,10]} b_k(\\mathbf{x}_k,t)\\geq 5.17\\cdot 10^{-4}$ implying $\\mathbf{x}_k \\models \\phi_k, k \\in \\mathcal{V}$. Additionally, it holds that $\\inf_{t\\in [0,10]} b_c(\\mathbf{x},t)\\geq 0.0234$. Hence, $\\rho^{\\phi}(\\mathbf{x},0)\\geq 0.005$.\n\n\\section{Conclusions}\nIn this work a global STL formula is decomposed to a set of local STL tasks whose satisfaction depends on an a-priori chosen subset of agents. The predicate functions of the new formulas are chosen as functions of the infinity norm of the agents' states. A convex optimization problem is, then, designed for optimizing their parameters towards increasing the volume of their zero level-sets. Two alternatives are proposed for defining the local STL tasks in both of which the interval of satisfaction corresponding to the eventually formulas is considered a designer's choice. Future work will consider a more sophisticated framework for choosing the interval of satisfaction of the formulas aiming at increasing the total robustness of the task.\n\n\n\n \n \n \n \n \n\n\n\n\n\n\n\n\n\n\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro} \nThe replicator equation is the most widely applied dynamics among evolutionary game models. It also gives the first dynamical model in evolutionary game theory \\cite{TJ}, thereby establishing an important connection to theoretical explanations of animal behaviors \\cite{MP}. For this application, the derivation of the equation considers a large well-mixed population. The individuals implement strategies from a finite set $S$ with $\\#S\\geq 2$, such that the payoff to $\\sigma\\in S$ from playing against $\\sigma'\\in S$ is $\\Pi(\\sigma,\\sigma')$. In the continuum, the density $X_\\sigma$ of strategy $\\sigma$ evolves with a per capita rate given by the difference between its payoff \n\\begin{align}\\label{def:Fsigma}\nF_{\\sigma}(X)=\\sum_{\\sigma'\\in S}\\Pi(\\sigma,\\sigma')X_{\\sigma'}\n\\end{align}\nand the average payoff of the population, where $X=(X_{\\sigma'};\\sigma'\\in S)$.\n Hence, the vector density process of strategy obeys the following replicator equation:\n\\begin{align}\\label{eq:replicator}\n\\dot{X}_\\sigma=X_\\sigma\\left(F_\\sigma(X)-\\sum_{\\sigma''\\in S}F_{\\sigma''}(X)X_{\\sigma''}\\right),\\quad \\sigma\\in S.\n\\end{align}\nThis equation is a point of departure for studying connections between the payoff matrix $(\\Pi(\\sigma,\\sigma'))_{\\sigma,\\sigma'\\in S}$, and the equilibrium states of the model by methods from dynamical systems. See \\cite{Cressman,HS} for an introduction and more properties. The replicator equation also arises from the Lotka--Volterra equation of ecology and Fisher's fundamental theorem of natural selection \\cite{HS,SS}. \n\nIn this paper, we consider the stochastic evolutionary game dynamics in large finite structured populations. Our goal is to prove that the vector density processes of strategy converge to the replicator equation. In this direction, one of the major results in the biological literature is the convergence to the replicator equation on large random regular graphs~\\cite{ON}. The authors further conjecture that their approximations extend to more general graphs \\cite[Section~5]{ON}. To obtain the proofs, we view the model as a perturbation of the voter model, since this viewpoint has made possible several mathematical results for it (e.g. \\cite{CDP:2,C:BC, CD,CMN,ALCFYN,C:EGT}). Our starting point here is the method in \\cite{C:EGT}, extended from \\cite{CCC,CC}, for proving the diffusion approximations of the time-changed density processes of strategy under \\emph{weak selection}. In that context, the corresponding perturbations away from the voter model use strengths typically given by $w=\\mathcal O(1\/N)$, where $N$ is the population size. \n\nThe questions from \\cite{ON} nevertheless concern very different properties. The crucial step of the method in \\cite{C:EGT} develops along the equivalence of probability laws in the limit between the evolutionary game model and the voter model. Now, this property breaks down for nontrivial parameters according to the limiting equation from \\cite{ON}; distributional limits of the density processes under the evolutionary game and the voter model degenerate to delta distributions of distinct deterministic functions as solutions of differential equations. As will be explained in this introduction, the convergence to the replicator equation also requires the different range of perturbation strengths $w$ satisfying $1\/N\\ll w\\ll 1$. This stronger perturbation implies weaker relations between the two models, and thus, calls for perturbation estimates of the evolutionary game model by the voter model generalizing those in \\cite{C:EGT}. With this change of perturbation strengths, the choice of time changes for the density processes and the characterization of coefficients of the limiting equation are the further tasks to be settled.\n\nBefore further explanations of the main results of \\cite{ON,C:EGT}, let us specify the evolutionary game model considered throughout this paper. First, to define spatial structure, we impose directed weights $q(x,y)$ on all pairs of sites $x$ and $y$ in a given population of size $N$. We assume that $q$ is an irreducible probability transition kernel with a zero trace $\\sum_{x}q(x,x)=0$. The perturbation strength $w>0$ defines the selection intensity of the model in the following form: for an individual at site $x$ using strategy $\\xi(x)$, its interactions with the neighbors determine the fitness as the sum $1-w+w\\sum_{y}q(x,y)\\Pi\\big(\\xi(x),\\xi(y)\\big)$, where $\\xi(y)$ denotes the strategy held by the neighbor at $y$. Under the condition of positive fitness by tuning the selection intensity appropriately, the death-birth updating requires that in a transition of state, an individual is chosen to die with rate $1$. Then the neighbors compete to fill in the site by reproduction with probability proportional to the fitness. Although the main results of this paper extend to include mutations of strategies, we relegate this additional mechanism until Section~\\ref{sec:mainresults}. \n\nBesides the case of mean-field populations, the density processes of strategy play a significant role in the biological literature for studying equilibrium states of spatial evolutionary games. Here and throughout this paper, we refer the density of $\\sigma$ under population configuration $\\xi$ to the weighted sum $\\sum_x \\pi(x)\\mathds 1_\\sigma\\big( \\xi(x)\\big)$, where $\\pi$ is the stationary distribution associated with the transition probability $q$. For such macroscopic descriptions of the model, the critical issue arises from the non-closure of the stochastic equations. The density processes are projections of the whole system, and in general, the density functions are not Markov functions in the sense of \\cite{RP}. More specifically, for the evolutionary game with death-birth updating introduced above, the microscopic dynamics from pairwise interactions determine the densities' dynamics. It is neither clear how to reduce the densities' dynamics analytically in the associated Kolmogorov equations. See \\cite[Section~1]{C:EGT} for more details on these issues and the physics method discussed below. \n\nOne of the main results in \\cite{ON} shows that for selection intensities $w\\ll 1$, the \\emph{expected} density processes on large random $k$-regular graphs for any integer $k\\geq 3$ approximately obey the following extended form of the replicator equation :\n\\begin{align}\\label{eq:replicatorext}\n\\dot{X}_\\sigma=wX_\\sigma\\left(F_\\sigma(X)+\\widetilde{F}_\\sigma(X)-\\sum_{\\sigma''\\in S}F_{\\sigma''}(X)X_{\\sigma''}\\right),\\quad \\sigma\\in S.\n\\end{align}\nHere, $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ are linear functions in $X=(X_{\\sigma'};\\sigma'\\in S)$ such that the constant coefficients are explicit in the payoff matrix and the graph degree $k$. See \\cite[Equations (22) and (36)]{ON}. Note that \\eqref{eq:replicatorext} underlines the nontrivial effect of spatial structure, since the coefficients are very different from those for the replicator equation \\eqref{def:Fsigma} in mean-field populations. For the derivation of \\eqref{eq:replicatorext}, \\cite{ON} applies the physics method of pair approximation. It enables the asymptotic closure of the equations of the density processes by certain moment closure approximations and circumvents the fundamental issues discussed above. Moreover, based on computer simulations from \\cite{OHLN}, the authors of \\cite{ON} conjecture that the approximate replicator equation for the density processes applies to many non-regular graphs, provided that the constant graph degree $k$ in the coefficients of the replicator equation is replaced by the corresponding average degree. In approaching this conjecture, it is still not clear on how the average degrees of graphs enter. The method in this paper does not extend to this generality either. On the other hand, even within the scope of large random regular graphs, the constant graph degrees and the locally regular tree-like property seem essential in \\cite{ON}. We notice that locally tree-like spatial structures are known to be useful to pair approximations in general \\cite{SF}. \n\nIn the case of two strategies, the supplementary information (SI) of \\cite{OHLN} shows that the density processes of a fixed one approximate the Wright--Fisher diffusion with drift. The derivation also applies pair approximations on large random regular graphs, although it is noticeably different from the derivation in \\cite{ON} for the replicator equation on graphs. (A slow-fast dynamical system for the density and a certain rapidly convergent local density is considered in \\cite[SI]{OHLN}.) It is neither clear how to justify the derivation in \\cite[SI]{OHLN} mathematically. On the other hand, the diffusion approximations of the density processes can be proven on large finite spatial structures subject to appropriate, but general, conditions \\cite[Theorem~4.6]{C:EGT} that include random regular graphs as a special case. See \\cite{CFM06,CFM08} for mathematical investigations of moment closure in other spatial biological models and some general discussions, among other mathematical works in this direction.\n\nThe method in \\cite{C:EGT} begins with the aforementioned asymptotic equivalence of probability laws via perturbations for $w=\\mathcal O(1\/N)$ (not just equivalence of laws of the density processes). The $1\/N$-threshold is sharp such that the critical case yields nontrivial drifts of the limiting diffusions. This relation reduces the convergence of the game density processes to a convergence problem of the voter model. For the latter, fast mixing of spatial structure ensures approximations of the coalescence times in the ancestral lineages by analogous exponential random variables from large mean-field populations. This method goes back to \\cite{Cox}. Moreover, for the voter density processes, the relevant coalescence times can be reduced to the meeting times for two independent copies of the stationary Markov chains over the populations. The almost exponentiality of hitting times \\cite{Aldous:AE, AB,AF:MC} applies to these times and leads to the classical diffusivity in the voter density processes in general spatial populations \\cite{CCC,CC}. The first moments of these meeting times are also used to time change the densities for the convergence. \n\nBesides the methods, the convergence results in \\cite[Theorem~4.6]{C:EGT} for the game density processes under the specific setting of large random regular graphs and the payoff matrices for prisoner's dilemma games are closely related to the replicator equation on graphs from \\cite{ON}. See \\eqref{prisoner} for these payoff matrices and \\cite{C:MT} for the exact asymptotics $N(k-1)\/[2(k-2)]$, $N\\to\\infty$, of the expected meeting times on the large random $k$-regular graphs. In this case, the diffusion approximations in \\cite[SI]{OHLN} hold to the degree of matching constants if the time changes are formally undone \\cite{C:MT}. (See also \\cite[Remark~3.1]{C:MT} for a correction of inaccuracy in \\cite{C:EGT} on passing limits along random regular graphs.) This standpoint extends to a recovery of the replicator equation on graphs from \\cite{ON} by a similar formal argument. It shows that these results in \\cite{ON,OHLN}, both due to pair approximations, are algebraically consistent with each other. See the end of Section~\\ref{sec:mainresults} for details and the second main result discussed below for further comparison. In addition to its own interest, the replicator equation on graphs concerns a unified characterization of the evolutionary game within an enlarged range of selection intensities as mentioned above. \n\nThe main results of this paper obtain the convergence to the replicator equation under the above specific setting, in addition to extensions to general spatial populations and payoff matrices. Multiple strategies and mutations are allowed. See Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric}. For the extended context, the first main result [Theorem~\\ref{thm:main} (1${^\\circ}$)] proves the convergence of the vector density processes of strategy under the following assumptions. We require that the stationary distributions associated with the spatial structures are asymptotically comparable to the uniform distributions (see \\eqref{cond:pi}), and these spatial structures allow for suitable time changes of the density processes and suitable selection intensities (Definition~\\ref{def:admissible}). Here, for the typical eligible populations, the time changes can range in $1\\ll\\theta \\ll N$. The selection intensities are \\emph{of the inverse order} so that $1\/N\\ll w\\ll1 $. Then the precise limiting equation is given by \\eqref{eq:replicatorext}, with the selection intensity $w$ replaced by a constant $w_\\infty$ as a limit of the parameters. The proof also determines $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ for \\eqref{eq:replicatorext}:\n\\begin{align}\nF_\\sigma(X)&=\\overline{\\kappa}_{0|2|3}\\sum_{\\sigma'\\in S}\\Pi(\\sigma,\\sigma')X_{\\sigma'},\\label{F1}\\\\\n\\begin{split}\n\\widetilde{F}_\\sigma(X)&=(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})\\Pi(\\sigma,\\sigma)\\\\\n&\\quad +\\sum_{\\sigma'\\in S}({\\overline{\\kappa}}_{(0,3)|2}-{\\overline{\\kappa}}_{0|2|3})[\\Pi(\\sigma,\\sigma')-\\Pi(\\sigma',\\sigma)]X_{\\sigma'}\\\\\n&\\quad -(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})\\sum_{\\sigma'\\in S}\\Pi(\\sigma',\\sigma')X_{\\sigma'}.\\label{F2}\n\\end{split}\n\\end{align}\nHere, $\\overline{\\kappa}_{(2,3)|0}$, ${\\overline{\\kappa}}_{(0,3)|2}$, and $\\overline{\\kappa}_{0|2|3}$ are nonnegative constants defined by the asymptotics of some coalescent characteristics of the spatial structures. See Section~\\ref{sec:slow} for the definitions of these constants. \n\nThe first main result [Theorem~\\ref{thm:main} (1${^\\circ}$)] has the meaning of spatial universality as the diffusion approximations of the voter model and the evolutionary game in \\cite{CCC,CC,C:EGT}, although it does not recover the explicit equations obtained in \\cite{ON} on large random regular graphs under general payoff matrices. The conditions do not require convergence of local geometry as in the large discrete tori and large random regular graphs. The spatial structures can remain sparse in the limit, which is in stark contrast to the usual assumptions for proving scaling limits of particle systems. The locally tree-like property usually assumed in pair approximations is not required either. Based on these properties, the first main result [Theorem~\\ref{thm:main} (1${^\\circ}$)] gives an answer in the positive for the conjecture in \\cite{ON} to the degree of using constants that may depend implicitly on the space: The approximations of the expected density processes by the replicator equation extend to many non-regular graphs, whenever the initial conditions converge deterministically. \n \nTo further the formal comparison mentioned above with the approximate Wright--Fisher diffusion from \\cite[SI]{OHLN}, the second main result [Theorem~\\ref{thm:main} (2${^\\circ}$)] considers one additional aspect for the convergence of the density processes. In this part, the normalized fluctuations are proven to converge to a vector centered Gaussian martingale [Theorem~\\ref{thm:main} (2${^\\circ}$)]. The quadratic covariation is the Wright--Fisher diffusion matrix in the limiting densities $X$: $\\int_0^{ t} X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s$, $\\sigma,\\sigma'\\in S$, where $\\delta_{\\sigma,\\sigma'}$ are the Kronecker deltas. For the case of only two strategies on large random regular graphs, this covariation formally recovers the approximate Wright--Fisher diffusion term from \\cite[SI]{OHLN}. Note that this result and the convergence to the replicator equation do not imply the diffusion approximations of the density processes. \n\nIn the rest of this introduction, we explain the proof of the first main result. Its investigation raises all the central technical issues pointed out above. First, the lack of an asymptotic equivalence of probability laws is resolved via the populations' microscopic dynamics driving the density processes. Duhamel's principle replaces the pathwise, global change of measure method in \\cite{C:EGT} and shows the irrelevance of selection intensities in the microscopic dynamics (Proposition~\\ref{prop:duhamel}). This approach then links to the decorrelation proven in \\cite[Section~4]{CCC} for some ``local'' meeting time distributions, from the ancestral lineages, driving the dynamics of the voter density processes. Here, local meeting times refer to those where the initial conditions of the Markov chains are within fixed numbers of edges. \n\nThe decorrelation property from \\cite{CCC} shows that the probability distributions of those particular local meeting times for general populations converge to nontrivial convex combinations of the delta distributions at zero and infinity. The time scales are slower than those for the diffusion approximations. In particular, the exponential distribution is not just absent. No distributions with a nonzero mass between zero and infinity arise in the limit. Informally speaking, the decorrelation occurs at time scales between the period when details of the spatial structures dominate and the period when the almost exponentiality \\cite{Aldous:AE, AB,AF:MC} plays a role. To us, this presence of multiple time scales in the evolutionary dynamics is reminiscent of the slow-fast dynamical system in \\cite[SI]{OHLN}.\n\nFor the convergence to the replicator equation, the choice of the time changes for the densities and the characterization of the limiting equation use the decorrelation from \\cite{CCC} and its extensions. First, the time changes can only grow slower than those for the diffusion approximations since the limiting trajectories are less rougher. This requirement relates the convergence to the decorrelation. We are now interested in proving the best possible range of growing time changes for the decorrelation, not just using the particular ones from \\cite{CCC}. After all, in \\cite{ON}, the replicator equation is expected to be present within the broad range $w\\ll 1$, and our argument requires the selection intensities to be of the inverse order of the time changes. Moreover, the application of Duhamel's principle mentioned above leads to the entrance of various local meeting times more than those for the voter densities in \\cite{CCC}. Simultaneous decorrelation in these local meeting times is essential for getting a deterministic limiting differential equation: This property involves asymptotic path regularity of the density processes. The constant coefficients in \\eqref{F1} and \\eqref{F2} also arise as the weights at infinity in the limiting local meeting time distributions for the typical eligible populations. See Sections~\\ref{sec:slow} and \\ref{sec:eqn} for the related proofs. \\medskip \n\n\\noindent {\\bf Organization.} Section~\\ref{sec:mainresults} introduces the evolutionary game model and the voter model analytically and discusses the main results (Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric}). In Section~\\ref{sec:dynamics}, we define the voter model and the evolutionary game model as semimartingales and briefly explain the role of the coalescing duality. In Section~\\ref{sec:slow}, we quantify the time changes in proving the main results and characterize the coefficients of the limiting equation. Section~\\ref{sec:eqn} is devoted to the main arguments of the proofs of Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric}. Finally, Section~\\ref{sec:coal} presents some auxiliary results for coalescing Markov chains.\\medskip\n\n\\noindent {\\bf Acknowledgments}\nThe author would like to thank Lea Popovic for comments on earlier drafts and Sabin Lessard for pointing out several references from the literature. Support from the Simons Foundation before the author's present position and from the Natural Science and Engineering Research Council of Canada is gratefully acknowledged. \n\n\n\n\n\\section{Main results}\\label{sec:mainresults}\nIn this section, we introduce the stochastic spatial evolutionary game with death-birth updating in more detail. A discussion of the main results of this paper then follows. To be consistent with the viewpoint of voter model perturbations and the neutral role of the voter model, strategies will be called {\\bf types} in the rest of this paper. The settings here and in the next section are adapted from those in \\cite{CCC,CC,C:EGT} to the context of evolutionary games with multiple types. \n\nRecall that a discrete spatial structure considered in this paper is given by an irreducible, reversible probability kernel $q$ on a finite nonempty set $E$ such that ${\\sf tr}(q)=\\sum_{x\\in E}q(x,x)=0$. Write $N=\\#E$ and $\\pi$ for the unique stationary distribution of $q$. The interactions of individuals are defined by a payoff matrix $\\Pi=(\\Pi(\\sigma,\\sigma'))_{\\sigma,\\sigma'\\in S}$ of real entries. Fix $\\overline{w}\\in (0,\\infty)$ such that\n\\begin{align}\\label{def:wbar}\nw+w\\sum_{z\\in E}q(y,z)|\\Pi\\big(\\xi(y),\\xi(z)\\big)|<1,\\quad \\forall\\;w\\in [0,\\overline{w}],\\;y\\in E.\n\\end{align}\nThen the following perturbed transition probability is used to update types of individuals due to interactions:\n\\begin{align}\nq^w(x,y,\\xi)&\\stackrel{\\rm def}{=} \\frac{q(x,y)\\left[(1-w)+w\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big)\\right]}{\\sum_{y'\\in E}q(x,y')\\left[(1-w)+w\\sum_{z\\in E}q(y',z)\\Pi\\big(\\xi(y'),\\xi(z)\\big)\\right]}.\\label{def:qw}\n\\end{align}\nWith these updates and the updates based on a mutation measure $\\mu$ on $S$, two types of configurations $\\xi^{x,y},\\xi^{x|\\sigma}\\in S^E$ result. They are obtained from $\\xi\\in S^E$ by changing only the type at $x$ such that $\\xi^{x,y}(x)=\\xi(y)$ and $\\xi^{x|\\sigma}(x)=\\sigma$. Hence, the evolutionary game $(\\xi_t)$ is a Markov jump process with a generator given by\n\\begin{align}\\label{def:Lw}\n\\begin{split}\n\\mathsf L^{w} H(\\xi)=&\\sum_{x,y\\in E}q^w(x,y,\\xi)[H(\\xi^{x,y})-H(\\xi)]\\\\\n&+\\sum_{x\\in E}\\int_S [H(\\xi^{x|\\sigma})-H(\\xi)]\\d \\mu(\\sigma),\\quad H:S\\to {\\Bbb R}.\n\\end{split}\n\\end{align}\nThe first sum on the right-hand side of \\eqref{def:Lw} governs changes of types due to selection, and the second sum is responsible for mutations. Given $\\xi\\in S^E$ and a probability distribution $\\nu$ on $S^E$ as initial conditions, we write $\\P^w_\\xi$ and ${\\mathbb E}^w_\\xi$, or $\\P^w_\\nu$ and ${\\mathbb E}^w_\\nu$, under the laws associated with $\\mathsf L^w$. For $w=0$, the generator $\\mathsf L^{w}$ is reduced to the generator $\\mathsf L$ of the multi-type voter model with mutation, and the notation $\\P$ and ${\\mathbb E}$ is used.\n\n\n\nThe object in this paper is the vector density processes $p(\\xi_t)=(p_\\sigma(\\xi_t);\\sigma\\in S)$ for the evolutionary game with death-birth updating. Here, the density function of $\\sigma\\in S$ is given by\n\\begin{align}\\label{def:density}\np_\\sigma(\\xi)=\\sum_{x\\in E}\\mathds 1_\\sigma\\circ\\xi(x)\\pi(x),\n\\end{align}\nwhere $f\\circ \\xi(x)=f(\\xi(x))$. Under $\\P^w$, $p_\\sigma(\\xi_t)$ admits a semimartingale decomosition:\n\\begin{align}\\label{density:dynamics}\np_\\sigma(\\xi_t)=p_\\sigma(\\xi_0)+A_\\sigma(t)+M_\\sigma(t),\n\\end{align}\nwhere $A_\\sigma(t)=\\int_0^t \\mathsf L^w p_\\sigma(\\xi_s)\\d s $. In the sequel, we study the convergence of the vector density processes and the martingales $M_\\sigma$ separately, along an appropriate sequence of discrete spatial structures $(E_n,q^{(n)})$ with $N_n=\\#E_n\\to\\infty$. \\medskip\n\n\n\\noindent {\\bf Convention for superscripts and subscripts.} Objects associated with $(E_n,q^{(n)})$ will carry either superscripts ``$(n)$'' or subscripts ``$n$'', although additional properties may be assumed so that these objects are not just based on $(E_n,q^{(n)})$. Otherwise, we refer to a fixed spatial structure $(E,q)$. \\hfill $\\blacksquare$\\medskip\n\n\n\nFor the main theorem, we choose parameters as time changes for the density processes, mutation measures, and selection intensities. The choice is according to the underlying discrete spatial structures. We use $\\nu_n(\\mathds 1)=\\sum_{x\\in E_n}\\pi^{(n)}(x)^2$ and the first moment $\\gamma_n$ of the first meeting time of two independent stationary rate-$1$ $q^{(n)}$-Markov chains. The other characteristic of the spatial structure considers the mixing time $\\mathbf t^{(n)}_{\\rm mix}$ of the $q^{(n)}$-Markov chains and the spectral gap $\\mathbf g_n$ as follows. Recall that the semigroup of the continuous-time rate-$1$ $q^{(n)}$-Markov chain is given by $({\\rm e}^{t(q^{(n)}-1)};t\\geq 0)$. With\n\\begin{align}\\label{def:dE}\nd_{E_n}(t)=\\max_{x\\in E_n}\\big\\|{\\rm e}^{t(q^{(n)}-1)}(x,\\cdot)-\\pi^{(n)}\\big\\|_{\\rm TV}\n\\end{align}\nfor $\\|\\cdot\\|_{\\rm TV}$ denoting the total variation distance, we choose\n\\begin{align}\n\\label{def:tmix}\n\\mathbf t^{(n)}_{\\rm mix}=\\inf\\{t\\geq 0;d_{E_n}(t)\\leq (2{\\rm e})^{-1}\\}.\n\\end{align}\nThe spectral gap $\\mathbf g_n$ is the distance between the largest and second largest eigenvalues of $q^{(n)}$. \n\n\n\\begin{defi}\\label{def:admissible}\nFor all $n\\geq 1$, let $\\theta_n\\in (0,\\infty)$ be a time change, $\\mu_n$ a mutation measure on $S$, and $w_n\\in [0,\\overline{w}]$. The sequence $(\\theta_n,\\mu_n,w_n)$ is said to be {\\bf admissible} if all of the following conditions hold. First, $(\\theta_n)$ satisfies\n\\begin{align}\\label{cond1:thetan}\n\\lim_{n\\to\\infty}\\theta_n=\\infty,\\quad \\lim_{n\\to\\infty}\\frac{\\theta_n}{\\gamma_n}<\\infty,\\quad \n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-t\\theta_n}= 0,\\quad \\forall\\;t\\in (0,\\infty),\n\\end{align}\nand at least one of the two mixing conditions holds: \n\\begin{align}\\label{cond2:thetan}\n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mathbf g_n \\theta_n}=0\\quad\\mbox{or}\\quad \\lim_{n\\to\\infty}\\frac{\\mathbf t^{(n)}_{\\rm mix}}{\\theta_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]=0,\n\\end{align}\nwhere $\\log^+\\alpha=\\log (\\max\\{\\alpha,1\\})$. Second, we require the following limits for $(\\mu_n)$ and $(w_n)$: \n\\begin{align}\n& \\lim_{n\\to\\infty} \\mu_n(\\sigma)\\theta_n=\\mu_\\infty(\\sigma)<\\infty,\\quad \\forall\\;\\sigma\\in S;\\label{def:mun}\\\\\n& \\lim_{n\\to\\infty}\\ w_n=0,\\quad \n\\lim_{n\\to\\infty}\\frac{w_n\\theta_n }{2\\gamma_n\\nu_n(\\mathds 1)}=w_\\infty<\\infty,\\quad \\limsup_{n\\to\\infty}w_n\\theta_n<\\infty.\n\\label{def:wn}\n\\end{align}\n\\end{defi}\n\n\n\nAnother condition of the main theorem requires that $\\sup_nN_n\\max_{x\\in E_n}\\pi^{(n)}(x)<\\infty$, which implies $\\gamma_n\\geq \\mathcal O(N_n)$ (see \\eqref{ergodic} or \\cite[(3.21)]{CCC} for details). In this context, the admissible $\\theta_n$ has the following effects. If $\\lim_{n}\\theta_n\/\\gamma_n\\in (0,\\infty)$, the time-changed density processes $p_1(\\xi_{\\theta_nt})$ of the voter model converge to the Wright--Fisher diffusion \\cite{CCC,CC}. Moreover, the density processes of the evolutionary game converge to the same diffusion but with a drift \\cite[Theorem~4.6]{C:EGT}. These diffusion approximations hold under the mixing conditions slightly different from those in \\eqref{cond2:thetan}. Therefore, assuming $\\lim_{n}\\theta_n\/\\gamma_n=0$ in \\eqref{cond1:sn} has the heuristic that the time-changed density processes have paths less rougher in the limit, and so, do not converge to diffusion processes. Note that this variation of time scales can be contrasted with, e.g., the context considered in \\cite{EN:80} where, among other results, the discrete processes converge to the equilibrium states of the limiting process due to faster time changes.\n\nThe other conditions for the admissible sequences mainly consider the typical case of ``transient'' spatial structures. The kernels are characterized by the condition $\\sup_n \\gamma_n\\nu_n(\\mathds 1)<\\infty$ \\cite[Remark~2.4]{CCC}. In this case, \\eqref{cond1:thetan} can be satisfied by any sequence $(\\theta_n)$ such that $1\\ll \\theta_n\\ll N_n$, and \\eqref{def:mun} and \\eqref{def:wn} allow for nonzero $\\mu_\\infty$ and $w_\\infty$. The somewhat tedious condition in \\eqref{def:wn} simplifies drastically, and we get $N_n^{-1}\\ll w_n\\ll 1$ when $\\lim_n w_n\\theta_n$ is nonzero. As for the mixing conditions in \\eqref{cond2:thetan}, they can pose severe limitations if the spatial structures are ``recurrent'' ($\\sup_n \\gamma_n\\nu_n(\\mathds 1)=\\infty$). In this case, we may not be able to find admissible sequences such that $w_\\infty>0$, so that the limiting equation to be presented below only allows constant solutions in the absence of mutation. \nFor example, the two-dimensional discrete tori satisfy $\\gamma_n\\nu_n(\\mathds 1)\\sim C\\log N_n$, $\\mathbf t^{(n)}_{\\rm mix}\\leq \\mathcal O(N_n)$ and $\\mathbf g_n=\\mathcal O(1\/N_n)$. See \\cite{Cox} and \\cite[Theorem~10.13 on p.133, Theorem~5.5 on p.66 and Section~12.3.1 on p.157]{LPW}. We can choose $\\theta_n=N_n(\\log\\log N_n)^2$ to satisfy \\eqref{cond1:thetan} with $\\lim_n\\theta_n\/\\gamma_n=0$, and the first mixing condition in \\eqref{cond2:thetan}. But now the admissible $(w_n)$ only gives $w_\\infty=0$. We notice that a similar restriction is pointed out in \\cite{Cox:Feller} on the low density scaling limits of the biased voter model, where the limit is Feller's branching diffusion with drift. \n\nFrom now on, we write $\\pi_{\\min}=\\min_{x\\in E}\\pi(x)$ and $\\pi_{\\max}=\\max_{x\\in E}\\pi(x)$ for the stationary distribution $\\pi$ of $(E,q)$. The main theorem stated below shows a law of large numbers type convergence for the density processes and a central limit theorem type convergence for the fluctuations. These two results do not combine to give the diffusion approximation of the density processes proven in \\cite{C:EGT}. \n \n\n\n\\begin{thm}\\label{thm:main}\nLet $(E_n,q^{(n)})$ be a sequence of irreducible, reversible probability kernels defined on finite sets with $N_n=\\#E_n\\to\\infty$. Assume the following conditions:\n\\begin{enumerate}\n\\item [\\rm (a)] Let $\\nu_n$ be a probability measure on $S^{E_n}$ such that $\\nu_n(\\xi;p(\\xi)\\in \\cdot)$ converges in distribution to a probability measure $\\overline{\\nu}_\\infty$ on $[0,1]^S$.\n\\item [\\rm (b)] It holds that \n\\begin{align}\\label{cond:pi}\n0<\\liminf_{n\\to\\infty}N_n\\pi^{(n)}_{\\min}\\leq \\limsup_{n\\to\\infty}N_n\\pi^{(n)}_{\\max}<\\infty.\n\\end{align}\n\\item [\\rm (c)] The limits in \\eqref{def:|kell1|ell2} and \\eqref{def:|||} defining the nonnegative constants $\\overline{\\kappa}_{(2,3)|0}$, $\\overline{\\kappa}_{(0,3)|2}$ and ${\\overline{\\kappa}}_{0|2|3}$ exist. These constants depend only on space. \n\\item [\\rm (d)] We can choose an admissible sequence $(\\theta_n,\\mu_n,w_n)$ as in Definition~\\ref{def:admissible} such that $\\lim_n\\theta_n\/\\gamma_n=0$.\n\\end{enumerate}\nThen the following convergence in distribution of processes holds:\n\\begin{enumerate}\n\\item [\\rm (1${^\\circ}$)] The sequence of the vector density processes $\\big(p(\\xi_{\\theta_nt}),\\P^{w_n}_{\\nu_n}\\big)$ converges to the solution $X$ of the following differential equation with the random initial condition $\\P(X_0\\in \\cdot)=\\overline{\\nu}_\\infty$:\n\\begin{align}\n\\begin{split}\\label{p1:lim}\n\\dot{X}_\\sigma&=w_\\infty X_\\sigma\\left(F_\\sigma(X)+\\widetilde{F}_\\sigma(X)-\\sum_{\\sigma''\\in S}F_{\\sigma''}(X)X_{\\sigma''}\\right)\\\\\n&\\quad\\; +\\mu_\\infty(\\sigma)(1-X_\\sigma)-\\mu_\\infty(S\\setminus\\{\\sigma\\}) X_\\sigma ,\\quad \\sigma\\in S,\n\\end{split}\n\\end{align}\nwhere $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ are linear functions in $X$ defined by \\eqref{F1} and \\eqref{F2}. \nMoreover, the sum of the ${\\overline{\\kappa}}$-constants in $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ is nontrivial to the following degree: \n\\begin{align}\\label{kappa:>}\n(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})+ \\overline{\\kappa}_{0|2|3}+({\\overline{\\kappa}}_{(0,3)|2}-{\\overline{\\kappa}}_{0|2|3})\\in (0,\\infty).\n\\end{align}\n\n\\item [\\rm (2${^\\circ}$)] Recall the vector martingale defined by \\eqref{density:dynamics}, and set $M^{(n)}_\\sigma(t)=(M_\\sigma(\\theta_nt);\\sigma\\in S)$ under $\\P^{w_n}_{\\nu_n}$. If, moreover, $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$ holds, then $(\\gamma_n\/\\theta_n)^{1\/2}M^{(n)}$ converges to a vector centered Gaussian martingale with quadratic covariation $(\\int_0^tX_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s;\\sigma,\\sigma'\\in S) $. \n\\end{enumerate}\n\\end{thm}\n\n\n\n\n\n\nWe present the proof of Theorem~\\ref{thm:main} in Section~\\ref{sec:eqn}. The existence of the limits in condition (c) is proven in Proposition~\\ref{prop:kell}. See Lemma~\\ref{lem:tight} for the additional condition in Theorem~\\ref{thm:main} (2${^\\circ}$).\n\n\n\nTo illustrate Theorem~\\ref{thm:main}, we consider the generalized prisoner's dilemma matrix in the rest of this section. The matrix is for games among individuals of two types : \n\\begin{align}\\label{prisoner}\n\\Pi=\n\\bordermatrix{~ & 1& 0 \\cr\n 1 & b-c & -c \\cr\n 0 & b & 0 \\cr}\n\\end{align}\nfor real entries $b,c$. (The usual prisoner's dilemma matrix requires $b>c>0$.) The proof of the following corollary also appears in Section~\\ref{sec:eqn}.\n\n\\begin{cor}\\label{cor:symmetric}\nLet conditions {\\rm (a)--(d)} of Theorem~\\ref{thm:main} be in force and $\\Pi$ be given by \\eqref{prisoner}. If, moreover, $q^{(n)}$ are symmetric ($q^{(n)}(x,y)\\equiv q^{(n)}(y,x)$) and \n\\begin{align}\\label{cond:q2}\n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1)\\pi^{(n)}\\{x\\in E_n;q^{(n),2}(x,x)\\neq q^{(\\infty),2}\\}=0\n\\end{align}\nfor some constant $q^{(\\infty),2}$, then the differential equation for $X_1=1-X_0$ takes a simpler form:\n\\begin{align}\\label{eq:X1}\n\\dot{X}_1=w_\\infty(bq^{(\\infty),2}-c)X_1(1-X_1)+\\mu_\\infty(1)(1-X_1)-\\mu_\\infty(0) X_1.\n\\end{align}\n\\end{cor}\n\\medskip\n\n\n\nCorollary~\\ref{cor:symmetric} applies to large random $k$-regular graph for a fixed integer $k\\geq 3$, with $q^{(\\infty),2}=1\/k$ and $\\gamma_n\/N_n\\to (k-1)\/[2(k-2)]$ (see \\eqref{MUU:RG} and the discussion there). Additionally, $(\\theta_n)$ can be chosen to be any sequence such that $1\\ll \\theta_n\\ll N_n$, and $(w_n)$ can be any such that $(w_n\\theta_n)$ converges in $[0,\\infty)$. See \\cite{C:MT} and Section~\\ref{sec:rrg}. (More precisely, the application needs to pass limits along subsequences, since these graphs are randomly chosen.) Assume the absence of mutation. Then in this case, one can \\emph{formally} recover the replicator equation \\eqref{eq:X1} from the drift term of the approximate Wright--Fisher diffusion in \\cite[SI]{OHLN} as follows. For the density process $p_1(\\xi_t)$ under $\\P^{w_n}$, that drift term reads\n\\begin{align}\\label{drift}\nw_n\\cdot \\frac{(k-2) (b-ck)}{k(k-1)}p_1(\\xi_t)[1-p_1(\\xi_t)].\n\\end{align}\nNote that $\\gamma_n\\approx N_n(k-1)\/[2(k-2)]$ as mentioned above and the choice in \\eqref{def:wn} of $w_n$ gives $w_n \\approx w_\\infty 2\\gamma_nN_n^{-1}\/\\theta_n$. By using these approximations and multiplying the foregoing drift term by $\\theta_n$ as a time change, we get the approximate drift $w_\\infty (b\/k-c)p_1(\\xi_{\\theta_nt})[1-p_1(\\xi_{\\theta_nt})]$ of $p_1(\\xi_{\\theta_nt})$. This approximation recovers \\eqref{eq:X1}. The same formal argument can be used to recover the noise coefficient in Theorem~\\ref{thm:main} (2${^\\circ}$). See also \\cite[Remark~4.10]{C:EGT} for the case of diffusion approximations.\n\n\n\n\n\\section{Semimartingale dynamics}\\label{sec:dynamics}\nIn this section, we define the voter model and the evolutionary game model as solutions to stochastic integral equations driven by point processes. Then we view these equations in terms of semimartingales and identify some leading order terms for the forthcoming perturbation argument. We recall the coalescing duality for the voter model briefly at the end of this section. \n\n\nFirst, given a triplet $(E,q,\\mu)$, an equivalent characterization of the corresponding voter model is given as follows. Introduce independent $(\\mathscr F_t)$-Poisson processes $\\{\\Lambda(x,y);x,y\\in E\\}$ and $\\{\\Lambda^\\sigma_t(x);\\sigma\\in S,x\\in E\\}$ such that\n\\begin{align}\n\\begin{split}\\label{rates}\n\\Lambda_t(x,y)& \\quad\\mbox{with rate}\\quad {\\mathbb E}[\\Lambda_1(x,y)]=q(x,y)\\quad\\mbox{and}\\\\\n\\Lambda^\\sigma_t(x)&\\quad \\mbox{with rate}\\quad {\\mathbb E}[\\Lambda^\\sigma_1(x)]=\\mu(\\sigma),\\quad x,y\\in E,\\;\\sigma\\in S.\n \\end{split}\n\\end{align}\nThese jump processes are defined on a complete filtered probability space $\\big(\\Omega,\\mathscr F,(\\mathscr F_t),\\P\\big)$. Then given an initial condition $\\xi_0\\in S^E$, the $(E,q,\\mu)$-voter model can be defined as the pathwise unique $S^E$-valued solution of the following stochastic integral equations \\cite{CDP,MT}: for $x\\in E$ and $\\sigma\\in S$, \n\\begin{align}\n\\begin{split}\n\\mathds 1_\\sigma\\circ \\xi_t(x)&=\\mathds 1_\\sigma\\circ \\xi_0(x)+\\sum_{y\\in E}\\int_0^t [\\mathds 1_\\sigma\\circ \\xi_{s-}(y)-\\mathds 1_\\sigma\\circ \\xi_{s-}(x)]\\d \\Lambda_s(x,y)\\\\\n&{\\quad \\,}+\\int_0^t \\mathds 1_{\\sigma_{S\\setminus\\{\\sigma\\}}}\\circ \\xi_{s-}(x)\\d \\Lambda^\\sigma_s(x)-\\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}\\int_0^t\\mathds 1_\\sigma\\circ \\xi_{s-}(x)\\d \\Lambda^{\\sigma'}_s(x).\n\\label{eq:voter}\n\\end{split}\n\\end{align}\nHence, the type at $x$ is replaced and changed to the type at $y$\nwhen $\\Lambda(x,y)$ jumps, \nand the type seen at $x$ is $\\sigma$ right after $\\Lambda^\\sigma(x)$ jumps.\n\n\n\n\nRecall that the rates of the evolutionary game are defined by \\eqref{def:qw}. With the choice of $\\overline{w}$ from \\eqref{def:wbar}, $q^w(x,y,\\xi)>0$ if and only if $q(x,y)>0$. Hence, Girsanov's theorem for point processes \\cite[Section~III.3]{JS} can be applied to change the intensities of the Poisson processes $\\Lambda(x,y)$ to $q^w(x,y,\\xi)$ such that under a probability measure $\\P^w$ equivalent to $\\P$ on $\\mathscr F_t$ for all $t\\geq 0$, \n\\begin{align}\\label{mg:hat}\n\\widehat{\\Lambda}_t(x,y)\\stackrel{\\rm def}{=}\\Lambda_t(x,y)-\\int_0^t q^w(x,y,\\xi_s)\\d s\\quad\\&\\quad \\widehat{\\Lambda}_t^\\sigma(x)\\stackrel{\\rm def}{=}\\Lambda^\\sigma_t(x)-\\mu(\\sigma)t\n\\end{align}\nare $(\\mathscr F_t,\\P^w)$-martingales. See \\cite[Section~2]{C:EGT} for the explicit form of $D^w$ when $S=\\{0,1\\}$. Since all of $\\widehat{\\Lambda}(x,y)$ and $\\widehat{\\Lambda}^\\sigma(x)$ do not jump simultaneously under $\\P^w$ by the absolute continuity with respect to $\\P$, the product of any distinct two of them has a zero predictable quadratic variation \\cite[Theorem~4.2, Proposition~4.50, and Theorem~4.52 in Chapter~I]{JS}.\n\n\nThe point processes defined above now allows for straightforward representations of the dynamics of the density processes. By (\\ref{eq:voter}), \n\\begin{align}\n\\begin{split}\np_\\sigma(\\xi_t)\n&=p_\\sigma(\\xi_0)+\\sum_{x,y\\in E}\\pi(x)\\int_0^t \\big[\\mathds 1_{\\sigma}\\circ\\xi_{s-}(y)-\\mathds 1_\\sigma\\circ\\xi_{s-}(x)\\big]\\d\\Lambda_s(x,y)\\\\\n&\\quad +\n\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x)\\d \\Lambda^\\sigma_s(x)\\\\\n&\\quad -\\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{\\sigma}\\circ\\xi_{s-}(x)\\d \\Lambda^{\\sigma'}_s(x).\\label{dynamics:p1}\n\\end{split}\n\\end{align}\nTo obtain the limiting semimartingale for the density processes, we use the foregoing equation to derive the explicit semimartingale decompositions of the density processes. \n\nTo obtain these explicit decompositions, first, note that the dynamics of $p_\\sigma(\\xi_t)$ under $\\P^w$ relies on various kinds of frequencies and densities as follows. For all $x\\in E,\\xi\\in S^E$ and $\\sigma,\\sigma_1,\\sigma_2\\in S$, we set\n\\begin{align}\\label{def:Well}\n\\begin{split}\nf_\\sigma(x,\\xi)&=\\sum_{y\\in E}q(x,y)\\mathds 1_{\\sigma}\\circ \\xi(y),\\\\\n f_{\\sigma_1\\sigma_2}(x,\\xi)&= \\sum_{y\\in E}q(x,y)\\mathds 1_{\\sigma_1}(y)\\sum_{z\\in E}q(y,z)\\mathds 1_{\\sigma_2}\\circ\\xi(z),\\\\\nf_{\\bullet\\sigma}(x,\\xi)&= \\sum_{y\\in E}q(x,y)\\sum_{z\\in E}q(y,z)\\mathds 1_{\\sigma}\\circ \\xi(z),\\quad \n\\overline{f}(\\xi)=\\sum_{x\\in E}\\pi(x)f(x,\\xi).\n\\end{split}\n\\end{align} \nTo minimize the use of the summation notation, we also express these functions in terms of stationary \\emph{discrete-time} $q$-Markov chains $\\{U_\\ell;\\ell\\in \\Bbb Z_+\\}$ and $\\{U'_\\ell;\\ell\\in \\Bbb Z_+\\}$ with $U_0=U_0'$ such that conditioned on $U_0$, the two chains are independent. Additionally, let $(U,U')\\sim \\pi\\otimes \\pi$ and $(V,V')$ be distributed as\n\\begin{align}\\label{def:VV'}\n \\P(V=x,V'=y)=\\frac{\\nu(x,y)}{\\nu(\\mathds 1)},\\quad x,y\\in E,\n\\end{align}\nfor $\\nu(x,y)=\\pi(x)^2q(x,y)$ and $\\nu(\\mathds 1)=\\sum_{x,y}\\nu(x,y)=\\sum_x \\pi(x)^2$. (When $q$ is symmetric, $\\nu(\\mathds 1)$ reduces to $N^{-1}$.) For example, $\\overline{f_{\\sigma_1}f_{\\sigma_2\\sigma_3}}={\\mathbb E}[\\mathds 1_{\\sigma_1}\\circ \\xi(U_1')\\mathds 1_{\\sigma_2}\\circ \\xi(U_1)\\mathds 1_{\\sigma_3}\\circ \\xi(U_2)]$. We also set\n\\begin{align}\\label{def:p10}\np_{\\sigma\\sigma'}(\\xi)={\\mathbb E}[\\mathds 1_\\sigma\\circ \\xi(V)\\mathds 1_{\\sigma'}\\circ \\xi(V')].\n\\end{align}\n\n\n\nSecond, we turn to algebraic identities that determine the leading order terms for the forthcoming perturbation arguments. For $w\\in [0,\\overline{w}]$, the kernel $q^w$ defined by \\eqref{def:qw} can be expanded to the second order in $w$ as follows:\n\\begin{align}\nq^w(x,y,\\xi)&=q(x,y)\\frac{1-wB(y,\\xi)}{1-wA(x,\\xi)}\\notag\\\\\n&=q(x,y)+\\sum_{i=1}^\\infty w^iq(x,y)[A(x,\\xi)-B(y,\\xi)]A(x,\\xi)^{i-1}\\notag\\\\\n&=\nq(x,y)+wq(x,y)[A(x,\\xi)-B(y,\\xi)]+w^2q(x,y)R^w(x,y,\\xi),\\label{qwq:exp}\n\\end{align}\nwhere \n\\begin{align*}\nA(x,\\xi)&=1-\\sum_{z\\in E}q(x,z)\\sum_{z'\\in E}q(z,z')\\Pi\\big(\\xi(z),\\xi(z')\\big),\\\\\nB(y,\\xi)&=1-\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big),\n\\end{align*}\nand $R^w$ is uniform bounded in $w\\in [0,\\overline{w}],x,y,\\xi,(E,q)$.\n\n\n\\begin{lem}\\label{lem:D}\nFor all $\\xi\\in S^E$ and $\\sigma\\in S$,\n\\begin{align}\n\\overline{D}_\\sigma(\\xi)&\\!\\stackrel{\\rm def}{=} \\sum_{x,y\\in E}\\pi(x)\\big[\\mathds 1_\\sigma\\circ\\xi(y)-\\mathds 1_\\sigma\\circ\\xi(x)\\big]q(x,y)[A(x,\\xi)-B(y,\\xi)]\\label{eq:Dsigma0}\\\\\n&=\\sum_{\\stackrel{\\scriptstyle \\sigma_0,\\sigma_3\\in S}{ \\sigma_0\\neq\\sigma}}\\Pi(\\sigma,\\sigma_3)\\overline{f_{\\sigma_0}f_{\\sigma\\sigma_3}}(\\xi)-\\sum_{\\stackrel{\\scriptstyle \\sigma_2,\\sigma_3\\in S}{\\sigma_2\\neq\\sigma}}\\Pi(\\sigma_2,\\sigma_3)\\overline{f_{\\sigma}f_{\\sigma_2\\sigma_3}}(\\xi).\\label{eq:Dsigma}\n\\end{align}\nIn particular, if $\\Pi$ is given by \\eqref{prisoner},\nthen \n\\begin{align}\\label{eq:Dsigma1}\n\\overline{D}_1(\\xi)=b\\overline{f_{1}f_{ \\bullet 0}}(\\xi)-b\\overline{f_{10}}(\\xi)-c\\overline{f_1f_{0}}(\\xi).\n\\end{align}\n\\end{lem}\n\\begin{proof}\nBy using the reversibility of $q$ and taking $y$ in \\eqref{eq:Dsigma0} as the state of $U_0$ in the sequence $\\{U_\\ell\\}$ defined above, we can compute $\\overline{D}_\\sigma$ as\n\\begin{align}\n\\begin{split}\\label{Dbar:cal}\n\\overline{D}_\\sigma(\\xi)\n&=-\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(y)q(x,y)\\sum_{z\\in E}q(x,z)\\sum_{z'\\in E}q(z,z')\\Pi\\big(\\xi(z),\\xi(z')\\big)\\\\\n&{\\quad \\,}+\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(y)q(x,y)\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big)\\\\\n&{\\quad \\,}+\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(x)q(x,y)\\sum_{z\\in E}q(y,z)\\Pi\\big(\\xi(y),\\xi(z)\\big)\\\\\n&{\\quad \\,}-\\sum_{x,y\\in E}\\pi(x)\\mathds 1_\\sigma\\circ\\xi(x)q(x,y)\\sum_{z\\in E}q(x,z)\\sum_{z'\\in E}q(z,z')\\Pi\\big(\\xi(z),\\xi(z')\\big)\n\\end{split}\\\\\n&=-{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]+{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{2})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\label{Dbar:cal1}\\\\\n&=-{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\mathds 1_\\sigma\\circ\\xi(U_2)\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&\\quad -{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\mathds 1_{S\\setminus \\{\\sigma\\}}\\circ\\xi(U_2)\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&\\quad +{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{2})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&={\\mathbb E}\\left[\\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi(U_{0})\\mathds 1_\\sigma\\circ\\xi(U_{2})\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right]\\notag\\\\\n&\\quad -{\\mathbb E}\\left[\\mathds 1_\\sigma\\circ\\xi(U_{0})\\mathds 1_{S\\setminus \\{\\sigma\\}}\\circ\\xi(U_2)\\Pi\\big(\\xi(U_2),\\xi(U_3)\\big)\\right].\\notag\n\\end{align}\nHere, we use the reversibility of $q$ with respect to $\\pi$ to cancel\nthe last two terms in \\eqref{Dbar:cal} and write the first term in \\eqref{Dbar:cal} as the first term in \\eqref{Dbar:cal1}. \nSee \\cite[Lemma~1 on p.8]{CMN} for the case of two types. \n\n\nThe proof of \\eqref{eq:Dsigma1} appears in \\cite[Lemma~7.1]{C:EGT}. Now \\eqref{eq:Dsigma} allows for a quick proof:\n$\\overline{D}_1(\\xi)=(b-c)\\overline{f_0f_{11}}-c\\overline{f_0f_{10}}-b\\overline{f_1f_{01}}$. Then we use the identities $\\overline{f_0f_{11}}+\\overline{f_0f_{10}}=\\overline{f_0f_1}$, $\\overline{f_0f_{11}}+\\overline{f_0f_{01}}=\\overline{f_0f_{\\bullet 1}}$, and $\\overline{f_0f_{01}}+\\overline{f_0f_{01}}=\\overline{f_{01}}$. This calculation will be used in the proof of Corollary~\\ref{cor:symmetric}.\n\\end{proof}\n\nWe are ready to state the explicit semimartingale decompositions of the density processes and identify the leading order terms. \nFrom \\eqref{dynamics:p1}, \\eqref{qwq:exp} and the martingales in \\eqref{mg:hat}, we obtain the following decompositions extended from \\eqref{density:dynamics}:\n\\begin{align}\\label{psigma:dec}\np_\\sigma(\\xi_t)=p_\\sigma(\\xi_0)+A_\\sigma(t)+M_\\sigma(t)=p_\\sigma(\\xi_0)+I_\\sigma(t)+R_\\sigma(t)+M_\\sigma(t),\n\\end{align}\nwhere\n\\begin{align}\nI_\\sigma(t)&=w\\int_0^t \\overline{D}_\\sigma(\\xi_s)\\d s+\\int_0^t \\Bigg(\\mu(\\sigma)\\sum_{\\sigma'\n\\in S\\setminus\\{\\sigma\\}}p_{\\sigma'}(\\xi_s)-\\mu(S\\setminus\\{\\sigma\\}) p_\\sigma(\\xi_s)\\Bigg)\\d s,\n\\label{def:I}\\\\\nR_\\sigma(t)&=w^2\\sum_{x,y\\in E}\\pi(x)\\int_0^t \\big[\\mathds 1_\\sigma\\circ\\xi_{s}(y)-\\mathds 1_\\sigma\\circ\\xi_{s}(x)\\big]q(x,y)R^w(x,y,\\xi_s)\\d s,\\label{def:R}\\\\\n\\begin{split}\nM_\\sigma(t)&=\\sum_{x,y\\in E}\\pi(x)\\int_0^t \\big[\\mathds 1_\\sigma\\circ\\xi_{s-}(y)-\\mathds 1_\\sigma\\circ\\xi_{s-}(x)\\big]\\d\\widehat{\\Lambda}_s(x,y) \\\\\n&\\quad +\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x)\\d \\widehat{\\Lambda}^\\sigma_s(x)\\\\\n&\\quad -\\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}\\sum_{x\\in E}\\pi(x)\\int_0^t \\mathds 1_{\\sigma}\\circ\\xi_{s-}(x)\\d \\widehat{\\Lambda}^{\\sigma'}_s(x).\\label{def:M}\n\\end{split}\n\\end{align}\nBy \\eqref{mg:hat}, the predictable quadratic variations and covariations of $M_\\sigma$ and $M_{\\sigma'}$, for $\\sigma\\neq \\sigma'$, are \n\\begin{align}\n\\begin{split}\\label{def:}\n \\langle M_\\sigma,M_{\\sigma}\\rangle_t\n&=\\sum_{x,y\\in E}\\pi(x)^2\\int_0^t \\big\\{\\mathds 1_\\sigma\\circ\\xi_{s}(y)[1-\\mathds 1_{\\sigma}\\circ \\xi_s(x)]\\\\\n&\\hspace{-.5cm} +[1-\\mathds 1_{\\sigma}\\circ \\xi_s(y)]\\mathds 1_\\sigma\\circ\\xi_{s}(x)\\big\\} q^w(x,y,\\xi_{s})\\d s\\\\\n&\\hspace{-.5cm} +\\sum_{x\\in E}\\pi(x)^2\\int_0^t\\big[ \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x)\\mu(\\sigma)+\\mathds 1_\\sigma\\circ\\xi_{s}(x)\\mu\\big(S\\setminus\\{\\sigma\\}\\big)\\big] \\d s,\n\\end{split}\\\\\n\\begin{split}\\label{def:}\n\\langle M_\\sigma,M_{\\sigma'}\\rangle_t\n&=-\\sum_{x,y\\in E}\\pi(x)^2\\int_0^t \\big[\\mathds 1_\\sigma\\circ\\xi_{s}(y)\\mathds 1_{\\sigma'}\\circ\\xi_s(x)\\\\\n&\\quad +\\mathds 1_\\sigma\\circ\\xi_s(y)\\mathds 1_{\\sigma'}\\circ\\xi_{s}(x)\\big] q^w(x,y,\\xi_{s})\\d s\\\\\n& \\quad -\\sum_{x\\in E}\\pi(x)^2\\int_0^t\\big[ \\mathds 1_{S\\setminus\\{\\sigma\\}}\\circ\\xi_{s-}(x) \\mathds 1_{\\sigma}\\circ\\xi_{s-}(x)\\mu(\\sigma)\\\\\n&\\quad +\\mathds 1_{S\\setminus\\{\\sigma'\\}}\\circ\\xi_{s-}(x) \\mathds 1_{\\sigma'}\\circ\\xi_{s-}(x)\\mu(\\sigma')\\big] \\d s.\n\\end{split}\n\\end{align}\nIn Section~\\ref{sec:eqn}, the above equations play the central role in characterizing the limiting density processes.\n\n\nFor this study, we apply the coalescing duality between $(E,q,\\mu)$-voter model and the coalescing rate-$1$ $q$-Markov chains $\\{B^x;x\\in E\\}$, where $B^x_0=x$. These chains move independently before meeting, and for any $x,y\\in E$, $B^x=B^y$ after their first meeting time\n$ M_{x,y}=\\inf\\{t\\geq 0;B^x_t=B^y_t\\}$. In the absence of mutation, the duality is given by\n\\begin{align}\\label{dual:1}\n{\\mathbb E}\\left[\\prod_{i=1}^n \\mathds 1_{\\sigma_i}\\circ \\xi_0(B^{x_i}_t)\\right]={\\mathbb E}_{\\xi_0}\\left[\\prod_{i=1}^n \\mathds 1_{\\sigma_i}\\circ\\xi_t(x_i)\\right] \n\\end{align}\nfor all $\\xi_0\\in S^E$, $\\sigma_1,\\cdots,\\sigma_n\\in S$, distinct $x_1,\\cdots,x_n \\in E$ and $n\\in \\Bbb N$. See the proof of Proposition~\\ref{prop:mutation} for the foregoing identity and the extension to the case with mutations. \n\nWithout mutation, the density process is a martingale under the voter model by \\eqref{density:dynamics}, and it follows from \\eqref{def:M} and \\eqref{def:} that, for any $\\sigma\\neq \\sigma'$, \n\\begin{align}\\label{eq:p1p0-voter}\n{\\mathbb E}_\\xi^0[p_\\sigma(\\xi_{t})p_{\\sigma'}(\\xi_{t})]=p_\\sigma(\\xi)p_{\\sigma'}(\\xi)-\\nu(\\mathds 1)\\int_0^t {\\mathbb E}_\\xi^0[p_{\\sigma\\sigma'}(\\xi_{s})+p_{\\sigma'\\sigma}(\\xi_s)]\\d s.\n\\end{align}\nFor the present problem, the central application of this dual relation is the foregoing identity \\cite{CCC}. Let the random variables defined below \\eqref{def:Well} to represent frequencies and densities be independent of the coalescing Markov chains. Then the foregoing equality implies that\n\\begin{align}\\label{ergodic}\n\\P(M_{U,U'}>t)=1-\\nu(\\mathds 1)-2\\nu(\\mathds 1)\\int_0^t \\P(M_{V,V'}>s)\\d s,\\quad \\forall\\;t\\geq 0.\n\\end{align}\nSee \\cite[Corollary~4.2]{CCC} and \\cite[Section~3.5.3]{AF:MC}.\nThis identity for meeting times has several important applications to the diffusion approximation of the voter model density processes. See \\cite[Sections~3 and 4]{CCC} and \\cite{CC}. \n\n\n\n\n\n\\section{Decorrelation in the ancestral lineage distributions}\\label{sec:slow}\nThis section is devoted to a study of degenerate limits of meeting time distributions. Here, we consider meeting times defined on a sequence of spatial structures $(E_n,q^{(n)})$ as before. According to the coalescing duality, these distributions are part of the ancestral line distributions of the voter model, and by approximation, the ancestral line distributions of the evolutionary game. On the other hand, these meeting times encode the typical local geometry of the space, but in a rough manner. With the study of these distributions, the main results of this section (Propositions~~\\ref{prop:sn-selection} and \\ref{prop:kell}) extend to the choice of appropriate time scaling constants and the characterization of the limiting density processes. These properties are crucial to the forthcoming limit theorems. \n\n\nOur direction in this section can be outlined in more detail as follows.\nRecall the auxiliary random variables defined below \\eqref{def:Well}, which are introduced to represent frequencies and densities. Under mild mixing conditions similar to those in \\eqref{cond2:thetan} with $\\gamma_n$ replaced by $\\theta_n$ \n and the condition $\\nu_n(\\mathds 1)\\to 0$, the sequence $\\P^{(n)}(M_{V,V'}\/\\gamma_n\\in \\cdot)$ is known to converge. The limiting distribution is a convex combination of the delta distribution at zero and an exponential distribution. Moreover, one can choose \\emph{some} $s_n\\to\\infty$ such that $s_n\/\\gamma_n\\to0 $ and the following $t$-independent limit exists:\n\\begin{align}\\label{cond:kappa0}\n\\overline{\\kappa}_0\\,\\stackrel{\\rm def}{=}\\,\n\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{V,V'}>s_n t),\\quad\\forall\\;t\\in (0,\\infty)\n\\end{align}\nwith $\\overline{\\kappa}_0=1$. See \\cite[Corollary~4.2 and Proposition~4.3]{CCC} for these results. As an extension of this existence result, our first goal in this section is to introduce \\emph{sufficient} conditions for these sequences $(s_n)$. Specifically, we require that the limit \\eqref{cond:kappa0} exists with $\\overline{\\kappa}_0\\in (0,\\infty)$. See Section~\\ref{sec:dec1}. The following is enough for the existence and the applications in the next section. \n\n\n\\begin{defi}\\label{def:slow}\nWe say that $(s_n)$ is a {\\bf slow sequence} if \n\\begin{align}\\label{cond1:sn}\n\\lim_{n\\to\\infty}s_n=\\infty,\\quad \\lim_{n\\to\\infty}\\frac{s_n}{\\gamma_n}=0,\\quad \n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-ts_n}= 0,\\quad \\forall\\;t\\in (0,\\infty),\n\\end{align}\nand at least one of the two mixing conditions holds: \n\\begin{align}\\label{cond2:sn}\n\\lim_{n\\to\\infty}\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mathbf g_n s_n}=0\\quad\\mbox{or}\\quad \\lim_{n\\to\\infty}\\frac{\\mathbf t^{(n)}_{\\rm mix}}{s_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]=0.\n\\end{align}\n\\end{defi}\n\n\n\n\nOur second goal is to extend the existence of the limit \\eqref{cond:kappa0} to the existence of analogous time-independent limits for other meeting time distributions: for integers $\\ell\\geq 1$, $\\ell_0,\\ell_1,\\ell_2\\geq 0$ with $\\ell_0,\\ell_1,\\ell_2$ all distinct, and all $t\\in (0,\\infty)$, \n\\begin{align}\\label{def:kell}\n\\overline{\\kappa}_\\ell &\\,\\stackrel{\\rm def}{=}\\,\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>s_n t);\\\\\n\\label{def:|kell1|ell2}\n\\overline{\\kappa}_{(\\ell_0,\\ell_1)|\\ell_2}&\\,\\stackrel{\\rm def}{=}\\,\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}\\big(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}>s_n t\\big);\\\\\n\\label{def:|||}\n\\overline{\\kappa}_{\\ell_0|\\ell_1|\\ell_2}&\\,\\stackrel{\\rm def}{=}\\,\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}\\big(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}>s_n t,M_{U_{\\ell_0},U_{\\ell_2}}>s_n t\\big).\n\\end{align}\nThe extension to $\\overline{\\kappa}_1$ is straightforward if we allow passing limits along subsequences. Indeed,\nit follows from the definition of $\\{U_\\ell\\}$ and $(V,V')$ that\n\\begin{align}\\label{ineq:MUVcompare}\n\\frac{\\pi_{\\min}}{\\pi_{\\max}}\\P(M_{V,V'}\\in \\Gamma)\\leq \\P(M_{U_0,U_1}\\in \\Gamma)\\leq \n\\frac{\\pi_{\\max}}{\\pi_{\\min}}\\P(M_{V,V'}\\in \\Gamma),\\quad\\forall\\; \\Gamma\\in \\mathscr B({\\Bbb R}_+).\n\\end{align}\nHence, by taking a subsequence of $(E_n,q^{(n)})$ if necessary,\n\\eqref{cond:kappa0} and condition (a) of Theorem~\\ref{thm:main} imply the existence of the limit $\\overline{\\kappa}_1$. \n\n\nIn Section~\\ref{sec:higher-order}, we prove the existence of the other limits $\\overline{\\kappa}_\\ell$, $\\ell\\geq 2$. \nMore precisely, we prove tightness results as in the case of $\\overline{\\kappa}_1$ so that the limits may be passed along subsequences. We also prove that the limits $\\overline{\\kappa}_\\ell$, $\\ell\\geq 2$, are in $(0,\\infty)$. Note that in proving these results, we do not impose convergence of local geometry as in the case of discrete tori or random regular graphs. \n\n\n\n\\subsection{Mixing conditions for local meeting times}\\label{sec:dec1}\nTo apply mixing conditions to meeting times, first, we recall some basic properties of the spectral gap and the mixing time for the product of the continuous-time $q$-Markov chains. Note that by coupling the product chain with initial condition $(x,y)$ after the two coordinates meet, we get the coalescing chain $(B^x,B^y)$ defined before \\eqref{eq:p1p0-voter}. \n\n\nNow, the discrete-time chain for the product chain has a transition matrix such that each of the coordinates is allowed to change with equal probability. Hence, the spectral gap is given by $\\widetilde{\\mathbf g}=\\mathbf g\/2$ \\cite[Corollary~12.12 on p.161]{LPW}.\nIf $(\\widetilde{q}_t)$ denotes the semigroup of the product chain, then\n\\begin{align}\\label{product:bdd}\n\\sup_{(x,y)\\in E\\times E}\\big\\|\\widetilde{q}_t\\big((x,y),\\cdot\\big)-\\pi\\otimes \\pi\\big\\|_{\\rm TV}\\leq 2d_{E}(t),\n\\end{align}\nwhere $d_E$ is the total variation distance defined by \\eqref{def:dE}. Additionally, it follows from the definition \nthe mixing time in \\eqref{def:tmix} that \n\\begin{align}\\label{ineq:tmix}\nd_E(k\\mathbf t_{\\rm mix})\\leq {\\rm e}^{-k},\\quad \\forall\\;k\\in \\Bbb N\n\\end{align}\n\\cite[Section~4.5 on p.55]{LPW}. By the last two displays, the analogous mixing time $\\widetilde{\\mathbf t}_{\\rm mix}$ of the product chain satisfies \n\\begin{align}\\label{compare:mix} \n \\widetilde{\\mathbf t}_{\\rm mix}\\leq 3\\mathbf t_{\\rm mix}.\n\\end{align}\nWe are ready to prove the first main result of Section~\\ref{sec:slow}. Note that under the condition $\\sup_nN_n\\pi^{(n)}_{\\max}<\\infty$ (see the discussion below \\eqref{def:wn}), the first condition in \\eqref{cond2:sn} implies the first one in \\eqref{sn:lim}. \n\n\n\n\\begin{prop}\\label{prop:sn-selection}\nSuppose that $(s_n)$ satisfies \\eqref{cond1:sn} and at least one of the following mixing conditions:\n\\begin{align}\\label{sn:lim}\n\\lim_{n\\to\\infty}\\mathbf g_n s_n=\\infty\\quad\\mbox{or}\\quad \\lim_{n\\to\\infty}\\frac{\\mathbf t^{(n)}_{\\rm mix}}{s_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]=0.\n\\end{align}\nThen \\eqref{cond:kappa0} holds with $\\overline{\\kappa}_0=1$. \n\\end{prop}\n\\begin{proof}\nWrite $f_n(t)=\\P^{(n)}(M_{U,U'}>t)$ and $g_n(t)=\\P^{(n)}(M_{V,V'}>t)$. The required result is proved in two steps. \\medskip \n\n\n\\noindent {\\bf Step 1.} We start with a preliminary result: for all $t_0\\in[0,\\infty)$ and $\\mu\\in(0,\\infty)$,\n\\begin{align}\\label{eq:Lap_nun}\n&\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\n\\int_0^\\infty {\\rm e}^{-\\mu t}g_n\\big(s_n(t+t_0)\\big)\\d t=\\frac{1}{\\mu}.\n\\end{align}\n\n\nTo obtain \\eqref{eq:Lap_nun}, first, we derive a representation of the integrals in \\eqref{eq:Lap_nun} by $f_n(t)$. \nNote that \\eqref{ergodic} under the $q^{(n)}$-chain takes the following form:\n\\[\nf_n(s_nt)=\n1-\\nu_n(\\mathds 1)-2\\nu_n(\\mathds 1)s_n\\int_0^t g_n(s_ns)\\d s,\\quad t\\geq 0.\n\\]\nHence, for any fixed $0\\leq t_0<\\infty$,\n\\begin{align}\\label{delta:f}\nf_n(s_n(t+t_0))-f_n(s_nt_0)=-2\\nu_n(\\mathds 1)s_n\\int_{0}^{t} g_n\\big(s_n(s+t_0)\\big)\\d s,\\quad t\\geq 0.\n\\end{align}\nTaking Laplace transforms of both sides of the last equality, we get, for $\\mu>0$,\n\\begin{align}\n\\int_{0}^\\infty {\\rm e}^{-\\mu t}\\big[f_n\\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]\\d t&=-2\\nu_n(\\mathds 1)s_n\\int_0^\\infty {\\rm e}^{-\\mu t}\\int_0^{t}g_n\\big(s_n(s+t_0)\\big)\\d s\\d t\\notag\\\\\n&=-\\frac{2\\nu_n(\\mathds 1)s_n}{\\mu}\\int_0^\\infty {\\rm e}^{-\\mu t}g_n\\big(s_n (t+t_0)\\big)\\d t,\\notag\n\\end{align}\nwhere the last integral coincides with the integral in \\eqref{eq:Lap_nun}. \n\n\nNext, rewrite the last equality as\n\\begin{align}\n&\\quad 2\\gamma_n\\nu_n(\\mathds 1)\n\\int_0^\\infty {\\rm e}^{-\\mu t}g_n\\big(s_n (t+t_0)\\big)\\d t\\notag\\\\\n&= -\\frac{\\gamma_n\\mu}{s_n} \\int_0^\\infty {\\rm e}^{-\\mu t}\\big[f_n \\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]\\d t\\notag\\\\\n\\begin{split}\n&=-\\frac{\\gamma_n\\mu}{s_n} \\int_0^\\infty {\\rm e}^{-\\mu t}\\Big\\{\\big[f_n \\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]-\n[{\\rm e}^{-s_n(t+t_0)\/\\gamma_n}-{\\rm e}^{-s_nt_0\/\\gamma_n}]\n\\Big\\}\\d t \\\\\n&\\quad +\\frac{ {\\rm e}^{-s_nt_0\/\\gamma_n}}{\\mu+s_n\/\\gamma_n}.\\label{eq:sn-selection0}\n\\end{split}\n\\end{align}\nThe last term tends to $1\/\\mu$ since $s_n\/\\gamma_n\\to 0$. To take the limit of the integral term in \\eqref{eq:sn-selection0}, we use the first mixing condition in \\eqref{cond2:sn}. In this case, a bound for exponential approximations of the distributions of $M_{U,U'}$ \\cite[Proposition~3.23]{AF:MC} gives\n\\begin{align}\\label{eq:sn-selection1}\n\\begin{split}\n&\\left|\\frac{\\gamma_n\\mu}{s_n} \\int_0^\\infty {\\rm e}^{-\\mu t}\\Big\\{\\big[f_n \\big(s_n(t+t_0)\\big)-f_n(s_nt_0)\\big]-\n[{\\rm e}^{-s_n(t+t_0)\/\\gamma_n}-{\\rm e}^{-s_nt_0\/\\gamma_n}]\n\\Big\\}\\d t\\right|\\\\\n&\\leq \\frac{2}{\\widetilde{\\mathbf g}_n s_n}\\xrightarrow[n\\to\\infty]{} 0\n\\end{split}\n\\end{align}\nnow that $\\widetilde{\\mathbf g}_n=\\mathbf g_n\/2$.\nAlternatively, by a different bound from \\cite[Theorem~1.4]{Aldous:AE}, the foregoing inequality holds with the bound replaced by \n\\begin{align}\\label{eq:sn-selection2}\n\\frac{C_{\\ref{eq:sn-selection2}}\\widetilde{\\mathbf t}^{(n)}_{\\rm mix}}{s_n}\\big[1+\\log^+(\\gamma_n\/\\widetilde{\\mathbf t}^{(n)}_{\\rm mix})\\big]\n\\leq \\frac{C_{\\ref{eq:sn-selection2}}\\cdot 3\\mathbf t^{(n)}_{\\rm mix}}{s_n}\\big[1+\\log^+(\\gamma_n\/(3\\mathbf t^{(n)}_{\\rm mix}))\\big]\n\\end{align}\nby \\eqref{compare:mix}, the monotonicity of $x\\mapsto x(1+\\log (x^{-1}\\vee 1))$ on $(0,\\infty)$, where $C_{\\ref{eq:sn-selection2}}$ is independent of the $q^{(n)}$-chains.The last term in \\eqref{eq:sn-selection2} tends to zero by the second mixing condition in \\eqref{cond2:sn}. \n\nFinally, we apply \\eqref{eq:sn-selection1} and \\eqref{eq:sn-selection2} to \\eqref{eq:sn-selection0}. Since the last term in \\eqref{eq:sn-selection0} tends to $1\/\\mu$, we have proved \\eqref{eq:Lap_nun}. \\medskip\n\n\n\\noindent {\\bf Step 2.} We are ready to prove the existence of the limit in \\eqref{cond:kappa0} and its independence of $t$.\nFirst, note that since $g_n$ is decreasing, we have\n\\[\n2\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mu t}g_n\\big(s_n(t+t_0)\\big)\\leq \\frac{1}{t}2\\gamma_n\\nu_n(\\mathds 1)\\int_0^t {\\rm e}^{-\\mu s}g_n\\big(s_n(s+t_0)\\big)\\d s,\\quad\\forall\\;t,t_0\\in (0,\\infty),\n\\]\nwhereas the last integral is bounded by the same integral with the upper limit $t$ of integration replaced by $\\infty$. By the \\eqref{eq:sn-selection0} and the convergence proven for it in the preceding step, the last inequality implies that $t\\mapsto 2\\gamma_n\\nu_n(\\mathds 1){\\rm e}^{-\\mu t}g_n(s_nt)$, $n\\geq 1$, are uniformly bounded on $[a,\\infty)$, for any $a\\in (0,\\infty)$. Hence, by Helly's selection theorem, every subsequence of $\\{t\\mapsto 2\\gamma_n\\nu_n(\\mathds 1)g_n(s_nt)\\}$ has a further subsequence, say indexed by $n_j$, such that for some left-continuous function $g_\\infty$ on $(0,\\infty)$,\n\\begin{align}\\label{def:ginfty}\n\\lim_{j\\to\\infty}2\\gamma_{n_j}\\nu_{n_j}(\\mathds 1)g_{n_j}(s_{n_j}t)= g_\\infty(t),\\quad \\forall\\; t\\in (0,\\infty). \n\\end{align}\nMoreover, this convergence holds boundedly on compact subsets of $ (0,\\infty)$ in $t$.\n\nTo find $g_\\infty$, note that, as in \\eqref{eq:sn-selection1} and \\eqref{eq:sn-selection2}, either of the mixing conditions \\eqref{cond2:sn} implies that for fixed $0t)&={\\rm e}^{-2t}\\P(U_0\\neq U_\\ell)+\\int_0^t 2{\\rm e}^{-2(t-s)}\\P(M_{U_0,U_{\\ell+1}}>s)\\d s\\\\\n&\\quad -\\int_0^t 2{\\rm e}^{-2(t-s)}\\sum_{x,y\\in E}\\pi(x)q^{\\ell}(x,x)q(x,y)\\P(M_{x,y}>s)\\d s.\n\\end{split}\n\\end{align}\n\\end{lem}\n\\begin{proof}\nSince $M_{x,x}\\equiv 0$ and $(U_0,U_\\ell)$ is independent of the meeting times, conditioning on $(U_0,U_\\ell)$ gives $\\P(M_{U_0,U_\\ell}>t)=\\P(M_{U_0,U_\\ell}>t,U_0\\neq U_\\ell)$. Conditioning on the first update time of $(B^{U_0},B^{U_\\ell})$, which is an exponential variable with mean $1\/2$, yields\n\\begin{align}\\label{eq:st1}\n\\P(M_{U_0,U_\\ell}>t)&={\\rm e}^{-2t}\\P(U_0\\neq U_\\ell)+\\int_0^t 2{\\rm e}^{-2(t-s)}\\P(U_0\\neq U_{\\ell},M_{U_0,U_{\\ell+1}}>s)\\d s.\n\\end{align}\nHere, the initial condition $(U_0,U_{\\ell+1})$ in the last term follows from transferring the first transition of state of $(B^{U_0},B^{U_\\ell})$ to the initial condition. We also use the stationarity of $\\{U_\\ell;\\ell\\geq 0\\}$ when that first transition is made by $B^{U_0}$. To rewrite the integral term in \\eqref{eq:st1}, note that \n\\begin{align*}\n\\P(U_0\\neq U_\\ell,U_0=x,U_{\\ell+1}=y)\n&=\\P(U_0=x,U_{\\ell+1}=y)-\\P(U_0=U_\\ell,U_0=x,U_{\\ell+1}=y)\\\\\n&=\\pi(x)q^{\\ell+1}(x,y)-\\pi(x)q^{\\ell}(x,x)q(x,y)\n\\end{align*}\nso that\n\\begin{align}\\label{eq:st3}\n\\begin{split}\n\\P(U_0\\neq U_{\\ell},M_{U_0,U_{\\ell+1}}>s)=&\\P(M_{U_0,U_{\\ell+1}}>s)\\\\\n&-\\sum_{x,y\\in E}\\pi(x)q^{\\ell}(x,x)q(x,y)\\P(M_{x,y}>s).\n\\end{split}\n\\end{align}\nApplying \\eqref{eq:st3} to (\\ref{eq:st1}) yields\n (\\ref{eq:shifttime0}). \n\\end{proof}\n\n\n\nWe are ready to prove the existence of the limits in \\eqref{def:kell} and \\eqref{def:|kell1|ell2}.\n\n\\begin{prop}\\label{prop:kell}\nFor any sequence $(s_n)$ satisfying \\eqref{cond1:sn}, we have the following properties:\n\\begin{enumerate}\n\\item [\\hypertarget{prop:kell1}{\\rm (1${^\\circ}$)}] For any integer $\\ell\\geq 2$, every subsequence of $(E_n,q^{(n)})$ contains a further subsequence such that the limit in \\eqref{def:kell}\nexists in $[\\overline{\\kappa}_1,\\ell\\overline{\\kappa}_1]$ and is independent of $t\\in (0,\\infty)$.\n\n\\item [\\hypertarget{prop:kell2}{\\rm (2${^\\circ}$)}] Without taking any subsequence, \\eqref{def:kell} holds for $\\ell=2$ with $\\overline{\\kappa}_2=\\overline{\\kappa}_1$. \n\\item [\\hypertarget{prop:kell3}{\\rm (3${^\\circ}$)}] Suppose that \\eqref{cond:q2} holds for some constant $q^{(\\infty),2}$.\nThen without taking any subsequence, \\eqref{def:kell} holds $\\overline{\\kappa}_3=(1+q^{(\\infty),2})\\overline{\\kappa}_1$.\n\n\\item [\\hypertarget{prop:kell4}{\\rm (4${^\\circ}$)}] \nFor all distinct nonnegative integers $\\ell_0,\\ell_1,\\ell_2$, it holds that \n\\[\n{\\overline{\\kappa}}_{(\\ell_1,\\ell_2)|\\ell_0}+{\\overline{\\kappa}}_{(\\ell_0,\\ell_1)|\\ell_2}-{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2}={\\overline{\\kappa}}_{|\\ell_2-\\ell_0|},\n\\] \nprovided that all of the limits defining these constants exist.\n\n\\end{enumerate}\n\\end{prop}\n\\begin{proof} \n(1${^\\circ}$) To lighten notation in the rest of this proof but only in this proof, write $A_\\ell=\\P(U_0\\neq U_\\ell)$, $J_\\ell$ for $M_{U_0,U_\\ell}$, \n\\[\nB_\\ell=\\sum_{x,y\\in E}\\pi(x)q^\\ell(x,x)q(x,y),\n\\]\nand $K_\\ell$ for the first meeting time for the pair of coalescing Markov chains where the initial condition is distributed independently as $B_\\ell^{-1}\\pi(x)q^\\ell(x,x)q(x,y)$ provided that $B_\\ell\\neq 0$. We set $K_\\ell$ to be an arbitrary random variable.\n\n\n\nFix an integer $\\ell\\geq 1$. If $\\mathbf e$ is an independent exponential variable with mean $1$, then \\eqref{eq:shifttime0} can be written as \n\\[\n\\P(J_\\ell>t)=A_\\ell\\P(\\tfrac{1}{2} \\mathbf e>t)+\\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e>t,\\tfrac{1}{2} \\mathbf e\\leq t)-B_\\ell\\P(K_{\\ell}+\\tfrac{1}{2} \\mathbf e>t,\\tfrac{1}{2} \\mathbf e\\leq t).\n\\]\nAfter rearrangement, the foregoing equality yields\n\\begin{align*}\n\\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e>t)\n&=\\P(J_\\ell>t)+B_\\ell\\P(K_{\\ell}+\\tfrac{1}{2} \\mathbf e>t)+(1-A_\\ell-B_\\ell)\\P(\\tfrac{1}{2} \\mathbf e>t).\n\\end{align*}\nHence, for all left-open intervals $\\Gamma\\subset (0,\\infty)$, \n\\begin{align}\\label{id:JK}\n\\begin{split}\n&\\quad\\, \\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+(A_\\ell+B_\\ell)\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma)\\\\\n&=\\P(J_\\ell\\in \\Gamma)+B_\\ell\\P(K_{\\ell}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma).\n\\end{split}\n\\end{align}\nSince $q^{\\ell}(x,x)\\leq 1$,\nwe have $B_\\ell\\P(K_\\ell\\in \\cdot)\\leq \\P(J_1\\in \\cdot)$, and so, the foregoing identity gives\n\\begin{align}\\label{id:JK1}\n\\begin{split}\n&\\quad \\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+(A_\\ell+B_\\ell)\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma)\\\\\n&\\leq \\P(J_\\ell\\in \\Gamma)+\\P(J_{1}+\\tfrac{1}{2} \\mathbf e\\in \\Gamma)+\\P(\\tfrac{1}{2} \\mathbf e\\in \\Gamma).\n\\end{split}\n\\end{align}\n\nWe are ready to prove the required result. \nFor any $0s_n a)\\leq (\\ell+1) \\overline{\\kappa}_1,\\quad \\forall\\;t\\in (0,\\infty).\n\\end{align}\nOn the other hand,\nsince $\\P(J_{\\ell+1}+\\tfrac{1}{2} \\mathbf e\\in \\cdot)+|A_\\ell+B_\\ell-1|\\P(\\tfrac{1}{2}\\mathbf e\\in \\cdot)\\geq \\P(J_\\ell\\in \\cdot)$ by \\eqref{id:JK}, it follows from \\eqref{def:kell} with $\\ell=1$ and an argument similar to the one leading to \\eqref{eq:kell1} that \n\\begin{align}\n\\liminf_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(J_{\\ell+1}>s_n t)\\geq \\overline{\\kappa}_1>0,\\quad\\forall\\;t\\in (0,\\infty).\\label{eq:kell4}\n\\end{align}\n\n\n\nCombining \\eqref{eq:kell3} and \\eqref{eq:kell4}, we deduce that for fixed $t_0\\in (0,\\infty)$, any subsequence of the numbers\n$2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(J_{\\ell+1}>s_nt_0)$ has a further subsequence that converges in $[\\overline{\\kappa}_1,(\\ell+1)\\overline{\\kappa}_1]$. By \\eqref{eq:kell1}, this limit extends to the existence of the limit of the corresponding subsequence of \n$2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(J_{\\ell+1}>s_nt)$ \nfor any $t\\in (0,\\infty)$, and all of these limits for different $t$ are equal.\nWe have proved \\eqref{def:kell}. \\medskip \n \n\n\\noindent (2${^\\circ}$) Note that $B_1^{(n)}=0$ since ${\\sf tr}(q^{(n)})=0$ by assumption. \nThen an inspection of \\eqref{eq:kell0} shows the second limit superior on the right-hand side there can be dropped. The rest of the argument in (2${^\\circ}$), especially \\eqref{eq:kell3} and \\eqref{eq:kell4}, can be adapted accordingly to get\nthe required identity. \\medskip\n \n\\noindent (3${^\\circ}$) The proof is done again by improving the argument for \\eqref{eq:kell3} and \\eqref{eq:kell4}, but now using \\eqref{id:JK} with $\\ell=2$. In doing so, we also use the following implication of \\eqref{cond:q2}:\n\\[\n\\lim_{n\\to\\infty}\\sup_{s\\geq 0}\\gamma_n\\nu_n(\\mathds 1)\\big|B^{(n)}_2\\P^{(n)}(K_2>s)-q^{(\\infty),2}\\P^{(n)}(J_1>s)\\big|=0,\n\\]\nwhich follows since the distributions of $K_2$ and $J_1$ differ by the initial conditions. \\medskip \n\n\\noindent (4${^\\circ}$) By the definitions in \\eqref{def:kell}--\\eqref{def:|||}, we have\n\\begin{align*}\n&{\\quad \\,} ({\\overline{\\kappa}}_{(\\ell_1,\\ell_2)|\\ell_0}-{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2})+({\\overline{\\kappa}}_{(\\ell_0,\\ell_1)|\\ell_2}-{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2})+{\\overline{\\kappa}}_{\\ell_0|\\ell_1|\\ell_2}\\\\\n&=\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}\\leq s_n t,M_{U_{\\ell_0},U_{\\ell_2}}>s_n t)\\\\\n&\\quad\\, +\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1) \\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_1}}\\leq s_n t,M_{U_{\\ell_1},U_{\\ell_2}}> s_n t,M_{U_{\\ell_0},U_{\\ell_2}}>s_n t)\\\\\n&\\quad \\,+\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_1}}>s_n t,M_{U_{\\ell_1},U_{\\ell_2}}>s_nt,M_{U_{\\ell_0},U_{\\ell_2}}>s_nt)\\\\\n&=\\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_{\\ell_0},U_{\\ell_2}}>s_n t)={\\overline{\\kappa}}_{|\\ell_2-\\ell_0|}.\n\\end{align*}\nHere, the next to the last equality follows since on $\\{M_{U_{\\ell_0},U_{\\ell_2}}>s_n t\\}$, we cannot have both $M_{U_{\\ell_0},U_{\\ell_1}}\\leq s_n t$ and $M_{U_{\\ell_1},U_{\\ell_2}}\\leq s_n t$ by the coalescence of the Markov chains, and the last equality follows from the stationarity of the chain $\\{U_\\ell\\}$. The proof is complete.\n\\end{proof}\n\n\n\n\nWe close this subsection with another application of Lemma~\\ref{lem:MT}. It will be used in Section~\\ref{sec:eqn}.\n\n\\begin{prop}\\label{prop:Mcompare}\nLet $s_0\\in (2,\\infty)$.\nFor all integers $\\ell\\geq 1$ and all $t\\in (0,\\infty)$, it holds that \n\\begin{align}\\label{ineq:Cell}\n\\int_0^{t} \\P(M_{U_0,U_\\ell}>s_0s)\\d s\\leq \\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\int_0^{2j t} \\P(M_{U_0,U_1}>s_0s)\\d s,\n\\end{align}\nwhere $\\prod_{k=i}^ja_k\\equiv 1$ for $js_0s)\\d s\\\\\n&\\leq \n\\P(M_{U_0,U_\\ell}>s_0 r)+\\int_0^{ r}2s_0 {\\rm e}^{-2s_0 ( r-s)}\\P(M_{U_0,U_1}>s_0s)\\d s.\n\\end{split}\n\\end{align}\nThe assumption $s_0\\in (2,\\infty)$ gives $t\\leq 2t(1-s_0^{-1})$, and so, for $h$ nonnegative and Borel measurable, \n\\begin{align}\n\\begin{split}\\label{Mcompare:2}\n &{\\quad \\,} (1-{\\rm e}^{-4t})\\int_0^{t}h(s_0 s)\\d s\\leq \\int_0^{2t(1-s_0^{-1})}h(s_0 s)(1-{\\rm e}^{-4t})\\d s\\\\\n &\\leq \\int_0^{2t} h(s_0 s)\\big(1-{\\rm e}^{-2s_0(2t-s)}\\big)\\d s=\\int_0^{2t}\\int_0^r 2s_0 {\\rm e}^{-2s_0(r- s)}h(s_0 s)\\d s\\d r \\\\&\\leq \\int_0^{2t} h(s_0s)\\d s.\n \\end{split}\n\\end{align}\nIntegrating both sides of \\eqref{Mcompare:1} over $[0,2t]$ and applying the first and last inequalities in \\eqref{Mcompare:2} give\n\\begin{align}\n&{\\quad \\,}(1-{\\rm e}^{-4t})\n\\int_0^{t} \\P(M_{U_0,U_{\\ell+1}}>s_0 s)\\d s\\notag\\\\\n&\\leq \n\\int_0^{2t} \\P(M_{U_0,U_\\ell}>s_0 s)\\d s+\\int_0^{2t} \\P(M_{U_0,U_1}>s_0 s)\\d s \\notag\\\\\n&\\leq \\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+2}t}\\big)^{-1}\\int_0^{2^{j+1} t} \\P(M_{U_0,U_1}>s_0s)\\d s +\\int_0^{2t} \\P(M_{U_0,U_1}>s_0 s)\\d s\\notag\\\\\n&\\leq \\sum_{j=2}^{\\ell+1} \\prod_{k=2}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\int_0^{2^{j} t} \\P(M_{U_0,U_1}>s_0s)\\d s +\\int_0^{2t} \\P(M_{U_0,U_1}>s_0 s)\\d s,\\label{Mcompare:final}\n\\end{align}\nwhere the second inequality follows from induction. Dividing both sides of \\eqref{Mcompare:final} by $(1-{\\rm e}^{-4t})$ proves \\eqref{ineq:Cell} for $\\ell$ replaced by $\\ell+1$. Hence, \\eqref{ineq:Cell} holds for all $\\ell\\geq 1$ by induction. \n\\end{proof}\n\n\n\\section{Convergence of the vector density processes}\\label{sec:eqn}\nWe present the proofs of Theorem~\\ref{thm:main} and Corollary~\\ref{cor:symmetric} in this section. The key result is Proposition~\\ref{prop:duhamel} where we reduce the evolutionary game model to the voter model. {\\bf Throughout this section, \n conditions (a)--(d) of Theorem~\\ref{thm:main} are in force.} \n \n \n \n The other settings for this section are as follows. First, we write $I^{(n)}_\\sigma=I_\\sigma(\\theta_n t)$ for the process $I_\\sigma(t)$ defined by \\eqref{def:I}, when the underlying particle system is based on $(E_n,q^{(n)})$. This notation extends to the other processes in the decompositions \\eqref{psigma:dec} by using the same time change. Next, recall that $S$ denotes the type space. We will mostly consider $(\\sigma_0,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ such that $\\sigma_0\\neq \\sigma_2$. These triplets fit into the context of \\eqref{eq:Dsigma}, from which we will prove the limiting replicator equation in Theorem~\\ref{thm:main}. Additionally, given an admissible sequence $(\\theta_n,\\mu_n,w_n)$ such that $\\lim_n \\theta_n\/\\gamma_n=0$, we can choose a slow sequence $(s_n)$ (recall Definition~\\ref{def:slow}) such that \n\\begin{align}\n\\lim_{n\\to\\infty}\\frac{s_n}{\\theta_n}=0.\\label{sn:adm}\n\\end{align} \n\n\n\n\\subsection{Asymptotic closure of equations and path regularity}\\label{sec:closure}\nWe begin by showing that the leading order drift term $I_\\sigma^{(n)}$ in \\eqref{psigma:dec} can be asymptotically closed by the vector density process $(p_{\\sigma}(\\xi_{\\theta_nt});\\sigma\\in S)$. By \\eqref{def:I}, this term takes the following explicit form: \n\\begin{align}\\label{def:In}\n\\begin{split}\nI^{(n)}_\\sigma(t)&=w_n\\theta_n\\int_0^t \\overline{D}_\\sigma(\\xi_{\\theta_n s})\n\\d s\\\\\n&+\\int_0^t \\Bigg(\\theta_n\\mu_n(\\sigma)[1-p_{\\sigma}(\\xi_{\\theta_ns})]-\\theta_n\\mu_n(S\\setminus\\{\\sigma\\}) p_\\sigma(\\xi_{\\theta_ns})\\Bigg)\\d s.\n\\end{split}\n\\end{align}\nSpecifically, in terms of the explicit form of $\\overline{D}_\\sigma$ in \\eqref{eq:Dsigma}, our goal is to prove that \n\\begin{align}\\label{closure}\n\\lim_{n\\to\\infty}\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^{w_n}_{\\xi}\\left[\\left|\\int_0^tw_n\\theta_n \\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{\\theta_ns})-w_\\infty Q_{\\sigma_0,\\sigma_2\\sigma_3}\\big(p(\\xi_{\\theta_ns })\\big)\\d s \\right|\\right]=0,\n\\end{align}\nwhere $\\sigma_0\\neq \\sigma_2$, $w_\\infty$ is defined by \\eqref{def:wn}, and $Q_{\\sigma_0,\\sigma_2\\sigma_3}(X)$ is a polynomial in $X=(X_\\sigma)_{\\sigma\\in S}$ defined by\n\\begin{align}\n\\begin{split}\n\\textcolor{black}{Q_{\\sigma_0,\\sigma_2\\sigma_3}(X)}&\\stackrel{\\rm def}{=}\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}({\\overline{\\kappa}}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})X_{\\sigma_0}X_{\\sigma_2}\\\\\n&\\quad +\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}(\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3})X_{\\sigma_0}X_{\\sigma_2}\\\\\n&\\quad +\\overline{\\kappa}_{0|2|3}X_{\\sigma_0}X_{\\sigma_2}X_{\\sigma_3}.\n\\end{split}\n\\label{def:Qsigma}\n\\end{align}\nThe choice of $Q_{\\sigma_0,\\sigma_2\\sigma_3}$ is due to the proof of Lemma~\\ref{lem:RW}.\n\nThe proof of \\eqref{closure} begins with an inequality central to the proof of \\cite[Theorem~2.2]{CCC}, which goes back to \\cite{CMP} and is also central to the proof of \\cite[Lemma~4.2]{CC}. This inequality is presented in a general form for future references. In what follows, we write $a\\wedge b$ for $\\min\\{a,b\\}$. \n\n\n\\begin{prop}\\label{prop:L2}\nGiven a Polish space $E_0$ and $T\\in (0,\\infty)$, let $(X_t)_{0\\leq t\\leq T}$ be an $E_0$-valued Markov process with c\\'adl\\'ag paths. Let $f$ and $g$ be bounded Borel measurable functions defined on $E_0$. Suppose that $x\\mapsto {\\mathbb E}_x[f(X_t)]$ is Borel measurable, and for some bounded decreasing function $a(t)$, \n\\begin{align}\\label{def:a(t)}\n\\sup_{x\\in E_0}{\\mathbb E}_x[|f(X_t)|]\\leq a(t),\\quad \\forall\\;t\\in [0,T]. \n\\end{align}\nThen for all $0<2\\deltas_0 s)\\d s\\\\\n\\leq & \nC_{\\ref{UVbdd:1}}\\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right) \\left[\\ell t+\\min\\Bigg\\{ \\frac{1}{\\mathbf g_ns_0}, \\frac{\\mathbf t^{(n)}_{\\rm mix}}{s_0}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]\\Bigg\\}\\right],\\label{UVbdd:1}\n\\end{split}\n\\end{align}\nwhere $C_{\\ref{UVbdd:1}}$ is a universal constant.\n\\end{lem}\n\\begin{proof}\nBy \\eqref{ineq:MUVcompare} and Proposition~\\ref{prop:Mcompare}, we obtain the following inequality:\n\\begin{align}\n&{\\quad \\,} 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{t}\\P^{(n)}(M_{U_0,U_\\ell}>s_0 s)\\d s\\notag\\\\\n&\\leq\\frac{\\gamma_n}{s_0}\\cdot \\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right)\n2s_0\\nu_n(\\mathds 1)\\int_0^{2\\ell t}\\P^{(n)}(M_{V,V'}>s_0 s)\\d s\\notag\\\\\n\\begin{split}\n&=\\frac{\\gamma_n}{s_0} \\cdot \\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right)\\\\\n&\\quad \\times \\left[\\P^{(n)}(M_{U,U'}>0)-\\P^{(n)}(M_{U,U'}>2\\ell s_0 t)\\right]\\label{UVbdd:00}\n\\end{split}\\\\\n\\begin{split}\n&\\leq C_{\\ref{UVbdd:0}}\\frac{\\gamma_n}{s_0}\\cdot \\Bigg(\\sum_{j=1}^\\ell \\prod_{k=1}^{j-1} \\big(1-{\\rm e}^{-2^{k+1}t}\\big)^{-1}\\Bigg)\\left(\\frac{\\pi^{(n)}_{\\max}}{\\pi^{(n)}_{\\min}}\\right)\\\\\n&{\\quad \\,}\\times \\left[\\big(1-{\\rm e}^{-2\\ell s_0t\/\\gamma_n}\\big)+\\min\\Bigg\\{ \\frac{2}{\\widetilde{\\mathbf g}_n\\gamma_n}, \\frac{\\widetilde{\\mathbf t}^{(n)}_{\\rm mix}}{\\gamma_n}\\big[1+\\log^+(\\gamma_n\/\\widetilde{\\mathbf t}^{(n)}_{\\rm mix})\\big]\\Bigg\\}\\right] \\label{UVbdd:0}\n\\end{split}\n\\end{align}\nfor a universal constant $C_{\\ref{UVbdd:0}}$. Here, \\eqref{UVbdd:00} follows from \\eqref{ergodic}, and \\eqref{UVbdd:0} follows from the exponential approximation of $M_{U,U'}$ as in the proof of Proposition~\\ref{prop:sn-selection}. Recall the reduction of mixing of products chains to mixing of the coordinates as used in that proposition, and the inequality $1-{\\rm e}^{-x}\\leq x$ holds for all $x\\geq 0$. Hence, we obtain \\eqref{UVbdd:1} from \\eqref{UVbdd:0}. The proof is complete. \n\\end{proof}\n\n\n\n\n\\begin{prop}\\label{prop:duhamel}\nFix $(\\sigma_0,\\sigma_1,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ such that $\\sigma_0\\neq \\sigma_2$ and $\\sigma_0\\neq \\sigma_1$. \\medskip \n\n\\noindent {\\rm (1${^\\circ}$)}\nFor any $w\\in [0,\\overline{w}]$ and $t\\in(0,\\infty)$, the following estimates of the evolutionary game by the voter model holds: for some constant $C_{\\ref{LwL:0}}$ depending only on $\\Pi$,\n\\begin{align}\n&\\quad \n\\sup_{\\xi\\in S^{E}}\\Big|{\\mathbb E}^{w}_\\xi\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{t})\\big]-{\\mathbb E}^0_\\xi\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{t})\\big]\\Big|\\notag\\\\\n\\begin{split}\n&\\leq C_{\\ref{LwL:0}} w\\int_0^t \\P(M_{U_0,U_2}>s)\\d s\n+C_{\\ref{LwL:0}}w\\mu(\\mathds 1)\\int_0^t\\int_0^s\\P(M_{U_0,U_2}>r)\\d r\\d s;\\label{LwL:0}\n\\end{split}\\\\\n&\\quad \n\\sup_{\\xi\\in S^{E}}\\Big|{\\mathbb E}^{w}_\\xi\\big[\\,\\overline{ f_{\\sigma_0\\sigma_1}}(\\xi_{t})\\big]-{\\mathbb E}^0_\\xi\\big[\\,\\overline{ f_{\\sigma_0\\sigma_1}}(\\xi_{t})\\big]\\Big|\\notag\\\\\n\\begin{split}\n&\\leq C_{\\ref{LwL:0}} w\\int_0^t \\P(M_{U_0,U_1}>s)\\d s\n+C_{\\ref{LwL:0}}w\\mu(\\mathds 1)\\int_0^t\\int_0^s\\P(M_{U_0,U_1}>r)\\d r\\d s.\\label{LwL:001}\n\\end{split}\n\\end{align}\n \\medskip \n\n\\noindent {\\rm (2${^\\circ}$)} For any admissible sequence $(\\theta_n,\\mu_n,w_n)$ and $T\\in(0,\\infty)$, it holds that \n\\begin{align}\\label{LwL:1}\n\\begin{split}\n&\\lim_{n\\to\\infty}\\int_0^T\\sup_{\\xi\\in S^{E_n}}\\left|{\\mathbb E}^{w_n}_{\\xi}\\left[w_n\\theta_n \\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{2s_n t})\\right]-{\\mathbb E}^{0}_{\\xi}\\left[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{2s_n t})\\right]\\right|\\d t=0.\n\\end{split}\n\\end{align}\n\\end{prop}\n\\begin{proof}\n(1${^\\circ}$) Recall that the generator of $\\mathsf L^w$ of the evolutionary game is given by \\eqref{def:Lw}, and $\\mathsf L=\\mathsf L^0$ denotes the generator of the voter model. By Duhamel's principle \\cite[(2.15) in Chapter~1]{EK:MP},\n\\begin{align}\\label{expansion}\n{\\rm e}^{t\\mathsf L^w}H={\\rm e}^{t\\mathsf L }H+\\int_0^t{\\rm e}^{(t-s)\\mathsf L^w}(\\mathsf L^w-\\mathsf L){\\rm e}^{s\\mathsf L}H\\d s .\n\\end{align}\nHere, it follows from \\eqref{def:Lw} that\n\\begin{align}\n(\\mathsf L^w-\\mathsf L)H_1(\\xi)&=\\sum_{x,y\\in E}[q^w(x,y,\\xi)-q(x,y)][H_1(\\xi^{x,y})-H_1(\\xi)].\n\\label{LwL1}\n\\end{align}\nTo apply \\eqref{expansion} and \\eqref{LwL1}, we choose $H=\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}$ and $H_1={\\rm e}^{s\\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}$. The following bound will be proved in Proposition~\\ref{prop:mutation} (2${^\\circ}$):\n\\begin{align}\\label{claim:w0}\n\\begin{split}\n&\\sup_{\\xi\\in S^E}|{\\rm e}^{s\\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi^x)-{\\rm e}^{s\\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi)|\\\\\n&\\leq \n\\sum_{\\ell\\in \\{0,2,3\\}}4\\P(M_{U_0,U_2}>s,B^{U_\\ell}_{s}= x)\\\\\n&\\quad +\\sum_{\\ell\\in \\{0,2,3\\}}4\\mu(\\mathds 1)\\int_0^s \\P(M_{U_0,U_2}>r,B^{U_\\ell}_s=x)\\d r.\n\\end{split}\n\\end{align}\n\n\n\nTo bound $(\\mathsf L^w-\\mathsf L){\\rm e}^{s\\mathsf L}H=(\\mathsf L^w-\\mathsf L)H_1$ in the expansion \\eqref{expansion}, notice that \n\\begin{align}\\label{qwq:bdd}\n|q^w(x,y,\\xi)-q(x,y)|\\leq C_{\\ref{qwq:bdd}}wq(x,y)\n\\end{align}\nby \\eqref{qwq:exp} for some $C_{\\ref{qwq:bdd}}$ depending only on $\\Pi$. Putting \\eqref{LwL1}, \\eqref{claim:w0} and \\eqref{qwq:bdd} together, we get\n\\begin{align*}\n&\\quad \\sup_{\\xi\\in S^E}| (\\mathsf L^w-\\mathsf L){\\rm e}^{s \\mathsf L}\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi)|\\\\\n&\\leq C_{\\ref{qwq:bdd}} 4w\\sum_{\\ell\\in \\{0,2,3\\}}\\sum_{x,y\\in E}q(x,y)\\P(M_{U_0,U_2}>s,B^{U_\\ell}_s=x)\\\\\n&{\\quad \\,} +C_{\\ref{qwq:bdd}} 4w\\mu(\\mathds 1)\\sum_{\\ell\\in \\{0,2,3\\}}\\sum_{x,y\\in E}q(x,y)\\int_0^s \\P(M_{U_0,U_2}>r,B^{U_\\ell}_s=x)\\d r\\\\\n&\\leq C_{\\ref{qwq:bdd}}12 w\\P(M_{U_0,U_2}>s)+C_{\\ref{qwq:bdd}}12 w\\mu(\\mathds 1)\\int_0^s \\P(M_{U_0,U_2}>r)\\d r.\n\\end{align*}\nSince ${\\rm e}^{(t-s)\\mathsf L^w}$ is a probability, the required inequality in \\eqref{LwL:1} follows upon applying the foregoing inequality to \\eqref{expansion}. We have proved \\eqref{LwL:0}. The proof of \\eqref{LwL:001} is almost the same if we use Proposition~\\ref{prop:mutation} (3${^\\circ}$) instead of Proposition~\\ref{prop:mutation} (2${^\\circ}$). The details are omitted. \\medskip \n\n\\noindent (2${^\\circ}$) By the first limit in \\eqref{def:wn} and \\eqref{LwL:0}, it is enough to show that all of the following limits hold:\n\\begin{align}\n\\label{problem:wn1}\n&\\lim_{n\\to\\infty}\nw_n(2s_n)\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{T} \\P^{(n)}(M_{U_0,U_2}>2s_n s)\\d s=0;\\\\\n&\\lim_{n\\to\\infty}\n[w_n(2s_n)+1]\\cdot \\mu_n(\\mathds 1)(2s_n)\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{T}\\P^{(n)}(M_{U_0,U_2}>2s_n s)\\d s=0\\label{problem:wn2}.\n\\end{align}\n(The limit \\eqref{problem:wn2} is stronger than needed but is convenient for the other proofs below.)\n\nTo get \\eqref{problem:wn1}, first, note that by \\eqref{cond2:sn}, \\eqref{sn:adm} and the limit superior in \\eqref{def:wn},\n\\begin{align}\\label{problem:wn1-1}\n\\lim_{n\\to\\infty}w_n(2s_n)\\cdot \\left[T+\\min\\Bigg\\{ \\frac{1}{\\mathbf g_n(2s_n)}, \\frac{\\mathbf t^{(n)}_{\\rm mix}}{2s_n}[1+\\log^+(\\gamma_n\/\\mathbf t^{(n)}_{\\rm mix})]\\Bigg\\}\\right]=0.\n\\end{align}\nWe get \\eqref{problem:wn1} from applying \\eqref{cond:pi} and \\eqref{problem:wn1-1} to \\eqref{UVbdd:1} with $s_0=2s_n$.\nFor \\eqref{problem:wn2}, \n $\\lim_n\\mu_n(\\mathds 1)(2s_n)=0$ by \\eqref{def:mun} and \\eqref{sn:adm}. The limit superior in \\eqref{def:wn}\n and \\eqref{sn:adm} give $\\limsup_n w_n(2s_n)<\\infty$. These two properties are enough for \\eqref{problem:wn2}.\nThe proof is complete. \n\\end{proof}\n\nTo satisfy \\eqref{def:a(t)} under the setting of \\eqref{setup}, we consider the sum of the right-hand side of \\eqref{LwL:0}, with $t$ replaced by $\\theta_n t$, and $\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^0_\\xi\\big[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{\\theta_n t})\\big]$. Moreover, this supremum can be bounded by using \\eqref{ineq:mutation} and $\\P(M_{U_0,U_2}>\\theta_n t)$, thanks to duality and the choice $\\sigma_0\\neq \\sigma_2$. Therefore, given $T\\in(0,\\infty)$, we set $a(t)=a_n(t)=\\sum_{\\ell=1}^3 a_{n,\\ell}(t)$ for $t\\in [0,T]$, where\n\\begin{align}\\label{def:an(t)}\n\\begin{split}\na_{n,\\ell}(t)&\\stackrel{\\rm def}{=} C_{\\ref{def:an(t)}} \\cdot \n \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot w_n\\theta_n\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T \\P^{(n)}(M_{U_0,U_2}>\\theta_n s)\\d s\\\\\n&{\\quad \\,} +C_{\\ref{def:an(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_n t) \\\\\n&{\\quad \\,} +C_{\\ref{def:an(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot (w_n\\theta_n+1)\\cdot \n\\mu_n(\\mathds 1)\\theta_n\\\\\n&\\quad \\quad \\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_ns)\\d s\n\\end{split}\n\\end{align}\nand $C_{\\ref{def:an(t)}}$ depends only on $(\\Pi,T)$. \nFor any $n\\geq 1$, $t\\mapsto a_{n}(t)$ is bounded and decreasing on $[0,T]$, and\n\\begin{align*}\n\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^{w_n}_\\xi\\left[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{\\theta_nt})\\right]&\n\\leq a_n(t),\\quad \\forall\\;t\\in [0,T].\n\\end{align*}\nHence, the conditions of $a_n(t)$ required in Proposition~\\ref{prop:L2} hold. \n\n\nFor the proof of \\eqref{closure}, the next step is to show that under the setting of \\eqref{setup} and the above choice of $a(t)=a_n(t)$, the right-hand side of \\eqref{ineq:L2} vanishes as $n\\to\\infty$. For the first term on the right-hand side of \\eqref{ineq:L2}, proving $\\int_0^{\\delta_n}a_n(t)\\d t$ amounts to proving $\\int_0^{\\delta_n}a_{n,\\ell}(t)\\d t\\to 0$ for all $1\\leq \\ell\\leq 3$. For the latter limits, note that $\\delta_n\\to 0$ by \\eqref{sn:adm}. Also, a slight modification of the proofs of \\eqref{problem:wn1}--\\eqref{problem:wn2} shows that for the right-hand side of \\eqref{def:an(t)}, the first and last terms in are bounded in $n$, and the second term satisfies \n\\begin{align*}\n&\\quad \\lim_{n\\to\\infty}2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{\\delta_n}\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_n s)\\d s\\\\\n&=\\lim_{n\\to\\infty}\\frac{2s_n}{\\theta_n}\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^{1}\\P^{(n)}(M_{U_0,U_\\ell}>2s_n s)\\d s=0.\n\\end{align*}\nFor the second term in \\eqref{ineq:L2}, it is enough to show that $a_n(\\delta_n)$'s are bounded. From the above argument for the first term in \\eqref{ineq:L2}, this property follows if we use the second limit in \\eqref{def:wn} and note that\n\\[\n2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>\\theta_n t)|_{t=\\delta_n}=2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>2s_n )\\xrightarrow[n\\to\\infty]{} \\overline{\\kappa}_\\ell,\n\\]\nwhere the limit follows from Proposition~\\ref{prop:sn-selection} and Proposition~\\ref{prop:kell}. To use these propositions precisely, passing the foregoing limit actually requires that given any subsequence of $(E_n,q^{(n)})$, a suitable further subsequence is used. To lighten the exposition, we continue to suppress similar uses of subsequential limits. \n\n\nFor the third term in \\eqref{ineq:L2}, note that $\\delta_n\\to 0$ by \\eqref{sn:adm}, and the $g_n$'s in \\eqref{setup} are uniformly bounded in $n$. The last term in \\eqref{ineq:L2} is the major term. By \\eqref{setup} and Proposition~\\ref{prop:duhamel} (2${^\\circ}$), it remains to prove\n\\begin{align}\\label{voter:density}\n\\lim_{n\\to\\infty}\\sup_{\\xi\\in S^{E_n}}\\big|{\\mathbb E}^{0}_\\xi[w_n\\theta_n\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_{2s_n})]-w_\\infty Q_{\\sigma_0,\\sigma_2\\sigma_3}\\big(p(\\xi)\\big)\\big|=0.\n\\end{align}\nFor the next lemma, recall that the total variation distance $d_E$ and the spectral gap $\\mathbf g$ are defined at the beginning of Section~\\ref{sec:mainresults}. Also, here and in what follows, we use the shorthand notation ${\\mathbb E}[Z;A]={\\mathbb E}[Z\\mathds 1_A]$. \n\n\n\n\\begin{lem}\\label{lem:RW}\nFix $(\\sigma_0,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ such that $\\sigma_0\\neq \\sigma_2$. \\medskip \n\n\\noindent {\\rm (1${^\\circ}$)}\nGiven any $0s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad+\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad -\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad +\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\\\\n&\\quad\\quad -\\P(M_{U_0,U_2}>s,M_{U_2,U_3}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)p_{\\sigma_3}(\\xi)\n\\Big|\\\\\n&\\leq C_{\\ref{ineq:RW}}\\sum_{\\ell=1}^3\\Gamma_\\ell(s,t),\\label{ineq:RW}\n\\end{split}\n\\end{align} \nwhere $C_{\\ref{ineq:RW}}$ is a universal constant and\n\\begin{align}\n\\begin{split}\\label{ineq:Well_TV}\n\\Gamma_\\ell(s,t)&\\stackrel{\\rm def}{=} \\P(M_{U_0,U_\\ell}\\in (s,t])\\\\\n&{\\quad \\,}+\\min\\left\\{\\sqrt{\\frac{\\pi_{\\max}}{\\nu(\\mathds 1)}}{\\rm e}^{-\\mathbf g(t-s)},\\P(M_{U_0,U_\\ell}>s)d_E(t-s)\\right\\}\\\\\n&{\\quad \\,} + \\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{U_0,U_\\ell}>t)+\n\\mu(\\mathds 1)\\int_0^t\\P(M_{U_0,U_\\ell}>r)\\d r.\n\\end{split}\n\\end{align}\n{\\rm (2${^\\circ}$)} The limit in \\eqref{voter:density} holds. \n\\end{lem}\n\n\n\n\\begin{proof}\n(1${^\\circ}$) First, we consider the case that there is no mutation. Roughly speaking, the method of this proof is to express ${\\mathbb E}_\\xi^0\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\big]$ in terms of coalescing Markov chains before any two coalesce. This way we can express the coalescing Markov chains as independent Markov chains and compute the asymptotics of ${\\mathbb E}_\\xi^0\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\big]$ by the ${\\overline{\\kappa}}$-constants defined in \\eqref{def:kell}--\\eqref{def:|||}. This idea goes back to \\cite[Proposition~6.1]{CCC}.\n\n\nNow, by duality and the assumption $\\sigma_0\\neq \\sigma_2$, it holds that \n\\begin{align}\n&\\quad\\,{\\mathbb E}_\\xi^0\\big[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\big]\\notag\\\\\n&={\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\n\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t)\\mathds 1_{\\sigma_3}\\circ \\xi(B^{U_3}_t)]\\notag\\\\\n&=\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}\\leq t,M_{U_0,U_3}>t]\\notag\\\\\n&\\quad +\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}\\leq t]\\notag\\\\\n&\\quad +{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t)\\mathds 1_{\\sigma_3}\\circ \\xi(B^{U_3}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}>t]\\notag\\\\\n&=\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}{\\rm I}+\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}{\\rm II}+{\\rm III}.\\label{def:III}\n\\end{align}\nWe can further write ${\\rm I}$ and ${\\rm II}$ as\n\\begin{align}\n\\begin{split}\\label{def:Iterm}\n{\\rm I}&={\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_0,U_3}>t]\\\\\n&\\quad \\,-{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}>t]\\\\\n&={\\rm I'}-{\\rm I''},\n\\end{split}\\\\\n\\begin{split}\\label{def:IIterm}\n{\\rm II}&={\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t]\\\\\n&\\quad\\, -{\\mathbb E}[\\mathds 1_{\\sigma_0}\\circ \\xi(B^{U_0}_t)\\mathds 1_{\\sigma_2}\\circ \\xi(B^{U_2}_t);M_{U_0,U_2}>t,M_{U_2,U_3}> t,M_{U_0,U_3}> t]\\\\\n&={\\rm II'}-{\\rm I''}.\n\\end{split}\n\\end{align}\nWe estimate ${\\rm I'},{\\rm I''}, {\\rm II'}$ and ${\\rm III}$ below, using the property that the coalescing Markov chains move independently before meeting.\n\n\n\nFirst, for $0s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\big|\\\\\n&\\leq{\\mathbb E}\\Big[\\left|{\\rm e}^{(t-s)(q-1)}\\mathds 1_{\\sigma_0}\\circ\\xi(B_s^{U_0}){\\rm e}^{(t-s)(q-1)}\\mathds 1_{\\sigma_2}\\circ\\xi(B_s^{U_2})-p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\right|\\\\\n&\\quad \\quad ;M_{U_0,U_2}>s,M_{U_0,U_3}>s\\Big]\\\\\n&{\\quad \\,}+ \\P(M_{U_0,U_2}\\in (s,t])+\\P(M_{U_0,U_3}\\in (s,t]).\n\\end{split}\n\\end{align}\nOn the event $\\{M_{U_0,U_2}>s,M_{U_0,U_3}>s\\}$, $(B^{U_0}_r)_{0\\leq r\\leq s}$ and $(B^{U_2}_r)_{0\\leq r\\leq s}$ are independent $q$-Markov chains and each chain is stationary by the assumption on $\\{U_\\ell\\}$. Since $p_\\sigma(\\xi)=\\sum_x\\mathds 1_\\sigma\\circ \\xi(x)\\pi(x)$, the expectation in \\eqref{def:I1} can be estimate as in the proof of \\cite[Proposition~6.1]{CCC}. We get\n\\begin{align}\n&{\\quad \\,}\\big|{\\rm I}'-\\P(M_{U_0,U_2}>s,M_{U_0,U_3}>s)p_{\\sigma_0}(\\xi)p_{\\sigma_2}(\\xi)\\big|\\notag\\\\\n&\\leq \n\\min\\left\\{2\\sqrt{\\frac{\\pi_{\\max}}{\\nu(\\mathds 1)}}{\\rm e}^{-\\mathbf g(t-s)},4\\P(M_{U_0,U_2}>s,M_{U_0,U_3}>s)d_E(t-s)\\right\\}\\notag\\\\\n&{\\quad \\,}+ \\P(M_{U_0,U_2}\\in (s,t])+\\P(M_{U_0,U_3}\\in (s,t]).\\notag\n\\end{align}\nSimilar estimates apply to the other terms ${\\rm I''}$, ${\\rm II}'$ and ${\\rm III}$ in \\eqref{def:III}, \\eqref{def:Iterm} and \\eqref{def:IIterm}. \n\n\nApplying all of these estimates to \\eqref{def:III} proves \\eqref{ineq:RW} when there is no mutation. The additional terms in \\eqref{ineq:RW} arise when we include mutation and use \\eqref{ineq:mutation} again.\\medskip\n\n \n\\noindent (2${^\\circ}$) Recall the second limit in \\eqref{def:wn} and \\eqref{problem:wn2}.\nThen by \\eqref{ineq:RW} and \\eqref{ineq:Well_TV} with $t=2s_n$ and $s=s_n$, it suffices to show all of the following limits:\n\\begin{align}\n&\\lim_{n\\to\\infty} 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}\\in (s_n,2s_n])=0,\\quad 1\\leq \\ell\\leq 3;\\label{problem:wn4}\\\\\n&\\lim_{n\\to\\infty}\\big(1-{\\rm e}^{-2\\mu_n(\\mathds 1)\\cdot (2s_n)}\\big)\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>2s_n),\\quad 1\\leq \\ell\\leq 3;\\label{problem:wn5}\\\\\n&\\lim_{n\\to\\infty} \\Gamma_{n,\\ell}=0,\\quad 1\\leq \\ell\\leq 3,\\label{problem:wn6}\n\\end{align}\nwhere $\\Gamma_{n,\\ell}$ is given by the minimum of the following two terms:\n\\begin{align}\\label{def:Gamma}\n\\gamma_n\\nu_n(\\mathds 1)\\cdot \\sqrt{\\frac{\\pi^{(n)}_{\\max}}{\\nu_n(\\mathds 1)}}{\\rm e}^{-\\mathbf g_ns_n},\\quad\n 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{U_0,U_\\ell}>s_n)d_{E_n}(s_n) .\n\\end{align}\n\nTo see \\eqref{problem:wn4}, we simply use Propositions~\\ref{prop:sn-selection} and \\ref{prop:kell}. The limit in \\eqref{problem:wn5} follows from the same propositions, in addition to \\eqref{def:mun} and \\eqref{sn:adm}. For \\eqref{problem:wn6}, we consider the following two cases. When $\\Gamma_n$ is given by the first term in \\eqref{def:Gamma}, the required limit holds by \\eqref{cond:pi} and the first limit in \\eqref{cond2:sn}. When $\\Gamma_n$ is given by the other term in \\eqref{def:Gamma}, we first use Propositions~\\ref{prop:sn-selection} and \\ref{prop:kell}. Then note that the second limit in \\eqref{cond2:sn} implies $\\lim_{n}\\mathbf t^{(n)}_{{\\rm mix}}\/s_n=0$, and so, $\\lim_nd_{E_n}(s_n)=0$ by \\eqref{ineq:tmix}. We have proved \\eqref{problem:wn6}. The proof is complete.\n\\end{proof}\n\nUp to this point, we have proved the asymptotic closure of equation in the sense of \\eqref{closure}. Note that under \\eqref{setup}, the convergence of the last term in \\eqref{ineq:L2} also contributes to asymptotic path regularity of the density processes. \n\nThe next lemma proves the asymptotic path regularity more explicitly as tightness in the convergence results of Theorem~\\ref{thm:main}. The limit of the normalized martingale terms in Theorem~\\ref{thm:main} (2${^\\circ}$) is also proven. Here, recall that the density processes satisfy the decompositions in \\eqref{density:dynamics}. From now on, $\\xrightarrow[n\\to\\infty]{\\rm (d)}$ refers to convergence in distribution as $n\\to\\infty$. \n\n\\begin{lem}\\label{lem:tight}\nFix $\\sigma\\in S$. \n\n\\begin{enumerate}\n\\item [\\rm (1${^\\circ}$)] \nThe sequence of laws of $I_\\sigma^{(n)}$ as continuous processes under $\\P^{w_n}_{\\nu_n}$ is tight.\n\n\n\\item [\\rm (2${^\\circ}$)] \nThe sequence of laws of $\\mathds 1_{\\{w_n>0\\}}w_n^{-1}R_\\sigma^{(n)}$ as continuous processes under $\\P^{w_n}_{\\nu_n}$ is tight.\n\n\\item [\\rm (3${^\\circ}$)] The sequence of laws of $M_\\sigma^{(n)}$ as continuous processes under $\\P^{w_n}_{\\nu_n}$ converges to zero in distribution. \n\\end{enumerate}\nIf, in addition, $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$, then the following holds.\n\\begin{enumerate}\n\\item [\\rm (4${^\\circ}$)] The sequence of laws of \n\\begin{align}\\label{def:product}\n\\left(\\left(\n\\frac{\\gamma_n}{\\theta_n}\\right)^{1\/2}M_\\sigma^{(n)}(t),\\;\\;\n\\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_\\sigma^{(n)}\\rangle_{t}-\\int_0^t p_\\sigma(\\xi_{\\theta_ns})[1-p_{\\sigma}(\\xi_{\\theta_ns})]\\d s;t\\geq 0\\right)\n\\end{align}\nas processes with c\\`adl\\`ag paths under $\\P^{w_n}_{\\nu_n}$ is $C$-tight, and the second coordinates \nconverge to zero in distribution as processes. Moreover, for all $T\\in(0,\\infty)$,\n\\begin{align}\\label{ineq:L2-bdd}\n\\sup_{n\\geq 1}\\sup_{t\\in [0,T]}\\sup_{\\xi\\in S^{E_n}}\\frac{\\gamma_n}{\\theta_n}{\\mathbb E}^{w_n}_{\\xi}\\big[M^{(n)}_\\sigma(t)^2\\big]<\\infty.\n\\end{align}\n\\item [\\rm (5${^\\circ}$)] For any $\\sigma'\\in S$ with $\\sigma'\\neq \\sigma$, the sequence \n\\begin{align}\\label{def:product1}\n\\left(\n\\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_{\\sigma'}^{(n)}\\rangle_{t}+\\int_0^t p_\\sigma(\\xi_{\\theta_ns})p_{\\sigma'}(\\xi_{\\theta_ns})\\d s;t\\geq 0\\right)\n\\end{align}\nunder $\\P^{w_n}_{\\nu_n}$ converges to zero in distribution as processes. \n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n(1${^\\circ}$) First, we show a bound for $\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}_{\\xi}^{w_n}[|I^{(n)}_\\sigma(\\theta)|]$ explicitly in $\\theta$. By \\eqref{eq:Dsigma0},\n\\begin{align}\\label{Dsigma:22}\n|\\overline{D}_\\sigma(\\xi)|\\leq C_{\\ref{Dsigma:22}}\\sum_{\\stackrel{\\scriptstyle \\sigma'\\in S}{\\sigma'\\neq \\sigma}}\\overline{f_{\\sigma\\sigma'}}(\\xi)\n\\end{align}\nfor some constant $C_{\\ref{ineq:Itheta}}$ depending only on $\\Pi$ and $\\#S$. Indeed, for $q(x,y)>0$, $\\mathds 1_\\sigma\\circ\\xi(y)-\\mathds 1_\\sigma\\circ\\xi (x)\\neq 0$ implies that either $\\xi(x)$ or $\\xi(y)$ is $\\sigma$ but not both. By \\eqref{def:In} and \\eqref{Dsigma:22}, \n\\begin{align}\\label{In:continuity}\n\\begin{split}\n\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}_{\\xi}^{w_n}[|I^{(n)}_\\sigma(\\theta)|]&\\leq C_{\\ref{Dsigma:22}}\\sum_{\\stackrel{\\scriptstyle \\sigma'\\in S}{\\sigma'\\neq \\sigma}}\\int_0^\\theta \\sup_{\\xi\\in S^{E_n}}{\\mathbb E}^{w_n}_{\\xi}\\left[w_n\\theta_n\\overline{f_{\\sigma\\sigma'}}(\\xi_{\\theta_n s})\\right]\\d s\\\\\n&\\quad +2\\mu_n(\\mathds 1)\\theta_n\\cdot \\theta.\n\\end{split}\n\\end{align}\nFurthermore, we can bound the expectations on the right-hand side of \\eqref{In:continuity} by using an analogue of the $a_n(t)$ in \\eqref{def:an(t)}, but now involving only the meeting time $M_{V,V'}$. Specifically, given $T\\in(0,\\infty)$, the following inequality holds for all $t\\in [0,T]$:\n\\begin{align}\\label{def:an'(t)}\n\\begin{split}\n&{\\quad \\,}{\\mathbb E}^{w_n}_{\\xi}\\left[w_n\\theta_n\\overline{f_{\\sigma\\sigma'}}(\\xi_{\\theta_n t})\\right]\\\\\n&\\leq C_{\\ref{def:an'(t)}} \\cdot \n \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot w_n\\theta_n\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T \\P^{(n)}(M_{V,V'}>\\theta_n s)\\d s\\\\\n&{\\quad \\,} +C_{\\ref{def:an'(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\P^{(n)}(M_{V,V'}>\\theta_n t) \\\\\n&{\\quad \\,} +C_{\\ref{def:an'(t)}} \\cdot \\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot (w_n\\theta_n+1)\\cdot\n\\mu_n(\\mathds 1)\\theta_n\\\\\n&{\\quad \\,}\\quad \\cdot 2\\gamma_n\\nu_n(\\mathds 1)\\int_0^T\\P^{(n)}(M_{V,V'}>\\theta_ns)\\d s\n\\end{split}\n\\end{align}\nand $C_{\\ref{def:an'(t)}}$ depends only on $(\\Pi,T,\\sup_n\\pi^{(n)}_{\\max}\/\\pi^{(n)}_{\\min})$. To see \\eqref{def:an'(t)}, we combine \\eqref{LwL:001} and \\cite[Proposition~3.2]{CC} and then use \\eqref{cond:pi} and \\eqref{ineq:MUVcompare} to reduce probabilities of $M_{U_0,U_1}$ to probabilities of $M_{V,V'}$.\n\n\nNext, we show that \n\\begin{align}\\label{ineq:Itheta}\n\\lim_{\\theta\\searrow 0}\\limsup_{n\\to\\infty}\\sup_{\\xi\\in S^{E_n}}{\\mathbb E}_\\xi^{w_n}[|I^{(n)}_\\sigma(\\theta)|]=0.\n\\end{align}\nFirst, \\eqref{def:mun} readily gives the required limit of the last term of \\eqref{In:continuity}. We focus on the sum of integrals on the right-hand side of \\eqref{def:mun}. For each of these integrals, note that the first and last terms in \\eqref{def:an'(t)} are uniformly bounded in $n$ as in the case of \\eqref{def:an(t)}. The integral over $t\\in [0,\\theta]$ of the second term in \\eqref{def:an'(t)} satisfies \n\\begin{align}\\label{MVV':conv-tight}\n\\lim_{\\theta\\searrow 0}\\limsup_{n\\to\\infty}\\frac{w_n\\theta_n}{2\\gamma_n\\nu_n(\\mathds 1)}\\cdot \\frac{2\\gamma_n}{\\theta_n}\\nu_n(\\mathds 1)\\int_0^{\\theta\\theta_n} \\P^{(n)}(M_{V,V'}> s)\\d s=0\n\\end{align}\nby the second limit of \\eqref{def:wn}, \\eqref{ergodic}, and \\eqref{fnsn:est} with $s_n$ replaced by $\\theta_n$ and with $t_2=\\theta$ and $t_1=0$ since $(\\theta_n)$ is also a slow sequence. (The use of $M_{V,V'}$ allows us to circumvent Lemma~\\ref{lem:UVbdd} due to the explosion of the bound in \\eqref{UVbdd:1} as $t\\to 0$.) We have proved \\eqref{ineq:Itheta}.\n\n\n\nFinally, the required tightness follows from \\eqref{ineq:Itheta}, the strong Markov property of the particle system and Aldous's criterion for tightness \\cite[Proposition~VI.4.5 on p.356]{JS}. The detail is similar to the proof of \\cite[Theorem~5.1 (1)]{CCC}. \\medskip\n\n\\noindent {\\rm (2${^\\circ}$)} Recall the equation \\eqref{def:R} of $R_\\sigma$, and the explicit form of $R^w$ can be read from \\eqref{qwq:exp}. Then by the same reason for \\eqref{Dsigma:22}, the coefficient of $R_\\sigma$ satisfies \n\\begin{align}\\label{bdd:R}\n\\sum_{x,y\\in E}\\pi(x)|\\mathds 1_\\sigma\\circ\\xi(y)-\\mathds 1_\\sigma\\circ\\xi (x)|q(x,y)|R^w(x,y,\\xi)|\\leq C_{\\ref{bdd:R}}\\sum_{\\sigma'\\in S}\\overline{f_{\\sigma\\sigma'}}(\\xi),\n\\end{align}\nwhere $C_{\\ref{bdd:R}}$ depends only on $\\Pi$ and $\\#S$. From \\eqref{bdd:R},\n the argument in (1${^\\circ}$) applies again. \n\\medskip\n\n\n\\noindent {\\rm (3${^\\circ}$)} The proof follows from a slight modification of the proof of (4${^\\circ}$) below even without the additional assumption $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$. \\medskip \n\n\\noindent {\\rm (4${^\\circ}$)}\nWe start with the convergence of the second coordinate in \\eqref{def:product}. Define a density function $\\widetilde{p}_{\\sigma}(\\xi)$ on $S^{E_n}$ such that the stationary weights $\\pi^{(n)}(x)$ in $p_\\sigma(\\xi)$ are replaced by $\\pi^{(n)}(x)^2\/\\nu_n(\\mathds 1)$. From \\eqref{def:}, the following equality holds under $\\P^{w_n}_{\\xi}$ for all $\\xi\\in S^{E_n}$:\n\\begin{align}\\label{eq:Mn}\n\\begin{split}\n&\\quad \\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_{\\sigma}^{(n)}\\rangle_t\\\\\n&=\\gamma_n\\nu_n(\\mathds 1)\\int_0^t \\sum_{\\sigma'\\in S\\setminus\\{\\sigma\\}}[p_{\\sigma'\\sigma}(\\xi_{\\theta_ns})+p_{\\sigma\\sigma'}(\\xi_{\\theta_ns})]\\d s\\\\\n&\\quad +\\gamma_n\\nu_n(\\mathds 1)\\int_0^t \\Big([1-\\widetilde{p}_{\\sigma}(\\xi_{\\theta_ns})]\\mu_n(\\sigma)+\\widetilde{p}_\\sigma(\\xi_{\\theta_ns})\\mu_n(S\\setminus\\{\\sigma\\})\\Big)\\d s\\\\\n&\\quad + \\frac{\\gamma_n\\nu_n(\\mathds 1)}{\\theta_n}\\cdot w_n\\theta_n\\int_0^{t} \\widetilde{R}^{(n)}_{w_n}(\\xi_{\\theta_n s})\\d s.\n\\end{split}\n\\end{align}\nHere, $\\widetilde{R}^{(n)}_{w_n}$ can be bounded in the same way as \\eqref{Dsigma:22}. Note that due to the use of $\\widetilde{R}^{(n)}_{w_n}$, we only involve the first term $q(x,y)$ in the expansion \\eqref{qwq:exp} of $q^{w_n}(x,y,\\xi)$. \n\nLet us explain how the required convergence of the second coordinate in \\eqref{def:product} follows \\eqref{eq:Mn}. First, since $\\lim_n \\gamma_n\\nu_n(\\mathds 1)\/\\theta_n=0$ by assumption, the bound for $\\widetilde{R}^{(n)}_{w_n}$ mentioned above and the proof of (1${^\\circ}$) show that the continuous process defined by the last integral of \\eqref{eq:Mn} converges to zero in distribution. Next, for all $0\\leq T_0}. The details are omitted. \n\\end{proof}\n\n\n\n\n\\subsection{The replicator equation and the Wright--Fisher fluctuations}\nIn this subsection, we complete the proof of Theorem~\\ref{thm:main} and give the proof of Corollary~\\ref{cor:symmetric}.\\\\\n\n\n\\begin{proof}[Completion of the proof of Theorem~\\ref{thm:main}]\nBy \\eqref{eq:Dsigma}, \\eqref{def:In}, \\eqref{closure} and Lemma~\\ref{lem:tight} (1${^\\circ}$)--(3${^\\circ}$), we have proved that the following vector process converges to zero in distribution:\n\\begin{align*}\n&p_\\sigma(\\xi_{\\theta_n t})-\\int_0^tw_\\infty\\Bigg(\\sum_{\\stackrel{\\scriptstyle \\sigma_0,\\sigma_3\\in S}{ \\sigma_0\\neq\\sigma}}\\Pi(\\sigma,\\sigma_3) Q_{\\sigma_0,\\sigma\\sigma_3}\\big(p(\\xi_{\\theta_n s})\\big)-\\sum_{\\stackrel{\\scriptstyle \\sigma_2,\\sigma_3\\in S}{\\sigma_2\\neq\\sigma}}\\Pi(\\sigma_2,\\sigma_3) Q_{\\sigma,\\sigma_2\\sigma_3}\\big(p(\\xi_{\\theta_n s})\\big)\\Bigg)\\d s\\\\\n&{\\quad \\,}-\\int_0^t \\Bigg(\\mu_\\infty(\\sigma)[1-p_{\\sigma}(\\xi_{\\theta_ns})]-\\mu_\\infty(S\\setminus\\{\\sigma\\}) p_\\sigma(\\xi_{\\theta_ns})\\Bigg)\\d s,\\quad \\sigma\\in S,\n\\end{align*}\nwhere the polynomials $Q_{\\sigma_0,\\sigma_2\\sigma_3}$ are defined in \\eqref{def:Qsigma}. Hence, the sequence of laws of $p(\\xi_{\\theta_nt})$ is $C$-tight, and $p(\\xi_{\\theta_nt})$ converges in distribution to $X(t)$ as processes, where $X$ is the unique solution to the following system:\n\\begin{align}\\label{p1:lim0}\n\\dot{X}_\\sigma=w_\\infty Q_\\sigma(X)+\\mu_\\infty(\\sigma)(1-X_\\sigma)-\\mu_\\infty(S\\setminus\\{\\sigma\\}) X_\\sigma,\\quad \\sigma\\in S,\n\\end{align}\nand the polynomial $Q_\\sigma(X)$ in \\eqref{p1:lim0} is given by\n\\begin{align}\\label{def:grandQ}\nQ_\\sigma(X)&=\\sum_{\\stackrel{\\scriptstyle \\sigma_0,\\sigma_3\\in S}{\\sigma_0\\neq \\sigma}}\\Pi(\\sigma,\\sigma_3)Q_{\\sigma_0,\\sigma\\sigma_3}(X)-\\sum_{ \\stackrel{\\scriptstyle \\sigma_2,\\sigma_3\\in S}{\\sigma_2\\neq \\sigma}}\\Pi(\\sigma_2,\\sigma_3)Q_{\\sigma,\\sigma_2\\sigma_3}(X).\n\\end{align}\n\n\nTo simplify \\eqref{def:grandQ} to the required form in \\eqref{p1:lim}, note that the constraints $\\sigma_0\\neq \\sigma$ and $\\sigma_2\\neq \\sigma$ in \\eqref{def:grandQ} can be removed from the definition of $Q_\\sigma(X)$ by cancelling repeating terms. In doing so, we extend the definition $Q_{\\sigma_0,\\sigma_2\\sigma_3}(X)$ to $\\sigma_0=\\sigma_2$ by the same formula in \\eqref{def:Qsigma}, but only in this proof. We also lighten notation by the following: $A=\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3}$ and $B=\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3}$ and $C=\\overline{\\kappa}_{0|2|3}$. Then by \\eqref{def:grandQ},\n\\begin{align*}\n&Q_\\sigma(X)=\\sum_{\\sigma_0,\\sigma_3\\in S}\\Pi(\\sigma,\\sigma_3)\\left.\\left(\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}AX_{\\sigma_0}X_{\\sigma_2}+\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}BX_{\\sigma_0}X_{\\sigma_2}+CX_{\\sigma_0}X_{\\sigma_2}X_{\\sigma_3}\\right)\\right|_{\\sigma_2=\\sigma}\\\\\n&\\quad \\;-\\sum_{\\sigma_2,\\sigma_3\\in S}\\Pi(\\sigma_2,\\sigma_3)\\left.\\left(\\mathds 1_{\\{\\sigma_2=\\sigma_3\\}}AX_{\\sigma_0}X_{\\sigma_2}+\\mathds 1_{\\{\\sigma_0=\\sigma_3\\}}BX_{\\sigma_0}X_{\\sigma_2}+CX_{\\sigma_0}X_{\\sigma_2}X_{\\sigma_3}\\right)\\right|_{\\sigma_0=\\sigma}\\\\\n&=X_\\sigma \\sum_{\\sigma_0\\in S}A\\Pi(\\sigma,\\sigma)X_{\\sigma_0}+X_\\sigma\\sum_{\\sigma_0\\in S}B\n\\Pi(\\sigma,\\sigma_0)X_{\\sigma_0}+X_\\sigma \\sum_{\\sigma_3\\in S}C\\Pi(\\sigma,\\sigma_3)X_{\\sigma_3}\\\\\n&\\quad \\;-X_\\sigma\\sum_{\\sigma_2\\in S}A\\Pi(\\sigma_2,\\sigma_2)X_{\\sigma_2}-X_\\sigma \\sum_{\\sigma_2\\in S}B\\Pi(\\sigma_2,\\sigma)X_{\\sigma_2}\\\\\n&\\quad -\nX_\\sigma\\sum_{\\sigma_2\\in S}\\Bigg(\\sum_{\\sigma_3\\in S}C\\Pi(\\sigma_2,\\sigma_3)X_{\\sigma_3}\\Bigg)X_{\\sigma_2}\\\\\n&=X_\\sigma\\Bigg(A\\Pi(\\sigma,\\sigma)+\\sum_{\\sigma'\\in S}B\n[\\Pi(\\sigma,\\sigma')-\\Pi(\\sigma',\\sigma)]X_{\\sigma'}+\\sum_{\\sigma'\\in S}C\\Pi(\\sigma,\\sigma')X_{\\sigma'}\\Bigg)\\\\\n&\\quad -X_\\sigma \\sum_{\\sigma'\\in S}\\Bigg(A\\Pi(\\sigma',\\sigma')+\\sum_{\\sigma''\\in S}C\\Pi(\\sigma',\\sigma'')X_{\\sigma''}\\Bigg)X_{\\sigma'}.\n\\end{align*}\nNote that we have used the property $\\sum_{\\sigma}X_{\\sigma}=1$ in the last two equalities. The last equality is enough for the required form in \\eqref{p1:lim} upon recalling \\eqref{p1:lim0} and involving the polynomials $F_\\sigma(X)$ and $\\widetilde{F}_\\sigma(X)$ in \\eqref{F1} and \\eqref{F2}. Moreover, \\eqref{kappa:>} holds by Proposition~\\ref{prop:sn-selection}, \\eqref{ineq:MUVcompare}, and Proposition~\\ref{prop:kell} (1${^\\circ}$) and (4${^\\circ}$). \n\nFor the proof of (2${^\\circ}$), notice that by (1${^\\circ}$) and Lemma~\\ref{lem:tight} (4${^\\circ}$)--(5${^\\circ}$), the following convergence of matrix processes holds:\n\\[\n\\left(\\frac{\\gamma_n}{\\theta_n}\\langle M_\\sigma^{(n)},M_{\\sigma'}^{(n)}\\rangle_t\\right)_{\\sigma,\\sigma'\\in S}\\xrightarrow[n\\to\\infty]{\\rm (d)}\\left(\\int_0^t X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s\\right)_{\\sigma,\\sigma'\\in S}.\n\\]\nBy this convergence and \\eqref{ineq:L2-bdd}, the standard martingale problem argument shows that every weakly convergent subsequence of $((\\gamma_n\/\\theta_n)^{1\/2} M_\\sigma^{(n)};\\sigma\\in S)$ converges to a continuous vector $L_2$-martingale $(M_\\sigma^{(\\infty)};\\sigma\\in S)$ with a quadratic variation matrix given by $(\\int_0^t X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s;\\sigma,\\sigma'\\in S)$. See \\cite[Proposition~1.12 in Chapter~IX on p.525]{JS} and the proof of \\cite[Theorem~1.10 in Chapter~XIII on pp.519--520]{RY}. Hence, the limiting vector martingale $(M_\\sigma^{(\\infty)};\\sigma\\in S)$ is a Gaussian process with covariance matrix $(\\int_0^t X_\\sigma(s)[\\delta_{\\sigma,\\sigma'}-X_{\\sigma'}(s)]\\d s;\\sigma,\\sigma'\\in S)$ \\cite[Exercise (1.14) in Chapter~V on p.186]{RY}. Moreover, by uniqueness in law of this Gaussian process, the convergence holds along the whole sequence of the vector martingale $(\\gamma_n\/\\theta_n)^{1\/2}M^{(n)}$. The proof is complete.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:symmetric}]\nWrite $u$ for $X_1$. In this case, $X_0=1-u$ and the polynomial $Q_1(X)$ defined by \\eqref{def:grandQ} simplifies to\n\\[\nQ_1=(b-c)Q_{0,11}-cQ_{0,10}-bQ_{1,01}=(Q_{0,11}-Q_{1,01})b-(Q_{0,11}+Q_{0,10})c.\n\\]\nBy \\eqref{def:Qsigma}, the coefficient of $c$ is given by \n\\begin{align}\\label{coeff:c}\n\\begin{split}\n-Q_{0,11}(X)-Q_{0,10}(X)&=-(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})(1-u)u-\\overline{\\kappa}_{0|2|3}(1-u)u^2\\\\\n&\\quad -(\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3})(1-u)u-\\overline{\\kappa}_{0|2|3}(1-u)^2u,\n\\end{split}\n\\end{align}\nand the coefficient of $b$ is\n\\begin{align}\\label{coeff:b}\n\\begin{split}\nQ_{0,11}(X)-Q_{1,01}(X)&=({\\overline{\\kappa}}_{(2,3)|0}-{\\overline{\\kappa}}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}u^2(1-u)u^2\\\\\n&\\quad -(\\overline{\\kappa}_{(0,3)|2}-\\overline{\\kappa}_{0|2|3})(1-u)u-\\overline{\\kappa}_{0|2|3}(1-u)u^2.\n\\end{split}\n\\end{align}\n\n\n\nThese two coefficients can be simplified by using the definition of $Q_{\\sigma_0,\\sigma_2\\sigma_3}$ and Proposition~\\ref{prop:kell} (4${^\\circ}$), if we follow the algebra in the proof of Lemma~\\ref{lem:D} that simplifies \\eqref{eq:Dsigma} to \\eqref{eq:Dsigma1}. For example, a similar argument as in the proof of Lemma~\\ref{lem:RW} shows that \\eqref{voter:density} holds with \n\\[\nQ_{0,01}(X)=(\\overline{\\kappa}_{(0,2)|3}-\\overline{\\kappa}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}(1-u)^2u,\n\\]\nand so\n\\begin{align*}\nQ_{0,11}(X)+Q_{0,01}(X)&=(\\overline{\\kappa}_{(2,3)|0}-\\overline{\\kappa}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}(1-u)u^2\\\\\n&{\\quad \\,} +(\\overline{\\kappa}_{(0,2)|3}-\\overline{\\kappa}_{0|2|3})(1-u)u+\\overline{\\kappa}_{0|2|3}(1-u)^2u \\\\\n&=(1-u)u{\\overline{\\kappa}}_{3},\n\\end{align*}\nwhere the last equality follows from Proposition~\\ref{prop:kell} (4${^\\circ}$). In this way, we can obtain from \\eqref{coeff:c} and \\eqref{coeff:b} that $Q_1(X)=[({\\overline{\\kappa}}_3-{\\overline{\\kappa}}_1)b-{\\overline{\\kappa}}_2 c](1-u)u$. Moreover, by Proposition~\\ref{prop:kell}, we can pass limit along the whole sequence to get this limiting polynomial $Q_1(X)$.\n \\end{proof}\n \n \n \n \n\\section{Further properties of coalescing lineage distributions}\\label{sec:coal}\n\n\\subsection{A comparison with mutations}\nIn this section, we prove some auxiliary results for the proof of Theorem~\\ref{thm:main}. The next proposition estimates the voter model $(\\xi_t)$ under $\\P^0$ by its selection mechanism, that is, by the updates from $\\{\\Lambda(x,y);x,y\\in E\\}$. The proof extends \\cite[Proposition~3.2]{CC}. Recall the notation in Section~\\ref{sec:dynamics} for the coalescing Markov chains. \n\n\\begin{prop}\\label{prop:mutation}\n{\\rm (1${^\\circ}$)} Let $f:S\\times S\\times S\\to [-1,1]$ be a function such that $f(\\sigma,\\sigma,\\cdot)=0$ for all $\\sigma\\in S$. \nThen for all $t\\in (0,\\infty)$ and $x,y,z\\in E$,\n\\begin{align}\\label{ineq:mutation}\n\\begin{split}\n& \\sup_{\\xi\\in S^E}\\Big|{\\mathbb E}^0_\\xi\\big[f\\big(\\xi_t(x),\\xi_t(y),\\xi_t(z)\\big)\\big]-{\\mathbb E}\\big[f\\big(\\xi(B^x_t),\\xi(B^y_t),\\xi(B^z_t)\\big)\\big]\\Big|\\\\\n&\\quad \\leq \\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{x,y}>t)+\\mathds 1_{x\\neq y}\\big(1-{\\rm e}^{-\\mu(\\mathds 1)t}\\big)\\P(M_{x,z}\\wedge M_{y,z}>t)\\\\\n&\\quad \\quad +2\\mu(\\mathds 1)\\int_0^t \\P(M_{x,y}>s)\\d s+\\mathds 1_{x\\neq y}\\mu(\\mathds 1)\\int_0^t \\P(M_{x,z}\\wedge M_{y,z}>s)\\d s.\n\\end{split}\n\\end{align}\n{\\rm (2${^\\circ}$)} For all $(\\sigma_0,\\sigma_2,\\sigma_3)\\in S\\times S\\times S$ with $\\sigma_0\\neq \\sigma_2$, $t\\in(0,\\infty)$ and $x\\in E$,\n\\begin{align}\\label{claim:w0-mutation}\n\\begin{split}\n&{\\quad \\,}\\sup_{\\xi\\in S^E}\\big|{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]\\big|\\\\\n&\\leq \\sum_{\\ell\\in \\{0,2,3\\}}4 \\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)\\\\\n&\\quad +\\sum_{\\ell\\in \\{0,2,3\\}} 4\\mu(\\mathds 1)\\int_0^t \\P(M_{U_0,U_2}>s,B^{U_\\ell}_t=x)\\d s.\n\\end{split}\n\\end{align}\n\\noindent {\\rm (3${^\\circ}$)} For all $(\\sigma_0,\\sigma_1)\\in S\\times S$ with $\\sigma_0\\neq \\sigma_1$, $t\\in(0,\\infty)$ and $x\\in E$,\n\\begin{align}\\label{claim:w0-mutation1}\n\\begin{split}\n&{\\quad \\,}\\sup_{\\xi\\in S^E}\\big|{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0\\sigma_1}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{ f_{\\sigma_0\\sigma_1}}(\\xi_t)\\right]\\big|\\\\\n&\\leq \\sum_{\\ell\\in \\{0,1\\}}4 \\P(M_{U_0,U_1}>t,B^{U_\\ell}_t=x)\\\\\n&\\quad +\\sum_{\\ell\\in \\{0,1\\}} 4\\mu(\\mathds 1)\\int_0^t \\P(M_{U_0,U_1}>s,B^{U_\\ell}_t=x)\\d s.\n\\end{split}\n\\end{align}\n\\end{prop}\n\\mbox{}\n\nThe proof of this proposition extends the proof of \\cite[Proposition~3.2]{CC} and is based on the pathwise duality between the voter model and the coalescing Markov chains. The relation follows from time reversal of the stochastic integral equations in Section~\\ref{sec:mainresults} of the voter model. More specifically, for fixed $t\\in (0,\\infty)$, we define a system of coalescing $q$-Markov chains $\\{B^{a,t};a\\in E\\}$ such that in the absence of mutation, $B^{a,t}$ traces out the time-reversed ancestral line that determines the type at $(a,t)$ under the voter model. For example, if $s$ is the last jump time of $\\{\\Lambda_r(a,b);b\\in E,r\\in (0,t]\\}$ and $\\Lambda(a,c)$ causes this jump, the state of $B^{a,t}$ stays at $a$ before transitioning to $B^{a,t}_{t-s}=c$. Similarly, with the Poisson processes $\\Lambda^\\sigma$ driving the mutations, we can define $e(a,t)$ and $M(a,t)$ for the time and the type from the first mutation event on the trajectory of $B^{a,t}$, with $e(a,t)=\\infty$ if there is no mutation. Since $e(a,t)>t$ if and only if $e(a,t)=\\infty$, we have\n\\begin{align}\\label{prob:dual}\n\\xi_t(a)=M(a,t)\\mathds 1_{\\{e(a,t)\\leq t\\}}+\\xi\\big(B^{a,t}_t\\big)\\mathds 1_{\\{e(a,t)>t\\}},\\quad\\forall\\;a\\in E,\\quad\\mbox{$\\P^0_\\xi$-a.s.}\n\\end{align}\nMore details can be seen by modifying the description in \\cite[Section~6.1]{CC}. In the absence of mutation, this relation between the duality and the stochastic integral equations is known in \\cite{MT}. \n\n\nWe also observe two identities for the probability distributions of the mutation times $e(a,t)$'s when we condition on $\\mathscr G\\stackrel{\\rm def}{=}\\sigma(\\Lambda(a,b);a,b\\in E)$. Let $x,y\\in E$. Write $0=J_0t\\}.\n\\end{split}\n\\end{align}\nThen consider the corresponding differences for the left-hand side of \\eqref{ineq:mutation}:\n\\begin{align}\\label{def:Deltaj}\n\\Delta_j={\\mathbb E}^0_\\xi\\big[f\\big(\\xi_t(x),\\xi_t(y),\\xi_t(z)\\big);A_j\\big]-{\\mathbb E}\\big[f\\big(\\xi(B^x_t),\\xi(B^y_t),\\xi(B^z_t)\\big);A_j\\big],\\quad 1\\leq j\\leq 4.\n\\end{align}\nLet $\\mathbf e_1$ and $\\mathbf e_2$ be i.i.d. exponential random variables with mean $1\/\\mu(\\mathds 1)$. It follows from \\eqref{eq:e2} and the independence between selection and mutation that\n\\begin{align}\\label{ineq:Delta1}\n|\\Delta_1|&\\leq \\P(\\mathbf e_1\\wedge \\mathbf e_2\\leq t)\\P(M_{x,y}>t)= \\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{x,y}>t),\\\\\n\\label{ineq:Delta2}\n|\\Delta_2|&\\leq\\int_0^t \\P(t\\geq M_{x,y}>s)\\P(\\mathbf e_1\\wedge \\mathbf e_2\\in \\d s)\\leq 2\\mu(\\mathds 1)\\int_0^t \\P(M_{x,y}>s)\\d s.\n\\end{align}\nOn $A_3$, $B^{x,t}_t=B^{y,t}_t$ by coalescence, and hence, $\\xi_t(x)=\\xi_t(y)$ by \\eqref{prob:dual}. It follows from the assumption on $f$ that both of the expectations defining $\\Delta_3$ are zero. \n\nTo bound $\\Delta_4$, fix $z\\in E$ and partition $A_4$ into the following four sets:\n\\begin{align*}\nA_{41}&=\\{e(x,t)\\wedge e(y,t)>t, e(z,t)\\leq tt, e(z,t)\\leq M_{x,z}\\wedge M_{y,z}\\leq t\\},\\\\\nA_{43}&=\\{e(x,t)\\wedge e(y,t)>t,M_{x,z}\\wedge M_{y,z}< e(z,t)\\leq t\\},\\\\\nA_{44}&=\\{e(x,t)\\wedge e(y,t)>t,e(z,t)>t\\}.\n\\end{align*}\nThen define $\\Delta_{4k}$ for $1\\leq k\\leq 4$ as in \\eqref{def:Deltaj} by replacing $A_j$ with $A_{4k}$. By \\eqref{eq:e1} and similar arguments for \\eqref{ineq:Delta1} and \\eqref{ineq:Delta2}, we get\n\\begin{align}\\label{ineq:Delta412}\n\\begin{split}\n|\\Delta_{41}|&\\leq \\mathds 1_{x\\neq y}\\big(1-{\\rm e}^{-\\mu(\\mathds 1)t}\\big)\\P(M_{x,z}\\wedge M_{y,z}>t),\\\\\n|\\Delta_{42}|&\\leq \\mathds 1_{x\\neq y}\\mu(\\mathds 1)\\int_0^t \\P(M_{x,z}\\wedge M_{y,z}>s)\\d s,\n\\end{split}\n\\end{align}\nwhere the use of the indicator function $\\mathds 1_{x\\neq y}$ follows from the assumption of $f$. For $\\Delta_{43}$, it is zero because $A_{43}=\\varnothing$. Indeed, on $\\{M_{x,z}\\wedge M_{y,z}< e(z,t)\\leq t\\}$, either $e(x,t)\\leq t$ or $e(y,t)\\leq t$ since either $e(x,t)=e(z,t)$ (if $M_{x,z}\\wedge M_{y,z}=M_{x,z}$) or $e(y,t)=e(z,t)$ (if $M_{x,z}\\wedge M_{y,z}=M_{y,z}$). Hence, $\\{M_{x,z}\\wedge M_{y,z}< e(z,t)\\leq t\\}$ does not intersect $\\{e(x,t)\\wedge e(y,t)>t\\}$. Finally, $\\Delta_{44}=0$ by \\eqref{prob:dual} now that the random variables being taken expectation are actually equal. \n\nIn summary, we have proved that $\\Delta_3=\\Delta_{43}=\\Delta_{44}=0$. In addition, $\\Delta_{1}$, $\\Delta_2$, $\\Delta_{41}$ and $\\Delta_{42}$ satisfy \\eqref{ineq:Delta1}, \\eqref{ineq:Delta2} and \\eqref{ineq:Delta412}. We have proved \\eqref{ineq:mutation}.\\medskip \n\n\n\\noindent (2${^\\circ}$) For the left-hand side of \\eqref{claim:w0-mutation}, we use \\eqref{prob:dual} to write\n\\begin{align}\n&{\\quad \\,}{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]\\notag\\\\\n\\begin{split}\\label{A1-A4:0000}\n&={\\mathbb E}\\Bigg[\\prod_{j\\in \\{0,2,3\\}}\\mathds 1_{\\sigma_j}\\Big(M(U_j,t)\\mathds 1_{\\{e(U_j,t)\\leq t\\}}+\\xi^x(B^{U_j,t}_{t})\\mathds 1_{\\{e(U_j,t)> t\\}}\\Big)\\\\\n&{\\quad \\,}-\\prod_{j\\in \\{0,2,3\\}}\\mathds 1_{\\sigma_j}\\Big(M(U_j,t)\\mathds 1_{\\{e(U_j,t)\\leq t\\}}+\\xi(B^{U_j,t}_{t})\\mathds 1_{\\{e(U_j,t)> t\\}}\\Big)\\Bigg].\n\\end{split}\n\\end{align}\nMutation neglects the role of the initial condition. \nHence, to get a nonzero value for the difference inside the foregoing expectation, we cannot have $e(U_j,t)\\leq t$ for all $j\\in \\{0,2,3\\}$.\n In this case, at least one of the sums\n$\\mathds 1_{\\sigma_j}\\circ \\xi(B^{U_j,t}_t)+\\mathds 1_{\\sigma_j}\\circ \\xi^x(B^{U_j,t}_t)$, $j\\in \\{0,2,3\\}$, has to be nonzero. We must have $B^{U_j,t}_t=x$ for some $j\\in \\{0,2,3\\}$. By bounding the indicator functions associated with $\\sigma_3$ by $1$, we obtain from \\eqref{A1-A4:0000} that \n\\begin{align}\n&{\\quad \\,}\\big|{\\mathbb E}^0_{\\xi^x}\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]-{\\mathbb E}^0_\\xi\\left[\\,\\overline{f_{\\sigma_0} f_{\\sigma_2\\sigma_3}}(\\xi_t)\\right]\\big|\\notag\\\\\n&\\leq \\sum_{\\ell\\in \\{0,2,3\\}}\\left({\\mathbb E}_{\\xi^x}+{\\mathbb E}_{\\xi}\\right)\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\xi_t(U_j);B^{U_\\ell,t}_t=x\\Bigg],\\quad\\forall\\;x\\in E.\\label{A1-A4:main1}\n\\end{align}\n\n\nThe method in (1${^\\circ}$) now enters to remove mutations in each of the two expectations in the $\\ell$-th summand of \\eqref{A1-A4:main1}. For $\\eta\\in S^E$, we consider\n\\begin{align}\\label{A1-A4-diff}\n{\\mathbb E}_\\eta\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\xi_t(U_j);B^{U_\\ell,t}_t=x\\Bigg]-{\\mathbb E}\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\eta(B^{U_j,t}_t);B^{U_\\ell,t}_t=x\\Bigg]\n\\end{align}\nand use only the partition in \\eqref{A1-A4} with $x=U_0$ and $y=U_2$. In this case, on $A_4$, the two products of the indicator functions in \\eqref{A1-A4-diff} are equal. Since $\\sigma_0\\neq \\sigma_2$ ensures that the second expectation in \\eqref{A1-A4-diff} can be bounded by $\\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)$, \\eqref{A1-A4-diff} and a slight extension of \\eqref{ineq:Delta1} and \\eqref{ineq:Delta2} give\n\\begin{align}\n&{\\quad \\,}{\\mathbb E}_\\eta\\Bigg[\\prod_{j\\in \\{0,2\\}}\\mathds 1_{\\sigma_j}\\circ \\xi_t(U_j);B^{U_\\ell,t}_t=x\\Bigg]\\notag\\\\\n&\\leq \\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)+\\big(1-{\\rm e}^{-2\\mu(\\mathds 1)t}\\big)\\P(M_{U_0,U_2}>t,B^{U_\\ell}_t=x)\\notag\\\\\n&{\\quad \\,}+2\\mu(\\mathds 1)\\int_0^t \\P(M_{U_0,U_2}>s,B^{U_\\ell}_t=x)\\d s,\\quad \\forall\\;\\eta\\in S^E,\\;x\\in E.\\label{A1-A4:main2}\n\\end{align}\nThe required inequality \\eqref{claim:w0-mutation} now follows from \\eqref{A1-A4:main1} and \\eqref{A1-A4:main2}.\\medskip \n\n\\noindent {\\rm (3${^\\circ}$)} The proof of \\eqref{claim:w0-mutation1} is almost the same as the proof of \\eqref{claim:w0-mutation} and is omitted. \n\\end{proof}\n\n\\subsection{Full decorrelation on large random regular graphs}\\label{sec:rrg}\nIn this subsection, we give a different proof of the explicit form of \\eqref{cond:kappa0} by using the graphs' local convergence. Throughout the rest of this subsection, we use the graph-theoretic terminologies from \\cite{Bollobas,Chung}.\n\nWe start with the definition of the random regular graphs. Fix an integer $k\\geq 3$. Choose a sequence $\\{N_n\\}$ of positive integers such that $N_n\\to\\infty$ and $k$-regular graphs (without loops and multiple edges) on $N_n$ vertices exist. The existence of $\\{N_n\\}$ follows from the Erd\\H{o}s--Gallai necessary and sufficient condition. Then the random $k$-regular graph on $N_n$ vertices is the graph $G_n$ chosen uniformly from the set of $k$-regular graphs with $N_n$ vertices. We assume that the randomness defining the graphs is collectively subject to the probability $\\mathbf P$ and the expectation $\\mathbf E$. \n\nFor applications to the evolutionary dynamics, we need two properties of random walks on the random graphs. See \\cite[Section~3]{C:MT} and the references there for more details. First, the random walks are asymptotically irreducible in the following sense:\n \\begin{align}\\label{prob:comp}\n\\mathbf P(G_n\\mbox{ has only one connected component})\\to 1\\quad\\mbox{ as $n\\to\\infty$.}\n\\end{align} \nThis property follows since the $\\mathbf P$-probability that $G_n$ has a nonzero spectral gap tends to one \\cite{Friedman, Bordenave}. See \\cite[Lemma~1.7 (d) on pp.6--7]{Chung} for connections between graph spectral gaps and numbers of connected components. Second, $G_n$ for large $n$ is locally like the infinite $k$-regular tree $G_\\infty$ in the following sense. Write $q^{(n),\\ell}(x,y)$ for the $\\ell$-step transition probability of random walk on $G_n$. For any $n, r\\in \\Bbb N$, write $\\mathcal T_n(r)$ for the set of vertices $x$ in $G_n$ such that the subgraph induced by vertices $y$ with $d(x,y)s_n''t)=1,\\quad\\forall\\;t\\in (0,\\infty); \\quad \\mathbf P\\mbox{-a.s.}\n\\end{align}\nand so, by \\eqref{MUU:RG} and \\cite[Proposition~4.3 (2)]{CCC}, \\eqref{gamma} with $s_n''$ replaced by $s_n$ holds. We obtain \\eqref{ass:sigman} from this limit and \\eqref{gamma0}. The proof is complete.\n\\end{proof}\n\n\\begin{rmk}\nMcKay \\cite[Theorem~1.1]{McKay} derives the limiting spectral measures of large random regular graphs. There the randomness of graphs only plays the role of inducing asymptotically deterministic properties. For the present case, we could have worked with given sequences of $k$-regular graphs and obtained the same limit if the graphs have spectral gaps bounded away from zero and are locally tree-like. (Dropping the locally tree-like assumption calls for a different evaluation of the limit.) We choose to work with the above context to explain how the randomness of graphs should be handled for the convergence of the evolutionary game model. \\hfill $\\blacksquare$\n\\end{rmk}\n\n\n \n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\n\n\\section{Methods}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.95\\textwidth]{workflow_v2}\n\t\\centering\n\t\\caption{Overall workflow of GraphMapper. It is composed of three stages: shape initialization, shape refine and relationship modeling.}\n\t\\label{fig:workflow}\n\\end{figure}\n\n\\subsection{Primitive Graph}\nThe primitive graph is a homogeneous undirected graph $G=\\{V, E\\}$. Here, $V$ is a $L \\times C$ matrix representing $L$ geometric primitives and their $C$ dimensional properties. Point and line segment are the two basic types of primitives used in this study. Typical property $C$ includes coordinates, direction, and image features. $E$ is an $L \\times L$ adjacency matrix representing a set of relationships between primitives in $L$. Topology reconstruction can be represented by estimating a primitive graph with point or line segment primitives and a pairwise connectivity matrix as the relationship matrix. Shape regularization can be represented by the consistency between primitive's directions and their relationships. Further discussion on primitive configuration for building and road mapping can be found in Section \\ref{section:apply}.\n\n\\subsection{Primitive Initialization}\nThe goal of the primitive initialization module is to propose a reasonable initial shape for following shape modeling stages. We first employ feature extractor $f_{imEnc}$ to extract high resolution image features $F$ of input image $I^{H\\times W}$. Similar to \\cite{Bai_2017_CVPR,Girard_2021_CVPR}, an offset head $f_{off}$ is added in parallel to semantic segmentation head $f_{seg}$ for improved awareness of shape prior for feature extractor. A sampler $f_{sample}$ is applied to the predicted segmentation mask to extract a set of initial primitives. Initial primitives should be sparse to reduce the percentage of simple relationships.\n\n\\paragraph{Sampler.}\nFor polygons, $f_{sample}$ traces polygons from the predicted mask. Douglas-Pecker algorithm \\cite{douglas_1973} is applied with a tight error threshold (1 pixel) on the traced contours to remove redundant points. The points or line segments in the simplified contour can be used as primitives. For poly-lines, Sat2Graph \\cite{He_2020_ECCV} predicts evenly spaced points on line segments of poly-lines by segmentation. Local maximum points are sampled and sparsified using NMS (Non-Maximum Suppression) \\cite{Papandreou_2017_CVPR}. We found this method tends to miss points that are distant from junctions\/overlays. We resolve this by sampling additional points from the road segmentation mask outputted from $f_{seg}$. To learn more semantics in road topology, junctions, and overlays are segmented in additional to evenly spaced points on line segments using $f_{kp}$. Our full sampling scheme takes a stratified approach: sample key points from every non-background channel of $f_{kp}$ output and $f_{seg}$ independently; combine all sampled key points with priority order: junctions > overlays > other segmented key points > points sampled from $f_{seg}$ so that topologically significant points are kept over non-significant points.\n\n\\subsection{Primitive Refinement}\nPrimitive refinement improves the accuracy of $V$, which can reduce the ambiguity of relationship learning. We found it essential for accurate relationship prediction, especially for small structures. Primitive refinement network takes image feature $F$ and initial primitives $V$ as inputs and outputs the deformation of primitives' coordinates and directions (point tangent direction or line segment direction). Image features at primitives' locations are pooled (point interpolation for point primitives; LOI \\cite{Zhou_2019_ICCV} for line segment primitives). The sin-cos position encoding from $f_{pos}$ and the pooled image features are concatenated and fed into a multi-layer multi-head-attention (MHA) network $f_{GL\\_SH}$ \\cite{carion_2020_ECCV} to generate global shape contextualized primitive features. Primitive deform head $f_{voff}$ and primitive direction head $f_{vdir}$ take this feature as input and each outputs a $L\\times 2$ matrix representing the offset and direction of each primitive. \n\n\\subsection{Relationship modeling}\nThe goal of relationship modeling is to compute the pairwise relationship matrix of the refined primitives $V'$. This stage uses the same network structure as the primitive refinement network, except for the relationship classifier $f_{vRel}$, which takes concatenated primitive pair features to classify the pair's relationships. \n\nIn road mapping, the optimal connectivity between two primitives depends on the existence of other primitives\nIt requires sorting primitives in an embedding space, where connected primitives are expected to be closer than disconnected primitives. Inspired by popular contrastive learning methods \\cite{NEURIPS2020_f3ada80d,Bardes2021,Chen_2021_CVPR}, L2 normalization is used to improve embedding quality, which we found is essential for accurate topology reconstruction. Pairwise L2-normalized features are concatenated and fed into classifier $f_{vRel}$ for relationship classification. \n\nStandard cross-entropy loss (Eq. \\ref{eq:loss_vrel}) is used to train relationship modeling network. In Eq. \\ref{eq:loss_vrel}, $C$ is the set of possible relationships, $\\hat{E}^{(c)}_{i,j}$ and $E^{(c)}_{i,j}$ are the predicted probability and ground truth probability of primitive pair $i,j$ at relationship $c$. Compared to using feature correlation for relationship classification, cross-entropy works slightly better through our experiments. Cross entropy also allows a consistent network structure for both shape regularization and topology reconstruction. Additional contrastive loss on feature distance is explored, but not shown to improve performance in our study.\n\n\\begin{equation}\n\t\\mathcal{L}_{vRel} = -\\frac{1}{(L-1)L}\\sum_{i=1}^{L}\\sum_{j=1,j\\neq i}^{L}\\sum_{c\\in C} E_{i,j}^{(c)}log(\\hat{E}^{(c)}_{i,j})\n\t\\label{eq:loss_vrel}\n\\end{equation}\n\n\\paragraph{Shape Regularization.} Shape regularization is modeled by adding a direction-relation consistency loss:\n\\begin{equation}\n\t\\begin{aligned}\n\tL_{consistency} &= \\frac{1}{|V|^2-|V|}\\sum_{c\\in C\\setminus 0}\\hat{E}^{(c)} * [\\cos(2A_c) - \\cos(2mod(V_{dir}' - V_{dir}'^T+2\\pi, \\pi))]^2\n\t\\end{aligned}\n\t\\label{eq:loss_reg}\n\\end{equation}\n, where $mod$ term represents the direction difference between primitives pairs normalized to range [0, $\\pi$); $A_c$ is the perfect angle for relation $c$ ($A_c=0$ for relationship between a primitive and itself). Eq. \\ref{eq:loss_reg} computes the mean squared error of surrogate angles weighted by relation probability. Surrogate angle will be explained in the following sub-section. It enables the network to directly generate regularized shapes without any post-processing. \n\n\n\\subsection{Training Tasks and Objectives} \\label{section:apply}\nGraphMapper is trained with multiple losses. Common losses used by shape regularization and topology reconstruction include:\n\\begin{itemize}\n\t\\item Cross entropy loss for image feature extraction and segmentation sub-net (Eq. \\ref{eq:loss_seg}):\n\t\\begin{equation}\n\t\t\\mathcal{L}_{seg} = -\\frac{1}{|I|}\\sum_{i\\in I}\\sum_{c\\in \\{0,1\\}}y_i^{(c)}log(\\hat{y}_i^{(c)}),\n\t\t\\label{eq:loss_seg}\n\t\\end{equation}\n\twhere $y$ and $\\hat{y}$ are $f_{seg}$ predicted segmentation prob and ground truth segmentation mask.\n\t\\item L2 loss for offset branches $f_{off}$ (Eq. \\ref{eq:loss_off}):\n\t\\begin{equation}\n\t\t\\mathcal{L}_{off} = \\sum_{i\\in I_{sub}}{\\|\\hat{o_i} - o_i \\|_2}, \n\t\t\\label{eq:loss_off}\n\t\\end{equation}\n\twhere $I_{sub}$ is the subset of pixels in a buffer region of boundaries or center lines of segmentation mask $y$. $\\hat{o}_i$ is the predicted offset by $f_{off}$.\n\t\\item Surrogate L2 loss for direction branch $f_{vdir}$ (Eq. \\ref{eq:loss_dir}). The discontinuity of rotation angles can lead to unstable learning \\cite{Zhou_2019_CVPR}, we regress a surrogate angle which is 2 times the actual angle.\n\t\\begin{equation}\n\t\t\\mathcal{L}_{vDir} = \\frac{1}{|V|}\\sum_{v\\in V}(\\cos(2V_{dir}') - \\cos(2V_{dir}^T))^2\n\t\t\\label{eq:loss_dir}\n\t\\end{equation}\n\t\\item Bi-projection loss $L_{bp}$ \\cite{CHEN2020114} is used for deformation $f_{vOff}$ (Eq. \\ref{eq:loss_voff}). It first matches the vertices in ground truth to its nearest predictions; the rest of the predicted vertices are matched to its nearest projection in ground truth shape. \n\t\\begin{equation}\n\t\t\\mathcal{L}_{vOff} = L_{bp}(V', V)\n\t\t\\label{eq:loss_voff}\n\t\\end{equation}\n\t\n\\end{itemize}\n\n\\paragraph{Building Mapping.} \nLine segments of simplified building contours are used as primitives. Relationship classification predicts whether two consecutive lines are inline. Finally, the total training loss is a weighted sum of ($\\mathcal{L}_{seg}, \\mathcal{L}_{off}, \\mathcal{L}_{vOff}, \\mathcal{L}_{vDir}, \\mathcal{L}_{vRel}, \\mathcal{L}_{consistency}$).\n\nTo reconstruct the building polygon, the resulting line segments in $V'$ are rotated around their center to the direction estimated in the shape refinement stage. For each pair of neighboring line segments: merge if parallel (angle < parallel angle threshold) and close-by (nearest point pair distance > shortest edge length), connecting near-ends if parallel and not close-by; extend to the intersection if not parallel. The shortest edge length can be set according to the requirements in mapping accuracy. The parallel angle threshold is set to 30\\textdegree.\n\n\\paragraph{Road Network Mapping.}\nRoad network reconstruction uses the point as primitives and connectivity as the pairwise relationship. Pixels within a buffer distance (3 meters) to the center-line are considered as the road surface. The final training loss for road network reconstruction is a weighted sum of ($\\mathcal{L}_{seg}, \\mathcal{L}_{off},\\mathcal{L}_{kp}, \\mathcal{L}_{vOff}, \\mathcal{L}_{vDir}, \\mathcal{L}_{vRel}$). Only point pairs within a certain distance are used for relationship classification loss computation, to balance the ratio of positive and negative point pairs.\n\nThe road network graph is reconstructed by connecting each point to its nearest $N$ neighbors in embedding space, where $N$ is 3 for junction and 2 for others. No post-processing is required.\n\n\\section{Introduction}\nMaps are vectorized and simplified representations of the real world. It is important for urban planning, navigation, disaster recovery, and large-scale surveys and census. In less developed regions, efficient map production is vital for combating poverty. Traditionally, map production fully relies on manual labeling, which is time-consuming and expensive. With the increasingly available high-resolution satellite images and increasing needs in global monitoring, environment protection, and urbanization, it is desired to produce large-scale vector maps with efficient methods.\n\nA variety of techniques have been developed to automatically extract vector maps from satellite images. These methods often follow a \"segmentation and modeling\" paradigm: using semantic segmentation to extract the target mask and shape modeling algorithms to extract vector representations from the mask. Semantic segmentation of satellite images has improved dramatically in recent years \\cite{MARMANIS2018158,Zhou_2018_CVPR_Workshops,Chen_2018_CVPR,Batra_2019_CVPR} with the advances in deep learning. State-of-the-art methods \\cite{He_2020_ECCV,Girard_2021_CVPR} still rely on heuristic rules optionally with optimization for shape modeling. These methods often require careful parameter tuning for practical use, which has limited their ability to handle various environments in large-scale mapping. Additionally, state-of-the-art methods are often designed for a specific type of mapping target, and multiple methods are needed for comprehensive vector mapping of multiple target types, which further increases parameter tuning effort. Finally, the separation of semantic segmentation and shape modeling causes error accumulation, which leads to performance dropping.\n\nEnd-to-end shape modeling methods use data-driven regularization and shape priors instead of heuristics, which is potentially more applicable in large-scale scenarios compared to heuristics-based methods. State-of-the-art end-to-end shape modeling methods formulate shape prediction as a point sequence prediction problem. Recurrent neural networks (RNNs) are used to predict points one by one in sequence \\cite{Castrejon_2017_CVPR, Acuna_2018_CVPR, CHEN2020114, Zorzi_2020_ICPR}. Unfortunately, these methods are difficult to train and practically outperformed by state-of-the-art heuristics-based methods.\nTo address these problems, we propose a unified framework for end-to-end vector mapping from satellite images with a novel generic shape representation named \"primitive graph\" that describes both geometric primitives and pairwise relationships between them. Compared to existing unified shape representations \\cite{Li_2019_ICCV} that only work for coordinates learning, such a representation enables explicit joint learning of geometry, shape regularization, and topology in one model. \n\nAccordingly, we propose a generic network named GraphMapper to progressively reconstruct primitive graphs. The overall workflow is shown in Fig. \\ref{fig:workflow}. Multi-head attention networks are used to encode global-shape-context for primitives, which encodes object-level shape context or image-level shape context to support primitive-level learning. One of the main challenges in relationship matrix prediction is the ambiguity in dynamic matching between predicted primitives and ground truth shapes for loss computation. By progressively refining the coordinates before relationship learning, the dynamic matching ambiguity is reduced along the training process, which we demonstrate is a crucial strategy of the proposed approach.\n\nGraphMapper enables shape regularization through learning. Specifically, reconstruction of shape geometry and regularization requires accurate coordinate regression and faithful parallel\/orthogonal relationships prediction. Primitive graph jointly explores the geometry regression and relationship classification and benefits both tasks by imposing inherent consistency.\nUnlike existing methods where shape regularization is enforced in a post-optimization stage, we define shape regularization as consistency between geometry and relationships in a primitive graph during training, thereby avoids tedious parameter tuning.\n\nAs for topological reconstruction, primitive graph provides compact and straightforward representation by directly representing topological connectivity as pairwise relationship.\nWe observe the challenge of accurate classification of connectivity relationships, whose performance is sensitive to probability threshold of predicted scores. Therefore, we consider topology reconstruction as a hidden space sorting problem, i.e., primitives are embedded into a space where connected primitives are closer in feature distance; topology is reconstructed by connecting every primitive to a certain number of nearest primitives in hidden feature space. We found this strategy significantly improves performance.\n\nWe mainly solve two major problems in vector mapping: building footprint mapping and road network mapping. Building footprint mapping requires regularized building footprint polygons, while road network mapping requires topologically correct and smooth poly-line graphs. We demonstrate that GraphMapper can adapt to both tasks well by simple reconfiguration. With simple post-processing or no post-processing, our method outperforms state-of-the-art methods by 10\\% in building footprint mapping and 8\\% in road network mapping.\n\\section{Experiments}\n\n\\subsection{Datasets And Metrics}\n\\paragraph{Building} (1) CrowdAI Mapping Challenge Dataset \\cite{mohanty2020deep} (CrowdAI dataset): it contains 341438 annotated aerial images of size 300 $\\times$ 300 pixels. The official train-valid-test splits are used\n\nThe commonly used mIOU (Mean Intersection Over Union) and AP (Average Precision) in semantic segmentation tasks cannot describe the cleanness of predictions at boundaries. Mean Max Tangent Angle Error (MMTE) \\cite{Girard_2021_CVPR} is adopted to evaluate the correctness of extracted vector shapes of buildings. MMTE computes the average max angle error of all line segments of each building over the entire dataset.\n\n\\paragraph{Road Network} (1) SpaceNet road dataset \\cite{vanetten2019spacenet}: it contains 2549 satellite images of size 1300 $\\times$ 1300 pixels with resolution around 0.3m. This dataset is challenging due to the diverse scenarios from 5 cities around the globe. The train-val-test splits used by Sat2Graph \\cite{He_2020_ECCV} are used, which uses 80\\% for training, 15\\% for testing, and 5\\% for validation. (2) City-Scale Dataset \\cite{He_2020_ECCV}: it contains 180 tiles of size 2000 $\\times$ 2000 with 1 meter spatial resolution. This dataset covers 20 U.S. cities with less diversity compared to the SpaceNet road dataset. The ground truth vector annotations were collected from OpenStreetMap \\cite{Mordechai_2008}. Follow \\cite{He_2020_ECCV}, images in both datasets are resized to 1 meter spatial resolution.\n\nRoad network topology is evaluated using TOPO \\cite{Biagioni_2012_TRR} and Average Path Length Similarity (APLS) \\cite{vanetten2019spacenet}. TOPO measures the similarity of sub-graphs randomly sampled from the inferred graph and ground truth graph within a certain distance of a seed location. The similarity of sub-graphs is quantified as positive or negative. Average precision, recall, and F1 scores are reported on randomly sampled seed points. APLS measures the difference of the shortest path between sampled point pairs on the inferred graph and ground truth graph. It sums up the path difference for all paths in the graph. \n\n\\subsection{Implementation Details}\n\\paragraph{Network structure.} We use the panoptic segmentation FPN \\cite{Kirillov_2019_CVPR} with ResNet101 backbone \\cite{He_2016_CVPR} pre-trained on Imagenet \\cite{ILSVRC15} as the multi-scale image feature extractor. Predictors $f_{off}, f_{seg}, f_{kp}$ share the same structure of 2$\\times$(Conv-BN-ReLU)-2$\\times$ConvTrans. The last two Transpose Convolution layers use stride $2$ and kernel size $3$ to generate output the same size of input image.\n\nFollowing the same setting in \\cite {carion_2020_ECCV}, position encoding, image features, segmentation logits, and key point logits are concatenated and fed into $f_{GL\\_SH}$. The pooled features of primitives are compressed using a 1x1 convolution projection layer to 256 channels. $f_{GL\\_SH}$ uses three MHA layers proposed in \\cite{Vaswani_2017_NIPS} for global shape feature encoding which uses 256-dimensional internal representations and output 256-dimensional features. $f_{voff}, f_{vdir}$ use the same structure as $f_{off}$. The kernel size of convolution layers in predictors is set to 1$\\times$1 for road topology reconstruction because of the unordered nature of point primitives. For building, line segment primitives are ordered in sequence, the kernel sizes are set to 3$\\times$3 to better use context information.\n\n\\paragraph{Training and testing.} Shape initialization stage is first trained without the shape modeling stages to provide reasonable initial shapes for shape modeling stages. Adam optimizer \\cite{adam} is used with batch size 32 and initial learning rate 1e-3. The learning rate will decrease 10 times twice when training loss plateaus. After the shape initialization stage is trained, all stages are trained together using Adam\\cite{adam} with batch size 1 and initial learning rate 1e-4. Weight decay is disabled. Training stops when the validation score stops increasing. Input crop size is 300x300 for building datasets and 448x448 for road datasets. Standard data augmentation techniques are used, including random rotation between [-30\\textdegree, 30\\textdegree], flipping, color jittering, and rescale between [0.7, 1.5]. The model with the best validation score is selected for testing. No test-time augmentation is used.\n\n\nOur implementation is based on Detectron2 \\cite{wu2019detectron2} and Pytorch \\cite{NEURIPS2019_bdbca288}. All experiments are conducted on a work station equipped with 1 Intel(R) Xeon(R) Gold 6278C CPU @ 2.60GHz and 4 NVIDIA Tesla T4 GPUs.\n\n\\subsection{Benchmark results}\n\n\\paragraph{Building footprint extraction.} The performance of our model and state-of-the-art methods are reported in Tab. \\ref{tab:building_main}. GraphMapper outperforms Frame Field Learning in both datasets by a large margin. As shown in Fig. \\ref{fig:building_main}, GraphMapper generates visually natural yet well-regularized shapes without parallel or perpendicular rectification. Small errors in semantic segmentation can be corrected by shape modeling. In Fig. \\ref{fig:building_main}(4), some gross errors in semantic segmentation are also corrected by shape modeling.\n\n\\begin{table}[ht]\n\t\\caption{Building evaluation results.}\n\t\\centering\n\t\\input{resource\/table_building_main.tex}\n\t\\label{tab:building_main}\n\\end{table}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{building_main_vis.png}\n\t\\centering\n\t\\caption{Example building footprint extraction results. Top row is the simplified segmentation contour, bottom row is the GraphMapper output. (1) Round corners are corrected; (2) small errors in semantic segmentation are corrected by shape modeling; (3) irregular shape can be modeled correctly; (4) large segmentation errors are corrected by shape modeling.}\n\t\\label{fig:building_main}\n\\end{figure}\n\n\\paragraph{Road network extraction.} Evaluation results are reported in Tab. \\ref{tab:road_main}. With no post-processing, GraphMapper achieves superior performance on both TOPO and APLS compared to state-of-the-art methods. TOPO F1 is improved by 6-8 in absolute value compared to Sat2Graph\\cite{He_2020_ECCV}. GraphMapper is showing to fix the topologically disconnected scenarios in segmentation mask (Fig. \\ref{fig:road_main}(a,b)). Incorrectly predicted road surface key points (blue) and road junction points (red) are not showing to affect the shape modeling output (Fig. \\ref{fig:road_main}(a,b)). It suggests that the network is learning more complicated rules than relying on the predicted key point category for topology reconstruction.\n\\begin{table}[t]\n\t\\caption{Comparison of road TOPO and APLS evaluation metric.}\n\t\\centering\n\t\\input{resource\/table_road_main.tex}\n\t\\label{tab:road_main}\n\\end{table}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.9\\textwidth]{road_main_vis.png}\n\t\\centering\n\t\\caption{Qualitative result of road network reconstruction. (1) Extracted road network visualized in yellow lines; (2) segmentation mask of (1); (3) a zoomed-in view of the red box in (1), blue points are predicted road surface points, red points are predicted road junctions; (4) segmentation mask of (3).}\n\t\\label{fig:road_main}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{minipage}[b]{.45\\textwidth}\n\t\\includegraphics[width=\\textwidth]{road_overlay_vis.png}\n\t\\centering\n\t\\caption{Qualitative results at road junctions and overlays.}\n\t\\label{fig:junction_and_overlays}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[b]{.45\\textwidth}\n\\includegraphics[width=0.9\\textwidth]{sort_vs_cls_sensitivity.png}\n\t\\centering\n\t\\caption{Threshold sensitivity for embedding space sorting and connectivity classification.}\n\t\\label{fig:sort_vs_cls}\n\\end{minipage}\n\\end{figure}\n\n\\subsection{Ablation Study and Discussion}\n\n\\paragraph{Shape embedding normalization and sorting.} The usefulness of shape embedding normalization and hidden-space sorting-based topology reconstruction are tested on the City-Scale dataset. The results are reported In Tab. \\ref{tab:road_ablation}. Shape embedding normalization improves road-mapping performance by 4.6 in TOPO F1 and 5.6 in APLS. Sorting in embedding space is shown to outperform standard relationship classification (-sort) and is more robust to connectivity threshold as shown in Fig. \\ref{fig:sort_vs_cls}.\n\\begin{table}\n\\begin{minipage}[t]{.49\\textwidth}\n\\centering\n\\caption{Ablation study on City-Scale dataset.}\n\\input{resource\/table_road_ablation.tex}\n\\label{tab:road_ablation}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{.49\\textwidth}\n\\centering\n\\caption{Direction accuracy of different stages on CrowdAI dataset.}\n\\input{resource\/table_building_ablation.tex}\n\\label{tab:building_ablation}\n\\end{minipage}\n\\end{table}\n\n\\paragraph{Shape regularization.} As the ground truth shape of buildings are regularized shapes manually created, MMTE can be used as a good indicator of shape regularity. Increasing direction-topology consistency loss weight from 0.001 to 0.1 is showing to to increase MMTE by 0.7\\textdegree (Tab. \\ref{tab:building_main}). The increase of shape regularity hurts mIOU in our experiments. Similar behavior has been reported in previous studies \\cite{CHEN2020114,Girard_2021_CVPR}. This may be related to the misalignment noise in the ground truth. \n\n\\paragraph{Accuracy of direction.} The MMTE of line direction of different stages are reported in Tab. \\ref{tab:building_ablation}. The final polygon uses the predicted direction in refine stage. It has MMTE 26.7\\textdegree, which is significantly better than the line direction derived from refined polygons which have MMTE 35.9\\textdegree. The big difference in MMTE is surprising as deformation and direction regression are coupled tasks sharing the same MHA shape feature encoder. It could be caused by the relatively accurate line direction and less accurate line location in training annotations. \n\n\\paragraph{Parameter tuning and sensitivity.} GraphMapper requires little parameter tuning for shape post-processing. In our building shape post-processing step, a shorted line length term is used to control the output simplicity. This term is not data-dependent but task-dependent. The value should be set according to the mapping task requirement in map accuracy. For road extraction, no post-processing is needed. GraphMapper is less sensitive to point sampling density compared to Sat2Graph \\cite{He_2020_ECCV} during our test. A possible reason is that GraphMapper is trained to work with the preset point density.\n\n\n\\paragraph{Limitations and future study.} \\label{limitations}\nOur primitive sampling method may remove important points when multiple roads are densely overlaid, which can cause missing connections at road overlays. Line segment prediction methods such as \\cite{Zhou_2019_ICCV} can be used to directly sample line segments as initial primitives. We leave this for future study. \n\nThe quality of dynamic ground truth generation is directly related to the network's performance. Incorrect matching for relationship classification is not rare in our study. Methods that do not rely on complicated dynamic ground truth generation will be our focus in the following studies.\n\n\n\\section{Conclusion}\nWe propose GraphMapper, an end-to-end model for unified vector mapping from satellite images based on primitive graphs. By converting vector mapping tasks into primitive graph estimation tasks, it can handle various topology reconstruction and shape regularization tasks. With simple post-processing or no post-processing, GraphMapper achieved state-of-the-art performance in both building footprint and road network mapping. This makes it easy to deploy for map production systems. We hope this can increase the mapping capability of less developed regions. \n\n\n\\section{Related Works}\n\\label{sec:relatedwork}\n\\paragraph{Building footprint mapping.} Modern building footprint extraction methods typically use semantic segmentation neural networks to extract building masks from satellite images. Shape rectification rules such as main direction alignment \\cite{MSBuilding}, parallel and perpendicular \\cite{sirko2021continentalscale} rectification are applied to the contours of segmented building masks to extract vector maps. Rule-based shape modeling methods are widely used in practice due to their simplicity and reasonable performance on simple buildings. Instead of using heuristic rules, Active Contour Model (ACM) or similar contour optimization techniques are developed to encourage building contours to be consistent with predicted energy fields, such as boundary probability, distance, and direction \\cite{Marcos_2018_CVPR, Cheng_2019_CVPR,Girard_2021_CVPR}. With energy field predictors highly coupled with semantic segmentation, the predicted directions or boundary probabilities are often highly consistent with segmentation masks, which limited their impact on output shapes. Several terms are used to balance shape smoothness and consistency to image semantics. There is no systematic approach to tuning these terms. ASIP \\cite{Li_2020_CVPR} learns to approximate shapes with low complexity polygons through a split and merge strategy. It can generate simple but not regularized shapes for vector mapping.\n\nAnother stream of work takes end-to-end approaches. PolygonRNN \\cite{Castrejon_2017_CVPR} first proposed to use LSTM (Long-Short-Term-Memory) to recurrently generate the points of a polygon in sequence. It defines a coarse grid on an image and classifies within which grid the next point should be located. To reduce the number of possible point locations, the grid is set to have low resolution, which limited the location accuracy of predicted polygons. PolygonRNN++ \\cite{Acuna_2018_CVPR} further improved PolygonRNN by predicting location offset in the grid. Polymapper \\cite{Li_2019_ICCV} improved on PolygonRNN by using ConvLSTM \\cite{NIPS2015_07563a3f} with additional global boundary mask and vertex mask for point prediction. Compared to state-of-the-art heuristics-based methods, RNN-based methods still lacks performance and are more difficult to train. \n\nInstead of recurrently generating shapes, PolygonCNN \\cite{CHEN2020114} and Polygon Transformer \\cite{Liang_2020_CVPR} predict point deformation of polygon contours. Polygon Transformer \\cite{Liang_2020_CVPR} uses Attention Network to compute shape features for deformation prediction of instance contours. These models are shown to improve instance segmentation boundaries. Similar to Polygon Transformer, we also use a transformer to encode global shape features, but with a few critical differences: our method generalizes to all different kinds of primitives, while Polygon Transformer only uses point; we aim at generic shape regularization and topology reconstruction, while Polygon Transformer is designed for semantic segmentation. \n\n\\paragraph{Road network mapping.}\nThe key challenge of road network mapping is topology reconstruction under scenarios such as shadows, tree blocking, or complex junctions and overlays. One stream of work focus on improving road segmentation, such as using CNN \\cite{Mnih_2010}, adding more context information modeling structure \\cite{Zhou_2018_CVPR_Workshops}, training with auxiliary perception loss \\cite{Mosinska_2018_CVPR}, self-supervised pre-training \\cite{singhBMVC18overhead}, and multi-stage road segment connection refinement \\cite{Batra_2019_CVPR}. Another stream of work focus on improving topology from imperfect segmentation results. \\citet{Mattyus_2017_ICCV} uses a binary decision classifier to predict the correctness of connections of nearby road endpoints from their image features. Local road structure and shape prior are not exploited. Graph-based methods \\cite{Bastani_2018_CVPR, Tan_2020_CVPR, Li_2019_ICCV} are able to use both shape prior and image information by iterative searching of the next point in the road graph, using CNNs or CNN-RNN structure. Compared to graph-based methods, our method generates a road graph in one forward run. This allows easier integration of global shape information and shapes regularization into the shape modeling process.\n\nRecently, Sat2Graph \\cite{He_2020_ECCV} achieved the state-of-the-art performance by connecting predicted key points along road direction using heuristic rules. Uniformly distributed points on road segments are predicted in a multi-task learning CNN together with point direction and road surface mask. The road network is reconstructed by connecting each key point to neighboring key points near its predicted directions. A list of shape post-processing steps with additional tunable parameters is performed to reduce loops and false connections. A fan radius threshold and fan spread angle threshold need to be tuned for connectivity estimation. We found Sat2Graph sensitive to these parameters and the accuracy of direction prediction, while our method does not need complicated parameter tuning or post-processing.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction to Belle at KEKB}\nKEKB~\\cite{bib:KEKB} is an asymmetric-energy $e^+e^-$ collider operating at and near $\\Upsilon(4S)$ mass peak. As the only detector installed in KEKB, Belle detector has a good performance on momentum and vertex resolution, $K\/\\pi$ separation etc. A detailed description of the Belle detector can be found elsewhere~\\cite{bib:BelleDetector}. It has been ten years since the final full data set ($\\sim$1 ${\\rm ab^{-1}}$) was accumulated, however, fruitful results on physics are lasting to be produced. Here we select some recent charm results from Belle to present in this proceedings. \n\n\\section{Charm-mixing parameter $y_{CP}$ in $D^0\\toK_S^0 \\omega$}\nThe mixing parameter $y_{CP}$ is measured in $D^0$ decays to the $CP$-odd final state $K_S^0\\omega$ for the first time~\\cite{bib:ycp_D0ToKsOmega}.\nConsidering mixing parameters $|x|$ and $|y|\\ll1$, the decay-time dependence of $D^0$ to a $CP$ eigenstate is approximately exponential, $d\\Gamma\/dt \\propto e^{- \\Gamma(1+\\eta_{f} y_{cp})t}$ where $\\eta_f=+1$ ($-1$) for $CP$-even (-odd) decays.\nAlong with the decay rate in flavored eigenstate decays $d\\Gamma\/dt \\propto e^{-\\Gamma t}$, the $y_{CP}$ is determined by the decay proper-time value with the formula $y_{CP}=1-\\frac{\\tau(D^{0}\\toK^{-}\\pi^{+})}{\\tau(D^{0}\\toK_S^0\\omega)}$, where $D^{0}\\toK^{-}\\pi^{+}$ is the chosen normalization mode with flavor eigenstate final state. \n\n Based on the full Belle data sample of 976 $\\rm fb^{-1}$, we obtain 91 thousands of $D^{0}\\toK_S^0\\omega$ and 1.4 millions of reference mode $D^{0}\\toK^{-}\\pi^{+}$ in $M-\\Delta M$ signal region, where $M$ is the invariant mass of reconstructed $D^{0}$ and $\\Delta M$ is the mass difference of reconstructed $D^{*+}$ and $D^{0}$. \nUsing unbinned maximum-likelihood fits for lifetime on these two samples with high purities, the proper decay-time of $D^{0}$ is determined as $\\tau_{K_S^0\\omega}=(410.47\\pm 3.73)$ fs and $\\tau_{K\\pi}=(406.53\\pm 0.57)$ {\\rm fs}, as shown in Fig.~\\ref{fig:ycp}. \nThus, we calculate $y_{CP}=(0.96\\pm0.91\\pm 0.62^{+0.17}_{-0.00})\\%$, where the first uncertainty is statistical, the second is systematic due to event selection and background, and the last is due to possible presence of CP-even decays in the data sample. \nThis $y_{CP}$ result is consistent with the world average value. \nIn the future, comparing more precise measurements of $y_{CP}$ with that of $y$ may test the SM precisely or reveal new physics effects in the charm system.\n\n\\begin{figure}[!htpb]\n \\begin{centering}\n \\begin{overpic}[width=0.49\\textwidth,height=0.48\\textwidth]{tau_ycp1.png}\n \\put(20, 64){\\large(a)}\n \\end{overpic}~~\n \\begin{overpic}[width=0.49\\textwidth,height=0.48\\textwidth]{tau_ycp2.png}\n \\put(20, 64){\\large(b)}\n \\end{overpic}\n \\vskip-5pt\n \\caption{\\label{fig:ycp} The fit of $D^0$ proper lifetime: (a) $D^0\\toK_S^0\\omega$ and (b) $D^0\\toK^{-}\\pi^{+}$. The dashed red curves are the signal contribution, and the shaded surfaces beneath are the background estimated from $M-\\Delta M$ sidebands.}\n \\end{centering}\n\\end{figure}\n\n\n\n\n\\section{Dalitz-plot analysis of $D^{0}\\toK^{-}\\pi^{+}\\eta$ decays}\nThe understanding of hadronic charmed-meson decay is theoretically challenging due to the significant non-perturbative contributions, and input from experimental measurements thus plays an important role. A Dalitz-plot analysis of $D^0\\toK^{-}\\pi^{+}\\eta$ is performed for the first time at Belle based on 953 ${\\rm fb^{-1}}$ of data~\\cite{bib:PRD102012002}. \nUsing a $M$-$Q$ two-dimensional fit where $M$ is the invariant-mass of reconstructed $D^0$ meson, $M=M(K^{+}\\pi^{-}\\eta)$, and $Q$ is the released energy of $D^{*+}$ decay, $Q=M(K^{-}\\pi^{+}\\eta\\pi_s)-M-m_{\\pi_s}$, a signal yield of $105\\,197\\pm990$ is obtained in the signal region of $1.85~{\\rm GeV}\/c^2 < M < 1.88~{\\rm GeV}\/c^2$ \nand $5.35~{\\rm MeV}\/c^2 < Q < 6.35~{\\rm MeV}\/c^2$ with a high purity $(94.6\\pm0.9)\\%$. \nThe Dalitz plot is well described by a combination of the six resonant decay channels $\\bar{K}^{*}(892)^0\\eta$, $K^{-} a_0(980)^+$, $K^{-} a_2(1320)^+$, $\\bar{K}^{*}(1410)^0\\eta$, $K^{*}(1680)^-\\pi^{+}$ and $K_2^{*}(1980)^-\\pi^{+}$, together with $K\\pi$ and $K\\eta$ S-wave components, as shown in Fig.~\\ref{fig:nominal}.\nThe dominant contributions to the decay amplitude arise from $\\bar{K}^{*}(892)^{0}$, $a_0(980)^{+}$ \nand the $K\\pi$ S-wave component. The $K\\eta$ S-wave component, including $K_0^{*}(1430)^{-}$, \nis observed with a statistical significance of more than $30\\sigma$, and the decays \n$K^{*}(1680)^{-}\\toK^{-}\\eta$ and $K^{*}_2(1980)^{-}\\toK^{-}\\eta$ are observed for the first time \nand have statistical significances of $16\\sigma$ and $17\\sigma$, respectively. \n\n\\begin{figure}[!htpb]\n \\begin{centering}\n \\begin{overpic}[width=0.45\\textwidth]{exp_dlz_withcolor.eps}\n \\put(21, 65){\\large(a)}\n \\end{overpic}%\n \\begin{overpic}[width=0.42\\textwidth]{dlz_m2ksp0_lass.eps}\n \\put(22, 68){\\large(b)}\n \\end{overpic}\\\\\n \\begin{overpic}[width=0.42\\textwidth]{dlz_m2p0et_lass.eps}\n \\put(22, 68){\\large(c)}\n \\end{overpic}%\n \\begin{overpic}[width=0.42\\textwidth]{dlz_m2kset_lass.eps}\n \\put(22, 68){\\large(d)}\n \\end{overpic}%\n \\vskip-5pt\n \\caption{\\label{fig:nominal} The Dalitz plot of $D^{0}\\toK^{-}\\pi^{+}\\eta$ in (a) $M$-$Q$ signal region $1.85~{\\rm GeV}\/c^2 < M < 1.88~{\\rm GeV}\/c^2$ \nand $5.35~{\\rm MeV}\/c^2 < Q < 6.35~{\\rm MeV}\/c^2$, and projections on (b) $m_{K\\pi}^2$, (c) $m_{\\pi\\eta}^2$ and (d) $m_{K\\eta}^2$. In projections the fitted contributions of individual components are shown, along with contribution of combinatorial background (grey-filled) from sideband region.}\n \\end{centering}\n\\end{figure}\n\nWe extract the signal yield from the $D^0$ invariant mass distribution in $1.78~{\\rm GeV}\/c^2T_f$), 54(3) ($11$~K$\\sim T_f$), 45(1) ($4.2$~K, sharp), and 30(2)~\\textmu s ($4.2$~K, broad). $T_2$ drops sharply at $T_f$=11.1~K and does not recover even at base temperature, as is the case for other frustrated systems with slow low-temperature dynamics.\nIn order to quantify the loss of NMR intensity below $T_f$, all data were normalized by the extrapolated $I(0)T$ at 30~K [Fig.~R2(b)]. The resulting $I(0)T$ are 0.66(2), 0.16(0), and 0.75(4) at 11.1~K, and for the sharp and broad component at 4.2~K, respectively. This indicates that 91~\\% of the intensity is conserved at the base temperature. Finally, we note that the ratio of the intensities corresponding to the two components is estimated to be $1:4.7$ at the base temperature.\n\n\\begin{figure*}[htbp]\n\t\\includegraphics[width=0.75\\linewidth]{SupMat_IntegInt2.eps}\n \\caption{(a) $\\tau$ dependence of spectra at 4.2~K. (b) $\\tau$ dependence of the integrated intensity of NMR spectra. }\n \\label{IntegInt}\n\\end{figure*}\n\t\n\t\t\n\\subsection*{Neutron and X-ray diffraction}\nThe sample used in the synchrotron X-ray powder diffraction experiment was enclosed in a silica capillary of $3$ mm diameter, and loaded into a $^4$He flow cryostat with a base temperature of 5 K. Neutron powder diffraction patterns were measured on 5.8 g of powder loaded in an $8$ mm diameter vanadium can. \n\nThe measured diffraction patterns were analysed by means of Rietveld refinement. Part of one refined neutron diffraction pattern is presented in Fig.~\\ref{NDpattern}, and parameters showing the goodness of fit for all detector banks and temperatures are presented in Tab.~\\ref{fit_params}. Due to the fact that the peak profile of WISH is difficult to describe with the commonly used back-to-back exponential function and high-$Q$ parts of the patterns suffer from significant peak overlap, we present the weighted $R$ parameters for the Rietveld refinement and LeBail profile matching (representing the best possible fit for the given set of profile parameters) along with the expected $R$ value based on the data set's statistics. The strain ($S_{hkl}$) parameters, whose temperature dependence is presented in main body of this work, were extracted from the SXRPD data, due to the superior resolution of these measurements. On the other hand, $B_{iso}$ and the fractional $x$ coordinate of chromium were derived from refinements of neutron powder diffraction data. The refined structural parameters are shown in Table~\\ref{Rietv_params}, where position values are retrieved from SXRD patterns and $B_{iso}$ parameters come from NPD data.\n\n\t\\begin{figure*}[htbp]\n\t\\includegraphics[width=0.8\\linewidth]{FigS2.eps}%\n\t\\caption{Measured time-of-flight neutron diffraction pattern (black points) at $1.5$~K and its refinement (red line). The data was measured on banks 5 and 6 of the WISH diffractometer in backscattering geometry.}\n\t\\label{NDpattern}\n\t\\end{figure*}\n\t\n\t\t\\begin{table*}\n\t\\caption{Parameters showing the goodness of fit for high angle detector banks of WISH. Where $R_{Rwp}$ and $R_{Lwp}$ are weighted $R$ paraneters for Rietveld and LeBail refinements respectively and $R_{exp}$ is $R$ expected.\\label{fit_params}}\n\t\\begin{ruledtabular}\n\t\\begin{tabular}[c]{cccc}\n $T$ (K)\t& Banks 5 \\& 6 \t\t\t\t\t\t\t\t\t& Banks 4 \\& 7\t\t\t\t\t\t\t\t& Banks 3 \\& 8\t\t\t\t\t\t\t\t\\\\ \\hline\n 1.5\t\t& $R_{Rwp}$=7.11 $R_{Lwp}$=6.32 $R_{exp}$=0.64\t\t& $R_{Rwp}$=6.86 $R_{Lwp}$=6.13 $R_{exp}$=0.65\t& $R_{Rwp}$=6.32 $R_{Lwp}$=5.81 $R_{exp}$=0.67\t\\\\\n 6\t\t& $R_{Rwp}$=7.15 $R_{Lwp}$=6.37 $R_{exp}$=0.66\t\t& $R_{Rwp}$=6.88 $R_{Lwp}$=6.15 $R_{exp}$=0.68\t& $R_{Rwp}$=6.34 $R_{Lwp}$=5.83 $R_{exp}$=0.69\t\\\\\n 11\t\t& $R_{Rwp}$=7.98 $R_{Lwp}$=7.12 $R_{exp}$=0.57\t\t& $R_{Rwp}$=7.53 $R_{Lwp}$=7.16 $R_{exp}$=0.58\t& $R_{Rwp}$=6.85 $R_{Lwp}$=6.80 $R_{exp}$=0.60\t\\\\\n 15\t\t& $R_{Rwp}$=7.87 $R_{Lwp}$=7.09 $R_{exp}$=0.56\t\t& $R_{Rwp}$=7.75 $R_{Lwp}$=7.24 $R_{exp}$=0.40\t& $R_{Rwp}$=7.37 $R_{Lwp}$=6.98 $R_{exp}$=0.41\t\\\\\n 30\t\t& $R_{Rwp}$=7.96 $R_{Lwp}$=7.19 $R_{exp}$=0.39\t\t& $R_{Rwp}$=7.88 $R_{Lwp}$=7.39 $R_{exp}$=0.40\t& $R_{Rwp}$=7.51 $R_{Lwp}$=7.10 $R_{exp}$=0.41\t\\\\\n\t\\end{tabular}\n\t\\end{ruledtabular}\n \\end{table*}\n\t\n\t\\begin{table*}\n\t\\caption{Structural parameters obtained by refinement of neutron and synchrotron X-ray powder diffraction data at 6 K.\\label{Rietv_params}}\n\t\\begin{ruledtabular}\n\t\\begin{tabular}[c]{ccccccc}\n Atom\t& Wyckoff position\t& $x$\t\t\t& $y$\t\t\t& $z$\t\t\t& Occupancy\t& $B_{iso}$ (\\AA$^2$)\t\\\\ \\hline\n Li1\t& $4a$\t\t\t\t& 0\t\t\t\t& 0\t\t\t\t& 0\t\t\t\t& 0.994(2)\t& 2.175(171)\t\t\t\t\\\\\n Ga1\t& $4d$\t\t\t\t& 0.75\t\t\t& 0.75\t\t\t& 0.75\t\t\t& 0.932(6)\t & 0.406( 60)\t\t\t\t\\\\\n Ga2\t& $4a$\t\t\t\t& 0\t\t\t\t& 0\t\t\t\t& 0\t\t\t\t& 0.006(4)\t& 0.406( 60)\t\t\t\t\\\\\n Li2\t& $4d$\t\t\t\t& 0.75\t\t\t& 0.75\t\t\t& 0.75\t\t\t\t& 0.002(4)\t& 2.175(171)\t\t\t\t\\\\\n In\t& $4d$\t\t\t\t& 0.75\t\t\t& 0.75\t\t\t& 0.75\t\t\t& 0.066(6)\t& 0.406( 60)\t\t\t\t\\\\\n Cr\t& $16e$\t\t\t\t& 0.37185(4)\t\t& 0.37185(4)\t\t& 0.37185(4)\t\t& 4\t\t\t& 0.385( 36)\t\t\t\t\\\\\n O1\t& $16e$\t\t\t\t& 0.13702(14)\t& 0.13702(14)\t& 0.13702(14)\t& 4\t\t\t& 0.509( 20)\t\t\t\t\\\\\n O2\t& $16e$\t\t\t\t& 0.61802(16)\t& 0.61802(16)\t& 0.61802(16)\t& 4\t\t\t& 0.509( 20)\t\t\t\t\\\\\n\t\\end{tabular}\n\t\\end{ruledtabular}\n \\end{table*}\n\n\\subsection*{Magnetic diffuse scattering and reverse Monte Carlo refinement}\n\nTo extract the diffuse magnetic scattering from the neutron powder diffraction data, a flat background $C$ ($C\\simeq 0.3$) was subtracted from the data such that the mean intensity at $Q<0.4$ \\AA$^{-1}$ was zero. This choice was justified by the fact that the high temperature diffuse scattering in the first Brillouin zone is zero (indeed, polarized neutron scattering on the $x=1$ compound, where the high temperature scattering is similar, also suggests this is the case \\cite{nilsen15}), and that the WISH instrumental background is flat when using a V can and radial oscillating collimator. The nuclear Bragg peaks were removed from the ranges where the build-up of diffuse scattering was observed. RMC refinements were performed on boxes of spins containing $6\\times 6\\times 6$ unit cells at low temperature and $3\\times 3\\times 3$ unit cells at $30$~K, and the simulation was repeated 10 times for every temperature point to ensure a stable solution. The optimal value of the Monte Carlo weight parameter \\cite{paddison13} for all temperature datasets was determined to be $W=0.6$ for the low-$T$ Ising-type collinear spin models and $W=1$ for the high-$T$ Heisenberg-type spin models.\n\nThe analysis of the RMC-refined spin configurations consisted of several steps. Firstly, for each simulation box, the normalized real space spin-spin correlation function was evaluated:\n\\begin{widetext}\n\\begin{equation}\n\\langle S_0 \\cdot S_i\\rangle\/S(S+1)=\\langle S(0) \\cdot S(r)\\rangle\/S(S+1)=\\frac{1}{n(r)S(S+1)}\\sum^N_j\\sum^{Z_{jk}(r)}_k\\mathbf{S}_j(0)\\cdot \\mathbf{S}_k(r),\n\\end{equation}\n\\end{widetext}\nwhere $\\mathbf{S}_i$ is the vector of the $i$-th spin in the simulation box, $N$ is the total number of sites inside the box, and $Z_{ij}(r)$ is the number of spins in the coordination shell at distance $r$. This was done using the SPINCORREL application in the SPINVERT suite \\cite{paddison13}. Despite the noticeable differences between the form of the diffuse scattering at high and low temperatures, it is difficult to distinguish the differences between the real space spin-spin correlations corresponding to the fits of those datasets. This is most likely due to the dominant contribution of the broad Coulomb liquid-like component to both patterns. Following evaluation of $\\langle S_0 \\cdot S_i\\rangle$, the single crystal patterns were reconstructed in the $(hk0)$ and $(hhl)$ planes using SPINDIFF, also part of the SPINVERT suite. This allowed for a more diagnostic comparison with theoretical calculations, and permitted identification of the Coulomb-like and short-range ($\\mathbf{k}=(001)$) ordered components of the scattering.\n\nThe third step, the calculation of color populations and correlations, applied only to the low-temperature collinear spin configurations. In the case of a single tetrahedron, the bilinear-biquadratic model yields six possible magneto-structural ground state configurations, three of which are independent with respect to a global rotation of spins. These may be given a color $c=\\{R,G,B\\}$ according to the direction of the ferromagnetic bond (which corresponds to the long bond in the tetragonally distorted tetrahedron), as explained in the main text and Ref. \\cite{tchernyshyov02}. In the pure bilinear-biquadratic model on the pyrochlore lattice, the colors in the low-temperature nematic state are uncorrelated, whereas in the bilinear-biquadratic model with long-range couplings, they are fully ordered. By calculating the correlations between the colors, we may thus effectively separate the correlations due to the nematic and short-range ordered components identified in the second step. Since the diamond lattice formed by the tetrahedra is bipartite, a color correlation function (order parameter) which distinguishes between same (\\textit{e.g.} all R) and different color (\\textit{e.g.} RG) orders is considered sufficient:\n\\begin{equation}\n\\langle C_0C_i\\rangle=\\langle C(0)C(r)\\rangle=\\frac{N_s^b(r)-\\bar{p}_{sd}N_d^b(r)}{Z_{tot}^b}\n\\end{equation}\nwhere $N_s^b(r)$ and $N_d^b(r)$ are the number of bonds of the same and different color at distance $r$, respectively, $Z_{tot}^b$ is the total number of bonds for that distance, and \n\\begin{equation}\n\\bar{p}_{sd}=\\frac{\\sum_{c=c\\prime}(N_c^t)^2\/N_{tot}^2}{\\sum_{c\\neq c\\prime}N_c^tN_{c^\\prime}^t\/N_{tot}^2}\n\\end{equation}\nis the mean ratio of the probabilities of finding the same color tetrahedron to a different color tetrahedron on a particular bond. The final term is required as the populations of the colors in the simulation box are generally not equal. This correlation function generates $\\langle C_0C_i\\rangle=1$ for all shells for a same color order, and an alternation between $\\pm1$ for a different color order.\n\n\n\\subsection*{Spectrum simulation from RMC spin map}\n\nThe $^7$Li NMR spectrum of the low temperature phase was simulated from the spin configurations obtained by RMC. The shape of the powder averaged NMR spectra of magnetic substances is mainly determined by magnitude of the internal field at the nuclear positions. To obtain an internal field distribution at each $^7$Li site from a given spin map, we summed up the classical dipole field and transfer hyperfine field from Cr$^{3+}$ spins within a 100~{\\AA} radius and from nearest-neighbor Cr$^{3+}$ sites, respectively. The transfer hyperfine coupling constant is 0.10~T\/$\\mu_{B}$ from the 12 nearest-neighbor Cr$^{3+}$ spins, as estimated from a $K$-$\\chi $ plot above 100~K. Figure~\\ref{Bhist} shows a histogram of magnitude of the internal field $B_\\mathrm{int}$ calculated from the spin map with ordered moments of 1~$\\mu_{B}$ per Cr$^{3+}$ ion.\n\n\t\\begin{figure}[htbp]\n\t\\includegraphics[width=0.8\\linewidth]{FigS4.eps}%\n\t\\caption{Histogram of magnitude of the internal field at Li sites. The size of ordered moment in the RMC spin map is taken as 1~$\\mu_{B}$ per Cr$^{3+}$ ion.}\n\t\\label{Bhist}\n\t\\end{figure}\n\n\t\\begin{figure}[htbp]\n\n\t\\includegraphics[width=0.9\\linewidth]{SimulatedSp2}%\n\t\\caption{Experimental NMR spectrum of the ordered phase at 4.2~K with pulse separation $\\tau=7$~\\textmu s(black curve) and simulated spectrum from RMC spin map (red curve). The unit of the horizontal axis of the experimental one is changed to magnetic field by dividing frequency by the gyromagnetic ratio of $^{7}$Li. The origin of the horizontal axis corresponds to the center of gravity of both spectra.}\n\t\\label{SimulatedSp}\n\t\\end{figure}\n\t\t\nWhen the external field $B_\\mathrm{ext}$ is sufficiently larger than $B_\\mathrm{int}$, a rigid AF spin arrangement producing a single value of $B_\\mathrm{int}$ yields a rectangular NMR spectrum bounded by $|B_\\mathrm{int}-B_\\mathrm{ext}| \\leq B \\leq B_\\mathrm{int}+B_\\mathrm{ext}$ for powder samples~\\cite{yamada86} . We obtained our simulated NMR spectrum by piling up rectangle distributions whose half widths were $B_\\mathrm{int}$, the horizontal axis of Fig.~\\ref{Bhist}. The height of each rectangle was normalized so that its area was proportional to the vertical value of each point in Fig.~\\ref{Bhist}.\n\nThe experimental NMR spectrum at 4.2~K and a simulation from an RMC spin configuration at 1.5~K are shown in Fig~\\ref{SimulatedSp}; the vertical and horizontal scales of the simulated spectrum are both adjusted to match the broad component of the experimental spectrum. While the simulated spectrum appears triangular, the experimental one clearly consists of two Gaussian components with different linewidth. This discrepancy could be due to a variety of factors, including the difficulty of fitting the $\\mathbf{k}=(001)$ scattering near nuclear positions, the different temperatures of the NMR ($4.2$~K) and neutron ($1.5$~K) measurements, and the inhomogenous decay of the NMR intensity due to slow fluctuations.\n\nBased on the horizontal axis scaling factor mentioned above, the moment size of Cr$^{3+}$ spins in the wide component may be estimated to be 1.3 $\\mu_B$ at 4.2~K. The moment size in the sharper component is more difficult to estimate. However, because the perfect two-color orders produce either no field distribution or a relatively narrow rectangular spectrum (the FWHM is 0.1~T at most for a full moment $gS=3~\\mu_{B}$) depending on the color configurations, this contribution can readily be associated with two-color order domains.\n\n\\subsection*{Magnetic susceptibility}\n\nThe zero field cooled (ZFC) and field cooled (FC) magnetic susceptibility curves for $x=0.05$ (Fig.~\\ref{Chi}) show a clear splitting, indicating spin freezing below $T_f$; hence, the nematic spin freezing scenario appears likelier than the pure nematic transition. Unlike a regular spin glass, the splitting persists up to $5$~T. This robustness towards applied magnetic field is also observed in materials like SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ and Y$_2$Mo$_2$O$_7$ \\cite{gardner99,silverstein14}, where the ground states are characterised by flat energy landscapes with shallow minima. Such energy landscapes, and as a result, behaviors of the ZFC and FC susceptibility, have also been predicted for the pyrochlore lattice with bilinear-biquadratic interactions and bond disorder \\cite{shinaoka14}.\n\n\t\\begin{figure\n\t\\includegraphics[width=\\linewidth]{FigS5}%\n\t\\caption{Magnetic susceptibility: zero field cooled(ZFC) and field cooled(FC) measured at various values of magnetic field.}\n\t\\label{Chi}\n\t\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\subsection{Attributed Graphs}\n\\label{sec:attributedG}\n\nAs shown in Table \\ref{tab:methods}, for attributed graphs, five kinds of attributes have been considered for CS, which are keywords, locations, temporal information, profile, and influence values. However, the semantics of these attributed communities are different. Moreover, the problem definitions are also different. Therefore, it may not make sense to compare them under the same metrics.\n\nFor location, temporal information, and profile-based attributed graphs, only the $k$-core model has been studied on these graphs, which have been discussed and compared extensively in Section \\ref{sec:kcoreDiscuss}.\nFor influence value-based graphs, the meanings of influences are very different.\nIn $k$-core-based CS solutions \\cite{Li:vldb:2015,Li:VLDBJ:2017,chen2016efficient,Bi:2018,Li:SIGMOD:2018,ding2018search}, the influence values are associated to graph vertices, denoting their influence or importance.\nIn $k$-truss-based CS solutions \\cite{Zheng:IS:2017}, the influence values are associated to graph edges, representing the influence or importance of edges.\nIn $k$-clique-based CS solutions \\cite{kclique2017}, the influence values are also associated to graph edges, but they are probability values, meaning how likely a vertex is influenced by another vertex.\nMeanwhile, none of these influence value-based graphs has been investigated with at least two different cohesiveness metrics, so we do not compare solutions for influence value-based graphs in this paper.\nIn the following, we mainly focus on comparing and analyzing CS solutions on keyword-based attributed graphs.\n\nFor keyword-based attributed graphs, there are two representative studies,\nnamely ACQ \\cite{Fang:VLDB:2016,Fang:VLDBJ:2017} and ATC \\cite{Huang:2017:ATC}. Generally, both of them seek to find a densely connected community containing query vertex(es) with similar query keywords, but ACQ adopts the $k$-core model, while ATC uses the $k$-truss model.\nFrom the discussions in Section~\\ref{sec:analysis}, we infer that\nthe community of ATC\nis more structurally cohesive, but may take higher computational cost.\nBesides, in terms of keyword cohesiveness, ACQ model in Section~\\ref{sec:kcoreKeyword} imposes a strict homogeneity constraint, requiring that each vertex shares same query attributes in the community; ATC model in Section~\\ref{sec:ktrussInfluence} uses an attribute score function to quantify the query keyword coverage and allows missing some query keywords in the community.\n\nIn \\cite{Huang:2017:ATC}, Huang et al. empirically compared the community quality and efficiency of ACQ and ATC. \nThey used 13 real graphs with ground-truth communities. For each graph, they ran 200 CS queries.\nSpecifically, for each query, they randomly selected a ground-truth community, and then randomly selected a vertex from the community as the query vertex. After that, they ran ACQ and ATC with the same parameters, i.e., $k$=4 and two query keywords which are selected from the community.\nThe results are consistent with the discussions above. Specifically, ATC achieves higher average $F_1$ score values than ACQ on all the datasets, which means that it is more accurate to search communities. On the other hand, in terms of efficiency, ACQ consistently outperforms ATC on all the datasets, and is up to two orders of magnitude faster than ATC. \n\\subsection{Query Biased Density-Based Community Search}\n\\label{sec:bias}\n\nIn \\cite{Wu:VLDB:2015}, Wu et al. proposed the query biased density as the goodness function for CS. Before introducing the query biased density, the authors presented a vertex weighting scheme, which ensures that vertices far away from the query vertices will have large weights, resulting in high penalties to be included in the community. To assign each vertex $u$ a weight $r(u)$ w.r.t a set $Q$ of query vertices, they adopted the penalized hitting probability, which can be computed by random walk. Then, the query biased vertex weight of vertex $u$, $\\pi(u)$, can be defined as the reciprocal of $r(u)$, i.e., $\\pi(u)$=$1\/r(u)$.\n\nBased on the weights, the authors defined the {\\it query biased density} of a graph $S$ as $\\rho(S)$=$\\frac{e(S)}{\\pi(S)}$, where $e(S)$ is the sum of edges weights and $\\pi(S)$ is the sum of query biased weights for vertices in $S$. After that, the authors proposed and studied the problem of finding the query biased densest subgraph $S$ from a graph $G$ (or QDS problem), which theoretically guarantees that QDS is a connected subgraph and contains $Q$.\n\nClearly, if $\\pi(u)$=1, the query biased density degenerates to the classical edge-density (i.e., $\\frac{e(S)}{|S|}$), and accordingly the QDS problem is reduced to the problem of densest subgraph discovery \\cite{goldberg1984finding}. This also implies that after weighting $\\pi(u)$, it forces the global densest subgraph shift to the neighborhood of the query vertices.\n\nUnfortunately, the QDS problem is computationally intractable. To improve efficiency, the authors introduced two variants of the QDS problem by removing constraints that $S$ is connected and $Q$ is included in $S$, respectively. They showed that these variants can be solved in polynomial time and the results can be used to find an optimized solution for the QDS problem.\n\n\\subsection{Community Detection}\n\\label{sec:CD}\n\nBelow, we review representative CD studies on undirected graphs, directed graphs, and attributed graphs.\n\n\\subsubsection{Undirected Graphs}\n\\label{sec:CDSimple}\n\nA large number of studies aim to detect communities from simple graphs, and we can classify these studies based on the techniques they use. Some representative classes are as follows, to name a few:\n\\begin{enumerate}\n \\item community quality optimization-based methods (e.g., modularity \\cite{newman2004fast});\n \\item clustering methods (e.g., $k$-means \\cite{tang2009scalable}, spectral clustering \\cite{von2007tutorial});\n \\item graph partitioning methods (e.g., Metis \\cite{karypis1995metis});\n \\item embedding-based methods (e.g., DeepWalk \\cite{perozzi2014deepwalk}, \\cite{li2018community});\n \\item random walk-based methods (e.g., \\cite{pons2005computing});\n \\item label propagation-based methods (e.g., \\cite{gregory2010finding});\n \\item information diffusion-based methods (e.g., \\cite{hajibagheri2012community});\n \\item statistic inference-based models (e.g., \\cite{hastings2006community});\n \\item deep learning-based methods (e.g., \\cite{Yang:2016});\n \\item centrality-based methods (e.g., \\cite{community-phy2004});\n \\item locality sensitive hashing-based methods (e.g., \\cite{macropol2010scalable});\n \\item physics-based methods (e.g., Potts low \\cite{wu1982potts});\n \\item local metric-based methods (e.g., $k$-plex \\cite{conte2018d2k});\n \\item multi-commodity flow-based methods (e.g., \\cite{leighton1988approximate});\n \\item hybrid-based methods (e.g., \\cite{henderson2010hcdf}).\n\\end{enumerate}\n\nFor a detailed survey of CD, please refer to the following survey and empirical evaluation papers:\n\\cite{CD:Survey:2009,Yang2010,community-phy2010,papadopoulos2012community,danon2005comparing,gulbahce2008art,CD:Survey:2011,CD:Survey:2011a,CD:Survey:2013,CD:Survey:2017,CD:Survey:2014-overlap,CD:Survey:2015-multilayer,CD:Survey:2018-dynamic,CD:Survey:2010-compare,CD:Survey:2014-evaluate,CD:Survey:2015-truth}.\nAlthough these CD solutions are able to discover communities from networks, they may not well satisfy the desirable factors of CS on big graphs as we discuss in Section \\ref{sec:intro}, because most of them often use a global predefined criterion for generating communities and cannot find communities in an online manner.\n\n\n\\subsubsection{Directed Graphs}\n\\label{sec:CDDirected}\n\nIn recent years, a number of studies have investigated CD on directed graphs. Here are some representative studies, to name a few.\nIn~\\cite{prl2008}, Leicht et al. extended the concept of modularity maximization~\\cite{newman2004fast}, which was originally designed for undirected graphs, for detecting community structure in directed networks that makes explicit use of information contained in edge directions. In \\cite{flake2000}, Flake et al. identified communities from websites network, which can be considered as directed graphs.\nIn~\\cite{CDBenchmark}, Lancichinetti et al. introduced new benchmark graphs to test CD methods on directed\nnetworks.\nIn~\\cite{pre2010}, Kim et al. also proposed a new modularity metric for CD on directed networks.\nIn~\\cite{sdm2010}, Yang et al. developed a new stochastic block model for CD on directed networks.\nIn~\\cite{yang2014detecting}, Yang et al. presented algorithms for detecting communities from both directed and undirected networks.\nNing et al. \\cite{ning2016local} studied local community extraction in directed networks.\nA recent survey can be found in~\\cite{CDSurvey}.\n\n\\subsubsection{Keyword-Based Attributed Graphs}\n\\label{sec:CDKeyword}\n\nTo identify communities from keyword-based attributed graphs, recent works~\\cite{attr-vldb2009,SDM2013,WSDM2013,cheng2012clustering,huang2012semi,attr-www2013} often use clustering techniques. Zhou et al. \\cite{attr-vldb2009} computed vertices' pairwise similarities using both links and keywords, and then clustered the graph.\nSubbian et al. \\cite{SDM2013} explored noisy labeled information of graph vertices for finding communities.\nQi et al. \\cite{WSDM2013} dynamically maintained communities of moving objects using their trajectories.\nRuan et al.~\\cite{attr-www2013} developed a method {\\tt CODICIL}, which augments the original graph by creating new edges based on content similarity, and then performs clustering on the new graph. Huang et al.~\\cite{huang2016attributed} investigate community detection and community search in attributed networks, respectively in terms of global network-wide analysis and ego-centric personalized analysis aspects. \n\nAnother common approach is based on topic models. In~\\cite{attr-topic-kdd2008,attr-topic-icml2009}, the {\\tt Link-PLSA-LDA} and {\\tt Topic-Link LDA} models jointly model vertices' content and links based on the {\\tt LDA} model. In~\\cite{attr-topic-sigmod2012}, the attributed graph is clustered based on probabilistic inference. In~\\cite{attr-topic-www2012}, the topics, interaction types, and the social connections are considered for discovering communities. {\\tt CESNA}~\\cite{yang2013community} detects overlapping communities by assuming communities ``generate'' both the link and content. A discriminative approach~\\cite{attr-kdd2009} has also been considered for community detection.\nHowever, computing pairwise similarity among vertices is very costly, and thus they are questionable for performing online CS queries.\n\n\n\n\\subsubsection{Location-Based Attributed Graphs}\n\\label{sec:CDLocation}\n\nThe problem of CD on location-based attributed graphs (or geo-social networks) \\cite{barthelemy2011} has been extensively studied \\cite{girvan2002,guo2008,pnas2011,KDD2013,IJGIS2015}. In \\cite{girvan2002}, Girvan et al. introduced the geo-community, which is a graph of intensely connected vertices being loosely connected with others, but it is more compact in space.\nGuo et al.~\\cite{guo2008} proposed the average linkage (ALK) measure for clustering objects in spatially constrained graphs.\nIn~\\cite{pnas2011}, Expert et al. uncovered communities from spatial graphs based on modularity maximization.\nIn~\\cite{KDD2013}, Shakarian et al. used a variant of Newman-Girvan modularity to mine the geographically dispersed communities.\nIn~\\cite{IJGIS2015}, Chen et al. proposed a method using modularity maximization for detecting communities from geo-social networks.\n\n\\subsubsection{Temporal Graphs}\n\\label{sec:CDTemporal}\n\nMany recent studies aim to detect communities from temporal graphs. In \\cite{zhou2007discovering}, Zhou et al. studied CD over a temporal heterogeneous social network consisting of authors, document content, and the venues.\nIn \\cite{liu2014persistent}, Liu et al. studied persistent community detection for identifying communities that exhibit persistent behavior over time.\nIn \\cite{angadi2015overlapping}, Angadi et al. detected communities from dynamic networks where data arrives as a\nstream to find the overlapping vertices in communities.\nIn \\cite{bazzi2016community}, Bazzi et al. investigated the detection of communities in temporal multi-layer networks.\nIn \\cite{ditursi2017local}, DiTursi et al. proposed a filter-and-verify framework for community detection in dynamic networks.\nIn \\cite{kuncheva2017multi}, Kuncheva et al. presented a method by using spectral graph wavelets to detect communities in temporal graphs.\nFor more related studies, please refer to survey papers \\cite{CD:Survey:2018-dynamic,tamimi2015literature}.\n\n\\subsection{C-Explorer}\n\\label{sec:cexplorer}\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width=3.32in]{figures\/cexplorerMain}\n \\caption{Interface of C-Explorer~\\cite{Fang:demo:2017}.\\label{fig:ce-ui}}\n\\end{figure}\n\nC-Explorer is a web-based system that enables community retrieval in a simple, online, and interactive manner. The key features of C-Explorers are as follows:\n\nFirst, it implements several typical CS algorithms on simple undirected graphs and keyword-based attributed graphs, including {\\tt Global} and {\\tt Local} (see Section~\\ref{sec:kcoreSimple}), ACQ algorithm (see Section~\\ref{sec:kcoreKeyword}). In addition, a CD algorithm called {\\tt CODICIL}~\\cite{attr-www2013} is included.\n\nSecond, it offers a user-friendly facility that enables online visualization of communities. Fig. \\ref{fig:ce-ui} shows the user interface of C-Explorer configured to run on the DBLP bibliographical network. On the left panel, a user inputs the name of an author (e.g., ``jim gray\") and the minimum degree of each vertex in the community she wants to have. The user can also indicate the labels or keywords related to her community. Once she clicks the ``Search\" button, the right panel will display a community of Jim Gray. The user can further click on one of the vertices (e.g., Michael Stonebraker), and continue to examine its community.\n\nThird, it allows users to compare the communities retrieved by various CS and CD algorithms, in terms of community quality and statistics.\n\nFinally, it provides a list of API functions so that other CS and CD algorithms can be plugged in. For public users, they can easily plug their own algorithms into C-Explorer using these API functions.\n\n\n\n\\subsection{Cohesive SubGraph Discovery}\n\\label{sec:cohesive}\n\nIn this section, we review studies on cohesive subgraph discovery. Notice that CD is one kind of cohesive subgraph discovery, but the latter one is more general.\n\n\\subsubsection{Simple Graphs}\n\\label{sec:cohSimple}\n\nFor simple graphs, typical cohesive subgraph models are\n$k$-core \\cite{md1983,kcore2003},\n$k$-truss \\cite{saito2008extracting,cohen2008trusses,zhang2012extracting},\n$k$-clique \\cite{kclique,article05clique},\nand $k$-ECC \\cite{gibbons1985algorithmic,hu2016querying}, as discussed in Section \\ref{sec:pre}.\nTo compute these subgraphs, there are many efficient in-memory algorithms (e.g., $k$-core \\cite{kcore2003}, $k$-truss \\cite{wang2012truss}, $k$-clique \\cite{mauro2018}, and $k$-ECC \\cite{zhou2012finding,Chang:SIGMOD:2013,akiba2013linear}).\nFor graphs that are too large to be kept in memory, there are also some disk-based and parallel algorithms.\nFor example, in \\cite{cheng2011efficient,wen2018efficient}, \\cite{wang2012truss,khaouid2015k}, and \\cite{cheng2012fast}, disk-based algorithms for computing $k$-core, $k$-truss, and $k$-clique are developed, respectively; in \\cite{montresor2013distributed} and \\cite{chen2014distributed}, parallel algorithms for computing $k$-core and $k$-truss are proposed, respectively. In addition, to maintain $k$-core and $k$-truss for dynamic graphs, some efficient algorithms are developed in \\cite{li2014efficient,sariyuce2016incremental,zhang2017fast} and \\cite{zhou2014efficient}, respectively.\n\nBesides, there are many other cohesive subgraph models and the representatives are as follows.\nIn \\cite{seidman1978graph}, Seidman proposed the $k$-plex model (which is introduced in Section~\\ref{sec:kclique}).\nIn \\cite{matsuda1999classifying}, Matsuda et al. introduced the concept of quasi-clique model.\nIn \\cite{zhang2018discovering}, Zhang et al. proposed the ($k$, $s$)-core, which considers both user engagement and tie strength.\nIn \\cite{sariyuce2016fast}, the authors proposed the concept of nucleus, which is a generalization of $k$-core and $k$-truss.\nIn \\cite{zhao2012large}, Zhao et al. introduced the mutual-friend subgraph.\nIn \\cite{wang2010triangulation}, Wang et al. proposed the DN-Graphs by considering vertices' common neighbors.\nIn \\cite{Chang:SIGMOD:2013}, Chang et al. studied the problem of enumerating $k$-ECCs in a graph for a given $k$.\nIn \\cite{zhu2018diversified}, Zhu et al. introduced the notion of coherent cores on multi-layer graphs.\nIn addition, Goldberg et al. \\cite{goldberg1984finding} discovered the densest subgraph, Galbrun et al. \\cite{Galbrun:2016} studied the top-$k$ densest subgraphs, Tsourakais et al. \\cite{tsourakakis2013denser} computed the quasi-clique-based dense subgraphs, and Qin et al. \\cite{Qin:2015} proposed and studied the problem of finding top-$k$ locally densest subgraphs.\n\n\n\\subsubsection{Attributed Graphs}\n\\label{sec:cohAttributed}\n\nFor attributed graphs, in addition to CD methods, there are also many studies of finding cohesive subgraphs.\nIn \\cite{denian:2012}, Yang et al. studied the socio-spatial group query which finds a group of users that are cohesively linked and close to the rally point in a geo-social network.\nIn \\cite{Zhang:VLDB:2017}, Zhang et al. studied the problem of finding ($k$, $r$)-cores on attributed graph and for a specific ($k$, $r$)-core, each vertex has at least $k$ neighbors, and the attribute similarity of each pair of vertices is at least $r$.\nIn \\cite{Chen:VLDB:2018}, Chen et al. studied the problem of ($k$, $d$)-MCC (maximum co-located community) search on geo-social network, where a ($k$, $d$)-MCC is a connected $k$-truss and for any two vertices, their distance is at most $d$.\nIn addition, Wu et al. \\cite{wu2015finding} studied the problem of finding the densest connected subgraph from the dual network, which can be considered as an attributed graph.\n\n\\section{Comparison Analysis}\n\\label{sec:compare}\n\nRecall that in the last subsections of Sections \\ref{sec:kcore}, \\ref{sec:ktruss}, \\ref{sec:kclique}, and \\ref{sec:kecc}, we have compared and analyzed the CS solutions using $k$-core, $k$-truss, $k$-clique, and $k$-ECC, respectively.\nIn this section, we would like to further compare these CS solutions across different metrics. Due to the space limitation, we are unable to compare all the surveyed 27 CS problems as well as their solutions. In the following, we mainly compare the representative CS problems and solutions on simple graphs and attributed graphs respectively, while other solutions can be considered as either their variants or less representative studies.\n\n\n\\input{simpleG}\n\n\\input{attributedG}\n\n\\section{Conclusion}\n\\label{sec:conclude}\n\nIn this paper, we conduct an extensive survey on the topic of community search over large graphs. We systematically review over 30 research articles, which focus on the topic of community search, published between 2010 and 2019.\nWe first analyze and compare different community cohesiveness metrics. Then, we classify studies about CS according to these metrics, and for each class of works, we review and discuss the representative studies on different types of graphs. Furthermore, two systems that are customized for the purpose of community search are discussed. Finally, we point out a list of future research topics as well as challenges. In summary, our survey provides an overview of the start-of-the-art research achievements on the topic of community search, and it will give researchers a thorough understanding of community search.\n\n\\section{Community Search Systems}\n\\label{sec:demo}\n\nRecently, many graph processing systems have been developed \\cite{Batarfi:2015}. Generally, they can be classified into two groups. The first group (e.g., GraphX \\cite{gonzalez2014graphx} and Pregel \\cite{malewicz2010pregel}) aims to provide a platform for supporting general graph tasks (e.g., computing PageRank scores).\nThe second group is customized for specific graph tasks. For example, in \\cite{fei2013expfinder}, Fan et al. developed a graph system, called Expfinder, for finding experts in social networks; in \\cite{jayaram2015viiq}, a system called VIIQ is developed for interactive graph query formulation; in \\cite{yi2017autog}, AutoG shows an interactive system to facilitate graph query formulation. However, none of them can be readily used for CS. To address this issue, recently some systems have been developed for searching, visualizing, and analyzing communities in large graphs. Below, we introduce two systems, namely C-Explorer \\cite{Fang:demo:2017} and VizCS \\cite{Huang:ICDE:2018}.\n\n\\input{cexplorer}\n\\input{vizcs}\n\n\\section{Future Work}\n\\label{sec:future}\n\nRecall that in Table~\\ref{tab:methods}, the cohesiveness metrics are orthogonal to graph types,\nso if a metric has not been studied for a particular type of graphs,\nthen it is a future research direction to study CS by applying the metric on this type of graphs.\nApart from this, we present a number of promising future directions as follows.\n\n\\subsection{Optimization for Query Parameters}\n\\label{sec:queryPara}\n\nMost existing CS queries require users to input some parameters, in addition to the query vertex. A typical parameter is the integer $k$ \\cite{KDD2010,local2014,barbieri2015efficient}, which controls the structure cohesiveness of returned communities. For attributed graphs, existing works also require users to input some parameters related to attributes. For example, in ACQ~\\cite{Fang:VLDB:2016} and ATC~\\cite{Huang:2017:ATC}, a set of query keywords are required.\nAlthough these parameters provide strong flexibility and personalization for the query, it may not be easy for users to set proper values for these parameters. For example, if the integer $k$ is too large, a false query may incur, i.e., the query returns empty result. On the other hand, if $k$ is too small (e.g., $k$=1 or 2), the returned community may contain too many vertices, which may make the community meaningless.\n\nUnfortunately, most existing CS works assume that users can input proper values for these parameters. This assumption, however, is too strong, especially when users do not know much about the underlying network. To suggest query parameters, a possible research direction is to exploit historical query logs and suggest some values of parameters automatically \\cite{baeza2004query,marcel2011survey}. Another direction is to study how to use crowdsourcing platforms (e.g., AMT~\\cite{amt}) to facilitate query suggestions.\n\n\\subsection{More Cohesiveness Metrics}\n\\label{sec:futureMetric}\n\nAs aforementioned, in CS solutions, a community is required to satisfy certain cohesiveness metrics. Essentially, the cohesiveness metrics formally define the communities, so they play crucial roles in CS.\n\nFor structure cohesiveness, there are many other cohesiveness models (see Section~\\ref{sec:cohesive}) which have not been used for CS. Thus, it would be interesting to study CS using these models. For example, in \\cite{sariyuce2016fast,sariyuce2015finding}, the authors have proposed the concept of nucleus, which is a generalization of $k$-core and $k$-truss.\n\nFor attribute-based cohesiveness, as discussed in Section~\\ref{sec:cohesive}, there are some studies finding cohesive subgraphs from attributed graphs. Thus, it is of interest to extend them for CS on attributed graphs.\nBesides, each existing CS solution only focuses on one particular type of attribute (e.g., keyword). This, however, may be problematic for many real applications because a real graph often involves multiple types of attributes.\nThus, it is desirable to study how to perform CS by considering multiple types of attributes.\n\n\\subsection{Other Types of Graphs}\n\\label{sec:futureGraph}\n\nIn recent years, many novel network models have been developed and the representative ones are as follows:\n\\begin{itemize}\n \\item {\\it Public-private network~\\cite{Archer2017,huang2018pp,chierichetti2015efficient}}. In a public-private network (e.g., Facebook), there is a public graph $G$, containing a set of vertices and a set of edges that are visible to all users of the network. In particular, each vertex $u$ is associated with a private graph $G_u$, where vertices of $G_u$ are vertices from the public graph $G$, and $G_u$ is only known to $u$.\n \\item {\\it Uncertain graph~\\cite{hu2017embedding,li2018edbt,huang2016truss}}. In many real applications (e.g., biology), the graph data are often noisy, inexact, and inaccurate, and they can be modeled as uncertain graphs, where each edge is associated with a value denoting its existence probability.\n \\item {\\it Signed graph~\\cite{yang2007community}}. A signed graph is a graph whose edges carry signs. For example, in social networks, the relationship of two users is either positive (e.g., friendship) or negative (e.g., hostility). Thus, users' relationship can be modeled as a signed graph.\n \\item {\\it Multi-dimensional graphs~\\cite{fang2014detecting}}. In many scenarios, a graph often contains various types of edges, which represent various types of relationships between entities. Such graphs are often called multi-dimensional graph, or multi-layer graphs or multi-view graphs.\n \\item {\\it Heterogeneous information network (HIN)~\\cite{shi2017survey,Hu:HIN:2019}}. HINs are networks with multiple typed objects and multiple typed links denoting different relations\n\\end{itemize}\n\nTo our best knowledge, there is no prior research about CS on these graphs. Thus, it is still an open problem of how to perform CS on these graphs.\n\n\\subsection{Real Big Graphs}\n\\label{sec:futureEff}\n\nMost existing CS studies assume that the graphs can be kept in the memory of a single machine. The graphs used for experimental evaluation are often million-scale, and only a few of them \\cite{Fang:TKDE:CSD,Li:vldb:2015} are able to process billion-scale graphs. However, in many real applications (e.g., Facebook), the graphs may involve billions of vertices and edges \\cite{li2015walking}. As a result, existing CS solutions may fail to process such real big graphs within reasonable time cost. Hence, how to efficiently perform online CS on such big graphs is a challenging task.\n\nFor big graphs that cannot be kept by a single machine, some possible research directions are as follows.\nFirst, we can consider developing query algorithms based on distributed computation platforms (e.g., GraphX \\cite{gonzalez2014graphx}), which are able to process big graphs in a cluster.\nSecond, to save memory space, we may keep the graph data on disk and design I\/O-efficient query algorithms.\n\n\\subsection{An Online Repository for Codes and Datasets}\n\\label{sec:opensource}\n\nFor most of surveyed CS studies, their codes of algorithms and datasets are not publicly available. Thus, it is desirable to build an online repository to keep these codes and datasets. The major benefits of doing this are two-fold: First, for researchers, the codes and datasets can serve as a benchmark for comparison studies. Second, practitioners can easily plug these CS solutions into their applications without re-implementation.\n\n\\subsection{Graph Keyword Search}\n\\label{sec:GKS}\n\nGenerally, graph keyword search \\cite{wang2010survey,keyword-yu-2009} aims to find a tree or a subgraph, which contains a set of query keywords, from a large graph $G$. Earlier studies often output a tree structure.\nIn~\\cite{keyword-icde2002}, Bhalotia et al. developed a backward algorithm for finding Steiner trees.\nIn~\\cite{keyword-icde2007}, Ding et al. proposed a dynamic programming algorithm finding Steiner trees.\nIn~\\cite{golenberg2008keyword}, Golenberg et al. presented a novel algorithm which produces Steiner trees with polynomial delay.\nIn~\\cite{keyword-vldb2005}, Kacholia et al. proposed a bidirectional search algorithm, and He et al.~\\cite{he2007blinks} improved it by introducing a new index structure.\n\nRecently, some solutions have output subgraphs.\nIn \\cite{keyword-sigmod2008}, Li et al. proposed to find $r$-radius Steiner graphs that contain query keywords.\nQin et al. \\cite{keyword-icde2009} proposed to find multi-centered subgraphs that contain query keywords within a given distance. Kargar et al. \\cite{keyword-vldb2011} studied the $r$-clique which is a set of vertices that cover query keywords and satisfy the distance constraint.\n\nHowever, these works are substantially different from CS queries on keyword-based attributed graphs. First, they do not specify query vertices as required by CS queries. Second, the tree or subgraph produced do not guarantee structure cohesiveness. Third, their solutions do not ensure strong keyword cohesiveness.\n\n\\subsection{Graph Pattern Matching (GPM)}\n\\label{sec:GPM}\nFor simple graphs, the problem of GPM is NP-complete \\cite{cook1971complexity} and it has been studied extensively under different settings:\n(1) in main memory \\cite{ullmann1976algorithm,chiba1985arboricity}. For example, Ullmann \\cite{ullmann1976algorithm} proposed a backtracking algorithms.\n(2) in external memory, Chu et al. \\cite{chu2011triangle} and Hu et al. \\cite{hu2014efficient} studied triangle counting; in \\cite{qiao2017subgraph}, a novel GPM solution based on graph compression is presented.\n(3) in distributed platforms, both DFS-style approaches \\cite{afrati2013enumerating,park2016pte}and BFS-style approaches \\cite{lai2015scalable,lai2016scalable} are developed. The DFS-style approaches avoid intermediate results by using one-round computation, while BFS-style approaches shuffle a large number of intermediate results.\n\nFor attributed graphs, there are also many studies. Tong et al. \\cite{GPM-KDD2007} studied the use of lines, loops and stars for finding the matched subgraphs; Zou et al. \\cite{zou2009distance} developed a novel GPM solution based on distance join;\nFan et al. \\cite{GPM-VLDB2010} studied GPM by using bounded simulation; in \\cite{GPM-PVLDB2015}, GPM has been studied for finding graph association rules; in \\cite{GPM-ICDE2012}, Cheng et al. studied the problem of top-$k$ GPM. Recently, Fang et al. have studied a variant of the GPM problem on spatial databases \\cite{Fang:ICDE:SPM,Fang:ICDE:DEMO}, and it aims to find spatial objects that are matched with a given pattern.\nHowever, GPM is different with CS because (1) it often focuses on small patterns, so it cannot generate large communities; and (2) the subgraphs of GPM solutions often do not have guarantee on strong structure cohesiveness. Other related topics include node similarity computation \\cite{yu2013more,yu2012space}, keyword search \\cite{yuan2017keyword,yuan2013efficient}, and uncertain subgraph search \\cite{yua,yuan2012efficient}.\n\n\\subsubsection{Geo-Social Group Queries with Minimum Acquaintance Constraint (GSGQs)}\n\\label{sec:gsgq}\n\nThe GSGQ is defined formally as follows:\n\n\\begin{problem}[GSGQ]\n\\label{prob:gsgq}\nGiven a geo-social network $G(V$, $E)$, a vertex $q\\in V$, a positive integer $k$ and a spatial constraint $\\Lambda$, return a subgraph $G_q\\subseteq G$, and the following properties hold:\n\\begin{enumerate}\n \\item \\textbf{Connectivity}. $G_q$ is connected and contains $q$;\n \\item \\textbf{Structure cohesiveness}. $\\forall v\\in G_q$, $deg_{G_q}(v)\\geq k$;\n \\item \\textbf{Spatial cohesiveness}. $G_q$ satisfies constraint $\\Lambda$.\n \\item \\textbf{Maximality constraint}. There exists no other subgraph $G_q'$ satisfying properties above and $G_q\\subset G_q'$.\n\\end{enumerate}\n\\end{problem}\n\nIn Problem~\\ref{prob:gsgq}, for spatial constraint $\\Lambda$, Zhu et al. \\cite{zhu2017geo} considered three kinds of constraints:\n\\begin{enumerate}\n \\item $\\Lambda$ is a spatial rectangle for containing $G_q$;\n \\item $\\Lambda$ is a circle centered at $q$ with radius less than the distance from $q$ to its $k$-th nearest vertex in $G_q$ ($G_q$ may contain more than $k$+1 vertices);\n \\item $\\Lambda$ is a circle satisfying Constraint 2 and $G_q$ contains exactly $k$+1 vertices.\n\\end{enumerate}\n\nBy using an R-tree index \\cite{guttman1984r}, a GSGQ with the first constraint can be answered in $O(n+m)$ time; when the second constraint is imposed, a GSGQ can be solved in $O(n(n+m))$ time; when the third constraint is applied, a GSGQ takes $O(C_k^{n-1}(m+n))$ time.\n\nTo improve efficiency, they proposed the social-aware R-trees (or SaR-tree) index, which incorporates both vertices' spatial locations and social relations. It is built based on the concept of core bounding rectangle (CBR), which projects the minimum degree constraint on the spatial layer. Specifically, the CBR of a vertex $v$ is a rectangle containing $v$, inside which any vertex group with $v$ does not satisfy the minimum degree constraint.\n\nUnlink classical R-tree, each entry of an SaR-tree refers to two pieces of information, i.e., a set of CBRs and a minimum bounding rectangle (MBR). Perceptually, a CBR bounds a group of vertices from the social perspective, while an MBR bounds vertices from the spatial perspective. As such, SaR-tree gains power for both social-based and spatial-based pruning. In addition, they developed a variant of SaR-tree, called SaR*-tree, which optimizes the group of spatial objects to minimize the disk I\/O cost. Based on these indexes, they developed efficient algorithms for answering GSGQs with different spatial constraints.\n\n\\section{Introduction}\n\\label{sec:intro}\n\nWith the rapid development of information technologies, various big graphs are prevalent in many real applications (e.g., social media and knowledge bases). An important component of these graphs is the network community. Essentially, a community is a group of vertices which are densely connected internally. For example, in Facebook, communities consist of users that are with strong friendship \\cite{acquisti2006imagined}; on the World Wide Web, communities contain web sites which share similar topics \\cite{broder2000graph}; in protein-protein interaction networks \\cite{article05clique} and metabolic networks \\cite{guimera2005functional}, communities correspond to functionality modules.\nRetrieving communities from a network is a fundamental problem in network science, and it can be applied to many real-life applications. Here are some typical applications, to name a few:\n\\begin{itemize}\n \\item {\\it Event organization.} A social event (e.g., a party or a conference) often involves a group of users and its organization can benefit from communities. For example, to hold a cocktail part, a user can find his community, i.e., a group of researchers, each of which is well acquainted.\n \\item {\\it Friend recommendation.} Many social media platforms (e.g., Facebook) often maintain a friendship network. To suggest candidate friends to a specific user $u$, intuitively we can recommend $u$ those who are in $u$'s community but are not yet $u$'s friends.\n \\item {\\it Protein complex identification.} In biology, proteins interact with each other and a gene is often regulated by a set of proteins. To study a gene, a biologist may focus on a set of proteins that highly interact with each other, which is a community of proteins.\n \\item {\\it Advertisement in e-commence.} Users of the same community often share similar interests. To push advertisements for a user $u$, we may find her community first and then select advertisements that are checked by members of her community.\n\\end{itemize}\n\nOwing to the importance of communities, how to effectively and efficiently find communities from large graphs is an important research topic in the era of big data. With a careful observation on these applications, we identify a list of factors that the community retrieval solutions should satisfy:\n\\begin{itemize}\n \\item {\\it High efficiency.} For many real applications (e.g., event organization), the communities often need to be retrieved in real-time, based on query requests. Thus, the community retrieval solutions should be able to respond in real-time.\n \\item {\\it High scalability.} Nowadays, many real networks contain millions or billions of vertices. As a result, the solutions should be scalable to real big graphs.\n \\item {\\it High personalization.} In practice, for large networks, people usually are interested in communities of some specific users, rather than all the users. Thus, the solutions should allow users to specify query vertices. Moreover, some personalized requirements on structures (and attributes) could be imposed.\n \\item {\\it High quality.} The vertices in the communities retrieved should be cohesively linked. Moreover, the communities should be easy for interpretation.\n \\item {\\it Support for dynamic graphs.} Since real networks often involve as the time goes on, the solutions should be able to adapt for the dynamic changes easily.\n\\end{itemize}\n\nTowards the goals above, recently a large group of research works, called community search (CS),\nhave been proposed \\cite{Huang:ICDE:2017}. Generally, the goal of CS is to search high-quality communities in an online manner, based on a query request. Specifically, given a vertex $q$ of a graph $G$, it aims to find a community, or a dense subgraph, which contains $q$ and satisfies the properties: (1) {\\it connectivity}, i.e., vertices in the community are connected; and (2) {\\it cohesiveness}, i.e., vertices in the community are intensively linked to each other w.r.t. a particular goodness metric \\cite{KDD2010,KDD2010,local2014,barbieri2015efficient,online-sigmod2013}. The metric is often defined by using some classical subgraph cohesiveness metrics such as:\n\\begin{itemize}\n \\item $k$-core. The $k$-core~\\cite{md1983,kcore2003} is the largest subgraph of $G$, in which each vertex's degree is at least $k$ within the subgraph.\n \\item $k$-truss. The $k$-truss~\\cite{cohen2008trusses,k-truss2014} is the largest subgraph of $G$ in which every edge is contained in at least ($k-2$) triangles within the subgraph.\n \\item $k$-clique. A $k$-clique~\\cite{kclique} is a set of $k$ vertices of $G$ such that each pair of vertices has an edge.\n \\item $k$-ECC. A $k$-ECC ($k$-edge connected component)~\\cite{gibbons1985algorithmic} is a subgraph of $G$ such that after removing any $k$--1 edges, it is still connected.\n\\end{itemize}\n\n\\begin{figure}[]\n\\centering\n\t\\includegraphics[width=0.6\\columnwidth]{figures\/intro}\n\\caption{An example of community search.}\n\\label{fig:intro}\n\\end{figure}\n\n\nLet us illustrate CS by an example. Consider the graph with ten vertices in Fig. \\ref{fig:intro}, and CS solutions \\cite{KDD2010,local2014,barbieri2015efficient}, which are based on the $k$-core model. Let $q$=$A$. Then, the induced subgraph of vertices \\{$A$, $B$, $C$, $D$\\} will be returned as the community. Note that the subgraph forms a $k$-core with $k$=3, since each vertex's degree is 3 within the subgraph, and it is also the core attaining the maximum value of $k$.\n\nIn the literature, there is a highly related group of research works, called community detection (CD) \\cite{CD:Survey:2009,CD:Survey:2011,CD:Survey:2011a,CD:Survey:2013,CD:Survey:2017}.\nGenerally, it has similar goals with CS, but there are three key differences:\n(1)\tThe problem definitions are different. CS aims to search communities regarding a set of query vertices and some query parameters, while CD often detects all communities in the graph.\n(2)\tThe criteria of defining communities are different. In CS, the criteria of defining communities are based on query parameters given by the users. In other words, communities are retrieved depending on user-defined parameters. In contrast, CD methods often use the same global criterion to detect communities by partitioning the entire graph.\nFor example, in Fig.~\\ref{fig:intro}, if $q$=$A$, CS solutions \\cite{KDD2010,local2014} will find the community \\{$A$, $B$, $C$, $D$\\}, and if $q$=$E$, they will find the community \\{$A$, $B$, $C$, $D$, $E$\\}. In contrast, if using a CD method (e.g., the spectral clustering \\cite{von2007tutorial}) with setting the number of communities to 3, we will obtain three communities, each of which forms a connectivity component, where $B$ and $E$ are in the same community.\n(3)\tThe algorithms are different. As shown in existing studies, CS solutions can search communities efficiently in an online manner, while CD solutions are often time consuming and unscalable to big graphs. Moreover, CS queries can often be supported by indexes and handle dynamic graphs easily.\nThus, compared to CD solutions, CS solutions can better satisfy factors aforementioned.\n\n\\begin{table*}[ht]\n \\centering\n \\caption {Classification of works of community search (``P.\" means Problem).}\n \\label{tab:methods}\n \\begin{tabular}{c|c|c|c|c|c|c}\n \\hline\n \\multirow{2}*{\\textbf{Metric}}\n & \\multirow{2}*{\\textbf{Simple graphs}}\n & \\multicolumn{5}{|c}{\\textbf{Attributed graphs}}\n \\\\\n \\cline{3-7}\n & & \\textbf{Keyword} & \\textbf{Location} & \\textbf{Temporal} & \\textbf{Influence (weight)} & \\textbf{Profile}\\\\\n \\hline\\hline\n $k$-core\n & \\tabincell{c}{\\cite{KDD2010,local2014,barbieri2015efficient,Fang:TKDE:CSD}\\\\\n (P. \\ref{prob:kcoreGlobal}, \\ref{prob:kcoreLocal}, \\ref{prob:kcoreGlobalSize}, \\ref{prob:kcoreMinSize}, \\ref{prob:CSD})}\n & \\tabincell{c}{\\cite{Fang:VLDB:2016,Fang:VLDBJ:2017}\\\\\n (P. \\ref{prob:acq})}\n & \\tabincell{c}{\\cite{Fang:VLDB:2017,Fang:TKDE:SAC,wangkai2018,zhu2017geo}\\\\\n (P. \\ref{prob:sac}, \\ref{prob:rbkcore}, \\ref{prob:gsgq})}\n & \\tabincell{c}{\\cite{Li:ICDE:2018}\\\\\n (P. \\ref{prob:timecs})}\n & \\tabincell{c}{\\cite{Li:vldb:2015,Li:VLDBJ:2017,chen2016efficient,zheng2017querying,Bi:2018,Li:SIGMOD:2018} \\\\\n (P. \\ref{prob:nic}, \\ref{prob:skyCS})}\n & \\tabincell{c}{\\cite{Yankai18}\\\\\n (P. \\ref{prob:PCS})}\n \\\\\n \\hline\n $k$-truss\n & \\tabincell{c}{\\cite{k-truss2014,Akbas:VLDB:2017,huang2015approximate}\\\\\n (P. \\ref{problem:TTC}, \\ref{prob:ctc})}\n & \\tabincell{c}{\\cite{Huang:2017:ATC}\\\\\n (P. \\ref{prob:atc})}\n & --\n & --\n & \\tabincell{c}{\\cite{Zheng:IS:2017}\\\\\n (P. \\ref{problem:WTC})}\n & --\n \\\\\n \\hline\n $k$-clique\n & \\tabincell{c}{\\cite{online-sigmod2013,kclique2018,yang2011social,wang2017query}\\\\\n (P. \\ref{prob:kcliquesearch}, \\ref{prob:densest}, \\ref{prob:sgq}, \\ref{prob:mckpq})}\n & --\n & --\n & \\tabincell{c}{\\cite{kclique2017}\\\\\n (P. \\ref{prob:infCS})}\n & --\n & --\n \\\\\n \\hline\n $k$-ECC\n & \\tabincell{c}{\\cite{Chang:SIGMOD:2015,hu2016querying,hu2017querying}\\\\\n (P. \\ref{prob:keccMax}, \\ref{prob:keccMinimum}, \\ref{prob:keccMinimal})}\n & --\n & --\n & --\n & --\n & --\n \\\\\n \\hline\n Others & \\multicolumn{6}{l}{\n local modularity: \\cite{clauset2005finding,Luo2006ELC}\n query biased density: \\cite{Wu:VLDB:2015}\n pagerank: \\cite{andersen2006communities,Kloumann2014} (P. \\ref{prob:PPR})\n neighbors: \\cite{Mehler2009}\n }\\\\\n \\hline\n \\end{tabular}\n \\vspace{-0.1in}\n\\end{table*}\n\n\nAlthough there are many CS solutions, they deal with different types of graphs and formulate communities in different manners. Meanwhile, there is a lack of systematic survey of CS solutions. Thus, it is desirable to organize these works and understand how well they perform in terms of efficiency and quality. To this end, in this paper we will provide a thorough review of these works. We will also compare different CS solutions so that readers can better understand the state-of-the-art, and point out directions for future study.\n\nAs shown in Table~\\ref{tab:methods}, we classify CS solutions into five categories such that solutions in each category (except the last category) adopt the same structure cohesiveness metric. Moreover, for works in each category, we further partition them into two groups, where the first group focuses on simple graphs while the second group targets attributed graphs. Note that the IDs of CS problems are also included in the brackets of Table~\\ref{tab:methods}.\nFor simple graphs, CS solutions search communities purely based on link information, while for attributed graphs, CS solutions often consider both links and attributes.\nWe remark that these cohesiveness metrics are orthogonal to graph types. This implies that if a metric has not been studied for a particular type of graphs, then it is a possible future research direction to study CS by applying the metric on this type of graphs.\n\nIn summary, our main contributions are as follows:\n\\begin{itemize}\n \\item First, we provide a systematic classification of studies on CS. Specifically, we classify these studies according to the community cohesiveness metrics. For each class of works, we review the representative studies on different types of graphs.\n \\item Second, we perform a thorough analysis and comparison of different community cohesiveness metrics. Moreover, we analyze and compare CS solutions on simple graphs and attributed graphs.\n \\item Third, we offer insightful suggestions for future study on CS. This may give researchers new to CS an understanding of the recent development of CS, as well as a good starting point to work in this field.\n\\end{itemize}\n\nThe rest of this paper is organized as follows. In Section~\\ref{sec:pre}, we introduce and discuss community cohesiveness metrics. In Sections~\\ref{sec:kcore}, \\ref{sec:ktruss}, \\ref{sec:kclique}, \\ref{sec:kecc}, and \\ref{sec:other}, we extensively discuss CS solutions in each category. We also present two CS systems in Section \\ref{sec:demo}. We review the related work in Section~\\ref{sec:related}. Finally, we present a list of future topics in Section~\\ref{sec:future} and conclude in Section~\\ref{sec:conclude}.\n\n\\subsection{Discussions}\n\\label{subsec:kclique_discussion}\n\nIn this section, we survey the CS solutions \\cite{online-sigmod2013,kclique2018,yang2011social,wang2017query,kclique2017} using $k$-clique model. We can divide them into two groups, where the first group \\cite{online-sigmod2013,kclique2018,yang2011social,wang2017query} focuses on simple graphs, while the second group \\cite{kclique2017} is developed for attributed graphs.\nIn the first group, the first one \\cite{online-sigmod2013} uses quasi-clique model, the second one \\cite{kclique2018} adopts $k$-clique model, and the last two \\cite{yang2011social,wang2017query} are based on $k$-plex model.\nHowever, to our best knowledge, there is no systematic study to compare the goodness of different $k$-clique based models in real-life applications, which is crucial for researchers and practitioners to choose desirable models in practice. Moreover, there is no investigation on the trade-off between the computing time complexity and the flexibility of these models. It will be interesting to fill these two gaps in the future study.\n\n\\subsection{Most Influential Community Search}\n\\label{subsec:kcliqueInfluence}\n\nIn~\\cite{kclique2017}, Li et al. proposed the problem of most influential community search,\nwhich aim to find the most influential cohesive subgraph.\nThe concept of $kr$-clique community (Definition~\\ref{def:krclique}) is proposed to capture the cohesiveness of a set of vertices. In addition to cohesiveness, authors also considered the influence of the community.\nFollowing the popular Linear Threshold (LT) model~\\cite{DBLP:journals\/tkde\/LeeC15},\nthe aggregate influence probability of a community $C$ w.r.t a vertex $v$,\ndenoted by $Pr(v|C)$, is defined as follows:\n$$\nPr(v|C) = 1 - \\prod_{u \\in C} (1 - P_{u \\rightarrow v})\n$$\nwhere $P_{u \\rightarrow v}$ is the probability that $v$ is influenced by $u$.\nNote that there is a influence probability $P_{uv}$ for each edge ($u$, $v$) in $G$,\nand $P_{u \\rightarrow v}$ is computed by multiplying the influence\nof the edges along the maximum influence path~\\cite{DBLP:journals\/tkde\/LeeC15}\nfrom $u$ to $v$.\nGiven a probabilistic threshold $\\Delta$,\nthe influence score of the community $C$ is the number of vertices in $G \\setminus C$\nwith aggregate influence not less than $\\Delta$, denoted by $score(C)$.\nBelow is the problem definition.\n\n\\begin{problem}\n\\label{prob:infCS}\nGiven a simple graph $G$ where each edge has an influence probability, the problem of the most influential community search is to find a maximal $kr$-clique community with the highest influence score.\n\\end{problem}\n\nIt is shown in~\\cite{kclique2017} that the problem is NP-hard because of the clique computation.\nA baseline solution is to access the vertices by their individual influence and\ncompute the maximal $kr$-clique for each vertex.\nTo improve efficiency, a tree structure named $C$-Tree is proposed such that any $kr$-clique community\ncan be generated efficiently. Four efficient search algorithms are developed to\nsignificantly prune the search space based on the $kr$-clique constraints and the influence scores.\n\n\\subsection{Location-Based Attributed Graphs}\n\\label{sec:kcliqueSimple}\n\n\\subsection{Simple Undirected Graphs}\n\\label{sec:kcoreSimple}\n\n\\section{$K$-Clique-Based Community Search}\n\\label{sec:kclique}\n\nIn this section, we survey CS solutions that use $k$-clique or its variants to capture the structure cohesiveness.\nWe first briefly introduce the $k$-clique model and its variants in Section~\\ref{subsec:kclique_def}.\nThen, we present CS solutions using $k$-clique component and $k$-plex models in Sections~\\ref{subsec:kplex} and \\ref{subsec:kplex}. After that, we discuss the most influential CS using $k$-clique in Section \\ref{subsec:kcliqueInfluence}. Finally, we discuss these studies in Section~\\ref{subsec:kclique_discussion}.\n\n\\subsection{$K$-Clique and Its Variants}\n\\label{subsec:kclique_def}\n\nRecall that by Definition \\ref{def:kclique}, a $k$-clique is a complete graph with $k$ vertices where there is an edge between every pair of vertices. The $k$-clique model has been widely used for the overlapping community detection (e.g.,~\\cite{article05clique,DBLP:journals\/bioinformatics\/AdamcsekPFDV06}).\nAs the condition of $k$-clique is strict, some relaxed variants\nsuch as $\\gamma$-quasi-$k$-clique~\\cite{brunato2007effectively,online-sigmod2013} and\n$k$-plex~\\cite{seidman1978graph}, are proposed to identify cohesive subgraphs.\nBelow are detailed definitions.\n\n\\begin{definition}[$\\gamma$-quasi-$k$-clique~\\cite{brunato2007effectively,online-sigmod2013}]\n\\label{def:quasikclique}\nA $\\gamma$-quasi-$k$-clique is a graph with $k$ vertices and at least $\\lfloor \\gamma\\frac{k(k-1)}{2} \\rfloor$ edges, where $0\\leq\\gamma\\leq1$.\n\\end{definition}\nWhen $\\gamma = 1$, the corresponding $\\gamma$-quasi-$k$-clique is a $k$-clique.\nWe can tune the desired cohesiveness of the $k$ vertices by varying $\\gamma$ value.\n\n\\begin{definition}[$k$-plex~\\cite{seidman1978graph}]\n\\label{def:kplex}\nA graph $G(V,E)$ is a $k$-plex, if for each vertex $v\\in V$, $v$ has at least $|V|-k$ neighbors in $G$, where $1\\leq k\\leq |V|$.\n\\end{definition}\n\nWhen $k$=1, the $k$-plex is exactly a $k$-clique. Clearly, by setting a smaller value of $k$, we can obtain a more cohesive $k$-plex. The problem of finding a $k$-plex from a given graph for an integer $k$ is NP-hard \\cite{balasundaram2011clique}.\n\nAnother way to relax the constraint of $k$-clique is to consider the connection of two vertices.\n\n\\begin{definition}[$kr$-clique~\\cite{kclique2017}]\n\\label{def:krclique}\nGiven a graph $G$ and two integers $k$ and $r$, a $kr$-clique $S$ is an induced subgraph of $G$\nsuch that: (1) the number of vertices in $S$ is at least $k$;\nand (2) any two vertices in $S$ can reach each other within $r$ hops.\n\\end{definition}\n\nClearly the problem of finding $kr$-clique is NP-hard because $kr$-clique is a $k$-clique when $r$=1.\n\n\\subsection{$K$-Clique-Based Community Search}\n\\label{subsec:clique_community}\nIn Section~\\ref{subsub:kcliquecom}, we introduce the seminar work on overlapping community detection~\\cite{article05clique},\nin which the $k$-clique component is proposed.\nSection~\\ref{subsub:kcliquecomsearch} presents the community search algorithm based on the relaxation of $k$-clique component, while Section~\\ref{subsubsec:densest_clique_community} studies the densest $k$-clique community search.\n\n\\subsubsection{$K$-Clique-Based Community}\n\\label{subsub:kcliquecom}\n\nIn~\\cite{article05clique}, Palla {\\emph et al.} showed that that many real networks are characterized by well-defined\noverlapping communities. For instance, a person may belong to three different communities related to school, hobby and family. For a given graph $G$, a \\textit{k-clique graph} $G_k$ can be derived where each node is a $k$-clique\nin $G$ and there is an edge if two nodes ($k$-cliques) are adjacent,\ni.e., they share $k-1$ vertices in $G$.\nThen the $k$-clique communities are the union of all adjacent $k$-cliques, which are defined as follows.\n\n\\begin{definition} [$k$-clique component]\nLet $C$ denote a connected component in the $k$-clique graph, then a \\textit{k-clique component} is the union of all $k$-cliques represented by vertices in $C$.\n\\end{definition}\n\nOne may explore the communities of the graph based on the $k$-cliques and their adjacency, and a graph vertex may belong to several communities.\nEfficient $k$-clique component detection algorithm is presented in~\\cite{DBLP:journals\/bioinformatics\/AdamcsekPFDV06}.\nParticularly, considering that each $k$-clique must be contained by at least one maximal clique,\nthey first identify all maximal cliques of the network and then\nenumerate the communities by carrying out a standard component analysis of the clique overlap matrix.\n\n\n\n\\subsubsection{$K$-Clique-Based Community Search}\n\\label{subsub:kcliquecomsearch}\n\nIn~\\cite{online-sigmod2013}, Cui {\\emph et al.} showed that there are two shortcomings in the $k$-clique community model:\n(1) there are overwhelming number of $k$-cliques communities in real-life graphs;\nand (2) the $k$-clique constraint and the definition of adjacent (i.e., sharing $k$-1 common vertices) are not flexible in practice.\nTo address these two shortcomings, they proposed an online community search (OCS) problem.\nInstead of enumerating all communities, they focused on the search of the communities containing a given query vertex $q$. They relaxed the $k$-clique adjacent from $k-1$ common vertices to $\\alpha$ vertices, namely $\\alpha$-adjacency.\nThey also relaxed $k$-clique model to $\\gamma$-quasi-$k$-clique model (Definition~\\ref{def:quasikclique}).\nBy doing this, the $k$-clique components in the $k$-clique communities are relaxed to the $\\gamma$-quasi-$k$-clique components. Below is the formal problem definition.\n\n\\begin{problem}\n[($\\alpha$, $\\gamma$)-OCS]\n\\label{prob:kcliquesearch}\nGiven an undirected simple graph $G(V,E)$, a query vertex $q \\in V$, and an integer $k$,\nan integer $\\alpha \\leq k-1$, and a real value $\\gamma$ with $0 \\leq \\gamma \\leq 1$,\nfind all $\\gamma$-quasi-$k$-clique components containing query vertex $q$.\n\\end{problem}\n\nClearly, a $k$-clique component search is a special case of ($\\alpha$, $\\gamma$)-OCS\nwith $\\alpha=k-1$ and $\\gamma = 1$. By reducing to $k$-clique decision problem, it is shown in~\\cite{online-sigmod2013} that the ($\\alpha$, $\\gamma$)-OCS problem is $\\#P$-Complete. It is shown that the density of each community in ($\\alpha$, $\\gamma$)-OCS is at least $2 \\max\\{0, \\min \\{ f(1), f(\\alpha) \\} \\}$\nwhere $f(x) = \\frac{ \\gamma {k \\choose 2} {k-x \\choose 2}}{x}$.\nBoth exact and approximate solutions are proposed in~\\cite{online-sigmod2013}.\nA naive algorithm for exact solution is to enumerate all $\\gamma$-quasi-$k$-cliques containing the query vertex $q$,\nand then compute the $\\gamma$-quasi-$k$-clique components based on the $\\alpha$-adjacency.\nTo avoid enumerating cliques belonging to none of the valid communities, a new computing framework is proposed\nto check the adjacency when a clique is discovered. By carefully maintaining the visit status of each clique, authors further optimize the searching cost.\nAuthors also proposed an approximate solution.\nTo reduce the search space, the approximate algorithm only enumerates an unvisited clique which contains\nat least one new vertex not contained by any existing community.\nA heuristic is proposed to choose a vertex sequence such that the resulting clique sequence is short, leading to a good approximation solution.\n\n\\subsubsection{Densest Clique Percolation Community Search}\n\\label{subsubsec:densest_clique_community}\n\nFollowing the $k$-clique community model in~\\cite{article05clique}, Yuan et al.\nstudied the problem of densest clique percolation community search~\\cite{kclique2018},\nwhere a \\textit{k-clique percolation community} (KCPC) is a \\textit{k-clique component} in~\\cite{article05clique}.\nIn particular, they aimed to find the $k$-clique percolation community with the maximum $k$ value\nthat contains a given set of query vertices.\n\n\\begin{problem}\n\\label{prob:densest}\nGiven an undirected simple graph $G(V,E)$ and a set of query vertex $Q \\subseteq V$,\nthe problem of the \\textit{\\underline{d}{ensest} \\underline{c}lique \\underline{p}ercolation \\underline{c}ommunity} (DCPC) search is to find the $k$-clique component with the maximum $k$ value that contains all the vertices in $Q$.\n\\end{problem}\n\nFig. \\ref{fig:densest_community} in~\\cite{kclique2018}\nillustrates a part of the collaboration network in DBLP,\nin which each vertex represents an author and each edge indicates the co-author relationship between two authors.\n$G_1$ is a 4-clique percolation community as it is a maximal union of five adjacent $4$-cliques: $\\{v_{14}, v_{15}, v_{16}, v_{17}\\}$, $\\{v_{14}, v_{15}, v_{16}, v_{18}\\}$, $\\{v_{14}, v_{15}, v_{17}, v_{18}\\}$, $\\{v_{14}, v_{16}, v_{17}, v_{18}\\}$, $\\{v_{15}, v_{16}, v_{17}, v_{18}\\}$, and any two $4$-cliques share 3 nodes. Similarly, $G_2$ is also a 4-clique percolation community. $G_1$ overlaps $G_2$ with nodes ${v_{14}, v_{15}}$. Given a query $q = \\{v_9, v_{18}\\}$, the densest clique percolation community of $q$ is the $3$-clique percolation community $G_3$ since $G_3$ is the $k$-clique percolation community with maximum $k$ value that contains $v_9$ and $v_{18}$.\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/kcpc}\n\\caption{Illustrating DCPC search \\cite{kclique2018}.}\n\\label{fig:densest_community}\n\\end{figure}\n\nA baseline solution is to start from the maximal possible $k$ value\nand check if there is a KCPC by applying the $k$-clique component detection algorithm in~\\cite{article05clique}.\nIf there is no KCPC detected, the $k$ value will be decreased by one until a KCPC is detected.\nTo efficiently support online DCPC search, an index-based approach is developed in~\\cite{kclique2018}.\nParticularly, based on the observation that a $k$-clique component can be treated as a union of \\textit{maximal} cliques, they take maximal cliques as building blocks of $k$-clique components and propose a tree-structure named\n\\textit{clique adjacency tree} which can efficiently identify the $k$-clique components for a given $k$ value.\nThe authors further developed a new tree-structure named \\textit{ordered adjacency tree}\nsuch that only the subtrees related to the query vertices will be explored.\nTogether with maximal cliques and their inverted indexes, a compact index structure named DCPC-Index\nis proposed to support efficient DCPC queries.\n\n\\subsection{$K$-Plex-Based Community Search}\n\\label{subsec:kplex}\n\n\n\\input{sgq}\n\\input{mckpq}\n\n\\input{kcliqueInfluence}\n\\input{kcliqueDiscuss}\n\n\\subsection{Directed Graphs}\n\\label{sec:kcoreDirected}\n\nA directed graph is a graph $G(V,E)$, which contains a set of vertices $V$ and a set of directed edges $E$.\nThe in-degree and out-degree of a vertex $v$ in $G$, denoted by $deg_G^{in}(v)$ and $deg_G^{out}(v)$, are the number of its in-neighbors and out-neighbors, respectively. The minimum in-degree and out-degree of the graph $G$ are denoted by $\\delta_{in}(G)$ and $\\delta_{out}(G)$ respectively.\nFig. \\ref{fig:CSDEg}(a) depicts a directed graph with nine users.\n\nA straightforward method of performing CS on directed graph is to ignore the directions and then use the method {\\tt Global} in Section \\ref{sec:sizeUnbounded} to find the community. In Fig. \\ref{fig:CSDEg}(a), if we let $q$=Jack, then we will find a community with members \\{{\\tt Jack}, {\\tt Jeff}, {\\tt Bob}, {\\tt Tom}, {\\tt Tim}, {\\tt Jim}\\}. However, {\\tt Tim} has no in-neighbors and {\\tt Jim} has no out-neighbors in the community, which implies their interactions with other members are quite weak.\n\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c c}\n \\begin{minipage}{3.7cm}\n\t\\includegraphics[width=3.7cm]{figures\/CSDIntro}\n \\end{minipage}\n &\n \\begin{minipage}{3.8cm}\n\t\\includegraphics[width=3.8cm]{figures\/CSDGraph}\n \\end{minipage}\n \\\\\n (a) a directed graph\n &\n (b) illustrating D-cores\n\\end{tabular}\n\\caption{Two directed graphs~\\cite{Fang:TKDE:CSD}.}\n\\label{fig:CSDEg}\n\\end{figure}\n\nIn~\\cite{Fang:TKDE:CSD}, Fang et al. extended the minimum degree measure for directed graphs, and study the problem of \\underline{C}ommunity \\underline{S}earch on \\underline{D}irected graph (or CSD problem), based on the D-core, also called ($k$, $l$)-core \\cite{Dcore2014}.\n\n\\begin{definition}[($k$, $l$)-core~\\cite{Dcore2014}]\n\\label{def:Dcore}\nGiven a directed graph $G(V,E)$ and two non-negative integers $k$ and $l$, the $(k,l)$-core is the maximum subgraph $C$ of $G$ such that $\\delta_{in}(C)\\ge k$ and $\\delta_{out}(C) \\ge l$ .\n\\end{definition}\n\n\\begin{problem}[CSD]\n\\label{prob:CSD}\nGiven a directed graph $G(V,E)$, two positive integers $k$ and $l$, and a query vertex $q$, return a connected subgraph $G_q\\subseteq G$, such that it contains $q$ and $\\forall v\\in G_q$, $\\delta_{in}(G_q)\\geq k$ and $\\delta_{out}(G_q)\\geq l$.\n\\end{problem}\n\nFig. \\ref{fig:CSDEg}(b) shows a directed graph with its D-cores. Let $q$=$B$, $k$=2, and $l$=2. Then, the subgraph of \\{$A$, $B$, $C$\\} is the returned community for $B$.\n\nSimilar to {\\tt Global}, a simple solution to the CSD problem is to peel vertices iteratively until each remaining vertex satisfies the in-degree and out-degree constraints. As a result, its time complexity is $O(m+n)$, which may be inefficient for large graphs.\nTo improve efficiency, Fang et al.~\\cite{Fang:TKDE:CSD} proposed an index-based method. Specifically, it first performs D-core decomposition (i.e., computing all the ($k$, $l$)-cores), then organizes these cores in an index with a 2-dimensional table, and finally answers queries using the index.\n\nTo keep all D-cores, a simple method takes $O(n^3)$ space since $k$, $l$$\\leq$$n$--1 and each D-core takes $O(n)$ space. To alleviate this issue, three methods are proposed. For ease of exposition, let $V_{i,j}$ denote the set of vertices in ($i$, $j$)-core.\nThe first one exploits the nested property of D-cores, i.e., for any $l\\geq0$, we have ($k$, $l$+1)-core $\\subseteq$ ($k$, $l$)-core, so if ($k$, $l$+1)-core has been kept, we only need to keep vertices $V_{k,l}\\backslash V_{k,l-1}$ for the ($k$, $l$)-core. As a result, for any $k$, it takes $O(n)$ space to keep all ($k$, $l$)-cores (0$\\leq{l}$$\\leq$$n$), so the overall space cost is $O(m)$.\n\nThe second method relies on a key observation that for any $k,l\\geq0$, we have both ($k$+1, $l$)-core $\\subseteq$ ($k$, $l$)-core and ($k$, $l$+1)-core $\\subseteq$ ($k$, $l$)-core. After keeping ($k$, $l$+1)-core and ($k$+1, $l$)-core, for ($k$, $l$)-core, if $|V_{k+1,l}|\\geq|V_{k,l+1}|$, we only keep $V_{k,l}\\backslash V_{k+1,l}$; otherwise, we keep $V_{k,l}\\backslash V_{k,l+1}$. Thus, it takes less space than the first method. For the third method, after keeping ($k$, $l$+1)-core and ($k$+1, $l$)-core, it only keeps vertices $V_{k,l}\\backslash(V_{k+1,l}\\cup V_{k,l+1})$ for the ($k$, $l$)-core and takes the least space cost.\n\nIn addition, although the community $G_q$ of a CSD query is a connected subgraph, it may not be a strongly connected component (SCC)~\\cite{hopcroft1983data} (i.e., each vertex of the SCC is reachable from each other vertex). To tackle this issue, a variant of the CSD problem is to find a community, which not only satisfies the minimum degree constraints, but also is an SCC. The CSD algorithms can be extended for solving this variant \\cite{Fang:TKDE:CSD}.\n\n\\subsection{Discussions}\n\\label{sec:kcoreDiscuss}\n\nIn this section, we review CS studies that use the $k$-core model. For simple graphs, we can divide them into two groups, where the first group \\cite{KDD2010,local2014,barbieri2015efficient} focuses on undirected graphs while the second group \\cite{Fang:TKDE:CSD} only considers directed graphs. In particular, for the first group, the first work \\cite{KDD2010} returns the maximal $k$-core containing the query vertex, while communities of the other two studies \\cite{local2014,barbieri2015efficient} may not be the maximal $k$-core or with size constraints.\n\nFor attributed graphs, all the corresponding CS studies take both link relationship and attributes into consideration, because the attributes often make communities more meaningful and easy for interpretation. As a result, the solutions for different attributed graphs are different. Generally, both online and index-based algorithms are developed for CS on these graphs. The index-based algorithms run faster, but incur an offline computational cost.\n\nIn practice, the query users can select the CS solutions based on the graph models since the community models are formulated based on the graph models. For example, for keyword-based attributed graphs, ACQ can be considered.\nMeanwhile, if the CS queries are executed with high frequency, the index-based algorithms should be better choices as they are faster, although they have to build the index in an offline manner.\n\n\\subsection{Influence Value-Based Attributed Graphs}\n\\label{sec:kcoreInfluence}\n\n\\subsubsection{Single-dimensional Influential CS}\n\nLi et al. \\cite{Li:vldb:2015} proposed the influential CS problem. They considered an undirected graph $G(V,E)$ with vertex set $V$ and edge set $E$. Each vertex $v \\in V$ is associated with a weight $w_u$ indicating the influence (or importance) of $u$. Without loss of generality, they assumed that the weight vector $W=(w_1, w_2, \\cdots, w_n)$ forms a total order, i.e., for any two vertices $v_i$ and $v_j$, if $i \\ne j$, then $w_i \\ne w_j$.\n\n\\begin{definition} [Influence value of a subgraph]\n \\label{def:influvalue} \\\\ Given an undirected graph $G(V, E)$ and an induced subgraph $H(V_H, E_H)$ of $G$, the influence value of $H$ denoted by $f(H)$ is defined as the minimum weight of the vertices in $H$, i.e., $f(H) = \\mathop {\\min }\\nolimits_{u \\in {V_H}} \\{ {w_u}\\} $.\n\\end{definition}\n\n\\begin{definition} [$k$-influential community]\n \\label{def:influcore} Given an undirected graph $G=(V, E)$ and an integer $k$. A $k$-influential community is an induced subgraph $H^k=(V_H^k, E_H^k)$ of $G$ that meets all the following constraints.\n \\begin{enumerate}\n \\item \\textbf{Connectivity.} $H^k$ is connected;\n \\item \\textbf{Cohesiveness.} Each vertex $u$ in $H^k$ has degree at least $k$;\n \\item \\textbf{Maximal structure.} There is no other induced subgraph ${\\tilde H}$ such that (1) ${\\tilde H}$ satisfies connectivity and cohesiveness constraints, (2) ${\\tilde H}$ contains $H^k$, and (3) $f({\\tilde H}) = f(H^k)$.\n \\end{enumerate}\n\\end{definition}\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.7\\hsize]{figures\/lu-influential.pdf}\n\\caption{An example of influential CS (the numbers denote the weights) \\cite{Li:vldb:2015}.}\n\\label{fig:lu-influential}\n\\end{center}\n\\end{figure}\n\n\nConsider the graph shown in Fig.~\\ref{fig:lu-influential}. Suppose, for instance, that $k=2$, then by definition the subgraph induced by vertex set $\\{v_{12}, v_{13}, v_{14}, v_{15}\\}$ is a $2$-influential community with influence value $12$, as it meets all the constraints in Definition~\\ref{def:influcore}. Note that the subgraph induced by vertex set $\\{v_{12}, v_{14}, v_{15}\\}$ is not a $2$-influential community. This is because it is contained in a $2$-influential community induced by vertex set $\\{v_{12}, v_{13}, v_{14}, v_{15}\\}$ whose influence value equals its influence value, thus fail to satisfy the maximal structure constraint.\n\n\\begin{problem} [Top-$r$ $k$-influential CS problem \\\\(TIC)] Given a graph $G(V, E)$ and two parameters $k$ and $r$, the problem is to find the top-$r$ $k$-influential communities with the highest influence value.\n\\end{problem}\n\n\n\n\\begin{definition} [Non-contained $k$-influential community]\n \\label{def:noncontaincore} Given a graph $G(V, E)$ and an integer $k$. A non-contained $k$-influential community $H^k=(V_H^k, E_H^k)$ is a $k$-influential community that meets the following constraint.\n \\begin{itemize}\n \\item \\textbf{Non-containment.} $H^k$ cannot contain a $k$-influential community ${\\bar H}^k$ such that $f({\\bar H}^k) > f(H^k)$.\n \\end{itemize}\n\\end{definition}\n\nConsider the graph shown in Fig.~\\ref{fig:lu-influential}. Assume that $k=2$. By Definition~\\ref{def:noncontaincore}, we can see that the subgraphs induced by $\\{ v_{3},$ $v_{4},$ $v_{5}\\}$, $\\{v_8,$ $v_9,$ $v_{11}\\}$ and $\\{ v_{13},$ $v_{14},$ $v_{15}\\}$ are non-contained $2$-influential communities. However, the subgraph induced by $\\{ v_{12}, v_{13}, v_{14}, v_{15}\\}$ is not a non-contained $2$-influential community, because it includes a $2$-influential community (the subgraph induced by $\\{ v_{13},$ $v_{14},$ $v_{15}\\}$) with a larger influence value.\n\n\n\\begin{problem} [Top-$r$ non-contained $k$-influential \\\\ CS problem]\n\\label{prob:nic}\nGiven a graph $G(V, E)$ and parameters $k$ and $r$, find the top-$r$ non-contained $k$-influential communities with the highest influence value.\n\\end{problem}\n\n\n\\noindent\\textbf{$\\bullet$ Online search algorithms.}\nAn online search algorithm is proposed in \\cite{Li:vldb:2015} to compute the top-$r$ (non-contained) $k$-influential communities given graph $G$ and parameters $r$ and $k$. The algorithm first computes the $k$-core $C$ of $G$, and then iteratively updates $C$ by removing vertices from $C$ until $C$ becomes empty. In each iteration, a vertex $u$ with the smallest influence value is removed from $C$. After $u$ is removed, the algorithm further removes those vertices that do not belong to the $k$-core from $C$ by invoking a DFS procedure. For each iteration, the connected component that vertex $u$ belongs to forms a $k$-influential community. The $k$-influential communities obtained by the last $r$ iterations are the top-$r$ $k$-influential communities. If after deleting a certain $u$, the vertices in the whole connected component that $u$ belongs to are removed in the DFS procedure, then the corresponding connected component is a non-contained $k$-influential community. In this way, we can obtain the top-$r$ non-contained $k$-influential communities. The algorithm runs in $O(m+n)$ time using $O(m+n)$ space.\n\nThe above algorithm needs to compute all (non-contained) $k$-influential communities before obtaining the top-$r$ (non-contained) $k$-influential communities which is costly when the graph is large and $r$ is small. Therefore, Chen et al. \\cite{chen2016efficient} proposed a backward search algorithm to obtain the top-$r$ (non-contained) $k$-influential communities. The general idea is as follows. Instead of deleting the vertex with the smallest influence value each time, the backward search algorithm initializes an empty vertex set $C$ and inserts into $C$ the vertex with the largest influence value in each iteration. After a vertex $u$ with the largest influence value is inserted, if the core number of $u$ in the subgraph induced by $C$ is no smaller than $k$, the connected component containing $u$ in the subgraph induced by $C$ represents a $k$-influential community. The algorithm can terminate once $r$ $k$-influential communities are reported. The top $r$ non-contained $k$-influential communities can be computed in a similar way by checking whether each $k$-influential community is a non-contained $k$-influential community before reporting the community.\n\nThe online search algorithms in \\cite{Li:vldb:2015} and \\cite{chen2016efficient} need to access the whole graph to obtain the top-$r$ (non-contained) $k$-influential communities. To solve this issue, Bi et al. \\cite{Bi:2018} proposed a local search algorithm. Let $G_{\\geq \\tau}$ be the subgraph of $G$ induced by all vertices with weights at least $\\tau$, the authors proved that \\textit{if the subgraph $G_{\\geq \\tau}$ of $G$ contains at least $r$ $k$-influential communities, then the top-$r$ $k$-influential communities in $G_{\\geq \\tau}$ is the query result}. The goal is to find the smallest subgraph $G_{\\geq \\tau^*}$ of $G$ containing at least $r$ $k$-influential communities. The general idea is as follows. The algorithm starts with a large $\\tau$, and iteratively decreases the value of $\\tau$ until reaching the target value. For each $\\tau$, only the vertices with weights no smaller than $\\tau$ need to be accessed. The authors proved that the time complexity of the algorithm is linear to the size of the smallest subgraph $G_{\\geq \\tau^*}$ that an online search algorithm without indexes needs to access to correctly compute the top-$r$ $k$-influential communities. Thus the algorithm is instance-optimal. Their algorithm can be easily extended to solve Problem \\ref{prob:nic}.\n\n\n\\vspace*{0.1cm}\\noindent\\textbf{$\\bullet$ An index-based algorithm.}\nIn \\cite{Li:vldb:2015}, an index, called ICP-Index, is presented for solving Problem \\ref{prob:nic}.\nThe index is designed based on the observation that \\textit{for each $k$, the $k$-influential communities form an inclusion relationship}. Based on such an inclusion relationship, all the $k$-influential communities can be organized by a tree-shape structure. The index includes such tree structures for all possible $k$ values. In addition, instead of keeping the whole community for each tree node, a compression method is proposed to make the ICP-Index compact. Specifically, for each non-leaf node in the tree which corresponds to a $k$-influential community, the index only stores the vertices of the $k$-influential communities that are not included in their sub-$k$-influential communities. The same idea is recursively applied to all the non-leaf nodes of the tree following a top-down manner. For each leaf node which corresponds to a non-contained $k$-influential community, the index stores all the vertices of that non-contained $k$-influential community. Using the ICP-Index, the query can be answered efficiently because each node in the tree corresponds to a $k$-influential community and each leaf-node in the tree corresponds to a non-contained $k$-influential community. In \\cite{Li:vldb:2015}, the authors proved that the ICP-Index can be constructed in $O(m^{1.5})$ time using $O(m+n)$ space.\n\n Consider the graph shown in Fig.~\\ref{fig:lu-influential}. Let us consider the case of $k=2$. Clearly, the entire graph is a connected $2$-core, so it is a $2$-influential community. Therefore, the root node of the tree corresponds to the entire graph. After deleting the smallest-weight vertex $v_1$, we get three $2$-influential communities which are the subgraphs induced by the vertex sets $\\{v_3, v_4, v_5\\}$, $\\{v_6, \\cdots, v_{11}\\}$, and $\\{v_{12}, \\cdots, v_{15}\\}$ respectively. Thus, we create three child nodes for the root node which correspond to the three $2$-influential communities respectively. Since $v_1$ and $v_2$ are not included in these three $2$-influential communities, we store them in the root node. The same idea is recursively applied in all the three $2$-influential communities.\n %\n\n %\n Fig.~\\ref{fig:lu-icpindex} shows the tree organization for all $k$ for the graph shown in Fig.~\\ref{fig:lu-influential}.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.95\\hsize]{figures\/lu-icpindex.pdf}\n\\caption{Tree organization of all the $k$-influential communities (the ICP-Index) \\cite{Li:vldb:2015}.} \\label{fig:lu-icpindex}\n\\end{center}\n\\end{figure}\n\n\n\\noindent\\textbf{$\\bullet$ An I\/O efficient algorithm.} An I\/O efficient algorithm to compute the top-$r$ (non-contained) $k$-influential communities is presented in \\cite{Li:VLDBJ:2017}. It assumes that all vertices of the graph can be stored in the main memory.\nThe key idea of the algorithm is that it computes the $k$-influential communities following the decreasing order of their weights, and the communities (as well as the edges in community) with large weights can be safely deleted without affecting the correctness of the algorithm to compute the tree vertices with small weights. Specifically, let $w(e)=\\min\\{w_u, w_v\\}$ be the weight of an edge $e=(u, v)$. The algorithm first sorts the edges in a non-increasing order of their weights using the standard external-memory sort algorithm (we can use the vertex ID to break ties). Then, following this order, the algorithm loads the edges into the main memory up to the memory limit. Subsequently, the algorithm invokes an in-memory algorithm to compute the influential communities in the main memory. After that, the algorithm deletes the computed influential communities as well as the associated edges from the main memory, and then sequentially loads new edges into the main memory until reaches the memory limit. The algorithm iteratively performs this procedure until all the edges are scanned. Note that in each iteration, the algorithm only works on a partial graph, which is loaded in the main memory.\n\nAs an example, consider the graph shown in Fig.~\\ref{fig:lu-influential}. Suppose $k=2$ and the memory can hold at most $10$ edges. The partial graph loaded into memory in the first three iterations for the algorithm is shown in Fig.~\\ref{fig:lu-partialgraph}\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{figures\/lu-partialgraph.pdf}\n\\end{center}\n\\caption{Partial graphs in the memory ($k=2$, memory can hold at most 10 edges) \\cite{Li:VLDBJ:2017}.}\n\\label{fig:lu-partialgraph}\n\\end{figure}\n\n\n\\noindent\\textbf{$\\bullet$ Center-core CS.} Another model to capture the influence of vertices is called the centre-core community search, which is studied by Ding et al. \\cite{ding2018search}. The model uses $k$-core to qualify the dense structure for the community and uses coreness to evaluate the vertex influence. Given a query vertex $q$ and an integer $k$, the center-core community is a connected component of the maximal $k$-core containing the query vertex $q$ and the coreness of vertices in the community is no less than $q$. In addition, the community excludes those vertices with coreness equal to $q$ but cannot be reached from $q$ via vertices with the same coreness with $q$. An online search algorithm and an index based algorithm are proposed in \\cite{ding2018search} to compute the center-core community.\n\n\n\n\\subsubsection{Multi-dimensional Influential CS}\n\nIn \\cite{Li:SIGMOD:2018}, Li et al. studied the multi-dimensional influential CS. It deals with a multi-valued graph $G(V,E,X)$ where $V$ and $E$ denote the set of vertices and edges respectively, and $X$ ($|X|=n$) is a set of $d$-dimensional vectors. In a multi-valued graph, each vertex $v\\in V$ is associated with a $d$-dimensional real-valued vector denoted by $X_v$ $=$ $(x_1^v,$ $\\cdots,$ $x_d^v)$, where $X_v \\in X$ and $x_i^v \\in \\mathbb{R}$. Suppose without loss of generality that on the $x_i$ dimension, $x_i^v$ for all $v \\in V$ form a strict total order, i.e., $x_i^v \\ne x_i^u$ for any $u \\ne v$. It is important to note that if this assumption does not hold, we can easily construct a strict total order by using the vertex identity to break ties for any $x_i^v = x_i^u$. The $d$-dimensional vector $X_v$ represents the values of the vertex $v$ w.r.t. $d$ different numerical attributes. The model studied in \\cite{Li:SIGMOD:2018}, called the skyline community search, is based on the one-dimensional influential community model proposed in \\cite{Li:vldb:2015}. The authors defined the value of $H$ on the $x_i$ dimension (for $i$=1, 2, $\\cdots$, $d$) as ${f_i}(H) \\triangleq \\mathop {\\min }\\nolimits_{v \\in {V(H)}} \\{ x_i^v\\}$.\n\n\\begin{definition}\n \\label{def:communitydominate} Let $H(V_H, E_H)$ and $H^\\prime(V_{H^\\prime}, E_{H^\\prime})$ be two subgraphs of a multi-valued graph $G$. If $f_i(H) \\le f_i(H^\\prime)$ for all $i=1, \\cdots, d$, and there exists $f_i(H) < f_i(H^\\prime)$ for a certain $i$, we call that $H^\\prime$ dominates $H$, denoted by $H \\prec H^\\prime$.\n\\end{definition}\n\n\n\\begin{definition}\n \\label{def:skylinecommunity} Given a multi-valued graph $\\small G(V, E, X)$ and an integer $k$. A skyline community with a parameter $k$ is an induced subgraph $H(V_H, E_H, X_H)$ of $G$ such that it satisfies the following properties.\n\n \\begin{enumerate}\n\\vspace*{-0.1cm}\n \\item \\textbf{Cohesive property.} $H$ is a connected $k$-core;\n \\item \\textbf{Skyline property.} There does not exist an induced subgraph $\\small H^\\prime$ of $G$ such that $\\small H^\\prime$ is a connected $k$-core subgraph and $\\small H \\prec H^\\prime$;\n \\item \\textbf{Maximal property.} There does not exist an induced subgraph $\\small H^\\prime$ of $G$ such that (1) $\\small H^\\prime$ is a connected $k$-core subgraph, (2) $H^\\prime$ contains $H$, and (3) $f_i(H^\\prime)=f_i(H)$ for all $i=1,\\cdots, d$.\n \\end{enumerate}\n\n\\end{definition}\n\n\n\\begin{problem} [Skyline CS problem]\n\\label{prob:skyCS}\nGiven a multi-valued graph $G(V, E, X)$ and an integer $k$, the problem is to find all the skyline communities from $G$ with the parameter $k$. More formally, let $\\cal H$ be the set of all connected $k$-core subgraphs in $G$. We aim to compute a subset $\\cal R$ of $\\cal H$ which is defined as:\n\n{\\scriptsize \\[{\\cal R} \\triangleq \\{ H\\in {\\cal H}|\\neg \\exists {H^\\prime, H^{\\prime\\prime}}\\in {\\cal H}:H \\prec {H^\\prime}, H \\subset {H^{\\prime\\prime}} \\wedge f(H) = f({H^{\\prime \\prime}})\\},\\]}\n\n\\noindent where {$H \\subset {H^{\\prime\\prime}}$} denotes that {$H$} is a subgraph of {${H^{\\prime\\prime}}$} and {$ H\\ne {H^{\\prime\\prime}}$}, and {$f(H) = f({H^{\\prime \\prime}})$} means that {$f_i(H)=f_i({H^{\\prime \\prime}})$} for $i=1, \\cdots, d$.\n\\end{problem}\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\hsize]{figures\/lu-skyline.pdf}\n\\caption{An example of a multi-valued graph \\cite{Li:SIGMOD:2018}.} \\label{fig:lu-skyline}\n\\end{center}\n\\end{figure}\n\nConsider the graph shown in Fig.~\\ref{fig:lu-skyline}. The left panel is a graph with 6 vertices, and the right panel shows the values of these vertices in three different dimensions. Suppose for instance that $k=2$. Then, by Definition~\\ref{def:skylinecommunity}, $H_1=\\{v_1, v_2, v_3\\}$ is a skyline community with values $f(H_1)=(8, 14, 3)$, because there does not exist a connected $2$-core subgraph that can dominate it, and it is also the maximal subgraph that satisfies the cohesive and skyline properties. Similarly, $H_2=\\{v_2, v_4, v_5, v_6\\}$ is a skyline community with $f(H_2)=(6,8,4)$. The subgraph $H_3=\\{v_4, v_5, v_6\\}$ is not a skyline community, because it is contained in $H_2=\\{v_2, v_4, v_5, v_6\\}$ which has the same $f$ values as $H_3$. The subgraph $H_4=\\{v_2, v_3, v_4, v_5, v_6\\}$ is not a skyline community, as $f(H_4)=(6,8,3)$ is dominated by $H_1$ and $H_2$.\n\nIn \\cite{Li:SIGMOD:2018}, the authors first developed an efficient algorithm, called SkylineComm2D, to find all the skyline communities in the 2D case, i.e., $d=2$. The time complexity of SkylineComm2D is $O(s(m+n))$ where $s$ denotes the number of 2D skyline communities (i.e., the answer size), and the space complexity of SkylineComm2D is $O(m+n+s)$, which is linear w.r.t. the graph and answer size. To handle the high-dimensional case (i.e., $d \\ge 3$), the authors proposed a space-partition algorithm to find the skyline communities efficiently. Two novel features of the space-partition algorithm are that (1) its worst-case time complexity is dependent mainly on the answer size, thus it is very efficient when the answer size is not very large; and (2) it is able to progressively output the skyline communities during the execution of the algorithm, and thus it is useful for applications that only require part of skyline communities.\n\n\\subsection{Keyword-Based Attributed Graphs}\n\\label{sec:kcoreKeyword}\n\nA keyword-based attributed graph is an undirected graph $G(V,E)$, with vertex set $V$ and edge set $E$. Each vertex $v \\in V$ is associated with a set of keywords, $W(v)$. The keyword-based attributed graphs are prevalent in social media, bibliographical networks, and knowledge bases. In Fig. \\ref{fig:kcoreACQEg}(a), a keyword-based attributed graph is depicted. For example, vertex $A$ has a set of keywords $\\{w,x,y\\}$.\nIn \\cite{Fang:VLDB:2016,Fang:VLDBJ:2017,fang2017workshop,Shang2017}, CS on keyword-based attributed graphs has been studied extensively.\n\n\\begin{problem}[ACQ \\cite{Fang:VLDB:2016}]\n\\label{prob:acq}\nGiven a keyword-based attributed graph $G(V,E)$, a positive integer $k$, a vertex $q \\in V$ and a set of keywords $S\\subseteq W(q)$, return a set $\\mathcal {G}$ of subgraphs of $G$, such that $\\forall G_q \\in \\mathcal {G}$, the following properties hold:\n\\begin{enumerate}\n \\item \\textbf{Connectivity}. $G_q$ is connected and contains $q$;\n \\item \\textbf{Structure cohesiveness}. $\\forall v\\in G_q$, $deg_{G_q}(v)\\geq k$;\n \\item \\textbf{Keyword cohesiveness}. The size of $L(G_q, S)$ is maximal, where $L(G_q, S)=\\cap_{v \\in G_q}(W(v)\\cap S)$ is the set of keywords shared in $S$ by all vertices of $G_q$.\n\\end{enumerate}\n\\end{problem}\n\nFor example, in Fig. \\ref{fig:kcoreACQEg}(a), if $q$=$A$, $k$=2 and $S$=$\\{w$, $x,y\\}$, then the output of Problem~\\ref{prob:acq} is the subgraph of $\\{A,C,D\\}$, with a shared keyword set $\\{x,y\\}$, meaning that these vertices share the keywords $x$ and $y$.\n\nThe subgraph $G_q$ is called an {\\it attributed community} (or AC) of $q$, and $L(G_q, S)$ is the {\\it AC-label} of $G_q$. In Problem~\\ref{prob:acq}, the first two properties ensure the structure cohesiveness.\nProperty 3 enables the retrieval of communities whose vertices have common keywords in $S$.\nIt requires $L(G_q, S)$ to be maximal, because it aims to find the AC(s) only containing the most related vertices, in terms of the number of common keywords. In Fig. \\ref{fig:kcoreACQEg}(a), if we use the same query\n($q$=$A$, $k$=2, $S$= $\\{w,x,y\\}$),\nwithout the ``maximal'' requirement, we can obtain communities such as $\\{A,B,E\\}$ (which share no keywords), $\\{A,B,D\\}$, or $\\{A,B,C\\}$ (which share 1 keyword). Note that there does not exist an AC with AC-label being exactly $\\{w$, $x,y\\}$.\n\nTwo outstanding features of ACQ are as follows:\n(1) {\\it Ease of interpretation.}\nAn AC contains tightly-connected vertices with similar contexts or backgrounds. Thus, an ACQ user can focus on the common keywords or features of these vertices, i.e., the AC-labels facilitate understanding of the vertices that form the AC.\n(2) {\\it Personalization.} The user of an ACQ can control the semantics of the AC, by specifying a set of $S$ of keywords. Intuitively, $S$ decides the meaning of the AC based on the user's need.\n\nThe ACQ problem is challenging. A simple method to answer an ACQ runs three steps. First, all non-empty subsets of\n$S$, $S_1,S_2,\\cdots$, $S_{2^l-1}$ ($l$=$|S|$),\nare enumerated. Then, for each subset $S_i$(1$\\leq i\\leq2^l-$1), it checkes whether there is a subgraph which satisfies the first two properties. Finally, it outputs the subgraphs having the most shared keywords. However, since there are exponential number of subsets, it is impractical for large graphs. To alleviate this issue, the authors observed the {\\it anti-monotonicity} property, which states that given a set $S$ of keywords, if it appears in every vertex of an AC, then for every subset $S'$ of $S$, there exists an AC in which every vertex contains $S'$. Based on this property, many subsets of $S$ can be pruned, and thus faster online query algorithms can be developed.\n\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c c}\n \\begin{minipage}{4.0cm}\n\t\\includegraphics[width=4.0cm]{figures\/kcoreGraphACQ}\n \\end{minipage}\n &\n \\begin{minipage}{3.8cm}\n\t\\includegraphics[width=3.379cm]{figures\/ck-tree}\n \\end{minipage}\n \\\\\n \n (a) a graph $G$\n &\n \n (b) CL-tree index of $G$\n\\end{tabular}\n\\caption{An example for illustrating ACQ \\cite{Fang:VLDB:2016}.}\n\\label{fig:kcoreACQEg}\n\\end{figure}\n\nAn index, called {\\tt CL-tree}, is proposed for organizing the vertex keyword data in a hierarchical structure. The CL-tree has the same tree structure as {\\tt ShellStruct} (see Section~\\ref{sec:sizeUnbounded}), but for each node $p$, it maintains an additional inverted list such that for each keyword $e$ that appears in the vertices of $p$, a list of IDs of vertices which contain $e$ is stored.\nSince each graph vertex and each keyword appear only once, the space cost of keeping such an index is $O({\\widehat l}\\cdot n)$, where $\\widehat l$ denotes the average size of $W(v)$ over $V$. As a result, the space cost is linear to the size of $G$. As shown in~\\cite{Fang:VLDB:2016}, the {\\tt CL-tree} structure can be built level by level in a bottom-up manner and it takes linear time cost, i.e., $O(m\\cdot\\alpha(n))$. In addition, index maintenance algorithms for the {\\tt CL-tree} are developed \\cite{Fang:VLDBJ:2017}. Fig. \\ref{fig:kcoreACQEg}(b) presents the {\\tt CL-tree} index for the graph in Fig. \\ref{fig:kcoreACQEg}(a).\n\nBased on the {\\tt CL-tree}, two incremental algorithms (from examining smaller candidate keyword sets to larger ones) and one decremental algorithm (from examining larger candidate keyword sets to smaller ones) are developed. For each candidate keyword set, they check whether there is a connected $k$-core containing $q$, and finally return the one with largest keyword set.\n\n\\subsection{Location-Based Attributed Graphs}\n\\label{sec:kcoreLocation}\n\nA location-based attributed graph, also called geo-social network, is an undirected graph $G(V,E)$ with vertex set $V$ and edge set $E$. For each vertex $v\\in V$, it has a location position $(v.x, v.y)$, where $v.x$ and $v.y$ denote its positions along $x$- and $y$-axis in a two-dimensional space. Geo-social networks widely exist in many location-based services, including Twitter, Facebook, and Foursquare \\cite{armenatzoglou2013general,fang2014detecting,fang2016scalable}.\nIn Fig. \\ref{fig:SACEg}(a), a geo-social network with ten vertices is depicted.\n\n\\begin{table}[h]\n \\centering\n \\caption {CS works on geo-social networks.}\n \\label{tab:CSLocation}\n \\begin{tabular}{c|l}\n \\hline \\textbf{CS query} & \\textbf{Spatial cohesiveness}\\\\\n \\hline\\hline\n SAC search \\cite{Fang:VLDB:2017,Fang:TKDE:SAC} & smallest minimum covering circle\\\\\n \\hline\n RB-$k$-core search \\cite{wangkai2018} & radius-fixed covering circle\\\\\n \\hline\n GSGQ \\cite{zhu2017geo} & rectangle, center-fixed circles\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\nThree kinds of CS queries have been studied on geo-social networks, namely {\\it spatial-aware community (SAC) search} \\cite{Fang:VLDB:2017}, {\\it radius-bounded $k$-core (RB-$k$-core) search} \\cite{wangkai2018}, and {\\it geo-social group queries with minimum acquaintance constraint (GSGQ)} \\cite{zhu2017geo}.\nGenerally, they all require that the communities are structurally and spatially cohesive. For structure cohesiveness, they all adopt the $k$-core model, but for spatial cohesiveness, they use different constraints, as outlined in Table~\\ref{tab:CSLocation}. In SAC search, the community is in the smallest minimum covering circle (MCC); in RB-$k$-core search, the community is in a circle with radius less than an input threshold; in GSGQ, the community is in a given rectangle or circle centered at the query vertex.\n\n\\input{sac}\n\n\\input{rbkcore}\n\n\\input{gsgq}\n\n\\subsection{Profile-Based Attributed Graphs}\n\\label{sec:kcoreProfile}\n\nA profiled-based attributed graph, or profiled graph, is an undirected graph $G(V,E)$ with vertex set $V$ and edge set $E$, in which each vertex is associated with {\\it profile}. The profile of a vertex $v\\in V$ is a set of keywords $T(v)$ that are arranged in a hierarchical manner, called a P-tree. Typical such attributes are users' affiliation, expertise, locations, etc. Profiled graphs are prevalent in knowledge bases, and social media.\n\nFig. \\ref{fig:profiledGraph}(a) depicts a profiled graph. For instance, vertex $D$ has a hierarchically organized profile that describes his expertise in Computer Science (e.g., abbreviation AI means Artificial Intelligence) by following the \\emph{ACM Computing Classification System (CCS)}~\\footnote{ACM CCS: http:\/\/www.acm.org\/publications\/class-2012}.\n\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c}\n \\begin{minipage}{6.4cm}\n\t\\includegraphics[width=6.4cm]{figures\/PCSGraph}\n \\end{minipage}\n \\\\\n \\begin{minipage}{7.2cm}\n\t\\includegraphics[width=7.2cm]{figures\/PCSResults}\n \\end{minipage}\n\\end{tabular}\n\\caption{A profiled graph and two PC's~\\cite{Yankai18}.}\n\\label{fig:profiledGraph}\n\\end{figure}\n\nChen et al.~\\cite{Yankai18} investigated the problem of {\\it profiled community search} (or PCS) on profiled graphs. To capture the profile-based cohesiveness, they introduced the concept of ``maximal common subtree'', which describes the commonality of vertices' profile.\n\n\\begin{definition}[Maximal common subtree]\n\\label{df:MaximalTree}\nGiven a profiled graph $G$, the maximal common subtree of $G$, denoted by $\\mathcal M$($G$), holds the properties:\n(1) $\\forall v \\in G$, $\\mathcal M$($G$) $\\subseteq T(v)$;\n(2) there exists no other common subtree $\\mathcal {M'}$($G$) such that $\\mathcal {M}$($G$) $\\subseteq \\mathcal {M'}$($G$).\n\\end{definition}\n\n\\begin{problem}[PCS]\n\\label{prob:PCS}\nGiven a profiled graph $G(V,E)$, a positive integer $k$, a query node $q\\in G$, find a set $\\mathcal {G}$ of graphs, such that $\\forall G_q \\in \\mathcal {G}$, following properties hold:\n\\begin{enumerate}\n\\item \\textbf{Connectivity.} $G_q$ is connected and contains $q$;\n\\item \\textbf{Structure cohesiveness.} $\\forall v\\in G_q$, $deg_{G_q}(v)\\geq k$;\n\\item \\textbf{Profile cohesiveness.} There exists no other $G'_q \\subseteq G$ satisfying the above two constraints, such that $\\mathcal M(G_q) \\subseteq \\mathcal M(G'_q)$.\n\\item \\textbf{Maximal structure.} There exists no other subgraph $G'_q$ satisfying the above properties, such that $G_q \\subset G'_q$ and $\\mathcal M(G_q)$ = $\\mathcal M(G'_q)$;\n\\end{enumerate}\n\\end{problem}\n\nThe subgraph $G_q$ is called a {\\it profiled community} (or PC). In Problem~\\ref{prob:PCS}, the first two properties guarantee the structure cohesiveness. The {\\it profile cohesiveness} captures the maximal shared profile among all vertices in $G_q$. The {\\it maximal structure} property aims to retrieve all qualified vertices in the community.\nFor instance, in Fig. \\ref{fig:profiledGraph}(a), if $q$=D, $k$=2, then two PC's and their maximal common subtrees are respectively shown in Fig. \\ref{fig:profiledGraph}(b) and (c).\nThese two common subtree sufficiently reflects the ``theme'' of the community. For example, in the PC grouped by vertices \\{B, C, D\\}, all the researchers involved share interest in ML (i.e., Machine Learning) and Artificial Intelligence, whereas for the other PC, the researchers are all interested in other research domains.\n\nThe PCS problem is technically challenging, because the number of subtrees of a P-tree could be exponentially large, and thus enumerating all of them is impractical. To answer the PCS query efficiently, Chen et al.~\\cite{Yankai18} introduced the anti-monotonicity property, based on which the query can be performed much faster. To further improve efficiency, they developed the \\emph{CP-tree} index, which systematically organizes all the graph vertices and their P-trees into a compact tree structure. The CP-tree index enables the development of two fast PC discovery algorithms.\n\n\n\\subsection{Undirected Graphs}\n\\label{sec:kcoreSimple}\n\nAn undirected graph, denoted by $G(V,E)$, contains a set $V$ of vertices and a set $E$ of edges. Existing CS works on simple undirected graphs can be classified as {\\it size-unbounded} and {\\it size-bounded} CS, where the former one has no constraint on the size of the community and the latter one imposes constraint on the community size.\n\n\\input{sizeUnbounded}\n\\input{sizeBounded}\n\n\n\\noindent\\textbf{Remark.} Some other factors, such as distances among vertices~\\cite{KDD2010} and local distance dynamics~\\cite{khop2017,khop2018}, have also been considered for CS on simple graphs. Due to the space limitation, we skip the details.\n\n\\subsection{Temporal Graphs}\n\\label{sec:kcoreTemporal}\n\nLi et al. \\cite{Li:ICDE:2018} studied the persistent community search problem in a temporal graph. A temporal graph is an undirected graph $G(V,E)$ with vertex set $V$ and edge set $E$. Each edge $e\\in E$ is a triplet $(u,v,t)$ where $u$, $v$ are vertices in $V$, and $t$ is the interaction time between $u$ and $v$. For a temporal graph $G$, the projected graph denoted by $G_p$ over the time interval $[t_s, t_e]$ is defined as $G_p=(V, E, [t_s, t_e])$, where $V=V(G)$ and $E=\\{(u,v)|(u, v, t)\\in E(G), t \\in [t_s, t_e]\\}$. Fig.~\\ref{fig:lu-temporal} (b) illustrates the projected graph of the temporal graph in Fig.~\\ref{fig:lu-temporal} (a) over the interval $[1, 8]$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.9\\columnwidth]{figures\/lu-temporal.pdf}\n\\caption{A temporal graph and the projected graph \\cite{Li:ICDE:2018}.}\n\\label{fig:lu-temporal}\n\\end{center}\n\\end{figure}\n\n\\begin{definition}\n \\label{def:maximalcoreinterval} \\textbf{(Maximal $(\\theta, k)$-persistent-core interval)} Given a temporal graph ${G}=({V}, {E})$ and parameters $\\theta>0$ and $k>0$, an interval $[{t_s}, {t_e}]$ with ${t_e}-{t_s} \\ge \\theta$ is called a maximal $(\\theta, k)$-persistent-core interval for $G$ if and only if the following two conditions hold. (1) For any $t \\in [t_s, t_e-\\theta]$, the projected graph of $G$ over the interval $[t, t+\\theta]$ is a connected $k$-core subgraph. (2) There is no super-interval of $[t_s, t_e]$ such that (1) holds.\n\\end{definition}\n\n\n\\begin{definition} [Core persistence]\n\\label{def:corepersistence}\nLet $T$ $=$ $\\{[t_{s_1},$ $t_{e_1}],$ $\\cdots,$ $[t_{s_r},$ $t_{e_r}]\\}$ be the set of all maximal $(\\theta, k)$-persistent-core intervals of $G$. Then, the core persistence of $G$ with parameters $\\theta$ and $k$, denoted by $F(\\theta, $ $k,$ $G)$, is defined as\n \\begin{equation*}\n \\label{eq:corepersistent}\nF(\\theta ,k, G) = \\left\\{ \\begin{gathered}\n \\sum\\limits_{i = 1}^r {(t_{{e_i}} - t_{{s_i}})} - (r - 1)\\theta , \\quad if\\ T \\ne \\emptyset \\hfill \\\\\n 0\\quad otherwise \\hfill \\\\\n\\end{gathered} \\right.\n\\end{equation*}\n\\end{definition}\n\n\n\\begin{definition} [$(\\theta, \\tau)$-persistent $k$-core]\n \\label{def:globalcore} Given a temporal graph $G$, parameters $\\theta$, $\\tau$, and $k$, a $(\\theta, \\tau)$-persistent $k$-core is an induced temporal subgraph $C=(V_C, E_C)$ that meets the following properties.\n \\begin{enumerate}\n \\item \\textbf{Persistent core property.} $F(\\theta, k, C) \\ge \\tau$; \n \\item \\textbf{Maximal property.} There does not exist an induced temporal subgraph $C^\\prime$ that contains $C$ and also satisfies the persistent core property.\n \\end{enumerate}\n\\end{definition}\n\n\\begin{problem}\n\\label{prob:timecs}\n\\textbf{(The persistent community search \\\\ problem)} Given a temporal graph $G$, parameters $\\theta$, $\\tau$ and $k$, the persistent community search problem aims to find the largest $(\\theta, \\tau)$-persistent $k$-core in $G$.\n\\end{problem}\n\n\nConsider the temporal graph $G$ in Fig.~\\ref{fig:lu-temporal}(a). Assume that $\\theta$=3 and $k$=2. We can see that there is no maximal $(3, 2)$-persistent-core interval for the entire graph $G$. There is a maximal $(3, 2)$-persistent-core interval $[1, 5]$ for the subgraph $C$ induced by vertices $\\{v_1, v_2, v_3\\}$. This is because $[1, 5]$ is the maximal interval such that in any $3$-length subinterval of $[1, 5]$, the vertices $\\{v_1, v_2, v_3\\}$ form a connected $2$-core. Let $\\tau = 4$, we can see that the subgraph $C$ induced by vertices $\\{v_1, v_2, v_3\\}$ is a $(3, 4)$-persistent $2$-core. Because $F(3, 2, C)$=4, which is no less than $\\tau$; and $C$ is the maximal subgraph that meets such a persistent core property.\n\nAs shown in \\cite{Li:ICDE:2018}, the persistent community search problem is NP-hard. Therefore, a prune-and-search approach is proposed in \\cite{Li:ICDE:2018}. In the pruning phase, a temporal graph reduction algorithm is designed by decomposing the whole time span of the temporal graph into several meta-intervals, each of which has some properties to prune vertices. In the search phase, a branch and bound algorithm with several pruning rules are proposed to find the maximum $(\\theta, \\tau)$-persistent $k$-core.\n\n\\section{$K$-Core-Based Community Search}\n\\label{sec:kcore}\n\nIn this section, we review CS works that use the $k$-core as structure cohesiveness metric. We classify these works into several groups according to the types of graphs, namely undirected graphs, directed graphs, and attributed graphs including keyword-based, location-based, temporal, influence value-based, and profile-based graphs, and then discuss them respectively.\n\n\\input{kcoreSimple}\n\\input{kcoreDirected}\n\\input{kcoreKeyword}\n\\input{kcoreLocation}\n\\input{kcoreTemporal}\n\\input{kcoreInfluence}\n\\input{kcoreProfile}\n\\input{kcoreDiscuss}\n\n\\subsection{Discussions}\n\\label{kecc:discuss}\n\nIn this section, we review two CS studies that adopt the $k$-ECC model as the community cohesiveness metric. The first one \\cite{Chang:SIGMOD:2015} aims to find the maximum SMCS, while the second one \\cite{hu2016querying,hu2017querying} tries to find the minimum SMCS. In terms of efficiency, the maximum SMCS can be computed more efficiently. For example, by using the MST index \\cite{Chang:SIGMOD:2015}, it can be computed in the optimal time cost. Nevertheless, the maximum SMCS may have size much larger than that of the minimum or minimal SMCS's. This also implies that for practitioners, they have to choose the specific algorithm, based on their specific requirements on community sizes and efficiency.\n\nWe remark that these two CS studies mainly focus on simple graphs. It is not clear how to adapt for them for other kinds of graphs, such as directed graphs and attributed graphs. Thus, an interesting future topic is to investigate how to perform CS on other kinds of graphs by adopting the $k$-ECC model.\n\n\\subsection{Maximum SMCS}\n\\label{sec:maxSMCS}\n\nIn \\cite{Chang:SIGMOD:2015}, Chang et al. computed the maximum SMCS for a set of query vertices $Q$, which is defined as follows.\n\n\\begin{problem}\n\\label{prob:keccMax}\nGiven an undirected simple graph $G(V,E)$, and a set of query vertices $Q\\subseteq V$, return a subgraph $H(V_H,E_H)$ of $G$, such that\n\\begin{enumerate}\n \\item $V_H$ contains $Q$;\n \\item $\\lambda(H)$ is maximized;\n \\item There exists no other subgraph $H'$ satisfying the above properties, such that $H\\subset H'$.\n\\end{enumerate}\n\\end{problem}\n\n\\begin{figure}[ht]\n\\centering\n\\hspace*{-.3cm}\n\\begin{tabular}{c c}\n \\begin{minipage}{5.7cm}\n\t\\includegraphics[width=5.7cm]{figures\/keccGraph}\n \\end{minipage}\n &\n \\begin{minipage}{2.4cm}\n\t\\includegraphics[width=2.4cm]{figures\/keccIndex}\n \\end{minipage}\n \\\\\n (a) a graph\n &\n (b) the index\n\\end{tabular}\n\\caption{An example for illustrating maximum SMCS \\cite{Chang:SIGMOD:2015}.}\n\\label{fig:keccMaxEg}\n\\end{figure}\n\nFor example, consider the graph in Fig. \\ref{fig:keccMaxEg}(a). Let $Q$=$\\{v_1$,$v_4\\}$. Then, for this query we will return the subgraph $g_1$, and its connectivity is $\\lambda(g_1)$=4.\n\nA basic solution of Problem \\ref{prob:keccMax} is to sequentially enumerate all the maximal $k$-ECCs by varying $k$ from $|V|$ to 1, and stops when the first $k$-ECC which contains $Q$ is found. Then, the first $k$-ECC is returned as the community.\nIn the literature, there are two efficient $k$-ECC enumeration algorithms. One is based on graph decomposition \\cite{Chang:SIGMOD:2013}, while the other one is based on the random contraction \\cite{akiba2013linear}. As shown in \\cite{Chang:SIGMOD:2015}, the basic solution takes $O(|V|\\cdot h\\cdot l\\cdot |E|)$ time if the first $k$-ECC enumeration algorithm is adopted, or $O(|V|\\cdot t\\cdot |E|)$ time if the second one is used, where $h$ and $l$ are bounded by small constants for real graphs, and $t$=$O(log^2\\cdot |V|)$. Obviously, both of them are inefficient for large graphs.\n\nTo improve the query efficiency, Chang et al. proposed a novel compact index structure, which allows the query can be answered in optimal time cost, i.e., the time cost is linear to the size of $H$. The index is built based on a key observation that for any pair of vertices $u$ and $v$ in $H$, their connectivity $\\lambda(u,v)$ is at least $\\lambda(H)$. This implies, if the connectivity of each pair of vertices in $G$ is preserved, then the query can be answered in linear time cost, because we can first get $\\lambda(H)$ by checking the connectivity of vertex pairs in $Q$, and then find $H$ by traversing the connected edges whose connectivity are at least $\\lambda(H)$.\n\nTo preserve all the connectivity information of $G$, Chang et al. developed the concept of {\\it connectivity graph} $G_c$ for the graph $G$, which has the same sets of vertices and edges with $G$, and for each edge $(u,v)\\in G_c$, it is associated with a connectivity value denoting the edge-connectivity between vertices $u$ and $v$ in $G$. Then, the maximum spanning tree (MST) of $G_c$ is the index structure built for $G$. For example, Fig. \\ref{fig:keccMaxEg}(b) presents the index structure for the graph in Fig. \\ref{fig:keccMaxEg}(a). The index can be built by first constructing the connectivity graph $G_c$ and then computing the MST from $G_c$. Clearly, the space cost of the MST is $O(|V|)$ since it has $|V|$ vertices and at most $|V|$--1 edges.\n\nBased on the index MST, Chang et al. proposed an efficient query algorithm to solve Problem \\ref{prob:keccMax}. Specifically, it first computes $\\lambda(H)$ by using the MST, and then finds the maximum SMCS by collecting the subtree of MST, whose edges have connectivity values being at least $\\lambda(H)$. By using the technique of lowest common ancestor (LCA), the query can achieve a time cost of $O(|H_V|)$, which is optimal since outputting the vertex set of $H$ takes $O(V_H)$ time.\n\nIn addition, the authors studied a variant of Problem \\ref{prob:keccMax} by imposing an additional constraint, which requires the number of vertices in $H$ is at least $L$, where $L$ is a parameter specified by the user. It can also be solved in optimal time cost with the index MST.\n\n\\subsection{Minimum and Minimal SMCS's}\n\\label{sec:minSMCS}\n\nIn~\\cite{hu2017querying}, Hu et al. found that although the maximum SMCS has a high cohesiveness (i.e., high {\\it connectivity}), the size of maximum SMCS's are often extremely large and complex. For example, on the DBLP bibliographical network that contains $803K$ vertices and $3.2M$ edges, the average number of vertices in a maximum SMCS is over $400K$. This not only hinders the analysis of the SMCS structure, but also makes it difficult to be used in real situations. To remedy this issue, Hu et al. examined the discovery of an SMCS that has a small number of vertices. Particularly, they studied the minimum SMCS and minimal SMCS problems:\n\n\\begin{problem}[Minimum SMCS]\n\\label{prob:keccMinimum}\nGiven an undirected simple graph $G(V,E)$, and a set of query vertices $Q\\subseteq V$, return a subgraph $H(V_H,E_H)$ of $G$, such that\n\\begin{enumerate}\n \\item $V_H$ contains $Q$;\n \\item $\\lambda(H)$ is maximized;\n \\item $|H_V|$ is minimized.\n\\end{enumerate}\n\\end{problem}\n\n\\begin{problem}[Minimal SMCS]\n\\label{prob:keccMinimal}\nGiven an undirected simple graph $G(V,E)$, and a set of query vertices $Q\\subseteq V$, return a subgraph $H(V_H,E_H)$ of $G$, such that\n\\begin{enumerate}\n \\item $V_H$ contains $Q$;\n \\item $\\lambda(H)$ is maximized;\n \\item There exists no other subgraph $H'\\subset H$ satisfying the above properties.\n\\end{enumerate}\n\\end{problem}\n\nObviously, a minimum SMCS is also a minimal SMCS, and both of them are much smaller than the maximum SMCS. For example, on the DBLP network, their average sizes are less than $0.23K$, while the average size of maximum SMCS is over $400K$. We illustrate these three kinds of SMCS in Fig. \\ref{fig:example_smcs}.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.32\\textwidth]{.\/figures\/example_smcs.pdf}\n\\caption{The maximum SMCS ($G_3$), minimum SMCS ($G_2$), and minimal SMCS's ($G_1$ and $G_2$) for query Q=\\{$f$\\}~\\cite{hu2017querying}.}\n\\label{fig:example_smcs}\n\\end{figure}\n\nIn \\cite{hu2017querying}, Hu et al. showed that the minimum SMCS problem is APX-hard, since it is a generalization of the \\textsc{Steiner Tree} problem (see Section \\ref{sec:sizeBounded}). Furthermore, unless P=NP, there does not exist any polynomial-time algorithm that approximates the minimum SMCS problem within any constant ratio. Therefore, it is not only intractable to obtain a minimum SMCS, but also hard to get its approximate version in an accurate manner. To trade off the efficiency and result quality, Hu et al. \\cite{hu2017querying} focused on the minimal SMCS problem.\n\nA naive solution for Problem \\ref{prob:keccMinimal} is to first adopt the solution in~\\cite{Chang:SIGMOD:2015} to compute the maximum SMCS $G'$, and then iteratively refine $G'$ to ensure its minimality. While this solution is simple, it has a high time complexity, since the cost of testing the minimality of an SMCS is high.\nTo achieve higher efficiency, Hu et al. proposed an {\\it Expand-Refine} framework to find a minimal SMCS, which consists of three steps. First, the Steiner-connectivity of the query vertex set $Q$ (i.e., the maximum $\\lambda(H)$) is computed. Then, in the \\emph{Expand} step, through local expansion of vertices starting from vertices in $Q$, a subgraph $H'$ of $G$ with connectivity being $\\lambda(H)$ is obtained. In the \\emph{Refine} step, an algorithm is proposed to remove vertices based on the dependence of vertices on their minimal SMCS's. As a result, the minimal SMCS problem can be solved in a polynomial time cost, i.e., $O(t\\cdot\\ h\\cdot\\ l\\cdot |E|)$, where $t\\textless|H_V|$, and $h$ and $l$ are usually bounded by small constants.\nBesides, to further improve the efficiency, the authors relaxed the constraints from two perspectives, namely connectivity and minimality, and computed the approximate SMCS with theoretical guarantee.\n\nIn addition, for an important special case with only one query vertex (i.e., $|Q|$=1), Hu et al. developed a customized algorithm for it. The main idea is to keep the processing information related to the current query in a small cache structure, and use these information to answer the subsequent queries. As a result, it performs faster than the solution above.\n\n\n\\section{$K$-ECC-Based Community Search}\n\\label{sec:kecc}\n\nIn this section, we review CS studies \\cite{Chang:SIGMOD:2015,hu2016querying} that use the $k$-ECC model as the community structure cohesiveness. Given a graph $G$ and a set $Q$ of vertices, their general goals are to find a subgraph $H$ of $G$, which contains $Q$ and has the maximum edge-connectivity, also called the Steiner Maximum-Connected Subgraph (SMCS). Their difference is that one maximizes the size of $H$ \\cite{Chang:SIGMOD:2015}, while the other one tries to minimize the size of $H$ \\cite{hu2016querying}.\n\n\\input{keccMax}\n\\input{keccMin}\n\\input{keccDiscuss}\n\\subsection{Discussions}\n\\label{sec:ktrussDiscuss}\n\n\nGenerally, the $k$-truss-based CS solutions on simple graphs can be divided into two groups, where the first group \\cite{k-truss2014,Akbas:VLDB:2017} computes the $k$-truss community, while the second group \\cite{huang2015approximate} aims to find closest communities. In the first group, Akbas et al. \\cite{Akbas:VLDB:2017} improved the efficiency of \\cite{k-truss2014} by developing a novel index.\nFor attributed graphs, there are two CS solutions, which consider keywords \\cite{Huang:2017:ATC} and influence values \\cite{Zheng:IS:2017} respectively. For all these studies above, both online and index-based algorithm are developed.\n\nFor practitioners, to perform CS, we would like to offer some suggestions:\n(1) We should figure out the type of graph (e.g., simple graphs and attributed graphs) in the application.\n(2) For simple graphs, there are two community models, i.e., triangle-connected model and closest model. Generally, the triangle-connected model \\cite{k-truss2014,Akbas:VLDB:2017} is suitable for one single query vertex to discover all overlapping communities containing it, while the closest model \\cite{huang2015approximate} is suitable to discover one closest community containing multiple query vertices, which is not strict to one query vertex. Moreover, triangle connectivity is weaker than the optimization metric of minimum diameter. According to our experience in the real-world applications, the discovered closest community has smaller graph size than triangle-connected truss community.\n(3) For triangle-connected model \\cite{k-truss2014,Akbas:VLDB:2017}, the index-based algorithm in \\cite{Akbas:VLDB:2017} is faster than that in \\cite{k-truss2014}.\n\n\n\\subsection{Weight-Based Attributed Graphs}\n\\label{sec:ktrussLocation}\nIn this section, we consider an undirected weighted graph $G=(V, E, W)$, where the weight of $e$ is denoted by $w(e) \\in W$, representing the importance between vertices $u$ and $v$. Weighted graphs naturally exist in the real-world applications. For instance, in the collaboration network, the edge weights may represent the number of co-authored articles between two authors. Fig. \\ref{fig.WTC} depicts an undirected weighted graph $G$, e.g., edge $(q, s_1)$ has a weight of 0.8. Taking the edge weights into consideration, community search on weighted graphs can find communities capturing more semantics. Zheng et al. \\cite{Zheng:IS:2017} proposed a model of weighted truss community (WTC):\n\n\\begin{figure}[t]\n\\small\n\\vskip -0.1in\n\\centering\n\\includegraphics[width=0.60\\linewidth]{.\/figures\/WTC.pdf}\n\\caption{An example of weighted truss community search.}\n\\label{fig.WTC}\n\\end{figure}\n\n\n\n\n\n\n\\begin{definition}[Weighted Truss Community]\n\\label{prob:wtc}\nGiven an undirected weighted graph $G$=($V$, $E$, $W$), and a positive integer $k$, a weighted $k$-truss community is an induced subgraph $H \\subseteq G$, the following properties hold:\n\\begin{enumerate}\n \\item \\textbf{Connectivity}. $\\forall e_1, e_2\\in E(H)$, $e_1$ and $e_2$ are triangle connected in $H$;\n \\item \\textbf{Cohesiveness}. $\\forall e\\in E(H)$, $\\sup_H(e) \\geq k-2$;\n \\item \\textbf{Maximal Structure}. $H$ is a maximal induced subgraph that satisfies Properties 1 and 2\n\\end{enumerate}\n\\end{definition}\n\nIn the weighted $k$-truss community model, Property 1 adopts the same constraint of triangle connectivity as other $k$-truss community models \\cite{k-truss2014}; Property 2 requires the community to satisfy the structure of $k$-truss; Property 3 can guarantee the property of maximal structure in the weighted $k$-truss community. Given a weighted truss community $H$, the community weight of $H$ is defined as $w(H)=\\min_{e\\in E(H)} w(e)$. To discover the communities with large weights, Zheng et al. \\cite{Zheng:IS:2017} investigated the problem of weighted truss community (WTC) search.\n\n\\begin{problem}\n[WTC search]\n\\label{problem:WTC}\nGiven an undirected weighted graph $G(V,E,W)$, and parameters $k$ and $r$, find the top-$r$ weighted $k$-truss communities $H$ with the largest weights $w(H)$.\n\\end{problem}\n\nConsider a weighted graph $G$ in Fig. \\ref{fig.WTC}, $k=5$, and $r=1$. The community $C_1$ shown in Fig. \\ref{fig.WTC} has the weight $w(C_1)=0.8$, which is larger than the weight of community $C_2$ as $w(C_2)=0.2$. Thus, $C_1$ is the answer of WTC search with the largest weight.\n\n\nStraightforward to enumerate all weighted $k$-truss communities to find the $r$ communities with the largest community weights is impractical in large graphs. To speed up the search efficiency, an index structure called KEP-Index is designed. KEP-Index is built upon the observation that all the communities can be organized into a tree-shaped structure. This is because all the weighted $k$-truss communities from a partial order relationship for each value of $k$. By indexing all the pre-computed weighted $k$-truss communities in a tree-shaped structure, WTC search can be done in the linear time w.r.t. the answer size, which is optimal.\n\n\\subsection{Keyword-Based Attributed Graphs}\n\\label{sec:ktrussInfluence}\nIn this section, we introduce a \\truss-based community search model on a keyword-based attributed graph where vertices are associated with a set of keywords. Huang and Lakshmanan \\cite{Huang:2017:ATC} proposed an attribute-driven truss community model, denoted by ATC, which finds the densely inter-connected communities containing query vertices with similar query attributes. ATC is equipped with two key components of \\kdtruss and an attribute score function.\n\nTo capture dense cohesiveness and low communication cost, ATC builds upon a notion of dense and tight substructure called \\kdtruss. A \\kdtruss requires that every edge is contained at least $(k-2)$ triangles, and the communication cost between the vertices of $H$ and the query vertices is no greater than $d$. By definition, the cohesiveness of a \\kdtruss increases with $k$, and its proximity to query vertices increases with decreasing $d$. For instance, $H$ in Fig. \\ref{fig.subcom}(b) for $V_q=\\{q_1, q_2\\}$ is a \\kdtruss with $k=4$ and $d=2$.\n\n\\begin{figure}[t]\n\\centering\n{\n\\subfigure[\\small{An attributed graph $G$}]{\\includegraphics[width=0.54\\linewidth]{.\/figures\/ATC-graph.pdf} \n\\subfigure[$H$]{\\includegraphics[width=0.43\\linewidth]{.\/figures\/DB_DM.pdf} }\n}\n\\caption{An example of attributed truss community search with query vertices $V_q=$ $\\{q_1,$$ q_2\\}$ and query attributes $W_q=\\{$ `DB', `DM'$\\}$. Here, $k=4$.}\n\\label{fig.subcom\n\\end{figure}\n\n\nTo measure the goodness of an attributed community w.r.t. attribute coverage and correlation, an attribute score function is developed for ATC. Let $\\keyw(H, W_q)$ be the attribute score of community $H$ w.r.t. query attributes $W_q$. Then, $\\keyw(H,W_q) = \\sum_{w\\in W_q} \\frac{\\score(H,w)^2}{|V(H)|}$, where $\\score(H,w) = |V_w\\cap V(H)|$ is the number of vertices covering query attribute $w$. The function $\\keyw(H,W_q)$ satisfies three important properties as follows. \\underline{Property 1}: The more query attributes that are covered by some vertices of $H$, the higher score of $\\keyw(H, W_q)$. The rationale is obvious; \\underline{Property 2}: The more vertices that contain an attribute $w\\in W_q$, the higher the contribution of $w$ should be toward the overall score $\\keyw(H, W_q)$. The intuition is that attributes that are covered by more vertices of $H$ signify homogeneity within the community w.r.t. shared query attributes; \\underline{Property 3}: The more vertices of $H$ that are irrelevant to the query, the lower the score $\\keyw(H, W_q)$. The more query attributes a community has that are shared by more of its vertices, the higher its attribute score. For example, consider the query $Q=(\\{q_1\\}, \\{$`DB', `DM'$\\})$ on the running example graph of Fig. \\ref{fig.subcom}(a). Intuitively, we can see that $H$ has 5 vertices covering `DB' and `DM' each and also has the highest attribute score (i.e., $\\keyw(H, W_q) = \\frac{5^2}{8} + \\frac{5^2}{8}=6.25$), which is the attributed truss community. On the other hand, the induced subgraph of $G$ by vertices $\\{q_1, q_2, v_1, v_2, v_3\\}$ and $\\{q_1, q_2, v_4, v_5, v_6\\}$ are mainly focused in one area (`DB' or `DM'), achieving the score of $5.8$.\n\n\nBased on the \\kdtruss and $\\keyw(H, W_q)$, Huang et al. \\cite{Huang:2017:ATC} studied the ATC problem.\n\n\\begin{problem}\n[ATC search]\n\\label{prob:atc}\nGiven a graph $G$, a query $Q = (V_q, W_q)$, and two numbers $k$ and $d$, return an attributed truss community (ATC) $H$, satisfying the following properties:\n\\begin{enumerate}\n \\item $H$ is a \\kdtruss containing $V_q$.\n\n \\item $H$ has the maximum attribute score $\\keyw(H, W_q)$ among all subgraphs satisfying property 1.\n \n\n\\end{enumerate}\n\\end{problem}\n\n\n\n\n\nTheoretical proofs show that ATC search is NP-hard \\cite{Huang:2017:ATC}, which shows the challenging for computation. To help efficiently processing of ATC queries, \\cite{Huang:2017:ATC} presents a greedy algorithmic framework for finding an ATC in a top-down search manner. The general ideas of this algorithm has three steps. First, it finds the maximal \\kdtruss of original graph $G$ as a candidate. Second, it iteratively removes vertices with the smallest ``attribute marginal gains'' from the candidate graph, and maintains the remaining graph as a \\kdtruss, until no longer possible. The removed vertices have the smallest contribution to attribute score function $\\keyw(H, W_q)$. Finally, it returns a \\kdtruss with the maximum attribute score among all generated candidate graphs as the answer. If there exists more than one \\kdtruss with the maximum attribute scores, the algorithms just outputs one answer.\n\nTo further improve the search efficiency while ensuring high quality, a novel index called attributed-truss index (\\ati) is developed. The \\ati consists of two components: structural trussness and attribute trussness, which maintain known graph structure and attribute information. \\ati can quickly identify a good candidate of $(k,d)$-truss to the answer. In addition, another technique of local exploration is applied for efficiently detecting a small neighborhood subgraph around query vertices, which tends to be densely and closely connected with the query attributes.\n\n\n\n\\subsection{Location-Based Attributed Graphs}\n\\label{sec:ktrussLocation}\n\n\\subsection{Simple Graphs}\n\\label{sec:ktrussLocation}\nIn a simple and undirected graph $G(V,E)$, triangle-connected $k$-truss community model proposed by Huang et al. \\cite{k-truss2014}, finds all communities containing a query vertex. We first introduce the definitions of $k$-truss and triangle connectivity, and then present the model below.\n\nA $k$-truss is the largest subgraph $H$ of $G$ such that every edge is contained in at least $k-2$ triangles in $H$, i.e., $\\forall e\\in E$, its support $sup(e, H)\\geq k-2$ by Definition~\\ref{def:ktruss}. However, $k$-truss may be disconnected with several components in a graph, which is similar with $k$-core. Consider the graph $G$ in Fig. \\ref{fig.4-truss}. There exist two components in the shaded regions to form the 4-truss of $G$, which are obviously disconnected. Disconnected subgraphs are insufficient to define a cohesive and meaningful community.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{.\/figures\/4-truss.pdf}\n\\caption{Example of 4-truss with 2 disconnected components.}\n\\label{fig.4-truss}\n\\end{figure}\n\nTo address the disconnectivity problem of $k$-truss, \\emph{triangle connectivity} is imposed on top of the $k$-truss in \\cite{k-truss2014}. Given two triangles $\\triangle_1$ and $\\triangle_2$ in $G$, $\\triangle_1$ and $\\triangle_2$ are said to be adjacent if they share a common edge. Then, for two edges $e_1, e_2\\in E$, $e_1$ and $e_2$ are \\emph{triangle connected} if they either belong to the same triangle, or are reachable from each other through a series of adjacent triangles. In other words, $\\exists \\triangle_1, \\triangle_2$ such that $e_1 \\in \\triangle_1$, $e_2\\in \\triangle_2$, then either $\\triangle_1=\\triangle_2$, or $\\triangle_1$ is triangle connected with $\\triangle_2$. Based on the $k$-truss and triangle connectivity, the problem of triangle-connected truss community (TTC) search is formulated as follows.\n\n\n\\begin{problem}\n[TTC search]\n\\label{problem:TTC}\nGiven an undirected simple graph $G(V,E)$, a query vertex $q\\in V$, and an integer $k \\geq 2$, return all subgraphs $H\\subseteq G$ satisfies the following three properties:\n\\begin{enumerate}\n\\item \\textbf{Structure Cohesiveness.} $H$ contains the query vertex $q$ such that $\\forall e\\in E(H)$, $sup(e, H)$ $\\geq (k-2)$;\n\n\\item \\textbf{Triangle Connectivity.} $\\forall e_1, e_2\\in E(H)$, $e_1$ and $e_2$ are triangle connected;\n\n\\item \\textbf{Maximal Subgraph.} $H$ is the maximal subgraph of $G$ satisfying Properties 1 and 2. \n\n\\end{enumerate}\n\\end{problem}\n\n\n\nTTC model imposes the triangle connectivity requirement in Property 2 to ensure the discovered communities are connected. This requirement also allows the query vertex to participate in multiple overlapping communities. For example, consider the graph $G$ in Fig. \\ref{fig.ttc}(a), a query vertex $q$, and parameter $k=5$. Two triangle-connected 5-truss communities $C_1$ and $C_2$ containing vertex $q$ are shown in Fig. \\ref{fig.ttc}(b). As the edges in $C_1$ cannot reach the edges in $C_2$ through adjacent triangles, $C_1$ and $C_2$ cannot merge as one large community. This is reasonable, as there are few connections between the two vertex sets $\\{s_1,s_2,s_3,s_4\\}$ and $\\{x_1,x_2,x_3,x_4\\}$.\n\n\n\n\n\\begin{figure} [t]\n\\vskip -0.1in\n\\centering \\mbox{\n\\subfigure[Graph $G$]{\\includegraphics[width=0.67\\linewidth]{.\/figures\/IndexGraph-New.pdf}} \\hskip 0.1in\n\\subfigure[TTCs]{\\includegraphics[width=0.28\\linewidth]{.\/figures\/TTC.pdf} }\n}\n\\caption{An example of TTC search. Here, $k=5$.}\\vskip -0.1in\n\\label{fig.ttc\n\\end{figure}\n\n\n\nThanks to $k$-trusses, truss-based community model inherits several good structural properties of\n$k$-trusses \\cite{k-truss2014}, such as $(k-1)$-edge-connected, bounded diameter and hierarchical\nstructure. Specifically, the diameter of a $k$-truss community $H$ with $|V(H)|$ vertices is no larger than $\\lfloor\\frac{2|V(H)|-2}{k}\\rfloor$ \\cite{cohen2008trusses}. Small diameter has been considered as an important feature of a good community in \\cite{Edachery99graphclustering}. Second,\na $k$-truss community is ($k-1$)-edge-connected \\cite{cohen2008trusses}, i.e., the community keeps connected whenever fewer than $k-1$ edges are deleted \\cite{DBLP:books\/fm\/GareyJ79}. Third, truss-based communities have a strong decomposability for analyzing large-scale networks at different levels of granularity.\n\nTo tackle the problem of TTC search, there exists one online search algorithm \\cite{k-truss2014}, and two index-based search algorithms, which are respectively based on TCP-index \\cite{k-truss2014} and EquiTruss \\cite{Akbas:VLDB:2017}. In the following, we briefly introduce the key ideas of these algorithms one by one.\n\n\\noindent\\textbf{$\\bullet$ Online search algorithm \\cite{k-truss2014}.} Huang et al. \\cite{k-truss2014} proposed an online query algorithm to process a TTC query on a graph $G$. The algorithm firstly applies the truss decomposition \\cite{wang2012truss} on graph $G$ to compute the trussness of all edges in $G$. By the community definition, it starts from the query vertex $q$ and checks an incident edge of $(q, v)\\in E$ with trussness $\\tau((q,v))\\geq k$ to search triangle-connected truss communities. It explores all edges that are triangle-connected to $(q, v)$ and having trussness no less than $k$ in a BFS manner. This process iterates until all incident edges of $q$ have been processed. Finally, a set of $k$-truss communities containing $q$ are returned.\n\nHowever, this online search algorithm may incur a large number of wasteful edge accesses on checking disqualified edges, which is inefficient.\n\n\\noindent\\textbf{$\\bullet$ TCP-index based search algorithm \\cite{k-truss2014}.} To avoid the computational issues mentioned above, Huang et al. \\cite{k-truss2014} designed a Triangle Connectivity Preserving index (TCP-index). TCP-Index preserves the truss number and triangle adjacency relationship in a compact tree-shape index, and supports the query of $k$-truss community in linear time with respect to the community size, which is optimal. Given a graph $G$, it needs to construct a TCP-index for each vertex in $G$, which is denoted as ${\\cal T}_x$. Take a vertex $x$ as an example for TCP-index construction.\nEssentially, $T_x$ is the maximum spanning forest of $G_x$, where $G_x$ is the induced subgraph of $G$ by vertex set of $x$'s neighbors as $N(x)$. For each edge $(y,z)\\in E(G_x)$, a weight $w(y,z)=\\min\\{\\tau((x,y)), \\tau((x,z)),$ $ \\tau((y,$ $z))\\}$ is assigned to it, which indicates that $\\triangle_{xyz}$ can appear only in $k$-truss communities where $k\\leq w(y,z)$.\nFig. \\ref{fig.truss-index} presents a TCP-index $T_{q}$ for vertex $q$ in graph $G$ shown in Fig. \\ref{fig.ttc}(a). Vertices $x_1$, $x_2$, $x_3$ and $x_4$ are connected via the weighted edges of 5, indicating these vertices present in a triangle-connected 5-truss community.\n\n\n\n\\begin{figure} [t]\n\\vskip -0.1in\n\\centering \\mbox{\n\n\\includegraphics[width=0.60\\linewidth]{.\/figures\/TCP.pdf}\n}\n\\caption{TCP-index $T_{q}$ for vertex $q$ of $G$ in Figure~\\ref{fig.ttc}(a).}\\vskip -0.1in\n\\label{fig.truss-index\n\\end{figure}\n\n\n\n\nBased on the TCP-index, an efficient query processing algorithm is developed for CTC search. Assume that we want to query 5-truss communities containing a query vertex $q$ in $G$ in Fig. \\ref{fig.ttc}(a), we first visit an incident edge on $q$, say $(q, x_1)$, where $\\tau((q,x_1))=5$. From TCP-index ${\\cal T}_q$ in Fig. \\ref{fig.truss-index}, we retrieve the vertex set $\\{x_1,x_2,x_3,x_4\\}$ belong to the same 5-truss community. Since ${\\cal T}_q$ is a spanning forest, which does not keep all the edges between the vertices, the query processing algorithm then performs the reverse operations on the TCP-index for each vertex $x_1, x_2, x_3,x_4$ and gets the complete 5-truss community.\n\nRemarkably, the TCP-index supports the $k$-truss community query in the optimal time, which accesses each edge in the answer community exactly twice \\cite{k-truss2014}.\nMeanwhile, the TCP-index can be constructed in $O($ $\\sum_{(u,v)\\in E}$ $\\min\\{deg_G(u), deg_G(v)\\})$ time and stored in $O(m)$ space.\n\n\n\n\\noindent\\textbf{$\\bullet$ EquiTruss-index based search algorithm \\cite{Akbas:VLDB:2017}.} To further improve efficiency, Akbas and Zhao \\cite{Akbas:VLDB:2017} proposed a novel indexing technique of $k$-truss equivalence, to represent the triangle connectivity and $k$-truss cohesiveness in the triangle-connected truss communities.\n\n\n\nWe introduce the definition of $k$-truss equivalence as follows. Given two edges $e_1$, $e_2 \\in E$, $e_1$ and $e_2$ are $k$-truss equivalence, if and only if (1) $\\tau(e_1)=\\tau(e_2)=k$, and (2) $e_1$ and $e_2$ are triangle-connected via a series of triangles in a $k$-truss.\n\nThe index of EquiTruss, a summarized graph $\\mathcal{G}=(\\mathcal{V}, \\mathcal{E})$, is constructed based on $k$-truss equivalence. According to $k$-truss equivalence, all edges of a given graph $G$ are partitioned into a series of mutually exclusive equivalence classes. Each class represents a TTC. A super-node $E_i\\in \\mathcal{V}$ represents a distinct equivalence class $C_i$ where $e\\in G$, and a super-edge $(E_i, E_j)\\in \\mathcal{E}$ , where $E_i, E_j\\in \\mathcal{V}$, indicates that the two equivalence classes are triangle-connected; that is, there exists two edges $e_1\\in E_i$ and $e_2\\in E_j$, s.t., $e_1$ and $e_2$ are $k$-truss triangle adjacent. Note that EquiTruss is a community-preserving graph summary, where all triangle-connected $k$-truss communities are comprehensively recorded in the super-nodes, and the triangle connectivity across different communities is exactly encoded in super-edges. In this way, EquiTruss keeps records of all the information critical to community search. Moreover, each edge $e$ is recorded in exactly one super-node, which represents its $k$-truss equivalence class, $C_e$. Compared with TCP-Index, which may redundantly maintain an edge in multiple maximum spanning forests, EquiTruss is significantly more succinct and space-efficient \\cite{Akbas:VLDB:2017}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{.\/figures\/EquiTruss.pdf}\n\\caption{EquiTruss index for graph $G$ in Fig. \\ref{fig.ttc}(a).}\n\\label{fig.equitruss}\n\\end{figure}\n\nFor example, Fig. \\ref{fig.equitruss} shows an EquiTruss index for graph $G$ in Fig. \\ref{fig.ttc}(a). It has 5 super-nodes representing the $k$-truss equivalence classes for edges in $G$, as tabulated in Fig. \\ref{fig.equitruss}. The super-node $E_2$ represents a 5-truss community with 10 edges: all these 10 edges are triangle connected, and belong to the 5-truss. In addition, there exist 5 super-edges in EquiTruss, which represents the triangle connectivity between super-nodes (triangle-connected $k$-truss communities).\n\n\nThe EquiTruss-index based community search algorithm is described as follows. Finding triangle-connected communities containing vertex $q$ can be carried out directly on EquiTruss, without the access to graph $G$. First, the algorithm finds all super-nodes containing $q$. A hash structure can help quick identification of such super-nodes. Next, starting from these super-nodes, we can traverse $\\mathcal{G}$ in a BFS manner. For each unvisited neighboring super-nodes $E^*$ with $\\tau(E^*)\\geq k$, the edges within $E^*$ will be included into the $k$-truss community. The algorithm outputs all the discovered communities containing $q$. Consider the graph $G$ in Fig. \\ref{fig.ttc}(a), $k=5$ and query vertex $q$. Based on the EquiTruss index Fig. \\ref{fig.ttc}(a), we first find two super-nodes $E_2$ and $E_4$ containing $q$ with trussness no less than $5$. Super-nodes $E_2$ and $E_4$ are disconnected via any super-edges. Then, $E_2$ and $E_4$ can be respectively output as two communities. Compared to TCP-index, EquiTruss-index based query processing only needs to access each edge exactly once, which is more efficient \\cite{Akbas:VLDB:2017}.\n\n\n\n\n\\subsection{Closest Truss Community Search}\nIn this section, we introduce a new truss-based community model for multiple query vertices.\nAlthough the triangle-connected \\truss community model works well to find all overlapping communities containing a single query vertex $q$, it may fail to discover any community for multiple query vertices, due to the strict requirement of triangle connectivity constraint. For example, for the graph $G$ in Fig. \\ref{fig.ctc-community}(a) and query vertices $Q = \\{v_4, q_3, p_1\\} $, the above \\truss community model cannot find a qualified community for any $k$, since the edges $(v_4, q_3)$ and $(q_3, p_1)$ are not triangle connected in any \\truss. To address this limitation, Huang et al. \\cite{huang2015approximate} studied the problem of closest truss community (CTC) search for multiple query vertices as follows.\n\n\n\\begin{figure}[t]\n\\small\n\\vskip -0.1in\n\\centering\n\\includegraphics[width=1.0\\linewidth]{.\/figures\/diam_example.pdf}\n\\vskip -0.1in\n\\caption{Closest truss community example.}\n\\label{fig.ctc-community}\n\\end{figure}\n\n\n\n\\begin{problem}\n\\label{prob:ctc}\n[CTC search] Given a graph $G$ and a set of query vertices $Q$, return a subgraph $H\\subseteq G$\nas a closest truss community (CTC), satisfying the following two properties:\n\\begin{enumerate}\\label{def.ctc}\n \\item Connected \\truss. $H$ is containing $Q$ and a connected $k$-truss with the largest $k$, i.e., $Q\\subseteq $ $H$ and $\\forall e\\in E(H)$, $sup(e, H) \\geq k-2$;\n\n \n \\item Smallest Diameter. $H$ is a subgraph of smallest diameter satisfying Property 1.\n \n \n\n\\end{enumerate}\n\\end{problem}\n\nProperty 1 requires that the closest community contains the query vertices $Q$ which are densely connected. In addition, to ensure every vertex included in the community is close to query vertices and other vertices in the community, Property 2 uses graph diameter to measure the closeness of all vertices in the community. Moreover, the CTC model can avoid the free rider effect issue, that is, vertices far away from query vertices and irrelevant to them are included in the detected community \\cite{huang2015approximate}.\n\nConsider the graph $G$ in Fig. \\ref{fig.ctc-community}(a), and $Q=\\{q_1, q_2, q_3\\}$; the subgraph in the region shaded gray is a 4-truss containing $Q$ with the largest trussness, and has a diameter of 4. Fig. \\ref{fig.ctc-community}(b) shows another 4-truss containing $Q$ but not $p_1, p_2, p_3$, and its diameter is 3. It can be verified that this is indeed the CTC, which is the 4-truss containing the query vertices $Q$ with the smallest diameter.\n\n\n\n\n\n\nThe problem of CTC search is very challenging. A connected $k$-truss with the largest $k$ containing query vertices can be found in polynomial time. However, finding such a $k$-truss with the minimum diameter is NP-hard \\cite{huang2015approximate}. Moreover, it is even hard to approximate the \\ctcp within a factor better than 2. Here, the approximation is with regard to the minimum diameter.\n\n\nTo find the closest truss community, a simple but effective greedy algorithm is proposed in \\cite{huang2015approximate}. The method uses a greedy strategy for finding a CTC that delivers a 2-approximation to the optimal solution, thus essentially matching the lower bound. Here is an overview of this algorithm. First, given a graph $G$ and query vertices $Q$, we find a maximal connected \\ktruss, denoted $G_0$, containing $Q$ and having the largest trussness. As $G_0$ may have a large diameter, we iteratively remove vertices far away from the query vertices, while maintaining the trussness of the remainder subgraph at $k$. Actually, this algorithm can find a connected $k$-truss with the largest $k$ containing query vertices, which achieves the smallest query distance in optimal. According to the inequality of query distance and graph diameter, this answer is a 2-approximation to CTC \\cite{huang2015approximate}.\n\nIn order to improve the efficiency of CTC search, Huang et. al proposed two new techniques of bulk deletion and local exploration. One of them is based on bulk deletion of vertices far away from query vertices. This speeds up the pruning process, by deleting at least $k$ vertices in batch, to achieve quick termination while sacrificing some approximation ratio. Second, they also propose a heuristic strategy of local exploration to quickly find the closest truss community in the local neighborhood of query vertices. The key idea is as follows. It first forms a Steiner tree to connect all query vertices, and then expand the Steiner tree to a $k$-truss with the largest $k$ by involving the local neighborhood of the query vertices. Finally, to reduce the diameter, it iteratively removes the furthest vertices from this $k$-truss using the bulk deletion.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{$K$-Truss-Based Community Search}\n\\label{sec:ktruss}\n\nIn this section, we review CS works that use the $k$-truss as structure cohesiveness metrics, including triangle-connected truss community \\cite{k-truss2014,Akbas:VLDB:2017}, closest truss community \\cite{huang2015approximate}, attribute-driven truss community \\cite{Huang:2017:ATC}, and weighted truss community \\cite{Zheng:IS:2017}. In the following, we will introduce the community models, and compare their algorithms and applications.\n\n\n\n\\input{ktrussSimple}\n\\input{ktrussKeyword}\n\\input{ktrussInfluence}\n\\input{ktrussDiscuss}\n\n\n\n\n\n\\subsection{Local Modularity-Based Community Search}\n\\label{sec:localModu}\n\nGenerally, studies of local modularity-based CS follow Problem~\\ref{prob:kcoreGlobal} with a local modularity-based goodness function $f$. Two typical such functions are as follows.\n\n\\noindent$\\bullet$ \\textbf{Boundary-based local modularity \\cite{clauset2005finding}.}\nAssume we have a simple undirected graph $G$ and three sets of vertices, i.e., $\\mathcal C$, $\\mathcal U$, $\\mathcal B \\in G$. The known set $\\mathcal C$ contains vertices in the known proportion of the community;\nthe unknown set $\\mathcal U$ is a set of vertices that are adjacent to vertices in $\\mathcal C$;\nand the boundary set $\\mathcal B$ is a subset of $\\mathcal C$, which contains vertices having neighbors in $\\mathcal U$.\n\nBy considering all the edges linked to sets $\\mathcal B$ and $\\mathcal C$, Clauset et al. \\cite{clauset2005finding} defined the local modularity of $C$ as $f(\\mathcal C)$=$I\/T$, where $I$ is the number of edges with no end vertex in $\\mathcal U$, and $T$ is the number of edges with at least one end vertex in $\\mathcal B$. Intuitively, a good community has a sharp boundary, which means that there are few connections from its boundary set $\\mathcal B$ to the unknown set $\\mathcal U$, resulting in a higher value of $f(\\mathcal C)$.\n\nTo uncover a community, Clauset et al. developed an algorithm that works in vertex-at-a-time manner. Let $q$ be a source (seed) vertex. Initially, it lets $\\mathcal C$=$\\{q\\}$ and puts $q$'s neighbors into set $\\mathcal U$. At each step, it adds to $\\mathcal C$ the neighboring vertex that results in the largest increase of the local modularity. This process continues until it has agglomerated either a given number of vertices $k$, or it has discovered the entire enclosing component, whichever happens first. As a result, its time complexity is $O(k^2d)$, where $d$ is the mean degree and $k$ is the number of vertices to be explored.\n\n\\noindent$\\bullet$ \\textbf{Subgraph degree-based local modularity \\cite{Luo2006ELC}.}\nGiven a subgraph $\\mathcal C$ of a graph $G$, Luo et al \\cite{Luo2006ELC} defined its indegree, $ind(\\mathcal)$, as the number of edges within $\\mathcal C$, and its out-degree, $outd(\\mathcal C)$, as the number of edges that connect $\\mathcal C$ to the remaining part of $G$. Then, they defined the subgraph modularity of $S$ as $f(\\mathcal C)$=$ind(\\mathcal)\/outd(\\mathcal)$. Clearly, its value will increase if $\\mathcal C$ has more internal edges and fewer external edges.\n\nTo find a community, Luo et al. proposed an algorithm consisting of an addition step and a deletion step. Initially, $\\mathcal C$ contains a seed vertex $q$ and its neighbors are in a set $\\mathcal N$. In the addition step, it iteratively adds vertices from $\\mathcal N$ to $\\mathcal C$ that result in the greatest increase of $f(\\mathcal C)$, until a certain number of neighbors have been in the subgraph. In the deletion step, it iteratively removes vertices in $\\mathcal C$ that result in the increase of $f(\\mathcal C)$ but not separating $\\mathcal C$. The addition and deletion steps will be repeated until no vertex is added to $\\mathcal C$. Note that there is no guarantee whether $q$ will be in the returned community as it may be removed during the deletion step. It has the same time complexity as the algorithm for the boundary-based local modularity.\n\n\\section*{Acknowledgments}\nWe would like to thank Jiafeng Hu and Kai Wang for their helpful discussions, Dan Yin for the proof-reading, and Jinbin Huang for conducting experimental comparisons.\nXin Huang is supported by the NSFC Project No. 61702435, and Hong Kong General Research Fund (GRF) Project No. HKBU 12200917.\nLu Qin is supported by DP160101513.\nYing Zhang is supported by FT170100128 and DP180103096.\nWenjie Zhang is supported by DP180103096.\nReynold Cheng is supported by the Research Grants Council of Hong Kong (RGC Projects HKU 17229116 and 17205115) and HKU (Projects 102009508 and 104004129).\nXuemin Lin is supported by 2019DH0ZX01, 2018YFB1003504, NSFC61232006, DP180103096 and DP170101628.\n\\blfootnote{*For lack of space, we use abbreviations for the names of major conferences and\njournals in database and data mining areas (e.g., we use ``PVLDB\" to mean ``Proceedings of the VLDB Endowment\").\nFor other venues, we use full names.}\n\n\\bibliographystyle{abbrv}\n\n\\subsubsection{Maximum $k$-Plex Community Query (MCKPQ)}\n\\label{sec:mckpq}\n\nIn \\cite{wang2017query}, Wang et al. proposed and studied the maximum $k$-plex community query (MCKPQ):\n\n\\begin{problem}[MCKPQ]\n\\label{prob:mckpq}\nGiven a simple undirected graph $G(V,E)$, a set of query vertices $Q\\in V$, an integer $k$,\nreturn a subgraph $G_Q(V_Q,E_Q)\\subseteq G(V,E)$ such that the following properties hold:\n\\begin{enumerate}\n \\item \\textbf{Connectivity}. $G_Q$ is connected and contains $Q$;\n \\item \\textbf{Structure cohesiveness}. $G_Q$ is a $k$-plex;\n \\item \\textbf{Maximal structure}. There exists no other $G_Q'\\subseteq G$ satisfying the above properties and $G_Q\\subset G_Q'$.\n\\end{enumerate}\n\\end{problem}\n\nA good property of MCKPQ is that the communities returned by an MCKPQ can avoid the free rider effect, which has been introduced and discussed in Section~\\ref{sec:ktruss}. Nevertheless, the MCKPQ problem is very computationally challenging, because it is NP-complete, which can be proved by a reduction from the $k$-plex problem \\cite{balasundaram2011clique}. Moreover, it is hard to approximate for MCKPQ problem in polynomial time within a factor $n^{1-\\epsilon}$.\n\nA basic solution to the MCKPQ problem is to use the generate-and-verify method, which enumerates all the $k$-plexes in the whole search space, and then returns the one with the largest size. Obviously, this method is too expensive and impractical for large graphs. To alleviate this issue, Wang et al. developed a more advanced method based on the branch-and-bound paradigm with some effective pruning criteria and a heuristic method which performs fast but has no theoretical guarantee \\cite{wang2017query}. We skip the details due to space limitation.\n\n\\subsubsection{Neighbors Expansion-Based Community Search}\n\\label{sec:neighbors}\n\nIn \\cite{Mehler2009}, Mehler et al. presented a neighbor expansion method to discover the community from representative seeds. Specifically, given a graph $G(V,E)$ and a set $S$ of seed vertices, it repeatedly identifies the optimal ``next\" vertex $v$, which is not in the community $C$ (initially $C$=$S$) but linked with vertices of $C$, based in some manner on the number or strength of $v$'s neighbors who had previously been identified as community members.\nDetails of vertex selection criteria and stopping rules of the expansion process are introduced as follows.\n\n\\noindent$\\bullet$ \\textbf{Selection criteria.} Mehler et al. proposed to assign a score to each vertex in the graph and select the highest-scoring outside vertex to join the community. The score assignment criteria are as follows:\n\\begin{itemize}\n \\item {\\it neighbor count:} the number of $v$'s neighbors in $C$;\n \\item {\\it juxtaposition count:} consider the weights of edges when counting the number of $v$'s neighbors in $C$;\n \\item {\\it neighbor ratio:} normalize vertices' degrees and count the degree-normalized neighbors in $C$;\n \\item {\\it juxtaposition ratio:} consider the weights of edges when computing the neighbor ratio;\n \\item {\\it binomial probability:} compute the binomial probability that $v$ is in $C$, given its neighbor count.\n\\end{itemize}\n\n\\noindent$\\bullet$ \\textbf{Stopping rules.} The authors proposed to reserve some fraction of seed vertices as validation members, and then monitor the frequency with which these validation members are incorporated into the community, during the expansion process. In the first phase, when community members are identified with high precision, we expect to add a new validation member with frequency equal to the fraction of community comprised by the validation set. After leaving the natural boundaries of the neighborhood, we expect to rediscover validation members according to their frequency in the entire graph.\nAs a result, we can find the stopping vertex as the one that best splits the validation interval (i.e., the difference between the discovery times of the $i$th and ($i-1$)-st validation members) into two groups.\n\n\\subsection{Discussions}\n\\label{sec:othersDiscuss}\n\nIn this section, we review CS studies that do not rely on metrics introduced in Section \\ref{sec:pre}, which are often referred as local community detection. These studies mainly focus on simple undirected graphs, and uncover the communities by seed expansion using link-based metrics, such as modularity, density, pagerank, etc. Unlike CS studies introduced before, these works often rely on good seed selection algorithms \\cite{moradi2014local} and assume that there are some ground truth communities. In other words, they might not aim to search communities in an online manner over big graphs, based on a query request. As a result, some of them may cost high running time for searching communities. Consequently, an interesting research direction is to develop index-based solutions for supporting efficient online CS queries using these metrics. Moreover, it would be interesting to study how to apply them for CS on attributed graphs.\n\n\\section{Other Metrics-Based Community Search}\n\\label{sec:other}\n\nIn this section, we review a particular kind of community search, namely local community detection, which takes an input vertex as a seed and expands the community from the seed according to a specific goodness function. The representative goodness functions are local modularity \\cite{clauset2005finding,Luo2006ELC}, query biased density \\cite{Wu:VLDB:2015}, personalized pagerank \\cite{Kloumann2014}, and neighbor expansion \\cite{Mehler2009}.\n\n\\input{localModu}\n\\input{bias}\n\\input{pagerank}\n\\input{neighbors}\n\\input{othersDiscuss}\n\n\\subsection{Personalized PageRank-Based Community Search}\n\\label{sec:pagerank}\n\nIn \\cite{Kloumann2014}, Kloumann et al. studied the use of personalized PageRank (PPR) model for identifying the community of a set of seed vertices $Q$. We first introduce the PageRank model: suppose there are an infinite number of surfers walking on a graph. If at a certain timestamp a surfer is staying at vertex $i$, at the next timestamp she goes to a random neighbor vertex $j$. As time goes on, the expected percentage of surfers at each vertex $i$ converges (under certain conditions) to a limit $r(i)$, called PageRank score of vertex $i$. Since $r(i)$ is independent of the distribution of starting vertices, it reflects the global importance of the vertex $i$.\n\nNotice that $r(i)$ is computed with no preference for any particular vertices. However, in reality, for a particular user, some vertices, denoted by a set $Q$, may be more interesting than others, and they could be considered as the {\\it preferred vertices}. To incorporate preferences of $Q$ into the model above, we can make a modification: at each step, a surfer jumps back to a vertex in $Q$ with probability $c$, and with probability ($1-c$) continues forth along a neighbor. The limit distribution of surfers in this model would favor vertices in $Q$ and vertices which are close to $Q$. The modified model is also called PPR model. Clearly, if we let $Q$ be a set of query vertices, the vertices whose limit probabilities are highest can be considered as $Q$'s community members.\n\nNow we formally introduce the PPR model. Consider a graph $G$ and let $deg_G(i)$ denote the degree of vertex $i$ and $\\mathbf A$ be the adjacent matrix of $G$, i.e., $A_{i,j}$=$\\frac{1}{deg_G(i)}$ if vertex $i$ is linked to vertex $j$, where $deg_G(i)$ is the degree of vertex $i$. The {\\it preference vector} $\\mathbf u$ is defined over the seed vertices such that $|\\mathbf u|$=1 and $u(i)$=$\\frac{1}{|Q|}$ if the $i$-th vertex is in $Q$. Then, the PPR equation is\n${\\mathbf v}$= $(1-c){\\mathbf {Av}}+c\\mathbf u$,\nwhere $c\\in(0, 1]$ is the decay factor and a typical value of $c$ is 0.10 \\cite{Kloumann2014}. The solution $\\mathbf v$, called PPR vector, is a steady-state distribution of surfers.\n\n\\begin{problem}\n\\label{prob:PPR}\nGiven a graph $G(V,E)$, a set of query vertices $Q\\subseteq V$, and an integer $k$, return a set $C$ of vertices, such that\n\\begin{enumerate}\n \\item $Q\\subseteq C$;\n \\item $C$ contains $k$ vertices, whose corresponding values in the PPR vector w.r.t $Q$ are the highest;\n\\end{enumerate}\n\\end{problem}\n\nIn the literature \\cite{andersen2006communities,Kloumann2014}, many efficient PPR algorithms have been developed, and thus can be applied to CS. We skip the details due to space limitation.\n\n\\section{Preliminaries}\n\\label{sec:pre}\n\nIn this section, we first formally introduce the commonly-used community cohesiveness metrics, and then compare their cohesiveness and computational efficiency.\n\n\\subsection{Cohesiveness Metrics}\n\\label{sec:definitions}\n\nFor ease of exposition, we consider a simple undirected graph $G(V,E)$, with vertex set $V$ and edge set $E$. Let $n$ and $m$ be the corresponding sizes of $V$ and $E$. The degree of a vertex $v$ of $G$ is denoted by $deg_G(v)$.\n\n\\noindent\\underline{$\\bullet$ \\textbf{$k$-core.}} We introduce its formal definition as follows.\n\n\\begin{definition}[$k$-core~\\cite{md1983,kcore2003}]\n\\label{def:kcore}\nGiven an integer $k$ ($k\\geq 0$), the $k$-core of $G$,\ndenoted by $H_{k}$, is the largest subgraph of $G$, such that $\\forall v \\in H_k$, $deg_{H_k}(v) \\geq k$.\n\\end{definition}\n\nWe say that $H_k$ has an order of $k$. Notice that $H_k$ may not be a connected graph~\\cite{kcore2003}. Observe that $k$-$core$s are ``nested''~\\cite{kcore2003}: given two positive integers $i$ and $j$, if $i$24 hours \\\\\n \\hline\n Livejournal &\n 85s &\n 854s &\n 1726s &\n $>$24 hours \\\\\n \\hline\n Wise &\n 553s &\n 5764s &\n 32221s &\n $>$24 hours \\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\nBased on the comparison analysis above, we would like to make some suggestions:\n(1) For small or moderate-size graphs, $k$-clique and $k$-truss not only achieve higher cohesiveness but also reasonable efficiency.\n(2) For large graphs, $k$-core and $k$-ECC should be better choices since they can be computed more efficiently.\n(3) For graphs with higher clustering coefficient which can be decomposed into more triangles, $k$-truss is preferable.\n(4) For some special graphs (e.g., bipartite graphs), there may not exist any triangles and thus the $k$-truss model may not work.\n\n\\subsubsection{Radius-Bounded $k$-core Search}\n\\label{sec:rbkcore}\n\nProblem~\\ref{prob:rbkcore} defines the radius-bounded $k$-core search.\n\n\\begin{problem}[RB-$k$-core search]\n\\label{prob:rbkcore}\nGiven a geo-social network $G(V,E)$, a positive integer $k$, a radius $r$ and a vertex $q\\in V$, return all the subgraphs $G_q\\subseteq G$, and the following properties hold:\n\\begin{enumerate}\n \\item \\textbf{Connectivity}. $G_q$ is connected and contains $q$;\n \\item \\textbf{Structure cohesiveness}. $\\forall v\\in G_q$, $deg_{G_q}(v)\\geq k$;\n \\item \\textbf{Spatial cohesiveness}. The MCC of vertices in $G_q$ has a radius $r' \\leq r$;\n \\item \\textbf{Maximality constraint}. There exists no other subgraph $G_q'$ satisfying properties above and $G_q\\subset G_q'$.\n\\end{enumerate}\n\\end{problem}\n\nSimilar to SAC search, it adopts the MCC, but imposes a constraint on its radius. To solve Problem \\ref{prob:rbkcore}, Wang et al. proposed three algorithms. The first one, denoted by {\\tt TriV}, is a triple-vertex-based algorithm, which is also based on the observation that a spatial circle can be determined by three points on its boundary~\\cite{mcc-ts}. It proposes to generate all the candidate circles containing $q$ at first and then compute the maximum $k$-core for the subgraphs contained in the candidate circles with radius $r' \\leq r$. The time complexity of {\\tt TriV} is $O(mn^3)$, since there are $O(n^3)$ candidate circles in the worst case and each circle needs $O(m)$ time to verify.\n\nTo reduce the number of candidate circles, a binary-vertex-based algorithm {\\tt BinV} is proposed. In {\\tt BinV}, only the circles with radius $r'$=$r$ are generated and for each candidate circle, its arc passes a pair of vertices in $G$. In this manner, for each pair of vertices, at most two circles are generated. As a result, it reduces the number of candidate circles from $O(n^3)$ to $O(n^2)$.\n\nTo further improve the efficiency, a rotating-circle-based algorithm {\\tt RotC} is proposed to reuse the intermediate computation results in the process of finding RB-$k$-cores. Fixing each vertex $v \\in V$ as a pole, {\\tt RotC} generates the candidate circles in a rotating way so that the computation cost can be shared among the adjacent circles. In addition, the authors also proposed several pruning techniques to early terminate the processing of invalid candidate circles.\n\n\n\\section{Related Work}\n\\label{sec:related}\n\nIn this section, we review related studies, including community detection, cohesive subgraph discovery, graph keyword search, and graph pattern matching.\n\n\\subsection{Community Detection}\n\\label{sec:CD}\n\nBelow, we review representative CD studies on undirected graphs, directed graphs, and attributed graphs.\n\n\\subsubsection{Undirected Graphs}\n\\label{sec:CDSimple}\n\nA large number of studies aim to detect communities from simple graphs, and we can classify these studies based on the techniques they use. Some representative classes are as follows, to name a few:\n\\begin{enumerate}\n \\item community quality optimization-based methods (e.g., modularity \\cite{newman2004fast});\n \\item clustering methods (e.g., $k$-means \\cite{tang2009scalable}, spectral clustering \\cite{von2007tutorial});\n \\item graph partitioning methods (e.g., Metis \\cite{karypis1995metis});\n \\item embedding-based methods (e.g., DeepWalk \\cite{perozzi2014deepwalk}, \\cite{li2018community});\n \\item random walk-based methods (e.g., \\cite{pons2005computing});\n \\item label propagation-based methods (e.g., \\cite{gregory2010finding});\n \\item information diffusion-based methods (e.g., \\cite{hajibagheri2012community});\n \\item statistic inference-based models (e.g., \\cite{hastings2006community});\n \\item deep learning-based methods (e.g., \\cite{Yang:2016});\n \\item centrality-based methods (e.g., \\cite{community-phy2004});\n \\item locality sensitive hashing-based methods (e.g., \\cite{macropol2010scalable});\n \\item physics-based methods (e.g., Potts low \\cite{wu1982potts});\n \\item local metric-based methods (e.g., $k$-plex \\cite{conte2018d2k});\n \\item multi-commodity flow-based methods (e.g., \\cite{leighton1988approximate});\n \\item hybrid-based methods (e.g., \\cite{henderson2010hcdf}).\n\\end{enumerate}\n\nFor a detailed survey of CD, please refer to the following survey and empirical evaluation papers:\n\\cite{CD:Survey:2009,Yang2010,community-phy2010,papadopoulos2012community,danon2005comparing,gulbahce2008art,CD:Survey:2011,CD:Survey:2011a,CD:Survey:2013,CD:Survey:2017,CD:Survey:2014-overlap,CD:Survey:2015-multilayer,CD:Survey:2018-dynamic,CD:Survey:2010-compare,CD:Survey:2014-evaluate,CD:Survey:2015-truth}.\nAlthough these CD solutions are able to discover communities from networks, they may not well satisfy the desirable factors of CS on big graphs as we discuss in Section \\ref{sec:intro}, because most of them often use a global predefined criterion for generating communities and cannot find communities in an online manner.\n\n\n\\subsubsection{Directed Graphs}\n\\label{sec:CDDirected}\n\nIn recent years, a number of studies have investigated CD on directed graphs. Here are some representative studies, to name a few.\nIn~\\cite{prl2008}, Leicht et al. extended the concept of modularity maximization~\\cite{newman2004fast}, which was originally designed for undirected graphs, for detecting community structure in directed networks that makes explicit use of information contained in edge directions. In \\cite{flake2000}, Flake et al. identified communities from websites network, which can be considered as directed graphs.\nIn~\\cite{CDBenchmark}, Lancichinetti et al. introduced new benchmark graphs to test CD methods on directed\nnetworks.\nIn~\\cite{pre2010}, Kim et al. also proposed a new modularity metric for CD on directed networks.\nIn~\\cite{sdm2010}, Yang et al. developed a new stochastic block model for CD on directed networks.\nIn~\\cite{yang2014detecting}, Yang et al. presented algorithms for detecting communities from both directed and undirected networks.\nNing et al. \\cite{ning2016local} studied local community extraction in directed networks.\nA recent survey can be found in~\\cite{CDSurvey}.\n\n\\subsubsection{Keyword-Based Attributed Graphs}\n\\label{sec:CDKeyword}\n\nTo identify communities from keyword-based attributed graphs, recent works~\\cite{attr-vldb2009,SDM2013,WSDM2013,cheng2012clustering,attr-www2013,huang2016attributed} often use clustering techniques. Zhou et al. \\cite{attr-vldb2009} computed vertices' pairwise similarities using both links and keywords, and then clustered the graph.\nSubbian et al. \\cite{SDM2013} explored noisy labeled information of graph vertices for finding communities.\nQi et al. \\cite{WSDM2013} dynamically maintained communities of moving objects using their trajectories.\nRuan et al.~\\cite{attr-www2013} developed a method {\\tt CODICIL}, which augments the original graph by creating new edges based on content similarity, and then performs clustering on the new graph.\n\nAnother common approach is based on topic models. In~\\cite{attr-topic-kdd2008,attr-topic-icml2009}, the {\\tt Link-PLSA-LDA} and {\\tt Topic-Link LDA} models jointly model vertices' content and links based on the {\\tt LDA} model. In~\\cite{attr-topic-sigmod2012}, the attributed graph is clustered based on probabilistic inference. In~\\cite{attr-topic-www2012}, the topics, interaction types, and the social connections are considered for discovering communities. {\\tt CESNA}~\\cite{yang2013community} detects overlapping communities by assuming communities ``generate'' both the link and content. A discriminative approach~\\cite{attr-kdd2009} has also been considered for community detection.\nHowever, computing pairwise similarity among vertices is very costly, and thus they are questionable for performing online CS queries.\n\n\n\n\\subsubsection{Location-Based Attributed Graphs}\n\\label{sec:CDLocation}\n\nThe problem of CD on location-based attributed graphs (or geo-social networks) \\cite{barthelemy2011} has been extensively studied \\cite{girvan2002,guo2008,pnas2011,KDD2013,IJGIS2015}. In \\cite{girvan2002}, Girvan et al. introduced the geo-community, which is a graph of intensely connected vertices being loosely connected with others, but it is more compact in space.\nGuo et al.~\\cite{guo2008} proposed the average linkage (ALK) measure for clustering objects in spatially constrained graphs.\nIn~\\cite{pnas2011}, Expert et al. uncovered communities from spatial graphs based on modularity maximization.\nIn~\\cite{KDD2013}, Shakarian et al. used a variant of Newman-Girvan modularity to mine the geographically dispersed communities.\nIn~\\cite{IJGIS2015}, Chen et al. proposed a method using modularity maximization for detecting communities from geo-social networks.\n\n\\subsubsection{Temporal Graphs}\n\\label{sec:CDTemporal}\n\nMany recent studies aim to detect communities from temporal graphs. In \\cite{zhou2007discovering}, Zhou et al. studied CD over a temporal heterogeneous social network consisting of authors, document content, and the venues.\nIn \\cite{liu2014persistent}, Liu et al. studied persistent community detection for identifying communities that exhibit persistent behavior over time.\nIn \\cite{angadi2015overlapping}, Angadi et al. detected communities from dynamic networks where data arrives as a\nstream to find the overlapping vertices in communities.\nIn \\cite{bazzi2016community}, Bazzi et al. investigated the detection of communities in temporal multi-layer networks.\nIn \\cite{ditursi2017local}, DiTursi et al. proposed a filter-and-verify framework for community detection in dynamic networks.\nIn \\cite{kuncheva2017multi}, Kuncheva et al. presented a method by using spectral graph wavelets to detect communities in temporal graphs.\nFor more related studies, please refer to survey papers \\cite{CD:Survey:2018-dynamic,tamimi2015literature}.\n\n\\subsection{Cohesive SubGraph Discovery}\n\\label{sec:cohesive}\n\nIn this section, we review studies on cohesive subgraph discovery. Notice that CD is one kind of cohesive subgraph discovery, but the latter one is more general.\n\n\\subsubsection{Simple Graphs}\n\\label{sec:cohSimple}\n\nFor simple graphs, typical cohesive subgraph models are\n$k$-core \\cite{md1983,kcore2003},\n$k$-truss \\cite{saito2008extracting,cohen2008trusses,zhang2012extracting},\n$k$-clique \\cite{kclique,article05clique},\nand $k$-ECC \\cite{gibbons1985algorithmic,hu2016querying}, as discussed in Section \\ref{sec:pre}.\nTo compute these subgraphs, there are many efficient in-memory algorithms (e.g., $k$-core \\cite{kcore2003}, $k$-truss \\cite{wang2012truss}, $k$-clique \\cite{mauro2018}, and $k$-ECC \\cite{zhou2012finding,Chang:SIGMOD:2013,akiba2013linear}).\nFor graphs that are too large to be kept in memory, there are also some disk-based and parallel algorithms.\nFor example, in \\cite{cheng2011efficient,wen2018efficient}, \\cite{wang2012truss,khaouid2015k}, and \\cite{cheng2012fast}, disk-based algorithms for computing $k$-core, $k$-truss, and $k$-clique are developed, respectively; in \\cite{montresor2013distributed} and \\cite{chen2014distributed}, parallel algorithms for computing $k$-core and $k$-truss are proposed, respectively. In addition, to maintain $k$-core and $k$-truss for dynamic graphs, some efficient algorithms are developed in \\cite{li2014efficient,sariyuce2016incremental,zhang2017fast} and \\cite{zhou2014efficient}, respectively.\n\nBesides, there are many other cohesive subgraph models and the representatives are as follows.\nIn \\cite{seidman1978graph}, Seidman proposed the $k$-plex model (which is introduced in Section~\\ref{sec:kclique}).\nIn \\cite{matsuda1999classifying}, Matsuda et al. introduced the concept of quasi-clique model.\nIn \\cite{zhang2018discovering}, Zhang et al. proposed the ($k$, $s$)-core, which considers both user engagement and tie strength.\nIn \\cite{sariyuce2016fast}, the authors proposed the concept of nucleus, which is a generalization of $k$-core and $k$-truss.\nIn \\cite{zhao2012large}, Zhao et al. introduced the mutual-friend subgraph.\nIn \\cite{wang2010triangulation}, Wang et al. proposed the DN-Graphs by considering vertices' common neighbors.\nIn \\cite{Chang:SIGMOD:2013}, Chang et al. studied the problem of enumerating $k$-ECCs in a graph for a given $k$.\nIn \\cite{zhu2018diversified}, Zhu et al. introduced the notion of coherent cores on multi-layer graphs.\nIn addition, Goldberg et al. \\cite{goldberg1984finding} and Fang et al. \\cite{fang2019efficient} discovered the densest subgraph, Galbrun et al. \\cite{Galbrun:2016} studied the top-$k$ densest subgraphs, Tsourakais et al. \\cite{tsourakakis2013denser} computed the quasi-clique-based dense subgraphs, and Qin et al. \\cite{Qin:2015} studied the problem of finding top-$k$ locally densest subgraphs.\n\n\n\\subsubsection{Attributed Graphs}\n\\label{sec:cohAttributed}\n\nFor attributed graphs, in addition to CD methods, there are also many studies of finding cohesive subgraphs.\nIn \\cite{denian:2012}, Yang et al. studied the socio-spatial group query which finds a group of users that are cohesively linked and close to the rally point in a geo-social network.\nIn \\cite{Zhang:VLDB:2017}, Zhang et al. studied the problem of finding ($k$, $r$)-cores on attributed graph and for a specific ($k$, $r$)-core, each vertex has at least $k$ neighbors, and the attribute similarity of each pair of vertices is at least $r$.\nIn \\cite{Chen:VLDB:2018}, Chen et al. studied the problem of ($k$, $d$)-MCC (maximum co-located community) search on geo-social network, where a ($k$, $d$)-MCC is a connected $k$-truss and for any two vertices, their distance is at most $d$.\nIn addition, Wu et al. \\cite{wu2015finding} studied the problem of finding the densest connected subgraph from the dual network, which can be considered as an attributed graph.\n\n\\subsection{Graph Keyword Search}\n\\label{sec:GKS}\n\nGenerally, graph keyword search \\cite{wang2010survey,keyword-yu-2009,yuan2017keyword,yuan2013efficient} aims to find a tree or a subgraph, which contains a set of query keywords, from a large graph $G$. Earlier studies often output a tree structure.\nIn~\\cite{keyword-icde2002}, Bhalotia et al. developed a backward algorithm for finding Steiner trees.\nIn~\\cite{keyword-icde2007}, Ding et al. proposed a dynamic programming algorithm finding Steiner trees.\nIn~\\cite{golenberg2008keyword}, Golenberg et al. presented a novel algorithm which produces Steiner trees with polynomial delay.\nIn~\\cite{keyword-vldb2005}, Kacholia et al. proposed a bidirectional search algorithm, and He et al.~\\cite{he2007blinks} improved its efficiency by introducing a new index structure.\n\nRecently, some solutions have output subgraphs.\nIn \\cite{keyword-sigmod2008}, Li et al. proposed to find $r$-radius Steiner graphs that contain query keywords.\nQin et al. \\cite{keyword-icde2009} proposed to find multi-centered subgraphs that contain query keywords within a given distance. Kargar et al. \\cite{keyword-vldb2011} studied the $r$-clique which is a set of vertices that cover query keywords and satisfy the distance constraint.\n\nHowever, these works are substantially different from CS queries on keyword-based attributed graphs. First, they do not specify query vertices as required by CS queries. Second, the tree or subgraph produced do not guarantee structure cohesiveness. Third, their solutions do not ensure strong keyword cohesiveness.\n\n\\subsection{Graph Pattern Matching (GPM)}\n\\label{sec:GPM}\nFor simple graphs, the problem of GPM is NP-complete \\cite{cook1971complexity} and it has been studied extensively under different settings:\n(1) in main memory \\cite{ullmann1976algorithm,chiba1985arboricity}. For example, Ullmann \\cite{ullmann1976algorithm} proposed a backtracking algorithms.\n(2) in external memory, Chu et al. \\cite{chu2011triangle} and Hu et al. \\cite{hu2014efficient} studied triangle counting; in \\cite{qiao2017subgraph}, a novel GPM solution based on graph compression is presented.\n(3) in distributed platforms, both DFS-style approaches \\cite{afrati2013enumerating,park2016pte}and BFS-style approaches \\cite{lai2015scalable,lai2016scalable} are developed. The DFS-style approaches avoid intermediate results by using one-round computation, while BFS-style approaches shuffle a large number of intermediate results.\n\nFor attributed graphs, there are also many studies. Tong et al. \\cite{GPM-KDD2007} studied the use of lines, loops and stars for finding the matched subgraphs; Zou et al. \\cite{zou2009distance} developed a novel GPM solution based on distance join;\nFan et al. \\cite{GPM-VLDB2010} studied GPM by using bounded simulation; in \\cite{GPM-PVLDB2015}, GPM has been studied for finding graph association rules; in \\cite{GPM-ICDE2012}, Cheng et al. studied the problem of top-$k$ GPM. Recently, Fang et al. have studied a variant of the GPM problem on spatial databases \\cite{Fang:ICDE:SPM,Fang:ICDE:DEMO}, and it aims to find spatial objects that are matched with a given pattern.\nHowever, GPM is different with CS since (1) it often focuses on small patterns, so it cannot generate large communities; and (2) the subgraphs of GPM solutions often do not guarantee strong structure cohesiveness. Other related topics include subgraph search \\cite{yua,yuan2012efficient}.\n\n\n\n\\subsubsection{Spatial-Aware Community (SAC) Search}\n\\label{sec:sac}\n\nThe MCC and SAC search are defined as follows. Note that the notion of MCC has been widely adopted to describe a set of spatially compact objects \\cite{mcc-ts,mcc-sigmod}.\n\n\\begin{definition}[MCC]\n\\label{def:mcc}\nGiven a set $S$ of vertices with locations, the MCC of $S$ is the spatial circle, which contains all the vertices in $S$ with the smallest radius.\n\\end{definition}\n\n\\begin{problem}[SAC search]\n\\label{prob:sac}\nGiven a geo-social network $G(V,E)$, a positive integer $k$ and a vertex $q\\in V$, return a subgraph $G_q\\subseteq G$, and the following properties hold:\n\\begin{enumerate}\n \\item \\textbf{Connectivity}. $G_q$ is connected and contains $q$;\n \\item \\textbf{Structure cohesiveness}. $\\forall v\\in G_q$, $deg_{G_q}(v)\\geq k$;\n \\item \\textbf{Spatial cohesiveness}. The MCC of vertices in $G_q$ satisfying Properties 1 and 2 has the smallest radius.\n\\end{enumerate}\n\\end{problem}\n\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c c}\n \\begin{minipage}{3.8cm}\n\t\\includegraphics[width=3.8cm]{figures\/SACgraph}\n \\end{minipage}\n &\n \\begin{minipage}{3.8cm}\n\t\\includegraphics[width=3.8cm]{figures\/SACAppInc}\n \\end{minipage}\n \\\\\n (a) a graph\n &\n (b) Illustrating {\\tt AppInc}\n\\end{tabular}\n\\caption{Illustrating SAC search \\cite{Fang:VLDB:2017}.}\n\\label{fig:SACEg}\n\\end{figure}\n\nA subgraph satisfying properties 1 and 2 is a~\\emph{feasible} solution, and the subgraph satisfying all the three properties is the \\emph{optimal} solution (denoted by $\\Psi$). The radius of the MCC containing $\\Psi$ is denoted by $r_{opt}$.\nIn Fig. \\ref{fig:SACEg}(a), the two circles denote the MCCs of $C_1$=$\\{Q, C, D\\}$ and $C_2$=$\\{Q, A, B\\}$. Let $q$=$Q$ and $k$=2. Then, $\\Psi$ contains vertex set $C_1$ with $r_{opt}$=1.5.\n\nThe SAC search problem is challenging. A basic exact approach takes $O(m\\times{n^3})$ time, which relies on an observation that a spatial circle can be determined by three points on its boundary~\\cite{mcc-ts}. This implies, we can enumerate all the three-vertex combinations, and for each combination we find a connected $k$-core in its circle, and finally get $\\Psi$. This approach, however, is impractical for large graphs due to its high complexity.\n\nTo improve efficiency, the authors resorted to approximation algorithms. The first one, called {\\tt AppInc}, returns the feasible solution in a circle $O(q,\\delta)$ which centers at $q$ and has the smallest radius $\\delta$, and it has an approximation ratio of 2. Here, the approximation ratio is defined as the ratio of the radius of MCC returned over $r_{opt}$. In Fig. \\ref{fig:SACEg}(b), let $q$=$Q$ and $k$=2. Then, {\\tt AppInc} returns the subgraph of $\\{A,B,Q\\}$.\n\nThe circle $O(q,\\delta)$ can also be approximated by performing binary search on the radius $\\delta$. As a result, we can get another approximation solution with ratio of (2+$\\epsilon_F$), where $\\epsilon_F\\geq0$ is an input parameter. To achieve an approximation ratio of (1+$\\epsilon_A$) where $0\\textless\\epsilon_A\\textless1$, the authors developed another algorithm, called {\\tt AppAcc}. It first locates the area containing the center of the circle of $\\Psi$, then approximates the center by splitting the area into small grids, and finally finds an approximation solution by using these grids. Overall, these approximation algorithms guarantee that the radius of the MCC of $\\Psi$ has an arbitrary expected approximation ratio. Based on {\\tt AppAcc}, an advanced exact algorithm is developed. An interesting observation is that there is a trade-off between the quality of results and efficiency, i.e., algorithms with lower approximation ratios tend to have higher complexities. In addition, the SACs can be returned in a continuous manner, as shown in \\cite{Fang:TKDE:SAC}.\n\n\\subsubsection{Social Group Query (SGQ)}\n\\label{sec:sgq}\n\nProblem~\\ref{prob:sgq} presents SGQ, which was designed for suggesting attendees in activity planning \\cite{yang2011social}.\n\n\\begin{problem}[SGQ]\n\\label{prob:sgq}\nGiven a simple undirected graph $G(V,E)$, an activity initiator $q\\in V$, three integers $p$, $s$, and $k$,\nreturn a set $F$ of vertices from $G$ such that the following properties hold:\n\\begin{enumerate}\n \\item $|F|$=$p$;\n \\item The length of the minimum distance path between $v$ and $q$, $d_{v,q}$, is at most $s$;\n \\item Each vertex $v\\in F$ is allowed to share no edges with at most $k$ other vertices in $F$;\n \\item The total social distance $\\Sigma_{v\\in F}d_{v,q}$ is minimized.\n\\end{enumerate}\n\\end{problem}\n\nIn Problem~\\ref{prob:sgq}, Property 1 controls the expected number of attendees in the activity;\nProperty 2 specifies a radius constraints which requires each attendee is close to $q$ in the graph $G$;\nProperty 3 requires that each attendee is acquainted with other attendees by following the $k$-plex model;\nProperty 4 ensures that the returned group is the most compact one among all the groups satisfying all the above properties.\n\nThe SGQ problem is computationally challenging because it is NP-hard, which can be proved by a reduction from the $k$-plex problem \\cite{balasundaram2011clique}. To answer SGQ, Yang et al. \\cite{yang2011social} proposed an efficient solution {\\tt SGSelect}. The idea is that we can first extract a subgraph $H\\subseteq G$ by using the radius constraint. Then, starting from $q$, we iteratively explore vertices in $H$ to derive the optimal solution. In each iteration, we can keep track of a set of vertices that satisfy the constraint of $k$, until the set has $p$ vertices. To further speedup this process, some effective pruning criteria have been developed. For example, to choose vertices, we can give high priorities for vertices that may significantly increase the total social distance. Also, during the search process, we can prune vertices which would not lead to eventual answer by considering the acquaintance constraint $p$ and social radius constraint $s$.\n\nIn addition, Yang et al. \\cite{yang2011social} studied another query, called social-temporal group query (STGQ), which generalizes SGQ by considering the available time of each candidate attendee. In specific, it finds a group of vertices satisfying:\n(1) all constraints in an SGQ; and\n(2) all the attendees are available in a time period $[t, t+{\\delta_t}]$, where $t$ is time slot and $\\delta_t$ is query parameter.\nThe STGQ problem is also NP-hard and some efficient solutions are developed. For details, please refer to \\cite{yang2011social}.\n\n\\subsection{Simple Graphs}\n\\label{sec:simpleG}\n\nIn this section, we compare representative CS problems for cohesiveness metrics studied on simple graphs, which are Problem \\ref{prob:kcoreGlobal} for $k$-core, Problem \\ref{problem:TTC} for $k$-truss, Problem \\ref{prob:densest} for $k$-clique, and Problem \\ref{prob:keccMax} for $k$-ECC.\nIn the following, we first compare these solutions in terms of the complexities and scalability of the state-of-the-art online algorithms, index construction complexities, index-based query algorithms, community cohesiveness, and support for overlapped CS as well as dynamic graphs. After that, we perform an experiment on real large graphs by using these CS algorithms, and compare their empirical performance.\n\nTo make a fair comparison, we consider a simple undirected graph $G(V,E)$, where $n$=$|V|$, $m$=$|E|$, and its arboricity is denoted by $\\alpha(G)$ ($\\alpha(G)$ is often much smaller than $\\sqrt{m}$). We use $h$ and $l$ to denote small values that can be bounded by small constants \\cite{Chang:SIGMOD:2015}.\nIn Table \\ref{tab:compare}, we compare these representative CS solutions on $G$. Note that to measure the strength of algorithm scalability and community cohesiveness, we use notation $\\bigstar$; that is, an algorithm with more $\\bigstar$ means that it has better scalability or cohesiveness. Meanwhile, if a CS solution returns only one community $C$, we denote its community edge number by $|E(C)|$. If multiple communities are returned, we use $C_i$ to denote the $i$-th (1$\\leq$$i$$\\leq$$r$) community, where $r$ is the total number of returned communities.\nWe use ``O\" and ``D\" to denote whether the solutions support overlapped CS and dynamic graphs respectively.\n\nIn addition, for the complexities of the $k$-clique-based algorithm, we adopt the notations in \\cite{kclique2018}, where $s$ is the average size of maximal cliques, $T$ is the time to enumerate all maximal cliques, $L$ is the number of maximal cliques, $p$ is the average number of maximal cliques a vertex is contained in, $Q$ is the number of maximal cliques containing at least one query vertex, and $g$ is the height of the index tree.\n\nFrom Table \\ref{tab:compare}, we can make the observations:\n\\begin{itemize}\n \\item For online query algorithms, in terms of query time complexity, we can rank them as: $k$-core $\\preceq$ $k$-ECC $\\preceq$ $k$-truss $\\preceq$ $k$-clique, which is consistent with the efficiency ranking relationship of these metrics in Section \\ref{sec:analysis}. As a result, the $k$-core-based algorithm achieves the highest scalability while the $k$-clique-based algorithm has the lowest scalability.\n \\item For index construction algorithms, the ranking relationship above still holds. For index-based query algorithms, most of them except $k$-clique have the optimal time complexity, which is linear to the community edge number (i.e., $|E(C)|$).\n \\item The community structure cohesiveness is in line with the cohesiveness of these four metrics.\n \\item The $k$-core and $k$-ECC-based solutions can only return one community for each query, while the other two solutions may return multiple overlapped communities containing the query vertex.\n \\item All algorithms support dynamic graphs where vertices and edges are inserted or deleted dynamically.\n\\end{itemize}\n\nNext, we empirically evaluate the performance of algorithms in Table \\ref{tab:compare}. The input of these algorithms except the $k$-truss-based one is a query vertex, and they aim to find communities containing the query vertex which will maximize the value of $k$. For the $k$-truss-based one (Problem \\ref{problem:TTC}), its input is a set of query vertices and an integer $k$. To make a fair comparison, we adapt its algorithm such that its input is a query vertex and the algorithm will maximize the value of $k$. To measure the quality of returned communities (subgraphs), we introduce four metrics, i.e., diameter, degree, density (i.e., the number of edges over the maximum number of possible edges in a graph), and clustering coefficient (CC). Generally, a lower value of diameter and higher values of degree, density, and CC mean the higher quality of the community.\n\nTo conduct the experiments, we use a real-world graph Google \\footnote{\\scriptsize{Available at \\url{http:\/\/snap.stanford.edu\/data\/index.html}}}, which contains 875,713 vertices and 5,105,039 edges. We randomly select 100 vertices from the graph as query vertices, perform CS queries using these vertices, compute the average running time and community quality, and report experimental results in Table \\ref{tab:exp}. Generally, the efficiency results in Table \\ref{tab:exp} are consistent with the complexity analysis in Table \\ref{tab:compare}. More specifically, we have:\n\n\\begin{itemize}\n \\item For online query algorithms, the $k$-core-based algorithm is the fastest. The $k$-truss and $k$-ECC-based algorithms have similar time cost. The $k$-clique-based algorithm takes the highest time cost.\n \\item To build indexes, the $k$-core-based algorithm is the fastest and the $k$-truss-based algorithm is slower than others.\n \\item For index space cost, the $k$-core-based index takes the least space, while the space cost of others is around or over an order of magnitude larger than that of $k$-core-based algorithm.\n \\item For index-based query algorithms, the $k$-core-based algorithm is slower than the $k$-truss-based algorithm (which also takes optimal query time cost), because its returned communities are larger than those of other algorithms. The $k$-clique-based algorithm is the slowest, as its complexity is higher than others.\n \\item In terms of community quality, the $k$-truss-based solution achieves the smallest diameter, highest density, and highest clustering coefficient, due to small and tight triangle-based community structure. The $k$-core-based algorithm achieves the highest degree, against other methods. The $k$-clique-based method achieves the smallest degree.\n \\item In line with Table \\ref{tab:compare}, the $k$-core and $k$-ECC-based solutions return one community, while $k$-truss-based and $k$-clique-based solutions respectively return 1.31 and 1.05 communities.\n\\end{itemize}\n\n\\subsubsection{Size-Bounded Community Search}\n\\label{sec:sizeBounded}\n\nOne drawback of Problem~\\ref{prob:kcoreGlobal} is that the returned subgraph may contain a large number of vertices. Notice that although {\\tt Local} may find communities which are smaller than those of {\\tt Global}, it does not have any guarantee on the sizes of the returned communities, which implies that the returned communities may still have very large sizes.\n\nFor many real applications, such as holding a cocktail part, they often require the size of the output community is less than a pre-specified upper bound. Thus, it is desirable to search communities with bounded-size. By imposing the size constraint, we obtain another problem:\n\n\\begin{problem}\n\\label{prob:kcoreGlobalSize}\nGiven an undirected simple graph $G(V,E)$, a set of query vertices $Q\\subseteq V$, a size constraint $k$, and a goodness function $f$, return a subgraph $H(V_H,E_H)$ of $G$, such that\n\\begin{enumerate}\n \\item $V_H$ contains $Q$;\n \\item $H$ is connected;\n \\item $|V_H|\\leq k$ ($H$ has at most $k$ vertices);\n \\item $f(H)$ is maximized among all feasible choices for $H$.\n\\end{enumerate}\n\\end{problem}\n\nUnfortunately, due to the size constraint, Problem~\\ref{prob:kcoreGlobalSize} is NP-hard~\\cite{KDD2010}.\nThis implies that an exact algorithm for solving Problem~\\ref{prob:kcoreGlobalSize} will take exponential time cost, and thus it is impractical for large graphs. To alleviate the computational issue, some heuristic algorithms are developed~\\cite{KDD2010}, and they are able to achieve reasonable efficiency, although they do not have any provable quality guarantee.\n\nTo further reduce the size of the returned community, Barbieri et al.~\\cite{barbieri2015efficient} proposed the {\\it minimum community search problem}, which aims to find a community that satisfies all the constraints of Problem~\\ref{prob:kcoreGlobal} and has the minimum number of vertices.\n\n\\begin{problem}\n\\label{prob:kcoreMinSize}\nGiven an undirected simple graph $G(V,E)$, a set of query vertices $Q\\subseteq V$, and a minimum degree based function $f$, let $H^*$ be the subgraph returned by {\\tt Global}. Find a subgraph $H$ of $G$, such that\n\\begin{enumerate}\n \\item $V_H$ contains $Q$;\n \\item $H$ is connected;\n \\item $f(H)$=$f(H^*)$;\n \\item the size of $H$ is the smallest.\n\\end{enumerate}\n\\end{problem}\n\nSimilar to Problem~\\ref{prob:kcoreGlobalSize}, Problem~\\ref{prob:kcoreMinSize} is also NP-hard. It can be proved by a reduction from the \\textsc{Steiner Tree} problem: given a graph $G(V,E)$ and a set of terminal vertices $T\\subseteq V$, find a connected subgraph $G'$ of $G$ such that it contains all the terminal vertices and has the minimum number of edges.\nNote that the most efficient algorithm~\\cite{kou1981fast} of \\textsc{Steiner Tree} problem achieves an approximation ratio of (2-2\/$|Q|$), and takes linear time cost by the Mehlhorn's implementation~\\cite{mehlhorn1988}.\n\nTo answer the query in Problem~\\ref{prob:kcoreMinSize}, Barbieri et al.~\\cite{barbieri2015efficient} proposed an algorithm, and it consists of two steps:\nFirst, it reduces the size of $H^*$ as much as possible using some local greedy search. Note that after the reduction, the subgraph $H^*$ is still a qualified community of Problem~\\ref{prob:kcoreGlobal}, but may have much smaller size.\nSecond, it finds a subgraph from $H^*$ by adopting the above approximation algorithm for the \\textsc{Steiner Tree} problem.\n\\subsubsection{Size-Unbounded Community Search}\n\\label{sec:sizeUnbounded}\n\nIn~\\cite{KDD2010}, Sozio et al. proposed and studied the problem of community search, defined as follows:\n\n\\begin{problem}\n\\label{prob:kcoreGlobal}\nGiven an undirected simple graph $G(V,E)$, a set of query vertices $Q\\subseteq V$, and a goodness function $f$, return a subgraph $H(V_H,E_H)$ of $G$, such that\n\\begin{enumerate}\n \\item $V_H$ contains $Q$;\n \\item $H$ is connected;\n \\item $f(H)$ is maximized among all feasible choices for $H$.\n\\end{enumerate}\n\\end{problem}\n\nHere, $f(H)$ is a general goodness function for measuring cohesiveness of the community $H$. Intuitively, the value of $f(H)$ should be larger, if $H$ is densely connected. There are many possible choices for $f$, and an outstanding one is defined based on the {\\it minimum degree}, i.e., $f(H)$=$\\min_{\\forall v\\in H} deg_H(v)$. The reasons why the minimum degree is a good metric for the community are three-fold:\nFirst, minimum degree is one of the most fundamental characteristics of a graph. For instance, it is adopted for describing the evolution of random graphs and graph visualization~\\cite{local2014}.\nSecond, it is often used to measure the cohesiveness of user groups in social media. In~\\cite{md1983}, Seidman et al. compared the minimum degree with many other metrics of cohesiveness (e.g., connectedness and diameter) and found that the minimum degree is indeed a good metric for social network analysis.\nThird, for community search tasks, Sozio et al.~\\cite{KDD2010} also showed that it is better than some other metrics, including the average degree and density.\nIn the following, we assume that the minimum degree metric is adopted in $f$.\n\nTo solve Problem~\\ref{prob:kcoreGlobal}, there are two online algorithms, which are based on global and local search~\\cite{KDD2010,local2014} respectively, and one index-based algorithm~\\cite{barbieri2015efficient}.\n\n\\noindent\\textbf{$\\bullet$ A global search algorithm.}\nSozio et al.~\\cite{KDD2010} proposed a greedy algorithm, which follows the peeling framework \\cite{charikar2000greedy} of computing the densest subgraphs~\\cite{goldberg1984finding} and removes vertices iteratively. Specifically, let $G_0$=$G$ and $G_t$ be the graph in $t$-th iteration ($1\\leq t\\textless n$). At the $t$-th ($1\\leq t \\textless n$) step, it removes the vertex which has the minimum degree in $G_{t-1}$ and obtain an updated graph $G_t$. The above operation iterates and stops at the $T$-th step, if either (1) at least one of the query vertices $Q$ has minimum degree in the graph $G_{T-1}$, or (2) the query vertices $Q$ are no longer connected. Let $G_t'$ be the connected component containing $Q$ in $G_t$. Then, the subgraph $G_O$=$\\arg\\max\\{f(G_t')\\}$ satisfies all the constraints in Problem~\\ref{prob:kcoreGlobal}.\n\nWe denote the algorithm above by {\\tt Global}, as it finds the community in a global manner. By using some special optimization techniques~\\cite{KDD2010,charikar2000greedy}, {\\tt Global} is able to achieve linear time and space complexities, i.e., $O(n+m)$. Note that the function $f(H)$ above can be generalized to any monotone function, and the corresponding problem can also be solved by {\\tt Global}~\\cite{KDD2010}.\n\nIt is easy to observe that since {\\tt Global} peels all the vertices with low degrees, the subgraph returned is the largest connected subgraph, in which each vertex has at least $k$ neighbors. As a result, the returned subgraph is a connected $k$-core containing $Q$, where $k$ equals to the minimum core number of vertices in $Q$.\n\n\\noindent\\textbf{$\\bullet$ A local search algorithm.}\nAccording to Problem~\\ref{prob:kcoreGlobal}, there may exist some subgraphs of $G_O$, which satisfy all the constraints and achieve the same value on the function $f$, but have smaller sizes. Thus, they can be considered the communities as well.\n\n\\begin{example}\n\\label{eg:kcoreGlobal}\nLet the graph be the one in Fig. \\ref{fig:kcoreEg}(a), $Q$=$\\{E\\}$. {\\tt Global} will return the subgraph of vertices $\\{A,B,C,D,E\\}$ as the community, and the value of function $f$ is 2.\nHowever, there are other three subgraphs, whose vertex sets are $\\{A,B,C,E\\}$, $\\{A,B,D,E\\}$, and $\\{A,B,E\\}$,\nalso satisfy the constraints of Problem~\\ref{prob:kcoreGlobal}, and their values on $f$ are 2. Thus, they can be considered as communities.\n\\end{example}\n\nIn \\cite{local2014}, Cui et al. proposed a local CS method, denoted by {\\tt Local}, which works in a local expansion manner and finds a community that may have smaller size than that of {\\tt Global}. Specifically, it assumes that there is only one query vertex $q$ (i.e., $Q$=$\\{q\\}$). {\\tt Local} consists of three steps: First, it expands the search space from $q$. Second, it generates a candidate vertex set $C$ in the search space. Third, it finds the community from $C$.\n\nThe key step is the second step, which works in an iterative manner. In each iteration, it selects the vertex that is the local optimal and adds it into the candidate set $C$. To decide the local optimal vertex, some heuristic criteria are adopted. One typical criterion is to select the vertex that leads to the largest increment of the function $f$; another one is to select the vertex which has the largest number of connections to vertices of the candidate set.\nThe iterations stop when the candidate set $C$ theoretically guarantees that it contains a community satisfying the constraints of Problem~\\ref{prob:kcoreGlobal}.\n\nLet $H$ and $H'$ denote the communities returned by {\\tt Global} and {\\tt Local} respectively. Then, we have $f(H')=f(H)$ and $H'\\subseteq H$. Besides, since in the worst case the candidate set $C$ could be the same as vertex set $V$, the time complexity of {\\tt Local} is the same as that of {\\tt Global}, but in practice for large graphs, the candidate set is often much smaller than the entire graph, and thus {\\tt Local} achieves higher efficiency.\n\n\\noindent\\textbf{$\\bullet$ An index-based algorithm.}\nIn~\\cite{barbieri2015efficient}, Barbieri et al. proposed an index structure, called {\\tt ShellStruct}, which organizes all the connected $k$-cores in an offline manner. Based on {\\tt ShellStruct}, Problem~\\ref{prob:kcoreGlobal} can be answered in optimal time cost, i.e., $O(|H_V|)$, where $H_V$ is the set of vertices in the returned community and it is the same with that of {\\tt Global}.\n\nThe index is built based on the key observation that cores are nested. That is, for any integer $0\\textless k\\leq k_{\\max}$, the $k$-core is contained by the ($k$--1)-core, where $k_{\\max}$ is the maximum core number. {\\tt ShellStruct} is a tree-like structure with $k_{\\max}$ levels. The root of the tree corresponds to the 1-core, and the $k$-th level keeps track of the information about the $k$-th core. In $k$-th level, each tree node, $p_k$, corresponds to a connected component $C_k$ of the $k$-core, and it keeps:\n\\begin{enumerate}\n \\item the set of ``children\" nodes, each of which corresponds to a connected component that is in the ($k$+1)-core and contained by $C_k$;\n \\item the set of vertices in $C_k$ but not in ($k$+1)-core.\n\\end{enumerate}\n\nIt is easy to observe that in {\\tt ShellStruct}, all the connected $k$-cores are well organized. The space cost is exactly $O(n)$ because each vertex appears only once. To build the index, Barbieri et al. proposed an index construction algorithm, which builds the tree level by level, starting from the root level. As a result, its time complexity is $O(n\\cdot k_{\\max} +m)$. We remark that a more efficient algorithm for building the same index is proposed in~\\cite{Fang:VLDB:2016}, which takes $O(m\\cdot\\alpha(n))$ time, where $\\alpha(n)$ is the inverse Ackermann function and it is less than 5 for all remotely practical values of $n$.\n\nBased on {\\tt ShellStruct}, a query algorithm is proposed. Specifically, it starts from the $l$-th level where $l$ is the maximum core number of vertices in $Q$ and checks its upper levels, until there is a connected component containing all the query vertices. By using the lowest-common-ancestor (LCA) data structure~\\cite{LCA1983}, the time cost of the query algorithm can be reduced to $O(|H_V|)$.\n\nIn Problem~\\ref{prob:kcoreGlobal}, the cohesiveness function is required to be maximized. However, for some applications, such as infectious disease control discussed in Section~\\ref{sec:intro}, this constraint may need to be relaxed so that vertices which have less connections with the query vertices can also be involved. Motivated by this, a variant of Problem~\\ref{prob:kcoreGlobal} is also studied in the literature~\\cite{local2014}:\n\n\\begin{problem}\n\\label{prob:kcoreLocal}\nGiven an undirected simple graph $G(V,E)$, a query vertex $q\\in V$, and a non-negative integer $k$, return a subgraph $H(V_H,E_H)$ of $G$, such that\n\\begin{enumerate}\n \\item $V_H$ contains $q$;\n \\item $H$ is connected;\n \\item for each vertex $v\\in H$, $deg_H(v)\\geq k$.\n\\end{enumerate}\n\\end{problem}\n\nIn Fig. \\ref{fig:kcoreEg}(a), let $q$=$A$ and $k$=2. Then, the subgraph of $\\{A,B,C,D,E\\}$ satisfies all the constraints, and thus is a community for Problem~\\ref{prob:kcoreLocal}. Note that if we maximize the minimum degree as required by Problem~\\ref{prob:kcoreGlobal}, we will return a smaller subgraph, i.e., $\\{A,B,C,D\\}$, since the minimum degree is 3.\nThe algorithms {\\tt Global} and {\\tt Local} can be easily adapted for answering the query of Problem~\\ref{prob:kcoreLocal}. For details, please refer to~\\cite{local2014}.\n\n\n\\subsection{VizCS}\n\\label{sec:vizcs}\nVizCS is an online query processing system for searching and visualizing communities in graphs \\cite{Huang:ICDE:2018}. VizCS exhibits four key innovative features as follows.\n\nFirst, VizCS adopts a triangle-connected truss community model for dynamic graphs where vertices\/edges undergo frequently insertions\/deletions \\cite{k-truss2014}. It provides the feature of CS over dynamic graphs, which can be uploaded with one file of graph updates by users\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width=2.5in]{figures\/VizCs.pdf}\n \\caption{Interface of VizCS~\\cite{Huang:ICDE:2018}.}\\label{fig:vizcs}\n\\end{figure}\n\nSecond, VizCS offers a user-friendly visual interface to formulate queries and a real-time response query processing engine. Fig. \\ref{fig:vizcs} shows an example query of author vertex $q$=``Jim Gray\" and parameter $k$=8. Thanks to efficient $k$-truss CS algorithms, the query results can be quickly obtained in real-time.\n\nThird, VizCS generates a community exploration wall by offering interactive community visualization, which facilitates users to in-depth understanding of the data. The community exploration wall uses graph visualization techniques to depict the community results and also presents informative features to users through various exploration channels, such as the profile search of community members by Google, structural statistic report, collaborator recommendation, and tag cloud. Fig. \\ref{fig:vizcs} shows the community exploration wall.\n\nLast but not least, VizCS is a CS platform that can visualize and compare different community results by various state-of-the-art algorithms and user-uploaded approaches. It benefits users to understand different models vividly and directly. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{Int}\n\nThe Korteveg--de Vries equation appears as a model for the propagation of weakly nonlinear dispersive waves in several fields. Among them there are gravity driven waves on a surface of an incompressible irrotational inviscid fluid \\cite{Whit,EIGR,Rem,DrJ,MS90,Ding}, ion acoustic waves in plasma \\cite{EIGR}, impulse propagation in electric circuits \\cite{Rem} and so on. \nIn the shallow water wave problem the KdV equation corresponds to the case when the bottom is even.\nThere have been many attempts to study nonlinear waves in the case of an uneven bottom because of its significance, for instance in such phenomena as tsunamis.\nAmong the first papers dealing with a slowly varying bottom are papers of Mei and Le M\\'ehaut\\'e \\cite{Mei} and Grimshaw \\cite{Grim70}. When taking an appropriate average of vertical variables one arrives at Green-Nagdi type equations \\cite{GN,Nad,Kim}.\nVan Groesen and Pudyaprasetya \\cite{G&P1,G&P2} studied uni-directional waves over a slowly varying bottom within the Hamilton approach, obtaining a forced KdV-type equation. An extensive study of wave propagation over an uneven bottom conducted before 2000 is summarized in Dingemans's monograph \\cite{Ding}.\nThe papers \\cite{Pel,Peli,Peli1} are examples of approaches that combine linear and nonlinear theories. The Gardner equation and the forced KdV equation, were also extensively investigated in this context, see, e.g., \\ \\cite{Grim,Smy,Kam}.\n\nIn previous papers, \\cite{KRR,KRI} we derived a new KdV-type equation containing terms which come directly from an uneven bottom. These terms, however, appear naturally only if Euler equations for the fluid motion are considered up to second order in small parameters, whereas the KdV equation is obtained in first order approximation. \nThere are no analytic solutions for the above equation. In \\cite{KRR,KRI} we presented several cases of numerical simulations for that equation obtained using the finite difference method (FDM) with periodic boundary conditions.\n\n\nIt was demonstrated in \\cite{DebP} that finite element method (FEM) describes properly the dynamics of the KdV equation \n(\\ref{kdvm}), which is the equation in a moving frame of reference. \n\nThe first aim of this paper is to construct an effective FEM method for solving higher order KdV equations, both with even bottom\nand uneven bottom.\nThe second goal is to compare the results obtained in this numerical scheme with some of the results obtained earlier using the finite difference method in \\cite{KRR} and in \\cite{KRI}.\n\nThe paper is organized as follows: In section \\ref{prel} we review the KdV equation (\\ref{kdv1}), the extended KdV equation (\\ref{etaab}) and KdV-type equation containing direct terms from bottom variation (\\ref{etaabd}), all expressed in scaled dimensionless variables. In section \\ref{numM} the construction of the numerical method for solving these equations within the FEM is described. Coupled sets of nonlinear equations for coefficients of expansion of solutions to these equations in a basis of piecewise linear functions are obtained. In section \\ref{nsym} several examples of numerical simulations are presented.\n\n\n\n\\section{Preliminaries}\\label{prel}\n\nExtended KdV type equations, derived by some of the authors in\n \\cite{KRR,KRI}, second order in small parameters, have the following form \n(written in scaled dimensionless coordinates, in a fixed coordinate system).\nFor the case with an uneven bottom\n\\begin{eqnarray} \\label{etaabd}\n\\eta_t &+&\n\\eta_x + \\alpha\\, \\frac{3}{2}\\eta\\eta_x +\\beta\\,\\frac{1}{6} \\eta_{3x} \\\\\n&-&\n\\frac{3}{8} \\alpha^2 \\eta^2\\eta_x \n+ \\alpha\\beta \\!\\left(\\!\\frac{23}{24}\\eta_x\\eta_{2x}\\!+\\!\\frac{5}{12}\\eta\\eta_{3x}\\! \\right)\\!+\\!\\frac{19}{360}\\beta^2\\eta_{5x} \\nonumber \\\\\n&+&\n\\beta\\delta\\left(-\\frac{1}{2\\beta}(h\\eta)_x +\\frac{1}{4} \\left(h_{2x}\\eta\\right)_x -\\frac{1}{4} \\left(h\\eta_{2x}\\right)_x\\right) =0.\n\\nonumber\n\\end{eqnarray}\nDetails of the derivation of the second order equation (\\ref{etaabd}) from the set of Euler equations with appropriate boundary conditions can be found in \\cite{KRR,KRI}.\nIn (\\ref{etaabd}), $\\eta(x,t)$ stands for a wave profile and $h=h(x)$ denotes a bottom profile. Subscripts are used for notation of partial derivatives, that is, for instance $\\eta_{2x}\\equiv \\frac{\\partial^2 \\eta}{\\partial x^2}$, and so on.\nSmall parameters ~$\\alpha,\\beta,\\delta$ are defined by ratios of the amplitude of the wave profile ~$a$, the depth of undisturbed water ~$h_0$, average wavelength ~$l$ and the amplitude of the bottom changes ~$a_h$\n\\begin{equation} \\label{smallp}\n \\alpha =\\frac{a}{h_0}, \\qquad \\beta=\\left(\\frac{h}{l}\\right)^2, \\qquad \\delta=\\frac{a_h}{h_0}.\n\\end{equation}\nFor details of the transformation of the original dimensional variables to the nondimensional, scaled ones used here, see, e.g.,\\ \\cite{KRR,KRI,BS}.\n\nIt should be emphasized that in equation (\\ref{etaabd}) all three terms originating from an uneven bottom are second order in small parameters. These terms appear from the boundary condition at the bottom which is already in second order with coefficient $\\beta\\delta$, see\nequation (5) in \\cite{KRI} or equation (10) in \\cite{KRR}.\nThen in the final second order equation (\\ref{etaabd}) we write them in the form $\\beta\\delta (\\cdot)$ in order to epmhasize that they all come from the second order perturbation approach. For details we refer to the mentioned papers. \n\nIn the case of an even bottom ($\\delta=0$) equation (\\ref{etaabd}) is reduced to the second order KdV type equation \n\\begin{eqnarray} \\label{etaab}\n\\eta_t &+& \\eta_x + \\alpha\\, \\frac{3}{2}\\eta\\eta_x \n+\\beta\\,\\frac{1}{6} \\eta_{3x} - \\frac{3}{8} \\alpha^2 \\eta^2\\eta_x\\\\\n\\! \\!&\\! \\!+ & \n \\alpha\\beta\\left(\\frac{23}{24}\\eta_x\\eta_{2x}+\\frac{5}{12}\\eta\\eta_{3x} \\right)+\\frac{19}{360}\\beta^2\\eta_{5x} = 0 \\nonumber \n\\end{eqnarray}\nand when $\\beta=\\alpha$ it becomes identical to Eq.~(21) in \\cite{BS}.\nEquation (\\ref{etaab}) was obtained earlier by Marchant and Smyth \\cite{MS90} and called the {\\em extended KdV} equation.\n\nLimitation to first order approximation in small parameters gives the KdV in a fixed system of coordinates \n\\begin{equation} \\label{kdv1}\n\\eta_t+\\eta_x + \\alpha\\, \\frac{3}{2}\\eta\\eta_x +\\beta\\,\\frac{1}{6} \\eta_{3x} =0.\n\\end{equation}\n\nThe standard, mathematical form of the KdV equation is obtained from (\\ref{kdv1}) by transformation to a moving reference frame. Substituting\n\\begin{equation} \\label{tr}\n\\bar{x} =\\sqrt{\\frac{3}{2}}(x-t), \\qquad \\bar{t}=\\frac{1}{4}\\sqrt{\\frac{3}{2}}\\,\\alpha\\,t, \\qquad u = \\eta,\n\\end{equation}\none obtains from (\\ref{etaab}) the equation\n\\begin{equation} \\label{kdvm}\nu_{\\bar{t}} + 6\\,u\\,u_{\\bar{x}} + \\frac{\\beta}{\\alpha}u_{3\\bar{x}\n =0,\n\\end{equation}\nor finally, when ~$\\beta=\\alpha$,\n\\begin{equation} \\label{kdvm1}\n u_{\\bar{t}} + 6\\,u\\,u_{\\bar{x}} + u_{3\\bar{x}}=0.\n\\end{equation}\n\nIn this paper we attempt to solve numerically the equation (\\ref{etaabd}) for several cases of bottom topography and different initial conditions. \nIn several points we follow the method applied by Debussche and Printems~\\cite{DebP}. However, the method is extended to higher order KdV type equations with plain bottom (\\ref{etaab}) and with bottom fluctuations (\\ref{etaabd}).\nFor both cases we work in a fixed reference system, necessary for a bottom profile depending on the position.\n\n\n\\section{Numerical method} \\label{numM}\n\nThe emergence of soliton solutions to the KdV equation was observed in numerics fifty years ago by \\cite{ZK}. Several numerical methods used for solving the KdV equation are discussed in \\cite{TaAb}. Among them are the finite difference explicit method \\cite{ZK}, the finite difference implicit method \\cite{Goda} and several versions of the pseudospectral method, as in \\cite{FoWhi}.\nIt is also worth mentioning papers using the FEM and Galerkin methods \\cite{BoCh,CuMa}. Most numerical applications use periodic boundary conditions, but\nthere exist also works that apply Dirichlet boundary conditions on a finite interval \\cite{SkKa,YiHu,YuSh}.\n\nThe authors are trying to construct a method which will be applicable not only for the numerical simulation of an evolution of nonlinear waves governed by equations (\\ref{etaabd}) or (\\ref{etaab}) but also for their stochastic versions. Such stochastic equations will be studied in the next paper.\nSince stochastic noise is irregular, solutions are not necessarily smooth, neither in time nor space. A finite element method (FEM) seems to be suitable for such a case. \n\n\\subsection{Time discretization} \\label{timd} \nWe have adapted the Crank--\\-Nicholson scheme for time evolution, beginning with the KdV equation (\\ref{kdv1}) in a fixed coordinate system. Note that $\\eta\\eta_x=\\frac{1}{2}(\\eta^2)_x$. Denote also $v:=\\eta_x$ and $w:=v_x$. Let us choose time step $\\tau$. Then the KdV equation (\\ref{kdv1}) in the Crank--Nicholson scheme can be written as a set of coupled first order differential equations\n\\begin{eqnarray} \\label{Cr}\n\\eta^{n+1}- \\eta^{n}+\\tau \\left(\\frac{\\partial }{\\partial x}\\eta^{n+\\frac{1}{2}} \\right.\\hspace{10ex}\\nonumber\\\\ \\left. \\hspace{2ex}\n+ \\frac{3\\alpha}{4} \n\\frac{\\partial }{\\partial x}(\\eta^{n + \\frac{1}{2}})^{2} + \\frac{\\beta}{6} w^{n + \\frac{1}{2}} \\right) &=& 0, \\\\\n\\frac{\\partial }{\\partial x}\\eta^{n + \\frac{1}{2}} &=& v^{n + \\frac{1}{2}}, \\hspace{4ex} \\nonumber\\\\\n\\frac{\\partial }{\\partial x}v^{n + \\frac{1}{2}} &=& w^{n + \\frac{1}{2}}, \\hspace{4ex}\\nonumber\n\\end{eqnarray}\nwhere \n\\begin{equation} \\label{e12a}\\def1.4{1.4} \n\\begin{array}{rcl}\n\\eta^{n+\\frac{1}{2}}&=&\\frac{1}{2}\\left( \\eta^{n+1} + \\eta^{n} \\right), \\\\\nv^{n+\\frac{1}{2}}&=&\\frac{1}{2}\\left( v^{n+1} + v^{n} \\right), \\\\\nw^{n+\\frac{1}{2}}&=&\\frac{1}{2}\\left( w^{n+1} + w^{n} \\right). \n\\end{array}\n\\end{equation}\n\nFor second order equations (\\ref{etaabd}) or (\\ref{etaab}) we need to introduce two new auxiliary variables: $p:=w_x$ and $q:=p_x$. \nNote that $\\eta^2\\eta_x=\\frac{1}{3}(\\eta^3)_x$, ~$\\eta_x\\eta_{2x}=\\frac{1}{2}(\\eta_x^2)_x= \\frac{1}{2}(v^2)_x$. Moreover, $\\eta_{5x}=q=p_x$ and\n$$\\frac{23}{24}\\eta_x\\eta_{2x}+\\frac{5}{12}\\eta\\eta_{3x}=\\frac{13}{48}(v^2)_x+\\frac{5}{12}(\\eta w)_x.$$\n\nThis setting allows us to write the Crank--\\-Nicholson scheme for (\\ref{etaab}) as the following set of first order equations\n\\begin{eqnarray} \\label{CrAB}\n\\eta^{n+1}- \\eta^{n}+ \\hspace{28ex}\\nonumber \\\\\n+\\tau \\frac{\\partial }{\\partial x} \\left[ \\eta^{n+\\frac{1}{2}} +\\frac{3\\alpha}{4} \n\\left(\\eta^{n + \\frac{1}{2}}\\right)^{2} \n + \\! \\frac{\\beta}{6} w^{n + \\frac{1}{2}} \\! \\right. \\nonumber \\\\ \\hspace{-3ex}\n-\\!\\frac{1}{8}\\alpha^2 \\left(\\eta^{n+\\frac{1}{2}}\\right)^3\n+ \\alpha\\beta\\left(\\!\\frac{13}{48}\\left(v^{n + \\frac{1}{2}}\\right)^2 \\right. \\\\ \\left. \\left.\\hspace{-3ex}\n\\!+\\!\\frac{5}{12}\\left(\\eta^{n + \\frac{1}{2}}w^{n + \\frac{1}{2}\\!}\\right)\\! \\right)\\! \n+ \\frac{19}{360}\\beta^2 \\left(q^{n + \\frac{1}{2}}\\right)\n \\right]\n&=& 0, \\nonumber\\\\\n\\frac{\\partial }{\\partial x}\\eta^{n + \\frac{1}{2}}-v^{n + \\frac{1}{2}} &=& 0, \\nonumber\\\\\n\\frac{\\partial }{\\partial x} v^{n + \\frac{1}{2}}-w^{n + \\frac{1}{2}} &=& 0, \\nonumber \\\\\n\\frac{\\partial }{\\partial x} w^{n + \\frac{1}{2}}-p^{n + \\frac{1}{2}} &=& 0, \\nonumber\\\\ \\frac{\\partial }{\\partial x} p^{n + \\frac{1}{2}}-q^{n + \\frac{1}{2}} &=& 0, \\nonumber \n\\end{eqnarray}\nwhere\n\\begin{equation} \\label{e12}\\def1.4{1.4} \n\\begin{array}{rcl}\np^{n+\\frac{1}{2}}&=&\\frac{1}{2}\\left( p^{n+1} + p^{n} \\right), \\\\\nq^{n+\\frac{1}{2}} &=&\\frac{1}{2}\\left( q^{n+1} + q^{n} \\right). \n\\end{array}\n\\end{equation}\n\nFor the second order KdV type equation with an uneven bottom (\\ref{etaabd}) the first equation in the set (\\ref{CrAB}) has to be supplemented by terms originating from bottom variations, yielding\n\\begin{eqnarray} \\label{CrABD}\n\\eta^{n+1}- \\eta^{n}+ \\hspace{28ex}\\nonumber \\\\\n+\\tau \\frac{\\partial }{\\partial x} \\left[ \\eta^{n+\\frac{1}{2}} +\\frac{3\\alpha}{4} \n\\left(\\eta^{n + \\frac{1}{2}}\\right)^{2} \n + \\! \\frac{\\beta}{6} w^{n + \\frac{1}{2}} \\! \\right. \\nonumber \\\\ \\hspace{-3ex}\n-\\!\\frac{1}{8}\\alpha^2 \\left(\\eta^{n+\\frac{1}{2}}\\right)^3\n+ \\alpha\\beta\\left(\\!\\frac{13}{48}\\left(v^{n + \\frac{1}{2}}\\right)^2 \\right. \n\\\\ \\left. \\hspace{-3ex}\n\\!+\\!\\frac{5}{12}\\left(\\eta^{n + \\frac{1}{2}}w^{n + \\frac{1}{2}\\!}\\right)\\! \\right)\\! \n+ \\frac{19}{360}\\beta^2 \\left(q^{n + \\frac{1}{2}}\\right) \\nonumber \n\\\\ \n\\frac{1}{4}\\beta\\delta \\left(-\\frac{2}{\\beta}\\left(h^{n + \\frac{1}{2}}\\eta^{n + \\frac{1}{2}\\!}\\right) \\right. \\hspace{5ex} \\nonumber \\\\ \\left. \\left.\n+\\eta^{n + \\frac{1}{2}}g^{n + \\frac{1}{2}} + h^{n + \\frac{1}{2}}w^{n + \\frac{1}{2}} \\right) \\right]\n&=& 0, \\nonumber\n\\end{eqnarray} \nwhere ~$g:=h_{xx}$.\n\nBelow we focus on the second order equations (\\ref{etaab}) and (\\ref{CrAB}), pointing out contributions from bottom variation later.\n\n\\subsection{Space discretization} \\label{spad}\nFollowing the arguments given by Debussche and Printems \\cite{DebP} we apply the Petrov-Galerkin discretization and finite element method. We use piecewise linear shape functions and piecewise constant test functions. We consider wave motion on the interval $x\\in [0,L]$ with periodic boundary conditions. Given $N\\in \\mathbb{N}$, then we use a mesh $M_{\\chi}$ of points $x_j= j\\chi$, $j=0,1,\\ldots,N$, where $\\chi =L\/N$. Let $V^1_{\\chi}$ which is a space of piecewise linear functions $\\varphi_j(x)$, such that $\\varphi_j(0)=\\varphi_j(L)$, defined as\n\\begin{equation} \\label{phi}\n\\varphi_{j}(x) = \\left\\{ \\begin{array}{lll}\n\\frac{1}{\\chi}(x-x_{j-1}) & \\mbox{if} & x \\in [x_{j-1},x_{j}] \\\\\n\\frac{1}{\\chi}(x_{j+1}-x) & \\mbox{if} & x \\in [x_{j},x_{j+1}] \\\\\n0 & & \\mbox{otherwise} . \\end{array} \\right. \n\\end{equation}\nAs test functions we have chosen the space of piecewise constant functions $\\psi_j(x)\\in V^0_{\\chi}$, where \n\\begin{equation} \\label{psi}\n\\psi_{j}(x) = \\left\\{ \\begin{array}{lll}\n1 & \\mbox{if} & x \\in [x_{j},x_{j+1})\\\\\n0 & &\\mbox{otherwise}. \\end{array} \\right.\n\\end{equation}\n\nAn approximate solution and its derivatives may be written as an expansion in the basis (\\ref{phi}) \n\\begin{equation} \\label{etfi}\\def1.4{1.4} \n\\begin{array}{lcl}\n\\eta_{\\chi}^{n}(x)& = &\\sum_{j=1}^{N}a_j^n\\,\\varphi_j(x),\\\\\n v_{\\chi}^{n}(x)& = &\\sum_{j=1}^{N}b_j^n\\,\\varphi_j(x),\\\\\n w_{\\chi}^{n}(x)& = &\\sum_{j=1}^{N}c_j^n\\,\\varphi_j(x),\\\\\n p_{\\chi}^{n}(x)& = &\\sum_{j=1}^{N}d_j^n\\,\\varphi_j(x),\\\\\n q_{\\chi}^{n}(x)& = &\\sum_{j=1}^{N}e_j^n\\,\\varphi_j(x),\n\\end{array} \n\\end{equation}\nwhere $a^n_j,b^n_j ,c^n_j ,d^n_j , e^n_j$ are expansion coefficients.\nTherefore, in a weak formulation we can write (\\ref{Cr}) as\n\\begin{eqnarray} \\label{CrABw}\n\\left(\\eta_{\\chi}^{n+1}- \\eta_{\\chi}^{n},\\psi_i\\right)+ \\tau \\left\\{ \\left(\\partial_x\\eta_{\\chi}^{n+\\frac{1}{2}},\\psi_i\\right) \\right. \\hspace{10ex} \\nonumber \\\\\n +\\frac{3\\alpha}{4} \\left(\n\\partial_x\\left(\\eta_{\\chi}^{n + \\frac{1}{2}}\\right)^{2},\\psi_i\\right) \n + \\frac{\\beta}{6}\\left(\\partial_x w_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\right) \\nonumber \\\\ \\hspace{-3ex}\n-\\frac{1}{8}\\alpha^2 \\left(\\partial_x\\left(\\eta_{\\chi}^{n+\\frac{1}{2}}\\right)^3,\\psi_i\\right)\\nonumber \\\\\n+ \\alpha\\beta\\left[\\frac{13}{48}\\left(\\partial_x\\left(v_{\\chi}^{n + \\frac{1}{2}}\\right)^2,\\psi_i\\right) \\right. \\\\ \\left.\\hspace{-3ex}\n+\\frac{5}{12}\\left(\\partial_x\\left(\\eta_{\\chi}^{n + \\frac{1}{2}}w_{\\chi}^{n + \\frac{1}{2}}\\right),\\psi_i\\right) \\right] \\nonumber \\\\ \\left.\n+ \\frac{19}{360}\\beta^2\\left( \\partial_x\\left(q_{\\chi}^{n + \\frac{1}{2}}\\right),\\psi_i\\right) \\right\\}\n &=& 0, \\nonumber\\\\\n\\left(\\partial_x\\eta_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\right)-\\left(\\!v_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\!\\right) &=& 0, \\nonumber\\\\ \n\\left(\\partial_x v_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\right)-\\left(\\!w_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\!\\right) &=&0, \\nonumber \\\\\n\\left(\\partial_x w_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\right)-\\left(\\!p_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\!\\right) &=& 0, \\nonumber\\\\ \n\\left(\\partial_x p_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\right)-\\left(\\!q_{\\chi}^{n + \\frac{1}{2}},\\psi_i\\!\\right) &=& 0, \\nonumber \n\\end{eqnarray}\nfor any $i=1,\\dots,N$, where for abbreviation $\\partial_x$ is used for $\\frac{\\partial }{\\partial x}$.\nIn (\\ref{CrABw}) and below scalar products are defined by appropriate integrals\n$$(f,g):=\\int_0^L f(x) g(x) dx.$$ \n\nIn the case of \nequation (\\ref{etaabd}), the first equation of the set (\\ref{CrABw}) has to be supplemented inside the bracket \\{ \\} by the terms\n\\begin{eqnarray} \\label{CrABdno}\n &+& \\frac{1}{4}\\beta\\delta \\left(\\partial_x\\left[-\\frac{2}{\\beta}\\left(h^{n + \\frac{1}{2}}\\eta_{\\chi}^{n + \\frac{1}{2}\\!}\\right) \\right.\\right. \\\\\n&&\\hspace{9ex} \\left.\\left. +\\eta_{\\chi}^{n + \\frac{1}{2}}g^{n + \\frac{1}{2}} + h^{n + \\frac{1}{2}}w_{\\chi}^{n + \\frac{1}{2}} \\right],\\psi_i\\right) . \\nonumber\n\\end{eqnarray}\n\nInsertion of (\\ref{etfi}) into (\\ref{CrABw}) yields a system of coupled linear equations for coefficients $a_j^n, b_j^n, c_j^n, d_j^n, e_j^n$. The solution to this system supplies an approximate solution to (\\ref{etaab}) given in the mesh points $x_j$.\n\n\\subsubsection{KdV equation}\nIn order to demonstrate the construction of the matrices involved we limit at this point our considerations to the first order equation (\\ref{kdv1}). It means that we drop temporarily in (\\ref{CrABw}) terms of second order,\nthat is, the terms with $\\alpha^2, \\alpha\\beta, \\beta^2$.\nEquations with $p$ and $q$ do not apply because $\\eta_{4x}$ and $\\eta_{5x}$ do not appear in (\\ref{kdv1}).\nThis leads to equations \n\\begin{eqnarray} \\label{matr1}\n \\sum_{j=1}^{N} (a^{n\\!+\\!1}_{j} \\!-\\! a^{n}_{j}) (\\varphi_{j},\\psi_{i}) \n\\!+\\! \\tau \\frac{1}{2} \\sum_{j=1}^{N}(b^{n\\!+\\!1}_{j} \\!+\\! b^{n}_{j}) (\\varphi_{j},\\psi_{i})\n & & \\nonumber \\\\\n\\!+\\! \\tau \\alpha \\frac{3}{16} \\sum_{j=1}^{N} \\sum_{k=1}^{N}(a^{n\\!+\\!1}_{j} \\!+\\! a^{n}_{j}) (a^{n\\!+\\!1}_{k} \\!+\\! a^{n}_{k}) \\hspace{10ex} & & \\\\ \n\\times \\,\n(\\varphi_{j}'\\varphi_{k} \\!+\\! \\varphi_{j}\\varphi_{k}',\\psi_{i}) \\hfill & & \\nonumber\\\\\n\\!+\\!\\tau \\beta \\frac{1}{12} \\sum_{j=1}^{N}(c^{n\\!+\\!1}_{j} \\!+\\! c^{n}_{j}) (\\varphi_{j},\\psi_{i}) &=&0 , \\nonumber \\\\\n\\sum_{j=1}^{N} \\left[(a^{n\\!+\\!1}_{j} \\!+\\! a^{n}_{j}) (\\varphi_{j}',\\psi_{i}) \\!-\\! (b^{n\\!+\\!1}_{j} \\!+\\! b^{n}_{j}) (\\varphi_{j},\\psi_{i}) \\right] &=&0 ,\\nonumber\\\\\n\\sum_{j=1}^{N} \\left[(b^{n\\!+\\!1}_{j} \\!+\\! b^{n}_{j}) (\\varphi_{j}',\\psi_{i}) \\!-\\! (c^{n\\!+\\!1}_{j} \\!+\\! c^{n}_{j}) (\\varphi_{j},\\psi_{i}) \\right] &=&0 . \\nonumber \n\\end{eqnarray}\nDefine \n\\begin{equation} \\label{defC}\\def1.4{1.4}\n\\begin{array}{ll}\nC^{(1)}_{ij}:=(\\varphi_{j},\\psi_{i}),& C^{(2)}_{ij}:=(\\varphi'_{j},\\psi_{i}), \\\\\n C^{(3)}_{ijk}:=(\\varphi'_{j}\\varphi_{k}+\\varphi_{j}\\varphi'_{k},\\psi_{i}), &\n\\end{array}\n\\end{equation} \nwhere $\\varphi'_j=\\frac{d\\varphi}{dx}(x_j)$. Simple integration shows that \n\\begin{equation} \\label{C1}\\def1.4{1.4}\nC^{(1)}_{ij}=\\left\\{ \\begin{array}{rl}\n\\frac{1}{2} \\chi &\\quad \\textrm{if} \\quad i=j \\vee i=j-1 \\\\\n0 & \\quad \\textrm{otherwise},\n\\end{array} \\right. \n\\end{equation} \n\\begin{equation} \\label{C2}\\def1.4{1.4}\nC^{(2)}_{ij}=\\left\\{ \\begin{array}{rl}\n-1&\\quad \\textrm{if} \\quad i=j \\\\\n 1&\\quad \\textrm{if} \\quad i=j-1 \\\\\n0 & \\quad \\textrm{otherwise}.\n\\end{array} \\right. \n\\end{equation} \n Similarly one obtains\n\\begin{equation} \\label{C3}\\def1.4{1.4}\nC^{(3)}_{ijk}= C^{(2)}_{ij}\\, \\delta_{jk}.\n\\end{equation} \nThe property (\\ref{C3}) reduces the double sum in the term with $\\tau\\alpha\\frac{3}{16}$ to the single one of the square of $(a^{n\\!+\\!1}_{j} \\!+\\! a^{n}_{j})$.\nInsertion of (\\ref{C1})--(\\ref{C3}) into (\\ref{matr1}) gives\n\\begin{eqnarray} \\label{matr2}\n \\sum_{j=1}^{N}\\left[ (a^{n+1}_{j} \\!-\\! a^{n}_{j}) C^{(1)}_{ij} \n\\!+\\! \\tau\\left( \\frac{1}{2} (b^{n+1}_{j} \\!+\\! b^{n}_{j}) C^{(1)}_{ij} \\right. \\right. \\hspace{4ex}\n \\!&\\! \\!&\\! \\\\ \\left. \\left.\n+ \\alpha \\frac{3}{16} (a^{n+1}_{j} \\!+\\! a^{n}_{j})^2 C^{(2)}_{ij} \n+\\beta \\frac{1}{12} (c^{n+1}_{j} \\!+\\! c^{n}_{j}) C^{(2)}_{ij} \\right) \\right]\n \\! \\!& = &\\! \\!0, \\nonumber \\\\\n\\sum_{j=1}^{N} \\left[(a^{n+1}_{j} + a^{n}_{j}) C^{(2)}_{ij} - (b^{n+1}_{j} + b^{n}_{j}) C^{(1)}_{ij} \\right] &\\! \\!= &\\! \\!0, \\nonumber\\\\\n\\sum_{j=1}^{N} \\left[(b^{n+1}_{j} + b^{n}_{j}) C^{(2)}_{ij} - (c^{n+1}_{j} + c^{n}_{j}) C^{(1)}_{ij} \\right] &\\! \\!=\\! \\!&\\! \\!0. \\nonumber \n\\end{eqnarray}\n\nDefine the 3$N$-dimensional vector of expansion coefficients \n\\begin{equation} \\label{vec1}\nX^n= \\left(\\!\\begin{array}{c}A^n\\\\ B^n\\\\ C^n \\end{array}\\!\\right),\n\\end{equation} \nwhere \n\\begin{equation} \\label{vec2}\nA^n= \\left(\\begin{array}{c} a_1^n\\\\ a_2^n\\\\ \\vdots \\\\ a_N^n \\end{array} \\right)\\!,~\nB^n= \\left(\\begin{array}{c} b_1^n\\\\ b_2^n\\\\ \\vdots \\\\ b_N^n \\end{array} \\right)\\!,~\nC^n= \\left(\\begin{array}{c} c_1^n\\\\ c_2^n\\\\ \\vdots \\\\ c_N^n \\end{array} \\right)\\!.\n\\end{equation}\nIn (\\ref{matr2}), $A^{n+1}, B^{n+1}, C^{n+1}$ represent the unknown coefficients and $A^{n}, B^{n}, C^{n}$ the known ones. Note that the system (\\ref{matr2}) is nonlinear. The single nonlinear term is quadratic in unknown coefficients. \nFor the second order equations (\\ref{etaab}) and (\\ref{etaabd}) there are more nonlinear terms.\n\nIn an abbreviated form the set (\\ref{matr2}) can be written as \n\\begin{equation} \\label{mv1}\nF_i(X^{n+1},X^{n}) = 0, \\quad i=1,2,\\ldots,3N.\n\\end{equation}\nSince this equation is nonlinear we can use the Newton method for each time step. That is, we find $ X^{n+1}$ by iterating the equation\n\\begin{equation} \\label{iter}\n(X^{n+1})_{m+1}=(X^{n+1})_{m} + J^{-1}(X^{n+1})_{m} = 0,\n\\end{equation}\nwhere $J^{-1}$ is the inverse of the Jacobian of $F(X^{n+1},X^{n})$ (\\ref{mv1}). Choosing $(X^{n+1})_{0} =X^{n}$ we\nobtain the approximate solution to (\\ref{mv1}), $(X^{n+1})_m$ in $m=3-5$ iterations with very good precision.\nThe Jacobian itself is a particular $(3N\\times 3N)$ sparse matrix with the following block structure\n\\begin{equation} \\label{J1}\nJ= \\left( \\begin{array}{ccc} (A_a) & (A_b) & (A_c) \\\\ (C2) & -(C1) & (0) \\\\ (0) & (C2) & -(C1) \\end{array} \\right),\n\\end{equation}\nwhere each block $(\\cdot )$ is a two-diagonal sparse $(N\\times N)$ matrix. The matrix $A_a$ is given by\n\\begin{equation} \\label{Aa}\nA_a\\!=\\! \\left(\\! \\!\\begin{array}{ccccccc} \na^{1}_{1} & 0 & 0 & \\cdots & 0 & a^{1}_{N-1} & a^{1}_{N} \\\\\na^{2}_{1} &a^{2}_{2} & 0 & \\cdots & 0 & 0 & a^{2}_{N} \\\\\n0 & a^{3}_{2} & a^{3}_{3} & 0 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & a^{N-3}_{N-4} & a^{N-3}_{N-3} & 0 & 0 \\\\\n0 & 0 & \\cdots & 0 & a^{n-2}_{N-3} & a^{N-2}_{N-2} & 0 \\\\\na^{N}_{1} & 0 & \\cdots & 0 & 0 & a^{N-1}_{N} & a^{n}_{N} \\!\n\\end{array} \\right).\n\\end{equation}\nIn (\\ref{Aa}) the nonzero elements of $A_a$ are given by \n\\begin{equation} \\label{Aai}\na^i_j=\\frac{\\partial\\, F_i}{\\partial\\, a^{n+1}_j}, \n\\end{equation}\nwhere $ F$ is given by (\\ref{mv1}). The\nelements in the upper right and lower left corners result from periodic boundary conditions.\nMatrices $A_b$ and $A_c$ have the same structure as $A_a$, with only elements $a^i_j$ having to be replaced by $b^i_j=\\frac{\\partial\\, F_i}{\\partial\\, b^{n+1}_j}$ and $c^i_j=\\frac{\\partial\\, F_i}{\\partial\\, c^{n+1}_j}$, respectively.\n\nMatrices $C1$ and $C2$ are constant. They are defined as\n\\begin{equation} \\label{Ck1}\nCk\\!=\\! \\left(\\! \\!\\begin{array}{ccccc} \nC^{(k)}_{11} & 0 & \\cdots & C^{(k)}_{11} & C^{(k)}_{1N} \\\\\nC^{(k)}_{21} & C^{(k)}_{22} & \\cdots & 0 & C^{(k)}_{2N} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & C^{(k)}_{N-1N-1} & 0 \\\\\nC^{(k)}_{N1} & 0 & \\cdots & C^{(k)}_{N-1N} & C^{(k)}_{NN} \\!\n\\end{array} \\right) ,\n\\end{equation}\nwhere ~$k=1,2$.\n\n\\subsubsection{Extended KdV equation (\\ref{etaab})} \\label{2nd}\n\nFor the second order equation (\\ref{etaab}) there are more nonlinear terms. These are terms with $\\alpha^2$ and $\\alpha\\beta$. According to the Petrov--Galerkin scheme we get for the term with $\\alpha^2$\n\\begin{eqnarray} \\label{u3}\n& \\displaystyle \\partial_x & \\left(\\eta^{n+\\frac{1}{2}}\\right)^{3} = \\frac{1}{8} \\left(\n\\partial_x\\sum_{j=1}^{N} \\left(a_{j}^{n+1} + a_{j}^{n}\\right)\\varphi_{j}\\right)^{3} \\nonumber \\\\\n& = & \\frac{1}{8} \\partial_x\\sum_{j=1}^{N} \\sum_{k=1}^{N} \\sum_{l=1}^{N} [a_{j}^{n+1}+a_{j}^{n}][a_{k}^{n+1}+a_{k}^{n}] [a_{l}^{n+1}+a_{l}^{n}] \\nonumber\\\\\n& & \\hspace{20ex} \\times \\varphi_{j}\\varphi_{k}\\varphi_{l} \\\\\n& = & \\frac{1}{8} \\sum_{j=1}^{N} \\sum_{k=1}^{N} \\sum_{l=1}^{N} [a_{j}^{n+1}+a_{j}^{n}][a_{k}^{n+1}+a_{k}^{n}] [a_{l}^{n+1}+a_{l}^{n}] \\nonumber\\\\\n& & \\hspace{10ex} \\times \\left(\\varphi'_{j}\\varphi_{k}\\varphi_{l} + \\varphi_{j}\\varphi'_{k}\\varphi_{l} + \\varphi_{j}\\varphi_{k}\\varphi'_{l} \\right). \\nonumber \n\\end{eqnarray}\nDenote \n\\begin{equation} \\label{C4}\nC^{(4)}_{ijkl}:= \\left(\\left[\\varphi'_{j}\\varphi_{k}\\varphi_{l} + \\varphi_{j}\\varphi'_{k}\\varphi_{l} + \\varphi_{j}\\varphi_{k}\\varphi'_{l} \\right],\\psi_i \\right).\n\\end{equation}\nAs with $C^{(3)}_{ijk}$ in (\\ref{C3}) the following property holds\n\\begin{equation} \\label{C4a}\nC^{(4)}_{ijkl} = C^{2}_{ij}\\,\\delta_{jk}\\,\\delta_{kl} .\n\\end{equation}\n\nIn a similar way, for terms with $\\alpha\\beta$ we obtain\n\\begin{eqnarray} \\label{v2}\n& \\displaystyle \\partial_x & \\left(v^{n+\\frac{1}{2}}\\right)^2 \\\\ \n& = & \\frac{1}{4}\n\\partial_x\\left( \\sum_{j=1}^{N} \\left(b_{j}^{n+1} + b_{j}^{n}\\right)\\varphi_{j} \\sum_{k=1}^{N} \\left(b_{k}^{n+1} + b_{k}^{n}\\right)\\varphi_{k} \\right) \\nonumber \\\\\n& = & \\frac{1}{4}\n\\sum_{j=1}^{N} \\sum_{k=1}^{N} [b_{j}^{n+1}+a_{j}^{n}][b_{k}^{n+1}+b_{k}^{n}] \\left( \\varphi'_{j}\\varphi_{k} + \\varphi_{j}\\varphi'_{k} \\right) \\nonumber \n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{u1u2}\n& \\displaystyle \\partial_x & \\left(\\eta^{n+\\frac{1}{2}}w^{n+\\frac{1}{2}}\\right) \\\\ \n& = & \\frac{1}{4}\n\\partial_x\\left( \\sum_{j=1}^{N} \\left(a_{j}^{n+1} + a_{j}^{n}\\right)\\varphi_{j} \\sum_{k=1}^{N} \\left(a_{k}^{n+1} + a_{k}^{n}\\right)\\varphi_{k} \\right) \\nonumber \\\\\n& = & \\frac{1}{4}\n\\sum_{j=1}^{N} \\sum_{k=1}^{N} [a_{j}^{n+1}+a_{j}^{n}][b_{k}^{n+1}+b_{k}^{n}] \\left( \\varphi'_{j}\\varphi_{k} + \\varphi_{j}\\varphi'_{k} \\right). \\nonumber \n\\end{eqnarray}\nThe scalar products appearing in the terms proportional to $\\alpha^2$ and $\\alpha\\beta$ are already defined: $\\left(\\left( \\varphi'_{j}\\varphi_{k} + \\varphi_{j}\\varphi'_{k} \\right),\\psi_i \\right)=C^{(3)}_{ijk}$.\n\nDue to properties (\\ref{C4a}) and (\\ref{C3}) triple and double sums reduce to single ones.\nWith these settings the second order KdV equation (\\ref{CrABw}) gives the following system of equations \n\\begin{eqnarray} \\label{matr3}\n \\sum_{j=1}^{N} \\left\\{ (a^{n+1}_{j} - a^{n}_{j}) C^{(1)}_{ij} \n+ \\tau \\left[ \\frac{1}{2} (b^{n+1}_{j} + b^{n}_{j}) C^{(1)}_{ij} \\right. \\right.\n\\hspace{2ex}\n & & \\\\\n+ \\left( \\alpha \\frac{3}{16} (a^{n+1}_{j} + a^{n}_{j})^2 \n+ \\beta \\frac{1}{12} (c^{n+1}_{j} + c^{n}_{j}) \\right. & & \\nonumber \\\\\n- \\alpha^2 \\frac{1}{64} \n(a^{n+1}_{j} + a^{n}_{j})^3 \n+ \\alpha\\beta \\frac{13}{192} (b^{n+1}_{j} + b^{n}_{j})^2 & &\\nonumber \\\\\n+\\alpha\\beta \\frac{5}{96} (a^{n+1}_{j} + a^{n}_{j}) (c^{n+1}_{j} + c^{n}_{j}) & &\\nonumber \\\\ \\left. \\left. \\left.\n+ \\beta^2 \\frac{19}{720}(e^{n+1}_{j} + e^{n}_{j}) \\right) C^{(2)}_{ij} \\right]\\right\\}\n&= & 0, \\nonumber \\\\\n\\sum_{j=1}^{N} \\left[(a^{n+1}_{j} + a^{n}_{j}) C^{(2)}_{ij} - (b^{n+1}_{j} + b^{n}_{j}) C^{(1)}_{ij} \\right] &=&0, \\nonumber\\\\\n\\sum_{j=1}^{N} \\left[(b^{n+1}_{j} + b^{n}_{j}) C^{(2)}_{ij} - (c^{n+1}_{j} + c^{n}_{j}) C^{(1)}_{ij} \\right] &=&0, \\nonumber \\\\\n\\sum_{j=1}^{N} \\left[(c^{n+1}_{j} + c^{n}_{j}) C^{(2)}_{ij} - (d^{n+1}_{j} + d^{n}_{j}) C^{(1)}_{ij} \\right] &=&0, \\nonumber\\\\\n\\sum_{j=1}^{N} \\left[(b^{n+1}_{j} + b^{n}_{j}) C^{(2)}_{ij} - (e^{n+1}_{j} \\!+\\! e^{n}_{j}) C^{(1)}_{ij} \\right] &=&0, \\nonumber \n\\end{eqnarray}\nwhere $i=1,2,\\ldots,N$.\n\nIn this case the vector of expansion coefficients $X^n$ is $5N$-dimensional \n\\begin{equation} \\label{vec3}\nX^n= \\left(\\!\\begin{array}{c}A^n\\\\ B^n\\\\ C^n \\\\ D^n\\\\ E^n\\end{array}\\!\\right),\n\\end{equation} \nwhere $A^n$, $B^n$ and $C^n$ are already defined in (\\ref{vec2}) and \n\\begin{equation} \\label{vec4}\nD^n= \\left(\\begin{array}{c} d_1^n\\\\ d_2^n\\\\ \\vdots \\\\ d_N^n \\end{array} \\right)\\!,~\nE^n= \\left(\\begin{array}{c} e_1^n\\\\ e_2^n\\\\ \\vdots \\\\ e_N^n \\end{array} \\right)\\!.\n\\end{equation}\nThe Jacobian becomes now $(5N\\times 5N)$ dimensional. Its structure, however, is similar to (\\ref{J1}), that is\n\\begin{equation} \\label{J2}\nJ= \\left( \\begin{array}{ccccc} (A_a) & (A_b) & (A_c) & (0) & (A_e) \\\\ (C2) & -(C1) & (0) & (0) & (0) \\\\ (0) & (C2) & -(C1) & (0) & (0) \\\\\n(0) & (0) &(C2) & -(C1) & (0) \\\\ (0) & (0) & (0) &(C2) & -(C1) \n\\\\\\end{array} \\right),\n\\end{equation}\nwhere the matrices $(A_a), (A_b), (A_c)$ are defined as previously and $(A_e)_{ij}=\n\\frac{\\partial F_i}{\\partial e^{n+1}_j}$. \nNow $F_i$ represents\nthe set (\\ref{matr3}) which contains four nonlinear terms.\n\n\n\\subsubsection{Extended KdV equation with uneven bottom} \\label{2ndb}\n\nFor the extended KdV with non-flat bottom we have to include into (\\ref{matr3}) three additional terms contained in the last line of formula (\\ref{etaabd}).\nExpanding the bottom function ${h(x)}$ and its second derivative $h_{2x}(x)$ in the basis $\\{\\varphi_j(x)\\}$\n\\begin{equation} \\label{h02}\nh(x)=\\sum_{j=1}^{N} H0_j \\varphi_j(x), \\quad h_{2x}(x)=\\sum_{j=1}^{N} H2_j \\varphi_j(x)\n\\end{equation}\nwe can write the terms mentioned above \nin the following form\n\\begin{eqnarray} \\label{heta}\n \\!&\\! \\! \\partial_x \\!&\\! \\left(h\\eta^{n+\\frac{1}{2}} \\right) \\\\\n \\!&\\! \\! = \\!&\\! \\frac{1}{2}\n\\sum_{j=1}^{N} \\sum_{k=1}^{N} H0_j\\,\\left(a_{k}^{n+1}+a_{k}^{n}\\right) \\left( \\varphi'_{j}\\varphi_{k} + \\varphi_{j}\\varphi'_{k} \\right), \\nonumber \\\\\n \\!&\\! \\! \\partial_x \\!&\\! \\left(h_{2x}\\eta^{n+\\frac{1}{2}} \\right) \\label{h2eta}\\\\ \n \\!&\\! \\! = \\!&\\! \\frac{1}{2}\n\\sum_{j=1}^{N} \\sum_{k=1}^{N} H2_j\\,\\left(a_{k}^{n+1}+a_{k}^{n}\\right) \\left( \\varphi'_{j}\\varphi_{k} + \\varphi_{j}\\varphi'_{k} \\right). \\nonumber \\\\\n \\!&\\! \\! \\partial_x \\!&\\! \\left(h\\eta_{2x}^{n+\\frac{1}{2}} \\right) \\label{heta2}\\\\ \n \\!&\\! \\! = \\!&\\! \\frac{1}{2}\n\\sum_{j=1}^{N} \\sum_{k=1}^{N} H0_j\\,\\left(c_{k}^{n+1}+c_{k}^{n}\\right) \\left( \\varphi'_{j}\\varphi_{k} + \\varphi_{j}\\varphi'_{k} \\right). \\nonumber \n\\end{eqnarray}\nSince \n$$\\left( \\left(\\varphi'_{j}\\varphi_{k} + \\varphi_{j}\\varphi'_{k} \\right),\\psi_i \\right)=C^{(3)}(i,j,k)=C^{(2)}(i,j)\\,\\delta_{jk},$$\nterms proportional to $\\beta\\delta$ can be reduced to single sums like those proportional to $\\alpha^2, \\alpha\\beta$ and $\\beta^2$ discussed in previous subsections. Finally\none obtains (\\ref{etaabd}) as a system of coupled nonlinear equations ($i=1,2,\\ldots,N$)\n\\begin{eqnarray} \\label{matr4}\n \\sum_{j=1}^{N} \\left\\{ (a^{n+1}_{j} - a^{n}_{j}) C^{(1)}_{ij} \n+ \\tau \\left[ \\frac{1}{2} (b^{n+1}_{j} + b^{n}_{j}) C^{(1)}_{ij} \\right. \\right.\n\\hspace{3ex}\n & & \\\\\n+ \\left( \\alpha \\frac{3}{16} (a^{n+1}_{j} + a^{n}_{j})^2 \n+ \\beta \\frac{1}{12} (c^{n+1}_{j} + c^{n}_{j}) \\right. & & \\nonumber \\\\\n- \\alpha^2 \\frac{1}{64} \n(a^{n+1}_{j} + a^{n}_{j})^3 \n+ \\alpha\\beta \\! \\left(\\! \\frac{13}{192} (b^{n+1}_{j} + b^{n}_{j})^2 \\right. & &\\nonumber \\\\ \\left.\n+ \\frac{5}{96} (a^{n+1}_{j} \\!+\\! a^{n}_{j}) (c^{n+1}_{j} \\!+\\! c^{n}_{j}) \\!\\right)\\!\n+ \\beta^2 \\frac{19}{720}(e^{n+1}_{j} + e^{n}_{j}) & &\\nonumber \\\\\n-\\frac{1}{4} \\delta H0_j\\,\\left(a_{k}^{n+1}+a_{k}^{n}\\right) +\\frac{1}{8} \\beta\\delta H2_j\\,\\left(a_{k}^{n+1}+a_{k}^{n}\\right) & &\\nonumber \\\\\n\\left. \\left. \\left. -\\frac{1}{8} \\beta\\delta H0_j\\,(c^{n+1}_{j} + c^{n}_{j})\n\\right) C^{(2)}_{ij} \\right] \\! \\right\\}\n&= & 0, \\nonumber \\\\\n\\sum_{j=1}^{N} \\left[(a^{n+1}_{j} + a^{n}_{j}) C^{(2)}_{ij} - (b^{n+1}_{j} + b^{n}_{j}) C^{(1)}_{ij} \\right] &=&0, \\nonumber\\\\\n\\sum_{j=1}^{N} \\left[(b^{n+1}_{j} + b^{n}_{j}) C^{(2)}_{ij} - (c^{n+1}_{j} + c^{n}_{j}) C^{(1)}_{ij} \\right] &=&0, \\nonumber \\\\\n\\sum_{j=1}^{N} \\left[(c^{n+1}_{j} + c^{n}_{j}) C^{(2)}_{ij} - (d^{n+1}_{j} + d^{n}_{j}) C^{(1)}_{ij} \\right] &=&0, \\nonumber\\\\\n\\sum_{j=1}^{N} \\left[(b^{n+1}_{j} + b^{n}_{j}) C^{(2)}_{ij} - (e^{n+1}_{j} \\!+\\! e^{n}_{j}) C^{(1)}_{ij} \\right] &=&0. \\nonumber \n\\end{eqnarray}\nIn this case the structures of the vector $X^n$ and all matrices remain the same as in (\\ref{vec3})--(\\ref{J2}). However the matrix elements in matrices $A_a$ and $A_c$ are now different to those in the previous subsection \\ref{2nd}, due to new terms in (\\ref{matr4})\n \n\\begin{figure}[tbh] \n\\resizebox{1.01\\columnwidth}{!}{\\includegraphics{2ndOrder_nofloor_nostoch.eps}}\n\\vspace{-5mm}\n\\caption{Time evolution of the initial KdV soliton according to the extended KdV equation (\\ref{etaab}). Profiles are obtained by numerical solution of the set of equations (\\ref{matr3}). Dashed lines represent the undisturbed fluid surface.} \n \\label{plaskie}\n\\end{figure} \n\n\n\\section{Numerical simulations} \\label{nsym}\n\nIt was demonstrated in \\cite{DebP} that the method described in the previous section works reasonably well for the KdV equation (\\ref{kdvm1}).\nOur aim was to apply the finite element method in order to numerically solve the second order equations with a flat bottom (\\ref{etaab}) and with an uneven bottom (\\ref{etaabd}). There exist two kinds of solutions to KdV equations: soliton (in general, multi-soliton) solutions and periodic solutions called cnoidal waves, see, e.g.\\ \\cite{Whit,Ding}. In subsections \\ref{nsyme} and \\ref{unb} we present some examples of numerical simulations for soliton solutions, whereas in the subsection \\ref{cno}, some examples for cnoidal solutions.\n\n\\subsection{Extended KdV equation (\\ref{etaab})} \\label{nsyme}\n\n In Fig.~1 we present several steps of the time evolution of the soliton wave (at $t=0$ it is the KdV soliton) according the the extended KdV equation (\\ref{etaab}) and numerical scheme (\\ref{matr3}).\nThe mesh size is $N=720$, with a time step $\\tau=\\chi^2$,\nand parameters $\\alpha=\\beta=0.1$. Plotted are the calculated profiles of the wave $\\eta(x,t_k)$ where $t_k=5\\cdot k$, \\linebreak$k=0,1,...,10$. \nIn order to avoid overlaps of profiles at different time instants each subsequent profile is shifted up by 0.15 with respect to the previous one. \nThis convention is used in Figs. \\ref{gauss} and \\ref{2gauss}, as well. Here and in the next figures the dashed lines represent the undisturbed fluid surface.\nAs the initial condition we chose the standard KdV soliton centered at $x_0=18$. That is, in the applied units, \n$\\eta(x,t=0) = \\textrm{sech}^2\\left[\\frac{\\sqrt{3}}{2}(x-x_0)\\right]$. Note, that since we use scaled variables and definition (\\ref{smallp}) the amplitude of the soliton is equal~1.\nIn Figs. \\ref{gauss}-\\ref{well} we use the same initial conditions.\n\nThe soliton motion shown in Fig.~1 is in\nagreement with the numerical results obtained with the finite difference method in \\cite{KRR,KRI}. With parameters $\\alpha=\\beta=0.1$ the resulting distortion of the KdV soliton due to second order terms in (\\ref{etaab}), (\\ref{matr3}) is \nin the form of a small amplitude wavetrain created behind the main wave.\n\n\\subsection{Uneven bottom \n} \\label{unb} \n\nWe may question whether the FEM numerical approach to the extended KdV (\\ref{matr4}) is precise enough to reveal the details of soliton distortion caused by a varying bottom. The examples plotted in Figs.~\\ref{gauss}-\\ref{well} show that it is indeed the case. \nIn all the presented calculations the amplitude of the bottom variations is $\\delta\\!=\\!0.2$.\nThe bottom profile is plotted as a black line below zero on a different scale than the wave profile.\n\n\\begin{figure}[bht]\n\\resizebox{1.01\\columnwidth}{!}{\\includegraphics{gauss_floor_nostoch.eps}}\n\\vspace{-5mm}\n\\caption{Time evolution of the initial KdV soliton governed by the extended KdV equation (\\ref{etaabd}) when the bottom has one hump. Here and in the following figures the dotted line shows the position of (the) undisturbed bottom.} \n \\label{gauss}\n\\end{figure}\n\nIn Fig.~\\ref{gauss} the motion of the KdV soliton over a wide bottom hump of Gaussian shape is presented. Here, the bottom function is $h(x)= \\delta \\exp(-(\\frac{x-36}{7})^2)$.\nIn the scaled variables the undisturbed surface of the water (dashed lines) is at $y=0$.\nThe soliton profiles shown in Fig.~\\ref{gauss} are almost the same as the profiles obtained with the finite differences method (FDM) used in \\cite{KRR,KRI}. \nThere are small differences due to smaller precision of our FEM calculations. The FEM allows for the use of larger time steps then FDM. However, in the FEM the computing time grows rapidly with the increase in the number $N$ of the mesh, since calculation of the inverse of the Jacobian $(5N\\times 5N)$ matrices becomes time consuming. \\vspace{1mm}\n\\begin{figure}[tbh]\n\\resizebox{1.01\\columnwidth}{!}{\\includegraphics{double_gauss_floor.eps}}\n\\vspace{-5mm}\n\\caption{Time evolution of the initial KdV soliton governed by the extended KdV equation (\\ref{etaabd}) when the bottom has two narrow humps.} \n \\label{2gauss}\n\\end{figure}\n\nFig.~\\ref{2gauss} displays the motion of the KdV soliton above a double humped Gaussian shaped bottom defined by $h(x)=\\delta[\\exp(-(\\frac{x-30}{6\\sqrt{2}})^2)+\\exp(-(\\frac{x-48}{6\\sqrt{2}})^2)$. \nHere both Gaussians are rather narrow and therefore distortions of the wave shape from the ideal soliton are smaller than those in Fig.~2. \\vspace{0.5ex}\n\n\\begin{figure}[bht]\n\\resizebox{1.01\\columnwidth}{!}{\\includegraphics{dno_ujemne.eps}}\n\\vspace{-5mm}\n\\caption{Time evolution of the initial KdV soliton governed by the extended KdV equation (\\ref{etaabd}) when the bottom has a well.} \n \\label{well}\n\\end{figure}\nIn Fig.~\\ref{well} we see the influence of a bottom well with horizontal size extending the soliton's wavelength. The bottom function is chosen as $h(x)=1-\\frac{\\delta}{2}[\\textrm{tanh}(x-28)+\\textrm{tanh}(44-x)]$ symmetric with respect to the center of the $x$ interval. \nFig.\\ \\ref{well} shows that during the motion above smooth obstacles two effects appear. First, some additional 'waves' of small amplitude, but moving faster than the main solitary wave appear. Second, a wave of smaller amplitude and smaller velocity appears behind the main wave. Both these properties were observed and described in detail in our previous paper \\cite{KRI}.\n\n\n\\subsection{Motion of cnoidal waves} \\label{cno} \n\nThe cnoidal solutions to KdV equation are expressed by the Jacobi elliptic {\\sf cn$^2$} function. \nThe explicit formula for cnoidal solutions is,\nsee, e.g.,\\ \\cite{Ding}:\n\\begin{equation} \\label{cnsol}\n\\eta(x,t) = \\eta_2 + H \\textrm{cn}^2\\left(\\left. \\frac{x-ct}{\\Delta} \\right\\vert m \\right),\n\\end{equation}\nwhere \n\\begin{equation} \\label{cnsol1}\n\\eta_2=\\frac{H}{m}\\left(1-m-\\frac{E(m)}{K(m)} \\right),\\quad \\Delta = h\\sqrt{\\frac{4 m h}{3 H}},\n\\end{equation}\nand\n\\begin{equation} \\label{cnsol2}\nc=\\sqrt{gh}\\left[1+\\frac{H}{mh} \\left(1-\\frac{m}{2} - \\frac{3 E(m)}{2 K(m)}\\right) \\right].\n\\end{equation}\nThe solution (\\ref{cnsol})-(\\ref{cnsol2}) is written in dimensional quantities, where $H$ is the wave height, $h$ is mean water depth, $g$ is the gravitational acceleration and $m$ is an elliptic parameter. $K(m)$ and $E(m)$ are complete elliptic integrals of the first kind and the second kind, respectively. The value of $m\\in [0,1]$ governs the shape of the wave. \n \nFor $m\\to 0$ the cnoidal solution converges to a cosine function.\nFor $m\\to 1$ the cnoidal wave forms peaked crests and flat troughs, such that for $m=1$ the distance between crests increases to infinity and the cnoidal wave converges to a soliton solution. \n\\vspace{0.5ex}\n\nFor\n(\\ref{etaabd}) and ({\\ref{etaab}}) we have to express the formulas (\\ref{cnsol})-(\\ref{cnsol2}) in dimensionless variables.\n\n\\begin{figure}[bht]\n\\resizebox{1.01\\columnwidth}{!}{\\includegraphics{soliton_cnoidalny_bez_dna.eps}}\n\\vspace{-5mm}\n\\caption{Time evolution of the initial KdV cnoidal wave governed by the extended KdV equation (\\ref{etaab}) and numerical scheme (\\ref{matr3}).} \n \\label{cn1}\n\\end{figure}\n\n\\begin{figure}[bht]\n\\resizebox{1.01\\columnwidth}{!}{\\includegraphics{soliton_cnoidalny_z_dnem.eps}}\n\\vspace{-5mm}\n\\caption{Time evolution of the initial KdV cnoidal wave governed by the extended KdV equation (\\ref{etaabd}). The bottom function is here $h(x)=\\frac{1}{2}[-\\textrm{tanh}(2(x-8.6)-\\frac{1}{2})+\\textrm{tanh}(2(x-66.5552)-\\frac{1}{2})]$.} \n \\label{cn2}\n\\end{figure}\n \n\\begin{figure}[tbh]\n\\resizebox{1.01\\columnwidth}{!}{\\includegraphics{podwojny_soliton_cnoidalny_z_dnem.eps}}\n\\vspace{-5mm}\n\\caption{Time evolution of the initial KdV cnoidal wave governed by the extended KdV equation (\\ref{etaabd}). The bottom function is here $h(x)=\\frac{1}{2}[-\\textrm{tanh}(2(x-13.3)-\\frac{1}{2})+\\textrm{tanh}(2(x-67)-\\frac{1}{2})]$.} \n \\label{cn3}\n\\end{figure}\n\nFig.\\ \\ref{cn1} shows the time evolution of the cnoidal wave according to the extended KdV equation (\\ref{etaab}), that is, the second order KdV equation with flat bottom. The parameters of the simulation are: $\\alpha=\\beta=0.14$,~$m\\!=\\!1\\!-\\!10^{-16}$. With this value of $m$ the wavelength of the cnoidal wave is equal to $d\\approx 75.1552$\ndimensionless units, and calculations were performed on the interval of that length, $x\\in[0,75.1552]$ with a periodic boundary condition. The mesh size was taken as $N=752$. \nThe initial position of the wave peak was chosen at the center of chosen interval, that is $x_0=37.5776$. The explicit form of the initial condition in this case was $\\eta(x,t=0)= -0.0189862 + 0.368486 \\,\\textrm{cn}^2\\left(\\left. \\frac{x-x_0}{1.90221} \\right\\vert m \\right) $. Profiles of the wave are plotted at time instants $t_k=10\\cdot k$, where $k=0,1,...,8$.\nSince the amplitudes of cnoidal waves are smaller than 1, the vertical shift for the sequential profiles in Figs.\\ \\ref{cn1}-\\ref{cn3} is chosen to be 0.075.\n\nIn Fig.\\ \\ref{cn2} we display the initially cnoidal wave moving over an extended, almost flat hump. In this simulation the value of parameters $\\alpha,\\beta,m$ and $x$ interval are the same as in the previous figure. Since we consider here the motion over an uneven bottom defined by the function $h(x)=\\frac{1}{2}[-\\textrm{tanh}(2(x-8.6)-\\frac{1}{2})+\\textrm{tanh}(2(x-66.5552)-\\frac{1}{2})]$ the evolution was calculated according to equation (\\ref{etaabd}) and numerical scheme (\\ref{matr4}). Profiles of the wave are plotted at time instants $t_k=10\\cdot k$, where $k=0,1,...,8$.\nFig.\\ \\ref{cn2} shows that during the\n wave motion over the obstacle a kind of slower wave with smaller amplitude is created following the main peak.\n\n\nIn Fig.\\ \\ref{cn3} we present the initially cnoidal wave moving over an extended, almost flat hump. In this simulation $m\\!=\\!1\\!-\\!10^{-8}$. The intial condition is given by \n$\\eta(x,t=0)= - 0.0359497 + 0.368486 \\, \\textrm{cn}^2(\\frac{x-x_0}{1.90221}|m)$ with $x_0=20.1571$.\nBecause $m$ is smaller than in the previous cases, the wavelength $d$ of the cnoidal wave is also smaller, $d\\approx 40.3241$.\nCalculations were made on the interval $x\\in [0,2d] $ with $N=807$. Profiles of the wave are plotted at time instants $t_k=10\\cdot k$, where $k=0,1,...,8$.\nFig.\\ \\ref{cn3} shows qualitatively similar features to those in Fig.\\ \\ref{cn2}.\n\n\n\\begin{figure}[bht]\n\\resizebox{1.0\\columnwidth}{!}{\\includegraphics{rms.eps}}\n\\vspace{-5mm}\n\\caption{Precision of numerical calculations for KdV equation in the fixed frame as a function of mesh size.} \n \\label{ff8}\n\\end{figure}\n\n\\subsection{Precision of numerical calculations}\n\n\nThe KdV equation (\\ref{kdvm}) or (\\ref{kdvm1}) is unique since it possesses an infinite number of invariants, see, e.g.,\\ \\cite{MGK,DrJ}. \nThe lowest invariant, \\linebreak\n$I_1\\!=\\!\\int_{-\\infty}^{+\\infty} \\eta dx$, \nrepresents the conservation law for the mass (volume) of the liquid. The second, $I_2\\!=\\!\\int_{-\\infty}^{+\\infty} \\eta^2 dx$, is related to\n momentum conservation, and the third, \\linebreak\n$I_3\\!=\\!\\int_{-\\infty}^{+\\infty} (\\eta^3-\\frac{1}{3}\\eta_x^2) dx$, is related to energy conservation.\nHowever, as pointed by \\cite{AbSe,AlKa,KRI2}, the relations between $I_2$ and momentum and $I_3$ and energy are more complex.\n\n \nApproximate conservation of these invariants serves often as a test\nof the precision of numerical simulations.\nHowever, this is not the case for the second order KdV type equations (\\ref{etaabd}) and (\\ref{etaab}). It was noted in \\cite{KRI2} that $I_1$ is an\ninvariant of equations \n(\\ref{etaabd}) and (\\ref{etaab}) but $I_2$ and $I_3$ are not invariants. Therefore, only $I_1$ can be used as a test for the precision of numerical calculations of waves moving according to the second order extended KdV equations. \nIn all the presented calculations the precision of the numerical values of $I_1$ was consistently high (the values \n\\frac{I_1(t)-I_1(0)}{I_1(0)}\\le 10^{-6}$). \n\n\nWave motion according to KdV and extended (second order) KdV equations is usually\ncalculated in the reference frame moving with the natural velocity $c=1$ in scaled dimensionless variables (in original variables $c=\\sqrt{gh}$). The KdV and extended KdV equations for \na moving reference frame are obtained by the transformation $\\hat{x}=(x-t),~~ \\hat{t}=t$ which removes the term $\\eta_x$ from the equation (\\ref{etaab}). Then the soliton velocity in the fixed frame is proportional to $1+\\frac{\\alpha}{2}$ whereas in the moving frame it is proportional to $\\frac{\\alpha}{2}$. Therefore, for value of $\\alpha=0.1$ the distance covered by a soliton in the moving frame is $\\frac{\\alpha}{2}\/(1+\\frac{\\alpha}{2})=\\frac{1}{21}$ times shorter than the distance covered in the fixed frame for the same duration. Then, with the same number of the mesh points $N$ the mesh size $\\chi$ can be more than 20 times smaller assuring a much higher precision of calculation in the moving frame at the same operational cost.\nFor instance \\cite{DebP} obtained a good precision for motion of KdV soliton with the FEM method using $N=200$, $\\chi=0.01$ and time step $\\tau=\\chi$ on the interval $x\\in[0,2]$.\n\nPrecision of FEM method in the fixed frame can be tested by calculation of a root mean square (RMS) of deviations of wave profile obtained numerically from those obtained from the analytic solution. Denote by \n$\\eta_i^{anal}(t)$ and $\\eta_i^{num}(t)$ the values of the solutions at given mesh point $i$ an time instant $t$, analytic and numerical, respectively. Then the RMS is expressed as \n\\begin{equation}\\label{var}\n\\textrm{RMS}(\\chi,t) = \\left(\\frac{1}{N}\\sum_{i=1}^{N} (\\eta_i^{anal}(t)-\\eta_i^{num}(t))^2\\right)^{1\/2}\n\\end{equation}\n\n We checked our implementation of the FEM on the interval $x\\in[0,20]$ using several different sizes $\\chi$ of the mesh and several time values. \nFig.\\ \\ref{ff8} displays the RMS (\\ref{var}) values for $t=10$. It shows that deviations from\nanalytic solution decrease substantialy with decreasing~$\\chi$. Small $\\chi$ assures a very high precision in numerical simulations, however, at the expense of large computation time.\nAnother tests (not shown here) in which $\\chi$ was fixed and RMS was calculated as a function of time showed that for $\\tau=\\chi^2$ RMS increases with time linerly and very slowly.\n \nWhen the bottom is not flat simulations {\\em have to be done in the fixed reference frame}. For our purposes we needed to choose the $x$ intervals of the order of 70 or 80. Even for $\\chi=0.1$ the size of Jacobian matrices (\\ref{J2}) reaches (4000$\\times$4000) and its inversion is time consuming. In a compromise between numerical precision and reasonable computing times we made our simulations with $\\chi=0.1$. This choice resulted in about one week of computing time for a single run on the cluster. \nIn spite of the insufficient precision the results presented in Figs. 1-7 reproduce details of evolution known from our previous studies, obtained with the finite difference method. These details, resulting from second order terms in extended KdV (\\ref{etaab}), are seen in Fig.~\\ref{plaskie} as a wavetrain of small amplitude created behind the main one (compare with Fig.~2 in \\cite{KRI}). A similar wavetrain behind the main one was observed in numerical simulations by \\cite{MS96}, see e.g.\\ Fig.~2 therein.\nFor waves moving with presence of bottom obstacle these secondary waves behind the main one are amplified by interaction with the bottom and new faster secondary waves appear (see, \ne.g., Figs.\\ 2-4). These effects were already observed by us, see Figs.\\ 6 and 7 in \\cite{KRI}.\n\n\n\\vspace{3mm}\n\\noindent {\\bf Conclusions}\\\\[2mm]\nThe main conclusions of our study can be summarized as follows.\n\\begin{itemize}\n\n\\item A weak formulation of the finite element method (FEM) for extended KdV equation \n(\\ref{etaab}) can be effectively used for numerical calculations of the time evolution of both soliton and cnoidal waves when calculations are done in a moving frame.\n\n\\item Since numerical calculations for equation (\\ref{etaabd}) have to be performed in a fixed frame, the presented FEM method is not as effective as the FDM method used by us in previous papers because the computer time necessary for obtaining sufficiently high precision becomes impractical. On the other hand, the presented results (though not as precise as FDM ones) exhibit all secondary structures generated by higher order terms of the equations.\n\n\\item First tests of numerical solutions to second order KdV type equations with a stochastic term seem to be very promising \\cite{KRSB2}. \n\n\\end{itemize}\n\nThe authors would like to thank anonymous referees for several helpful suggestions and remarks that affected the article content.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nGiven the increasing complexity and size of software projects, software development is a team effort rather than a one-person activity~\\cite{kraut1995coordination}. Working in a team requires social interactions and adequate communication, as the success of a software project depends on the quality of the collaboration. Research has shown that happy developers are more productive and solve problems better than dissatisfied ones~\\cite{graziotin2014happy,graziotin2015you}. Hence, project leaders and managers are interested in being aware of the temporary emotional shade in the team, which we refer to as mood. One method to gain an overview of the team mood on-the-fly is known as \\textit{sentiment analysis} which analyzes communication with respect to the transported polarity or other sentiments~\\cite{bakshi2016opinion}. So far, sentiment analysis has been frequently applied in software engineering, as a recent systematic literature study shows~\\cite{obaidi2021development}. Sentiment analysis is applied to a wide variety of data sources, including JIRA, GitHub, and Stack Overflow~\\cite{obaidi2021development}. However, all these data sources provide textual communication such as text-messages, comments, tickets, and the like. \n\nDespite the increasing amount of decentralized software development~\\cite{kuhrmann2018helena}, a lot of communication takes part in meetings. Based on the results of a fine-grained interaction analysis applied to 32 student software project meetings, Schneider et al.~\\cite{schneider2018positive} show the relevance of specific types of statements enabling to increase the positive team mood after the meeting. There are other attempts to analyze interactions in meetings, including a coding scheme adjusted for software projects~\\cite{prenner2018making,klunder2020you}, but these methods still require manual effort leading to subjective results. \n\nIn a first attempt to allow for meeting analysis in real-time providing objective results, we want to \\textit{apply sentiment analysis to verbal communication in meetings}. In this paper, as a first step, we present an approach that processes verbal communication to prepare a transcript that can be used as input for different sentiment analysis tools. We base our work on the sentiment analysis tool presented by Kl\u00fcnder et al.~\\cite{kluender2020identifying} and adjust it to be suitable for verbal communication. The preliminary application of the tool, which we refer to as the \\textit{SEnti-Analyzer}, to a student software project meeting provides two promising results:\n\n\\begin{itemize}\n \\item[(1)] Sentiment analysis can be applied to meetings, and\n \\item[(2)] The application of sentiment analysis to verbal communication is as meaningful as the application to textual communication.\n\\end{itemize}\n\nFurthermore, our exemplary application in the case study reveals interesting and relevant aspects of future work.\n\n\\textit{Context.} This paper is based on Herrmann's bachelor thesis~\\cite{herrmann2021automatic} entitled ``Automatic Classification of Statements in Meetings of Development Teams''. \n\n\\textit{Outline.} The rest of the paper is structured as follows: In Section~\\ref{sec:background}, we highlight related research and background details. Section~\\ref{sec:research} introduces the concept and its application in the case study. The results are presented in Section~\\ref{sec:results} and interpreted in Section~\\ref{sec:interpretation}. The paper is summarized in Section~\\ref{sec:conclusions}.\n\n\\section{Background and Related Work}\n\\label{sec:background}\nMeeting analysis and sentiment analysis have both been frequently applied to software projects. For example, Kl\u00fcnder et al.~\\cite{klunder2020you} elaborate the coding scheme \\textit{act4teams-SHORT}, which they derived from an established interaction analysis scheme in psychology. Using \\textit{act4teams-SHORT}, statements in a meeting can be categorized in one of eleven different categories such as ``naming problems'' or ``giving information''. According to the results of Kl\u00fcnder et al. \\cite{klunder2020you}, using this coding scheme and analyzing the resulting interactions in each category help identifying possible problematic behavior. Resolving this kind of behavior at early stages of a software project can lead to better overall team performance and project success \\cite{klunder2020you}. This categorization of high level interaction analysis can be traced back to basic low level sentiment analysis which finds more and more applications recently.\n\nThere are numerous different sentiment analysis tools available, even some especially related to software engineering~\\cite{calefato2018senti}. However, tools for languages differing from English are still rare. Kl\u00fcnder et al.~\\cite{kluender2020identifying} developed a classifier for German text messages from group chats of development teams that maps the input data to the polarity of the message, i.e., \\textit{positive}, \\textit{negative}, or \\textit{neutral}. This classifier is based on a trained classification model and defines a key part of the \\textit{SEnti-Analyzer} which we present in this paper. Calefato et al.~\\cite{calefato2018senti} present their tool \\textit{Senti4SD} to provide a sentiment analysis tool trained on the software engineering domain (using data from \\textit{Stack Overflow}), which does not result in the misclassification of the associated terminology. A similar approach is used by Islam et al.~\\cite{islam2017leveraging}, using \\textit{JIRA} issue comments for training their tool named \\textit{SentiStrength-SE}, and Ahmed et al.~\\cite{ahmed2017senticr}, who use code review comments for training their tool \\textit{SentiCR}.\n\nBesides sentiment analysis, our approach is related to previous research in speech recognition. \nAgarwal and Zesch~\\cite{agarwalzesch2019german} present their approach in training a German-language model for the \\textit{Mozilla DeepSpeech} framework, which also constitutes the foundation of our speech recognition. The framework provides a transcript that can be used as input for existing sentiment analysis tools. \n\nSpeech recognition has also been applied to meetings in software engineering with another focus:\nGall and Berenbach~\\cite{gall2006towards} present a framework recording requirements elicitation meetings on video, thereby collecting relevant information raised by stakeholders. Shakeri et al.~\\cite{abad2018elica} also strive to extract relevant information presented in elicitation meetings. Their tool \\textit{ELICA} collects knowledge and information related to requirements. This way, it helps analyzing the meeting outcome. \n\nIn this paper, we combine the approaches of sentiment analysis tools with automatic speech recognition to analyze verbal team communication in real-time. Our tool provides an overview of the distribution of sentiment categories for the recognized statements at the end of the meeting. This way the project manager can easily gain first direct feedback about the course of the meeting.\n\nAlthough both sentiment analysis and meeting analysis have been proven to be beneficial for software projects, to the best of our knowledge sentiment analysis has not yet been used for meeting analysis. \n\n\\section{Study Design}\n\\label{sec:research}\nIn the following, we present our research objective, the research questions, and the study. Our approach basically consists of two steps: (1) the transcription of a meeting and (2) the application of a sentiment analysis tool~\\cite{kluender2020identifying}. \n\n\\subsection{Research Objective and Research Questions}\nThe main objective of our research is to \\textit{analyze the sentiments transported in statements made in a meeting of a software project}. To reach this goal, we developed and evaluated a concept and a corresponding software tool, the so-called \\textit{SEnti-Analyzer}, which uses an audio stream of verbal (meeting) communication as input and predicts the polarity of each statement. We formulated the following research questions:\n\n\\begin{itemize}[leftmargin=.38in]\n \\item[RQ1:] How can automatic speech recognition and sentiment analysis be combined to analyze the statements in meetings of a software project? \n \\item[RQ2:] How do the automatically produced results differ from the subjective analysis of a human observer?\n\\end{itemize}\n\n\\subsection{Instrument Development}\nOur approach for the \\textit{SEnti-Analyzer} is to feed the users microphone input into our software, e.g., the microphone of a laptop placed in the middle of a conference table during the meeting. Note that multiple audio inputs (e.g., as in online conferences) are also possible. \n\nThe sequence of processing steps from the raw audio input to the resulting prediction of sentiment categories is visualized in Figure~\\ref{fig:processing}. After the meeting the user can stop the recording and instantly receives the transcript. This will be generated right during the meeting using the \\textit{Mozilla Deepspeech} framework, alongside with the German language models\\footnote{Note that the focus on German is due to the nature of the bachelor thesis as outlined in the introduction. Future work will focus on extending the approach to English.}. We use state-of-the-art voice activity detection to separate the audio stream into frames of statements. Stopping the recording also starts the application of natural language processing to the transcript. Once completed, the collected statements and corresponding metrics are fed into the sentiment analysis tool provided by Kl\u00fcnder et al.~\\cite{kluender2020identifying} which interprets the results. Finally, an output of classified interactions is presented to the user, e.g., by calculating the total and relative proportions of each category. These steps are fully automated, requiring the user only to start the tool and specifying the end of recording by a single key stroke. The potential for improvement offered by this preliminary tool is outlined in the end of the paper. \n\n\n\\begin{figure}[htbp]\n\\centering\n\n\\pgfdeclarelayer{background}\n\\pgfdeclarelayer{foreground}\n\\pgfsetlayers{background,main,foreground}\n\n\\tikzstyle{materia}=[draw, fill=gray!3, text width=6.0em, text centered, minimum height=1.5em,drop shadow]\n\\tikzstyle{etape} = [materia, text width=14em, minimum width=10em, minimum height=3.5em, rounded corners, drop shadow]\n\\tikzstyle{texto} = [above, text width=6em, text centered]\n\\tikzstyle{linepart} = [draw, thick, color=black!50, -latex', dashed]\n\\tikzstyle{line} = [draw, thick, color=black!80, -latex']\n\\tikzstyle{ur}=[draw, text centered, minimum height=0.01em]\n\n\\newcommand{1.3}{1.3}\n\\newcommand{1.5}{1.5}\n\n\\newcommand{\\etape}[2]{node (p#1) [etape]\n {#2}}\n\n\\newcommand{\\background}[5]{%\n \\begin{pgfonlayer}{background}\n \n \\path (#1.west |- #2.north)+(-0.5,0.25) node (a1) {};\n \n \\path (#3.east |- #4.south)+(+0.5,-0.25) node (a2) {};\n \n \\path[fill=gray!10,rounded corners, draw=black!50, dashed]\n (a1) rectangle (a2);\n \\path (#3.east |- #2.north)+(0,0.25)--(#1.west |- #2.north) node[midway] (#5-n) {};\n \\path (#3.east |- #2.south)+(0,-0.35)--(#1.west |- #2.south) node[midway] (#5-s) {};\n \\path (#3.east |- #2.north)+(0.7,0)--(#3.east |- #4.south) node[midway] (#5-w) {};\n \\end{pgfonlayer}}\n\n\\newcommand{\\transreceptor}[3]{%\n \\path [linepart] (#1.east) -- node [above]\n {\\normalsize #2} (#3);}\n\n\\begin{tikzpicture}[scale=0.7,transform shape]\n\n\\path \\etape{1}{\\large Microphone Access\\\\\\normalsize Or Pre-Recorded Audio File};\n\n\\path (p1.south)+(0.0,-1.5) \\etape{2}{\\large Voice Activity Detection\\\\\\normalsize Splitting Audio Into Statements};\n\\path (p2.south)+(0.0,-1.0) \\etape{3}{\\large Transcribing Statements\\\\\\normalsize Mozilla DeepSpeech};\n\n\\path (p3.south)+(0.0,-1.5) \\etape{4}{\\large Feature Extraction\\\\\\normalsize Natural Language Processing};\n\\path (p4.south)+(0.0,-1.0) \\etape{5}{\\large Model Based Classification\\\\\\normalsize Classification Algorithm};\n\n\\path (p5.south)+(0.0,-1.5) \\etape{6}{\\large Training Data\\\\\\normalsize Labeled Meeting Statements} (p6);\n\n\n\\path [line] (p1.south) -- node [above] {} (p2);\n\\path [line] (p2.south) -- node [above] {} (p3);\n\\path [line] (p3.south) -- node [above] {} (p4);\n\\path [line] (p4.south) -- node [above] {} (p5);\n\\path [line] (p6.north) -- (p5.south) {};\n\n\\background{p2}{p2}{p3}{p3}{bk1}\n\\background{p4}{p4}{p5}{p5}{bk2}\n\n\\path (bk1-w)+(+3.0,0) node (ur1)[] {\\large Audio Processing};\n\\path (bk2-w)+(+3.0,0) node (ur2)[] {\\large Sentiment Analysis};\n\\transreceptor{bk1-w}{}{ur1};\n\\transreceptor{bk2-w}{}{ur2};\n\\end{tikzpicture}\n\n\\caption{Simplified processing pipeline of the \\textit{SEnti-Analyzer}}\n\\label{fig:processing}\n\\end{figure}\n\n\\subsection{The Case Meeting}\n\\label{subsec:meeting}\nTo get a proof of concept for our \\textit{SEnti-Analyzer} we tested the tool on a student software project meeting. The Software Engineering Group at Leibniz University Hannover yearly hosts a student software project for students in their last year of the bachelor in computer science. This way students can gain insight and experience in the professional software development process. Five to ten students work together on a software project, which are mostly applications for real life local customers (such as the \\textit{Hannover Police Department} and the \\textit{Hannover Medical School}). The whole project lasts one semester (approx. 15 weeks) with weekly meetings both team-internal and with the customer(s). The project team that participated in the case study worked on the \\textit{VirtuHoS}-Project (Virtual House of Software), an application to virtually empathize the feeling of working together in an office with a decentralized development team. The team was tasked to create an editor for drawing a virtual office and creating the underlying semantic structure for further use by other groups using the Java programming language. Due to the ongoing Sars-CoV2 pandemic meetings could only be held virtually. For our case study, we recorded a 33-minute online meeting on 13th January 2021 in which all six team members participated. We collected written consent of each team member allowing us to use the recorded audio files for research purposes and for scientific publications. The team participated voluntarily in the study and the participation had no influence on passing the course, on grades, etc. \n\n\\subsection{Data Collection and Pre-Processing}\n\nThe meeting session was recorded digitally. We transcribed and classified the recordings by hand to get reasonable training data for the \\textit{SEnti-Analyzer}. The team members used \\textit{Discord} as their VoIP service and held the meeting together in a group call. A recording bot was used to record the meeting which enabled a multi-track recording separating each team member from another. At the end of the meeting, the team members were asked how they felt about the mood of the meeting (concerning the communication behavior). All team members agreed on the meeting communication being neutral to positive. A second prerecorded meeting from an older iteration of the student software project was also transcribed by hand increase the training set. The complete data set was then fed into the training function of the \\textit{SEnti-Analyzer}, which finds new solutions by hyperparameter search for the included metrics extracted by natural language processing. For this training process, we used 1000 generations in total using an (1 + 1) evolutionary algorithm, thus only introducing one new population per generation and minimizing run-time.\n\n\\subsection{Data Analysis}\nBoth transcripts were split into single statements, which were then manually fitted with training labels to create the training data. A special training script loaded the whole data set into the training function of the \\textit{SEnti-Analyzer} intending to learn a generalizing model. In total, our training data set consists of 712 manually transcribed and labeled statements, which follow the distribution shown in Table~\\ref{table:distributions}. To validate our results, we used Fleiss' $\\kappa$ as a statistical measure.\n\n\\section{Results}\n\\label{sec:results}\n\nBased on the distribution of sentiment classes in the training set, an accuracy of 77.5\\% would be possible by classifying each statement as neutral alone. Our model for the sentiment analysis tool provided by Kl\u00fcnder et al.~\\cite{kluender2020identifying} however reached an accuracy of 81.8\\% over the 712 statements from the training set, indicating a learning curve. The training of the model reached the peak fitness of 81.8\\% around the 800th to 900th generation. Out of a 10-minute audio file from the recorded meeting mentioned in subsection \\ref{subsec:meeting} our tool extracted 140 statements. The distribution of the classified statements is shown in Table~\\ref{table:distributions}.\n\n\n\\begin{table}[htbp]\n\\caption{Distributions of sentiment classes in training and test set}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline\nStatements in & \\textbf{Total} & \\textbf{Positive} & \\textbf{Neutral} & \\textbf{Negative}\\\\\n\\hline\\hline\n\\textbf{Training se\n} & 712 & 77 (10.8\\%) & 552 (77.5\\%) & 83 (11.7\\%)\\\\\n\\hline\n\\textbf{Test se\n} & 140 & 15 (10.7\\%)& 124 (88.6\\%)& 1 (0.7\\%)\\\\\n\\hline\n\\end{tabular}\n\\label{table:distributions}\n\\end{center}\n\\end{table}\n\n\\subsection{Comparing Test Set Results to Training Data}\nThe trained model seemingly performs and generalizes well and the distribution of sentiment classes differs from that of the training data. Especially the relative distribution of the categories \\textit{positive} and \\textit{negative} changed widely from the training data. As Table~\\ref{table:distributions} illustrates, the training set shows a virtually equal distribution of \\textit{positive} and \\textit{negative} statements (both within 11.2\\% $\\pm$ 0.5\\% compared to the total training set size). The classified test set on the other hand shows a divergent distribution of the three sentiment classes, with the relative share of \\textit{negative} statements decreased by 11\\%. The \\textit{neutral} class gains around 11.1\\%, with only the \\textit{positive} class staying at around the same percentage, only decreasing by 0.1\\%. However, the classification of our tool directly corresponds to the feedback received by the team members, who told us they perceived the meeting communication as neutral to positive.\n\n\\subsection{Transcription Quality}\nThe speech recognition system we used showed difficulties when exposed to indistinct pronunciation or the high pace of speech of some of the team members. Usually, only single word errors occurred, but sometimes, when the pace was just too high for the speech engine or something said was obscure the transcript would differ so much that one could no longer deduce the actual statement from it. However, one has to consider that the German speech models we used had a given word error rate (WER) of 12.8\\%. The English models offered directly by Mozilla on the DeepSpeech GitHub page are specified with a much lower WER of 5.97\\% (release 0.8.2). Therefore much better transcription performance for English audio can be expected from the \\textit{SEnti-Analyzer}. \n\n\\subsection{SEnti-Analyzer Compared to Manual Classification}\nTo further verify the quality of our results we manually picked 50 from the 140 statements, which most matched the actual said and classified them again by hand to compare our classifications against the \\textit{SEnti-Analyzer}. We did so according to our perception of the statement and the given context of previous statements, which the \\textit{SEnti-Analyzer} does not yet consider. Table~\\ref{table:results} compares the classifications taken by the \\textit{SEnti-Analyzer} to the manual classifications. \n\n\\begin{table}[htbp]\n\\caption{Comparison between classifications taken by the Senti-Analyzer and a human observer}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|}\n\\hline\nClassification & \\textbf{Positive} & \\textbf{Neutral} & \\textbf{Negative} \\\\\n\\hline\\hline\n\\textbf{Software} & 10 (20\\%) & 39 (78\\%) & 1 (2\\%) \\\\\n\\hline\n\\textbf{Manual} & 5 (10\\%) & 45 (90\\%) & 0 (0\\%) \\\\\n\\hline\n\\end{tabular}\n\\label{table:results}\n\\end{center}\n\\end{table}\n\nRemarkably, the classification of these statements by the \\textit{SEnti-Analyzer} is more scattered than the classification by hand. Using Fleiss' $\\kappa$, we calculated a $P_e$-value of $0.7282$ which means that the probability of a random match is 72.82\\%. This can be traced back to the many matching classifications of both the \\textit{SEnti-Analyzer} and the human observer in the category neutral, which reduces the value of $\\kappa$. The total calculated $\\kappa$-value is 0.56 and considered as an upper-moderate agreement according to the the scale provided by Landis and Koch~\\cite{landis1977measurement}. A $\\kappa$-value of 0.61 would already be considered as substantial agreement. \n\n\n\n\\subsection{Summary}\nRecapitulating, we fitted a model to our training data for the sentiment analysis tool provided by Kl\u00fcnder et al.~\\cite{kluender2020identifying} to use in conjunction with our \\textit{SEnti-Analyzer}. \\textbf{The \\textit{SEnti-Analyzer} automatically classified statements from an audio file in real-time producing results consistent with the feedback of the team members}.\nWe examined our results using the Fleiss' $\\kappa$ measure and obtained an \\textbf{upper-moderate agreement} as result.\n\n\\section{Interpretation}\n\\label{sec:interpretation}\nWe presented a concept which performs sentiment analysis on microphone audio in real time. We tested our concept on an exemplary tool and showed proof of concept on a real meeting from a student software project. This section discusses the findings with respect to the research question and threats to validity. In the end of this section, we point to future work. \n\n\\subsection{Answering the Research Questions}\nThe findings and results we obtained can be used to answer the research questions we formulated at the beginning the following way:\n\n\\begin{itemize}[leftmargin=.38in]\n \\item[RQ1:] We acquired proof of concept for our approach in the exemplary case study, and therefore successfully combined an automatic speech recognition system with our existing sentiment analysis tool~\\cite{kluender2020identifying}. The developed tool is just exemplary and can be improved in many ways. Nevertheless, we are confident that this approach delivers valuable results and will lead to more productive working environments within software project teams in the future. \n \n \\item[RQ2:] According to Fleiss' $\\kappa$, we reached a moderate agreement between the automatically produced results and the manual classification of a human observer. However, this outcome may be the result of numerous matching classifications in the \\textit{neutral} class leading to a high $P_e$-value and thus a lower $\\kappa$-value.\n\\end{itemize}\n\n\\subsection{Discussion}\n\nDespite the preliminary nature of our study, we can observe two remarkable findings:\n\n\\begin{itemize}\n \\item[(1)] Sentiment analysis can be applied to meetings, and\n \\item[(2)] The application of sentiment analysis to verbal communication seems to be as meaningful as the application to textual communication.\n\\end{itemize}\n\nThe first finding appears to be a rather weak and not surprising observation, but offers a lot of potential for future work. Despite the fact that we do not have evidence supporting finding (2), this directly emerges from finding (1) with the fact that sentiment analysis itself has potential to support the collaboration in a team~\\cite{obaidi2021development}. Therefore, it would be meaningful to start exploring the potential and to improve the polarity detection in meetings of software projects in a larger study. \n\nNevertheless, besides the threats to validity which we discuss in the following section, there are four aspects that should be considered in the context of our study: \n\n\n\\subsubsection{High Chance of Random Agreement and Fleiss' $\\kappa$}\nThe concept's applicability has been demonstrated by our exemplary tool, the \\textit{SEnti-Analyzer}, alongside with the application on a real-world student software project meeting. The \\textit{SEnti-Analyzer} provided feedback in line with the team's feedback after the meeting. For the chosen Fleiss' $\\kappa$, a higher $P_e$-value (chance of a random match) leads to a lower $\\kappa$-value. Unfortunately, due to the distribution of data in our test set, a lot of matching classifications in the class neutral occurred, thus leading to a high $P_e$-value of $0.7282$. The overall $\\kappa$-value with 0.56 was, therefore, lower than expected concerning the overall agreement of 88\\% (44 out of 50 total statements were classified identical both by the \\textit{SEnti-Analyzer} and by hand). A more equally distributed test set would lead to a much lower $P_e$-value and thus facilitate a higher $\\kappa$-value, and an even more meaningful result.\n\n\\subsubsection{Difficulty in Labeling Statements and Statement Context}\nKl\u00fcnder et al.~\\cite{kluender2020identifying} already noted how labeling statements represents a difficult task, especially for a single human. Everybody has his\/her own perception of the sentiment of a statement and, thus, two different people may choose a different sentiment class for the same statement. The training and test set which have been manually classified for our research were only labeled by a single person. The size of the training data set was also limited by this factor. Furthermore, the statements were labeled concerning the context of the whole conversation, while the \\textit{SEnti-Analyzer} (at the moment) only evaluates each statement on its own (without taking the context into account). Classifying the training and test sets by multiple persons choosing always the most voted class would reduce the impact of a single person's perception on the training data and results. The implementation of the concept of a statement context for the \\textit{SEnti-Analyzer} is also imaginable.\n\n\\subsubsection{Domain Specificity}\nCurrently, the domain specificity of our tool is only given by the training data (all of the statements were taken from conversations out of software project meetings). For the German language, there are currently only general sentiment lexicons available, and no tools that would enable a domain-specific sentiment analysis. To improve the domain specificity and further tailor the tool to a software engineering context, the integration of a sentiment analysis tool designed specifically for software engineering would be beneficial. For the English language, one such example would be Senti4SD by Calefato et al.~\\cite{calefato2018senti}.\n\n\\subsubsection{Impacts of the Sars-CoV2 Pandemic}\nBecause of the ongoing Sars-CoV2 pandemic, the meetings could only be held virtually which may have influenced our results. Some members lacked in audio quality and some had a notable level of background noise. Delays were also a problematic factor for the communication leading to multiple team members starting to talk at the same time or interrupting each other unintentionally. This would not happen that frequently in a real-world face-to-face meeting and a high-end audio setup in a conference room would provide better input audio quality enabling better results for the transcription of statements. The speech engine DeepSpeech is also constantly being improved, therefore future version upgrades may improve the transcription quality on their own.\n\n\\subsection{Threats to Validity}\nOur case study results are limited to the used sample and cannot be generalized for other meetings. In this section, we summarize the most relevant threats to validity probably impacting our results.\n\nWe applied the \\textit{SEnti-Analyzer} to a single meeting. The training set emerged from the manual labeling of two meetings. This small sample size was caused by the high effort of manually transcribing and labeling the meeting audio, and limits the statistical power of the results. In the same way, the used statistical measure, Fleiss' $\\kappa$, may be unsuitable regarding the high number of matching classifications in the class \\textit{neutral} by both observers (\\textit{SEnti-Analyzer} and classification by hand). This aspect alone had a high influence on the calculated $\\kappa$-value and the resulting strength of agreement. Because of the Sars-CoV2 pandemic, and the resulting curfew, the case study meeting had to be held online through VoIP software. Therefore, participants used their computers to attend the meeting. This influenced the experiment environment due to background noise, sounds from other rooms, noise from outside, and other static or interference noises. Delays over the VoIP also showed to be problematic while talking together, e.g., by cutting each other short unintentionally.\n\nThe student software project team recorded in the case study consisted of bachelor degree students in computer science. The prior knowledge about professional software development varied between team members. While some had already worked in private software corporations alongside their studies, others had programming knowledge only consisting of basic programming courses in computer science required to participate in the student software project. Therefore, the team members with less experience also did not use \\textit{JIRA} or \\textit{GitLab} prior to the software project. These variations may not influence the application of the \\textit{SEnti-Analyzer}, but should be taken into account when interpreting the results. \n\nThe \\textit{SEnti-Analyzer} is currently not capable of differentiating voices, resulting in a transcription that consists merely of a concatenation of all recognized statements, instead of offering dialogue-like structuring. For the transcription to work as intended, it is necessary that only one person talks at a time, or otherwise, the quality of the transcript will be compromised. The overall results are therefore limited by the currently free available transcription technology. Future research needs to focus on these issues. \n\n\n\\subsection{Future Work}\nTo reduce the possible impact of the threats to validity and to increase the reliability of our results, we propose the following steps for future work:\n\n\\begin{itemize}\n \\item[(1)] Adjust the tool to English: As a first step, we want to adjust the tool to be applicable to meetings conducted in English, as both audio speech recognition and sentiment analysis tools provide better results in English than in German. This also helps extending the training set for the tool, as labeled data sets in English are way more frequently available.\n \\item[(2)] Improve the reliability of the results: First and foremost, a (longitudinal) case study and a multi-case study are required to strengthen the results and to evaluate the usefulness of the application for the teams.\n \\item[(3)] Taking facial and the tone expressions into account: As verbal communication is not the only communication used in meetings (rolling eyes or getting loud, e.g., also transport a lot of information), it would be interesting to also consider gestures, facial expressions, etc. \n \\item[(4)] Increase the granularity of the results, e.g., by distinguishing between different categories as proposed by Kl\u00fcnder et al. \\cite{klunder2020affecting}. This helps pointing, for example, to destructive behavior which endangers project success by demotivating team members. \n\\end{itemize}\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nMeetings represent a valuable way to communicate within development teams and are essential for every software project. To make software project meetings more effective and productive, and thus increase the overall mood and satisfaction of the project team, automated interaction analysis can be used. As the first step to our long-term research goal of an automated fine-grained interaction analysis, we introduce an approach combining prior interaction analysis research with the latest open source speech recognition achievements. The \\textit{SEnti-Analyzer} processes meeting audio by cutting the conversation into single statements and transcribing them in real-time, before processing them using natural language processing.\nThe tool returns the classified statements and the overall meeting performance by showing the proportions of the sentiment classes \\textit{positive}, \\textit{negative}, and \\textit{neutral}. This way, the project manager can gain additional informative feedback tracing the course of the meeting with little to no effort. Further actions can be taken due to the given resulting feedback, to improve future meeting behavior and communication. \n\nIn a case study, we applied our tool to a real student software project meeting. The \\textit{SEnti-Analyzer} delivered results that directly corresponded with the feedback the team members gave themselves. Using our results we could also verify moderate agreement of the classifications taken by the \\textit{SEnti-Analyzer} in comparison to a human observer.\n\nOverall, we propose to keep pursuing research on interaction analysis in software development teams using known sentiment analysis methods and machine learning algorithms to further expand the established concepts by integrating other components such as speech or gesture recognition. Automating tools for ease of use is also an important factor to disseminate interaction analysis in software development. \n\n\\section*{Acknowledgment}\nThis research was funded by the Leibniz University Hannover as Leibniz Young Investigator Grant (Project \\textit{ComContA}, 2020--2022).\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Language Models Capture Embodied Commonsense}\n\\label{sec:exp:llm} \n\\input{sections\/lm_table_warp}\n\n\\textbf{Methods.} We evaluate \\texttt{CM} and \\texttt{ZS-MLM} using RoBERTa~\\cite{Liu2019} as our base LLM. We also compare these with GloVe-based~\\cite{pennington2014glove} embeddings, and a baseline that randomly ranks rooms (for \\texttt{OR} task) and receptacles (for \\texttt{ORR} task).\n\n\\noindent \\textbf{Evaluation.} We evaluate mean average precision (mAP) across objects to compare the ranked list of rooms\/receptacles obtained from our ranking module\nto the list of rooms\/receptacles deemed \\texttt{correct} by the human annotators. Recall from \\cref{sec:task:episodes}, for a given object, a receptacle is considered \\texttt{correct} when at least 6 annotators vote for it, and a room is considered \\texttt{correct} if it has at least one \\texttt{correct} receptacle within it. Higher AP score indicates \\texttt{correct} items are likely to ranked higher than the \\texttt{incorrect} items.\n\n\\noindent \\textbf{Results.} \\Cref{tab:orm} shows that \\texttt{RoBERTa+CM} outperforms \\texttt{ZS-MLM} by a large margin even when fintuned on a relatively small-sized training set (\\url{~}40\\% of total data, see Section~\\ref{sec:task:eval}). We find good transfer of results from \\texttt{val} to \\texttt{test} splits by \\texttt{RoBERTa+CM} method on both tasks demonstrating the better generalization capabilities of LLMs. Whereas, \\texttt{GloVe+CM} do not seem to transfer well for the \\texttt{ORR} task. Finally, notice that \\texttt{Random} baseline performs relatively well on room-matching (\\texttt{OR}) task, which is expected since there are ample of rooms with at least one correct receptacle for any given object. \n \n\\csubsection{Main Results for Housekeep\\xspace}\n\\label{sec:exp:main} \nWe utilize the best method from \\Cref{sec:exp:llm}, \\texttt{RoBERTa+CM} as scoring function within \\texttt{Ranker} module to continuously rerank (thus replan) newly discovered rooms and receptacles while exploring Housekeep\\xspace episodes.\n\n\\noindent \\textbf{Oracle Modules.} We show oracle agent's performance, by swappping \\texttt{Ranker} and \\texttt{Explore} modules with their oracle (perfect) counterparts. Oracle ranker uses the ground truth human preferences to rank the objects and receptacles found. Oracle exploration gives a complete map of the environment, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot agent knows all objects, receptacles and their respective locations. \n\n\\input{tables\/rearrangement}\n\n\\noindent \\textbf{Upper Bounds.} In \\Cref{tab:main:unseen}, we show results on both \\texttt{test-seen} and \\texttt{test-unseen} splits. Rows 1, 5 with oracle ranking and exploration denote the upper bounds achievable across all metrics. Note that Soft Object Success (\\texttt{SOS}) and Rearrangement Quality (\\texttt{RQ}) are not perfect since human agreement across correct receptacles is not 100\\%.\n\n\\noindent \\textbf{LLM-based Ranker, Compounding Errors.} Compared to oracle ranker (Row 1) language model (Row 3) impacts object success (\\texttt{OS}) by -56\\%, and episode success (\\texttt{ES}) by -96\\%. The dramatic drop in \\texttt{ES} is expected as Housekeep\\xspace is a multi-step problem with compounding errors between rearrangements. That means, with average 4 rearrangements necessary per episode and with \\texttt{OS} at $46\\%$, \\texttt{ES} will be $0.46^{4} \\approx 0.045$ as seen. We further analyze this in \\Cref{fig:part_succ} showing that \\texttt{ES@K} drops with each successive rearrangement attempt made.\n\n\\input{sections\/part_succ}\n\n\n\\noindent \\textbf{Frontier Exploration, Full baseline.} Using Frontier exploration (rows 1,2), \\texttt{OS} drops by $47\\%$. This drop in performance signifies the importance of task-driven exploration needed for Housekeep\\xspace to find \\texttt{misplaced} objects or \\texttt{correct} receptacles quickly. Finally, we evaluate the fully non-oracle baseline (row 4) which achieves a $30\\%$ object success rate. From rows 4 and 8, we see that \\texttt{OS} drops by $7\\%$, but \\texttt{SOS} drops only by $3\\%$ across seen vs unseen objects which supports our claim from Section~\\ref{sec:exp:llm} that LLMs can indeed serve as a generalizable planning module aligned with human preferences. \n\nWe put additional experiments analyzing the effect of exploration steps (\\texttt{n}$_e$), exploration strategies in \\Cref{sec:supp:add_exps}, and qualitative results in \\Cref{sec:supp:qual_analysis}.\n\n\n\\section{Approach}\n\\label{sec:supp:approach} \n\\subsection{LLM Ranking Module}\nIn Table~\\ref{tab:orm_details}, we provide the hyperparameters that we use to train the \\texttt{OR} and \\texttt{ORR} modules using the contrastive matching (\\texttt{CM}) strategy. \nEach method trained using \\texttt{CM} is trained on a single GPU for 1000 epochs and we choose the training checkpoint that gives the best mAP score (evaluated as in Section~\\ref{sec:exp:llm}) on the validation set. In the case of \\texttt{RoBERTa+CM}, we use the pretrained roberta-base model and average the last-layer hidden state at all positions (including the CLS token) to obtain the text embeddings.\n\\begin{table}[h!]\n \\centering\n \\caption{\\small{Hyperparameter choices for training the \\texttt{CM} modules}}\n \\setlength{\\tabcolsep}{10pt}\n \\begin{tabular}{c l c}\n \\toprule\n\t\t\\textbf{\\#} & \\textbf{Hyperparameter} & \\textbf{Value} \\\\\n\t\t\\midrule\n 1 & Embedding size & 768 (RoBERTa) \/ 300 (GloVe) \\\\\n 2 & MLP hidden dimension & 512 \\\\ \n 3 & MLP out dimension & 512 \\\\\n 4 & MLP hidden layers & 2 \\\\\n 5 & Batch size & 64 \\\\\n 6 & Optimizer & Adam \\\\\n 7 & Learning rate & 0.01 \\\\\n 8 & Weight decay & 0.2 \\\\\n \\bottomrule\n \\end{tabular}\n \\label{tab:orm_details}\n\\end{table}\n\n\\section{Agent}\n\\label{sec:supp:navpp} \n\nWe expand on low-level modules used in the agent for navigation and pick-place. %\n\n\\noindent \\textbf{\\texttt{Navigation (N): }} Indoor navigation between two points (aka PointNav) is a well-studied problem both in embodied AI~\\cite{Wijmans2020DDPPOLN, ZhaoICCV2021, Ye2020AuxiliaryTS} and classical robotics~\\cite{Chan2018Robust2I, 8968455, indoor-robot}. Our navigation module takes as input the allocentric map and a goal position (object, receptacle, or frontier), and executes a sequence of low-level base control actions to reach the goal.\\smallskip\n\n\\noindent \\textbf{\\texttt{Pick-Place (P): }} Recall from Section~\\ref{sssec:task} that to interact with an object, the agent invokes a discrete action that casts a ray, and if it intersects an object or receptacle within 1.5m of the agent, it picks or places the object.\nOur hierarchical baseline picks and places objects via the instance ID of an object or receptacle currently in the view of the agent.\nThe agent then orients itself to face the desired instance ID via look up\/down and turn left\/right actions.\nOnce the desired instance ID is within the agent's view, the agent calls the ray-cast interaction action.\nThe Pick-Place module fails if the agent is unable to view the object\/receptacle of interest or navigate to a place within interaction distance. \nHowever, we ensure all episodes are solvable by an oracle agent, so this does not occur in the episodes on which we run our hierarchical baseline.\nThe Pick-Place module can also fail to place an object on a receptacle if sufficient space is not available on the receptacle.\n\n\n\n\n\n\n\n\\subsection{Episode statistics}\n\\label{ssec:supp:rec_use} \n\n\\begin{figure}[h!]\n \\label{fig:episode_stats}\n \\centering\n \\begin{subfigure}[c]{0.6\\columnwidth}\n \n \\includegraphics[width=\\textwidth]{figures\/misplaced_high_lvl.jpg}\n \\caption{\n \\small\n Histogram of misplaced objects in episodes across different high-level object categories\n }\n \\label{fig:misplacements_per_high_level_cat}\n \\end{subfigure}\\\\\n \\begin{subfigure}[t]{0.45\\columnwidth}\n \\centering\n \\includegraphics[width=0.95\\textwidth]{figures\/misplaced_counts_per_split.jpg}\n \\captionsetup{width=0.95\\textwidth}\n \\caption{\n \\small\n Histogram showing percentage of train, val and test episodes with given number of misplaced objects\n }\n \\label{fig:misplacements_per_episode}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.45\\columnwidth}\n \\centering\n \\includegraphics[width=0.95\\textwidth]{figures\/starts_goals_per_room_agg_again.jpg}\n \\captionsetup{width=0.95\\textwidth}\n \\caption{\n \\small\n Histogram showing percentage of start and goal positions in each room\n }\n \\label{fig:start_goal_across_rooms}\n \\end{subfigure}\n \\caption{\n \\small\n \\textbf{Episode Statistics.} Analysis on misplaced objects in episodes and their start and goal positions\n }\n\\end{figure}\n\n \\begin{figure}[h!]\n \\begin{subfigure}[t]{0.45\\columnwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/all_distances.jpg}\n \\caption{\\small{Start to every goal}}\n \\label{fig:distance_to_all_goals}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.45\\columnwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/shortest_distances.jpg}\n \\caption{\\small{Start to closest goal}}\n \\label{fig:distance_to_closest_goal}\n \\end{subfigure}\n \\label{fig:geodesic}\n \\caption{\n \\small{Distribution of geodesic distance from start receptacle to (a) every goal (b) closest goal.}\n }\n \\end{figure}\n \n We analyze the generated train, val and test episodes. The val and test episodes include high-level categories already seen in train episodes as well as a few novel high-level categories (Figure~\\ref{fig:misplacements_per_high_level_cat}). Each episode in the train, val and test splits has $3-5$ misplaced objects. Our val and test episodes have slightly higher percentages of episodes with 4 or 5 misplaced objects compared to train episodes (Figure~\\ref{fig:misplacements_per_episode}). A large fraction of the misplaced objects in our episodes start in a bathroom, bedroom, kitchen or living room. A large number of goal receptacles for the misplaced objects are located in the kitchen~\\ref{fig:start_goal_across_rooms}. This is expected since a large number of misplaced objects in a household usually are food or cooking-related (see Figure~\\ref{fig:misplacements_per_high_level_cat}), and kitchens usually have a large number of receptacles.\n \n\\noindent \\textbf{Object-Receptacle Distances:} Next, we visualize the distribution of geodesic distances from object to correct receptacles across all misplaced objects in all episodes. The median distance in our test episodes is 5.36m (Figure~\\ref{fig:distance_to_all_goals}) and the median distance to the closest correct receptacle (out of the 3-5 mispalced) in the test episodes is 0.62m (Figure~\\ref{fig:distance_to_closest_goal}).\n\n\\subsection{Formal definitions of metrics}\n\\label{sec:supp:eval} \n\nIn \\Cref{sec:task:eval}, we informally described our evaluation metrics for Housekeep\\xspace. Here, we formally define the metrics for which more rigorous explanations are required.\n\nFor a given scene, ${\\mathcal{R}}$ and ${\\mathcal{O}}$ are the set of all receptacles and objects respectively. Given an object $o \\in {\\mathcal{O}}$, let $c_{or}$, $m_{or}$ respectively be the ratio of annotators who placed receptacle $r \\in {\\mathcal{R}}$ in \\texttt{correct} and \\texttt{misplaced} bins respectively. We call an object \\emph{correctly placed} if $c_{or} > 0.5$, and \\emph{misplaced} if $m_{or} > 0.5$, where both cannot be simultaneously true. We use:\n\\begin{compitemize}\n\\item ${\\mathcal{O}}_{m}$ for the set of objects which were \\emph{initially misplaced} in the episode.\n\\item ${\\mathcal{O}}_{i}$ for the set of objects which were \\emph{interacted} with by the agent during the episode.\n\\item ${\\mathcal{O}}_{mi}$ (${\\mathcal{O}}_{i} \\cup {\\mathcal{O}}_{m}$) for the set of objects \\emph{initially misplaced} or \\emph{interacted} with by the agent during the episode.\n\\end{compitemize}\n\nFinally, we define the final placement of the object $o$ at the end of the episode via a mapping function $\\Phi: {\\mathcal{O}} \\rightarrow {\\mathcal{R}}$. The receptacle on which an object $o \\in {\\mathcal{O}}$ is placed at the end of the episode is given by $\\Phi(o)$\n\nGiven the relative change in placement of objects between the start and end states of the episode (${\\mathcal{S}}_1$ vs ${\\mathcal{S}}_{T}$), we can formally write the rearrangement metrics as:\n\n\\begin{compenumerate}\n \\item \\textbf{Episode Success (\\texttt{ES})}: Strict binary (\\emph{all} or \\emph{none}) metric that is one if and only if all objects are correctly placed, \\texttt{ES}$=\\prod_{o \\in {\\mathcal{O}}} \\mathbbm{1}[{c_{o, \\Phi(o)} > 0.5]}$.\n \\item \\textbf{Object Success (\\texttt{OS})}: Fraction of the objects which were \\emph{initially misplaced} or \\emph{interacted} with by the agent placed correctly at end of the episode, \\texttt{OS}$=\\sum_{o \\in {\\mathcal{O}}_{mi}} \\mathbbm{1}[{c_{o, \\Phi(o)} > 0.5]}\/|{\\mathcal{O}}_{mi}|$.\n \\item \\textbf{Soft Object Success (\\texttt{SOS})}: The ratio of reviewers that agree that every object \\emph{interacted} with or \\emph{initially misplaced} is placed correctly averaged across all rearranged objects, \\texttt{SOS}$=\\sum_{o \\in {\\mathcal{O}}_{mi}} c_{o,\\Phi(o)}\/|{\\mathcal{O}}_{mi}|$.\n This metric is more lenient because it will be a non-zero number even if just one annotator thought the mapping $(o, \\phi(o))$ is correct.\n \\item \\textbf{Rearrange Quality (\\texttt{RQ})}: The normalized ranking in $(0, 1]$ (via mean reciprocal rank ~\\cite{mrr}) of the receptacle on which an object is placed, ranked among all correct receptacles of an object, if the object was correctly placed, 0 otherwise, averaged across all \\emph{initially misplaced} or \\emph{interacted} objects. \\texttt{RQ}$=\\sum_{o \\in {\\mathcal{O}}_{mi}} \\mathbbm{1}[c_{o, \\Phi(o)} > 0.5] mrr_{c_{o, \\Phi(o)}}.$ Intuitively, RQ will score higher those rearrangements that have a high overall rank in the human preferences dataset.\n\\end{compenumerate}\n\nTo formally define Pick and Place Efficiency (\\texttt{PPE}), one of our exploration metrics, we need a few extra definitions.\n\nWe define $N: {\\mathcal{O}}_i \\rightarrow \\{1, 2, \\cdots\\}$ to be a function mapping an object $o \\in {\\mathcal{O}}_{i}$ to the number of times it was \\emph{picked} or\\emph{ placed} by the agent. We similarly define $N_{min}: {\\mathcal{O}}_i \\rightarrow \\{0, 2\\}$ to be the minimum number of picks and places to place an object $o \\in {\\mathcal{O}}_{i}$ in a correct receptacle: it is 2 when $o \\in {\\mathcal{O}}_{m}$ and 0 otherwise.\n\n\\textbf{Pick and Place Efficiency (\\texttt{PPE})}: The minimum number of interactions needed to rearrange an object divided by the number of interactions the agent actually took to rearrange it if the object was placed in the correct receptacle by the agent at the end of the episode, and 0 if the object was in the incorrect receptacle at the end of the episode, averaged across all objects the agent \\emph{interacted} with. \\texttt{PPE} $= \\sum_{o \\in {\\mathcal{O}}_{i}} \\mathbbm{1}[c_{o, \\phi(o)} > 0.5] \\frac{N(o)}{N_{min}(o))} \/ |{\\mathcal{O}}_{i}|$\n\n\\section{Data Statistics}\n\\label{sec:supp:data_stat}\n\nIn this section we provide details about category level breakdown of objects and receptacles.%\n\n\n\\subsection{High-level Object and Receptacle Categories}\n\\label{ssec:supp:high_level}\nTable~\\ref{table:high_level_categories} details the high-level categorization and frequencies of object and receptacles. We also provide one example of every high-level category, and the original source of the data. We gather 2194 object and receptacle models from multiple sources after filtering objects that are not useful for the task. %\n\n\\noindent \\textbf{Object Filtering Details.} We used category-based filtering for ReplicaCAD, and AB datasets (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot sofa, bikes, etc) to remove unhelpful objects. Then, we removed objects if any of their dimensions exceeded 50 meters. We also used some manual filtering in order to remove very small objects (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot keychains).\n\\input{sections\/supp\/obj_rec_count_table}\n\n\\subsection{Low-level Object Categories}\n\\label{ssec:supp:low_level} \nTable~\\ref{table:object_categories} lists the object categories in each of the train, val-unseen and test-unseen splits. The train split has 8 high-level categories, val-unseen has 2 high-level categories and test-unseen split has 9 high-level categories.\n\\input{sections\/supp\/obj_cats_table}\n\n\n\\section{AMT Human Preferences Dataset}\n\\label{sec:supp:amt_study} \n\nIn this section, we provide more details on our AMT study interface and perform some analysis on the collected data. Our interface consists of an instructions section and is followed by the main task section. After completing the task, the participants are allowed to submit feedback on the interface and the task. The video at \\small{\\href{https:\/\/www.youtube.com\/watch?v=BcHmSzoNBYw}{https:\/\/www.youtube.com\/watch?v=BcHmSzoNBYw}} walks through our AMT data collection interface. %\n\n\\subsection{Participant Instructions}\n\\label{ssec:supp:amt_ins} \n\nBefore beginning the study, each participant is required to read the instructions section. We show the full set of instructions we used during data collection in Figure~\\ref{fig:amt_instructions}. In our instructions, we describe the tasks that need to be performed to successfully complete a HIT (Human Intelligence Task; an AMT term for a unique task instance). As part of a single HIT, the participants are required to complete 10 sub-tasks. For each sub-task, the participant is given an object, a room and a list of receptacles within the given room. The participant is required to classify these receptacles as \\texttt{correct}, \\texttt{misplaced} and \\texttt{implausible} locations. For the receptacles put into the \\texttt{correct} and \\texttt{misplaced} bins, the participant is also required to provide a relative ordering between receptacles.\n\nThe instructions section includes an interactive example that the participants can use to practice before they work on the actual tasks. As a part of our instructions, we provide multiple examples of valid responses. We ask the participants to assume the object is in its ``base\" state (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot utensils being clean, packaged food being unopened) before making their placement decisions. \n\n\\subsection{Task Interface}\n\\label{ssec:supp:amt_inter} \nWe now describe the task interface in detail. We use the same examples that were used to train the participants.\n\n\\noindent \\textbf{Task Start}: For each sub-task we display an object, a room name and four columns. We show all receptacles to be categorized in the first column, with empty correct and misplaced columns (ranked), and an empty implausible column. The object and receptacles are displayed as rotating animated GIFs. Figure~\\ref{fig:amt_task1} shows a screenshot of our task interface at the start of the task. In this example, the receptacles within the kitchen are to be classified as being the correct, misplaced and implausible locations for the alt shaker.\n\n\\begin{figure*}[t!]\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/amt_instructions.pdf}\n \\caption{\\small{AMT Instructions page describing the task with illustrative examples.}}\n \\label{fig:amt_instructions}\n\\end{figure*}\n\\clearpage\n\n\\begin{figure*}[t!]\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/amt_task1.pdf}\n \\caption{\\small{\n AMT starting interface for categorizing and ranking receptacles in the kitchen for a salt shaker.\n }}\n \\label{fig:amt_task1}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/amt_task1_response.pdf}\n \\caption{\\small{AMT Example 1: A sample response for salt shaker on receptacles in the kitchen provided as an example to the users.}}\n \\label{fig:amt_task2}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/amt_task2_response.pdf}\n \\caption{\\small{AMT Example 2: A sample response for clean fork on receptacles in the bathroom.}}\n \\label{fig:amt_task3}\n\\end{figure*}\n\n\\noindent \\textbf{Sample Response \\#1}: Figure~\\ref{fig:amt_task2} shows a sample response for the task in Figure~\\ref{fig:amt_task1}.\n\n\\noindent \\textbf{Sample Response \\#2}: Now consider the example in Figure~\\ref{fig:amt_task3}. Here the given object is fork and the given room is bathroom. Since any receptacle within the bathroom is unlikely to be a correct\/misplaced location for fork, all receptacles are placed under the Implausible column.\n\n\n\n\n\n\\subsection{Dataset statistics}\nWe collect 10 annotations for each object-room pair. We consider that a room-receptacle (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot kitchen-sink) is \\textit{selected} as being a correct\/misplaced location for a given object (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot sponge) if at least 6 annotators place the receptacle (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot sink) under the correct\/misplaced column when shown the given object-room pair (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot sponge-kitchen). Figure~\\ref{fig:selected_receptacles_hist} shows a histogram of objects across different numbers of room-receptacles selected as correct or misplaced. We see that fewer room-receptacles are selected as correct placement of objects while most receptacles are selected as incorrect. Additionally, for most objects (\\url{~}70\\%), annotators selected fewer than 20 receptacles across all rooms as correct. On the other hand, annotators tend to select 10-50 receptacles across all rooms as incorrect placements for most objects. This is also confirmed by Figure~\\ref{fig:selected_receptacles_boxplot}. It shows the distribution of the number of room-receptacles \\textit{selected} as being the correct and misplaced locations. \nMore receptacles are selected as locations where objects are misplaced compared to receptacles where objects are correctly placed.\n\\begin{figure}[h!]\n \\centering\n \\begin{subfigure}[t]{0.485\\columnwidth}\n \\includegraphics[width=\\textwidth,]{figures\/selected_placements_hist.pdf}\n \\caption{\n \\small\n Histogram of objects across different number of room-receptacles selected as correct or misplaced.\n }\n \\label{fig:selected_receptacles_hist}\n \\end{subfigure} \\hfill\n \\begin{subfigure}[t]{0.485\\columnwidth}\n \\includegraphics[width=\\textwidth,]{figures\/selected_placements_boxplot.pdf}\n \\caption{\n \\small\n Distribution per high-level category\n }\n \\label{fig:selected_receptacles_boxplot}\n \\end{subfigure}\n \\caption{\n \\small\n Number of room-receptacles selected as Correct and Misplaced.\n }\n\\end{figure}\n\n\n\\clearpage\n\\section{Housekeep\\xspace}\n\\label{sec:supp:hk} \n\n\\subsection{Episode Generation}\nAlgorithm~\\ref{alg:ep_gen} provides the logic used to generate an episode in Housekeep\\xspace. We start with an empty scene \\texttt{S} furnished with receptacles, AMT data \\texttt{D}, objects repository \\texttt{O}. Next, we filter objects by keeping only the ones that have at least one \\emph{correct} receptacle in the scene, and remove the others. After initializing an incorrectly placed object, we ensure that the agent is able to rearrange and place it on at least one of the \\emph{correct} receptacles. On the other hand, after initializing a correctly placed object, we just ensure that the agent is able to navigate to within grasping distance of it.\n\n\n\n\\label{sec:supp:episode_generation} \n\\input{sections\/dataset_algo}\n\n\n\\input{sections\/supp\/episode_statistics}\n\n\\input{sections\/supp\/eval}\n\n\\input{sections\/supp\/agent}\n\n\\input{sections\/supp\/approach}\n\n\\section{Additional Experiments}\n\\label{sec:supp:add_exps} \n\n\\subsection{Exploration Strategies}\n\\label{ssec:supp:expl_strats} \n\n\\input{tables\/exploration_strategy}\n\nIn \\Cref{sec:methods}, we discussed the \\texttt{Explore} module that used frontier exploration (\\texttt{FRT}). We evaluate 2 additional simple exploration strategies for a total of the following 3 strategies:\n\n\\begin{compitemize}\n \\itemsep0em \n \\item \\texttt{frontier}: Using the egocentric map we iteratively visit unexplored frontiers, frontiers are defined as the edges between known and unknown space. We keep our implementation details same as those used in~\\cite{ramakrishnan2020exploration}.\n \\item \\texttt{random}: Executes a random action in the navigator. \n \\item \\texttt{forward-right}: Executes the forward action until a collision occurs, then turns right. \n\\end{compitemize}\n\nAs we expect, from \\Cref{tab:expl:strat} we see that \\texttt{FRT} outperforms \\texttt{RND} and \\texttt{FWD} in \\texttt{OS}, exploration and efficiency metrics.\n\n\\subsection{Planner Ablations}\n\\label{ssec:supp:plan_abl}\n\n\\textbf{Rearrangement Ordering}: %\nIn \\Cref{sec:methods}, when discussing the \\texttt{Rearrange} submodule, we mentioned 3 key decisions in the submodule. One of them was the order in which misplaced objects are rearranged. In this section, we evaluate the following 4 ordering schemes:\n\n\\input{tables\/rearrangement_strategy}\n\n\\begin{compitemize}\n \\itemsep0em \n \\item \\texttt{score-diff}: We sort rearrangements in decreasing order of score difference between the current receptacle and best one. \n \\item \\texttt{obj-dist}: We sort rearrangements by the geodesic distance from agent to the object. \n \\item \\texttt{rearrange-dist}: We sort rearrangements by the geodesic distance required to execute the rearrangment.\n \\item \\texttt{disc-time}: We sort rearrangements by the time of discovery object. \n\\end{compitemize}\n\nIn \\Cref{tab:rearrange_strat}, we see that the \\texttt{DIS} rearrangement ordering performs slightly better than the other orderings. We choose this ordering to run our main experiments.\n\n\n\\noindent \\textbf{Exploration Steps}: %\nOne of the challenges in Housekeep\\xspace is balancing the exploration-exploitation trade-off; the agent must explore to find misplaced objects or suitable receptacles, but also must exploit its existing knowledge of where objects belong.\nThe exploration module in our hierarchical baseline has an adjustable parameter \\texttt{n}$_e$ that controls the number of steps at the beginning of the episode used for exploration.\nThis parameter thus controls how long the agent spends exploring versus rearranging objects according to a plan.\n\nWe find that fewer exploration steps is more effective. \nIf the agent spends too long exploring, then it will not have enough time to rearrange objects before the end of the episode.\n\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot when \\texttt{n}$_e=512$, our Object Coverage (\\texttt{OC}) is 80\\%, which is 4 points ahead of the next best \\texttt{n}$_e$. However, its Object Success (\\texttt{OS}) is the worst among the variants of \\texttt{n}$_e$ we evaluated.\nWe found the best number of exploration steps to be \\texttt{n}$_e=16$, achieving higher performance in terms of object success (\\texttt{OS}) than all \\texttt{n}$_e<16$ and \\texttt{n}$_e>16$.\n\n\\section{More Qualitative Analysis}\n\\label{sec:supp:qual_analysis} \n\n\\input{sections\/supp\/gantt_a}\n\n\\subsection{Agent states and scene layouts}\n\\label{ssec:supp:topdown}\n\n\\Cref{fig:gantt_2a_layouts} and \\Cref{fig:gantt_2b_layouts} contain similar plots to the ones in \\Cref{fig:gantt} that were discussed in \\Cref{sec:exp:qual_results}.\nIn particular, we notice that the layout of scene \\texttt{Beechwood\\_1} is significantly more complex than that of \\texttt{Benevolence\\_1}, which is the cause of the difference between their object discovery plots as discussed in \\Cref{sec:exp:qual_results}.\n\\input{sections\/supp\/gantt_b}\n\\pagebreak\n\\input{sections\/supp\/ego_video}\n\\input{sections\/supp\/orm_analysis}\n\n\\section{Egocentric rearrangement video}\n\nWe attach an egocentric video (\\small{\\href{https:\/\/www.youtube.com\/watch?v=XccBpQNGN1Q}{https:\/\/www.youtube.com\/watch?v=XccBpQNGN1Q}}) of the agent successfully rearranging all misplaced objects in an episode.\nThe 3 overlays on the left are, from top to bottom: the depth sensor, instance ID mask with semantic information, and the allocentric top-down occupancy map used by the \\texttt{Mapping} module (see \\Cref{sec:methods}).\nWe also include text logs at the bottom left, showing the object the agent is currently holding, the position and name of the object\/receptacle it is navigating towards, the action taken at each step, and whether it is exploring, navigating (rearranging) or picking\/placing.\n\nThe scene contains 4 misplaced objects: an Easter basket in the utility room table, an electronic adapter and a padlock on the dryer, and a toy vehicle on the sofa. The agent explores until 0:15. It then rearranges the Easter basket, the adapter and the padlock by moving them to a shelf. It completes this rearrangement phase at 1:41, after which it goes back to exploring until 2:07. It then moves the toy vehicle object to a nearby shelf, after which it explores for the remainder of the episode.\n\n\n\\section{Ranking module analysis}\nFor the main results in the paper (Table~\\ref{tab:orm} and Table~\\ref{tab:main:unseen}), we used \\texttt{RoBERTa+CM} as the scoring function. In this section, we analyze the design choices and the performance of our current ranking module.\n\\subsection{Ablations}\n\\input{sections\/supp\/orm_emb_ablation}\nIn Table~\\ref{tab:orm_features}, we analyze the effect of using different features as the language model text embedding. Our results in the paper use features that are globally averaged over all token positions of the language model (\\texttt{Avg-all}). We perform experiments using the features at CLS token (\\texttt{CLS}) and using features averaged at all positions except CLS token (\\texttt{Avg-all-exclude-CLS}). While the \\texttt{Avg-all-exclude-CLS} features perform close to \\texttt{Avg-all} features, using \\texttt{CLS} features results in poor performance on seen categories for \\texttt{OR} task.\n\n\\input{sections\/supp\/orm_lang_model_ablation}\nNext, we replace the embeddings from RoBERTa-base model with embeddings from GPT-2 and T5-base language models. Note that we use \\texttt{Avg-all} features for all language models. We find that using T5-base model results in superior performance on both \\texttt{OR} and \\texttt{ORR} tasks (Table~\\ref{tab:orm_lang_model}). The T5-base model has nearly double the number of parameters in RoBERTa-base model. We compare to T5-base model because the next smaller model, T5-small has 60 million parameters (half the number of parameters in RoBERTa-base).\n\n\\subsection{High-level category-wise performance}\nWe now analyze the performance of our \\texttt{RoBERTa+CM} scoring function across different high-level categories. We compute mAP scores for \\texttt{OR} and \\texttt{ORR} tasks (as in Section~\\ref{sec:exp:llm}) and average them per high-level object category. While the scoring function performs perfectly (mAP=1) on seen categories for the \\texttt{OR} task, the \\texttt{OR} task performance drops for unseen high-level categories categories (Figure~\\ref{fig:or_per_high_level}). In contrast, the mAP score is close to 0.8 for most seen and unseen high-level categories (Figure~\\ref{fig:orr_per_high_level}). The test-unseen high-level categories of fruit, furnishing and cosmetic have low mAP scores for both \\texttt{OR} and \\texttt{ORR} tasks.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{figures\/OR_per_high_level.pdf}\n \\caption{\\small{\\texttt{OR} performance of \\texttt{RoBERTa + CM} across different high-level categories}}\n \\label{fig:or_per_high_level}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{figures\/ORR_per_high_level.pdf}\n \\caption{\\small{\\texttt{ORR} performance of \\texttt{RoBERTa + CM} across different high-level categories}}\n \\label{fig:orr_per_high_level}\n\\end{figure}\n\n\n\\subsection{Generalization to unseen categories}\nIn Table~\\ref{tab:main:unseen}, we observed that the Object Success on unseen categories when using the language model-based ranking function is comparable Object Success on seen categories. We now provide qualitative examples showing the performance of our \\texttt{OR} and \\texttt{ORR} scoring functions on unseen categories. \n\nFigure~\\ref{fig:or_qual} shows the ranked list of rooms obtained for each object category using our \\texttt{OR} ranking function. We also indicate if the room is a valid room for the given object. Recall that a room is considered valid if it contains at least one receptacle that is deemed \\texttt{correct} by at least 6\/10 annotators. While the ranked lists for scissors (a tool) and large marker (stationery) have the valid rooms on top, a few valid rooms are further down in the list for banana (fruit category).\n\nFigure~\\ref{fig:orr_qual} shows the ranked list of receptacles with the room for the given object-room pair. These ranked lists are obtained using the \\texttt{ORR} ranking function. We indicate if the receptacle is a valid receptacle next to the receptacle's name. 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<{\\egroup}%\n}\n\\newcommand*\\rot{\\multicolumn{1}{R{30}{1em}}}\n\\usepackage{rotating}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzejbx b/data_all_eng_slimpj/shuffled/split2/finalzzejbx new file mode 100644 index 0000000000000000000000000000000000000000..95cd6c1a10bf8c6e0fda683abdee4c1c55b469bc --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzejbx @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n5G mobile networks promise to bring a new era of ultra high-speed communications that surpasses previous generations by several order of magnitudes in communication capacity \\cite{boccardi2014five}. One of the core technologies behind such a spectacular revolution is spatial devision multiple access (SDMA). SDMA enables massive Multi-Input-Multi-Output (MIMO) communication by providing an ability to focus energy on users' devices, empowering pushing the capacity of the network to such a immense boundaries required for 5G communications \\cite{agiwal2016next,hejazi2021dyloc}. Simultaneously, mobile mm-wave communication is enabled through 5G networks, that transform directional communication from a promising aspect of next generation networks, into a must-have feature \\cite{agiwal2016next,rappaport2013millimeter}. Mm-wave communication experiences huge attenuation in the open air, therefore the transmitted energy needs to be directed into narrow rays, to meet sufficient signal-to-noise-ratio (SNR) thresholds required at receivers \\cite{rappaport2019wireless}. In addition to 5G applications, DoA estimation is a required aspect of UAV-to-device and satellite-to-device high frequency and ultra high-speed communication \\cite{agrawal20165g}. Moreover, mm-wave and Terahertz radars used for autonomous driving exploit DoA estimation techniques to estimate angles of the objects around \\cite{dickmann2016automotive}. As directional communication has gained importance in new generation communications, DoA estimation has obtained gravity as an enabler of directional communication. To clarify this necessity, consider that any two devices that exploit directional antennas cannot communicate unless they ascertain in which direction they should send\/receive signals to\/from the other device. Moreover, this knowledge of angle (or position) of the other device should be maintained during the communication period otherwise the link will be disrupted \\cite{nitsche2014ieee}. \n\nThe most common DoA estimation techniques use directional antennas mounted on both the transmitter and the receiver to obtain the initial guess of the relative angle between two devices, this process is also referred as initial access (IA) \\cite{giordani2016initial}. To fulfil this strategy, the first device starts searching for the second device through a beam training protocol, until it finds the other device. Next, the second device repeats the same procedure until the link is established, at this point, they employ tracking techniques to maintain the directional connection between them \\cite{giordani2018tutorial}.\nAlthough such a strategy looks favorable for DoA estimation, it is highly probable that it does not work well when a large number of devices are packed into a specific area or in the presence of a strong multi-path between two device. Moreover, beams are most of the time busy with beam training\/tracking searches instead of transmission\/reception which reduces the communication capacity \\cite{giordani2019standalone}. In other words, using the same antenna for communication and direction-finding, requires using a common resource for two inherently antithetical task in terms of directional antenna requirements. Higher communication capacity requires highly directional antennas to reduce interference and to maximize signal power at the receiver, conversely, as antenna's beams become narrower the beam training\/tracking periods increase and consequently the overhead escalates which eventually reduces the effective communication capacity. To overcome deficiencies of such a strategy we propose to avoid using directional antennas for DoA estimation at both sides of the link, and estimate DoA based on measuring phase difference of arrival (PDoA) of signal between two antennas mounted on the device for multiple frequencies. Meanwhile, we can allocate a directional antenna exclusively for communication purposes. In our proposed strategy, we avoid spatial search to establish the link in the first place, on the other hand, we rely on the received signal in two omni-directional antennas. Subsequently, we amplify the attenuated received signal by a huge processing gain, then estimate DoAs of all of propagation paths between two devices.\nWe will show that exploiting our proposed technique, we can convert spatial search duration to a means to increase DoA estimation precision, and more importantly, we can allocate a specific highly directional antenna for communication, and consequently take advantage of the whole communication capacity such a directionality provides. \n\n\nIn our proposed technique, two antennas are mounted on the device with several mm gap between them, and the PDoA of signal measured through a novel technique named standing wave spectrometry for multiple frequencies. Standnig wave spectrometry is widely used in optical applications to measure phase difference between two rays at multiple frequecies of the optical spectrum \\cite{sabry2015monolithic}\\cite{wolffenbuttel2005mems}\\cite{jovanov2010standing}. To the best of authors knowledge, it is the first time that this technique is introduced for RF mm-wave applications. By applying spectrometry not only we can estimate the DoA of a signal precisely, but also we can estimate multi-path DoAs and the power of each path for a mm-wave propagation environment. Although the proposed approach is inherently a wide-band (WB) technique, it does not require ultra high speed sampling rates essential for must of WB techniques. Consequently, the proposed technique provides us with two main advantages: more data about the DoA of incoming signal, and reduced cost and complexity of the receiver. The first is obtained by discriminating between all incoming propagation paths between the source and the device. The second is secured by greatly reducing the complexity of the DoA estimation through simplification of the receiver by bypassing signal down-conversion and reducing the number of required antennas. Furthermore, we show that the proposed phase difference measurements equals to highly accurate measurement of time difference of arrival (TDoA) of signal between two antennas in the Fisher sense. Moreover, we will prove that the cramer-rao lower bound of error (CRLB) of DoA estimation using the proposed technique equals to a uniform linear array (ULA) that employs multiple antennas, in the Fisher sense. \n\n\n\n\\section{Related Works}\n\nDoA estimation techniques have plethora of applications in Radar, Sonar and Electronic Ware-fare (EW) literature. In these applications, DoA estimation is mainly used to find the relative direction between two objects. Primitive DoA estimation techniques use pencil beam antennas (e.g. dish antennas) along with mechanical actuators for steering the beam and spatial search \\cite{skolnik2001radar},\\cite{barshan1992bat},\\cite{poisel2012electronic},\\cite{hejazi2013lower},\\cite{hejazi2013new},\\cite{khalili2013secant}. More recent techniques, use beamforming techniques over array antennas to obtain narrow beams. In beamforming, input\/output of each antenna of an array, is multiplied by a weight (e.g. a phase shift) to form a desired beam shape. In beamforming, there is no need for mechanical steering, and beams can be steered electronically by changing weights of the antennas. Spatial scanning provided by beamforming proves to be much more faster than the mechanical scanning, moreover, can generate multiple beams simultaneously. Therefore, modern phased array radars can search the environment very fast, and can track and engage with multiple targets concurrently \\cite{mailloux2017phased}.\n\nRecently, DoA estimation also has gained attention as an enabler of ultra-high-speed (Multiple Gbps) directional communications between two devices or a base-station and multiple devices. 5G communication mainly utilizes advanced beamforming capabilities and array antennas for directional communications. 3 different architectures has been introduced for beamforing for 5G applications: 1-Analogue 2-Digital 3-Hybrid \\cite{kutty2015beamforming}. In Analogue beamforming, the beam is shaped via a single RF chain, and so only one beam can be shaped in each time slot. This structure is more power efficient compared to the two other architectures, however, is not as flexible as them in generating multiple beams. Digital beamforing, allocates a specific Rf chain and data-convertor for each antenna and potentially can generate several beams simultaneously. This structure is the most flexible one, however is very power hungry and complicated in comparison to other techniques \\cite{yang2018digital}. Hybrid beaforming scheme assigns multiple RF chains for antennas, while, the number of RF chains is less than the number of antennas. This type of beamforming is the most common scheme for 5G applications, since it can balance a trade-off between complexity, flexibility and power consumption \\cite{sohrabi2016hybrid,molisch2017hybrid}. All directional antennas powered by various beamforming architectures require spatial search to initiate a communication link . Giordani et. al showed that overhead caused by beam-training protocols heavily limits number of array elements at both base stations and user equipments, moreover, several milliseconds is required to establish a link between a base station and user equipment \\cite{lien20175g}. \n\nInterferometric wide-band DoA estimation, has been widely investigated in EW and lightning localization applications \\cite{mardiana2000broadband},\\cite{wu1995direction}, \\cite{hejazikookamari2018novel},\\cite{kookamari2017using},\\cite{hejazi2014sar},\\cite{hejazi2020tensor},\\cite{hejaziwireless},\\cite{joneidi2019large}. In this technique PDoA of signal between two antennas placed more than half-wavelength apart is measured. Since the phase difference is ambiguous and can represent several DoAs, a number of techniques has been introduced to disambiguate the phase. These techniques include: correlative interferometry (CORR), second order difference array (SODA), SODA-Base Inference (SBI) and Common Angle Search (CAS). CORR employs PDoAs between at least two pairs of antennas and compare measurements with a pre-prepaired database of measurements to determine DoA \\cite{kebeli2011extended}. SODA and SBI operate an additional antenna pair with less than half a wavelength gap between antennas to translate PDoA to an unambigeous DoA. SODA and SBI only works well when input SNR is high enough \\cite{mollai2018compact,mollai2019wideband}. CAS utilizes two or more antenna pairs and introduces the common angle recommanded by all PDoAs as the unambiguous DoA \\cite{searle2017disambiguation}. These techniques can estimate DoA very precisely in a wide-band frequency range, however, none of them can distinguish between DoAs, if two or more signals with differnet DoAs are received simultaneously at the antenna pairs. \n\nHere in section \\ref{Fisher}, we prove that phase interferometry meaurements (PIM) between two antennas equals to highly precise time difference of arrival (TDoA) measurements in the Fisher sense. Moreover, we demonstrate that DoA estimation using PIM between two antennas several wavelength apart equals to DoA estimation using a large ULA in the Fisher sense. Since PIMs represent ambigeous DoAs, we introduce phase spectrometry (PS) to disambiguate PDoAs in section \\ref{PS}. In contrast with Interferometric DoA estimation, we prove that PS can distinguish between multiple concurrent DoAs. Furthurmore, we introduce standing wave receiver (SWR) to extract PDoAs, which is much less complicated than beamforming receivers. We explain how SWR does not need any down-conversion or high sampling rates to extract PDoA. In section \\ref{DoARes} we investigate DoA estimation resolution provided by PS. Then we introduce two approaches to implement PS, one through a long wave-guide, another via employing a frequency code-book in section \\ref{longSW} and \\ref{FreqCB} respectively. In section \\ref{SNR}, we analyse SNR improvement caused by PS. Furthermore, we will show how the whole time required by directional techniques for spatial search can be effectively consumed in PS to improve DoA estimation precision. We discuss the ability of the proposed technique to identify DoA of signals from several devices in both uplink and downlink scenarios in section \\ref{scale}. Moreover, we introduce an alternative architecture of the technique that provides us with ultra-fast DoA estimation capability in section \\ref{UFast}. In Section \\ref{sim}, we examine PS performance via various simulations. Finally we conclude the paper in section \\ref{conc}. \n\n\\section{Phase Interferometry Measurements}\n\\label{Fisher}\nConsider 2 antennas with gap $D$ mounted on a device (Figure \\ref{PIMDEFGEO}), referred as phase interferometry array (PIA), both of them are receiving a signal emitted by a source $s(t)$. The signal is a monotone with carrier frequency $f_c$ \n\\begin{equation}\n s(t)=a \\: e^{j2\\pi f_c t}\\,,\n \\label{sigmod}\n\\end{equation}\nwhere $a$ is the amplitude of the signal. Both the first and the second antennas receive the signal, denoted by $s^{(1)}_{R}(t)$ and $s^{(2)}_{R}(t)$ respectively, with a relative delay $\\Delta (t)$ which results in a phase difference between two signals. We define phase interferometry measurements (PIM) as \n\\begin{equation}\n \\Delta \\phi = s^{(1)}_{R}(t)s^{*(2)}_{R}(t)=a^2_R \\: e^{j2\\pi f_c \\Delta (t)}+v_n=b e^{j2\\pi f_c \\Delta (t)}+v_n \\,,\n\\end{equation}\nwhere $a_R$ is the amplitude of the signal received at the PIA and $v_n$ is white noise, we also refer to $e^{j2\\pi f_c \\Delta (t)}$ as PDoA throughout this paper. In the next section we prove that PIM is equivalent to DoA estimation using a ULA in the Fisher sense. \n\\begin{figure}\n \\centering\n \\includegraphics[width=3in,height=2.3in]{PIMDEFGEO.png}\n \\caption{PIM illustration, two antennas implemented on a device receive a signal ($s(t)$) emitted by a source ($s^{(1)}_{R},s^{(2)}_{R}$). PIM is defind as the interaction of two signals $\\Delta \\phi = s^{(1)}_{R}s^{*(2)}_{R} $}\n \\label{PIMDEFGEO}\n\\end{figure}\n\\subsection{Fisher Information Matrix of PIM, TDoA \\& DoA}\nGiven noise is Gaussian and independent for each PIM, Fisher information matrix (FIM) of $\\Delta \\phi $ with respect to an arbitrary vector $\\boldsymbol{x}$ , e.g. unknowns to be estimated, can be derived as \\cite{farina1999target}\n\\begin{align}\n\\sum_{\\mathbb{P}} \\frac{1}{\\sigma^2} \\nabla_{\\boldsymbol{x}} \\Delta \\phi^{H} \\nabla_{\\boldsymbol{x}} \\Delta \\phi=&\\sum_{\\mathbb{P}} \\frac{1}{\\sigma^2} (-j2\\pi f_c \\nabla_{\\boldsymbol{x}} (\\Delta (t))^{H} be^{-j2\\pi f_c \\Delta (t)}) (j2\\pi f_c \\nabla_{\\boldsymbol{x}} \\Delta (t) be^{j2\\pi f_c \\Delta (t)}))= \\nonumber \\\\\n&\\sum_{\\mathbb{P}} \\frac{4b^2\\pi^2 f^2_c}{\\sigma^2} \\nabla_{\\boldsymbol{x}} \\Delta (t)^{H} \\nabla_{\\boldsymbol{x}} \\Delta (t) \\,,\n\\end{align}\nwhere $\\mathbb{P}$ is the set of all PIMs, and $\\nabla_{\\boldsymbol{x}}$ is the gradient operator with respect to (w.r.t) $x$. Therefore, FIM of PIM is exactly equals to the following observations, \n\\begin{equation}\n\\delta (t)=b\\Delta (t)+\\frac{v_s}{2\\pi f_c} \\,.\n\\label{delti}\n\\end{equation}\nwhere $\\delta (t)$ is an observation of TDoA of signal between two antennas. Therefore, PIM with additive white noise power $\\sigma^2$ equals to TDoA observations with additive white noise power $\\frac{\\sigma^2}{4\\pi^2 f^2_c}$ of the same PIA in the fisher sense. Assuming far field criteria is fulfilled \\cite{chen2002source}, we have\n\\begin{equation}\n\\delta (t)=b\\frac{D}{c} cos(\\theta_A)+\\frac{v_s}{2\\pi f_c} \\,,\n\\label{delti1}\n\\end{equation}\nwhere $c$ is the speed of light and $\\theta_A$ is DoA of signal and $D$ is the gap between two antennas. CRLB of $\\theta_A$ estimation based on measurements as of (\\ref{delti1}) can be derived as follows\n\\begin{equation}\n\\mathrm{CRLB}_{\\theta_{A}}=\\frac{\\frac{\\sigma^2_s}{b^2}}{(\\frac{D}{c}2\\pi f_c)^2 sin^2(\\theta_A)}=\\frac{\\frac{\\sigma^2_s}{b^2}}{(\\frac{2\\pi D}{\\lambda})^2 sin^2(\\theta_A)}\\,.\n\\label{CRMPIM}\n\\end{equation}\nNow lets take a look at CRLB of DoA estimation using a ULA in which antennas are placed half wavelength apart \\cite{penna2011bounds},\n\\begin{equation}\n\\mathrm{CRLB}_{\\theta_{A}}=\\frac{6 \\frac{\\sigma^2_s}{b^2}}{\\pi^2 m(m^2-1)sin^2(\\theta_A)} \\approx \\frac{6 \\frac{\\sigma^2_s}{b^2}}{\\pi^2 m^3 sin^2(\\theta_A)}\\,,\n\\label{CRBULZ}\n\\end{equation}\nwhere $m$ is the number of array elements. Given the same SNR, DoA estimation using PIM and a ULA array are equivalent in the Fisher sense when, \n\\begin{equation}\nm=(24)^{\\frac{1}{3}}(\\frac{D}{\\lambda})^{\\frac{2}{3}} \\approx 2.8845 (\\frac{D}{\\lambda})^{\\frac{2}{3}} \\,.\n\\label{m\/d}\n\\end{equation}\nFigure \\ref{mdlambda} illustrates \\eqref{m\/d}, as an example, DoA estimation using a PIA with $\\frac{D}{\\lambda}=200$ is equivalent to a ULA with 100 elements in the Fisher sense. Consequently, DoA estimation using PIM with gap $D$ between two antennas equals to DoA estimation exploiting a ULA with $m$ antennas placed half wavelength apart, in which $m$ obeys (\\ref{m\/d}). This could lead to a huge reduction in complexity of the antenna array required for high precision DoA estimation -that reduces the required number of antennas from $m$ to 2-; if so, why is it not a common DoA estimation technique now? it is because DoA estimation using PIM is ambiguous and there are a number of different DoAs that can be inferred from a specific PIM \\cite{vinci2011novel}; As $D$ increases CRLB decreases, however, ambiguity escalates. Moreover, DoA estimation using PIM is not capable of detecting and discriminating between multiple concurrent DoAs. In section \\ref{PS}, we propose a solution to estimate DoA using PIMs observed for multiple frequencies, instead of only measuring PIM for only a single frequency. We will see that this approach not only leads to PIM disambiguation, but also provides us with DoA estimation of all signal propagation paths between the source and the device. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=4.9in,height=3in]{mdlambda.png}\n \\caption{$m$ versus $\\frac{D}{\\lambda}$, where $m$ is the number of array elements of a ULA that is equivalent to (in the Fisher sense) a phase interferomery array (PIA) with gap $D$ between two antennas}\n \\label{mdlambda}\n\\end{figure}\n\\subsection{Relationship Between DoA Estimation Precision, Beam-width and Resolution}\n\\label{DoABW}\nIn this section, we explain why DoA estimation precision and antenna beam-width are not necessarily\ncoupled, which further proves that spatial division (SD) and IA can be considered and performed as two completely independent tasks. Referring to (\\ref{CRMPIM}) and (\\ref{CRBULZ}), CRLB of angle estimation precision is directly related to SNR, as SNR increases precision improves; in other words, we can obtain any arbitrary precision if SNR is high enough regardless of $m$ or $D$. Although SNR can be improved by increasing the number of antennas, in a ULA, it can also be improved by integration, which is the time interval we can coherently receive and integrate a signal. Equivalently, angle precision can be improved only by integration, which come at a time cost, regardless of $m$ or $D$. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=4.2in,height=3in]{FigSDMA.pdf}\n \\caption{Visualisation of spatial division concept. The antenna is able to discriminate between user 1 and user 2,3 because the angular distance between them are more than beam-width. While, it is not able to discriminate between user 2 and user 3, since their angular distance is less than the antenna beam-width.}\n \\label{SDMA}\n\\end{figure}\n\n\nNow let's take a look at angle resolution concept. Angle resolution help us to measure the capability of a technique to discriminate between multiple incoming signals from different DoAs. We define angle resolution as the minimum angular distance between two incoming signals that can be discriminated by a technique. Angle resolution is directly coupled with SD capability of a technique. A ULA can discriminate between two DoA if their angular distance is more than it's beam-width. Similarly, in the transmit mode, if the angular distance between two users is more than the beam-width and antenna sends signal to one of them, it causes much less interference for the second device compared to the situation where their angular distance are less than the beam-width (Figure \\ref{SDMA}). In a ULA, beam-width is merely determined by the number of array elements and equals to $\\frac{2}{m}$. Therefore, the SD capability of a ULA is solely governed by its number of array elements. \n\nFor ultra-fast mm-wave communication, devices has to be equipped with a highly directional antenna that enables SDMA. On the other hand, for initial access (IA), a good angle estimation is required. As we discussed earlier, an angle estimation with a desired precision can be obtained when SNR is high enough at the receiver. SNR and resolution are not two mutually-coupled aspects of a DoA estimation technique. Especially in the case of array antennas, resolution is governed by number of array elements, while DoA estimation precision is governed by SNR at the receiver. Consequently, we can seperate SD from IA, and dedicate a high processing gain technique for IA and a highly directional antenna for SD.\n\n\n\n\\section{Wide-band DoA Estimation Using Standing-wave Spectrometry}\n\\label{PS}\nIn this section we inaugurate a new idea to estimate DoA of a signal using PDoAs. Here, we propose the source emits a signal with several gigahertz bandwidth in mm-wave, in such a way that the receiver can detect and discriminate between all (line-of-sight (LoS) and none-line-of-sight (NLoS)) paths between the source and the receiver using our proposed PS technique. Now suppose there exists $N_{NL}+1$ paths, 1 LoS path and $N_{NL}$ NLoS paths, between the source and the device. Given the source emits a monotone signal as of (\\ref{sigmod}) with carrier frequency $f$ for the duration $T_p$, received signals at both antennas can be formulated as\n\\begin{equation}\n\\label{interefence}\n s^{(1)}_{R}(t) = \\!\\underbrace{a_0 e^{j 2\\pi f t}}_{\\text{LoS path}}\\!+\\!\\underbrace{\\sum_{k=1}^{N_{NL}}a_ke^{j 2\\pi f (t-t_k)}}_{\\text{NLoS paths}}+v_1(t)\\;\\;\\; and\\;\\;\\;\n s^{(2)}_{R}(t)=\\!\\underbrace{a_0 e^{j 2\\pi f (t-\\Delta{t_0})}}_{\\text{LoS path}}\\!+\\!\\underbrace{\\sum_{k=1}^{N_{NL}}a_ke^{j 2\\pi f (t-t_k-\\Delta{t_k}))}}_{\\text{NLoS paths}}+v_2(t)\\,.\n\\end{equation}\nwhere $t_k$ is the delay of signal arrival through NLoS path $k$ to the PIA w.r.t LoS path, and $\\Delta t_k$ and $a_k$ is TDoA of signal between two antennas and amplitude of received signal through path $k$, $k = 0,\\dots,N_{NL}$ (path 0 is the LOS path), respectively. Then, we guide the two received signals into a standing-wave wave-guide (SWWG) via two opposite directions (Figure \\ref{SWR}). Referring to \\cite{jovanov2010standing}, the first and the second paths of signal interact in the SWWG as \n\n\\begin{align}\n &s^{(1)}_{R}(t) e^{j\\beta(f) x} + s^{(2)}_{R} (t)e^{-j\\beta(f) x}=\\nonumber\\\\\n &e^{j 2\\pi f t} \\left(\\left(a_0 e^{j\\beta(f) x}+ a_0 e^{-j\\beta(f) x}e^{-j 2\\pi f (\\Delta{t_0})} \\right)+\\left(\\sum_{k=1}^{N_{NL}} a_k e^{-j 2\\pi f t_k} \\left(e^{j\\beta(f) x}+e^{-j\\beta(f) x} e^{-j 2\\pi f (\\Delta{t_k}))} \\right)\\right)\\right)= \\nonumber \\\\\n &e^{j 2\\pi f t}\\left(2a_0 e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}} 2 a_k e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right) \\,, \\nonumber \\\\ \n\\end{align}\n\nwhere $x$ is an arbitrary point along the SWWG, $L$ is the length of the wave-guide and $\\beta(f)=\\frac{2\\pi} {\\lambda_T}=2\\pi \\frac{f}{c_T}$, where $\\beta(f)$, $\\lambda_T$ and $c_T$ are phase constant, wavelength and phase velocity of electro-magnetive wave in the wave-guide, respectively \\cite{steer2019microwave}. As Figure \\ref{SWR} illustrates, using energy detectors along $x-axis$, we have\n\\begin{figure}\n \\centering\n \\includegraphics[width=6in,height=1.8in]{Figstand.pdf}\n \\caption{Standing-wave wave-guide. Two waves move in opposite directions interact to form a standing wave, the amplitude of the standing wave is sampled using a group of energy detectors (ED).}\n \\label{SWR}\n\\end{figure}\n\\begin{align}\n &E_{sw} (x,f) = \\left| 2a_0 e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}}2a_k e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right|^2 \\nonumber \\\\\n &= \\: 4 a^{2}_{0} cos^2 (\\beta(f) x + \\pi f \\Delta t_0) +\\sum_{k=1}^{N_{NL}} 4 a^{2}_{k} cos^2 (\\beta(f) x + \\pi f \\Delta t_{k})\\nonumber \\\\ \n &+\\sum_{k=1}^{N_{NL}} 8 a_0 a_k \\cos{(\\beta(f) x+ \\pi f \\Delta t_0)} \\cos{ (\\beta(f) x + \\pi f \\Delta t_k)} \\cos {(2\\pi f (t_k + \\frac{\\Delta t_k - \\Delta t_0}{2}))} \\nonumber\\\\ \n &+\\sum_{k=1}^{N_{NL}}\\sum_{l=K+1}^{N_{NL}} 8 a_l a_k \\cos{(\\beta(f) x+ \\pi f \\Delta t_l)} \\cos {(\\beta(f) x + \\pi f \\Delta t_k)} \\cos {(2\\pi f (t_k-t_l + \\frac{\\Delta t_k - \\Delta t_l}{2}))}\\,.\n \\label{Damaneh}\n\\end{align}\n\nwhere $E_{sw} (x,f)$ is the output of the ED located at $x$. Interestingly, as \\eqref{Damaneh} indicates, we could bypass down-conversion via mixing by using much simpler EDs. Now, suppose that input signal and its DoAs does not change during $T_p$, evidently, sampling rate after EDs can be as low as some $\\frac{1}{T_p}$, if energy detectors provide energy integration of the wave for the whole duration. To simplify (\\ref{Damaneh}), it is clear that $\\Delta t_k \\ll t_l$ and it is very probable that $\\Delta t_k \\ll t_k-t_l$; $ k = 0,\\dots,N, l = 1,\\dots,N$ \n\\footnote{The experimental results presented in \\cite{rappaport2013millimeter} shows that delays of paths in two urban environment of New York and Austin is an order of several tens of nano seconds, on the other hand, the TDoA of signal between two antennas is a fraction of a nano second if the gap between antennas does not exceed $30 cm$.} Regarding \\eqref{Damaneh}, the first and the second terms have \\emph{cos(.)} components with parameters $\\pi f \\Delta t_0$ and $\\pi f \\Delta t_k$ , while the third and the forth terms have \\emph{cos} components with parameters $\\pi f t_k$ and $\\pi f (t_k-t_l), k \\neq l$, respectively. Given we measure \\eqref{Damaneh} for multiple frequencies and $\\Delta t_k \\ll t_k-t_l$ for all $l,k$, applying Fourier transform over $E_{sw} (x,f)$ across $f$, the third and the forth terms of (\\ref{Damaneh}) can be filtered out using a simple low-pass filter \\footnote{In section \\ref{DoARes} we will show that, this filter can be the same as the matched filter applied for DoA detection.}. The remaining terms after low-pass filtering are denoted by $ \\hat{E}_{sw} (x,f)$\n\n\\begin{align}\n \\hat{E}&_{sw} (x,f) = \\: 4a_0^2 \\cos^2 (\\beta(f) x + \\pi f \\Delta t_0) +\\sum_{k=1}^{N_{NL}} 4 a^2_k \\cos^2 (\\beta(f) x + \\pi f \\Delta t_k) \\nonumber\\\\\n =2a_0^2& + 2 a^2 \\cos (2 \\beta(f) x + 2\\pi f \\Delta t_0)+ \\sum_{k=1}^{N_{NL}}2a^2_k + 2 a^2_k \\cos (2\\beta(f) x + 2\\pi f \\Delta t_k) \\,.\n \\label{phspecforf}\n\\end{align}\n\nThe number NLoS path from the source to the device are very few in mm-wave usually less than 3 path \\cite{heath2016overview}, so $N_{NL} \\le 3$. Here, if we estimate $\\theta_k,a_k$ for $k = 0,\\dots,N$, we can distinguish between all paths from the source to the device and determine signal received power from each path. In section \\ref{DoARes}, \\ref{longSW} and \\ref{FreqCB}, we will discuss two different techniques that that can be used to detect DoAs based on sampling \\eqref{phspecforf} in $f$-domain, and how the angular resolution that can be achived using PS. \n\nConsider that (\\ref{phspecforf}) is derived by assuming a monotone signal is transmitted by the source. \nNow lets assume, signal is not monotone and has bandwidth $B$, thus signal can be expressed as\n\n\n\n\\begin{equation}\n s(t) = \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a(f) e^{j2 \\pi f t} df\\,.\n \\label{wide-spread}\n\\end{equation}\n\nwhere $a_(f)$ is Fourier transform of $s(t)$. The received signals at the first and the second antennas can be expressed as\n\n\\begin{align}\n\\label{interefence1}\n &s^{(1)}_{R}(t)= \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_0(f) e^{j 2\\pi f t} df+\\sum_{k=1}^{N_{NL}}\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f)e^{j 2\\pi f (t-t_k)} df+v_1(t)\\nonumber \\\\\n &s^{(2)}_{R}(t)=\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}}a_0(f) e^{j 2\\pi f (t-\\Delta{t_0})} df+\\sum_{k=1}^{N_{NL}} \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f) e^{j 2\\pi f (t-t_k-\\Delta{t_k}))} df+v_2(t)\\,.\n\\end{align}\n\nAssuming a constant fading over $[f_c-\\frac{B}{2},f_c+\\frac{B}{2}]$, we can express $a_k(f)= \\alpha_k a(f)$, where $\\alpha_k$ denotes the attenuation of path $k$. Consequently, the two signals inside the SWWG can be formulated as \\cite{jovanov2010standing}\n\n\\begin{align}\n\\label{interefence1}\n &s^{(1)}_{R}(t,x)= \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_0(f) e^{j 2\\pi f t} e^{j \\beta(f) x} df+\\sum_{k=1}^{N_{NL}}\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f)e^{j 2\\pi f (t-t_k)} e^{j \\beta(f) x}df+v_1(t)\\nonumber\\\\\n &s^{(2)}_{R}(t,x)=\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}}a_0(f) e^{j 2\\pi f (t-\\Delta{t_0})} e^{-j \\beta(f) x} df+\\sum_{k=1}^{N_{NL}} \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f) e^{j 2\\pi f (t-t_k-\\Delta{t_k}))} e^{-j \\beta(f) x} df+v_2(t)\\,.\n\\end{align}\n Finally, the interaction between the two signals ($S_{int}$) in the SWWG can be formulated as \n \\small\n \\begin{align}\n &S_{int}(t,x)=s^{(1)}_{R}(t,x)+s^{(2)}_{R}(t,x)=\\nonumber\\\\\n &\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} \\left(\\left(a_0(f) e^{j\\beta(f) x}+ a_0(f) e^{-j\\beta(f) x}e^{-j 2\\pi f (\\Delta{t_0})} \\right)+\\left(\\sum_{k=1}^{N_{NL}} a_k(f) e^{-j 2\\pi f t_k} \\left(e^{j\\beta(f) x}+e^{-j\\beta(f) x} e^{-j 2\\pi f (\\Delta{t_k}))} \\right)\\right)\\right) e^{j 2\\pi f t} df= \\nonumber \\\\\n &\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} \\left(2a_0(f) e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}} 2 a_k(f) e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right)e^{j 2\\pi f t} df \\,. \n\\end{align}\n\\normalsize\nTherefore, the power spectral density of $S_{int}$ turns out to be \\cite{oppenheim2015signals}\n \\small\n\\begin{align}\n&\\mathscr{{E}}_{sw}(x,f)=\\lim_{T \\to +\\infty}\\mathscr{F} \\left\\{ \\frac{1}{T}\\int_{0}^{T}|S_{int}(t,x)|^2dt\\right\\}\\nonumber\\\\\n&=\\left|2a_0(f) e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}} 2 a_k(f) e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right|^2 ; \\: f \\in [f_c-\\frac{B}{2},f_c+\\frac{B}{2}]\\,.\n\\label{PSD}\n\\end{align}\n\\normalsize\n\n\n\n\nAs \\eqref{PSD} shows $\\mathscr{{E}}_{sw}$ exactly equals to \\eqref{Damaneh} for $f \\in [f_c-\\frac{B}{2},f_c+\\frac{B}{2}]$. Therefore, similar to the procedure of DoA estimation of a monotone signal, we can estimate all incoming signal DoAs and their power for a non-monotone signal using \\eqref{phspecforf}. Now, lets again consider \\eqref{phspecforf}, we can express $\\hat{E}_{sw} (x,f)$ as summation of two terms\n\\begin{align}\n &\\hat{E}_{sw} (x,f) = \\sum_{k=0}^{N_{NL}} 2a_k^2 +\\sum_{k=0}^{N_{NL}} 2 a^2_k \\cos (2\\beta(f) x + 2\\pi f \\Delta t_k) \\nonumber\\\\\n &=\\sum_{k=0}^{N_{NL}} 2a_k^2+\\sum_{k=0}^{N_{NL}} 2 a^2_k \\cos (\\frac{4\\pi}{c_T} fx + 2\\pi f \\Delta t_k) \\,.\n \\label{phspecforfsimp}\n\\end{align}\nInterestingly, factors $a_0,\\dots,a_{N_{NL}}$ and phases $2\\pi f \\Delta t_0,\\dots,2\\pi f \\Delta t_{{NL}}$, can simply be estimated by applying Fourier transform over $\\hat{E}_{sw} (x,f)$ across $x$. One useful example of signal as of (\\ref{wide-spread}) is multiple single tones (e.g. 30 monotones) around the center frequency; in section \\ref{FreqCB}, we show that this signal not only provides enough information to estimate all DoAs, but also enables integration to achieve very high SNRs, which results in very high precision DoA estimation.\n\n\\subsection{DoA Detection and Resolution}\n\\label{DoARes}\nAs we proved in the previous section, the interaction between two waves received at each antennas forms a standing-wave and its amplitude can be measured as of (\\ref{phspecforfsimp}) for each frequency $f$, measuring amplitude of the standing-wave, employing a group of EDs. Consider that \\eqref{phspecforfsimp} consists of $2 a^2_k \\cos (2\\beta(f) x + 2\\pi f \\Delta t_k)$ terms. Thus, estimating $\\Delta t_k$ and $\\alpha_k$ for $k = 0,\\dots,N_{NL}$ is equivalent to harmonic decomposition of \\eqref{phspecforfsimp} in $f$-domain. There are several techniques has been introduced for harmonic decomposition, such as Fourier transform, multiple signal classification (MUSIC) \\cite{schmidt1986multiple}, Pisarenco harmonic decomposition \\cite{pisarenko1973retrieval}, to name a few. Here for simplicity we only use a matched filter for DoA estimation. Given far-field assumption we have\n\n\\begin{equation}\n \\Delta t_k=\\frac{D\\cos{\\theta_k}}{c}\\,.\n\\end{equation}\n\nwhere $\\theta_k$ is DoA of path $k$. Therefore, DoAs can be estimated applying the following matched filter on (\\ref{phspecforf})\n\n\\begin{equation}\n h(\\theta,f,x)=e^{j2\\pi f \\Delta t_k}e^{j2\\beta(f)x}=e^{j2\\pi f \\frac{Dcos(\\theta)}{c}}e^{j2\\beta(f)x}=e^{j2\\pi f (\\frac{Dcos(\\theta)}{c}+\\frac{4\\pi x}{c_T})}\\,.\n \\label{matchedfilter}\n\\end{equation}\n(\\ref{matchedfilter}) shows that the matched filter is a single monotone in the $f$-domain. Moreover, as $\\Delta t_k$ increases, the matched filter represents a higher frequency signal in $f$-domain. Therefore, convolving (\\ref{matchedfilter}) with (\\ref{Damaneh}), the third and the forth terms of \\eqref{Damaneh} will be eliminated. To calculate the angular resolution of PS suppose two different paths with two different DoAs $\\theta_1,\\theta'_1$ arrive at PIA and we can completely discriminate between $\\theta_1$ and $\\theta_1'$ using matched filter in (\\ref{matchedfilter}), then we have \n\\begin{align}\n &\\int_{f-\\frac{B}{2}}^{f+\\frac{B}{2}} e^{j2\\pi f D \\frac{cos(\\theta_1)-cos(\\theta'_1)}{c}} df = 0 \\rightarrow BD\\frac{|cos(\\theta_1)-cos(\\theta'_1)|}{c}=k , k \\in \\mathbb{N} \\rightarrow \\frac{BD}{c} |\\theta_1-\\theta'_1||sin(\\theta_1)|\\approx k \\nonumber \\\\\n &\\rightarrow |\\theta_1-\\theta'_1| \\approx \\frac{ck}{BD |sin(\\theta)|}\n \\,.\n \\label{msdsds}\n\\end{align}\nTherefore the minimum possible angular distance between $\\theta_1$ and $\\theta_1'$ that can be resolved using our proposed technique (referred as DoA estimation resolution) can be approximated as\n\\begin{align}\n Res(\\theta) \\approx \\frac{c}{BD |sin(\\theta)|}\n \\,.\n \\label{Res}\n\\end{align}\nConsequently, DoA estimation resolution is determined merely by $BD$, which means as the gap between two antennas or the signal bandwidth increases the DoA resolution will increase. As we mentioned earlier, in this we mainly use marched filter for DoA detection for simplicity, however, since PDoAs are available in digital domain, future works may consider more complicated signal processing techniques for DoA estimation. Those techniques may result in much better angular resolution than match filtering.\n\n\\subsection{Frequency Resolution}\n\\label{longSW}\n\nConsidering $\\beta=2\\pi \\frac{f}{c_T}$, (\\ref{phspecforfsimp}) clarifies that angle and phase difference of PIMs for any arbitrary frequency inside $[f-\\frac{B}{2},f+\\frac{B}{2}]$ would be easily extracted by applying Fourier transform over $\\mathscr{{E}}_{sw}(x,f)$ across $x$, if we could measure $\\mathscr{{E}}_{sw}(x,f)$ for an infinite length. Unfortunately, in practice we can only measure $\\mathscr{{E}}_{sw}(x,f)$ for a limited length and it enforces a strong limitation on the frequency resolution of the Fourier transform. To calculate of resolution of FFT over $\\mathscr{{E}}_{sw}(x,f)$ across $x$, consider that if we have a signal for length $T$ (in time), the highest FFT resolution possible is $\\frac{1}{T}$ \\cite{oppenheim1999discrete}. Given SWWG length is $L$, referring to (\\ref{phspecforfsimp}), the frequency resolution ($\\delta(f)$) turns out to be\n\\begin{align}\n Res(f)=\\delta(f) \\rightarrow 2 \\frac{\\delta f}{c_T} L =1 \\rightarrow\n \\delta(f)=\\frac{c_T}{2L} \\,.\n \\label{Length}\n\\end{align}\nGiven $c_t\\approx c$, to reach a $1GHz$ frequency resolution we need a $15cm$ wave-guide and to reach a $100MHz$ resolution we need a $1.5m$ wave-guide. Such a long wave-guide may not be practical specially exploiting PCB or MMIC implementation since it results in a huge attenuation of the signal along the long wave-guide. Thus we may either employing alternative fabrication technologies or the following technique to resolve this issue. \n\\subsection{Frequency Swiping Interferometry (Frequecny Code-book)}\n\\label{FreqCB}\nInstead of spectrometry via a long wave-guide, we can sample PDoAs for a group of frequencies in $[f_c-\\frac{B}{2},f_c+\\frac{B}{2}]$ using a short wave-guide. To this end, we divide the frequency band into $S_f$ frequency steps (also referred as frequency code-book), each step is represented by a monotone (pilot), and measure (\\ref{phspecforf}) for each pilot. We also divide the whole PS duration into $S_f$ time slots and measure PDoA for each pilot at each time slot. Since we measure PDoA for a monotone in each time slot, our approach bypasses the need for a long SWWGs. Consider that, the number of pilots and the distance between them (in $f$-domain) should provide us enough information to detect all DoAs. Referring to (\\ref{phspecforf}), we measure $e^{j 2\\pi f \\Delta t_{in}}$ for each pilot, where $t_{in}$ can potentially changes between $[-\\frac{D}{c},\\frac{D}{c}]$, therefore we should sample the phase difference with at least $\\frac{c}{2D}$ rate (Nyquist rate) in the $f$-domain to capture all information regarding $\\Delta t_{in}$, thence, the code-book should contain at least \n\\begin{equation}\n \\mathrm{min} \\: S_f= \\frac{B}{\\frac{c}{2D}}=\\frac{2BD}{c}\\,,\n\\end{equation}\npilots (samples in $f$-domain). Consequently, we propose to establish a directional link between two devices, both devices should send pilots, so the other side can estimate DoAs of signal based on measuring PDoAs for all pilots. Using our proposed technique, there is no need for spatial search and all DoAs can be estimated via measuring PDoAs of pilots. Lets $f_0$ denotes the frequency of the first monotone and $\\Delta f_0 = \\frac{c}{2D}$ denotes the distance between pilots in $f-$domain. Thus, the vector of all measured phases for the frequency codebook ($\\boldsymbol{\\Delta \\phi}$) can be expressed as \n\n\\begin{align}\n&\\boldsymbol{\\Delta \\phi}=\n \\begin{bmatrix}\n e^{j 2\\pi f_0 \\Delta t_{in}} & e^{j 2\\pi (f_0+\\Delta f_0) \\Delta t_{in}} & \\dots & e^{j 2\\pi (f_0+(S_F-1)\\Delta f_0) \\Delta t_{in}} \n\\end{bmatrix} \n\\nonumber \\\\\n&=e^{j 2\\pi f_0 \\Delta t_{in}}\n\\begin{bmatrix}\n 1 & e^{j 2\\pi (\\Delta f_0) \\Delta t_{in}} & \\dots & e^{j 2\\pi ((S_F-1)\\Delta f_0) \\Delta t_{in}} \n\\end{bmatrix} \\,,\n\\label{PSULASIM}\n\\end{align}\nwhere $\\Delta t_{in}$ is the TDoA of signal between two antennas. \\eqref{PSULASIM} equals to the vector of phase differences measured by a ULA ($\\boldsymbol{\\Delta \\phi_{u}}$) with $S_F$ elements multiplied by $e^{j 2\\pi f_0 \\Delta t_{in}}$ \n\\begin{align}\n&\\boldsymbol{\\Delta \\phi_{u}}=\n \\begin{bmatrix}\n 1 & e^{j 2\\pi (\\Delta f_0)\\Delta t_{d}} & \\dots & e^{j 2\\pi ((S_F-1)\\Delta f_0) \\Delta t_{d}} \n\\end{bmatrix} \\,,\n\\label{ULAPSH}\n\\end{align}\nwhere $\\Delta t_{d}$ is the TDoA between of signal between two consecutive elements and $\\Delta f_0$ is the working frequency of ULA. In fact, we reconstruct a ULA that works at frequency $\\Delta f_0$ via PS that works at much higher frequency $f_0$ \\footnote{More interestingly, $f_0$ and $\\Delta f_0$ are independent. $f_{0}$ should be high enough to provide us with enough unused bandwidth required to emulate the ULA. Thus, PS is much more applicable in mm-Wave and Terahertz bands becuase large swaths of spectrum is available.}. As $B$ increases the number of pilots (equivalent to ULA elements) can increase and as $D$ increases $\\Delta f_0$ decreases and again we can increase the number of pilots which results in better angular resolution. In a ULA, usually PDoAs of \\eqref{ULAPSH} are compensated by phase shifters at each elements for different values of possible $\\Delta t_{d}$s to find the best match with $\\boldsymbol{\\Delta \\phi_{u}}$ and detect the DoA (i.e. the spatial search). In our technique, since we measure PDoAs using PS techniques we can find the incoming DoA by digital signal processing. In section \\ref{sim} we will show that output of matched filter of \\eqref{matchedfilter} applied on \\eqref{PSULASIM} is very similar to output of phase shifters applied on \\eqref{ULAPSH} (conventional beamforming). \\footnote{Throughout this work, we only consider a simple matched filter on \\eqref{PSULASIM} to detect DoAs. However, PS provides \\eqref{PSULASIM} in the digital domain, thus, much more complex signal processing techniques can be applied. Future works may consider various frequency sampling and corresponding array signal processing techniques to improve PS performance.} \n\\subsection{SNR Analysis}\n\\label{SNR}\n\nLong SWWGs is subject to suffering from a huge loss, specially in mmwave. Since SNR is an absolutely critical factor when we deal with millimeter waves, it is more practical not to attenuate the input signal in the receiver by employing long SWWGs. In this section we analyse SNR of the technique that employs a frequency code-book instead of a long SWWG. The block diagram of the receiver using the frequency code-book technique is depicted in Figure \\ref{BD}. As the figure illustrates, input signals pass through 3 stages until DoAs of signal are detected. Each stage may improves SNR. To measure how much the proposed receiver improves SNR we use the processing gain ($G_p$) metric \\cite{rouphael2009rf}. Procesing gain is defined as ratio of the SNR of a processed signal to the SNR of the input signal. $G_p$ of the whole receiver can be expressed as\n\n\\begin{equation}\n G_p(total)=G_p(stage-1)G_p(stage-2)G_p(stage-3)\\,. \n\\end{equation}\n\nNow lets calculate the $G_p$ for each stage. We ignore losses caused by hard-wares in our calculation. Consider a very basic formula that governs $G_p$ of any arbitrary process \\cite{dixon1994spread}\n \n\\begin{equation}\nG_p=\\frac{B_{rf}}{B_{info}}=B_{rf}T_{int}\\,,\n\\label{GP}\n\\end{equation}\n\nwhere $B_{rf}$ in input bandwidth, and $B_{info}$ is the information bandwidth and $T_{int}$ is the integration time. This formula states that you can improve SNR of the input signal by integration as long as noise of samples are independent, otherwise integration will amplify the noise the same as signal and SNR won't improve. To make it more clear, suppose that input signal bandwidth is $1 MHz$, and assume that it is sampled by $ 1MHz$ sampling rate. Then we integrate the signal coherently for $1 ms$, in other words, we integrate $1000$ samples of the signal coherently. Consequently, $G_p=1000=\\frac{1Mhz}{1Khz}=1Mhz*1ms$. If we sample the signal with a higher sampling rate, we will have more samples for integration, however, noise of samples are correlated and the integration won't result in higher SNRs. \nIn view of (\\ref{GP}), lets calculate $G_p$ for the first stage. Given each monotone of the code-book is received for $T_p$, assuming bandwidth of $B_{rf}$ for the BPF, $G_p$ of the first stage can be formulated as, \n\\begin{equation}\n G_p(stage-1)=B_{rf}T_p\\,.\n\\end{equation}\nConsider that the only information that each ED measures is the amplitude of the standing wave, which is constant during $T_p$, Therefore, the amplitude can be estimated by integrating the input signal for $T_p$.\nTo calculate $G_p$ for the next stage, consider that the wave-guide length is $L$ which is in order a wavelength, as we sample the standing wave through the wave-guide, it is equivalent to sample the standing wave in time with a rate more than $f_c$, since $B_{rf}$ is much less than $f_c$, noise of these samples are not independent and integration at the second stage won't result in any SNR improvement. \n\nAt the last stage we measure PDoAs for the frequency code-book in different time slots, therefore noise of phase difference measurements at each time slot is independent of all other time slots -even if frequencies of pilots at two different\ntime slots are the same-; therefore, PDoAs can be integrated over all the code-book's pilots and the processing gain of stage-3 can be expressed as \n \n\\begin{equation}\n G_p(stage-3)=S_f\\,.\n\\end{equation}%\nFinally, the total $G_p$ (processing gain) of all stages is \n\n\\begin{equation}\n G_p(total)=B_{rf}T_p S_f\\,.\n \\label{TPG}\n\\end{equation}\n\n$T_p S_f$ equals total time spent on receiving pilots by the receiver, in other words, using the proposed technique, we can make use of the whole duration of DoA estimation procedure to improve input SNR and consequently, improve DoA estimation precision. As we discussed earlier, directional techniques spend substantial amount of time for spatial search to find the other side of the link, moreover, both sides can not search for each other at the same time which further increases the spatial search duration. Contrarily, employing our proposed technique, both sides are able to search for the other side at the same time and can take advantage of the whole search duration to improve DoA estimation precision. \n\n\n\n \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=6in,height=3.5in]{Figure_BD.pdf}\n \\caption{Block diagram of our proposed DoA estimation technique. Input SNR is improved through stages 1 and 3. In stage-1, a monotone signal is received at two antennas and passes through a band-pass-filter (BPF) via each path. After amplification via a low-noise-amplifier (LNA) in each path, both signals enter a wave-guide to form a standing-wave. Amplitude of the standing-wave is measured by a group of energy detector (ED) sensors, which inherently are low-pass filters and therefore, improves the SNR. Then the amplitude is sampled and can be integrated during each monotone time-step ($T_p$). After sampling, signal passes through a phase detector. Finally, PDoAs measured for all frequencies of the code-book are used to estimate DoAs using a matched filter which improves SNR for the second time.}\n \\label{BD}\n\\end{figure}\n\\subsection{Uplink and Downlink DoA Estimation}\n\\label{scale}\nIn this section we are going to answer the following question: \"How does PS perform in the presence of multiple users? How many devices can find their relative angles simultaneously using PS?\" to answer these questions assume the following scenario: There is a base-station (BS) and $N_d$ devices around it in an environment, all devices require to estimate signal DoAs from the base-station (downlink), and the base-station requires to know DoAs of signals from devices (uplink). In downlink scenario, it is only required that BS sends one common code-book and all devices can find DoA of BS by measureing PDoAs of pilots of the common code-book. However, the uplink scenario is more complicated. If all devices send the same code-book it is impossible for the BS to distinguish between DoAs. Therefore, devices' code-books have to be orthogonal either in time or frequency. If the BS can split the code-book band ($B$) to $N_{rf}$ sub-bands and uses an exclusive SWR for each sub-band, it can estimate DoA from $N_{rf}$ devices simultaneously (Figure \\ref{Uplink}), since, $N_{rf}$ different frequency code-books can be processed simultaneously at the BS. Considering, the BS can be equipped by antennas with much larger $D$ and more complicated receivers than devices, the BS can estimate DoA from multiple devices simultaneously. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=7in,height=3.5in]{Uplink.pdf}\n \\caption{In uplink scenario, to be capable of discriminating between DoAs of multiple devices, the BS requires to be equipped with two filter-banks at both lines of it's PIA, and a separate SWR for each frequency sub-band.}\n \\label{Uplink}\n\\end{figure}\n\n\\subsection{Ultra-fast DoA Estimation}\n\\label{UFast}\nAs we discussed in section \\ref{FreqCB}, we suggest measuring PDoAs for multiple frequencies over multiple time slots to avoid using a large SWWG. In that architecture, we assumed we can only use a single SWR. Therefore, we have to measure PDoA of different pilots at different time slots. Nevertheless, instead of using a single SWR, it is possible to use a cascade of multiple SWRs, discriminating between multiple pilots using a filter bank, and find the PDoA for each monotone exploiting a specific SWR (the architecture is presented in Figure \\ref{Uplink}). Using such an architecture, we can estimate all incoming DoAs in a single time slot, without any negative impact on the processing gain and the DoA estimation precision. Such an ultra-fast DoA estimation has not been previously possible using directional antennas, since those techniques are bound to spatial search. Ultra-fast DoA estimation using PS requires more complex hardwares in comparison to the technique introduced in section \\ref{FreqCB}, which may make it overpriced or oversized to be implemented on commercial mobile phones. However, it may be very promising for applications such as radar, mm-wave network backhaul, UAV and satellite communications, where more complex and bulky hard-wares can be implemented on devices. \n\n\\section{Simulation Results}\n\\label{sim}\n\nIn this section, the perfromance of the proposed DoA estimation technique for different parameters is studied.\n \\subsection{Simulation Setup and Results}\n \n In the first simulation, we examine a basic scenario where a signal arrive at PIA through only one path, therefore, there is only one DoA to be estimated. We set $f_c=60 GHz$, $B=10GHz$, the steps of the codebook is 40 and pilots are selected equally spaced from $55 GHz$ to $65 GHz$ and $T_p=1 \\mu s$, $c=3 * 10^8 \\frac{m}{s}$, $D = 20 cm$, $L=2.5 mm$ and the number of EDs along the SWWG is set to 30. The received $SNR$ in each antenna is set to $20dB$ and the DoA of the signal is set to $60^o$. Figure \\ref{simfig1} shows the result of applying the matched filter of (\\ref{matchedfilter}) for differnt $\\theta$. As Figure \\ref{simfig1} illustrates the output shows a distinctive peak at $60^o$. Moreover, Figure \\ref{simfig1} illustrates that PS output pattern is similar to beam-pattern of a ULA with 13 elements. This result may seem contradictory to (\\ref{m\/d}), which indicates that FIM of angle estimation using PIMs equals to a FIM of a ULA with $m$ elements, in which $m$ obeys (\\ref{m\/d}), that results in $m=33$ applying the mentioned parameters. Keep in mind that, (\\ref{CRMPIM}) shows CRLB of angle estimation using PIMs if and only if signal from one source is received at PIA, on the other hand, Figure \\ref{simfig1} shows how PS can discriminate between two or more signals if they are originated from different DoAs. As (\\ref{CRMPIM}) indicates, this bound is only a function of $D$ and SNR, while (\\ref{Res}) shows that DoA estimation resolution is a function of $BD$, which means our technique can discriminate between two incoming DoAs if and only if $B$ is wide enough. \n \\begin{figure}\n \\centering\n \\includegraphics[width=5in,height=3.5in]{Fig11.png}\n \\caption{The matched filter of (\\ref{matchedfilter}) is applied to phase differences measured for 40 pilots of a frequency code-book that changes between $[55,65] GHz$ and the output is plotted for $\\theta$ between $[0,180]^o$ and is compared with a beam pattern of a ULA with 13 elements \\cite{er1990linear}. $DoA_{in}=60^o$, $f_c=60Ghz$, $SNR_{in}=20dB, \\frac{BD}{c}=6.67$. $D=20cm$ }\n \\label{simfig1}\n\\end{figure}\n \n In the following simulation we are going to study DoA estimation resolution of the technique. In this simulation, parameters are the same as the first simulation, unless, we assume that the signal received at PIA from two different paths and two different DoAs, we investigate whether the proposed technique can distinguish between these two DoAs or not. Figure \\ref{simfig2} shows the matched filter output for 4 different pairs of DoAs, the gap between 2 DoAs are $20^o,15^o,10^o,5^o$ respectively. As Figure \\ref{simfig2} illustrates, when the gap between two DoAs is $20^o$, two lobs regarding each DoA are completely separated and distinguishable. When the gap resuces to $15^o$, two lobs start merging together, however, two peaks regarding two DoAs are again distinguishable. As the gap further reduces to $10^o$, two lobes merges more and two peaks are hardly distinguishable. And finally when the gap reduces to $5^o$, two lobes completely merge together and two peaks are not distinguishable. With respect to (\\ref{Res}), the DoA resolution with $B=10Ghz$ and $D=20cm$ is approximated to be $17^o$. Since we calculate (\\ref{Res}) assuming matched filters of two DoAs are perpendicular to each other, which means that two lobes are completely separated, thus simulations results are in compliance with (\\ref{Res}). However, it seems that it is a strict metric for DoA resolution, to assume that two DoA are resolvable only if two lobes are completely separated. In practice, we may use $75\\%$ or $50\\%$ of (\\ref{Res}) as a more realistic metric of the resolution. In Figure \\ref{simfig4}, we illustrate matched filter main-lobe width and (\\ref{Res}) versus the parameter $\\frac{BD}{c}$, given $DoA=60^o$. Main-lobe width is defined as the gap between the minimum and the maximum $\\theta$ in which the matched filter output is closer than 3db to its peak. As Figure \\ref{simfig4} expresses, main-lobe width for $\\frac{BD}{c}=6.67$ is $8.5^o$ which is half of the figure calculated by (\\ref{Res}), moreover, this proportion between main-lobe width and (\\ref{Res}) almost holds for every $\\frac{BD}{c}$. Therefore we can use half of (\\ref{Res}) as the DoA estimation resolution if we consider the more practical main-lobe width metric. \n \n \\begin{figure}\n \\centering\n \\includegraphics[width=7in,height=5in]{fig2.pdf}\n \\caption{To analyse DoA estimation resoltion, the matched filter outputs are depicted for 4 different pairs of incoming DoAs : (a) $40^o,60^o$, (b) $45^o,60^o$, (c) $50^o,60^o$, (d) $55^o,60^o$. $SNR_{in}=20dB, \\frac{BD}{c}=6.67$. }\n \\label{simfig2}\n\\end{figure}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=4.5in,height=3in]{fig4.png}\n \\caption{DoA estimation resolution based on the main-lobe width metric and the meric introduced by (\\ref{Res}). $SNR_{in}=20dB$, $DoA=60^o$ }\n \\label{simfig4}\n\\end{figure}\nIn the next simulation we analyse the effect of input SNR on DoA estimation error for differnt values of $B$ and $D$. In this simulation, input SNR changes in the interval $[-15,20]dB$. To analyse the error we calculate the root-mean-square error (RMSE) for each input SNR, by repeating the simulation 1000 times and find the average of SE for each SNR. Figure \\ref{simfig3} illustrates RSME of DoA estimation error. As Figure \\ref{simfig3} illustrates angle estimation error depends on $BD$, as $BD$ and SNR increases, error declines. Similarly, Figure \\ref{CDF} shoes that error CDF of PIAs with equal $BD$ factors are roughly the same. This is consistent with our results on angle resolution. However, it may seems inconsistent with (\\ref{CRMPIM}), which indicates that CRLB of DoA estimation decrease in proportion to $D$ not $BD$, this is because we employ matched filter of (\\ref{matchedfilter}) to find the DoA. To improve the precision, future works may considering using the output of the matched filter only to disambiguate the phase to a valid TDoA and estimate DoA directly based on the TDoA.\n\nIn the next simulation we consider a scenario in which, frequency steps, band-width, antenna gap and integration time is strongly limited. In this scenario, the source can only send 4 pilots at $[59.5,59.83,60.16,60.5]GHz$, $D=1cm$, $T_p=100ns$ and the whole number of available time slots is $M$. The source send those four frequencies in $M$ time slots respectively and repeats sending them until covers the whole $M$ slots. Consequently, the integration time is $MT_p$ -the maximum integration time in this simulation is $16 \\mu s$-. We also assume there is only one incoming DoA at the PIA, since the PIA is not able to discriminate between two DoAs because of limited bandwidth and short antennas' gap. As Figure \\ref{tightcon} shows, the proposed technique is able to estimate DoA with RMSE less than $10^o$ if input SNR is high enough, for $M=160$ input SNR should be above $7dB$ and for $M=20$ input SNR should be above 16dB. Therefore, as input SNR levels decreases we should increase integration time of our technique to provide us with acceptable DoA estimation precision. \n\nIn the next simulation, parameters are the same, unless there is a NLoS path ($30^o$) besides the LoS ($90^o$) path with a power 15 dB less than LoS path. This simulation is consistant with the experimental results of \\cite{rappaport2013millimeter} on distribution of DoA paths between TX and RX in an urban environment in Brooklyn, New York. In This simulation integration time is set to $40 \\mu s$. As Figure \\ref{tightconSIR} shows existence of the second path does not have a considerable effect on RMSE of the proposed technique. Therefore, it seems that even a very simplified version of the proposed technique (narrow beam-width, short antenna gap) can be used in real world practical mm-wave DoA estimation applications.\n\nIn the next simulation, we investigate DoA estimation precision based on power of NLoS path. Given LoS path arrives at $90^o$ and NLoS path arrives at $30^o$ at the PIA, Figure \\ref{tightconSIR} depicts RMSE of DoA estimation versus power ratio of LoS path to NLoS path. we set the integration time to be $4 \\mu s$, since NLoS path can be considered as a coherent interference, thus SIR won't be improved by integration. Figure \\ref{tightconSIR} expresses that RMSE drops below $10^o$ when SIR is higher than $8dB$ and $5^o$ when SIR in higher than $12dB$. Referring to \\cite{rappaport2013millimeter}, the power of the strongest NLoS path expects to be more than 15dB weaker than the LoS path in a dense urban environment, therefore we expect that the proposed technique can estimate DoA of LoS path in an urban environment with error less than $3^{o}$ even when the available bandwidth is very limited (e.g. 1GHz) and the antenna gap is very short (1cm). Such a performance make PS a promising technique for beam initialization requirements of 5G networks, since the required band-width is easily accessible in mm-wave and the PIA size is very small that make it easily implementable on any device. \n\nIn the last simulation we compare the performance of PS technique with a ULA (beamforming) in terms of DoA estimation precision of a single incoming path. ULA exploits beamforming to steer its beam and compare received power from different angles to find DoA. Figure \\ref{ULAPS} depics RSME of DoA estimation for 3 PIAs with different values of $D$ and $B$ and 3 ULAs with different number of array elements. In this simulation, we suppose that ULA is able to integrate the received signal coherently for $T_p$, we also set $T_p=100ns$ and $M=200$, therefore the total integration time of the PIA is $20 \\mu s$ . The $B_{rf}$ for ULA and PIA is the same and is set to $100MHz$. As the figure illustrates, the performance of the PIA with $D=10cm$ and $B=10GHz$ is approximately equal to ULA with 20 antennas equally spaced with half wavelength gap (array aperture is 5cm) especially for SNR above -9 dB. Moreover, performance of ULA with 4 elements is close to PIA with $BD=10^8$. Consider that for SNRs above -3 dB, the RMSE is less than $5^o$ for an array with 4 elements, while, beam-width of the array is about $30^o$. If such wide beam antenna uses for communication, the angle estimation precision is much more than what is required. As we discussed in section \\ref{DoABW}, angle estimation precision and beamwidth are not coupled and there is no necessity for antennas of SDMA and IA tasks to be the same.\nMoreover, even when array aperture is small and the number of array elements is few, to obtain a DoA with desirable accuracy a long spatial search is required. For example, to reach an accuracy of $1^o$, any directional antenna with an arbitrary beam-width requires to search at least 180 points to cover a $180^o$ area, in a 2D scenario. On the other hand, to improve PS precision we can simply increase the gap between two antennas and therefore no more complex hardware is required. Furthermore, better precision with ULA requires narrower beams and consequently more time is needed for spatial search to perform the IA task. On the other hand, since no spatial search is required by PS technique, we can obtain an initial guess of DoA very fast, and gradually improve the precision of the estimation by improving SNR through integration. \n\n\\section{Conclusion}\n\\label{conc}\nIn this paper, we have introduced DoA estimation via SWR. We have shown that how SWR measures phase difference between two antennas for different frequencies named as PDoAs. We have considered two different implementation schemes for PS: 1- using a long wave-guide to measure amplitude of a standing wave, produced by interaction between two waves received at the two antennas 2- measuring the amplitude of the standing wave inside a short wave-guide for different frequencies of a frequency code-book at different time slots. Moreover, for the second scheme, we have explained that we can use a cascade of multiple PS receivers to measure PDoAs at different frequencies concomitantly. We have developed a signal processing method to extract multiple simultaneous DoAs from PDoAs. We have analyzed processing gain of the technique and discussed that we can take advantage of the required time for spatial search essential for directional techniques to improve DoA estimation precision in PS. Finally, we have analyzed that IA and SD tasks of mobile directional communication can be separated and performed via two dedicated antennas; IA can be performed by PS, SD can be by performed by an array. The separation between these two tasks, reduces delay and overhead and increases communication capacity. Our results have shown that, PS can perform similar to an array, while the required receiver is much less complex than the array receiver, and the spatial search required for DoA estimation can be bypassed. \n\n \\begin{figure}\n \\centering\n \\includegraphics[width=4.5in,height=3in]{FigSNRBD.png}\n \\caption{RSME of DoA estimation versus input SNR for differnt valus of $B$ and $D$. $DoA=60^o$, $S_f=40$,$B_{rf}T_p=100$.} \n \\label{simfig3}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\includegraphics[width=1.2\\textwidth]{CDFminus10dB.png}\n \\caption{SNR=-10dB}\n \\label{CDF1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\includegraphics[width=1.2\\textwidth]{CDFminus5dB.png}\n \\caption{SNR=-5dB}\n \\label{CDF2}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\includegraphics[width=1.2\\textwidth]{CDF0dB.png}\n \\caption{SNR=0dB}\n \\label{CDF3}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\includegraphics[width=1.2\\textwidth]{CDF5dB.png}\n \\caption{SNR=5dB}\n \\label{CDF4}\n \\end{subfigure}\n \\caption{CDF of angle estimation error for differnt values of $D$, $B$ and input SNR. $DoA=60^o$, $S_f=40$,$B_{rf}T_p=100$.}\n \\label{CDF}\n\\end{figure}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=4.5in,height=3in]{FigSNR.png}\n \\caption{RSME of DoA estimation. Source only transmits four pilots at $[59.5,59.83,60.16,60.5]GHz$, each in a time slot with duration $T_p$, source repeats emitting these monotones for $M$ time slots. $DoA=60^o$, $T_p=100ns$, $B_{rf}=100MHz$, $D=1cm$.} \n \\label{tightcon}\n\\end{figure}\n\n \\begin{figure}\n \\centering\n \n \\includegraphics[width=4.5in,height=3in]{FigSIR.png}\n \\caption{RSME of DoA estimation, signal receives at PIA via two paths, one LoS path ($90^o$) and one NLoS path ($30^o$) in the presence of coherent interference. Source only transmits four pilots at $[59.5,59.83,60.16,60.5]GHz$. $SIR=15dB$, $M=400$, $DoA=60^o$, $T_p=100ns$, $B_{rf}=100MHz$, $D=1cm$.} \n \\label{tightconSIR}\n\\end{figure}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=4.5in,height=3in]{FigvarSIR.png}\n \\caption{Signal receives at PIA via two paths, one LoS path ($90^o$) and one NLoS path ($30^o$), . $M=40$, $T_p=100ns$, $B_{rf}=100MHz$, $D=1cm$ and source frequency codebook is $[59.5,59.83,60.16,60.5]GHz$.} \n \\label{tightconvarSIR}\n\\end{figure}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=4.5in,height=3in]{BeamformingVsPS.png}\n \\caption{Comparing PS with ULA with half-wavelength gap between array elements in terms of DoA estimation precision. $M=200$, $T_p=100ns$, $B_{rf}=100MHz$} \n \\label{ULAPS}\n\\end{figure}\n\n\\newpage\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nLet $G$ be a residually finite group endowed with a word metric given by a finite generating set $X$.\nA subset $S \\subseteq G$ is \\emph{fully detected} by a group $Q$ if there exists a homomorphism $\\varphi: G \\to Q$ such that $\\varphi |_S$ is injective.\nFor a natural number $n$, set $\\G_G^X(n)$ to be the minimal order of a group $Q$ that fully detects the ball of radius $n$ in $G$ (first studied in \\cite{BM11}).\nThe \\emph{full residual finiteness growth of $G$ with respect to $X$}\nis the growth of the function $\\G_{G}^{X}$, that is, its equivalence\nclass under the\nequivalence relation defined by $f\\approx g$ if and only if there is a\nconstant $C$ so that $f(n) \\leq Cg(Cn)$ and $g(n)\\leq Cf(Cn)$ for all\nnatural numbers $n$.\nThe growth of $\\G_G^X$ is independent of choice of generating set $X$ (see Lemma \\ref{lem:containment}). Therefore\nfull residual finiteness growth is an invariant of a finitely\ngenerated group, and can be denoted simply $\\G_G$.\n\nThis article focuses on finitely generated nilpotent groups. \nWhile it is known that \\emph{word growth} (defined below after Theorem\n\\ref{MainTheorem}) has precisely polynomial growth over this class of\ngroups \\cite{Bass72}, computing other growth functions\nfor this class has proved to be a serious task. \nIndeed, even computing the answer for \\emph{subgroup growth} \\cite{MR1978431} in the two-generated free nilpotent case takes work; see \\cite{MR2342452}.\nThe main difficulty lies in that the structure of $p$-group quotients of a fixed finitely generated nilpotent group can depend heavily on the choice of the prime $p$.\nThat is, it is difficult to draw global behavior (behavior over all finite quotients) from local behavior (behavior over all finite quotients that are $p$-groups).\nMoreover, comparisons between full residual finiteness growth and\n word growth, which is to our knowledge the only nontrivial growth\n function known to have precisely polynomial growth over the class of\n nilpotent groups, do not allow one to immediately draw much information on $\\G_G^X$.\nIn fact, the growth of $\\G_G^X$ is often, but not always, strictly larger than the word growth of $G$ (see Theorem~\\ref{thm:abelian}).\nObtaining \\emph{sharp} control of full residual finiteness growth\nover this class requires new understanding of the structure theory of\nnilpotent groups.\n\nTo present our findings, we begin with some basic examples. In \\S\\ref{sec:examples}, we show that $\\G_G(n) = n^k$ for $G=\\mathbb{Z}^k$ and\n$\\G_G(n) = n^6$ for $G$ equal to the discrete Heisenberg group. The key\nproperty shared by these examples is that the center of $G$ is equal\nto the last term of the lower central series of $G$. In\nfact, we explicitly compute full residual\nfiniteness growths for all groups satisfying a slightly weaker condition.\nTo make this precise we introduce notation: for a nilpotent group $G$\nof class $c$ we denote by $\\gamma_c(G)$ the last nontrivial term of\nits lower central series, by $Z(G)$ its center, and by $\\dim(G)$ its\ndimension. (See \\S\\ref{sec:nilpotent} for more explicit definitions.)\n\n\\begin{introtheorem} \\label{theorem:zl} Let $G$ be a finitely generated\n nilpotent group of class $c$ with $[Z(G):\\gamma_c(G)] < \\infty$.\n Then\n $$\n \\G_G(n) \\approx n^{c \\dim(G)}.\n $$\n\\end{introtheorem}\n\nThe conclusion of Theorem \\ref{theorem:zl} does not generally hold when $[Z(G) :\n\\gamma_c(G)] = \\infty$. This is seen by taking $G$ to be the\ndirect product of the discrete Heisenberg group with $\\mathbb{Z}$, which\nsatisfies $\\G_G(n) \\approx n^7$ while $c=2$ and $\\dim(G) = 4$.\nGroups not satisfying the hypothesis of Theorem \\ref{theorem:zl} are\ngenerally more complicated than this example.\nFor instance, in Proposition \\ref{D22example} we provide an example of\na nilpotent group $\\Gamma$ of class $c=3$ with $\\dim(\\Gamma) = 8$ and\n$\\G_\\Gamma(n) \\approx n^{22}$ that does not split as a direct product.\n\nFor general nilpotent group $G$ of class $c$, we introduce methods to find an upper\nbound on the polynomial degree of $\\G_G$.\nDefine a {\\em terraced filtration} of $G$ to be a filtration $1 = H_0 \\leq\nH_1 \\leq \\dotsb \\leq H_{c-1}\\leq G$ where each $H_i$ is a maximal normal\nsubgroup of $G$ satisfying $H_i\\cap \\gamma_{i+1}(G) = 1$. \nEvery terraced filtration of $G$ gives an explicit polynomial upper bound\non growth of $\\G_G$.\n\n\\begin{introtheorem} \\label{MainTheorem}\nLet $G$ be a finitely generated nilpotent group of class $c$.\nSuppose $1=H_0\\leq H_1 \\leq \\dotsb H_{c-1}\\leq G$ is a terraced\nfiltration of $G$. Then\n\\[\n\\G_G(n) \\preceq n^{c \\dim(G) - \\sum_{i=1}^{c-1} \\dim(H_i)}.\n\\]\n\\end{introtheorem}\n\\noindent\nThis upper bound generally depends on the choice of terraced\nfiltration. See the comments following the proof of Theorem\n\\ref{MainTheorem} in \\S\\ref{MainProofSection} for an explicit example\ndemonstrating this dependence. It would be interesting to determine\nwhether the lowest upper bound obtained from a terraced filtration\nby Theorem \\ref{MainTheorem} is optimal.\n\nResults on distortion in nilpotent groups from Osin \\cite{MR1872804} and Pittet \\cite{Pittet97} play an important role in all of our proofs.\n\nWe also compare full residual finiteness growth to word growth.\nRecall that the {\\em word growth}, $w_G$, of a finitely generated group $G$ is the\ngrowth of the function $w_G^X(n) = \\left\\lvert B_G^X(n) \\right\\rvert$,\nwhich is independent of $X$.\nGromov \\cite{MR623534} has characterized nilpotent groups in the class\nof finitely generated groups as those for which $w_G$ is polynomial. \nBy applying this theorem, it is shown in \\cite{BM11} (see Theorem 1.3 there) \nthat full residual finiteness growth enjoys the same conclusion.\nIn spite of this similarity, these two growths rarely coincide.\nOur final result characterizes nilpotent groups for which full residual\nfiniteness growth equals word growth.\n\n\\begin{introtheorem} \\label{thm:abelian}\nLet $G$ be a finitely generated nilpotent group.\nThen $\\G_G \\approx w_G$ if and only if $G$ is virtually abelian.\n\\end{introtheorem}\n\\noindent\nIn \\S\\ref{sec:fullresidualfinitenessgrowth} we provide a geometric\ninterpretation of $\\G_G^X$. From this point of view, Theorem\n\\ref{thm:abelian} implies that virtually abelian groups are characterized in the\nclass of finitely generated nilpotent groups solely in terms of the\nasymptotic data of the Cayley graph.\nA non-normal version of full residual finiteness growth,\n the \\emph{systolic growth}, is studied in \\cite{YC14}.\nThere it is shown that systolic growth matches word growth if and only if the group is \\emph{Carnot}.\n\n\nThis paper is organized as follows: In \\S\\ref{BackgroundSection} we\npresent basic results on nilpotent groups and full residual finiteness\ngrowth, including important lemmas on\nword metric distortion of central subgroups of nilpotent groups\nfollowing from work of Osin \\cite{MR1872804} and Pittet \\cite{Pittet97}. In\n\\S\\ref{sec:abeliangroups} we prove Theorem \\ref{thm:abelian}. In\n\\S\\ref{sec:heisenberg} we compute the full residual\nfiniteness growth of the Heisenberg group and prove\nTheorem \\ref{theorem:zl}. \nIn \\S\\ref{MainProofSection} we give an illustrative example showing that the\nconclusion of Theorem \\ref{theorem:zl} does not hold in general, and\nprove Theorem \\ref{MainTheorem}.\n\nWe finish the introduction with a bit of history.\nThe concept of full residual finiteness growth was first studied by\nBen McReynolds and K.B. in \\cite{BM11}. The full\n residual finiteness growth of the discrete Heisenberg group is\n presented in \\cite{YC14}.\nCompare full residual finiteness growth to the concept of \\emph{residual finiteness growth}, which measures\nhow well individual elements are detected by finite quotients,\nappearing in \\cite{B09},\n\\cite{MR2583614}, \\cite{BM13}, \\cite{KM12}, \\cite{R12}, \\cite{BK12}, \\cite{KMS13}.\nAlso compare this with Sarah Black's \\emph{growth function} defined and studied in \\cite{MR1659911}. \nFull residual finiteness growth measures how efficiently the word growth function \ncan be recovered from Black's growth function. See remarks in \\cite{MR1659911} on p.\\ 406 before \\S 2 for further discussion.\n\n\\paragraph*{Acknowledgements}\n\nThe authors are grateful to Benson Farb for suggesting this pursuit.\nThe authors acknowledge useful conversations with Moon Duchin, Michael\nLarsen, Ben McReynolds, Christopher Mooney, and Denis Osin. The authors are\nfurther grateful to Benson Farb and Ben McReynolds for comments on\ndrafts of this paper.\nK.B. gratefully acknowledges support from the AMS-Simons Travel Grant Program.\nThe authors are very grateful to the excellent referee for comments\nand corrections that greatly improved the paper and for suggesting\nProposition \\ref{prop:Treduction}.\n\n\\section{Some background and preliminary results} \\label{BackgroundSection}\n\n\\subsection{Full residual finiteness growth} \\label{sec:fullresidualfinitenessgrowth}\n\nIn this subsection we give a geometric interpretation of full residual\nfiniteness growth for finitely presented groups.\n\nWrite $f \\preceq g$ to mean there exists $C$ such that $f(n) \\leq C g(Cn)$.\nWe write $f \\approx g$ if $f \\preceq g$ and $g \\preceq f$.\nRecall that the \\emph{growth} of a function $f$ is the equivalence class of $f$ with respect to $\\approx$.\n\nWe first prove a lemma that implies that the growth of the function\n$\\G_G^X$ defined in the introduction is independent of generating set\n$X$:\n\n\\begin{lemma} \\label{lem:containment}\nLet $G$ be finitely generated with finitely generated subgroup $H \\leq G$.\nFix finite generating sets $X$ and $Y$ for $G$ and $H$.\nThen $\\G^Y_H \\preceq \\G_G^X$.\n\\end{lemma}\n\n\\begin{proof}\nSince $H \\leq G$, there exists $C > 0$ such that any element in $Y$ can be written in terms of at most $C$ elements in $X$.\nThus, $B_H(n) \\subseteq B_G(Cn)$ for any $n > 1$.\nBecause any homomorphism from $G$ restricts to a homomorphism from\n$H$, this gives\n$$\n\\G_G^X(Cn) \\geq \\G_H^Y(n),\n$$\nas desired.\n\\end{proof}\n\nLemma \\ref{lem:containment} in particular implies that if $X$ and $Y$\nare two finite generating sets of a group $G$, then $\\G_G^X \\approx\n\\G_G^Y$. Let $\\G_G$ denote the equivalence class of $\\G_G^X$ with\nrespect to $\\approx$ for any finite generating set $X$ of $G$.\n\nWe now provide a geometric interpretation of $\\G_G$ in the case that\n$G$ is a finitely presented group.\nLet $G$ be a residually finite group with Cayley graph $\\Gamma$ with\nrespect to a finite generating set $S$.\nEach edge of $\\Gamma$ is labeled by the corresponding generator.\nFor a subset $X \\subseteq \\Gamma$, we set $\\partial X$ to be the collection of edges and vertices of $X$ each of which has closure not contained in the interior of $X$.\nLet $\\{ A_k \\}$ be an increasing sequence of finite connected subsets of $\\Gamma$ with \n$$\nA_{k+1} = \\partial A_{k+1} \\sqcup A_k.\n$$\nThen the sequence of subsets, $\\{A_k \\}$, is called a \\emph{growing\n sequence}. Let $B_{G}^S(n)$ denote the closed ball of radius $n$ in\nthe Cayley graph of $G$ with respect to the word metric induced by\n$S$. We will omit the $S$ from the notation when the generating set is\nunderstood and there is no chance for confusion. The prototypical\nexample of a growing sequence is the sequence that assigns to each\npositive integer $k$ the metric ball $B_{G}^S(k)$ in the Cayley graph\nof $G$ with respect to $S$.\n\nThe \\emph{geometric full residual finiteness growth of $\\Gamma$ with respect to\n $\\{ A_k \\}$} is the growth of the function, $\\G_{\\Gamma}^{\\{ A_k \\}}: \\mathbb{N} \\to \\mathbb{N}$, given by \n\\begin{eqnarray*}\nn \\mapsto \\min \\{ |Q| : \\text{ $Q$ is a group with $A_n$ isometrically} \\\\\n\\text{embedding in one of its Cayley graphs}\\}.\n\\end{eqnarray*}\n\nOur first lemma demonstrates that the growth of $\\G_G^{\\{A_k \\}}$ does\nnot depend on the growing sequence.\n\n\n\\begin{lemma} \\label{lem:indepofgrowingset}\nLet $\\{ X_k \\}$ and $\\{ Y_k \\}$ be two growing sequences for a finitely generated group $G$.\nThen $\\G_{G}^{\\{X_{k} \\}} \\approx \\G_{G}^{\\{ Y_{k} \\}}$.\n\\end{lemma}\n\n\\begin{proof}\nWe first assume that $\\{ X_k \\}$ and $\\{ Y_k \\}$ are growing sequences from the same Cayley graph realization of $G$.\nThen there exists $K \\in \\mathbb{N}$ such that\n$$\nY_1 \\subseteq X_K \\text{ and } X_1 \\subseteq Y_K.\n$$\nHence, $C_G^{\\{Y_k\\}} (n) \\leq C_G^{\\{X_{k}\\}}(K+i) \\text{ and } C_G^{\\{X_k\\}} (n) \\leq C_G^{\\{Y_{k}\\}}(K+i).$\nThus, we can assume that $\\{ X_k \\}$ and $\\{ Y_k \\}$ are the word metric $k$-balls of $G$ with respect to two different generating sets.\nIt is straightforward to see that there exists $C > 0$ such that $Y_{n} \\subseteq X_{Cn} \\subseteq Y_{C^2n}$ for every natural number $n$.\nHence,\n$$\n\\G_G^{\\{Y_k\\}} (n) \\leq \\G_G^{\\{X_k\\}} (Cn) \\leq \\G_G^{\\{Y_k\\}} (C^2 n),\n$$ as desired.\n\\end{proof}\n\nNext we show that the notions of full residual finiteness growth,\ngiven in the introduction, and geometric full residual\nfiniteness growth, given in this section, agree in the case that the group\n$G$ is finitely presented.\nIt would be interesting to determine if this equivalence holds for all finitely generated groups.\n\n\\begin{lemma} \\label{lem:cayleyvsgirth}\nLet $G$ be a finitely presented group.\nFor any generating set $X$ and growing sequence $\\{ A_k \\}$ we have\n\\[\\G^{\\{ A_k \\}}_G \\approx \\G^{X}_G.\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $X$ be a finite generating set for $G$ and let $R$ be the set of finite relations.\nIt is clear that $\\G^{\\{ A_k \\}}_G \\preceq \\G^{X}_G$. We show the reverse inequality.\nWe can, by Lemma \\ref{lem:indepofgrowingset}, suppose that the growing set $\\{ A_k \\}$ is simply the sequence $ \\{ B_G^X(k) \\}$.\nIt suffices, then, to show that there exists $N \\in \\mathbb{N}$ such that for any $n > N$ and any finite group, $Q$, with $B_G(n)$ isometrically embedding in a Cayley graph realization of $Q$, there exists a homomorphism\n$\\phi: G \\to Q$ with $\\phi |_{B_G(n)}$ being injective.\nSelect $N$ to be the maximal word length of any element in $R$.\nThen since $B_n$ isometrically embeds in a Cayley graph of $Q$, we see that there exists a generating set for $Q$ such that each relator $R$ is satisfied by this generating set.\nThis finishes the proof.\n\\end{proof}\n\nThe next lemma controls some of the full residual finiteness growth of a direct product of groups.\n\n\\begin{lemma} \\label{lem:directproducts}\nLet $G$ and $H$ be finitely generated groups.\nThen \n$$\n\\G_{G \\times H} \\preceq \\G_G \\cdot \\G_H.\n$$\n\\end{lemma}\n\n\\begin{proof}\n Fix generating sets $X$ and $Y$ for $G$ and $H$. Then $(X \\times \\{\n 1 \\}) \\cup ( \\{1\\} \\times Y )$ is a finite generating set for $G \\times H$.\n Note that\n $$\n B_{G\\times H} (n) \\subseteq (B_G (n) \\times \\{1\\}) (\\{1\\} \\times B_H(n)).\n $$\n Thus, if $Q_1$ is a quotient that fully detects $B_G(n)$ and $Q_2$ a quotient that fully detects $B_H(n)$, then $Q_1 \\times Q_2$ fully detects $B_{G \\times H}(n)$.\n We see then that $\\G_{G \\times H} \\preceq \\G_G \\G_H$, as desired.\n\\end{proof}\n\\noindent\nCan the conclusion of Lemma \\ref{lem:directproducts} be improved to $\\G_{G\\times H} \\approx \\G_G \\G_H$?\nThis can possibly be false: it is not even true that if $\\varphi : G \\to H$ is a surjective homomorphism, then $\\G_G(n) \\succeq \\G_H(n)$. Consider a free group mapping onto one of Kharlampovich-Sapir's solvable and finitely presented groups of arbitrarily large residual finiteness growth \\cite{KMS13}.\n\nFull residual finiteness growth is well-behaved under taking the\nquotient by a finite normal subgroup:\n\n\\begin{proposition} \\label{prop:Treduction}\n\tLet $G$ be a finitely generated residually finite group.\n\tLet $T$ be a finite normal subgroup of $G$.\n\tThen $\\Phi_G \\approx \\Phi_{G\/T}$.\n\\end{proposition}\n\n\\begin{proof}\n\tFix a generating set $X$ for $G$, and let $Y$ be the image of $X$ under the quotient map $G \\to G\/T$. Let $K$ be the largest length, with respect to $X$, of an element in $T$.\n\tWe first claim $\\Phi_G^X(K+n) \\geq \\Phi_{G\/T}^Y(n)$.\n\tLet $\\phi: G \\to Q$ be a finite quotient of minimal cardinality that fully detects $B_G^X(K+n)$. That is $|Q| = \\Phi_G^X(K+n)$.\n\tDefine $\\psi : G\/T \\to Q\/\\phi(T)$ by $gT \\mapsto \\phi(g) \\phi(T)$.\n\tLet $g \\in B_{G\/T}^Y(n) \\cap \\ker \\psi$. \n\tBy construction, we may lift $g$ to an element $\\tilde g \\in G$ such that $\\tilde g \\in B_{G}^X(n)$ and $\\phi(\\tilde g) \\in \\phi(T)$.\n\tThat is, there exists $t \\in T$, such that $\\phi(g) = \\phi(t)$, which gives\n\t$$\n\t\t\\phi(\\tilde g t^{-1}) = 1.\n\t$$\n\tIf $\\tilde g t^{-1} \\neq 1$, then this contradicts that $\\phi$ fully detects $B_G(K+n)$. Hence, $\\tilde g = t$, and so $\\ker \\psi \\cap B_{G\/T}^Y(K+n)$ is trivial. It follows that $\\psi$ fully detects $B_{G\/T}^Y(n)$, and so $\\Phi_G^X(K+n) \\geq \\Phi_{G\/T}^Y(n)$, as claimed.\n\n\tSince $G$ is residually finite and $T$ is finite, there exists a normal subgroup, $H$, such that $T\\cap H = 1$.\n\tTo finish, we claim that $\\Phi_{G}^X(n) \\leq [G:H]\\Phi_{G\/T}^Y(n)$.\n\tLet $\\psi : G\/T \\to Q$ be a quotient that fully detects $B_{G\/T}^Y(n)$, with $|Q| = \\Phi_{G\/T}^Y(n)$.\n\tLet $\\phi : G \\to Q$ be the natural map $G \\to G\/T \\to Q$.\n\tSet $N = \\ker \\phi \\cap H$.\n\tClearly, $[G: N] \\leq [G: \\ker \\phi] [G:H] = |Q| [G:H]$.\n\tMoreover, if $g \\in B_G^X(n) \\cap N$, then $g \\notin T$.\n\tHence, by the construction of $Y$, we have that $\\phi(g) \\neq 1$.\n\tIt follows that $G\/N$ fully detects $B_G^X(n)$, and so\n\t$\\Phi_{G}^X(n) \\leq [G:H]\\Phi_{G\/T}^Y(n)$, as desired.\n\\end{proof}\n\n\n We finish the section with a lemma that, in some restrictive cases, allows us to pass to finite-index subgroups.\n\n\\begin{lemma} \\label{lem:finiteindex}\nLet $G$ and $H$ be finitely generated nilpotent groups with $H$ normal\nsubgroup in $G$ of finite index.\nIf every normal subgroup of $H$ is normal in $G$, then $\\G_G \\approx \\G_H$.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lem:containment}, it suffices to show that $\\G_G \\preceq \\G_H$.\nFix generating sets for $G$ and $H$ so that $B_H(n) \\subseteq B_G(n)$\nfor all $n>0$.\nBecause $H$ is of finite index in $G$ and thus quasi-isometric to $G$, there exists $C > 0$ such that $H \\cap B_G(2n) \\subseteq B_H(Cn)$.\nLet $H\/K$ be a quotient of $H$ that fully detects $B_H(Cn)$.\nBy our assumption, $K$ is normal in $G$ so $G\/K$ is well-defined.\nThen any element in $B_G(2n)$ not in $H$ is mapped nontrivially onto $G\/K$.\nAnd since $H \\cap B_G(2n) \\subseteq B_H(Cn)$, it follows that $B_G(2n)$ is mapped nontrivially onto $G\/K$.\nThus, $B_G(n)$ is fully detected by $G\/K$, and so we are done.\n\\end{proof}\n\n\n \n\\subsection{Nilpotent groups} \\label{sec:nilpotent}\n\nIn this subsection we fix basic notation and present several lemmas that play important roles in our proofs.\nLet $G$ be a group. The \\emph{lower central series} $\\gamma_{k} (G) $ of $G $ is the sequence of subgroups defined by $\\gamma_{1}(G) = G$ and \n$$\\gamma_{k}(G) =[\\gamma_{k-1}(G),G].$$ \nFor any group $H$, let $Z(H)$ denote the center of $H$.\nThe \\emph{upper central series} $\\zeta_k (G) $ of $G $ is\ngiven by $\\zeta_{0}(G)=\\{e\\}$ and the formula \n$$\n\\zeta_{k}(G) \/ \\zeta_{k - 1} (G) =\nZ(G\/\\zeta_{k-1}(G)).\n$$ \nThe group $G$ is said to be \\emph{nilpotent} if $\\gamma_{k}(G) = 1$\nfor some natural number $k$. Equivalently, $G$ is nilpotent if and\nonly if it is an element of its upper central series. Moreover, $G$ is\nsaid to be \\emph{nilpotent of class $c$} if $\\gamma_c(G) \\neq 1$ and\n$\\gamma_{c+1}(G) = 1$.\n\nIf $G$ is a finitely generated nilpotent group, then the successive quotients of the upper central series of $G$ are abelian groups of finite-rank.\nThus, the upper central series has a refinement\n$$G = G_{1} \\ge G_{2} \\ge \\ldots G_{n+1} = 1,$$\nsuch that $G_{i} \/ G_{i+1} $ is cyclic for all $i = 1, \\ldots, n$. \nThe number of infinite cyclic factors in this series does not depend on the series and is called the \\emph{dimension} of $G$, denoted by $\\dim(G)$ \\cite[p. 16, Exercise 8]{MR713786}.\nLet this series be chosen so that $n$ is minimal.\nAn $n$-tuple of elements $(g_{1}, g_{2}, \\ldots , g_{n})\\in G^ n$\nis a \\emph{basis} for $G $ if $g_{i}\\in G_{i}$ and\n$G_i\/G_{i-1}=\\left $ for\neach $i = 1,\\ldots, n $.\nIn the case when $G_{i} \/ G_{i+1} $ is infinite for all $i = 1, \\ldots, n$ we call the $n$-tuple a \\emph{Malcev basis} for $G$.\n\n\nThe set of torsion elements $T$ in a finitely generated nilpotent\ngroup $G$ is a finite normal subgroup, and the quotient $G\/T$ is a\ntorsion-free nilpotent group \\cite[p. 13, Corollary 10]{MR713786}. A\ncorollary of Proposition \\ref{prop:Treduction} is that $G\/T$ has the\nsame full residual finiteness growth as $G$.\n\n\\begin{corollary}\\label{cor:tfreduction}\n If $G$ is a finitely generated nilpotent group and $T$ is the\n subgroup of torsion elements then $\\Phi_G \\approx \\Phi_{G\/T}$.\n\\end{corollary}\n\nWe recall a folklore result, used in the proof of the following\nlemmas.\n\n\\begin{lemma} \\label{lem:commutatorproduct}\n Suppose $G$ is a finitely generated nilpotent group of class $c$. The assignment $(x, y)\n \\mapsto [x,y]$ defines a homomorphism\n\\[\n\\left( \\zeta_k(G)\/\\zeta_{k-1}(G) \\right) \\times \\left( \\zeta_\\ell(G) \/\n \\zeta_{\\ell-1}(G) \\right) \\to \\zeta_{k+\\ell-c-1}(G) \/ \\zeta_{k+\\ell-c-2}(G).\n\\]\n\\end{lemma}\n\\begin{proof}\n This follows immediately from \\cite[Theorem 2.1]{MR2366181}, noting\n that the upper central series is a central filtration of $G$ when\n indexed so that the $i^{th}$ term of the filtration is\n $\\zeta_{c+1-i}(G)$.\n\\end{proof}\n\nIf $G$ is a group generated by a finite set $X$, for $g\\in G$ we use\n$\\| g \\|_X$ to denote the word length of $g$ with respect to $X$. \nLet $G$ be a finitely generated nilpotent group.\nThe following lemma is a consequence of well-known distortion estimates.\n\n\\begin{lemma} \\label{lem:distortion}\nLet $G$ be a nilpotent group of class $c$ generated by a finite set $X$.\nFix a positive integer $i$ and a generating set $X_i$ for $Z(G) \\cap \\gamma_i(G)$.\nThen there exists $C > 1$ such that for all $g \\in Z(G) \\cap \\gamma_i(G),$\n\\begin{equation} \\label{eq:distortion}\n\\|g \\|_{X} \\leq C \\| g \\|_{X_i}^{1\/i}. \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe first assume $G$ is torsion-free. \nFirst consider the case that $g = x^m$ for some $x\\in X_i$ and $m\\in\n\\mathbb{Z}\\setminus\\{0\\}$. \nAssume without loss of generality that $X_i$ is a basis for the free\nabelian group $Z(G) \\cap \\gamma_i(G)$, so that $\\| g \\|_{X_i} = \\left\\lvert\n m \\right\\rvert$.\nEmbed $G$ as a cocompact lattice in a simply-connected nilpotent Lie group $N$, which\nidentifies $Z(G) \\cap \\gamma_i(G)$ with a lattice in a\nsimply-connected central subgroup $Z\\leq N$. Fix any left-invariant\nRiemannian metric on $N$, which gives a norm $\\| \\cdot \\|_\\mf{n}$ on\n$\\mf{n}$, the Lie algebra of $N$. Consider the path $\\gamma : [0,\n\\left\\lvert m \\right\\rvert ] \\to Z$ defined so that $\\gamma(\n\\left\\lvert m \\right\\rvert ) = g$ and\n$\\gamma(t) = \\exp( t z )$ for some $z\\in \\mf{n}$. Note that $\\exp(z) =\nx$ if $m>0$ and $\\exp(z) = x^{-1}$ if $m<0$. In particular, $z$ does \\emph{not} depend on $m$.\nBy \\cite[Prop 4.1(1)]{Pittet97}, the length of $\\gamma$ is \n$\\| z \\|_\\mf{n} \\| g\\|_{X_i}$. \nThen applying\n\\cite[Prop 4.1(2)]{Pittet97} to the curve $\\gamma$, there is a\nconstant $C>0$ depending on $z$ so that\n\\begin{equation} \\label{pitteteqn}\nd_N(e, g) \\leq C \\|g\\|_{X_i}^{1\/i}.\n\\end{equation}\nThe quantity $d_N(e,g)$ is uniformly comparable to $\\| g \\|_X$, so\nthis proves the desired inequality for $g$ of the form $x^m$.\n\nNow for any $g\\in Z(G)\\cap \\gamma_i(G)$, write $g = \\prod_{j=1}^k\nx_j^{m_j}$ where $X_i = \\{x_1,\\dotsc, x_k\\}$. Let $C$ be the largest\nconstant appearing in equation \\ref{pitteteqn} as $x$ ranges over\n$x_1,\\dotsc, x_k$. Then there is some $D > 0$ so that\n\\begin{align*}\n \\| g \\|_X & \\leq \\sum_{j=1}^k \\| x_j^{m_j} \\|_X \\\\\n & \\leq C \\sum_{j=1}^k \\| x_j^{m_j} \\|_{X_i}^{1\/i} \\\\\n & \\leq C \\sum_{j=1}^k |m_j|^{1\/i} \\\\\n & \\leq C k \\left( \\sum_{j=1}^k | m_j | \\right)^{1\/i} \\\\\n & \\leq CkD \\| g \\|_{X_i}^{1\/i}.\n\\end{align*}\nThe last step follows because $Z(G) \\cap \\gamma_i(G)$ is abelian. \nThe penultimate step follows from the general fact that $(m_1^{1\/i} +\n\\dotsb + m_k^{1\/i})^i \\leq k^i (m_1 + \\dotsb + m_k)$ when $m_j \\geq 1$\nfor all $j$. This completes the proof in the case that $G$ is torsion-free. \n \n\nNow suppose $G$ is an arbitrary finitely generated nilpotent group.\nThere is a torsion-free normal subgroup $H$ of finite index in $G$. Fix a\ngenerating set $Y$ for $H$. The map $i : H \\to G$ is a quasi-isometry\nbecause $H$ is finite index in $G$. In fact, because distinct points\nin each of $G$ and $H$ are distance at least 1 and $i$ is injective,\nit is easy to check that $i$ is bi-Lipschitz. This means that there is\nsome $C\\geq 1$ so that:\n\\begin{enumerate}\n\t\\item For $g,h \\in H$, \n\t$$\n\t\t\\frac{1}{C} \\| g h^{-1} \\|_Y \\leq \\| g h^{-1} \\|_X \\leq C \\| g h^{-1} \\|_Y.\n\t$$\n\t\\item For every element $g \\in G$, there exists $h \\in H$ such that\n\t$$\n\t\t\\| h g^{-1} \\|_X \\leq C.\n\t\t$$\n\\end{enumerate}\n\nFix generating sets $X_i$ for $Z(G) \\cap \\gamma_i(G)$\nand $Y_i$ for $Z(H) \\cap \\gamma_i(H)$. \nWe claim that $Z(H) \\leq Z(G)$.\nIndeed, if not then there exists $h \\in Z(H)$, an integer $r \\geq 1$,\nand elements $x_1, \\ldots, x_r \\in G$ such that $h \\in \\zeta_{r+1}(G)\n\\setminus \\zeta_r(G)$ and\n$$\n[h, x_1, \\ldots, x_r] \\in Z(G) \\setminus \\{1\\}.\n$$\nSince $H$ has finite index in $G$ there exists $n \\in \\mathbb{N}$ such that $x_1^n \\in H$.\nBy Lemma \\ref{lem:commutatorproduct} we have \n\\[\n[h, x_1^n, \\ldots, x_r] = [h, x_1, \\ldots, x_r]^n.\n\\]\nSince $H$ is normal we have $[h, x_1, \\ldots, x_r] \\in H$.\nThis implies $[h, x_1^n, \\ldots, x_r] \\neq 1$ because $H$ is torsion-free.\nTherefore $[h,x_1^n]$ cannot be trivial, which contradicts the fact that $h \\in Z(H)$.\nBy the aforementioned claim, $Z(G) \\cap \\gamma_i(G)$ contains $Z(H) \\cap \\gamma_i(H)$ as a subgroup.\nIn fact, it is not hard to show that $Z(H) \\leq Z(G)$ and $\\gamma_i(H)\n\\leq \\gamma_i(G)$ are, in both cases, subgroups of finite index.\nHence, the inclusion\n$i_2 : Z(H) \\cap \\gamma_i(H) \\to Z(G) \\cap \\gamma_i(G)$ is a\nbi-Lipschitz quasi-isometry with constant $D\\geq 1$.\n\nNow select $C' > 1$ such that inequality \\ref{eq:distortion} holds for all\n$g \\in G$ that are finite order (again, there are only finitely many of them).\nNext, let $g$ be an infinite order element in $G$ with \n$$g \\in Z(G) \\cap \\gamma_i(G).$$\nWe can suppose, without loss of generality, that $X$ contains $X_i$.\nThen since $i_2$ is a $D$-quasi-isometry, there exists $h \\in Z(H) \\cap \\gamma_i(H)$ such that \n$$ \\| h g^{-1} \\|_{X} \\leq \\| h g^{-1} \\|_{X_i} \\leq D.$$\nSince $H$ is torsion-free, by enlarging $C$ if necessary we have\n$$\n\\| h \\|_Y \\leq C \\| h \\|_{Y_i}^{1\/i}.\n$$\nThus,\n\\begin{equation} \\label{QI1}\n\\| g \\|_X = \\| g h^{-1} h \\|_X \\leq \\| g h^{-1} \\|_X + \\| h \\|_X\n\\leq C + \\| h \\|_X.\n\\end{equation}\nAnd, further,\n\\begin{equation} \\label{QI2}\n\\| h \\|_X \\leq C \\| h \\|_Y \\leq C^2 \\| h \\|_{Y_i}^{1\/i}.\n\\end{equation}\nTo finish,\n\\begin{equation} \\label{QI3}\n\\| h \\|_{Y_i} \\leq D \\| h \\|_{X_i} = D \\| g g^{-1} h \\|_{X_i} \\leq D(\\| g \\|_{X_i} + D ).\n\\end{equation}\nThe desired inequality follows from equations \\ref{QI1}--\\ref{QI3}, as\nall additive constants can be absorbed into the multiplicative\nconstants.\n\\end{proof}\n\nNext, we show a technical lemma that will be important in our main proofs:\n\n\\begin{lemma} \\label{lem:technical}\nLet $G$ be a nilpotent group of class $c$ generated by a finite set\n$X$. Fix a number $0 0$ such that for any $g \\in \\zeta_i(G) \\setminus \\zeta_{i-1}(G)$, there exists $x_1, \\ldots, x_{i-1} \\in X$ such that for any $\\gamma \\in \\zeta_{i-1}(G)$,\n$$\n0 < \\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_Y \\leq C_i \\|g \\|_{Y_0}.\n$$\nIn fact, there is some $F_i > 0$ so that\n$$\n0 < \\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X} \\leq F_i \\|g \\|_{Y_0}^{1\/t}, \n$$\nwhere $t$ is the minimal $k$ satisfying $[g \\gamma, x_1, \\ldots, x_{i-1}] \\notin \\gamma_{k+1}(G)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $g \\in \\zeta_i(G) \\setminus \\zeta_{i-1}(G)$ be given.\nSince $G$ is nilpotent, there exists $x_1, \\ldots, x_{i-1} \\in X$ so that\n$$\n[g, x_1, \\ldots, x_{i-1}] \\in Z(G) \\setminus \\{1 \\}.\n$$\nNote that for any $x \\in \\zeta_i(G)$ we have that \n$$\n[x, x_1, \\ldots, x_{i-1}] \\in Z(G).\n$$\nWrite $g = \\prod_{i=1}^n g_i$ where $g_i \\in Y_0$ and $n$ is the word\nlength of $g$ with respect to $Y_0$.\nApplying Lemma~\\ref{lem:commutatorproduct} repeatedly gives\n\\begin{eqnarray*}\n[g, x_1, \\ldots, x_{i-1}] &=& [g_1 g_2 \\cdots g_n, x_1, \\ldots, x_{i-1}] \\\\\n&=& [g_1, x_1, \\ldots, x_{i-1}][g_2, x_1, \\cdots x_{i-1}] \\cdots [g_n, x_1, \\ldots, x_{i-1}].\n\\end{eqnarray*}\nSet $Y'$ to be $Y$ union the set of all elements of the form $[\\beta, \\alpha_1, \\alpha_2, \\ldots, \\alpha_{i-1}]$ where $\\beta \\in Y_0$ and $\\alpha_i \\in X$.\nNotice that $Y'$ does not depend on $g$. Further, by our above computation, we have\n$$\n\\| [g, x_1, \\ldots, x_{i-1}] \\|_{Y'} \\leq n.\n$$\nBecause $Y'$ is finite, $(Z(G), d_Y)$ is bi-Lipschitz equivalent to\n$(Z(G), d_{Y'})$.\nThis gives $C_i > 0$, depending only on $Y'$, such that\n\\begin{equation} \\label{assertionone}\n0 < \\| [g, x_1, \\ldots, x_{i-1} ] \\|_Y \\leq C_i \\|[g, x_1, \\ldots, x_{i-1}] \\|_{Y'}\n\\leq C_i n = C_i \\| g \\|_{Y_0}.\n\\end{equation}\nLet $\\gamma \\in \\zeta_{i-1}(G)$ be arbitrary.\nThen as $[g, x_1, \\ldots, x_{i-1}]$ and $[\\gamma, x_1, \\ldots, x_{i-1}]$ are central,\n$$\n[g, x_1, \\ldots, x_{i-1} ] = [g \\gamma, x_1, \\ldots, x_{i-1} ],\n$$\nso the proof of the first assertion is complete. \n\nFix generating sets $X_j$ for $\\gamma_j(G) \\cap Z(G)$ for each $1\\leq\nj \\leq c$. These sets can be chosen independently of $g$ and $i$.\nBy Lemma \\ref{lem:distortion}, for each $j$ we have that there exists $D_j > 1$ such that for all $w \\in \\gamma_j(G) \\cap Z(G)$\n$$\n\\|w \\|_X \\leq D_j \\| w \\|^{1\/j}_{X_j}.\n$$\nSet $D$ to be the maximal such $D_j$. Notice that $D$ only depends on $X$ and $G$.\nSince $\\gamma_t(G) \\cap Z(G)$ is a subset of the abelian group, $Z(G)$, we have that there exists $E > 1$, depending only on $Y$ and the selection of $X_j$, such that\n$$\n\\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X_t} \\leq E \\| [g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{Y} \\leq E^2 \\| [g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X_t}.\n$$\nCombining these inequalities with Inequality \\ref{assertionone} gives\n\\begin{eqnarray*}\n0 < \\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X} &\\leq& D \\| [g \\gamma, x_1, \\ldots, x_{i-1} ] \\|_{X_t}^{1\/t} \\\\\n&\\leq& E D \\| [g \\gamma, x_1, \\ldots, x_{i-1} ] \\|_{Y}^{1\/t} \\\\\n&\\leq& E D C_i \\| g \\|_{Y_0}^{1\/t} = F_i \\| g \\|_{Y_0}^{1\/t},\n\\end{eqnarray*}\nfor some constant $F_i$ that depends only on $i$ and our choice of generating sets, as desired.\n\\end{proof}\n\nFor any $g \\in G$ of infinite order, the \\emph{weight $\\nu_G(g)$ of $g$ in the group $G$} is the maximal $k$ such that $\\left< g \\right> \\cap \\gamma_k(G) \\neq \\{ 1 \\}$.\nIf $G$ is a group and $m$ a natural number, let $G^m$ denote the\nnormal subgroup of $G$ generated by all $m^{th}$ powers of elements of\n$G$. When $G$ is nilpotent we have $[G : G^m] < \\infty$ for any $m$\n(see, for instance, \\cite[p.\\ 20, Lemma 4.2]{MR0283083}).\nWe need the following technical result for Lemma \\ref{lem:torsionlengths}.\n\n\\begin{lemma}\n\t\\label{lem:distortion1}\n\tLet $G$ be a nilpotent group generated by a finite set $X$.\nFix a positive integer $i$.\nThen there exists a constant $C > 1$ such that for all $m \\in \\mathbb{N}$ and all $g \\in (Z(G) \\cap \\gamma_i(G))^m$ with $\\nu_G(g) = i$, we have \n$$\nm \\leq C \\| g \\|^i_X.\n$$\n\\end{lemma}\n\n\\begin{proof}\nSelect $Y = \\{ x_1, \\ldots, x_r \\}$ so that the image of $Y$ under\n$$\\pi : (Z(G) \\cap \\gamma_i(G)) \\to (Z(G) \\cap \\gamma_i(G)) \/ \\gamma_{i+1}(G)$$ \ngenerates a free abelian group of rank $r$, where $r$ is the rank of \n$(Z(G) \\cap \\gamma_i(G)) \/ \\gamma_{i+1}(G)$.\nSelect $N$ to be the order of $\\pi(Z(G) \\cap \\gamma_i(G))\/\\pi(\\left)$.\n\nLet $f$ be the projection $G \\to G\/\\gamma_{i+1}(G)$.\nWe apply \\cite[Theorem 2.2]{MR1872804} to the torsion-free subgroup $\\Pi = f(\\left< x_1, \\ldots,\n x_r \\right>) \\leq G\/\\gamma_{i+1}(G)$ to get $D > 1$ such that\n$$\n\\sup\\limits_{h \\in \\Pi \\cap B_{f(X)}(n)} \\| h \\|_{f(Y)} \\leq D n^{i}.\n$$\nThus, if $\\| h \\|_{f(X)} = n$, then $\\| h \\|_{f(Y)} \\leq D n^i = D (\\| h \\|_{f(X)})^i$.\nThat is, \n\\begin{equation} \\label{eqn:compareh}\n\\| h \\|_{f(Y)} \\leq D \\| h \\|_{f(X)}^i.\n\\end{equation}\n\nNow suppose $g$ is an element of $(Z(G) \\cap \\gamma_i(G))^m$ with $\\nu_G(g) = i$.\nSince $\\nu_G(g) = i$, we have that $\\pi(g)$ is infinite order.\nThus, we can write $g^N = h \\gamma$ where $h \\in \\left< Y \\right>$ and $\\pi(\\gamma)$ is trivial.\nThus, $\\gamma \\in \\gamma_{i+1}(G)$.\nThe map $f|_{ \\left< Y \\right> }$ is an injection, thus\n$$\n\\| f(h) \\|_{f(Y)} = \\| h \\|_{Y}.\n$$\nFinally, by the fact that $g^N \\equiv h \\mod \\gamma_{i+1}(G)$ and Inequality (\\ref{eqn:compareh}), we have\n\\begin{eqnarray*}\nN \\| g \\|_X &\\geq& \\| g^N \\|_X \\geq \\| f(g^N) \\|_{f(X)} \\\\\n&=& \\| f(h) \\|_{f(X)} \\geq D^{1\/i} \\| f(h) \\|_{f(Y)}^{1\/i}.\n\\end{eqnarray*}\nNotice that for any abelian group, $A$, and $\\ell \\in \\mathbb{N}$ we have\n$$A^{\\ell} = \\left< \\{ x^{\\ell} : x \\in A \\} \\right>.$$\nUsing additive notation, this becomes\n$$\nA^{\\ell} := \\left< \\{ \\ell x : x \\in A \\} \\right>\n= \\ell \\{ x : x \\in A \\} = \\ell A.\n$$\nSo we have\n$$mN A = m ( N A) = m \\{ n x : x \\in A \\}.$$\nTo apply this, note that $A = (Z(G) \\cap \\gamma_i(G))$ is an abelian group, as it is contained in the center.\nFor any element $y \\in N A$, by the definition of $N$, we have\n$f(y)$ is an element of $\\Pi$.\nThus, $f(m y) = m f(y)$ is an element of $\\Pi^m$, and so it follows that\n$$\nf(A^{mN}) \\leq \\Pi^m.\n$$\nIn particular, $g^N \\in (Z(G) \\cap \\gamma_i(G))^{Nm}$, so we have\n$$\nf(g^N) \\in \\Pi^m.\n$$\nSince $m f(Y)$ is a free basis for $\\Pi^m$, $f(Y)$ is a free basis for $\\Pi$, and $f(h) = f(g^N)$, we conclude that\n$$\n\\| f(h) \\|_{f(Y)} = \\| f(g^N) \\|_{f(Y)} \\geq m,\n$$\nso we are done.\n\\end{proof}\n\nWith the previous lemmas in hand, we finish with a proof that gives some control on the word lengths of elements in $G^m$.\n\n\\begin{lemma} \\label{lem:torsionlengths}\nLet $\\tilde G$ be a finitely generated nilpotent group of nilpotence $c$.\nThere exists $f \\in \\mathbb{N}$ such that $G = \\tilde G^f$ is a torsion-free characteristic subgroup of $\\tilde G$ of finite index.\nLet $g \\in G$, $X$, and $t \\in \\mathbb{N}$ be as in Lemma \\ref{lem:technical}.\nThen there exists $C > 1$, $M \\in \\mathbb{N}$, depending only on $G$, such that if $g \\in G^{Mm}$, we have that\n$$\n\\| g \\|_X \\geq C m^{1\/t}.\n$$\n\\end{lemma}\n\n\\begin{proof} \nSet $\\tau(\\tilde G)$ to be the set of all elements of finite order in $\\tilde G$.\nBy \\cite[p.\\ 13, Chapter 1, Corollary 10]{MR713786}, this is a finite characteristic subgroup of $\\tilde G$.\nSince $G$ is residually finite and $\\tau(H)$ is finite, there exists a finite $Q$ that fully detects $\\tau(\\tilde G)$. \nSet $f$ to be the exponent of $Q$ and set $G$ to be the characteristic\nfinite-index subgroup $\\tilde G^{f}$ \\cite[p.\\ 20, Lemma 4.2]{MR0283083},\nThen the map $\\tilde G \\to Q$ factors through $\\tilde G\/G$, and thus\n$\\tau(\\tilde G)$ is fully detected by $\\tilde G\/G$.\nSince $\\tau(\\tilde G)$ contains all the torsion elements in $\\tilde G$, it follows that $G$ is torsion-free.\n\nWe will show by induction on $d$ that for all $n > c$,\n\\begin{equation} \\label{eqn:claim}\n(\\zeta_{d}(G))^{(d)! \\cdots 2! n} \\cap Z(G) \\leq Z(G)^{n}.\n\\end{equation}\nThe base case $\\zeta_1(G) = Z(G)$ is immediate.\nFor the inductive step, set $M = (d)! (d-1)! \\cdots 2!$ and let $H =\n\\zeta_{d}(G) \\leq G$.\nLet $h \\in H^{Mn} \\cap Z(G)$.\nSince $h$ is in $H^{Mn}$ we can write\n$$\nh = g_1^{Mn} g_2^{Mn} \\cdots g_k^{Mn} \\in Z(G),\n$$\nwhere $g_1, \\ldots, g_k$ are elements in $H$.\n\nTo proceed, let $\\tau_n(x_1, x_2, \\ldots, x_k) = \\tau_n(\\overline{x})$ be the $n$th \\emph{Petresco word} \\cite[p. 40]{MR0283083}, which is defined by the recursive formula,\n$$\nx_1^n x_2^n \\cdots x_k^n = \\tau_1(\\overline{x})^n \\tau_2(\\overline{x})^{{n \\choose 2}} \\cdots \\tau_n(\\overline{x})^{n \\choose n-1}.\n$$\nBy the Hall-Petresco Theorem \\cite[p. 41, Theorem 6.3]{MR0283083}, we\nhave that $\\tau_n(H) \\subset \\gamma_{n}(H)$ for all $n \\in \\mathbb{N}$.\nThus, replacing $n$ with $Mn$ and using the Hall-Petresco Theorem, we get:\n$$\ng_1^{Mn} g_2^{Mn} \\cdots g_k^{Mn} = \\tau_1(\\overline{g})^{Mn} \\tau_2(\\overline{g})^{{{Mn} \\choose 2}} \\cdots \\tau_d(\\overline{g})^{{Mn} \\choose d}.\n$$\nBy the Hall-Petresco Theorem, $\\tau_k(\\overline{g}) \\in\n\\zeta_{d-1}(G)$ for all $k>1$, and by definition $h = g_1^{Mn} g_2^{Mn} \\cdots g_k^{Mn}\n\\in \\zeta_{d-1}(G)$. Therefore, because $G\/\\zeta_{d-1}(G)$ is torsion-free, $\\tau_1(\\overline{g})$ is in $\\zeta_{d-1}(G).$\nWe conclude that, for each $1 \\leq k \\leq d$, there exists $z_k \\in \\zeta_{d-1}(G)$ such that\n$$\n\\tau_k(\\overline{g})^{Mn \\choose k} = (z_k)^{\\frac{M}{(d)!}n} \\in \\zeta_{d-1}(G)^{\\frac{M}{(d)!}}.\n$$\nFurther,\n$$\n\\frac{M}{d!} = (d-1)! \\cdots 2!.\n$$\nHence, $h \\in (\\zeta_{d-1}(G))^{(d-1)! (d-2)! \\cdots 2!}$, so by the inductive hypothesis, we must have\n$h \\in Z(G)^{n}$, which completes the proof of equation \\ref{eqn:claim}.\n\nLet $D$ be the product of all finite order elements in $G\/ \\gamma_{n}(G)$ for all $n = 1, \\ldots, c$.\nSelecting $d = c$ in equation \\ref{eqn:claim} we get, for $M = (c)! (c-1)! \\cdots 2! D$, and $n > c$,\n\\begin{equation} \\label{eqn:tough}\nG^{Mn} \\cap Z(G) \\leq Z(G)^{D n}.\n\\end{equation}\n\nNow suppose $g \\in G^{Mm}$.\nBy Lemma \\ref{lem:technical}, there exists $x_1, \\ldots, x_{i-1} \\in X$ such that\n$[g,x_1, \\ldots, x_{i-1}] \\in \\gamma_t(G) \\cap Z(G)$.\nThus, as $G^{Mm}$ is normal, we have, by equation \\ref{eqn:tough},\n$[g,x_1, \\ldots, x_{i-1}] \\in Z(G)^{D m}$.\nHence, by our choice of $D$ and the fact that $Z(G)$ is a free abelian group, we have $\\nu_G([g,x_1, \\ldots, x_{i-1}]) = t$.\nThus, applying Lemma \\ref{lem:distortion1} gives $C_1 > 0$, depending only on $G$, such that\n$$\n\\| [g, x_1, \\ldots, x_{i-1}] \\|_X > C_1 m^{1\/t}.\n$$\nA simple counting argument gives a $C_2 > 0$, depending only on $G$, such that\n$$\n\\| g \\|_X \\geq C_2 \\| [g, x_1, \\ldots, x_{i-1}] \\|_X.\n$$\nThus, we have $C > 0$, depending only on $G$, such that\n$$\n\\| g \\|_X > C m^{1\/t},\n$$\nas desired.\n\\end{proof}\n\n\\section{Some examples and basic results} \\label{sec:examples}\n\n\\subsection{Abelian groups} \\label{sec:abeliangroups}\n\nIn this section we discuss some facts concerning abelian groups and present a proof of Theorem \\ref{thm:abelian}.\nThis begins with the simplest torsion-free group.\nFix $\\{ 1 \\}$ as the generating set $\\mathbb{Z}$.\nThen $B_\\mathbb{Z}(n) = \\{ -n, -n+1, \\ldots , n-1, n \\}$.\nClearly, $B_\\mathbb{Z}(n)$ is fully detected by $\\mathbb{Z}\/(2n+1) \\mathbb{Z}$.\nFurther, any quotient fully detecting $B_\\mathbb{Z}(n)$ has cardinality greater than $2n$.\nSo we get $\\G_\\mathbb{Z}(n) \\approx n$.\nThis result generalizes immediately to all torsion-free finitely\ngenerated abelian groups, and more generally to all finitely generated abelian groups.\n\n\\begin{corollary} \\label{prop:abelian}\nLet $A$ be a finitely generated abelian group.\nThen $\\G_A(n) \\approx n^{\\dim(A)}$.\n\\end{corollary}\n\n\\begin{proof}\nBy Corollary \\ref{cor:tfreduction}, we may assume $A$ is torsion-free.\nThe computation in this case is straightforward.\n\\end{proof}\n\nOne salient consequence of Corollary \\ref{prop:abelian} is that an\nabelian group's full residual finiteness growth $\\G_A$ matches its\nword growth $w_A$. We now prove Theorem~\\ref{thm:abelian}, which shows\nthat this property characterizes abelian groups in the class of\nnilpotent groups. It also demonstrates that although $\\G_G$ and $w_G$\nshare properties, they are seldom the same.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:abelian}]\nBy Corollary \\ref{cor:tfreduction}, we may assume that $G$ is torsion-free.\nLet's further assume $G$ is not abelian.\nFix a Malcev basis $x_1, \\ldots, x_k$ for $G$.\nFor every $n$, let $Q_n$ be a quotient fully detecting $B_G(n)$.\nLet $c$ be the nilpotent class of $G$.\nFix a tuple $(x_1, \\ldots, x_m)$ consisting of all the basis elements not in $\\zeta_{c-1}(G)$.\nWe claim that there exists $C >0$ such that for any $\\gamma \\in \\zeta_{c-1}(G)$, the image of\n$x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m} \\gamma$\nin $Q_{Cn}$ is nontrivial in $Q_{Cn}$ for any $|k_i| \\leq n^2$ with $\\sum_{i=1}^m |k_i| > 0$.\nIndeed, by the second assertion of Lemma \\ref{lem:technical} there exists $C > 0$ with\n$[x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m}, y_1, y_2, \\ldots, y_{c-1}]$\nbeing nontrivial and having word-length at most $Cn^{2\/{c}} \\leq Cn$ in $G$.\nThus, as nontrivial elements of $B_G(Cn)$ are nontrivial in $Q_{Cn}$,\nthe image of $x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m} \\gamma$ in $Q_{Cn}$\nis nontrivial, so the claim is shown.\n\nConsider the set \n$$\nB^+(n) := \\{ x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m} \\gamma : 1 \\leq k_i \\leq n^2, \\gamma \\in B_G(n) \\cap \\zeta_{c-1}(G) \\}.\n$$\nGiven any $x,y\\in B^+(n)$, the above claim implies that $y^{-1}x$ has\nnontrivial image in $Q_{Cn}$. It follows that $B^+(n)$ is fully detected by $Q_{Cn}$.\nOn the other hand, by comparing with the explicit calculations for\nword growth in \\cite{Bass72} and the appendix of \\cite{MR623534} we see that the set $B^+(n)$ has cardinality at least $n^m w_G(n)$.\nThus, we have\n$$\n\\G_G(n) \\succeq n^m w_G(n),\n$$\nas desired.\n\n\\end{proof}\n\n\\subsection{Some non-abelian groups} \\label{sec:heisenberg}\n\nWe begin this section with the simplest non-abelian example.\nRecall that the \\emph{discrete Heisenberg group} is given by\n$$H_3 = \\left< x, y, z : [x,y] = z , z \\text{ is central } \\right>.$$\n\n\\begin{proposition} \\label{prop:heisenberg}\nWe have $\\G_{H_3}(n) \\approx n^6$.\n\\end{proposition}\n\n\\begin{proof}\nLet $B_{H_3}(n)$ be the ball of radius $n$ in $H_3$ with respect to\nthe generating set $\\{x,y,z \\}$.\nAn exercise in the geometry of $H_3$ show that there is some $D>0$ so\nthat if $x^{\\alpha_1} y^{\\alpha_2} z^{\\alpha_3} \\in B_{H_3}(Dn)$ then\n$\\lvert \\alpha_i \\rvert \\leq\n n$ for $i=1,2$ and $\\lvert \\alpha_3 \\rvert \\leq n^2$. \nTherefore there is some $C>0$ so that $B_{H_3}(n)$ injects into the quotient\n$H_3 \/ H_3^{Cn^2}$, and so $\\G_{H_3}(n) \\preceq n^6$.\n\nNow note that $B_{H_3}(5n)$ contains $z^i$ for $-n^2 \\leq i \\leq n^2$, as\n$$\n[x^n,y^{j}] z^{k} = z^{nj + k},\n$$\nhas word length at most $5n$ for each $1 \\leq j, k \\leq n$.\nLet $Q_n$ be a quotient detecting $B_{H_3}(5n)$.\nConsider $w = x^a y^b x^c$.\nThen $[w,y] = z^{a}$ and $[w,x] = z^{-b}$. \nIf $w$ is trivial in $Q_n$ then both $[w,y]$ and $[w,x]$ are also trivial.\nIt follows that $w$ has nontrivial image in $Q_n$ for any values $0 <\na,b,c \\leq n^2$.\nThus, $|Q_n| \\geq n^6$, as desired.\n\\end{proof}\n\n\nWe now prove Theorem \\ref{theorem:zl} from the introduction, which\ngeneralizes the conclusion of Proposition \\ref{prop:heisenberg} to a\nlarge class of nilpotent groups.\n\nFor a finite $k$-tuple of elements $X = (x_1, \\ldots, x_k)$ from a group, we will use $B_X^+(n)$ to denote the set\n$$\nB_X^+(n) = \\{ x_1^{\\alpha_1} \\cdots x_k^{\\alpha_k} : 0 \\leq \\alpha_i \\leq n \\} \\subseteq G.\n$$\nNote that this is \\emph{not} generally the same as the semigroup ball\nof radius $n$.\n\n\\begin{proof} [Proof of Theorem \\ref{theorem:zl}]\n Lemma \\ref{lem:torsionlengths} demonstrates that $B_G(n)$ is fully detected by a quotient of\n the form $G \/ G^{Mn^c}$ for some $M>0$. We therefore have \n\\[\\G_G(n) \\preceq \\left(\\prod_{i=1}^c n^{\\dim(\\zeta_i(G)\/\\zeta_{i-1}(G))}\\right)^c = n^{\\dim(G) c}.\\]\n\nTo show the reverse inequality, we will show that for any positive integer $n$, there exists set of cardinality approximately $n^{\\dim(G)c}$ that is fully detected by any finite quotient of $G$ that realizes $\\G_G(n)$. To this end, for each $i$, equip $\\gamma_i(G)$ with a fixed generating set $X_i$.\n Let $Q$ be a quotient of $G$ that realizes $\\G_G(n)$. By \\cite[Theorem 2.2]{MR1872804}, for any generating\n set of $\\gamma_c(G)$ there is a constant $C> 0$ such that for every\n $h, h'$ in $\\gamma_c(G)$, we have\n$$\nd_{\\gamma_c(G)} (h,h') \\leq C [d_G(h,h')]^{c}.\n$$\nThus, the set $B_{\\gamma_c(G)}(n^c \/C)$ must inject into $Q$\nas it is contained in $B_G(n)$.\n\nTo continue, fix a basis $B = (g_1, \\ldots, g_k)$ obtained from the upper central series. \nFor any $i$, let \n\\[\nB_i = \\left\\{ g_{j}\\in B \\mid g_j \\in \\zeta_i(G) \\setminus\n \\zeta_{i+1}(G), g_j \\text{ nontorsion in } G\/\\zeta_i(G) \\right\\}.\n\\] \nSet $B^t$ to be the tuple consisting of elements from $B_i$ respecting the ordering of the basis.\nThat is, \n$$\nB^t = (g_{a_1}, \\ldots, g_{a_k}),\n$$\nwhere each entry is in some $B_i$ and $a_i < a_{i+1}$.\nWe claim that $B_{B^t}^+(D n^c)$ is fully detected by $Q$ for some $D > 0$.\nTo prove this claim, we will use the fact that if any element in the\nnormal closure of some $g\\in G$ has nontrivial image in $Q$, then $g$\nhas nontrivial image in $Q$.\nLet $x, y \\in B_{B^t}^+(n^c)$ be elements with $x \\neq y$.\nThere is some $i \\leq c$ so that $y^{-1}x \\in \\zeta_i(G) \\setminus \n\\zeta_{i-1}(G)$. \nSet \n\\[\nE = \\max \\{ |B_j| \\} \\cdot \\max \\{ \\| \\gamma \\|_{X_j} : \\gamma \\in\nB_j \\}.\n\\]\nThere is some $\\gamma \\in \\zeta_{i-1}(G)$ so that $\\| y^{-1}x \\gamma\n\\|_{X_i} \\leq E n^c$. \nThis statement follows by reducing the word $y^{-1} x$ to normal form with respect to the basis.\nLet $E_0$ be the largest constant $C_i$ output by Lemma\n\\ref{lem:technical} for $i = 1,\\dotsc, c$.\nBy Lemma \\ref{lem:technical},\n$$\n\\| [y^{-1}x , x_1, \\ldots, x_r ] \\|_{X_c} \\leq E_0 \\| x y^{-1} \\gamma \\|_{X_i} \\leq E_0 E n^c.\n$$\nIt follows then that the set $B_{B^t}^+(n^c\/ (C E_0 E) )$ is fully detected by $Q$.\nSet $D = 1\/(C E_0 E)$.\nBy the definition of a basis we have $|B_{B^t}^+( D n^c)| \\geq (D n)^{c \\dim(G)}$, so we get the desired inequality.\n\\end{proof}\n\n\n\n\n\\section{A general upper bound}\n\\label{MainProofSection}\n\nThe example $H_3 \\times \\mathbb{Z}$, which has full residual finiteness growth $n^7$, demonstrates that the conclusion of Theorem \\ref{theorem:zl} does not hold for any finitely generated nilpotent group.\nIn this section we prove Theorem \\ref{MainTheorem}, providing a\ntechnique that provides for {\\em any} finitely generated nilpotent group an\nexplicit upper bound of $\\G_G(n)$ of the form $n^d$. We first\nillustrate the technique in an example in Proposition\n\\ref{D22example}, where we show moreover that the upper bound is sharp\nin this example.\n\nLet $U_n$ denote the group of upper triangular unipotent matrices in\n$\\SL_n(\\mathbb{Z})$. \nFor $i\\neq j$, let $e_{i,j}$ denote the elementary matrix differing\nfrom the identity matrix only in that its $ij$-entry is 1.\nWe define the \\emph{coordinates} of the tuple $(x_1, \\ldots, x_k)$ to be the set\n$\\{ x_1, \\ldots, x_k \\}$.\nRecall that a {\\em terraced} filtration of $G$ is a filtration $1 = H_0 \\leq\nH_1 \\leq \\dotsb \\leq H_{c-1}\\leq G$ where each $H_i$ is a maximal normal\nsubgroup of $G$ satisfying $H_i\\cap \\gamma_{i+1}(G) = 1$.\n\n\n\\begin{proposition} \\label{D22example}\n Consider elementary matrices $x = e_{1,4}$ and $y = e_{1,5}$ in\n $U_5$. Define a normal subgroup $N = \\left< x, y \\right> \\leq\n U_5$ and set $\\Gamma = U_5 \/ N$. Then $\\G_{\\Gamma}(n) \\approx n^{22}.$\n\\end{proposition}\n\n\\begin{proof}\n Set $H_3 = \\Gamma$ and $H_2 = \\left< e_{1,2}, e_{1,3} \\right>$, and\n let $H_0 = H_1 = 1$. \n Note that $1=H_0\\leq H_1 \\leq H_2 \\leq \\Gamma$\n forms a terraced filtration of $\\Gamma$.\n Define two tuples of elements of $\\Gamma$ by $X_3 = (e_{1,3} ,\n e_{1,2})$ and $X_2 = (e_{2,5}, e_{2,4}, e_{3,5}, e_{2,3}, e_{3,4},\n e_{4,5})$. For each $i=2,3$, let $Y_i$ be the set of coordinates of\n $X_i$. Clearly $Y = Y_2 \\cup Y_3$ generates $\\Gamma$. \n\n To establish the upper bound, let $Q$ be a quotient of $\\Gamma$ detecting $B_\\Gamma(n)$.\n Each\n of $H_3^{n^3}$ and $H_2^{n^2}$ is normal in $\\Gamma$, so we can define a normal\n subgroup $N = H_3^{n^3} H_2^{n^2} \\leq \\Gamma$. \n A simple induction shows that if $g\\in B_\\Gamma(n)$ then $\\lvert\n g_{ij} \\rvert \\leq n^{j-i}$. In particular this implies that there\n is some $C>0$ so that $B_\\Gamma(Cn)$ is fully detected by\n $G\/N$. Since $\\lvert G\/N \\rvert \\approx n^{22}$, this establishes\n the desired upper bound on $\\G_\\Gamma(n)$.\n \n To establish the lower bound, define the \\emph{depth} of an element\n $\\gamma \\in \\Gamma$ to be the maximal $i$ with\n $\\gamma \\notin \\zeta_i(\\Gamma)$. Order the elements $Y$ in a tuple\n $(y_1, y_2, \\ldots, y_8)$ of non-increasing depth. Set $B^+(n)$ to\n be\n $$\n \\left\\{ \\prod_{i=1}^8 y_i^{\\alpha_i} : 0 \\leq \\alpha_i \\leq n^2 \\text{ if $y_i \\in Y_2$ and } 0 \\leq \\alpha_i \\leq n^3 \\text{ otherwise} \\right\\}.\n $$\n We claim that there exists $C >0$ such that any quotient $Q$ in which $B_\\Gamma(Cn)$ embeds restricts to $B^+(n)$ as an injection.\n This gives the desired lower bound, as $|B^+(n)| \\geq n^{22}$.\n To see this claim, let $x,y$ be distinct elements in $B^+(n)$.\n Set $i$ to be the depth of $y^{-1}x$.\n We break up the rest of the proof of this claim into cases depending on $i$.\n \n If $i = 0$, then\n $y^{-1} x$ is in the center of $\\Gamma$ and we have\n $$\n y^{-1} x = e_{1,2}^{a_1} e_{2,5}^{a_2},\n $$\n where $|a_1| \\leq n^2$ and $|a_2| \\leq n^3$.\n Note that $e_{1,2}\\in \\gamma_2(\\Gamma)$ and $e_{2,5}\\in \\gamma_3(\\Gamma)$.\n Applying Lemma \\ref{lem:distortion} twice, we have that\n $$\n \\| y^{-1} x\\|_\\Gamma \\leq \\| e_{1,2}^{a_1} \\|_\\Gamma + \\|\n e_{2,5}^{a_2} \\|_\\Gamma \\leq C n,\n $$\n for some $C > 0$, independent of $n$.\n Thus $y^{-1} x$ cannot vanish in any quotient that fully detects $B^+(Cn)$.\n \n If $i = 1$, then by definition, we may write\n $$\n y^{-1} x = e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma,\n $$\n where $\\gamma \\in \\zeta_i(\\Gamma)$, $|a_1| \\leq n^2$, $|a_2| \\leq n^3$, and $|a_3| \\leq n^3$.\n Since this $y^{-1}x$ is not in the center, there exists $z \\in Y$ such that \n $$\n [e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma, z] \\neq 1.\n $$\n This element is now in the center. Thus, by Lemma \\ref{lem:commutatorproduct}, we have\n $$\n [e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma, z]\n = \n [e_{1,2}, z]^{a_1}\n [e_{2,4}, z]^{a_2}\n [e_{3,5}, z]^{a_3}.\n $$\n Now by Lemma \\ref{lem:distortion} applied three times, we see that the word length of\n $[e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma, z]$\n is less than a constant multiple of $n$, where the constant does not depend on $n$.\n Thus $y^{-1} x$ cannot vanish in any quotient that fully detects $B^+(Cn)$ for some $C > 0$ independent of $n$.\n \n If $i = 2$, then by definition, we may write\n $$\n y^{-1} x = e_{2,3}^{a_1} e_{3,4}^{a_2} \\gamma,\n $$\n where $\\gamma \\in \\zeta_i(\\Gamma)$, $|a_1|, |a_2| \\leq n^3$.\n Suppose, without loss of generality, that $a_1 \\neq 0$.\n Then, using Lemma \\ref{lem:commutatorproduct}, we have that there exists $\\gamma' \\in \\gamma_1(\\Gamma)$ such that\n $$\n [y^{-1} x, e_{3,4}] = [e_{2,3}, e_{3,4}]^{a_1} [e_{3,4}, e_{3,4}]^{a_2} \\gamma' = e_{2,4}^{a_1} \\gamma'.\n $$\n Now it is clear that there exists $z \\in \\Gamma$ such that\n $$\n [[y^{-1} x, e_{3,r}], z] = [e_{2,4}^{a_1} \\gamma', z] \\neq 1.\n $$\n We can proceed as in case $i=1$ to achieve the desired conclusion.\n Indeed, Lemma \\ref{lem:distortion} applies, giving that $y^{-1} x$ is detected if $B(Cn)$ is fully detected for some constant $C > 0$ independent of $n$.\n That is, we cannot have $y^{-1}x =1$ in $Q$, if $Q$ detects $B_\\Gamma(Cn)$.\n The claim then follows, ending the proof.\n \n\\end{proof}\n\n\nWe now prove Theorem \\ref{MainTheorem}.\n\n\\begin{proof}[Proof of Theorem \\ref{MainTheorem}]\nLet $G$ be a finitely generated nilpotent group and suppose $1=H_0\n\\leq H_1 \\leq \\dotsb \\leq H_{c-1}\\leq G$ is a terraced filtration. Set\n$H_c = G$. \n\nChoose a basis $X_1$ of $H_1$. Inductively construct\ntuples $X_2,\\dotsc, X_c$ by setting $X_i$ to be a pull-back of a basis\nfor $H_i \/ H_{i-1}$.\nSet $Y_i$ to be the set of all coordinates of $X_i$ and $Y = \\cup_i Y_i$.\nIt is clear from the construction that $Y$ is generating set for\n$G$. Note also that for any $n \\in \\mathbb{N}$, the subgroup \n\\[\nN(n) = \\prod_{i=1}^c \\left<\n y^{n^k} : y \\in \\left< Y_1 \\cup Y_2 \\cup \\dotsb \\cup Y_k\n \\right> \\right>\n\\]\nis normal in $G$.\n\nWe now claim that there exists a constant $D \\in \\mathbb{N}$ so that for any\n$n \\in \\mathbb{N}$, the ball $B_Y(n)$ is detected by $G\/N(Dn)$.\nTo prove the claim, let $f, M \\in \\mathbb{N}$ be as in Lemma~\\ref{lem:torsionlengths}.\nThen $G^{fM}$ is torsion-free; let $K = G^{f M}$.\nFix a finite generating set $T$ for $K$.\nFor each $i$ and any $n\\in \\mathbb{N}$, Lemma \\ref{lem:torsionlengths} gives\nthat any element $g\\in K^n \\cap H_i$ has word length at least $C_i\nn^{1\/t_i}$ with respect to $T$.\nThus we have that there exists $D_0 > 0$ such that\n$B_T(D_0 n)$ is fully detected by $K\/N(fMn)$.\nFurther, since $K$ is of finite index in $G$, we have $D_1 > 1$ such that for any $g \\in K$,\n$$\n\\| g \\|_T \\leq D_1 \\| g \\|_Y \\leq D_1^2 \\| g \\|_T.\n$$\nTherefore, as $N(fMn)$ is contained in $K$, any singleton contained in $B_Y(n\/D_1)$ is fully detected by $G\/N(fMn)$ and so $B_Y(n\/(2 D_1))$ is fully detected by $G\/N(fMn)$.\nThis proves the claim, as we can select $D = 2D_1 fM$.\n\nWe will now demonstrate that the order of\n$G\/N(Dn)$ is dictated by a single polynomial of the form $n^b$ for \n\\[\nb = \\sum_{k=1}^c k\\cdot \\dim(H_k \/ H_{k-1}).\n\\]\nSet $G_k = H_k\/ H_{k-1}$. It is apparent from the definition of $N(Dn)$ the index of $N(Dn)$ in $G$ is bounded above by\n$$\n\\prod_{k=1}^{c} | G_k \/ G_k^{D^k n^{k}} |.\n$$\nBy the construction of $D$, the subgroup $G_k^D$ is torsion-free in $G_k$.\nThus, it is clear that $|G_k^D \/ G_k^{D^{k} n^{k}}|$ has order\n$D^{\\dim(G_k) (k-1)} n^{k \\dim(G_k)}$.\nThis gives an upper bound for the index of $N(Dn)$ in $G$ of the form\n$C_0 n^{\\sum_{k=1}^c k \\dim(G_k)}$, where $C_0 >0$ does not depend on $n$.\n\nOne can check that $b = c\\dim(G) - \\sum_{i=1}^{c-1}\\dim(H_i)$ using\nthe general fact that $\\dim(G\/H) = \\dim(G)- \\dim(H)$ for any finitely\ngenerated nilpotent group $G$ with normal subgroup $H$. This \ncompletes the proof since $G\/N(Dn)$ detects $B_Y(Cn)$.\n\\end{proof}\n\nWe conclude with an example that shows that the upper bound to $\\G_G$\ngiven by Theorem \\ref{MainTheorem} generally may depend on choice of\nterraced filtration. Consider the group\n$\\tilde G = U_3 \\times U_4 \\times U_5$, which is nilpotent of class\n$c=4$. There is an isomorphism\n$Z(\\tilde G) \\cong Z(U_3) \\times Z(U_4) \\times Z(U_5)$. Under\nidentifications $Z(U_3)\\cong Z(U_4) \\cong Z(U_5) \\cong \\mathbb{Z}$, define an\ninfinite cyclic subgroup\n\\[\nZ = \\{ (x,y,z)\\in Z(U_3)\\times Z(U_4)\\times Z(U_5) \\mid x=y=z \\} \\leq\nZ(\\tilde G).\n\\]\nLet $G = \\tilde G \/ Z$ and let $\\pi: \\tilde G \\to G$ be the quotient\nmap. Then $\\pi$ restricts to an isomorphism $Z(U_3)\\times Z(U_4) \\cong\nZ(G)$. Under this identification, the last term of the lower central\nseries of $G$ is\n\\[\n\\gamma_4(G) = \\{ (x,y)\\in Z(U_3)\\times Z(U_4) \\mid x=y \\}.\n\\]\nSince $\\gamma_3(G)$ contains the image of $Z(U_4)$, we see that\n$Z(G) \\leq \\gamma_3(G)$. Since $H\\cap\nZ(G)$ is nontrivial for any nontrivial normal subgroup $H\\leq G$, it follows that\n$H_2$ is trivial for any terraced filtration of $G$.\n\nNow define $H_0 = H_1 = H_2 = 1$ and $H_3 = \\pi(U_3)$, and\n$H_0' = H_1' = H_2' = 1$ and $H_3' = \\pi(U_4)$. It is easy to see that\nboth $\\pi(U_3)$ and $\\pi(U_4)$ are maximal normal subgroups of $G$\nwhose intersection with $\\gamma_4(G)$ is trivial. It follows from the\nabove comments that\n\\[\nH_0\\leq H_1 \\leq H_2 \\leq H_3 \\leq G \\quad \\text{ and } \\quad H_0'\n\\leq H_1' \\leq H_2' \\leq H_3' \\leq G\n\\]\nare terraced filtrations of $G$. However these filtrations give\ndifferent upper bounds for $\\G_G$ because $\\dim( \\pi(U_3) ) = 3$ while\n$\\dim( \\pi(U_4) ) = 6$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe gauge-gravity correspondence\n \\cite{Maldacena:1997re, Gubser:1998bc, Witten:1998qj}\nhas got a nonrelativistic version where\n strongly coupled quantum theories at critical points\n can be studied \\cite{Son:2008ye, Balasubramanian:2008dm, Herzog:2008wg, Hartnoll:2008vx, Hartnoll:2008kx, Maldacena:2008wh, Denef:2009tp, Kachru:2008yh, Singh:2009tq, Singh:2010rt, Balasubramanian:2010uk, Singh:2010zs, Singh:2012un, Narayan:2012hk, Singh:2013iba, Narayan:2012ks, Singh:2017wei, Singh:2018ibp, Mishra:2018tzj, Taylor:2015glc}.\nSome of these quantum systems involve strongly coupled fermions \n at finite density or it may simply be a gas of ultra-cold atoms \n\\cite{Son:2008ye, Balasubramanian:2008dm}. \nIn the studies involving \n`nonrelativistic' Schr\\\"odinger spacetimes the 4-dimensional\n spacetime geometry generally requires supporting\n Higgs like field such as massive vector field \n \\cite{Herzog:2008wg, Denef:2009tp, Son:2008ye} or a tensor field. \nThe spacetimes possessing a Lifshitz symmetry\n provide similar holographic dual description of nonrelativistic \nquantum theories living on their boundaries \\cite{Kachru:2008yh}, see \\cite{Taylor:2015glc} for a review. \n\nIn this work we shall mainly study entanglement entropy of the \nexcitations in asymptotically \n$Lif_4^{(a=2)}\\times S^1\\times S^5$ background. The latter is \na Lifshitz vacua in massive type IIA (mIIA) theory \\cite{Singh:2017wei, Singh:2018ibp} with dynamical exponent of time being $a=2$. The massive type IIA theory \\cite{ROMANS1986374} is a ten-dimensional maximal supergravity where the antisymmetric \ntensor field is explicitly massive. The theory also includes a positive cosmological constant related to mass parameter. Due to this structure the mIIA theory provides a unique setup to study Lifshitz solutions. Particularly the $Lif_4^{(2)}\\times S^1\\times S^5$\nsolution is a background generated by the bound state of $(F1,D2,D8)$ branes \\cite{Singh:2017wei}\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{sol2a9n}\n&&ds^2= L^2\\left(- {dt^2\\over z^4} +{dx_1^2+dx_2^2\\over z^2}+{dz^2\\over\nz^2} +{dy^2\\over q^2} + d\\Omega_5^2 \\right) ,\\nonumber\\\\\n&&e^\\phi=g_0 ,\n ~~~~~C_{(3)}= -{1 \\over g_0}\n{L^3\\over z^4}dt\\wedge dx_1\\wedge dx_2, \\nonumber\\\\ &&\nB_{(2)}= { L^2\\over q z^2}dt\\wedge dy \n\\eea \nThe metric and the form fields have explicit invariance under constant scalings (dilatation); \n$z\\to \\lambda z,~t\\to \\lambda^2 t,~\nx_i\\to \\lambda x_i, ~y\\to y$. The dynamical exponent of time is $2$ here.\nThe background describes a strongly coupled nonrelativistic \nquantum theory at the UV critical point. \n\\footnote{Analogous T-dual solution do also exist in type IIB theory with \nconstant axion flux switched \non \\cite{Balasubramanian:2010uk}} \n\nIt is worthwhile to study excitations of the\n$Lif_4^{(2)}\\times S^1\\times S^5$\n vacua as it immediately provides us a prototype \n$Lif_4^{(2)}$ background in four dimensions which is holographic dual to 3-dimensional Lifshitz theory on its boundary. The excitations would tell us how this Lifshitz theory behaves\nnear its critical point. Particularly we shall study a class of string like excitations which themselves form solutions of massive IIA sugra \nand explicitly involve $B$-field \\cite{Singh:2018ibp}. \nThese also induce running of dilaton as well. It is observed that the resulting RG flow in the deep IR can be described simply by ordinary type IIA theory. The reason for this is due to the fact that the contributions of massive stringy modes decouple from the low energy dynamics of the theory in the IR, far away from UV critical point \\cite{Singh:2018ibp}. \n\nIn this report we aim to study holographic entanglement entropy (HEE) \\cite{Ryu:2006bv, *Ryu:2006ef, *Hubeny:2007xt} of the excited Lifshitz subsystems which are either a disc or a strip in a perturbative framework. A critical observation is that for small sized systems the entanglement entropy density remains constant at first order. That is, \nthe first order contributions to the entropy density remain independent of the\nsize ($\\ell$) of the subsystem. This is a peculiarity and quite unlike relativistic CFTs where usually the entropy density (of excitations) is linearly proportional to the typical size of the subsystem \\cite{Bhattacharya:2012mi}. \nWe discover that the resolution lies in the nature\nof the chemical potential ($\\mu_E$) for the Lifshitz system. We gather evidence that \nsuggests that energy density (of excitations) falls off with the size of system as $\\propto 1\/\\ell^2$. Furthermore the $1\/\\ell^{2}$ dependence is exactly same as the entanglement temperature behaviour in the Lifshitz theory. \nNotwithstanding these peculiarities, \n the entropy of excitations consistently \nfollows the first law of entanglement\nthermodynamics \\cite{Bhattacharya:2012mi,Allahbakhshi:2013rda} up to first order.\n\\par In addition, we also carry out a calculation of entanglement entropy at second order for both disc and strip subsystems. Contributions arising at this order bestow an explicit $\\ell$ dependence upon the entropy. We argue how the first law can still be obeyed by modifying our chemical potential $(\\mu_E)$ and entanglement temperature $(T_E)$. A similar argument was put forward in \\cite{Mishra:2015cpa} for asymptotically AdS spacetime.\n\nThe unusual symmetry of Lifshitz spacetime makes it a good background to study novel features of entanglement in a non-relativistic quantum theory at zero temperature \\cite{Son:2008ye, Balasubramanian:2008dm, Kachru:2008yh}. It is well known that for such systems, e.g. a particle in a one-dimensional box the momentum of the particle scales with the length as $ p \\propto \\frac{1}{\\ell}$ and the energy $\\mathcal{E} \\propto \\frac{1}{\\ell^2}$; our calculations of entanglement entropy also support this explicit size dependence of energy, as shown in equation \\eqref{enr34}. We hope our work will help shed some light on holographic treatment of non-relativistic quantum systems at strong coupling that are often interesting in e.g. condensed matter theory.\n\n\\par The rest of the paper is organized as follows: in section \\ref{sec2} we review salient features of $Lif^{(2)}_4\\times S^1\\times S^5$ vacua with IR excitations in mIIa theory. The holographic entanglement entropy for a disc subsystem is calculated in section \\ref{sec3}. In section \\ref{sec4} we carry out similar analysis for strip subsystem at first and second orders, section \\ref{sec5} contains the conclusion.\n\n \n\\section{$Lif^{(2)}_4\\times S^1\\times S^5$ vacua and excitations}\\label{sec2}\n \n The massive type IIA supergravity theory is the only known maximal \nsupergravity in ten dimensions which allows massive string $B_{\\mu\\nu}$ \nfield and a mass dependent cosmological constant \\cite{ROMANS1986374}.\nThe cosmological constant \n generates a nontrivial potential term for the dilaton \nfield. The mIIA theory does not admit flat Minkowski \n solutions. \nNonetheless the theory gives rise to well known \nFreund-Rubin type vacua $AdS_4\\times S^6$ \\cite{ROMANS1986374}, \nthe supersymmetric domain-walls \nor D8-branes \\cite{Polchinski:1995mt, Bergshoeff:1996ui,Witten:2000mf, Hull:1998vy, Haack:2001iz}, \n $(D6,D8)$, $(D4,D6,D8)$ \nbound states \\cite{Singh:2001gt, Singh:2002eu} and Galilean-AdS geometries \\cite{Singh:2009tq, Singh:2010rt}. \n In all of these massive tensor field\nplays a key role.\nUnder the `massive' T-duality \\cite{Bergshoeff:1996ui} the D8-branes \n can be mapped over to the axionic D7-branes of type IIB string theory\nand vice-versa. \nThe $B$-field also plays important role in obtaining \nnon-relativistic Lifshitz solutions \\cite{Singh:2017wei, Singh:2018ibp}. \nThe latter solutions are of no surprise in mIIA theory,\nas an observed feature in four-dimensional AdS gravity theories\nhas been that in order to obtain non-relativistic \nsolutions one needs to include \nmassive (Proca) gauge fields in the gravity theory \\cite{Son:2008ye}. \nOther different situations where massless vector fields \ncan give rise to non-relativistic vacua, \n involve boosted black D$p$-branes\ncompactified along lightcone direction \\cite{Singh:2010zs, Singh:2012un}. These latter class of solutions are also called \nhyperscaling (or conformally) Lifshitz vacua \\cite{Narayan:2012hk}. \n\nParticularly the $a=2$ Lifshitz vacua with IR excitations in mIIA theory\ncan be written as \\cite{Singh:2018ibp} \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{sol2a9}\n&&ds^2= L^2\\left(- {dt^2\\over z^4h} +{dx_1^2+dx_2^2\\over z^2}+{dz^2\\over\nz^2} +{dy^2\\over q^2h} + d\\Omega_5^2 \\right) ,\\nonumber\\\\\n&&e^\\phi=g_0 h^{-1\/2},\n ~~~~~C_{(3)}= -{1 \\over g_0}\n{L^3\\over z^4}dt\\wedge dx_1\\wedge dx_2, \\nonumber\\\\ &&\nB_{(2)}= { L^2\\over q z^2}h^{-1}dt\\wedge dy \\ ,\n\\eea \nwhere the harmonic function $h(z)= 1+{z^2\\over z_{I}^2}. $\nThe parameter $z_I$ is related to the charge of the NS-NS \nstrings. \nThe excitations involve $g_{tt}$ and $g_{yy}$\n metric components, and leaving the $x_1,x_2$ \nplane (worldvolume directions of D2-branes) \nunaffected.\\footnote{\nHere $L={2\\over g_0 m l_s}$, \nand $m$ being the mass parameter in the mIIA action. (We would set $l_s=1$ and\n $g_0=1$.) \n The constant $q$ is a free (length) parameter \nand $g_0$ is weak string coupling.\nNote $L$ is dimensionless parameter, it\ndetermines overall radius of curvature of the spacetime.\n Therefore Romans' theory with \n $m \\ll{ 2\\over g_0 l_s}$ would be preferred here \nso that $L\\gg 1$ in the solutions \\eqn{sol2a9}, else\n these classical vacua cannot be trusted. Also, from the D8\nbrane\/domain-wall correspondence in \\cite{Bergshoeff:1996ui}, one typically\nexpects $m \\approx {g_0 N_{D8} \\over l_s}$, a value which is definitely\nwell within ${ 2\\over g_0 l_s}$ for a finite number of $D8$ branes, $N_{D8}$, \nin these backgrounds. }\n The excitations do also induce a running of dilaton field. \nThe $B_{ty}$ component of the string field \n is also coupled to the excitations. Since\n$h\\sim 1$ as $z\\to 0$, \nthese excitations form normalizable modes ($z_I$ would correspond to \nadding relevant operators in the boundary Lifshitz theory). \nThe solution \\eqn{sol2a9} asymptotically flows to weakly coupled \n regime in the UV (note that the string coupling,\\ $g_0<1$). \nWhile, in the\ndeep IR region, with $z\\gg z_I$ where $h\\approx {z^2\\over z_I^2}$,\nthe vacua is driven to another \nweakly coupled Lifshitz regime. For $z\\gg z_I$, \nthe IR geometry transforms to dilatonic\n $Lif_4^{(3)}\\times S^1\\times S^5$ solution.\nThis solution enables us to study the effect \nof the excitations in \n $a=2$ Lifshitz theory. Note the $z_I$ dependent excitations \nat zero temperature\nare mainly in the form of charge excitations, along with nontrivial \nentanglement chemical potential, as we would see next. \n \n\n\\section{ Entanglement of a disc subsystem}\\label{sec3}\n \nFor asymptotically AdS space-time dual to a CFT, the entanglement entropy can be calculated by the Ryu-Takayanagi formula \\cite{Ryu:2006bv, *Ryu:2006ef}. We assume the same is true for an asymptotically Lifshitz space-time, dual to a non-relativistic field theory with Lifshitz scaling symmetry. We consider a round disc of radius $\\ell$ at the center of the $x_1,x_2$ plane with its boundary identified with the corresponding boundary of $2d$ Ryu-Takayanagi surface lying inside the Lifshitz bulk geometry \\eqn{sol2a9}. We shall assume $y$ is a compactified direction \n\\begin{equation}\n\ty\\sim y +2\\pi r_y\\;.\n\\end{equation}\nIn radial coordinates $(r=\\sqrt{x_1^2+x_2^2})$ \nthe Ryu-Takayanagi area functional \\cite{Ryu:2006bv, Ryu:2006ef} \nfor static bulk surface is given by \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{areaint1}\n{\\cal A}_\\gamma = 2\\pi L^2 \\int^{z_\\ast}_{\\epsilon} dz \n {r\\sqrt{1+r'^2}\\over z^2} h^{1\\over2}\\,,\n\\eea\nwhere, $r'={dr\\over dz},~h(z)=\\left(1+ {z^2\\over z_I^2 }\\right)$ and \n$\\epsilon \\ll \\ell$ is UV cut-off \nof the Lifshitz theory. We need to extremize the area integral by solving the Euler-Lagrange equation for $r(z)$\n\\begin{multline}\n\t2zrr''h(z) - 4rr'^3h(z) - 4rr'h(z) - 2zr'^2h(z) -2zh(z) - zrr'^3h'(z) - zrr'h'(z) = 0\\,,\n\\end{multline}\nIt is impossible to analytically calculate the full area integral \\eqref{areaint1}. To facilitate our job, therefore, we restrict ourselves to small subsystems, with $\\ell\\ll z_I$. In this domain, we can make a perturbative expansion and obtain solutions order by order in the dimensionless ratio ${\\ell\\over z_I}$; such that $r(z)=r_{(0)}+ r_{(1)}+ \\cdots$, and correspondingly we would write $${\\cal A}_\\gamma ={\\cal A}_0 +{\\cal A}_1 +\\cdots\\;,$$ for small $\\ell$. Our immediate interest is in calculating terms up to leading order and \nfirst order only in the ${\\ell\\over z_I} $ expansion. \n\\par The equation at zeroth order is\n\\begin{equation}\n\tzr_{(0)}r_{(0)}^{\\prime \\prime} - 2r_{(0)}r_{(0)}^{\\prime 3} - 2r_{(0)}r_{(0)}^{\\prime} - zr_{(0)}^{\\prime 2} - z = 0\\,,\n\\end{equation}\nfor which $r_{(0)}=\\sqrt{\\ell^2-z^2}$ defines the extremal surface (half circle) \\cite{Ryu:2006ef, Blanco:2013joa} with the boundary conditions $r_{(0)}(0)=\\ell$, and $r_{(0)}(z_\\ast)=0$, where $z=z_*$ is the point of return that lies at $z_\\ast= \\ell$. One then finds that the area\n\n\\begin{align}\n{\\cal A}_0 &= 2\\pi L^2 \\int^{z_\\ast}_{\\epsilon} dz \\frac{r_{(0)}\\sqrt{1 + r_{(0)}^{\\prime 2}}}{z^2},\\nonumber \\\\ \n&= 2\\pi L^2 \\left(\\frac{\\ell}{\\epsilon} - 1 \\right).\n\\end{align}\n\n${\\cal A}_0$ being a ground state contribution it obviously remains independent of the parameter $z_I$ of the bulk geometry. This only means that there is no effect of excitations on the leading term. As explained in \\cite{Blanco:2013joa}, the first order contribution can be evaluated using only the tree level embedding function and is given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{area1}\n{\\cal A}_1 &=& \n2\\pi L^2 \\int^{z_\\ast}_{\\epsilon} dz r_{(0)} \n{\\sqrt{1+r_{(0)}'^2}\\over 2\\,z_I^2},\\nonumber\\\\\n&=& \\pi L^2 \\left({\\ell^2\\over z_I^2}\\right).\n\\eea\nFrom here the complete expression of entanglement entropy of \na disc shaped subsystem up to first\norder becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{gb12}\nS_E^{Disc}[\\ell,z_I]&\\equiv& {{\\cal A}_\\gamma \\over 4 G_{4}},\\nonumber\\\\\n&=& S_{E}^{(0)}+\n {\\pi L^2 \\over 4\\,G_{4}} \\left( \n {\\ell^2\\over z_I^2} \\right),\n\\eea\nwhere the Newton's constant in $4$D and $5$D are related to the 10-dimensional Newton's constant by $\\frac{1}{G_{4}} = \\frac{L\\,2\\pi r_y}{G_5}$ and $\\frac{1}{G_5}\\equiv {L^5 Vol(S^5)\\over G_{10}}$. We shall be using $G_4$ and $G_5$ back and forth in our calculation.\\\\\nThe ground state entropy contribution is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\nS_{E}^{(0)}= {\\pi L^2 \\over 2G_{4}} \\left( \n{\\ell\\over \\epsilon} -1\\right)\\ .\n\\eea\nThe equation \\eqref{gb12} is a meaningful expression for entanglement entropy only if we maintain $\\ell\\ll z_I$. The first order term explicitly depends on $z_I$, so small fluctuations of the bulk quantities, like $\\delta z_I$, would result in corresponding change in entropy. For a fixed size $\\ell$, one could express these variations of the entropy density as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{ent34}\n{\\delta s_E^{Disc}}\n={\\delta S_E^{Disc}\\over \n\\pi \\ell^2}\n= {L^2 \\over 4 G_4} \n \\delta \\left( {1\\over z_I^2} \\right),\n\\eea\nwhere $\\pi \\ell^2$ is the disc area.\nEquation \\eqref{ent34} provides a complete expression up to first order. \nAt second order the entropy\nwill receive new $z_I$ dependent contributions.\n\nNext, we note that\nthe right hand side of equation \\eqref{ent34} is actually independent of the disc size $\\ell$!\nOn first hand observation this appears very surprising because, according to the first law of \nentanglement thermodynamics \\cite{Bhattacharya:2012mi}, we expected that the entropy density of excitations would have had $\\ell^2$ dependence, namely in the form of inverse temperature (usually entanglement\ntemperature goes as $T_E^{-1}\\propto \\ell^a$; and the dynamical exponent of time in our Lifshitz background is $a=2$). Especially this aspect of the first law has been found to remain true in a variety of relativistic CFTs, where entanglement temperature is given by $T_E \\propto {1 \\over \\pi \\ell}$; see for example \\cite{Bhattacharya:2012mi, Allahbakhshi:2013rda, Mishra:2015cpa, Mishra:2016yor, Mishra:2018tzj, Ghosh:2017ygi, Bhattacharya:2019zkb}. What, then, is so different for the Lifshitz system described by equation \\eqn{ent34}? To understand this phenomenon we first need to get an estimate of the energy associated with the excitations in our system.\n\n\\subsection {Energy, winding charge and chemical potential}\n \nWe now turn to find the energy of excitations of the `massive strings' due to which we have a configuration in equation \\eqn{sol2a9},\nwhere we can express $B_{ty}\\simeq B_{ty}^{massive}\n+B_{ty}^{excitation}$.\nNote that we are treating $y$ as a compact direction.\nThe Scherk-Schwarz compactification \\cite{Scherk:1978ta, *Scherk:1979zr, Lavrinenko:1996mp} of the Lifshitz background \\eqn{sol2a9} on a circle \nalong $y$ gives rise to the following 1-form potential\n\\be\nA_{(1)}={L^2\\over q z^2}\\left(1+ {z^2\\over z_I^2}\\right)^{-1} dt.\n\\ee\nIt represents a gauge field in the lower dimensional supergravity whose only non-zero component is $A_t$. It can be determined from here that due to string excitations the net change in the $U(1)$ charge (due to winding strings) is\n\\be\\label{rho12}\n \\bigtriangleup \\rho \n= {N\\over V_2}= {\\bigtriangleup Q\\over 2\\pi r_y V_2}=\n{2 L\\over G_5 z_I^2},\\ee\nwhere $V_2$ is the area element of $x_1,x_2$ plane, see a calculation in the appendix. The entanglement chemical potential, with the prescription in \\cite{Mishra:2015cpa}, \ncan be obtained by measuring gauge field \nat the turning point, \n namely\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{chempotdef}\n\\mu_E \\equiv A_t|_{z=z_\\ast} = {L^2r_y \\over q z_\\ast^2}+ \\cdots \\,,\n\\eea \nwhere ellipses denote sub-leading terms which are not required at first order. This is a logical guess inspired by black hole thermodynamics, where the value of the one form at the black hole horizon is known to give the chemical potential conjugate to the U(1) charge. Even for backgrounds with non-relativistic conformal symmetry as considered in \\cite{Maldacena:2008wh}, the Kaluza-Klein gauge field measured at the horizon produces the correct thermal chemical potential. There's no horizon in our bulk space-time; instead, we use the critical point $z_*$ associated with the entanglement wedge.\n\nAt leading order we have $z_\\ast \\simeq \\ell$, hence essentially this thermodynamic variable gets uniquely fixed by the \nLifshitz ground state \\eqn{sol2a9n}. \nSo for small $\\ell ~ (> 0)$ the chemical potential remains quite important, and we obtain\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \n\\mu_E\\cdot \\bigtriangleup \\rho \\simeq \n{L^2 \\over \\pi G_4}{1\\over z_I^2\\ell^2}. \n\\eea\n There are no other excitations except the winding strings, the energy density due to\n the excitations can be estimated to be\n\\begin{equation} \\label{enr34}\n\\bigtriangleup {\\cal E}={\\cal E}-{\\cal E}_0 \\simeq \n\\frac{1}{2}\\mu_E \\bigtriangleup \\rho = \\frac{L^3 r_y}{q G_5} \\frac{1}{z_I^2 \\ell^2} = {L^2 \\over 2\\pi G_4}\\frac{1}{z_I^2 \\ell^2}\\,,\n\\end{equation}\n where ${\\cal E}_0$ is the (normalized) energy of the\nground state of our Lifshitz theory\\footnote{We do notice an explicit dependence of energy density on the system size; which is unlike relativistic CFT but is a familiar feature in non-relativistic theories, the particle in a box being an immediate example.}. This is the only meaningful deduction we can make from here, particularly in absence of a direct method to evaluate full stress-energy tensor of the Lifshitz theory.\\footnote{ There is an early work \\cite{Ross:2009ar} but it does not include dilatonic scalar field excitations like in our background. In contrast in asymptotically AdS spacetimes one knows how to obtain stress-energy tensor by doing Fefferman-Graham expansion near AdS boundary \\cite{Balasubramanian:1999re,*Kraus:1999di,*Bianchi:2001kw}. Perhaps something similar could also be done in the Lifshitz case involving dilaton field.} Assuming that the entanglement temperature of the 3-dimensional\n$a=2$ Lifshitz system faithfully behaves as \\cite{Bhattacharya:2012mi}\n \\be\\label{spheretemp1}\nT_E = {4\\over \\pi \\ell^2}\\,,\\ee\nwe determine that the ratio \n$${\\mu_E \\over T_E} = {\\pi L^2 r_y\\over 4 q}\\,,$$ \n is indeed independent of $\\ell$. \nEssentially this ratio seems to get uniquely fixed by the Lifshitz ground state \\eqn{sol2a9n} at the leading order. Note that the excitations seem to have no effect on it. The analysis also implies that the energy density and the entanglement temperature both fall off with the system size $\\ell$ at the same rate, and the ratio\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\n{\\bigtriangleup {\\cal E} \\over T_E}= \n{ \\pi L^3 r_y \\over 4 q G_5 z_I^2} \n\\equiv {1\\over 2} {k_E N\\over V_2}, \n\\eea\nstays fixed for small discs. However this ratio does depend on the excitations \nnamely through $z_I$. \nIn the second equality we have preferred to view dimensionless quantity\n $k_E={\\pi L^2 r_y\\over 8 q}$ as being \nanalogous to the Boltzmann constant in usual \nthermodynamics. (For example, we could have expressed total energy of disc as $\\bigtriangleup { E}= {1\\over 2} N k_E T_E $ with out affecting anything.) {\\it Hence it can be concluded that the entanglement\nentropy per unit disc area is fixed \nfor small discs of radii $\\ell\\ll z_I$}. It is \n also confirmed that the entropy of excitations \\eqref{ent34} \nfollows the first law relation \\cite{Bhattacharya:2012mi, Allahbakhshi:2013rda, Wong:2013gua, Pang:2013lpa, Mishra:2015cpa, Mishra:2016yor, Ghosh:2016fop, Ghosh:2017ygi, Bhattacharya:2019zkb}\n\\be\n\\delta s_E= {1 \\over T_E} (\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E\\delta \\Delta \\rho),\n\\ee\nunder infinitesimal changes in the bulk quantity, $\\delta z_I$.\n\nWe summarize our main observations at first order;\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\nT_E\\propto {1\\over \\ell^2}, ~~~~\n\\bigtriangleup s_E = \\text{Fixed}, ~~~~\n\\mu_E\\propto r_y T_E, ~~~~\n\\bigtriangleup {\\cal E}\\propto N T_E, ~~~~\n\\Delta \\rho= \\text{Fixed}, ~~~~\n\\eea\nat a given entanglement temperature.\n\n\\subsection{Entanglement entropy of a disc at second order}\nLet us now consider corrections to holographic entanglement entropy at next higher order. It is somewhat easier to calculate when one chooses $z(r)$ parameterization, so let us rewrite the integral as\n\\begin{equation}\n\t\\mathcal{A}_{\\gamma} = 2\\pi L^2\\int_{0}^{1}dr \\frac{r\\sqrt{1+z'^2}}{z^2}h^{\\frac{1}{2}}\\,,\n\\end{equation}\nwhere we rescaled $r$ and $z$ to the dimensionless variables $\\frac{r}{\\ell}$ and $\\frac{z}{\\ell}$. It suffices to obtain the embedding up to first order to get the entanglement at second order \\cite{Blanco:2013joa, Bhattacharya:2019zkb}. So, we expand $z(r)$ as $z(r) = z_{(0)} + z_{(1)} + \\cdots$, where $z_{(0)} = \\sqrt{1 - r^2}$ and $z_{(1)}$ satisfies the equation\n\\begin{equation}\\label{eqnsec}\n\tz_{(1)}'' + \\frac{1-2r^2}{r(1-r^2)}z_{(1)}'-\\frac{2}{(1-r^2)^2}z_{(1)} = \\frac{1}{\\sqrt{1-r^2}}\\,,\n\\end{equation}\nwith the boundary conditions: $z_{(1)}'(0) = 0$ and $z_{(1)}(\\ell) = 0$. One can check that a consistent solution to equation \\eqref{eqnsec} is\n\\begin{equation}\\label{embdsec}\n\tz_{(1)} = - \\frac{1-r^2-2\\sqrt{1-r^2}+2\\ln \\left(1+\\sqrt{1-r^2}\\right)}{2\\sqrt{1-r^2}}.\n\\end{equation}\nTherefore, the area integral now acquires a new contribution $\\mathcal{A}_\\gamma = \\mathcal{A}_0 + \\mathcal{A}_{1} + \\mathcal{A}_2$ where\n\\begin{equation}\\label{area2}\n\t\\mathcal{A}_2 = 2\\pi L^2\\frac{\\ell^4}{z_I^4}\\left(\\frac{5}{8} - \\ln 2\\right),\n\\end{equation}\nwhich is negative as expected. The area difference from pure $AdS$ at both orders is plotted in figure \\ref{spherearea} .\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale=0.75]{spherearea-eps-converted-to.pdf}\n\t\\caption{Area difference from $AdS$ ground state for spherical subsystem, the second order correction is negative. Plot drawn by choosing $z_I^2 = 2$ and $L = r_y = q = 1$.}\n\t\\label{spherearea}\n\\end{figure}\nTotal entropy of the disc at this order will be\n\\begin{equation}\\label{discent2}\n\tS_E^{(2)} = S_E^{(0)} + \\frac{\\pi L^2}{4 G_4}\\frac{\\ell^2}{z_I^2}\\left(1 + \\frac{\\ell^2}{z_I^2}\\left(\\frac{5}{4} - 2\\ln 2\\right)\\right).\n\\end{equation}\nSo that the variation of entropy density, at second order, becomes:\n\\begin{equation}\\label{entden2}\n\t\\delta s_E^{(2)} = \\frac{L^2}{4 G_4}\\left(1 + \\frac{\\ell^2}{z_I^2}\\left(\\frac{5}{2} - 4\\ln 2\\right)\\right)\\delta\\left(z_I^{-2}\\right).\n\\end{equation}\nAs previous, we wish to express \\eqref{entden2} as a `first law' like relationship. We find that one way to achieve this is to absorb all second order corrections to a modified temperature and chemical potential, this method was first used in \\cite{Mishra:2015cpa} although they worked with differences rather than variation as we do. To this end, we first note that the turning point $z_*$ should be corrected at $\\mathcal{O}\\left(\\frac{\\ell^2}{z_I^2}\\right)$ as\n\\begin{equation*}\n\tz_* \\equiv z(0) = \\ell + \\frac{\\ell^3}{z_I^2}\\left(\\frac{1}{2} - \\ln 2\\right).\n\\end{equation*}\nThe chemical potential, defined in equation \\eqref{chempotdef}, can be expressed including $\\mathcal{O}(\\frac{\\ell^2}{z_I^2})$ corrections as\n\\begin{align}\n\t\\mu_E^{(1)} \\simeq& \\frac{L^2r_y}{q\\ell^2}\\left(1+\\frac{\\ell^2}{z_I^2}\\left(\\frac{1}{2} - \\ln 2\\right)\\right)^{-2}\\left(1+\\frac{\\ell^2}{z_I^2}\\right)^{-1}\\,, \\nonumber \\\\\n\t=& \\frac{L^2r_y}{q\\ell^2}\\left(1 - \\frac{\\ell^2}{z_I^2}\\left(2 - 2\\ln 2\\right)\\right).\n\\end{align}\nSo we get\n\\begin{align*}\n\t\\mu_E^{(1)}\\delta\\Delta \\rho ~=&~ \\frac{2 L^3r_y}{qG_5\\ell^2}\\left(1 - \\frac{\\ell^2}{z_I^2}\\left(2 - 2\\ln 2\\right)\\right)\\delta\\left(z_I^{-2}\\right),\\\\\n\t=&~ \\frac{L^2}{\\pi G_4\\ell^2}\\left(1 - \\frac{\\ell^2}{z_I^2}\\left(2 - 2\\ln 2\\right)\\right)\\delta\\left(z_I^{-2}\\right),\n\\end{align*} \nwhile the energy remains the same as defined in \\eqref{enr34}. From equation \\eqref{entden2}, a bit of paperwork then leads to the following result\n\\begin{equation}\\label{spherelaw2}\n\t\\delta s_E^{(2)} = \\frac{1}{T_E^{(2)}}\\left(\\delta\\Delta\\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta\\Delta\\rho \\right),\n\\end{equation}\nwhere $T_E^{(2)}$ denotes the `entanglement temperature' at second order, which is given by\n\\begin{align}\\label{spheretemp2}\n\tT_E^{(2)} &= \\frac{\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta \\Delta \\rho}{\\delta \\Delta s_E^{(2)}},\\nonumber \\\\\n\t&=\\frac{\\frac{L^2}{\\pi G_4\\ell^2}\\left[1 - \\frac{\\ell^2}{z_I^2}\\left(1 - \\ln 2\\right) \\right]}{\\frac{L^2}{4 G_4}\\left[1 - \\frac{\\ell^2}{z_I^2}\\left(4\\ln 2 - \\frac{5}{2}\\right) \\right]},\\nonumber \\\\\n\t&\\simeq T_E^{(1)}\\left[1 + \\frac{\\ell^2}{z_I^2}\\left(5\\ln 2 - \\frac{7}{2}\\right) \\right],\n\\end{align}\nwhere $T_E^{(1)}$ stands for the first order temperature, defined in eqn. \\eqref{spheretemp1}. The term in parentheses is a negative number, so second order correction to `entanglement temperature' results in its sharper fall. See figure \\ref{fig1} for an illustration of this behaviour.\n\n\\begin{figure}[h!]\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{spheremu-eps-converted-to.pdf}\n\t\t\\caption{$\\mu_E$ vs. $l$}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{spheretemp-eps-converted-to.pdf}\n\t\t\\caption{$T_E$ vs. $l$}\n\t\\end{subfigure}\n\t\\caption{The unbroken and dashed curves display the behaviour of the uncorrected and corrected quantities, respectively; both the entanglement temperature and chemical potential decrease due to higher order corrections. The plots were drawn by setting $z_I^2 = 2$ and $L = r_y = q = 1$.}\n\t\\label{fig1}\n\\end{figure}\n\n\\par Some comments are in order to justify equation \\eqref{spherelaw2}, we have seen that for small enough subsystem size $(\\ell \\ll z_I)$, the change in entanglement entropy at first order in our perturbative calculation follows a relationship akin to the first law of thermodynamics. If one considers this relationship an actual `law' for entanglement entropy, one must find a consistent way to describe new contributions at higher orders. Equation \\eqref{spheretemp2} proposes that at second order, the chemical potential as well as the entanglement temperature should be corrected to keep the law intact. In fact, we expect this procedure to work at all higher orders. It could be thought that a more accurate measure of these quantities are obtained as one climbs the perturbation ladder.\n\\section{ Entanglement entropy of narrow strip}\\label{sec4}\n \nWe now consider a strip like subsystem with coordinate\nwidth $-\\ell\/2\\le x_1\\le \\ell\/2$, and the range of $x_2\\in[0,l_2]$, such that\n$l_2\\gg \\ell$. The straight line boundary of the two-dimensional strip\nis identified with the boundary of the RT surface in the bulk \nat constant time. \nThe area functional of this static surface is \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{str10}\n{\\cal A}_\\gamma=2 L^2 l_2 \\int^{z_\\ast}_{\\epsilon} dz \n{\\sqrt{1+x_1'^2}\\over z^2} h^{1\\over2}\\,. \n\\eea\nFor small width $\\ell\\ll z_I$, we \nmake a perturbative expansion of the integrand.\nThe extremal surface satisfies the following equation\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \nx_1'={z^2\\over z_\\ast^2} {1\\over \\sqrt{{h\\over h_\\ast}-{z^4\\over z_\\ast^4}}}\\,,\n\\eea\nwhere $h_\\ast\\equiv h(z_\\ast)$.\nWe have specific boundary conditions such that\nnear the spacetime boundary \n$x_{1}|_{z=0}=\\ell\/2$ and \nthe turning point is given by $x_{1}|_{z\\sim z_\\ast}=0$. \nThis leads to the first integral of the following type\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{str11} \n\\ell=2\\int^{z_\\ast}_0 dz{z^2\\over z_\\ast^2} {1\\over \\sqrt{{h\\over h_\\ast}-{z^4\\over z_\\ast^4}}}\\,,\n\\eea\nwhich gives rise to a perturbative expansion in ${z_\\ast \\over z_I}$ \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{str12}\n\\ell=z_\\ast\n\\left(b_0 +{z_\\ast^2 \\over2 z_I^2} I_1 + \\cdots\\right)\\,,\n\\eea\nwhere coefficients are expressible as Beta-functions $b_0=\n{1\\over 4} B\\left(\\frac{3}{4}, \\frac{1}{2}\\right)$ and $I_1= \n{1\\over 4}(B(\\frac{3}{4},-\\frac{1}{2}) -B(\\frac{5}{4},-\\frac{1}{2}))$.\nThe equation \\eqn{str12} can be inverted and expressed as a perturbative expansion of the turning point\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} z_\\ast=z_\\ast^{(0)} \\left(1 -{z_\\ast^{(0)2} \\over z_I^2} {I_1\\over2 b_0} + \\cdots \\right),\n\\eea\nwhere $z_\\ast^{(0)}\\equiv {\\ell \\over 2b_0}$ is the turning point in the \nabsence of excitations.\n\nThe leading area of strip can be evaluated using the tree level values\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\n{\\cal A}_0 &&= \n2 L^2 l_2 \\int^{z_\\ast^{(0)}}_{\\epsilon} dz \n{\\sqrt{1+x_{1(0)}'^2}\\over z^2}\\,,\\nonumber\\\\\n&&={2 L^2 l_2\\over z_\\ast^{(0)}} \\int^{1}_{\\epsilon \\over z_\\ast^{(0)}} \nd\\zeta {1\\over\\zeta^2 \\sqrt{1-\\zeta^4}}\\,,\n\\nonumber\\\\ &&\n=2 L^2 l_2 \\left({1\\over\\epsilon}-{2(b_0)^2\\over \\ell}\\right).\n\\eea\nwhile the first order contribution is evaluated as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{foa}\n{\\cal A}_1 &&= \n2 L^2 l_2 \\int^{z_\\ast}_{0} dz \n{\\sqrt{1+x_{1(0)}'^2}\\over 2 z_I^2}\\,,\\nonumber\\\\\n&& = L^2 l_2\\left({a_1 z_\\ast^{(0)} \\over z_I^2 }\\right).\n\\eea\nwhere the coefficient $a_1={1\\over 4} B\\left(\\frac{1}{4},\\frac{1}{2}\\right)$.\nThe entanglement entropy of small strip up to first order is then given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\nS_E^{strip} = \\frac{\\mathcal{A}_0 + \\mathcal{A}_1}{4G_5} = {L^2 l_2 \\over 2 G_{4}} \n\\left({1\\over \\epsilon} -{2b_0^2\\over \\ell}\n+{a_1\\over 4b_0}~\\frac{\\ell}{z_I^2} \\right).\n\\eea\n\nNow any small change in the bulk parameter \n($\\delta z_I$) will necessarily effect the entanglement entropy at first order.For a fixed width $\\ell$, we find the change in entropy per unit area of the strip as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{jh4}\n\\delta s_E^{strip} \\equiv {\\delta S_E^{strip}\\over l_2 \\ell}\n= {L^2 \\over 8 G_{4}} {a_1\\over b_0}\n\\delta \\left( {z_I^{-2}} \\right),\n\\eea\nwhich is complete expression up to first order. Once again we find that the right hand side is independent of $\\ell$, as it was also in the case of a disc. Following from the disc case in the previous section, the effective chemical potential for strip subregion is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\n\\mu_E=\n{L^2 r_y\\over q z_\\ast^2}\\simeq \n{4 b_0^2 L^2 r_y\\over q \\ell^2}\\,.\n\\eea\nFrom here and \\eqref{rho12}, let us define for the strip\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{enr35} \n\\bigtriangleup {\\cal E} \\equiv \\frac{1}{2}\n\\mu_E . \\bigtriangleup \\rho = {4 L^3r_y \\over G_5\\,q}\\frac{b_0^2}{z_I^2 \\ell^2 } = \\frac{2 L^2}{\\pi G_4}\\frac{b_0^2}{z_I^2\\ell^2}.\n\\eea\nThis is like the disc result in \\eqn{enr34}, $i.e.$ \n$\\bigtriangleup {\\cal E}\\propto T_E$.\nUsing \\eqn{enr35} we conclude that the entanglement\nentropy density \\eqn{jh4} of the strip subsystems \nalso conforms to the first law relation\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{flfo}\n\\delta s_E={1\\over T_E} \\left(\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E\\delta \\Delta \\rho\\right). \\eea\nwhere for the strip, entanglement temperature is\ndefined as $T_E={8 b_0^3 \\over a_1}{4\\over \\pi \\ell^2}$ \nin 3-dimensional Lifshitz theory.\n\n\\subsection{Strip entropy at second order}\n\nIt is instructive to find out the change in entanglement \nentropy at higher orders in $\\frac{\\ell^2}{z_I^2}$ and \ninterpret its thermodynamic property, here we include the \nresults at $\\mathcal{O}\\left(\\frac{\\ell^4}{z_I^4}\\right)$. \\\\ \n\\par The turning point $z_*$, as discussed before in \\eqn{str11} and \\eqn{str12}, could be related to the strip-width $\\ell$ as\n\t\\begin{equation}\\label{turnpt2}\n\t\tz_* = \n\\frac{z_*^{(0)}}{1 + \\frac{z_*^{(0) 2}}{2z_I^2} \\frac{I_1}{b_0} - \n\\frac{z_*^{(0) 4}}{8z_I^4} \\left(\\frac{I_2}{b_0} + \\frac{4I_1^2}{b_0^2}\\right)}\\,,\n\t\\end{equation}\n\twhere the new co-efficient $I_2$ can be expressed as: \n$I_2 = \\frac{1}{8}\\big(2B(\\frac{3}{4}, -\\frac{3}{2}) - \n3B(\\frac{5}{4}, -\\frac{3}{2})\\big)$. With the help of \n\\eqref{turnpt2}, the area integral \\eqn{str10} now reads\n \t\t$\\mathcal{A}_{\\gamma} = \\mathcal{A}_0 + \\mathcal{A}_1 + \\mathcal{A}_2$,\n where $\\mathcal{A}_0$ and \n$\\mathcal{A}_1$ are as obtained before.\n The second order contribution is\n\t\\begin{equation}\n\\mathcal{A}_2=\n- \\frac{2 L^2 l_2}{z_*^{(0)}} \\frac{z_*^{(0)4}}{8z_I^4}\\left(\\frac{4a_0I_1^2}{b_0^2} + \\frac{2I_1J_1}{b_0}\\right).\n\t\\end{equation}\n\tThe new coefficients introduced in above expression are listed below\n\t\\begin{align*}\n\t\ta_0 &= -\\frac{1}{4}B\\left(\\frac{3}{4}, \\frac{1}{2}\\right) = -b_0\\,, \\\\\n\t\tJ_1 &=\\frac{1}{4}\\left(B\\left(\\frac{3}{4}, -\\frac{1}{2}\\right) \n+ 3B\\left(\\frac{1}{4}, -\\frac{1}{2}\\right) \\right).\n\t\\end{align*}\n\nAfter some simplification the contribution to the area of the RT \nsurface at second order turns out to be\n\t\\begin{equation}\n\t\t\\mathcal{A}_2 = -\\frac{L^2 l_2\\ell}{64} \\frac{\\ell^2}{z_I^4}\\frac{1}{b_0^2}\\left(\\frac{a_1^2}{b_0^2} - 1 \\right).\n\t\\end{equation}\n\tThe coefficient $a_1$ has already been defined in eq. \\eqref{foa}. Hence, the total entanglement entropy density, at second order in perturbation theory, becomes\n\t\\begin{equation}\\label{kl23}\n\t\ts_E^{(2)} = s_E^{(0)} + \\frac{L^2}{8 G_4}\\frac{1}{z_I^2}\n\\frac{a_1}{b_0}\\left(1 - \\frac{\\ell^2}{z_I^2} \\frac{1}{32b_0^2}\\left(\\frac{a_1^2}{b_0^2} - 1\\right)\\right).\n\t\\end{equation}\n\tThe area difference including second order correction has been shown in figure \\ref{striparea}.\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.75]{striparea-eps-converted-to.pdf}\n\t\t\\caption{The area difference at first and second order of perturbation analysis for strip subsystem, plots drawn by choosing $z_I^2 = 2$ and $L=r_y=q=l_2=1$.}\n\t\t\\label{striparea}\n\t\\end{figure}\n\tTo write down the `first law' we need to rewrite the expression for $s_E^{(2)}$ in terms of variation in $\\mathcal{E}$ and $\\mu_E \\Delta \\rho$; recall that the chemical potential was defined as the value of the gauge potential at the turning point. Here, it is sufficient to compute $\\mu_E$ up to first order\n\t\\begin{equation*}\n\t\t\\mu_E^{(1)} \\simeq \\frac{L^2}{z_*^2}\\left(1 - \\frac{z_*^2}{z_I^2} \\right) = \\frac{L^2r_y}{qz_*^{(0)2}}\n\\left(1 + \\frac{z_*^{(0)2}}{z_I^2}(\\frac{I_1}{b_0^2} - 1) \\right).\n\t\\end{equation*}\n\tSo that,\n\t\\begin{align*}\n\t\t\\mu_E^{(1)}\\delta \\Delta \\rho ~=&~ \\frac{L^3r_y}{qG_5}\\frac{8b_0^2}{\\ell^2}\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{8b_0^2}\\left(\\frac{a_1}{b_0} - 3\\right) \\right]\\delta\\left(z_I^{-2}\\right),\\\\\n\t\t=&~ \\frac{L^2}{2\\pi G_4}\\frac{8b_0^2}{\\ell^2}\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{8b_0^2}\\left(\\frac{a_1}{b_0} - 3\\right) \\right]\\delta\\left(z_I^{-2}\\right).\n\t\\end{align*}\nA little effort, then, allows us to write\n\t\\begin{equation}\\label{law2}\n\t\t\\delta s_E^{(2)} = \\frac{1}{T_E^{(2)}}\\left(\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta \\Delta \\rho\\right).\n\t\\end{equation}\n\tHere, $T_E^{(2)}$ stands for the modified entanglement temperature at second order.\n\t\\begin{align}\\label{temp2}\n\t\tT_E^{(2)} &= \\frac{\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta \\Delta \\rho}{\\delta \\Delta s_E^{(2)}},\\nonumber \\\\\n\t\t\t\t &= \\frac{4}{\\pi \\ell^2}\\frac{8b_0^3}{a_1}\n\t\t\t\t\t\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{16b_0^2} \\left(\\left(\\frac{a_1}{b_0} - 3 \\right) + \\left(\\frac{a_1^2}{b_0^2} - 1\\right) \\right) \\right] \\nonumber \\\\\n\t\t\t\t &= T_E^{(1)}\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{16b_0^2}\\left(\\left(\\frac{a_1}{b_0} - 1\\right)\\left(\\frac{a_1}{b_0} + 2 \\right) - 2 \\right) \\right]\n\t\\end{align}\nWhere by $T_E^{(1)}$, we refer to the temperature at first order defined in equation \\eqref{flfo}, the numerical value of $\\frac{a_1}{b_0} \\approx 2.188$, so the correction at this order results in an increase of $T_E$, albeit by a tiny amount. The uncorrected and corrected temperatures are plotted in figure \\ref{fig2}.\n\n\\begin{figure}\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{Lif4mu-eps-converted-to.pdf}\n\t\t\\caption{$\\mu_E$ vs. $\\ell$}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{Lif4temp-eps-converted-to.pdf}\n\t\t\\caption{$T_E$ vs. $\\ell$}\n\t\\end{subfigure}\n\t\\caption{The unbroken and dashed curves display the behaviour of the uncorrected and corrected quantities, respectively; the entanglement temperature is found to increase due to higher order corrections while the chemical potential decreases.The plots were drawn by setting $z_I = 2$ and $L = r_y = q = G_5 = 1$.}\n\t\\label{fig2}\n\\end{figure}\n\n\\subsection{Numerical results for strip subsystem}\nWe end this section with a comparison of our perturbative results with some numerical analysis. For the numerical computation we chose $z_I = 4$ and used \\eqref{str11} to obtain corresponding lengths $\\ell$ of the sub-region for different choices of the turning point $z_{*}$. We also obtain the area difference $\\Delta \\cal{A}$ from \\eqref{str10} for the same $z_{*}$ values and plot the two sets against each other. The output is summarized in figure \\ref{stripnumeric}. From the graph we conclude that a second order perturbation series analysis is trustworthy for small strip-width.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale=0.5]{StripNumeric.pdf}\n\t\\caption{Numerical plot of area difference from $AdS$ ground state for strip subsystem and comparison with second order perturbation series analysis. The pre-factor in \\eqref{str10} was ignored in the plot.}\n\t\\label{stripnumeric}\n\\end{figure}\n\n\\section{Conclusion}\\label{sec5}\n\nThe Lifshitz background \n$Lif_4^{(2)}\\times {S}^1\\times S^5$ of the massive type IIA theory\nallows exact excitations which couple to massless modes of string in the IR. We calculated the entanglement entropy of the theory\nat the boundary of these spacetimes, both for strip as well as disc shaped systems. At leading order, we found that the entropy density of the excitations remains fixed and does not grow with $\\ell$, the subsystem size, so long as $\\ell\\ll z_I$. We find that this behaviour is consistent with the fact that energy density of the excitations itself behaves as $\\bigtriangleup{\\cal E} \\propto 1\/\\ell^2$, which is in agreement with $\\bigtriangleup{\\cal E}\n\\simeq \\frac{1}{2}\\mu_E \\bigtriangleup\\rho $. Note that the entanglement temperature itself goes as $T_E\\propto{1\\over \\ell^2}$.\n\nBut this entanglement behaviour is quite different in comparison to the relativistic CFTs, where the entropy density of excitations grows linearly with the subsystem size, while the energy density of excitations remains fixed. Nevertheless we have found that the first law of entanglement thermodynamics\n\\begin{equation}\n\t\\delta s_E = \\frac{1}{T_E}\\left(\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E\\delta \\Delta \\rho\\right),\n\\end{equation} \nholds good if we accept the hypothesis that the energy of a subsystem in the Lifshitz background \\eqref{sol2a9} is given by\n$$\\bigtriangleup{ E}\\simeq \\mu_E N \n\\simeq {1\\over 2} N k_E T_E\\,. $$\nOur results appear to indicate an \nequipartition nature of the entanglement thermodynamics for non-relativistic Lifshitz\nsystem. But this is perhaps true only for \nthe high entanglement temperature regime (i.e. small $\\ell\\ll z_I$).\n\\par Further, we studied what happens to the first law of entanglement if we assume it to remain valid beyond the leading order. There is lack of consensus on this aspect, despite there being enough evidence for it to be a natural feature at first order. We discussed how the first law could be extended up to second order by making use of appropriately modified chemical potential and entanglement temperature. We think this is necessary because otherwise, we need to look for a new quantity at each higher order to account for the corrections; while the entanglement entropy, like its thermal counterpart should depend only on the energy and conserved charges of the theory. Such redefinition should work at all orders, thereby allowing the `first law of entanglement thermodynamics' to be obeyed quite generally, irrespective of the degree of perturbation theory.\n\\par It would be interesting to obtain the HEE numerically for ball subsystems and compare with our perturbative results. This, however, involves solving boundary value problem and proves to be non-trivial. Another interesting problem is to consider shape dependence of holographic entanglement entropy in similar spirit to \\cite{Fonda:2015nma, Cavini:2019wyb}. We hope to return to these problems in future.\n\\vskip 0.5cm\n\\begin{acknowledgements} \nHS is thankful to the organisers of the \nAdS\/CFT@20 workshop at ICTS Bangaluru \nand the STRINGS-2019 at Brussels for the exciting meetings and warm hospitalities. SM would like to thank Aranya Bhattacharya for useful discussions and help with Mathematica.\n\\end{acknowledgements} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nPrimitive meteorites contain presolar dust grains, grains that formed \nin stellar outflows and supernova (SN) ejecta and survived not only a \nlong history as interstellar (IS) grains in the interstellar medium \n(ISM), but also the formation of the solar system and conditions in the \nmeteorites' parent bodies (Anders \\& Zinner 1993; Ott 1993). The stellar \norigin of these grains is indicated by their isotopic compositions, which \nare completely different from those of material in the solar system and \nreflect the compositions of their stellar sources. Their study in the \nlaboratory provides information on stellar nucleosynthesis, Galactic \nchemical evolution (hereafter GCE), physical and chemical properties \nof stellar atmospheres and ejecta, and conditions in the early solar \nsystem (Bernatowicz \\& Zinner 1997; Zinner 1998).\n\nThe following types of presolar grains have been identified to date: \ndiamond (Lewis et al. 1987), silicon carbide (SiC, carborundum) \n(Bernatowicz et al. 1987), graphite (Amari et al. 1990), aluminum \noxide (Al$_2$O$_3$, corundum), spinel (MgAl$_2$O$_3$) (Hutcheon et al. \n1994; Nittler et al. 1994) and silicon nitride (Si$_3$N$_4$) \n(Nittler et al. 1995). In addition, graphite and SiC contain tiny \nsubgrains of Ti carbide \n(graphite contains also Zr and Mo carbide) that were identified by \ntransmission electron microscopy (Bernatowicz et al. 1991, 1996; \nBernatowicz, Amari, \\& Lewis 1992). The carbonaceous phases diamond, \nSiC and graphite were discovered because they carry isotopically \nanomalous noble gases (Tang \\& Anders 1988a; Lewis, Amari, \\& Anders \n1990, 1994; Huss \\& Lewis 1994a,b; Amari, Lewis, \\& Anders 1995a); \nthey can be extracted from meteorites in almost pure form by chemical \nand physical processing (Tang et al. 1988; Amari, Lewis, \\& Anders 1994). \nThis results in relatively large amounts (micrograms) of samples that \ncan be analyzed in great detail. In contrast, presolar corundum and \nsilicon nitride have been discovered by ion microprobe isotopic \nmeasurements of individual grains from chemically resistant residues \nand only a limited number of grains ($\\sim$ 100 for corundum) have been \nmeasured to date.\n\nSilicon carbide, graphite, corundum, and Si$_3$N$_4$ grains are large \nenough (up to several $\\mu$m in diameter for corundum and Si$_3$N$_4$, \nup to $>$ 10 $\\mu$m for SiC and graphite) to be analyzed individually \nfor their elemental and isotopic compositions. The ion microprobe makes \nit possible to measure isotopic ratios of major and some minor elements \nin single grains down to $\\sim$ 0.5 $\\mu$m in diameter (e.g., Hoppe et al.\n1994, 1995; Huss, Hutcheon, \\& Wasserburg 1997; Travaglio et al. 1999). \nIon microprobe isotopic measurements have been made for C, N, O, Mg, Si, \nK, Ca, and Ti on many (for the major elements on thousands of) grains \n(see, e.g., Zinner 1998). Single grain isotopic measurements, albeit on \na limited number of grains, have also been made for Zr, Mo, and Sr by \nresonance ionization mass spectrometry (Nicolussi et al. 1997, 1998a,b,c; \nDavis et al. 1999; Pellin et al. 1999); and for He and Ne by laser \nextraction and noble gas mass spectrometry (Nichols et al. 1991, 1994; \nKehm et al. 1996). Diamonds, instead, are too small ($\\sim$ 2 nm) for \nsingle grain analysis and isotopic measurements have been made on ``bulk\nsamples'', collections of many grains. Bulk analyses have also been made \nin SiC and graphite of the noble gases (Lewis et al. 1994; Amari et al. \n1995a) and of trace elements such as Sr, Ba, Nd, Sm, and Dy in SiC (see, \ne.g., Anders \\& Zinner 1993; Hoppe \\& Ott 1997; Zinner 1998).\n\nBased on their isotopic compositions, several stellar sources have been \nidentified for presolar grains. Most corundum grains are believed to have \noriginated in low-mass red giant (RG) and asymptotic giant branch (AGB) \nstars. This identification rests on the O isotopic ratios and inferred \n$^{26}$Al\/$^{27}$Al ratios (Huss et al. 1994; Nittler et al. 1997; Nittler\n1997; Choi et al. 1998; Nittler \\& Alexander 1999a). Low-density graphite \ngrains, a subtype of SiC grains termed X grains, and Si$_3$N$_4$ grains have \nisotopic signatures indicative of a SN origin (Nittler et al. 1995; \nTravaglio et al. 1999). A SN origin has also been proposed for a few \ncorundum grains (Nittler et al. 1998; Choi et al. 1998). In addition, \na few rare SiC grains with large $^{30}$Si excesses and low \n$^{12}$C\/$^{13}$C and $^{14}$N\/$^{15}$N ratios are possibly of a nova origin\n(Gao \\& Nittler 1997).\n\nMost SiC grains, in particular the ``mainstream'' component, which \naccounts for $\\sim$ 93\\% of all SiC (Hoppe et al. 1994; Hoppe \\& Ott 1997), \nare believed to come from carbon stars, thermally pulsing (TP) AGB stars\nduring late stages of their evolution. \nThe best evidence for such an origin are the $s$-process (slow neutron \ncapture process) isotopic patterns displayed by the heavy elements Kr, \nSr, Zr, Mo, Xe, Ba, Nd, and Sm (Lewis et al. 1994; Hoppe \\& Ott 1997; \nGallino, Raiteri, \\& Busso 1993; Gallino, Busso, \\& Lugaro 1997; \nNicolussi et al. 1997, 1998a; Pellin et al. 1999) and the presence of \nNe-E(H), almost pure $^{22}$Ne (Lewis et al. 1990, 1994; Gallino et al. \n1990). The C and N isotopic compositions as well as $^{26}$Al\/$^{27}$Al\nratios of individual mainstream SiC grains are by and large consistent \nwith a carbon star origin (Hoppe et al. 1994; Huss et al. 1997; Hoppe \\& \nOtt 1997).\n\nIn contrast to C, N, Ne, Al and the heavy elements, the variations in the \nSi (and Ti) isotopic ratios measured in single mainstream grains cannot be \nexplained in terms of nucleosynthesis in AGB stars (Gallino et al. 1990, \n1994; Brown \\& Clayton 1992a). They have been interpreted to indicate that \nmany stellar sources (Clayton et al. 1991; Alexander 1993), whose \ninitial compositions vary because of GCE, contributed SiC grains to the \nsolar system (Gallino et al. 1994; Timmes \\& Clayton 1996; Clayton \\& \nTimmes 1997a). However, a fundamental problem with this interpretation \nis the fact that the metallicity implied by the Si isotopic compositions \nof the mainstream grains is higher than that of the sun. This would mean \nthat the grains are younger than the solar system. A solution to this \npuzzle has been proposed by Clayton (1997) who considered the possibility \nthat the sun and the AGB stars that were the sources of the mainstream \ngrains did not originate in the same Galactic region but changed their \npositions because of Galactic diffusion. Alexander \\& Nittler (1999), \non the other hand, used the Si and Ti isotopic compositions of the \nmainstream grains themselves (Figs. 3 and 4) to infer the metallicity of the \nISM at the time of solar system formation. From this exercise they \nconcluded that the sun has an atypical Si isotopic composition.\n\nIn this paper we will revisit the Si isotopic compositions of the \nmainstream SiC grains. In \\S 2 we will first describe the isotopic \nproperties of these grains in greater detail, discuss their AGB origin, \nand review previous attempts to understand their Si isotopic ratios. \nAfter presenting new calculations for the nucleosynthesis of Si (and Ti) \nin AGB stars (\\S 3), we will present a new approach to the problems \nof the distribution of Si isotopic ratios (\\S 4): instead of assuming an\naverage monotonic relationship of metallicity with time in the Galaxy, \nwe investigate how \nheterogeneities in the Si isotopic ratios could result from fluctuations \nin the contributions from various types of nucleosynthetic sources to \nthe low-mass stars that, in their AGB phase, produced the SiC grains. \nPreliminary accounts can be found in Lugaro et al. (1999a,b).\n\nGalactic chemical evolution models of the Galactic disk that, \nalthough in different ways, \ndeal with compositional inhomogeneities in the ISM have previously \nbeen presented by Malinie et al. (1993), Wilmes \\& K\\\"oppen (1995), \nCopi (1997) and van den Hoek \\& de Jong (1997). We \nwill show that our approach is consistent with the \nresults obtained by inhomogeneous GCE models, which, however, did \nnot address the evolution of isotopic compositions.\n\n\\section{Meteoritic SiC, the mainstream component and the Si isotope\npuzzle}\n\nAmong all presolar grains, SiC has been most widely studied because it \nis relatively abundant (6 ppm in the Murchison and in similar primitive \nmeteorites) and is present in various classes of meteorites (Huss \\& \nLewis 1995). Ion microprobe isotopic analyses of single grains have \nrevealed several distinct classes. This is shown in Figs. 1 and 2, which \ndisplay the C, N, and Si isotopic ratios. For historical reasons the Si \nisotopic ratios are expressed as $\\delta$-values, deviations in permil\n($^o\\!\/\\!_{oo}$) from the solar isotopic ratios of ($^{29}$Si\/$^{28}$Si)$_{\\odot}$ =\n0.0506331 and ($^{30}$Si\/$^{28}$Si)$_{\\odot}$ = 0.0334744 (Zinner, Tang, \\&\nAnders 1989):\n\n$\\delta$$^{29}$Si\/$^{28}$Si = [($^{29}$Si\/$^{28}$Si)$_{meas}$\/\n($^{29}$Si\/$^{28}$Si)$_{\\odot} -$ 1] $\\times$ 1000,\n\n$\\delta$$^{30}$Si\/$^{28}$Si = [($^{30}$Si\/$^{28}$Si)$_{meas}$\/\n($^{30}$Si\/$^{28}$Si)$_{\\odot} -$ 1] $\\times$ 1000. \n\nAccording to their C, N and Si isotopic compositions five different \ngroups of grains can be distinguished and these groups are indicated \nin the figures. Also indicated in the figures are the abundances of the \ndifferent groups. \nBy far the most common grains are the mainstream grains. It should \nbe noted that the frequency distributions of grains in the plots of \nFigs. 1 and 2 do not correspond to their abundances in meteorites, \nbut that rare grain types, located by automatic imaging in the ion \nprobe (Nittler et al. 1995; Amari et al. 1996) are over-represented. \nAn additional grain with extreme $^{29}$Si and $^{30}$Si excesses \nhas been found (Amari, Zinner, \\& Lewis 1999); its composition\n($\\delta$$^{29}$Si\/$^{28}$Si = 2678 $^o\\!\/\\!_{oo}$, $\\delta$$^{30}$Si\/$^{28}$Si =\n3287 $^o\\!\/\\!_{oo}$) lies outside the boundaries of the plot in Fig. 2. \n\nThe possible stellar sources of the different groups of SiC grains \nhave been discussed elsewhere (e.g., Zinner 1998). Here we wish to \nconcentrate on grains of the mainstream \ncomponent (Hoppe et al. 1994; Hoppe \\& \nOtt 1997). Their $^{12}$C\/$^{13}$C ratios lie between 15 and 100 and their \n$^{14}$N\/$^{15}$N ratios between the solar ratio of 272 and 10,000 (Fig. 1). \nTheir Si isotopic ratios plot along a line of slope 1.31 in a \n$\\delta$$^{29}$Si\/$^{28}$Si vs. $\\delta$$^{30}$Si\/$^{28}$Si 3-isotope \nplot (Fig. 3). Most grains show large $^{26}$Mg excesses attributed to the \npresence of $^{26}$Al ($T_{1\/2}$ = 7 $\\times 10^5$ yr), now extinct, at\nthe\ntime of their formation. Inferred $^{26}$Al\/$^{27}$Al ratios range up to \n10$^{-2}$ (Hoppe et al. 1994; Huss et al. 1997). Much more limited isotopic \ndata exist for Ti. In Fig. 4 the measurements by Hoppe et al. \n(1994) and Alexander \\& Nittler (1999) are plotted as\n$\\delta$$^{i}$Ti\/$^{48}$Ti values against the $\\delta$$^{29}$Si\/$^{28}$Si \nvalues of these grains. The correlation between the Ti ratios, especially\n$\\delta$$^{46}$Ti\/$^{48}$Ti, and \nthe Si isotopic ratios has already been noticed by Hoppe et al. \n(1994).\n\nThere are many pieces of evidence that indicate that mainstream \nSiC grains come from carbon stars. Carbon stars are TP-AGB stars \nwhose spectra are dominated by lines of C compounds such as C$_2$, \nCH, and CN, indicating that C $>$ O in their envelopes (Secchi 1868). \nThey become C-rich because of the recurrent third\ndredge up (TDU) episodes mixing with the envelope newly synthesized \n$^{12}$C~ from the He shell where it is produced by the triple-$\\alpha$ \nreaction \n(Iben \\& Renzini 1983). For high-temperature carbonaceous phases, \nsuch as SiC, to condense from a cooling gas the condition C $>$ O has \nto be satisfied (Larimer \\& Bartholomay 1979; Sharp \\& Wasserburg \n1995; Lodders \\& Fegley 1997a). Carbon stars experience substantial \nmass loss by stellar winds and have extended atmospheres with \ntemperatures of 1500 - 2000 K, at which SiC is expected to condense. \nIn fact, carbon stars are observed to have circumstellar dust \nshells that show the 11.3 $\\mu$m emission feature of SiC (Cohen 1984; \nLittle-Marenin 1986; Martin \\& Rogers 1987; Speck, Barlow, \\& \nSkinner 1997). Recently, Clayton, Liu, \\& Dalgarno (1999) proposed\nthat in a SN environment with high levels of ionizing $\\gamma$-rays carbon \ndust can condense from a gas of O $>$ C. However, even if this should \nbe possible, it would not apply to the expanding atmospheres of \ncarbon stars. In contrast to such atmospheres, the solar system \nis characterized by O $>$ C and phases such as SiC are not believed \nto be able to form under these conditions. This apparently is the \nreason why all SiC grains found in primitive meteorites are of presolar\norigin according to their isotopic compositions. This is in marked \ncontrast to presolar corundum grains, which make up only a small fraction\n($\\sim$1\\%) of all meteoritic corundum grains.\n\nIsotopic compositions of presolar grains are the most diagnostic \nindicators of their stellar sources. As already mentioned in \n\\S 1, the $s$-process patterns of the heavy elements exhibited \nby mainstream SiC grains constitute the most convincing argument \nfor their origin in carbon stars, which are the major source of the \n$s$-process elements in the Galaxy. The envelopes of these stars show \nlarge enhancements of typical $s$-elements such as Sr, Y, Zr, Ba, La, \nCe, and Nd (Smith \\& Lambert 1990). In SiC grains $s$-process patterns \nare seen in the elements Kr, Xe, Sr, Ba, Nd, Sm, and Dy (Lewis et al. \n1990, 1994; Ott \\& Begemann 1990a,b; Prombo et al. 1993; Richter, Ott, \n\\& Begemann 1993, 1994; Zinner, Amari, \\& Lewis 1991; Podosek et al. \n1999), which have been measured in bulk samples. Although these \nsamples were collections of all SiC grain types, there is little \ndoubt that the isotopic results must have been dominated by \ncontributions of mainstream grains.\n\nIn addition, isotopic analyses of Sr, Zr, and Mo by resonance \nionization mass spectrometry (RIMS) have been made on a limited \nnumber of single SiC grains. Although most of these measurements \nwere made on SiC grains for which no C and Si isotopic data had \nbeen obtained (Nicolussi et al. 1997, 1998a,b), for statistical \nreasons essentially all of these grains must have been mainstream \ngrains. They display characteristic $s$-process patterns with large \ndepletions in the $p$-only isotopes $^{84}$Sr, $^{92}$Mo, and \n$^{94}$Mo, and in the $r$-only isotope $^{100}$Mo. Large depletions \nare also seen in $^{96}$Zr, indicating that neutron densities in the \nstellar sources of these grains must have been low, compatible with \n$^{13}$C~ being the major neutron source in AGB stars (Gallino et al. \n1998a). The depletions in $^{96}$Zr are consistent with isotopic \nabundance data\nobtained by spectroscopic observations of the ZrO bandheads in AGB star \nenvelopes (Lambert et al. 1995). Recent RIMS analysis of Mo in SiC grains \nthat had been identified as mainstream grains on the basis of their \nC, N, and Si isotopic ratios confirmed that mainstream grains \nindeed carry $s$-process signatures in this element (Pellin et al. \n1999). Gallino et al. (1997) could successfully reproduce the measured \n$s$-process compositions of the heavy elements in mainstream SiC grains\nwith models of low-mass \nAGB stars of close-to-solar metallicity, in which neutrons are primarily \nproduced by the $^{13}$C~ source during the radiative interpulse period. In\naddition to the isotopic patterns of the heavy elements, the large \noverabundances of refractory $s$-process elements such as Zr, Y, Ba, \nand Nd in single SiC grains (Amari et al. 1995b) are further evidence \nfor an AGB-star origin (Lodders \\& Fegley 1995, 1997a,b, 1998).\n\nEvidence for a carbon star origin is also obtained from the light \nelements. It should be noted that as far as neutron-capture \nnucleosynthesis is concerned, because of their large abundance \nthe light elements (elements lighter than Fe) in AGB stars are \nconsidered to be neutron poisons for the synthesis of the \nheavy elements. However, because of their relatively small cross \nsections, they are only marginally affected by neutron capture. \nElements up to Mg are also affected by charged particle reactions \nwith H and He. The most important light-element signature of SiC \nis the Ne isotopic composition, which is dominated by $^{22}$Ne \n(Lewis et al. 1990, 1994). In fact, it has been the presence of \nthis Ne component, Ne-E(H), that, together with the so-called \nXe-S component (Srinivasan \\& Anders 1978; Clayton \\& Ward 1978) \nled to the isolation of presolar SiC (Tang \\& Anders 1988a). \nGallino et al. (1990, 1994) showed that Ne-E(H) matches the \npredicted isotopic composition of Ne in the He shell of AGB stars. \nAlmost all initial CNO nuclei are first converted to $^{14}$N during \nshell H burning and then to $^{22}$Ne via the chain $^{14}$N($\\alpha$,$\\gamma$)$^{18}$F($\\beta^+\\nu$)$^{18}$O~($\\alpha$,$\\gamma$)$^{22}$Ne~ in the \nHe shell during thermal pulses. \nAnother piece of evidence is obtained from the distribution of the \n$^{12}$C\/$^{13}$C ratios in mainstream grains that is very similar to that \nmeasured astronomically in carbon stars (Dominy \\& Wallerstein \n1987; Smith \\& Lambert 1990; see also Fig. 14 in Anders \\& Zinner \n1993).\n\nThe ranges of $^{12}$C\/$^{13}$C and $^{14}$N\/$^{15}$N ratios \nmeasured in mainstream grains roughly agree with the ranges predicted \nby theoretical \nmodels of AGB stars. Proton captures occurring in the deep envelope \nduring the main sequence phase followed by first (and second) \ndredge-up as well as shell He burning and the TDU during the \nTP-AGB phase affect the C and N isotopes in the envelope.\n$^{12}$C\/$^{13}$C ratios\npredicted by canonical stellar evolution models range from $\\sim$ 20 \nat first dredge-up in the RG phase to $\\sim$ 300 \nin the late TP-AGB phases (Iben 1977a; Bazan 1991; Gallino et al. \n1994). Predicted $^{14}$N\/$^{15}$N ratios are 600 - 1,600 (Becker \\& \nIben 1979; El Eid 1994), falling short of the range observed in the grains. \nHowever, the assumption of deep mixing (``cool bottom processing'' or \nCBP) of envelope material to deep hot regions in $M \\lesssim$ 2.5 \n$M_{\\odot}$~ stars during their RG and AGB phases (Charbonnel 1995; Wasserburg, \nBoothroyd, \\& Sackmann 1995; see also Langer et al. 1999a for \nrotationally induced mixing) results in partial H burning, with\nhigher $^{14}$N\/$^{15}$N and lower $^{12}$C\/$^{13}$C ratios in the envelope\nthan in canonical models (see also \nHuss et al. 1997). \nAs a matter of fact, CBP mechanisms have been introduced to explain the\nobserved \n$^{12}$C\/$^{13}$C ratios in RG stars of low mass \n(Gilroy 1989; Gilroy \\& Brown 1991; \nPilachowski et al. 1997), which are lower than those \npredicted by canonical models.\n\nIn contrast to the heavy elements, and the light elements C, N, \nNe and Al, the Si isotopic ratios of mainstream SiC grains cannot \nbe explained by nuclear processes taking place in a single star. \nIn Fig. 3 we plotted the Si isotopic data measured in SiC grains \nfrom three different size fractions isolated from the Murchison \ncarbonaceous (CM2) meteorite (Hoppe et al. 1994, 1996a) and data \nfrom the Orgueil (CI) meteorite (Huss et al. 1997). The three \nMurchison size fractions are KJE (0.5 - 0.8 $\\mu$m in diameter), \nKJG (1.5 - 3 $\\mu$m), and KJH (3 - 5 $\\mu$m) (Amari et al. 1994). \nOf the smallest Murchison grain size fraction KJE we plotted only \ndata points with errors smaller than 15 $^o\\!\/\\!_{oo}$. The distributions of \nSi isotopic ratios measured in SiC grains from other meteorites \nare very similar to that shown in Fig. 3 (Alexander 1993; Huss, \nFahey, \\& Wasserburg 1995; Gao et al. 1995). Also plotted in \nFig. 3 is the correlation line obtained from a fit to the grain \ndata. This line, which does not go through the solar composition \nbut passes slightly to the right of it, has a slope of 1.31 and \nan intercept of the ordinate at $\\delta$$^{29}$Si\/$^{28}$Si$_{int}$ \n= $-$ 15.9 $^o\\!\/\\!_{oo}$, a little different from the parameters determined from \nthe KJG and KJH dataset only (Hoppe et al. 1994).\n\nThere have been various attempts to explain the Si isotopic \ndistribution of the mainstream SiC grains. Zinner et al. (1989) \nalready realized that the scatter in the Si isotopic ratios \nindicate several stellar sources. Stone et al. (1991) first \nnoticed the correlation line of Si isotopic ratios in SiC grains \nfrom Orgueil and proposed an origin in a single AGB star \nwith mixing of two components but they did not address the question of \nhow the end components could be generated by nucleosynthetic processes \nin a single star. The only nuclear reactions in AGB stars that are \nbelieved to substantially affect the Si isotopes are neutron captures \nin the He shell. However, it has been determined early on (Gallino et \nal. 1990; Obradovic et al. 1991; Brown \\& Clayton 1992b), and will be \nseen in more detail in the \\S 3, that neutron captures shift the \nSi isotopic ratios along a line with a slope that varies, depending \non mass and metallicity, from 0.35 to 0.75 in a \n$\\delta$$^{29}$Si\/$^{28}$Si vs. $\\delta$$^{30}$Si\/$^{28}$Si \n3-isotope plot. Furthermore, in low-mass AGB stars of \nclose-to-solar metallicity predicted shifts of envelope material \nare only on the order of 20 $^o\\!\/\\!_{oo}$, an order of magnitude less than \nthe range seen in mainstream grains. Nuclear processes in a single \nAGB star therefore cannot produce the mainstream distribution and \nthis led to the conclusion that several stars with varying initial \nSi isotopic compositions must have contributed SiC grains to the \nsolar system (Clayton et al. 1991; Alexander 1993). The situation \nis similar for Ti. It should be emphasized that there is a \nfundamental qualitative difference between the isotopic compositions \nof C, N, Ne, and the heavy elements, and those of Si and Ti. While \nthe former are dominated by RG and AGB nucleosynthesis, the effect \nof stellar nucleosynthesis on the Si and Ti ratios is relatively \nsmall and cannot explain the compositions observed in grains; the \npresence of an extra component has to be invoked.\n\nBrown \\& Clayton (1992b, 1993) proposed a single-star model by \nconsidering Mg burning at elevated temperatures in the He-burning \nshell. In this model ($\\alpha$,n) reactions on Mg in a 5.5 $M_{\\odot}$~ \nAGB star produce a neutron-rich Si isotopic composition at the far \nend of the mainstream correlation line. Mixing with the original, \nclose to solar, composition in the envelope combined with variable \nmass loss from this star could lead to the observed distribution. \nHowever, to accomplish this, the temperature in the He shell \nhas to be raised by 10\\% above that produced by the standard AGB \nmodels (Iben 1977b). This leads to serious problems with other \nprocesses and with the general question of energy generation and \nstellar structure. Moreover, the Ti isotopic variations and \nespecially the correlation with the Si isotopic ratios (Fig. 4) \ncannot be explained in this way because the temperatures required \nfor Mg burning in the He shell do not affect at all the Ti \nisotopes, nor would the $s$-process isotopic signatures be compatible \nwith such a situation.\n\nThis leaves variations in the initial Si isotopic compositions of \nthe AGB stars that contributed SiC grains to the solar system as \nthe most likely explanation for the Si isotope distribution of \nmainstream grains. Variations of the initial Si compositions in \nturn are expected as the result of GCE. Low-mass stars, \nwhich became AGB stars at the end of their evolution and \ncontributed grains to the protosolar nebula, are likely to \nhave been born at different times before solar system formation \nand thus reflect the isotopic composition of the Galaxy in \ndifferent earlier epochs. An explanation of the Si isotopic \ncompositions in mainstream grains thus requires an understanding \nof the evolution of the Si isotopic ratios throughout Galactic \nhistory.\n\nGallino et al. (1994) approached this problem by assuming that \n$^{29}$Si and $^{30}$Si are mostly primary isotopes that are \nproduced by SNe of Type II, together with a major fraction of \n$^{28}$Si, and that substantial contributions to Galactic Si \nin the form of almost pure $^{28}$Si from SNe of Type Ia late \nin Galactic history determine the Si isotopic evolution reflected \nby the grains. \nThese authors also advanced a tentative interpretation of the \nTi isotopes. They noticed the correlation between the Ti and \nSi isotopic compositions (Fig. 4) and concluded that, as for \nSi, neutron-capture nucleosynthesis cannot explain the Ti \nisotopes nor the correlation with Si and they would have to be \ninterpreted within the framework of the chemical evolution of \nthe Galaxy.\n\nTimmes \\& Clayton (1996) and Clayton \\& \nTimmes (1997a,b) constructed a detailed model of the Galactic \nhistory of the Si isotopes that is based on the GCE model of Timmes, Woosley \n\\& Weaver (1995). The SN production yields were obtained from \nthe Type II SN models of Woosley \\& Weaver (1995, henceforth WW95) \nand from the popular W7 Type Ia SN model by Thielemann, Nomoto, \\& \nYokoi (1986). According to the WW95 models, $^{29}$Si and \n$^{30}$Si in Galactic disk stars are predominantly \nsecondary isotopes, i.e. their production in massive Type II SNe \nincreases with increasing metallicity, since it \nrequires the prior presence of primary isotopes such as $^{12}$C,\n$^{14}$N and $^{16}$O.\nAs a consequence, early SNIIe produced mostly pure $^{28}$Si, \nwhereas later SNIIe added more and more $^{29}$Si and $^{30}$Si \nto the ISM. This resulted in a continuous increase of the \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios throughout \nGalactic history and the distribution of the grains' Si isotopic ratios \napparently reflects this change. According to the Timmes \\& Clayton \nmodel, the spread in the isotopic compositions of the mainstream grains \ncorresponds to variations in the birth dates of their parent stars. \nSpecifically, the birth dates of the parents stars range over $\\sim$ \n5 Gyr (see Fig. 6 in Timmes \\& Clayton 1996).\n\nHowever, this model suffers from a fundamental problem. Because \nmost mainstream grains have $^{29}$Si\/$^{28}$Si and \n$^{30}$Si\/$^{28}$Si ratios that \nare larger than those of the solar system (Fig. 3), they are \ninferred to be younger than the sun. This absurd corollary of \nthe model led Clayton (1997) to consider the possibility that \nthe mainstream grains originated from stars that were born at \ndifferent Galactic radii than the sun. Indeed, according to \nWielen, Fuchs, \\& Dettbarn (1996), scattering \nby massive molecular clouds may lead to the diffusion of those stars \nfrom central metal-rich regions of the Galaxy (for which higher \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios are predicted \nthan those in the present solar neighborhood) to the region where \nthey, once they became AGB stars, shed their SiC grains into the \nprotosolar cloud. \nAlexander \\& Nittler (1999) took a different approach to resolve the \nparadox that the grains are apparently younger than the sun. They \nfitted the Si and Ti isotopic compositions of the mainstream grains \nto contributions from nucleosynthesis in the parent AGB stars and \nthe stars' original isotopic compositions. However, in contrast to \nthe Timmes \\& Clayton (1996) model, they tried to determine the \nevolution of the Si and Ti isotopes from the grain data themselves. \nFrom this fit they concluded that most mainstream grains \ndo not come from stars with higher than solar metallicities but that \nthe sun has an atypical Si isotopic composition.\n\nCommon to the GCE models listed above is that they predict a \nmonotonic relationship between the Si isotopic composition and \nGalactic time for the ISM at a given Galactic radius. In other \nwords, a given age of the Galaxy or, if different Galactic radii \nare considered, a given function of age and Galactic radius \ncorresponds to a given isotopic composition (this functional \nrelationship is implied in the Clayton \\& Timmes (1997a) model \nbut has never been explicitly worked out).\n\nHowever, one has to consider that the Si isotopic composition \nof the Galaxy, and in particular that of different regions where \nindividual low-mass stars are born, evolves as the result of \ncontributions from discrete stellar sources, mostly SNe. It \nis unlikely that these discrete contributions are instantly \nmixed with preexisting material so that the isotopic compositions \nof large Galactic regions are completely homogenized. The work \nof Edvardsson et al. (1993) and others has shown that stars \nfrom any given Galactic epoch and Galactic radius display a \nconsiderable spread in metallicity and, more generally, in elemental abundances. \nNumerous attempts have been made to explain these observations: \nstellar orbital diffusion, chemical condensation processes \nand\/or thermal diffusion in stellar atmospheres, incomplete mixing of \nstellar ejecta, sequential stellar enrichment and local infall of \nmetal-deficient gas (see van den Hoek \\& de Jong 1997 for review and \ndiscussion of these different attempts).\n\nThe study of meteorites provided ample evidence that the protosolar \nnebula was isotopically not homogenized. In addition to the survival \nof pristine stardust, isotopic anomalies are found in material that \napparently was processed in the solar nebula (see, e.g., Clayton, \nHinton, \\& Davis 1988; Lee 1988; Wasserburg 1987). Examples include \n$^{16}$O excesses (up to 5\\%) and large deficits and excesses of the \nn-rich isotopes of the Fe-peak elements such as $^{48}$Ca and $^{50}$Ti \nin refractory inclusions. These anomalies indicate the survival of \nisotopic signatures from different nucleosynthetic reservoirs. Another \nindication of local isotopic heterogeneity comes from the value of the \n$^{17}$O\/$^{18}$O ratio, which is 5.26 in the solar system but \n3.65 $\\pm$ 0.15 in diverse molecular clouds (Penzias 1981; Wannier \n1989; Henkel \\& Mauersberg 1993, see also discussion concerning \nthe $^{14}$N\/$^{15}$N ratio by Chin et al. 1999).\n\nWe therefore want to explore to what extent the Si isotopic \nspread of mainstream SiC grains can be explained by local \nheterogeneities in the regions from which the low-mass parent \nstars of the grains originally formed. Before doing this we \nwill examine in more detail the nucleosynthesis of the Si \nand Ti isotopes in AGB stars. However, because of the largely \nuncertain astrophysical origin of all Ti isotopes (see \nTimmes et al. 1995; Woosley 1996; Woosley et al. 1997), the \nsituation concerning the Ti isotopes in presolar SiC \ngrains and their interpretation in terms of SN contributions \nand GCE is a complicated issue in itself. It will be treated \nin a separate paper.\n\n\\section{Nucleosynthesis of the Si and Ti isotopes in AGB stars}\n\nDuring all the evolutionary phases of low-mass stars ($M <$ 10 $M_{\\odot}$) the\nmaximum temperature in the inner regions never reaches high enough values\nto allow the burning of any element heavier than He. \nConsequently, only the production of $^{12}$C~ and of $^{16}$O from\ninitial H or He nuclei is possible and, in particular, there are no \ncharged-particle interactions that involve the nucleosynthesis of \nSi and Ti.\nThe initial isotopic compositions of these elements can be nevertheless \nmodified by slow neutron capture (the $s$ process), which occurs\nin the tiny region between the H shell and the He shell (hereafter He\nintershell) during the AGB phase. \n\nAccording to the AGB models of low-mass ($M =$ 1.5 $-$ 3 $M_{\\odot}$) stars with \nmetallicities in the range from half solar to solar obtained with the FRANEC\ncode and discussed in detail \nby Straniero et al. (1997) and Gallino et al. (1998a), neutrons are \nreleased in the He intershell by two different sources, $^{13}$C~ and \n$^{22}$Ne. The maximum temperature achieved during the \nrecurrent thermal instabilities (or thermal pulses: TP) of the He \nshell is not high enough to consume $^{22}$Ne to an appreciable extent, so \nthat the $^{13}$C~ source has to play the major role. However, \nthe number of $^{13}$C~ nuclei left behind by the H-burning shell is too \nsmall to account for the $s$-element enhancements observed in carbon \nstars. A special mechanism has to be invoked in order to build up a \nsufficient amount of $^{13}$C~ in the He intershell. In AGB stars of mass \n$M \\ga 1.5$ $M_{\\odot}$, after a limited number \nof thermal pulses, soon after the quenching of a given instability, \nthe convective envelope penetrates into the top layers of\nthe He intershell and mixes with the envelope \nmaterial enriched in $^{12}$C~ and $s$-process elements. This TDU \nphenomenon leaves a \nsharp H\/He discontinuity, where some kind of hydrodynamical mixing, \npossibly driven by rotation, occurs \n(Herwig et al. 1997; Singh, Roxburgh, \\& Chan \n1998; Langer et al. 1999a,b). In these conditions, a small \namount of protons penetrates from the \nenvelope into the He intershell (see Gallino et al. 1998a for \ndiscussion). At H reignition, these protons are captured by the \nabundant $^{12}$C~ present in the intershell as \na consequence of partial He burning that occurred during the previous \nthermal instability. Consequently, a so-called $^{13}$C {\\it pocket} \nis formed in a small region at the top of the \nHe intershell. Before the onset of the next pulse, the progressive \ncompression and heating of these layers cause all $^{13}$C~ to burn\nradiatively in the interpulse period \nvia the $^{13}$C($\\alpha$,n)$^{16}$O~ reaction, at a temperature of around 8 keV. \nThe neutron exposure, or time integrated neutron flux \n$\\delta \\tau = \\int {\\it N}_{\\rm n}$ $v_{th}~ dt$, experienced\nduring the interpulse period may reach quite a \nhigh value, $\\delta \\tau_1 (8 {\\rm keV})$ of up to 0.4 mbarn$^{-1}$, \ndepending on the initial amount of $^{13}$C~. The maximum neutron density in this \nradiative phase remains low: $N_{{\\rm n}, max}$ $\\sim$ 10$^7$ n\/cm$^{3}$.\n\nThe material that experienced neutron captures in the $^{13}$C~ pocket \nis engulfed and diluted ($\\sim$1\/20) \nby the next growing convective thermal pulse, which extends over almost \nthe whole He intershell, and is mixed with material already $s$-processed \nduring the previous pulses, together with ashes of the H-burning \nshell. Among them are Si and Ti in their initial abundances. For advanced \npulses, the overlapping factor between subsequent pulses becomes $r$ \n$\\approx$ 0.4 \nand the mass of the convective pulse is slightly smaller than 10$^{-2}$ $M_{\\odot}$~ \n(Gallino et al. 1998a). A small neutron burst, at around 23 keV, is\nreleased in convective thermal pulses, during the latest phases of the\nAGB evolution, when the bottom temperature in the He shell is sufficiently\nhigh to marginally activate the $^{22}$Ne($\\alpha$,n)$^{25}$Mg~ reaction. The $^{22}$Ne is provided by\nthe \n$^{14}$N($\\alpha$,$\\gamma$)$^{18}$F($\\beta^+\\nu$)$^{18}$O~($\\alpha$,$\\gamma$)$^{22}$Ne~ chain starting from $^{14}$N present in the ashes of H burning.\nSome extra $^{14}$N derives from primary $^{12}$C that is dredged up into the\nenvelope and partly converted to $^{14}$N by H-shell burning. The neutron\nexposure provided by the $^{22}$Ne neutron source during the thermal \npulse is low, reaching at most\n$\\delta \\tau_2 (23 {\\rm keV}) =$ 0.03 mbarn$^{-1}$.\nHowever, the peak neutron density can reach 10$^{10}$ n\/cm$^{3}$.\n\nExposure to the two neutron fluxes is repeated through the \npulses with TDU. \nThe $^{13}$C-pocket features are kept constant pulse after pulse, \nwhile the small neutron exposure from the $^{22}$Ne neutron source \nduring thermal pulses increases \nwith pulse number, reflecting the slight increase of the strength of the \nthermal instability with core mass.\n\nWe have performed new calculations for the nucleosynthesis\ndue to neutron capture in AGB stars. \nThese calculations are based on the\nstellar models and the nuclear network described in detail by Gallino\net al. (1998a). We want to follow here in particular the modifications of \nthe Si and Ti isotope abundances arising by neutron capture \nin the intershell region during the neutron fluences in the \n$^{13}$C~ pocket and in the thermal pulses. Then we will follow the isotopic \ncompositions of Si and Ti in the envelope as they are modified \nduring the whole TP-AGB phase by the mixing of He intershell material \ndue to TDU episodes. The envelope itself is progressively eroded by stellar\nwinds and by the growth of the H-burning shell.\n\nIn Table 1 the Maxwellian averaged neutron capture cross sections (in the\nform of $\\sigma_{code}$=$\\left< \\sigma v \\right>\/v_{th}(30 {\\rm\nkeV})$, expressed in mbarn) are listed for selected isotopes. \nValues are reported for the two typical temperatures \n(8 keV and 23 keV) at which neutrons are released by the $^{13}$C~ and by the\n$^{22}$Ne~ neutron source, and for the standard temperature of 30 keV at \nwhich these cross sections are currently given in the literature. The\nquoted values have been \ntaken from the compilation by Beer, Voss, \\& Winters (1992), with three \nexceptions: $^{28}$Si, for which a renormalization to the 30 keV \nrecommended value of Bao \\& K\\\"appeler (1987) has been considered \n(Beer 1992, private communication), $^{150}$Sm, from Wisshak et al. (1993), \nand the $^{33}$S(n,$\\alpha$)$^{30}$Si cross section from Schatz et al. \n(1995). The use of $\\sigma_{code}$ demonstrates how the cross section\nat any given energy departs from the usual $1\/v$ rule. For a perfect \n$1\/v$ dependence, $\\sigma_{code}$ at different $kT$ values should remain\nconstant. In reality, with the exception of $^{49}$Ti, strong \ndepartures from this rule are shown by all Si \nand Ti isotopes. The $^{30}$Si abundance \nresulting from a given neutron exposure is strongly dependent on the\nreaction rate of $^{32}$S(n,$\\gamma$)$^{33}$S, where $^{33}$S is subsequently quickly transformed to $^{30}$Si via $^{33}$S(n,$\\alpha$)$^{30}$Si.\n\nThe neutron capture cross sections of two typical \nheavy $s$-only nuclei, $^{100}$Ru and $^{150}$Sm, have also been included\nin Table 1 \nin order to show that Si and Ti (this is also true for all \nother elements lighter that Fe) are not as much affected by neutron \ncapture as the heavier elements. The neutron capture cross sections of \nlight elements are much smaller (by as much as three \norders of magnitude) than \nthose of typical heavy isotopes. However, because of their large \ninitial abundances, isotopes lighter than $^{56}$Fe act as important \nneutron poisons for the build-up of the heavy elements. \nIn the last column, the relative uncertainties of the 30 keV cross sections are \nreported. The cross sections of all Si and Ti isotopes still \nsuffer from large uncertainties, around 10\\%, \nwhereas for many heavy isotopes recent experiments have achieved a \nprecision of the order of 1\\% (see K\\\"appeler 1999). \n\nTable 2 shows the production factors with respect to solar \nof the Si and Ti isotopes, as well as that of two \n$s$-only nuclei $^{100}$Ru and $^{150}$Sm, \nin the He intershell at different phases of the 15$^{\\rm th}$ thermal \npulse for an AGB star of 1.5 $M_{\\odot}$~ and solar metallicity, with the \nstandard choice of the $^{13}$C~ pocket (case ST of Gallino et al 1998a). \nColumns 2 and 3 show the effect of the $^{13}$C~ neutron source on the Si \nand Ti isotopic abundances inside the $^{13}$C~ pocket. The values of\ncolumn 4 were calculated at the time when the $s$-enriched pocket\nhas been engulfed by the growing convective pulse and diluted\nwith both $s$-processed material from\nthe previous pulse and material from the H-burning ashes, containing in\nparticular Si and Ti of initial composition. \nThe difference between the values of columns 4 and 5 \nexpresses the effect of the $^{22}$Ne neutron source activated during the\n15$^{\\rm th}$ pulse. Note how $^{100}$Ru and $^{150}$Sm production factors are up \nto three orders of magnitude larger than those of Si and Ti. \nFrom the results given in Table 2, it is easily recognized that \nneutron captures only marginally modify the initial Si and Ti \nisotopic compositions. $^{28}$Si tends to be slightly consumed (by \n15 $^o\\!\/\\!_{oo}$, and by 20 $^o\\!\/\\!_{oo}$, respectively) after both neutron exposures. \nActually, during \nthe high neutron exposure from the $^{13}$C~ source $^{29}$Si is more \nefficiently consumed (by a factor 1.6) than produced, because of the \nvery low cross section of $^{28}$Si. Also $^{30}$Si is consumed (by \na factor 2.2) during this phase. In contrast, the abundances of both \nneutron-rich Si isotopes grow during the thermal pulse, by factors\nof about 1.3 and 1.4, respectively, relative \nto their initial values in the convective \npulse. These features are mainly due to the fact that the neutron \ncapture cross sections strongly depart from the 1\/$v$ trend. Note \nthat $\\sigma_{code}(^{28}$Si) is almost an order of magnitude greater at\n23 keV than at 8 keV, which explains why in the TP phase this isotope is \ndestroyed to a larger extent than in the $^{13}$C~ pocket. \nAs a consequence, we observe the \ngrowth of $^{29}$Si during the TP. \n\nAmong the Ti isotopes, $^{50}$Ti is a neutron magic nucleus ($N$ $=$\n28) and its neutron capture cross section is very small compared to \nthose of the other Ti isotopes. It shows a strong departure from the \n1\/$v$ trend (see Table 1), being a factor of 4 greater at 23 keV than \nat 8 keV. As shown in Table 2, $^{50}$Ti accumulates during the \n$^{13}$C~ neutron exposure because of its very low neutron capture cross \nsection, which makes this isotope a bottleneck of the abundance flow. \n$^{49}$Ti is produced in both phases, by a larger factor during the \npulse, while $^{48}$Ti is consumed. Note that during the high neutron \nexposure by the $^{13}$C~ neutron source $^{48}$Ti is only marginally \nmodified, despite its relatively large cross section. This results \nfrom abundance flow starting at $^{40}$Ca, an isotope of large initial \nabundance and of cross section $\\sigma_{code}$(8 keV) $=$ 6.11 mbarn, \nwhich is consequently consumed (by a factor $\\approx$ 6) in this phase. \nThe $^{46}$Ti, $^{47}$Ti and $^{48}$Ti isotopes, similarly to the \nSi isotopes, suffer almost negligible variations. \n\nBecause of their relatively small cross sections, a behavior similar to \nthat of Si and Ti is shown by other light elements below Fe, among \nthem S and Ca. It should be emphasized that the final \nSi isotope composition mostly depends on the small neutron exposure by \nthe $^{22}$Ne neutron source in the convective pulse rather than on \nthe very large neutron exposure by the $^{13}$C~ neutron source taking \nplace in the tiny radiative $^{13}$C~ pocket. \n\nTable 2, column 6 shows the production factors in the envelope \nimmediately after the TDU that follows the\nquenching of the 15$^{\\rm th}$ thermal pulse. At this stage, the star has\nbecome a C star, with C\/O $=$ 1.3, and the isotopic composition of the\nenvelope results from the mixing of the $s$-processed and $^{12}$C-enriched \nmaterial cumulatively carried into the envelope by previous TDU episodes.\n\nPredictions for the Si and Ti isotopic compositions in the envelope of AGB\nstars of solar metallicity and initial mass of 1.5 and 3 $M_{\\odot}$~ during\nrepeated TDUs in the TP phase are shown in \nFig. 5. Fig. 6 reports predictions for the resulting Ti vs. Si correlation:\nas in Fig. 4, Ti ratios are plotted as function of the\n$^{29}$Si\/$^{28}$Si ratio. They are all reported in the form of \n$\\delta$-values for three different choices of the amount of $^{13}$C~ in the \nHe intershell. The standard case (ST) of Gallino et al. (1998a) \ncorresponds to an average mass fraction of $^{13}$C of 6 $\\times$ \n10$^{-3}$ distributed over a tiny layer of a few 10$^{-4}$ $M_{\\odot}$~ at the top \nof the He intershell, case d3 corresponds to the amount of case ST \ndivided by 3 and case u2 is an upper limit corresponding to the amount \nof case ST multiplied by 2. As already mentioned in Busso et al. (1999a) \n(see also Busso, Gallino, \\& Wasserburg 1999b), a spread in the $^{13}$C \namount in stars of different metallicities\nis required by spectroscopic observations of \n$s$-enhanced stars, and conceivably depends on the initial stellar mass or \nother physical characteristics (such as stellar rotation).\nThe measurements of Zr, Mo, and Sr isotopic ratios in individual SiC \ngrains have confirmed this spread in the $^{13}$C~ amount: all single \ngrain compositions can be matched by low-mass AGB models of about solar \nmetallicity if we consider different amounts of $^{13}$C~ (Gallino et al. \n1998b; Nicolussi et al. 1998b). Open symbols are for \nenvelopes with C\/O$>$1, the condition for SiC condensation. \nNote that case ST for solar metallicity best reproduces the $s$-process\nisotopic distribution of bulk SiC grains, which is slightly different from\nthe solar main component (for a general discussion see Gallino et al. 1997; \nBusso et al. 1999b).\n\nNot surprisingly the $^{29}$Si\/$^{28}$Si, $^{30}$Si\/$^{28}$Si, and \n$^{47}$Ti\/$^{48}$Ti ratios are only a few percent (up to 25 $^o\\!\/\\!_{oo}$, \n40 $^o\\!\/\\!_{oo}$, and \n14 $^o\\!\/\\!_{oo}$, respectively) higher than the corresponding solar ratios. The \n$^{46}$Ti\/$^{48}$Ti and $^{49}$Ti\/$^{48}$Ti ratios are \nup to 70 $^o\\!\/\\!_{oo}$~ and 200 $^o\\!\/\\!_{oo}$~ higher than the solar ratios. \nIn agreement with the results shown in Table 2, the\nonly ratio that is affected to a significant extent (up to 500 $^o\\!\/\\!_{oo}$~\nhigher than solar)\nis the $^{50}$Ti\/$^{48}$Ti ratio. Note that the \n$\\delta^{50}$Ti$\/^{48}$Ti values range from $+$100 $^o\\!\/\\!_{oo}$~ to $+$500 \n$^o\\!\/\\!_{oo}$, depending on the $^{13}$C~ amount. $^{50}$Ti is a\nmagic nucleus whose abundance is very sensitive to \nthe high neutron exposure in the $^{13}$C~ pocket. \nThe fact that the $^{50}$Ti\/$^{48}$Ti ratio is significantly changed during \nthe AGB phase is, in a way, consistent with the $^{50}$Ti\/$^{48}$Ti ratios \nmeasured in SiC grains. Model predictions do not reproduce the spread of the \nmeasured Si and Ti compositions, nor could they ever explain the \nnegative $\\delta$-values measured in some grains; \nthe calculated $^{50}$Ti\/$^{48}$Ti ratio, \nthough, reaches $\\delta$-values that are higher than those of \nall the other Si and Ti $\\delta$-values \nboth in AGB model predictions and in single \nSiC grain measurements (up to 300 $^o\\!\/\\!_{oo}$, \nsee Fig. 4). \n\nWe also investigated two other TP-AGB models: a case of $M=5$ $M_{\\odot}$~ of\nsolar metallicity (Fig. 7) and a case of $M=3$ $M_{\\odot}$~ of 1\/3 solar\nmetallicity (Fig. 8). An interesting feature is common to both models:\nthe maximum temperature at the bottom of the He convective shell is\nsomewhat higher than in the models described above. As a consequence, \nthe production factors for $^{29}$Si and\n$^{30}$Si, whose production is most sensitive to the $^{22}$Ne~ neutron \nsource (see Table 2), at the end of the 15$^{\\rm th}$ pulse reach 3.2 and \n5.9, respectively, for the $M=5$ $M_{\\odot}$~ star of solar metallicity, and 3.1 \nand 6.5, \nrespectively, for the $M$ $=$ 3 $M_{\\odot}$~ star of $Z$ $=$ $Z_{\\odot}$\/3. \n\nThis results in an increase of up to 80 $^o\\!\/\\!_{oo}$~ and 200 $^o\\!\/\\!_{oo}$~ in \n$\\delta^{29}$Si$\/^{28}$Si and $\\delta^{30}$Si$\/^{28}$Si, \nrespectively, in the envelope of the $M=5$ $M_{\\odot}$~ model (Fig. 7), \nand of up to 100 $^o\\!\/\\!_{oo}$~ and 200 $^o\\!\/\\!_{oo}$~ in the \n1\/3 $Z_{\\odot}$ model (Fig. 8). The largest $^{29}$Si and $^{30}$Si excesses\nmeasured in SiC are reproduced, however with a slope of about 0.5\nfor the mixing line, whereas the slope of the mainstream correlation line in \nthe Si 3-isotope plot is 1.31 (Fig. 4). \nNote the extremely high values \n(up to 2000 $^o\\!\/\\!_{oo}$) reached by $\\delta^{50}$Ti$\/^{48}$Ti in the last \ncase (Fig. 8). They\nresult from the fact that, in our AGB model, the neutron exposure in the\n$^{13}$C~ pocket is very sensitive to metallicity: it grows with\ndecreasing metallicity (see Gallino et al. 1999).\n\nAs for all the cases above, the initial isotopic composition of the star \nhas been assumed to be solar, including the $Z = Z_{\\odot}\/3$ case. \nIn principle, some enhancement for \nisotopes produced by $\\alpha$ captures (such as $^{16}$O, $^{20}$Ne, \n$^{24}$Mg, $^{28}$Si, $^{40}$Ca and $^{48}$Ti) as well as complex \nsecondary-like trends of many other nuclei should be taken into account \nin the initial composition of low-metallicity stars. This is a tricky point, \nfor these variations have to be deduced from GCE models together with \nspectroscopic observations and are, in many cases, not well defined. \nAn exercise of this kind, in connection with a possible explanation for \nthe Si isotopic composition of SiC of type Z, can be found in \nHoppe et al. (1997). For the AGB model of\n$Z$ $=$ $Z_{\\odot}$\/3, we made some tests by assuming a small enhancement of the \ninitial $^{28}$Si and $^{32}$S, as well as of other $\\alpha$-rich \nisotopes according to the spectroscopic evidence by\nEdvardsson et al. (1993), and small depletions in the initial\nabundance of the secondary-like isotopes $^{29,30}$Si. It turned out that\nthe resulting Si isotope composition in the He intershell as a consequence \nof neutron captures was quite insensitive to the above variations, being\ndominated by the most abundant $^{28}$Si. \n\nIt has to be remarked here that several features of the predicted Si and\nTi ratios (e.g., the slope in the Si 3-isotope plot) depend on the\nneutron capture cross sections which, for the Si as well as the\nTi isotopes, are still quite uncertain, as shown in\nthe last column of Table 1. New measurements are highly desirable for \nobtaining the best possible AGB model predictions. \n\n\\section{The stellar sources of Si and Galactic heterogeneity}\n\n\\subsection{Supernova sources}\n\nAs Timmes \\& Clayton (1996) have pointed out, SNe of Type II are \nthe dominant sources of Si in the Galaxy, especially in its early \nstages. At later Galactic times, SNe of Type Ia also contribute $^{28}$Si. \nThe SNII models of WW95 show that $^{28}$Si is a primary isotope \nwhereas $^{29}$Si and $^{30}$Si are predominantly secondary isotopes (see also \nTimmes \\& Clayton 1996 for details). This means that $^{28}$Si \ncan be synthesized in early Type II SNe from a pure H and He \ncomposition, whereas the production of $^{29}$Si and $^{30}$Si \nrequires the prior presence of primary isotopes such as $^{12}$C, \n$^{14}$N and $^{16}$O. As a \nconsequence, the $^{29,30}$Si\/$^{28}$Si ratios of the ejecta of \nSNIIe increase with the metallicity of the stars. While $^{28}$Si \nis the product of explosive O burning, both $^{29}$Si and $^{30}$Si \nare synthesized in a narrow region by explosive Ne burning. \nActually, $^{29}$Si production is restricted to the outer region \nof the Ne burning shell.\n\nFig. 9 shows the $^{29,30}$Si\/$^{28}$Si ratios (plotted as \n$\\delta$-values) of the averages of the yields of SNII models of \ndifferent metallicities. The cases of metallicity \n$Z$ = 0.1 $Z_{\\odot}$ and $Z$ = $Z_{\\odot}$ are taken from \nWW95, the cases $Z$ = 0.5 $Z_{\\odot}$ and \n$Z$ = 2 $Z_{\\odot}$ are from more recent, unpublished \ncalculations by Weaver \\& Woosley. To obtain the averages we took the \ninitial mass function for massive stars into account by weighing the \ncontributions from SNIIe of different masses according to $M^{-2.35}$ \nper unit mass interval, the Salpeter initial mass function. Fig. 9\nshows that in the WW95 models there exists a fairly good linear \nrelationship between the $^{29,30}$Si\/$^{28}$Si ratios of the \naverage SN yields and the metallicity, demonstrating the secondary \nnature of the heavy Si isotopes. \nThe GCE of the Si isotopes is thus believed to have progressed from small \n$^{29,30}$Si\/$^{28}$Si ratios at early Galactic times to larger and \nlarger ratios as the metallicity of the whole Galaxy increased and \nSNIIe of increasing metallicity contributed their Si to the ISM \n(Timmes \\& Clayton 1996).\n\nThis process is expected to have resulted in the Si isotopic ratios at the \ntime and place of solar formation. However, closer inspection of Fig. \n9 and especially Fig. 10, where the $\\delta$-values of the\n$^{29,30}$Si\/$^{28}$Si ratios of the averages of SNII yields are \nplotted in a Si 3-isotope plot, reveals that the SNII models by Weaver \n\\& Woosley do not exactly produce the solar Si isotopic composition. \nIt is evident that $^{29}$Si in the presently available \nmodels is under-produced and the isotopic evolution expected from SNII \ncontributions misses the solar isotopic composition (Fig. 10). This \nis a long-recognized problem: Type II SN models under-produce $^{29}$Si \nrelative to $^{30}$Si as compared to the solar isotopic ratio (Timmes \net al. 1995; Timmes \\& Clayton 1996; Thielemann, Nomoto, \\& Hashimoto \n1996; Nomoto et al. 1997). \nThis fact is also demonstrated by a comparison of model predictions and \nthe Si isotopic ratios of type X SiC, Si$_3$N$_4$, and low-density \ngraphite grains, all of which are believed to originate from Type II SNe \n(Nittler et al. 1995; Travaglio et al. 1999). The Si isotopic ratios \nof these grains have systematically higher $^{29}$Si\/$^{30}$Si ratios \nthan those predicted by SN models (Zinner et al. 1998; Travaglio et al. \n1999). In order to achieve the solar ratios, Timmes \\& Clayton (1996) \nproposed multiplying the $^{29}$Si yields of SNII models by a\nfactor \nof $\\sim$ 1.5. We will do likewise in this paper and multiply the \n$^{29}$Si yields by the same factor to obtain the best fit to the \nsolar isotopic ratios or to the grain data.\n\nIt should be mentioned that there still exist major problems associated \nwith the synthesis of the Si isotopes in massive stars. \nAs Arnett \\& Bazan (1997) pointed out, heterogeneous mixing \nbetween different layers during the late evolutionary stages \nmight have a major effect on the nucleosynthesis of \ncertain elements. Bazan \\& Arnett (1998) used a two-dimensional \nhydrodynamic code to investigate convective O-shell burning in a 20 \n$M_{\\odot}$~ star. They concluded that the results of these calculations differ \nin many ways from those of one-dimensional models and that corresponding \nchanges in the nucleosynthesis of Si during this stage are to be \nexpected. It remains to be seen whether full nucleosynthetic calculations \nin two- or three-dimensional models can shed light on the problem of \nrelative yields of the Si isotopes. Another problem is the relative \ncontribution of Type Ia and Type II SNe to the GCE of the heavy elements, \nin particular Fe. Whereas in the Timmes et al. (1995) GCE model Type II \nSNe were assumed to contribute 2\/3 of the Fe in the solar system, Woosley \net al. (1997) favored a more important role of Type Ia SNe, letting them \ncontribute as much as half of the solar Fe. This would indicate somewhat \nhigher contributions by Type Ia SNe to the Galactic $^{28}$Si relative\nto SNIIe. In addition to possible uncertainties in the nuclear physics \nand in the treatment of the various convective zones affecting the \nproduction of the three Si isotopes, problems are related to the \neffect of mass loss from the most massive stars and to Galactic \nenrichment by close binary massive stars (Woosley, Langer, \\& Weaver \n1993, 1995), to the effect of rotation (Heger, Langer, \\& Woosley 1999), \nand to the still uncertain development of the explosion (WW95, \nThielemann et al. 1996).\n\nWhile in Fig. 9 only averages of SNII models of different metallicities \nhave been plotted, it has to be realized that SN models of different masses \nyield very different Si isotopic ratios. In Fig. 10, in addition to\naverages, we also plotted the isotopic ratios of individual SNII models \nof different masses for the $Z$ = 0.1 $Z_{\\odot}$ and the \n$Z$ = $Z_{\\odot}$ case. The yields and Si isotopic ratios for the\n$Z_{\\odot}$ case are also given in Table 3. As can be seen, the isotopic \nratios of different mass SNIIe span a wide range. There are variations \nnot only in the $^{29,30}$Si\/$^{28}$Si ratios but also in the\n$^{29}$Si\/$^{30}$Si ratio. The last column in Table 3 shows the latter \nratio (already readjusted by augmenting the $^{29}$Si yield) for SNIIe of different\nmass. Note that the ratio is smaller than unity for most SNIIe of lower mass \nbut larger than unity for the two most massive models. In Fig. 11 we plotted \nagain the average Si isotopic ratios for the $Z_{\\odot}$ case together with\nthe averages for the SN models with masses $M$ $\\leq$ 25 $M_{\\odot}$~ and 30 $M_{\\odot}$~ \n$\\leq$ $M$ $\\leq$ 40 $M_{\\odot}$. This time the theoretical $^{29}$Si yield was\nincreased \nby the same factor 1.5 for Type II SNe of all masses in such a way that the \nweighted average ratios plot on the slope-one line or, in other words, \nthat the average $^{29}$Si\/$^{30}$Si ratio is solar. As can be seen, the\nlow-mass average falls slightly below the slope-one line through the origin \n(pure $^{28}$Si) and the solar isotopic composition and the high-mass \naverage falls above this line.\n\nWhile the evolution of the Si isotopes of the Galaxy as a whole and of \nthe average of material in an annulus of a given Galactic radius \nundoubtedly followed the slope-one line, we expect certain variations \nin the Si isotopic ratios even at a given time and a given Galactic radius \nin relatively small regions from which low-mass stars formed. The reason \nis that the addition of contributions from individual SNe, which are \nresponsible for the Si isotopic ratios of a certain region, is a stochastic \nprocess and we do not expect that material from these contributions is \ninstantly homogenized with preexisting material. Fluctuations result \nfrom the fact that individual SN sources have yields with very different \nSi isotopic compositions as is clearly shown in Figs. 10 and 11. In \naddition to the isotopic ratios of Type II SNe, in Figs. 10 and 11 \nas well as in Table 3 we show also the ratios of the W7 SNIa model \n(Thielemann et al. 1986; updated by Nomoto et al. 1997) and the SNIa model \noriginating from sub-Chandrasekhar (hereafter sub-Ch) white dwarfs accreting \nHe from \na binary companion (Woosley \\& Weaver 1994). These SN types are believed \nto be the major sources of Si in the Galaxy around the time of solar \nsystem formation (Timmes \\& Clayton 1996; Woosley et al. 1997). \n\nLet us now consider the effect on the Si isotopic ratios of the \nadmixture of material from one of these SN sources to material with a \ngiven isotopic composition (Fig. 11). Three-isotope plots such as those \nshown in Figs. 10 and 11 have the property that the isotopic composition \nof a mixture between two components lies on a straight line connecting \nthe isotopic ratios of the two components. For the sake of demonstration \nwe arbitrarily selected as starting composition the Si isotopic \ncomposition of the sun. Admixture of different SN sources (we chose \nSNIa W7, SNIa sub-Ch, and, again for the sake of demonstration, the low- and high-mass averages of the \n$Z_{\\odot}$ Type II SN models of WW95) will shift the starting \ncomposition in the directions of the arrows in the figure (see also Figs. \n8 of Timmes \\& Clayton 1996 for mixtures between average ISM and \nejecta from individual SNe. These authors also mentioned the possibility \nof reproducing a larger-than-unity slope for the Si isotopic ratios \nof the mainstream SiC grains but did not systematically develop the \nlocal heterogeneity picture as it is done in this work). Thus \nadmixture of material from a SNIa sub-Ch will shift the \ncomposition toward the origin (pure $^{28}$Si). We note that the average of\nthe high-mass SNII sources (with adjusted $^{29}$Si yield) lies above the\nslope-one line from the origin through the solar composition. This means \nthat admixture of material from these sources will shift, on average, \nthe original solar composition along a line with a slope larger than one. \nLikewise, because the average of the low-mass sources lies below the \nslope-one line, admixture from these sources will again, on average, \nresult in a shift along a line with a slope larger than one. The same \nis true, even though to a lesser extent, if material from the Type Ia \nW7 SN model (containing essentially pure $^{28}$Si) is added to the solar \ncomposition.\n\n\\subsection{Monte Carlo calculations}\n\nIf contributions from a limited number of such sources are \nconsidered, the resulting Si isotopic compositions will fluctuate from \none mix to the next because of the statistical nature of these \ncontributions. We have developed a simplified Monte Carlo (MC) model in \nwhich we add material from a limited number of discrete SN sources \nin a statistical way to material with an arbitrary (but reasonable) \nstarting isotopic composition in order to see whether the Si isotopic \ndistribution of the mainstream SiC grains can be explained as the \nresult of statistical fluctuations or, in other words, local \nheterogeneities in the regions where low-mass stars - as AGB stars \nthe sources of mainstream grain - were born. A detailed description of \nour MC model can be found in the Appendix.\n\nWe have performed different calculations with different assumed starting \ncompositions. As first test we took the average Si isotopic ratios of the \nmainstream SiC grains corrected for AGB contributions \n($\\delta$$^{29}$Si\/$^{28}$Si$_{mean}$ = 30.4 $^o\\!\/\\!_{oo}$~ and\n$\\delta$$^{30}$Si\/$^{28}$Si$_{mean}$ = 27 $^o\\!\/\\!_{oo}$~ - see Appendix) as starting \ncomposition. In other words, we considered the mean composition of the \nmainstream grains' parent stars as a possible ``standard'' composition \nof the ISM from which these stars were born. We then randomly added $N$~ \nSN contributions and randomly chose the sign of each contribution (i.e., \nthe sign of the parameter $a$, the constant factor by \nwhich the total mass ejected by each SN is multiplied) in order \nto simulate material that could \nhave seen more or less from each kind of SN source relative to the \nchosen standard ISM composition. In this way we generated 200 different \nmixtures, whose isotopic ratios are plotted Fig. 12a. The plot shows the \ncase for $N$~ = 100. \nFor $a$, the fraction taken from each SN \nsource, we obtained $a$ = 1.5 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$. Note that\nbecause \nwe computed abundances for each isotope $i$ in the form of mass fractions \n$X_i$ (see Appendix), the contributing terms $a \\times M_{ejected}$ do \nnot have a \ndimension and the parameter $a$ has the dimension of the inverse of a \nmass. As explained in the Appendix, \nbecause of the limited number (200) of cases, a certain range of the parameter \n$N$~ is expected to yield a good fit. Other similarly good matches\nare obtained for values of $N$~ ranging between $\\sim$ 50 and $\\sim$ 200, \nand $a$ accordingly from $\\sim$ 1.9 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ \nand $\\sim$ 0.95 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$. In Fig. 12a we also took \nthe modification of the Si isotopes by \nnucleosynthesis in the AGB parent stars into account, adding the average \nisotopic shift given above to the results of the Monte Carlo calculation. \nAs can be seen from the figure, the 200 different mixtures generated with \nthese parameters by MC in a random fashion match the distribution of the \ngrains surprisingly well. We note that the slope of the correlation line \nof the MC points is larger than unity but somewhat smaller than the slope \nof 1.31 of the mainstream correlation line. The slope of the MC \ncompositions reflects the distribution of the SN sources, mostly the \nSNII sources of solar metallicity. As has been already pointed out above \nin the discussion of Fig. 11, these sources are aligned with an average \nslope that is greater than one.\n\nIf we choose starting compositions different from the average of the mainstream \nSiC grains, it turns out that for a wide \nrange of starting Si isotopic ratios, as long as they are constrained to \nbe compositions expected for the Galactic evolution of the Si isotopes \n(i.e., compositions that in Si 3-isotope plots such as those in Figs. 10 \nand 11 lie on the slope-one line between the origin representing pure \n$^{28}$Si and the solar isotopic composition), values for the parameters \n$N$~ and $a$ can be found that let us achieve a good match with the Si \nisotope distribution of the mainstream SiC grains, albeit with different \nchoices of the parameters $N$~ and $a$ for each case. \nWe investigated three more cases for which we chose $a$ always to be \npositive. The results are shown in Figs. 12b, 12c, and 12d. In the \nfirst of these cases the starting isotopic composition is solar, in \nthe second case Si is depleted in the heavy isotopes by 100 $^o\\!\/\\!_{oo}$~ \n($\\delta$$^{29}$Si\/$^{28}$Si$_{init}$ = $-$ 100 $^o\\!\/\\!_{oo}$~ and\n$\\delta$$^{30}$Si\/$^{28}$Si$_{init}$ = $-$ 100 $^o\\!\/\\!_{oo}$) and \nin the third by 200 $^o\\!\/\\!_{oo}$. The best-fit parameters are $N$~ = 70 and \n$a$ = 1.7 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ for the first case, \n$N$~ = 420 and $a$ = 1.1 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ for the second case,\nand \n$N$~ = 600 and $a$ = 1.5 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ \nfor the third case. Also in these cases, we obtain good fits for a range \nof $N$~ and $a$ values.\n\nIt is clear that the addition of SN material will change the concentration \nof other elements as well. As the statistical nature of these additions \nresults in a range of Si isotopic ratios, we expect it to result in a \ncorresponding range of elemental ratios as well and these variations can \nbe compared with astronomical observations in stars. In Fig. 13a we\nplotted the scatter in elemental ratios obtained by the MC calculation \nfor the case with $\\delta$$^{29}$Si\/$^{28}$Si$_{init}$ =\n$\\delta$$^{30}$Si\/$^{28}$Si$_{init}$ = 0. In different Galactic \nregions elemental ratios relative to H are expected to be affected by newly \ninfalling gas and not only by the contributions from stellar nucleosynthesis. \nFor this reason we plotted ratios relative to Fe and normalized to the\nsolar system abundances (i.e., [Elem\/Fe] = \nlog[(Elem\/Fe)\/(Elem\/Fe)$_{\\odot}$]). The spreads in the theoretical \nelemental ratios are quite modest, especially if compared with ratios \nobserved in stars. Edvardsson et al. (1993) measured elemental abundances \nin a large number of stars from the Galaxy and concluded that stars \nfrom a given epoch (i.e. of a given age) and from a given Galactic \nradius show a considerable spread in metallicity. That this spread is \nnot simply the result of variations in the amount of newly infalling \nmaterial is shown by the fact that also elemental ratios between elements, \nin particular relative to Fe, show considerable variations (Fig. 13b). \nEdvardsson et al. (1993) pointed out that, despite observational errors \n(including a typical uncertainty of 1 - 2 Gyr in the age of dwarf stars), \nthese variations are intrinsic (see also Timmes et al. 1995). \nA comparison of Figs. 13a and 13b shows that the spread in elemental \nratios obtained from a model of statistical fluctuations in the \ncontributions from various SN sources is smaller than the spread \nobserved in stars. Since we do not know exactly how much of the spread \nin the Edvardsson et al. (1993) data is due to experimental errors, we \njust want to emphasize that the MC spread is not larger than that in \nstars. The models by Copi (1997) and van den Hoek \\& de Jong (1997), which \nmake use of stochastic approaches in the study of GCE, are able to \naccount for these elemental spreads. The mass of a well-mixed \nregion (a sort of mixing scale) that in the Copi (1997) model \nyields a good fit to the spread of the abundances \nof $\\alpha$-elements (such as Si) in stars \nis $M \\sim 10^5$ $M_{\\odot}$. If we interpret our constant \nparameter $a$ as the inverse of the total mass of the region in which the \nSN ejecta are expected to be well mixed, we find for this region a mass of $M \n= 1\/a \\sim 10^5$ $M_{\\odot}$, a number remarkably similar to that found by Copi (1997).\n\nWe conclude that local heterogeneities in Galactic regions that \ncan explain the variations in Si isotopic ratios observed in the \nmainstream SiC grains imply variations in elemental ratios that are \ncompatible with those observed in stars. In principle, \nsuch heterogeneities could be the cause of the mainstream isotopic \nvariations.\n\n\\subsection{Discussion}\n\nIt has to be emphasized that our model for explaining the Si isotopic\nvariations in mainstream SiC grains does not pretend to fully\nsimulate the isotopic compositions of the SiC mainstream grains. \nIt uses theoretical yields of the Si isotopes ejected from SNe that \nneeded adjustment to explain the composition of the solar system (see\nsection 4.1). In addition, the model is overly simplistic and at \nthis point should only be understood as a demonstration that local \nheterogeneities due to the statistical nature of SN contributions \ncan in principle successfully reproduce these variations. In reality \nthe situation is expected to be much more complicated:\n\n\\begin{itemize}\n\n\\item {1) There will be a statistical spread in the individual SN \ncontributions;}\n\\item {2) There will be a spread in the initial composition;} \n\\item {3) There will be contributions from SNe with a range of \nmetallicities;} \n\\item {4) There will be a range in ages of the AGB stars because \nof differences in their mass.}\n\n\\end{itemize}\n\nWe will discuss these points in turn. \n\n\\begin{itemize}\n\n\\item {1) We assumed that all the SN sources that add material to a given \nGalactic region contribute the same amount as expressed by a single \nvalue of the parameter $a$. In reality different SNe will contribute \ndifferent amounts and in some extreme cases one SN will completely \ndominate the local mix. In our MC calculations we have also assumed \ndifferent statistical distributions for the parameter $a$ and could \nachieve essentially the same final results as those shown in Figs. 12.}\n\n\\item {2) We have shown that different initial Si isotopic compositions can \nproduce distributions close to that of the mainstream grains if the \nparameters for the admixture of SN material (number of SN sources, \n$N$, and fraction of Si ejected by a SN, $a$) are chosen appropriately. \nIn reality we have to expect a whole range of initial compositions \nreflecting different times and different degrees of homogenization \nof matter in the Galaxy. We expect that material at a given Galactic \nradius is homogenized on a time scale of less than 10$^8$ yr, the period \nof Galactic rotation. Any complete homogenization will destroy the \nlocal heterogeneities in which we are interested. The real local \nisotopic compositions will represent some balance between heterogeneity \nand processes of homogenization. The overall result will be the GCE of \nthe elements and the isotopes, the overall trend being modified by \nlocal fluctuations.}\n\n\\item {3) In our model we have considered only Type II SNe of solar \nmetallicity. In reality there will be a range of metallicities. \nThis is for two reasons. First, we expect to encounter some range \nin age for AGB stars as will be discussed in the next section. \nSecond, if local regions are highly contaminated with previous \nSN contributions, new SNe from such regions will have higher-than-average \nmetallicities. According to the WW95 SNII models the addition of \nSNIIe ejecta of a given metallicity to an ISM parcel of the same \nmetallicity will result in higher $^{29}$Si\/$^{28}$Si and \n$^{30}$Si\/$^{28}$Si ratios than those of the starting material. This \nreflects the fact that $^{29}$Si and $^{30}$Si are secondary \nisotopes. The enrichment of the heavy Si isotopes in Type II SN \nejecta over the average ISM material has first been pointed out by \nClayton (1988) on the basis of an idealized GCE model. The enhancement of $^{29}$Si and $^{30}$Si in the SNII \nejecta over the starting composition is clearly seen in Fig. 11 for \nSNIIe of solar metallicity, where the average value plots to the upper right \nof the solar Si isotopic composition (this includes an assumed \nenhanced production of $^{29}$Si). However, a minimum metallicity is \nrequired for the contributing SNIIe in order to achieve the average Si isotopic \nratios of the protosolar nebula or those of the mainstream grains. We \nconclude from Fig. 9 that a metallicity of $Z$ $>$ 0.75 $Z_{\\odot}$ \nis required for the average Si isotopic composition of SNII \nejecta to be heavier than 150 $^o\\!\/\\!_{oo}$, the maximum of the \nmainstream grains. This, \nhowever, is a lower limit since in reality low-mass \nstars do not form from pure SN ejecta.}\n\n\\item {4) It has to be clear that the Si isotopic compositions of the \nmainstream grains reflect those of their parent stars at the time of their \nbirth and it is these compositions that we want to explain. However, it is \nalso clear that, depending on their mass, different stars were born at \ndifferent Galactic times, even if they all produced SiC grains at \nthe same time during their AGB phase (and even this last assumption is \nnot strictly valid because different SiC grains could have different IS \nlife times between their formation and the birth of the solar system). \nSo far we have made the implicit assumption that grains came only from \nstars in the 1.5 - 3 $M_{\\odot}$~ range when we computed the inferred Si isotopic \nratios of the mainstream grains without any AGB contributions (i.e., the \ninitial isotopic ratios of the parent stars). In the following subsection\nwe want to explore \nthis question in more detail.}\n\n\\end{itemize}\n\nThe processes leading to heterogeneity in the Si isotopic ratios \nin the ISM are much more complex than the simple mixing assumed in \nour MC model. However, and this is the most important conclusion of \nour tests, it is practically certain that the ISM at a given Galactic \ntime and at a given Galactic radius is not characterized by a unique \nSi isotopic composition but by a range of compositions. This \ndistribution of Si ratios will shift toward heavier isotopic \nratios during the evolution of the Galaxy. Note that presently we do not know \nthe exact distribution of the Si isotopes at a given Galactic time nor the \nrelationship between Galactic time and the mean of the Si isotopic ratios \nof the distributions. For the time being we assume that the \nlatter is the same \nas that of the Timmes \\& Clayton (1996) model, but this point will be discussed \nin more detail in \\S 4.5.\n\n\\subsection{The mass of the SiC parent stars}\n\nIn \\S 3 and Figs. 5 - 8 we showed that the shift in Si isotopic \ncompositions due to neutron capture in the He shell of AGB stars depends \non stellar mass and metallicity. As we demonstrated in that section, \nthe changes in the Si isotopic composition due to the $s$-process \nin AGB stars depend almost entirely on the small \nneutron exposure from the $^{22}$Ne source, with the $^{13}$C pocket \nhaving no influence, independent \nof the magnitude of its strength. The effects on stellar mass and \nmetallicity we are discussing in this section actually are the results \nof complete stellar evolutionary calculations of the AGB phases \nusing the FRANEC code. \nAt solar metallicity, from a 5 $M_{\\odot}$~ complete AGB evolutionary model we \nfind a somewhat higher maximum temperature at the bottom of the He thermal \npulses than in lower mass stars (1.5 $M_{\\odot}$~ to 3 $M_{\\odot}$). Because the $\\alpha$-capture reaction rate is proportional to \n$T^{21}$, this higher temperature \nincreases the efficiency of the neutron burst from the $^{22}$Ne \nsource, correspondingly changing the predicted final Si ratios \nas illustrated in the figures.\n\nThe same tendency is found in stellar evolutionary calculations \nwith the FRANEC code for AGB stars of different mass and\na metallicity of 1\/3 $Z_{\\odot}$, as shown in Table 4. The maximum\ntemperature at the bottom of the thermal pulse increases \nslightly from pulse to pulse, starting from about \n2.65 $\\times 10^8$ K for the pulse \nwhen TDU occurs for the first time. The temperature during the \npulse rises in a very rapid burst; subsequently the bottom \ntemperature decreases more \nor less exponentially from its maximum, with a total duration time \n(at $T >$ 2.5 $\\times 10^8$ K) of a few years.\n\nWhereas for 1.5 $M_{\\odot}$~ and 3 $M_{\\odot}$~ stars of \nsolar metallicity the maximum shifts in $\\delta$$^{30}$Si\/$^{28}$Si are \nonly 26 $^o\\!\/\\!_{oo}$~ and 37 $^o\\!\/\\!_{oo}$, respectively, the maximum shift for a 5 $M_{\\odot}$~ \nstar is 180 $^o\\!\/\\!_{oo}$. In Figs. 14a-d we plotted the results of the MC\ncalculations \nfor the $-$100 $^o\\!\/\\!_{oo}$~ case if we add the shifts expected for AGB stars of \nmasses 1.5 $M_{\\odot}$, 3 $M_{\\odot}$, and 5 $M_{\\odot}$~ with solar metallicity and of 3 $M_{\\odot}$~ \nwith $Z$ = 0.006. The ranges of shifts were added in a random, \nstatistical fashion in our MC test. As can be seen, only the 1.5 $M_{\\odot}$~ and \n3 $M_{\\odot}$~ stars of solar metallicity give results in reasonable agreement \nwith the grain data, while 5 $M_{\\odot}$~ stars as well as stars with $Z$ = 0.006 \nshift the Si isotopic compositions far to the right of the grain data \nand the solar composition. Especially for the low-metallicity case of \nFig. 14d the predicted shifts have a much wider spread. \nIt is a remarkable result of our heterogeneity model \nthat, without any AGB contributions, the solar composition is one of \nthe possible compositions and at the same time the mainstream data \ncan be reproduced if the AGB shifts are small, as for the 1.5 $M_{\\odot}$~ and \n3 $M_{\\odot}$~ star models of close-to-solar metallicity. This is not true anymore if the AGB shifts are as \nlarge as those for 5 $M_{\\odot}$~ stars.\n\nFrom the above discussion, it is reasonable, even if not proven, to assume \nthat mostly low-mass stars (with $M$ $\\leq$ 3 $M_{\\odot}$) contributed SiC to the solar system. Actually, \nthere are many pieces of evidence that indicate that this is indeed the \ncase:\n\n\\begin{itemize}\n\n\\item {1) Feast (1989) performed a study of the kinematics of peculiar red \ngiants including S, SC, and C stars. On the basis of 427 C stars he \nestimated their mean mass to be 1.6 $M_{\\odot}$. Although this estimate \nneeds to be improved, if SiC grains came from average \nC stars, they came from low-mass stars.}\n\n\\item {2) Another argument for low masses of carbon stars is based on a \ncomparison of the observed luminosities of AGB stars in the Magellanic \nclouds with predicted luminosities. Theory predicts intermediate-mass \nstars of 5 - 8 $M_{\\odot}$~ to have $M_v$ of less than - 6.5 but typical \nluminosities of S and C stars are much lower, indicating low-mass stars \n(Mould \\& Reid 1987; Frogel, Mould, \\& Blanco 1990; Van Loon et al. \n1998).}\n\n\\item {3) Another argument is based on theoretical predictions about the \noccurrence of hot bottom burning (HBB) in intermediate-mass (5 - 8 $M_{\\odot}$) \nstars. HBB takes place when the bottom layers of the convective envelope \nare hot enough for some proton capture nucleosynthesis to occur. In this \ncase, most $^{12}$C~ dredged up from the He \nshell during the TP-AGB phase is converted to $^{14}$N, \npreventing the star from becoming a carbon \nstar. There are several theoretical studies that indicate that HBB occurs in\nstars of $\\gtrsim$ 5 $M_{\\odot}$~ of solar metallicity and in stars with \n$\\gtrsim$ 4 $M_{\\odot}$~ of lower metallicity (Boothroyd, Sackmann, \\& Wasserburg \n1995; Forestini \\& Charbonnel 1997; Lattanzio et al. 1997). For solar \nmetallicity stars, the FRANEC code finds HBB in a 7 $M_{\\odot}$~ but not a 5 $M_{\\odot}$~ \nstar. The situation is complicated by the finding that if HBB stops while \nthermal pulses and TDUs continue in a star with mass loss, the star can \nbecome C-rich (Frost et al. 1998; Lattanzio \\& Forestini 1999). However, this happens only in \nlow-metallicity stars. Furthermore, if superwinds during the advanced \nAGB phase erode the envelope quickly, it is possible that TP \ncease before the star becomes C-rich. Thus, by and large \nit is not very likely that there are a substantial number \nof C-rich intermediate-mass stars that could have contributed SiC \nto the solar system.}\n\n\\item {4) We also obtain constraints on the mass and the metallicity of the \nparent stars from the isotopic compositions measured in presolar SiC \ngrains when these compositions are compared with model calculations:\n\n\\begin{itemize}\n\n\\item {i) It has already been pointed out that the heavy elements patterns \nmeasured in presolar SiC are well reproduced by models of neutron-capture \nnucleosynthesis in AGB stars (Gallino et al. 1997) . However, this \nagreement exists only for low-mass AGB stars of close-to-solar \nmetallicity and not for intermediate-mass stars, or of AGB stars of low\nmetallicity. A particularly \ndiagnostic isotopic ratio is the $^{96}$Zr\/$^{94}$Zr ratio. The large \n$^{96}$Zr depletions measured in mainstream SiC grains (Nicolussi et al. \n1997; Pellin et al. 1999) are well reproduced only with models of AGB \nstars of 1.5 - 3 $M_{\\odot}$~ and about solar metallicity (Gallino et al. 1998b), but \nhigher-mass stars and stars with low metallicity are predicted to \nproduce huge $^{96}$Zr excesses.} \n\n\\item {ii) Gallino et al. (1990) pointed out that the He and Ne isotopic \ndata of presolar SiC grains are best explained in terms of \nnucleosynthesis in low-mass AGB stars of close-to-solar metallicity \n(see their Fig. 1). Another important observation is that SiC grains \ndo not show large $^{25}$Mg excesses (within relatively large errors). \nThis again indicates low-mass stars in which $^{22}$Ne does not burn. \nIndeed, the FRANEC code yields $^{25}$Mg excesses of up to $\\sim$ 200 \n$^o\\!\/\\!_{oo}$~ in the envelope of 1.5 and 3 $M_{\\odot}$~ AGB stars of solar metallicity, \nwhereas predicted \nexcesses are an order of magnitude larger in the 5 $M_{\\odot}$~ model of $Z$ = \n0.02 and in the 3 $M_{\\odot}$~ model of $Z$ = 0.006.} \n\n\\item {iii) The situation is similar with regard to the \n$^{12}$C\/$^{13}$C ratios, where the observed range is best \nreproduced by low-mass AGB models of close-to-solar metallicity (Gallino et \nal. 1990; Bazan 1991). New results from the FRANEC code confirm \nthese earlier conclusions: the best agreement is obtained for 1.5 $M_{\\odot}$~ \n($^{12}$C\/$^{13}$C = 40 - 60) and 3 $M_{\\odot}$~ ($^{12}$C\/$^{13}$C = 90 - 100) AGB \nstars of solar metallicity, the 5 $M_{\\odot}$~ model of solar metallicity\nand the 3 $M_{\\odot}$~ model of low metallicity ($Z$ = \n0.006) yield much higher ratios ($^{12}$C\/$^{13}$C = 90 - 120 and \n100 - 700, respectively). In low-mass star models with $M \\lesssim$ 2.5 $M_{\\odot}$,\nthe presence of \ncold bottom processing (CBP) (Charbonnel 1995; Wasserburg et al. 1995;\nBoothroyd \\& Sackmann 1999) \nlowers the initial $^{12}$C\/$^{13}$C ratio at the beginning of the TP-AGB \nphase.} \n\n\\item {iv) Cold bottom processing is also important for the \n$^{14}$N\/$^{15}$N ratio. The high ratios observed in many individual \nmainstream SiC grains (Fig. 1) can only be explained by CBP (Huss et al. \n1997) operating in low-mass stars.}\n\n\\end{itemize}}\n\n\\item {5) A lower limit on the masses of carbon stars can be \nobtained from models with TDU. Existing models predict TDU only for \nstars with $M \\gtrsim$ 1.5 $M_{\\odot}$~ (Lattanzio 1989; \nStraniero et al. 1997; Gallino \net al. 1998a; Busso et al. 1999b). In the FRANEC \ncode the limit depends on the value of Reimer's parameter $\\eta$ used. \nFor a star of $M =$ 1.5 $M_{\\odot}$~ of solar metallicity the limit is \n$\\eta$= 0.3 for TDU to occur and for producing C\/O $>$ 1 in the \nenvelope during the advanced stages of the \nAGB phase. The fact that there is a \nminimum mass below which TDU does not occur is of great \nimportance, since because of it SiC grains cannot originate \nfrom long-lived stars of low mass and low metallicity. \nIt is worth noting that increasing the metallicity above \nsolar works against an AGB star to become C-rich. In our \ncalculations the star remains O-rich at $Z = 2 \\times$ $Z_{\\odot}$ \nfor $M = $ 1.5 $M_{\\odot}$, and already at $Z = 1.25 \\times$ \n$Z_{\\odot}$ for $M = $ 3 $M_{\\odot}$.}\n\n\\end{itemize}\n\nFrom all these considerations it appears that most presolar SiC grains \ncome from AGB stars of 1.5 - 3 $M_{\\odot}$~ and close-to-solar metallicity. Of \ncourse, it is not said that the mainstream SiC grains have to come from \ntypical carbon stars. It is possible that these very large grains \npreferentially originated from stars with very dense winds and thus from\nstars having masses at the upper end of the above range. However, stars \nwith masses much larger than 3 $M_{\\odot}$~ can definitely be excluded.\n\n\\subsection{Star lifetimes, grain lifetimes and Galactic chemical \nevolution}\n\nLet us now return to the question of lifetimes of the possible source \nstars for SiC grains. Whereas the calculated lifetime of a 5 $M_{\\odot}$~ star \nof solar metallicity is 1.1 $\\times 10^8$ yr (Schaller et al. 1992), \nthose of the 3 $M_{\\odot}$~ and 1.5 $M_{\\odot}$~ stars are 4.4 $\\times 10^8$ and \n2.9 $\\times 10^9$ yr, respectively. Especially the latter lifetime \nwould result in a non-negligible difference in the Si isotopic ratios \ndue to the overall temporal evolution of the Si isotopes in the Galaxy. \nAccording to the model of Timmes \\& Clayton (1996) a time difference of \n2.9 $\\times 10^9$ yr corresponds to a difference of 125 $^o\\!\/\\!_{oo}$~ in the \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios, which is almost the \nwhole range covered by the mainstream grains. This means that a star that\nwas born 2.9 $\\times 10^9$ yr before the sun should have \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios that are, on average, \n125 $^o\\!\/\\!_{oo}$~ smaller than the solar ratios. It also means that if stars of \ndifferent mass and therefore different lifetimes contributed SiC grains \nto the solar system, these grains are expected to have different Si isotopic \ncompositions. There are several factors \nthat play a role here. One is the initial mass function for stars. There \nare more stars of lower mass and we have already pointed out that the \nestimated mean mass of carbon stars is $\\sim$1.6 $M_{\\odot}$~ (Feast 1989). On the other hand, \nthe fact that SiC grains are relatively large \nsuggests that more massive AGB stars with very dense winds, meaning \nhigh mass loss at low speed, were selected as the SiC grains' parent \nstars.\n\nLet us consider two extreme cases. First we consider the case that all \nor most of the \nmainstream SiC grains came from AGB stars of approximately the same \nmass and therefore also the same time of formation. In this \ncase time differences do not play a role and \nthe distribution of the Si isotopic ratios of the mainstream grains \ncan in principle be explained as having an origin in \nlocal isotopic heterogeneities \ndue to the statistical nature of SN contributions to the ISM. This is \nschematically shown in Fig. 15a where the whole range of Si isotopic \ncompositions exhibited by the SiC mainstream grains is interpreted \nas the spread in Si isotopes that existed at the time of formation \nof the grains' parent stars. These parent stars have to have \napproximately the same mass but at this point it is not said whether \nit is low (1.5 $M_{\\odot}$) or high (3$M_{\\odot}$).\n\nThe second case to be considered is one in which AGB stars of \na {\\it considerable} mass range contributed the \nmainstream grains to the solar system. In this case the \nrange in Si isotopic shifts\ndue to the formation time difference between stars of 1.5 $M_{\\odot}$~ and of 3 $M_{\\odot}$~ \nis of the same order of magnitude \nas the spread of the mainstream grains. \nVariations in the Si \nisotopic ratios of individual grains are expected to arise from \nthe age differences \nof their parent stars (which in turn vary because of GCE) \nand local \nheterogeneities of the Si isotopes play a complementary role. \nThis situation is schematically depicted in \nFig. 15b, where the \nspread of the mainstream grains' Si isotopic ratios is interpreted \nas a superposition of local heterogeneity distributions representative \nof different Galactic times.\n\nAnother factor that plays a role here is the lifetime of the SiC grains \nin the ISM. This lifetime has to be added to the lifetime of the AGB \nparent stars in terms of time differences between the birth of these \nstars and the formation of the solar system and the implication of \nthese time differences for the Si isotopic ratios. Unfortunately, at \npresent we do not have any good direct measure of grain lifetimes. \nAttempts have been made to determine IS grain lifetimes from the \nmeasurement of cosmogenic $^{21}$Ne produced in the grains from the \nspallation of Si by Galactic cosmic rays (Tang \\& Anders 1988b; \nLewis et al. 1994). Estimates obtained in this way range up to \n1.3 $\\times 10^8$ yr. However, besides poor knowledge of the flux of \nGalactic cosmic rays, there are many other uncertainties associated \nwith this approach. Single grain measurements showed that only $\\sim$ \n5\\% of all SiC grains are rich in $^{22}$Ne (Nichols et al. 1991, 1992, \n1993). If one assumes that outgassing is the reason that the other \ngrains lack measurable amounts of $^{22}$Ne and that the same process \nremoved cosmogenic $^{21}$Ne from these grains, one arrives at much higher \nestimates for IS grain lifetimes. However, it is unclear that outgassing \nis indeed the cause for the large variations of $^{22}$Ne among single SiC \ngrains. Another problem is the determination of spallation recoil \nloss from the grains. From experimental measurements of spallation \nrecoil Ott \\& Begemann (1997, 1999) concluded that a determination of presolar \nexposure ages from cosmogenic $^{21}$Ne is not feasible. These authors \npropose the use of spallation Xe as more promising but before this is \ndone we do not have any reliable IS lifetimes for presolar grains. An \nalternative way is to use model ages derived from theoretical \ndestruction rates of IS grains by SN shocks and collisions (see, e.g., \nWhittet 1992; Jones et al. 1997). Estimates range up to $\\sim 10^9$ yr \nbut there are also large uncertainties in this approach.\n\n\\subsection{SiC grains and the Si isotopic composition of the sun}\n\nSo far we have discussed differences in the formation time of AGB \nstars of different masses that possibly contributed SiC grains to \nthe solar system. We have not discussed yet the relationship \nbetween the formation time of the grains' parent stars relative to \nthat of the solar system and implications for their relative Si \nisotopic compositions. In the Timmes \\& Clayton (1996) model the \nfact that most mainstream grains have isotopically heavier compositions \nthan the solar system (implying that they are younger) \nbut must have formed before the sun presents a fundamental problem. \nOur heterogeneity model alleviates this fundamental problem, because \nin principle it can explain the spread in the Si isotopic compositions \nof the mainstream grains as inhomogeneities of the Si isotopes in the \nISM at a given time (see Fig. 15a). However, the grains' parent stars \nmust have formed before the solar system and we must \ndiscuss the effect of this time difference on their Si isotopic \ncompositions.\n\nFor the sake of discussion we again consider two extreme cases. First we consider the case that most \nmainstream SiC grains came from AGB stars of 3 $M_{\\odot}$. The \nevolution time of such stars is 4.4 $\\times 10^8$ yr. According to \nTimmes et al. (1995) and Timmes \\& Clayton (1996), such a time \ndifference corresponds to a shift of \n19 $^o\\!\/\\!_{oo}$~ of the Si isotopic ratios. If we assume that the spread in \nSi isotopic ratios at the time of the birth of the parent stars \ncoincides with that of the mainstream grains, the Si isotopic distribution \nat the time of solar system formation 4.4 $\\times 10^8$ yr later is \nisotopically heavier by 19 $^o\\!\/\\!_{oo}$~ (Fig. 16a). This shift is \nrelatively small compared to the range of the mainstream grains. The Si \nisotopic composition of the sun, while falling at the outer edge of \nthis distribution, lies still within the range of compositions \nexpected to be present at the time of solar system formation. \nThe fact that the sun has a \ncomposition that differs from those of most of the grains, is not a \nfundamental problem in this case. It simply means that the sun, \nas many other SiC \nparent stars, has an unusual composition but one that is not incompatible with expectations.\n\nLet us next consider the other extreme, that all grains come from stars of \n1.5 $M_{\\odot}$. The evolution time of these stars is 2.9 $\\times 10^9$ yr. \nAccording to Timmes \\& Clayton (1996) this time difference corresponds to \na shift of the Si isotopic ratios by 125 $^o\\!\/\\!_{oo}$~. This means that the distributions \nof the Si isotopes at the time of star formation and at the time of solar \nsystem formation 2.9 $\\times 10^9$ yr later are shifted by this amount \nrelative to one another. This is shown in Fig. 16b where, again, we assume \nthat the Si isotopic distribution of the stars coincides with that of \nthe mainstream grains. This time, the inferred distribution \n2.9 $\\times 10^9$ yr later is shifted so much that it seems quite\nimpossible \nthat the solar system composition observed today can be explained as \nbeing part of this distribution. In other words, in the extreme case in which only stars of \n$M = $ 1.5 $M_{\\odot}$~ contributed SiC grains, we are faced with the \nsame fundamental problem as the Timmes \\& Clayton (1996) model, namely that \nthe actual solar system composition is much too light compared to the \ndistribution predicted for the time of solar system formation if the mainstream grains came from old stars. \n\nClayton (1997) has addressed this problem and \nhas proposed a solution in terms of a systematic difference in the \nGalactic radius at which the parent stars of the mainstream grains \non the one hand and the sun on the other hand formed. The parent stars \nare assumed to have formed at smaller Galactic radii where the \nmetallicity and $^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios are \nbelieved to be higher. His model involves the diffusion of stars from \nsmaller to larger Galactic radii due to scattering on IS clouds, \nfollowing the stellar \norbital diffusion model by Wielen, Fuchs, \\& Dettbarn (1996). Such a \nmodel had been proposed in order to explain the spread in elemental \nabundances observed in stars. However, a more detailed quantitative \ntreatment by Nittler \\& Alexander (1999b) shows that with reasonable \nassumptions a diffusion model cannot account for the isotopically heavy \nSi compositions of the grains relative to the sun. Furthermore, van \nden Hoek \\& de Jong (1997) pointed out that stellar orbital diffusion \ncannot sufficiently explain the elemental abundance variations. \nWhile we do not want to discard the orbital diffusion model, we \nhope that the heterogeneity explanation will give a more definitive answer \nto this important question. This would require a model that, \nby computing the Galactic evolution of the Si isotopes \nby taking into account incomplete mixing of \ndifferent stellar yields, overcomes the problems discussed in \n\\S 4.3 and all those connected \nto the overly simplistic nature of our approach.\n\nIt should be noted that in our estimates of the Si isotopic shifts \nassociated with time differences we have used the Si isotopic evolutions \nvs. time relationship given by Timmes et al. (1995) and \nTimmes \\& Clayton (1996). This relationship crucially depends on the \nrelative proportion in which Type Ia and Type II SNe contribute to the \nenrichment of the ISM in Si isotopes. \nTimmes et al. (1995) attributed a dominant role to Type II \nSNe by assuming that at the time of solar system formation they \ncontributed 2\/3 of the Fe. Woosley et al. (1997), on the other hand \nestimated that this fraction would be 1\/2. This would mean that \nthe $^{28}$Si contribution from Type Ia SNe is higher and therefore \nthe Si isotopes evolve more slowly toward heavier compositions. \nThis in turn would mean that a Si isotopic shift corresponding to \na given time difference (Fig. 16) is smaller than what we assumed. \n\nIn conclusion, there are still large uncertainties as to the masses \nof the parent AGB of the grains, the ISM life times of the grains, and the \ntime dependence of the evolution of the Si isotopes in the Galaxy. \nAll of these uncertainties have to be clarified before we can hope to \nsolve the problem of the difference of the Si isotopic compositions of \nthe mainstream SiC grains and that of the solar system.\n\n\\section{Conclusions}\n\nMainstream SiC grains are the major group of presolar SiC grains found in \nmeteorites. Although there \nis overwhelming evidence that mainstream grains have an origin in the \nexpanding atmospheres of AGB stars, their Si isotopic ratios show a \ndistribution (Fig. 3) that cannot be explained by nucleosynthesis in \nAGB stars. The theoretically predicted Si isotopic shifts in the \nenvelope of AGB stars are either much smaller than the range observed \nin the mainstream grains (for AGB models of $M$ = 1.5 and 3 $M_{\\odot}$~ and \nsolar metallicity) or (for AGB models of $M$ = 5 $M_{\\odot}$~ and solar \nmetallicity and $M$ = 3 $M_{\\odot}$~ and $Z$ = 0.006) show a slope of $\\sim$ 0.5 \ncorrelation between the $\\delta$$^{29}$Si\/$^{28}$Si and \n$\\delta$$^{30}$Si\/$^{28}$Si values instead of the slope 1.31 correlation \nline exhibited by the grains.\n\nThe distribution of the Si isotopic ratios of the mainstream grains \nhas previously been interpreted to be the result of GCE of the Si \nisotopes. In this interpretation the grains' parent stars are expected \nto have a range of different Si isotopic ratios if they were born at \ndifferent times. In this paper we proposed an alternative explanation \nfor the Si isotope distribution by invoking isotopic heterogeneities \ndue to the statistical nature of the contributions of a limited number \nof SN sources to the IS material from which the grains' parent stars \nformed. The Si isotopic ratios of the ejecta of possible SN sources, \nclassical Type Ia SNe, Type Ia SNe from sub-Ch white dwarfs, \nand Type II SNe of different masses, span a wide range. We developed a \nsimple Monte Carlo model in which contributions from these SN sources \nwere admixed in a random way to material with a given Si isotopic \ncomposition. As long as this composition lies on the theoretically \nexpected GCE line going through the solar Si isotopic composition, \nwe could show that, with the right choice of parameters, the \ndistribution of the Si isotopic ratios in the mainstream grains can \nbe successfully reproduced for a wide range of starting compositions. \nThe parameters to be adjusted are the total number of SN sources \nselected and the fraction of the material ejected from each SN that \nis admixed to the starting material. In addition, an adjustment of the SN \nyield of $^{29}$Si by a factor of 1.5 is necessary to achieve the \nSi isotopic ratios of the solar system. \nAstronomical observations of variations of \nelemental ratios in stars are compatible with the predictions \nfrom our MC model.\n\nThese results demonstrate that, in principle, the mainstream distribution \ncan be explained as the result of local fluctuation in the ISM due to \nthe admixture of material from a limited number of SN sources to the \npreexisting IS matter. If the AGB stars that contributed SiC grains to \nthe protosolar nebula were born within a short period of time (by having a \nnarrow range of masses and the grains having short IS lifetimes), such \nfluctuations must have been the dominant cause of the mainstream \ndistribution. If, however, the AGB parent stars had a large range of \nmasses and therefore a large range of lifetimes and\/or the grains themselves \nexperienced a large range of residence times in the ISM, the parent \nstars must have been born at different Galactic eras and their initial \nSi isotopic ratios must show considerable variations because of the \nvarying average composition of the ISM due to the GCE of the Si isotopes. \nIn this case we still expect that local fluctuations will be \nsuperimposed on these average compositions. To simulate these complex \nprocesses it will be necessary to apply to the Si isotopes a\nGalactic evolution model that is able to take both components properly \ninto account. \nHowever, a successful model would require knowledge of the mass \ndistribution of AGB stars that contributed SiC grains (at least in \nthe size range of the single grains whose data are plotted in Fig. 3) \nand the distribution of the IS lifetimes of these grains. Both pieces \nof information are presently unknown and we can only hope that further \nprogress in the study of the grains and their origin will get us \ncloser to an answer.\n\nWe are grateful to Peter Hoppe for providing isotopic data on the \nMurchison KJE size fraction, to Gary Huss for providing his Orgueil \ndata, and to Stan Woosley for providing unpublished results of the Si \nyields from his and Weaver's $Z$ = 0.5 $Z_{\\odot}$ and \n$Z$ = 2 $Z_{\\odot}$ supernova models. \nWe are deeply indebted to Oscar Straniero, Maurizio Busso, Alessandro \nChieffi and Marco Limongi for all \ntheir scientific input and thank Don Clayton for ideas and discussions. \nThe detailed and thoughtful review by Don Clayton substantially \ncontributed to the final version of this paper. \nML gratefully acknowledges the invaluable help of John Lattanzio. \nEZ deeply appreciates the hospitality extended to him by Roberto Gallino\nduring a visit to the Dipartimento di Fisica Generale of the University \nof Torino and the support for this visit provided by the Gruppo Nazionale di\nAstronomia del CNR. SA acknowledges the support for a visit to the same \nDepartment provided by the University of Torino.\nThis work was supported by an Overseas Postgraduate Research Scheme \naward (ML), NASA grant NAG5-8336 (SA and EZ) and by MURST Cofin98 \nProgetto Evoluzione Stellare (RG).\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\noindent\n\nThe isospin-symmetry violation in atomic nuclei is predominantly\ndue to the Coulomb interaction that exerts long-range polarizations\non neutron and proton states. To consistently take into account\nthis polarization, one needs to\nemploy huge configuration spaces. For that reason, an accurate description of isospin impurities in atomic nuclei, which\nis strongly motivated by the recent high-precision measurements of the $0^+\n\\rightarrow 0^+$ Fermi superallowed $\\beta$-decay rates, is difficult to\nbe obtained in shell-model\napproaches, and specific approximate methods are required.\\cite{[Orm95a],[Tow08]}\n\n\nThe long-range polarization effects can be included within the\nself-consistent mean-field (MF) or DFT\napproaches, which are practically the only microscopic frameworks\navailable for heavy, open-shell\nnuclei with many valence particles. These approaches, however, apart from\nthe {\\it physical\\\/} contribution to the isospin mixing, mostly caused by\nthe Coulomb field and, to a much lesser extent, by\nisospin-non-invariant components of the nucleon-nucleon force, also introduce the spurious isospin mixing due to the {\\it spontaneous\\\/} isospin-symmetry breaking.\\cite{[Eng70],[Cau80],[Cau82]}\n\n\nHereby, we present results on the isospin mixing and\nisospin symmetry-breaking corrections to the superallowed Fermi $\\beta$-decay\nobtained by using the newly developed isospin- and angular-momentum-projected DFT\napproach without pairing.\\cite{[Sat09],[Sat09a],[Sat10],[Sat10a]}\nThe model employs symmetry-restoration techniques to remove\nthe spurious isospin components and restore angular momentum symmetry, and\ntakes advantage of the natural ability of MF to describe self-consistently\nthe subtle balance between the Coulomb force making proton and\nneutron wave functions different and the isoscalar part of the strong\ninteraction producing the opposite effect.\n\nThe paper is organized as follows. In Sec.~\\ref{theo}, we describe the main theoretical\nbuilding blocks of the isospin- and angular-momentum-projected DFT.\nSection~\\ref{isomix} presents some preliminary applications of the formalism to the isospin symmetry-breaking corrections to the Fermi superallowed\n$\\beta$-decay matrix elements, whereas Sec.~\\ref{symm} discusses applications of the\nisospin-projected DFT to nuclear symmetry energy. The summary is contained in Sec.~\\ref{summary}.\n\n\n\\section{The projected DFT framework}\n\\label{theo}\n\n\n\n\n\nThe building block of the isospin-projected DFT is the Slater determinant,\n$|\\Phi\\rangle$, representing the self-consistent Skyrme-HF solution provided\nby the HF solver HFODD.\\cite{[Dob09d]} Self-consistency ensures that\nthe balance between the long-range Coulomb force and short-range strong\ninteraction, represented in our model by the Skyrme energy density functional (EDF), are properly taken\ninto account. The unphysical isospin mixing is taken care of by the\nrediagonalization of the entire Hamiltonian in the good isospin basis, $|T,T_z\\rangle$,\nas described in Refs.\\cite{[Sat10],[Sat10a]}\nThis yields the eigenstates:\n\\begin{equation}\\label{mix2}\n|n,T_z\\rangle\n= \\sum_{T\\geq |T_z|}a^n_{T,T_z}|T,T_z\\rangle\n\\end{equation}\nnumbered by an index $n$. The so-called isospin-mixing\ncoefficients (or, equivalently, isospin impurities)\nare defined for the $n-$th eigenstate as\n\\begin{equation}\n\\alpha_C^n = 1 - |a^n_{T,T_z}|_{\\text{max}}^2 ,\n\\end{equation}\nwhere $|a^n_{T,T_z}|_{\\text{max}}^2$ stands for the dominant amplitude in the wave function\n$|n,T_z\\rangle$.\n\nWithin the isospin- and angular-momentum-projected DFT, we\nuse the normalized basis of states $|I,M,K; T,T_z\\rangle$\nhaving both good angular momentum and good\nisospin.\\cite{[RS80]}\nHere, $M$ and $K$ denote the angular-mo\\-men\\-tum components\nalong the laboratory and intrinsic $z$-axes, respectively. The $K$ quantum\nnumber is not conserved. In order to avoid problems with overcompleteness of\nthe basis, the $K$-mixing is performed by rediagonalizing\nthe Hamiltonian in the so-called {\\it collective space}, spanned for each $I$\nand $T$ by the {\\it natural states\\\/}, $|IM;TT_z\\rangle^{(i)}$, as described\nin Refs.\\cite{[Dob09d],[Zdu07a]} Such a rediagonalization yields the\neigenstates:\n\\begin{equation} \\label{KTmix}\n|n; IM; T_z\\rangle =\n\\sum_{i,T\\geq |T_z|}\n a^{(n)}_{iIT} | IM; TT_z\\rangle^{(i)} ,\n\\end{equation}\nwhich are labeled by the index $n$ and by the conserved quantum numbers $I$, $M$, and\n$T_z=(N-Z)\/2$ [compare Eq.~(\\ref{mix2})].\n\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.54\\textwidth,clip]{kazi10_fig1.eps}\n\\caption[T]{\\label{fig1}\nThe absolute values of the norm kernels, $|{\\cal N}(\\beta_T; \\alpha, \\beta, \\gamma )|\n= |\\langle \\Phi | \\hat{R}(\\beta_T ) \\hat{R}(\\alpha, \\beta, \\gamma )\n|\\Phi\\rangle|$, for a state in $^{14}$N calculated with the SLy4 EDF, plotted versus the rotation angle in the isospace $\\beta_T$.\nThe solid curve, exhibiting the single singularity at $\\beta_T = \\pi $, corresponds to\nthe pure isospin-projected DFT theory, which is regular for all\nSkyrme-type functionals.\\protect\\cite{[Sat10]}\nThe dotted lines correspond to two fixed sets of the Euler\nangles in space, with $\\alpha =\\gamma \\approx 0.314$, and\n$\\beta \\approx 0.229$ (left curve)\nand $\\beta \\approx 1.414$ (right curve). The poles that appear\ninside the integration region, $0<\\beta_T<\\pi$, give rise to singularities in\n the projected DFT approach.}\n\\end{center}\\end{figure}\n\n\nThe isospin projection does not produce singularities in energy kernels; hence, it can be safely used with all commonly used EDFs.\\cite{[Sat10]} Coupling the isospin and angular-momentum\nprojections, however, leads to singularities in both the norm (see\nFig.~\\ref{fig1}) and energy kernels. This fact narrows the\napplicability of the model to Hamiltonian-driven EDFs which,\nfor Skyrme-type functionals, leaves only one option: the SV\nparametrization.\\cite{[Bei75]} The alternative would be to use an\nappropriate regularization scheme, which is currently under\ndevelopment.\\cite{[Lac09],[BD10]}\n\n\n\n\\section{Isospin-mixing and isospin-breaking corrections to superallowed\n$\\beta$-decay}\n\\label{isomix}\n\n\n\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.54\\textwidth,clip]{kazi10_fig2.eps}\n\\caption[T]{\\label{fig2}\nIsospin impurities in the ground states of $^{40}$Ca (upper panel) and\n$^{100}$Sn (lower panel), plotted as functions of the excitation energy of\nthe doorway state for a set of commonly used Skyrme EDFs.\\cite{[Ben03]} Results of the linear fits and the\ncorresponding regression\ncoefficients, $R$, are also shown.}\n\\end{center}\\end{figure}\n\n\nEvaluation of $\\alpha_C$ is\na prerequisite to calculate isospin corrections to reaction and decay rates.\nAs is well known,\\cite{[Aue83]} isospin impurities are\nthe largest in $N=Z$ nuclei, increase along the $N=Z$ line with increasing\nproton number, and are strongly quenched with increasing $|T_z|=|N-Z|\/2$.\nSuch characteristics were also early estimated based on the perturbation\ntheory\\cite{[Sli65]} or hydrodynamical model.\\cite{[Boh67]} Quantitatively,\nafter getting rid of the spurious mixing, which lowers the true $\\alpha_C$ by as\nmuch as 30\\%,\\cite{[Sat09a]}\nthe isospin impurity increases from a fraction of a percent in very light\n$N=Z$ nuclei to $\\sim$0.9\\% in $^{40}$Ca, and $\\sim$6.0\\% in $^{100}$Sn,\nas shown in Fig.~\\ref{fig2}. In the particular case of $^{80}$Zr, the\ncalculated impurity of 4.4\\% agrees well with the empirical value deduced from\nthe giant dipole resonance $\\gamma$-decay studies.\\cite{[Cam10a]} This makes us believe that our model is indeed capable of\ncapturing essential physics associated with the isospin mixing. Unfortunately,\ncurrent experimental errors are too large to discriminate between different\nparametrizations of the Skyrme functional. The variations between EDFs in Fig.~\\ref{fig2} result in $\\sim$10\\% uncertainty in calculated\nvalues of $\\alpha_C$.\n\n\nThe magnitude of theoretical $\\alpha_C$ is quite well correlated\nwith the excitation energy, $E_{T=1}$, of the $T=1$ doorway state,\nsee Fig.~\\ref{fig2}. However, in order to make a precise determination of $E_{T=1}$, spectroscopic quality EDFs are needed, and this is not yet the case.\\cite{[Kor08]} This explains why the values of $\\alpha_C$\ndo not correlate well with basic EDF characteristics, including the isovector and isoscalar effective mass,\nsymmetry energy, binding energy per particle, and\nincompressibility (see discussion in Ref.\\cite{[Sat10a]}).\n\n\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.54\\textwidth,clip]{kazi10_fig3.eps}\n\\caption[T]{\\label{fig3}\nValues of $|V_{ud}|$ deduced from the superallowed $\\beta$-decay\n(full circles) for three different sets of the $\\delta_C$ corrections calculated in:\nRef.\\protect\\cite{[Tow08]} (a); Ref.\\protect\\cite{[Lia09]} with NL3 and\nDD-ME2 Lagrangians (b); and in the present work (c).\nTriangles mark values of $|V_{ud}|$ obtained from the\npion-decay\\protect\\cite{[Poc04]} and neutron-decay\\protect\\cite{[Ams08]} studies,\nrespectively. The open circle shows the value deduced from the\n$\\beta$-transitions in $T=1\/2$ mirror nuclei.\\protect\\cite{[Nav09a]}\n}\n\\end{center}\\end{figure}\n\n\nIncreasing demand on precise values of isospin impurities has been\nstimulated by the recent high-precision measurements of superallowed\n$\\beta$-decay rates.\\cite{[Har05c],[Tow08]}\nA reliable determination of the corresponding\nisospin-breaking correction, $\\delta_C$,\nrequires the isospin- and angular-momentum-projected DFT.\\cite{[Sat10a]}.\nThis correction is obtained by calculating\nthe $0^+ \\rightarrow 0^+$ Fermi\nmatrix element of the isospin raising\/lowering operator $\\hat T_{\\pm}$ between\nthe ground state (g.s.) of the even-even nucleus $| I=0, T\\approx 1, T_z = \\pm 1 \\rangle$\nand its isospin-analogue partner in the $N=Z$ odd-odd nucleus, $|I=0, T\\approx\n1, T_z = 0 \\rangle$:\n\\begin{equation}\\label{fermime}\n|\\langle I=0, T\\approx 1,\nT_z = \\pm 1 | \\hat T_{\\pm} | I=0, T\\approx 1, T_z = 0 \\rangle |^2 \\equiv 2 (\n1-\\delta_C ).\n\\end{equation}\n\n\nTo determine the $|I=0, T\\approx 1, T_z = 0 \\rangle$ state\nin the odd-odd $N=Z$ nucleus, we\nfirst compute the so-called\nantialigned g.s.\\ configuration, $|\\bar \\nu \\otimes \\pi \\rangle$ (or $| \\nu\n\\otimes \\bar \\pi \\rangle$), by placing the odd neutron and the odd proton in\nthe lowest available time-reversed (or signature-reversed)\nHF orbits.\nThen, to correct for the fact that the antialigned\nconfigurations manifestly break the isospin symmetry,\\cite{[Sat10]} that is,\n$|\\bar \\nu \\otimes \\pi \\rangle \\approx \\frac{1}{\\sqrt 2} (|T=0 \\rangle + |T=1\n\\rangle )$, we apply the isospin and angular-momentum projections to create\nthe basis $|I,M,K,T,T_z=0 \\rangle$, in which the total Hamiltonian is rediagonalized (see Sec.~\\ref{theo}).\nA similar scheme is used to compute the\n$| I=0, T\\approx 1, T_z = \\pm 1 \\rangle$ states in the even-even nuclei.\n\n\nOur studies indicate\\cite{[Sat10a]} that to obtain a fair estimate\nof $\\delta_C$ for $A<40$ and $A>40$ nuclei, one needs to use large\nharmonic oscillator bases consisting of at least $N=10$ and 12\nfull shells, respectively. Even then, the results\nare subject to systematic errors due to the basis cut-off, which can be\nestimated to be $\\sim$10\\%.\nDespite the fact that not all $N=12$ calculations in heavy ($A> 40$) nuclei\nhave yet been completed, and that owing to the\n shape-coexistence effects, there are still some\nambiguities concerning the global minima, our preliminary results point\nto encouraging conclusions. Namely, the mean value of the structure-independent\nstatistical-rate function $\\bar{{\\cal F}}t$,\\cite{[Har05c]} obtained for 12 out of\n13 transitions known empirically with high precision (excluding the\n$^{38}$K$\\rightarrow$$^{38}$Ar case), equals $\\bar{{\\cal F}}t = 3069.4(10)$,\nwhich gives the value of the CKM matrix element equal to $|V_{ud}| = 0.97463(24) $.\nThese values match well those obtained by Towner and Hardy in their\nrecent compilation\\cite{[Tow08]} (see Fig.~\\ref{fig3}).\nBecause of a poor spectroscopic quality of the SV parameterization, the confidence\nlevel\\cite{[Tow10]} of our results is poor. Nevertheless, it should be\nstressed that our method is quantum-mechanically consistent (see\ndiscussion in Refs.\\cite{[Mil08],[Mil09]}) and contains no adjustable free parameters.\n\n\n\n\\section{Symmetry energy}\\label{symm}\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.46\\textwidth,clip]{kazi10_fig4.eps}\n\\caption[T]{\\label{fig4}\nTop: schematic illustration of the isospin-symmetry-breaking\nmechanism in MF of odd-odd $N=Z$ nuclei. Bottom:\n $E_{\\text{sym}}^{\\text{(int)}}$ in odd-odd $N=Z$ nuclei calculated with\nSLy4, SV, SLy4$_L$, and SkM$^*_L$ EDFs. See text for details.}\n\\end{center}\\end{figure}\n\n\n\nThe spontaneous violation of isospin symmetry in all but isoscalar MF configurations of\n$N=Z$ nuclei offers a way to study the nuclear symmetry\nenergy. The idea, which is schematically\nsketched in the upper portion of Fig.~\\ref{fig4}, invokes the mixed-symmetry\nantialigned $|\\bar\\nu \\otimes \\pi\\rangle$ (or $|\\nu \\otimes \\bar\\pi\\rangle$)\nconfiguration in an odd-odd $N=Z$ nucleus. By applying the isospin projection to the\nHF state $|\\bar\\nu \\otimes \\pi\\rangle$, one decomposes it into the isoscalar $T=0$ and isovector\n$T=1$ parts. As argued below, the magnitude of the splitting, $E_{\\text{sym}}^{\\text{(int)}}$,\ndepends on the isovector channel of a given EDF, i.e., its symmetry energy.\n\n\nFor the Skyrme-type EDFs, the symmetry energy\nin the nuclear matter limit can be decomposed as:\\cite{[Sat06w2]}\n\\begin{equation}\\label{nmsym}\na_{\\text{sym}} = \\frac{1}{8}\\varepsilon_{FG} \\left( \\frac{m}{m_0^\\star}\n\\right) +\n\\left[ \\left( \\frac{3\\pi^2}{2}\\right)^{2\/3} C_1^\\tau \\rho^{5\/3}\n+ C_1^\\rho \\rho \\right]\n\\equiv a_{\\text{sym}}^{\\text{(kin)}} + a_{\\text{sym}}^{\\text{(int)}}.\n\\end{equation}\nThe first term in Eq.~(\\ref{nmsym}) is associated with the isoscalar part of the nucleon-nucleon interaction\nand primarily depends on the mean single-particle level spacing at the Fermi energy. This term\nis scaled by the inverse isoscalar effective mass. The second\n(interaction) term, is related to the isovector part of the Skyrme-EDF:\n$\\delta {\\cal H}_{t=1} = C_1^\\rho \\rho_1^2 + C_1^\\tau \\rho_1 \\tau_1$\n(for definitions, see Ref.\\cite{[Ben03]} and references quoted therein).\n\n\nThe value of $E_{\\text{sym}}^{\\text{(int)}}$ appears to be mainly sensitive to the\ninteraction term, which is illustrated in Fig.~\\ref{fig4}.\nIndeed, despite the fact that\nSLy4 and SV EDFs have similar values of $a_{\\text{sym}}$ (equal\nto 32\\,MeV and 32.8\\,MeV, respectively), the\ncorresponding energy splittings $E_{\\text{sym}}^{\\text{(int)}}$ differ substantially.\nThe reduced values of $| E_{\\text{sym}}^{\\text{(int)}} |$ in SV\nare due to its small value of $a_{\\text{sym}}^{\\text{(int)}} = 1.4 $\\,MeV,\\footnote{\nThis small value shows how unphysical are the consequences of\nthe saturation mechanism built into SV through the strong\nmomentum dependence and results in an unphysically low\n isoscalar effective mass $m^*\/m\\approx 0.38$. Although SV has\na relatively reasonable global strength of the symmetry energy $a_{\\text{sym}}$, its physical origin is incorrect.}\n which is\nan order of magnitude smaller than the corresponding SLy4 value:\n $a_{\\text{sym}}^{\\text{(int)}} = 14.4$\\,MeV.\n\n\nAn interesting aspect of our analysis of\n$E_{\\text{sym}}^{\\text{(int)}}$ relates to its dependence on the\ntime-odd terms, which are poorly constrained for Skyrme EDFs.\nTo quantify this dependence, we have performed\ncalculations by using the SLy4$_L$ and SkM$^*_L$ functionals, which\nhave the spin coupling constants adjusted to the Landau parameters.\\cite{[Ben02],[Zdu05]}\nThese EDFs have different values of $a_{\\text{sym}}$ but the same\n$a_{\\text{sym}}^{\\text{(int)}} = 14.4$\\,MeV. The similarity of the calculated energy splittings shown in Fig.~\\ref{fig4} confirms that this quantity\nprimarily depends on the isovector terms of the functional. Moreover, its\nsignificant dependence on the time-odd terms opens up new options for adjusting\nthe corresponding coupling constants to experimental data. This will certainly require the simultaneous restoration of isospin and\nangular-momentum symmetries, as presented in this study.\n\n\n\n\n\\section{Summary}\\label{summary}\n\n\nIn summary, the isospin- and angular-momentum-projected DFT\ncalculations have been performed to estimate the isospin-breaking\ncorrections to $0^+ \\rightarrow 0^+$ Fermi superallowed $\\beta$-decays.\nPreliminary results for the average value of the nucleus-independent $\\bar{\\cal\nF}t = 3069.4(10)$ and the amplitude $|V_{ud}| = 0.97463(24) $ were found\nto be consistent with the recent estimates by Towner and Hardy,\\cite{[Tow08]}\nnotwithstanding a low spectroscopic quality of the Skyrme EDF SV used.\n\nApplicability of the isospin-projected DFT to analyze\nthe nuclear symmetry energy has also been discussed. It has been demonstrated\nthat the isospin projection offers a rather unique opportunity to study the\ninteraction part of the symmetry energy\nin the odd-odd $N=Z$ nuclei and that this quantity is influenced\nby time-odd fields of the energy density functional.\n\n\n\nThis work was supported in part by the Polish Ministry of Science\nunder Contract Nos.~N~N202~328234 and N~N202~239037, Academy of Finland and\nUniversity of Jyv\\\"askyl\\\"a within the FIDIPRO programme, and by the Office of\nNuclear Physics, U.S. Department of Energy under Contract Nos.\nDE-FG02-96ER40963 (University of Tennessee) and\nDE-FC02-09ER41583 (UNEDF SciDAC Collaboration).\nWe acknowledge the CSC - IT Center for Science Ltd, Finland for the\nallocation of computational resources.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfdtu b/data_all_eng_slimpj/shuffled/split2/finalzzfdtu new file mode 100644 index 0000000000000000000000000000000000000000..2abe21de4f9904ef1123a4ea84c92eb5277afb23 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfdtu @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nAlbregtsen \\& Maltby (\\cite{albregtsen1}, cf. Albregtsen \\& Maltby \n\\cite{albregtsen2}, Albregtsen et al. \\cite{albregtsen}) reported a dependence \nof umbral core brightness on the phase of the solar cycle based on 13 sunspots \nobserved at Oslo Solar Observatory. The umbral core is defined as the darkest \npart of the umbra. According to their findings, sunspots present in the early \nsolar cycle are the darkest, while as the cycle progresses spots have increasingly \nbrighter umbrae. Also, the authors did not find any dependence of this relation \non the size or the type of the sunspot. Following this discovery Maltby et al. \n(\\cite{maltby}) proposed three different semi-empirical model atmospheres \nfor the umbral core, corresponding to early, middle and late phases of the \nsolar cycle.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17cm]{6356fig1.eps} \n\\caption{Distribution of analysed sunspots over the ongoing solar cycle 23. \nThe grey solid line shows the International Sunspot Number (left scale). \nThe solid circles show the dates of observation and umbral radii of the analysed \nsunspots with umbral radii between 5 and 15 arc-sec, and the open circles show the \nsunspots with umbral radii less than 5 arc-sec or greater than 15 arc-sec (right-hand scale). \nNo sunspot in 1996 \\& 1997 fulfilled the selection criteria of $\\mu > 0.94$. \nThe hatched area marks the period when contact with SOHO was lost.}\n\\end{figure*}\n \nIn order to explain the umbral brightness variation with solar cycle \ntwo hypotheses have been put forward. Sch$\\mathrm{\\ddot{u}}$ssler \n(\\cite{schussler}) proposed that umbral brightness may be influenced \nby the age of the sub-photospheric flux tubes, whereas Yoshimura \n(\\cite{yoshimura}) suggested that the brightness of the umbra \ndepends on the depth in the convection zone at which the flux \ntube is formed. Confirmation of these results appears important for two reasons. \nFirstly, this is the only strong evidence for a dependence of local properties\nof the magnetic features on the global cycle. E. g., the facular contrast does not \ndepend on solar cycle phase (Ortiz et al. \\cite{ortiz}). Secondly, such a confirmation \nappears timely in the light of the recent paper by Norton \\& Gilman \n(\\cite{norton}), who reported a smooth decrease in umbral brightness from \nearly to mid phase in solar cycle 23, reaching a minimum intensity around \nsolar maximum, after which the umbral brightness increased again, based on the \nanalysis of more than 650 sunspots observed with the MDI instrument. \nThis decrease in brightness contradicts the results of Maltby et al. (\\cite {maltby}). \nAlso, the data used by Norton \\& Gilman (\\cite{norton}) were not corrected for stray \nlight and no sunspot size dependence of the brightness was discussed. \n\nIn this paper we investigate the dependence of umbral core brightness, \nas well as the mean umbral and penumbral brightness on the solar cycle and on the \nsize of the sunspot. In the following \nsection we describe the data selection. In the third \nsection we deal with the data correction for stray light and for the \ninfluence of Zeeman splitting of the nearby Ni~{\\scriptsize I} absorption line on \ncontinuum measurements. In sections 4 and 5 we present our results. We discuss our results \nand compare them with earlier findings in section 6.\n\\section{Data selection}\nContinuum full disk images recorded by the Michelson Doppler Imager (MDI; \nScherrer et al. \\cite{scherrer}) on board the SOHO spacecraft are used in \nthis analysis. The continuum images are obtained from five filtergrams \nobserved around the Ni~{\\scriptsize I} 6768 \\AA\\\/ mid-photospheric \nabsorption line with a spectral pass band of 94 m\\AA\\\/ each. The filtergrams \nare summed in such a way as to obtain the continuum intensity \nis free of Doppler cross talk at the 0.2\\% level. The \nadvantage of this data set is its homogeneity with no seeing fluctuations. \n\nWe selected 234 sunspots observed between March, 1998 and March, 2004. \nThe selected sunspots were located close to the \ndisk centre, i.e. for $\\mu > 0.94$, where $\\mu = \\cos \\theta $ and $\\theta$ is \nthe angle between the line-of-sight and the surface normal. Also, our analysis was \nmostly restricted to regular sunspots, this excluded complex sunspots having very \nirregular shape and multiple umbrae. By looking through the daily images and selecting \nthe sunspot when it was very close to the central meridian, we make sure that a \nparticular sunspot is included only once in our analysis during one solar rotation. \nOut of the selected sunspots, 164 sunspots have an umbral radius between 5\\arcsec\\\/ \nand 15\\arcsec. Even though all the 234 sunspots were used for the study of \nradius-brightness dependence, only the sunspots with umbral radius between 5\\arcsec\\\/ \nto 15\\arcsec\\\/ were used for the study of brightness dependence on solar cycle. \nThis is done in order to facilitate a direct comparison of our results with those \nof Maltby et al. (\\cite{maltby}). The data set covers most of solar cycle 23, \nalthough it does miss a few sunspots in the beginning of the cycle due to our \nselection criteria and the end of the cycle. \n\nFigure 1 shows the distribution of sunspots\\footnote{Sunspot numbers are \ncompiled by the Solar Influences Data Analysis Center \n(http:\/\/sidc.oma.be), Belgium} over cycle 23, \nused for the study of the solar cycle dependence of brightness (filled \ncircles) and those used only to determine the dependence on size (open circles). \nBy restricting the analysis to sunspots near the disk \ncentre, we avoid the suspected effect of centre-to-limb variation \non the umbral brightness (Albregtsen et al. \\cite{albregtsen}). \nThis is shown in Sect. 5. \n\nThe determination of umbral-penumbral and penumbral-quiet \nsun boundaries was carried out using the cumulative histogram \n(Pettauer \\& Brandt \\cite{pettauer}) \nof the intensity of the sunspot brightness and of the immediately surrounding quiet Sun \n(whose average is set to unity). This histogram was computed for 88 \nsymmetric sunspots scattered across the observational period and then \naveraged. This averaged cumulative histogram is shown in Fig. 2. Note that the \nhistogram is computed after stray light correction (see Sect. 3). The quiet \nSun corresponds to the steep rise around normalised intensity unity. The rise \nat around 0.6 corresponds to the penumbra, below that is the umbra. In order to\ndetermine the intensity threshold corresponding to the penumbra-photosphere and \numbra-penumbra boundaries linear fits to the flattest parts of the averaged histogram \nwere computed. The boundaries were chosen at the highest intensity \nat which the linear fit ceases to be a tangent to the histogram. \nThe reason why the penumbra is visible as a reasonably sharp drop, while the \numbra is not, is only partly due to the larger \nrange of intensity found in the umbra. It is mainly due to the large difference in \numbral brightness from spot to spot (see Sect. 4). From the average cumulative histogram\nit was found that values of 0.655 and 0.945 in normalised \nintensity correspond to umbral-penumbral, and penumbral-quiet sun boundaries, respectively. \nThese values were later used to determine the umbral and spot radius. All intensities are\nnormalised to quiet Sun values at roughly the same $\\mu$ value as the sunspot.\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig2.eps}} \n\\caption{Average cumulative intensity histogram used for obtaining the umbral-penumbral \nand penumbral-quiet Sun sunspot boundaries. Dotted lines are linear fits to the flattest \nparts of the histogram. Vertical dotted lines mark the values selected for umbral-penumbral \nand penumbral-quiet Sun boundaries.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig3.eps}} \n\\caption{Off-limb stray light profiles of SoHO\/MDI continuum images for 8 different years.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig4.eps}} \n\\caption{A typical stray-light fit to the observed limb profile. {\\bf Top:} the filled circles show the \nobserved average limb profile and the solid line shows the fit to the observed profile. \n{\\bf Bottom:} the residual after the fit.}\n\\end{figure}\n\\section{Data correction}\nBefore retrieving the brightness, we made a few corrections to the observed data.\nEven though the atmospheric seeing related blurring and distortions are absent, \nMDI continuum images are found to be contaminated by instrumental scattered light. \nBy checking the falloff of intensity just outside the solar limb, it was noticed \nthat the instrumental scattered light increased with the aging of the instrument, \nwhich we carefully correct for.\n\nAlso, continuum measurements in MDI are not carried out in a pure continuum spectral \nband. As described in the previous section, five filtergrams obtained around a \nNi~{\\scriptsize I} absorption line are used to compute the continuum intensities. \nWe investigate the effect of this absorption line in the vicinity of a filter \npass-band on observed continuum intensities. This is especially important in \nsunspots where the line profile changes due to the presence of a magnetic field. \nIn the following subsections we elaborate on these corrections.\n\n\\subsection{Stray light correction}\nIn order to remove the stray light, average radial profiles were obtained \nfrom the observed full-disk MDI continuum images. Figure 3 shows such intensity \nprofiles obtained from an MDI continuum image, averaged over the whole limb, \neach year just outside the solar disk. The gradual increase in scattered light \nwith time is clearly evident in the plot. These profiles were fitted to retrieve \nthe PSF (point-spread function) of the instrument \n(Mart\\' \\i nez Pillet \\cite{valentin}, Walton \\& Preminger \\cite {walton}). The \nradial profiles were generated using the spread function along with \nthe centre-to-limb variation (CLV). A fifth order polynomial is used to describe\nthe CLV. The initial values of the CLV coefficients were taken from \nPierce \\& Slaughter (\\cite {pierce}). The computed profiles were iteratively \nfitted to the observations by adjusting the coefficients of the PSF and CLV. \nA deconvolution of the observed image with the model PSF (generated from the \nfitted coefficients) is carried out to retrieve the original intensity. \n\nFigure 4 shows a typical fit to the observed radial profile and the difference \nbetween observed and fitted profiles. The spikes in the residual are due to the \nsharp change in intensity at the solar limb and are restricted to the points just \noutside and inside the Sun. Excluding these points, the residuals always lie \nbetween $\\pm 0.002$. \n\nWe tested our fitting procedure for the stray light correction \nusing MDI continuum images of the Mercury transit on 7th May, 2003. \nFigure 5 shows the observed and restored intensity for a \ncut across the solar disk through the Mercury image (at $\\mu=0.65$) whose expected full width \nat half maximum is 12\\arcsec, i.e. typical of the diameter of a sunspot umbra. \nIt is evident from the figure that while the intensity in the original cut \nthrough the Mercury image never drops below 16\\%, after the stray light removal \nthe intensity drops to very close to zero. More details of the stray light \ncorrection are given in Appendix A. \\\\\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig5.eps}} \n\\caption{Observed and restored intensity for a \ncut across the solar disk through the Mercury image.\nFilled circles show the observed intensity profile through the centre of the mercury image. \nThe solid line shows the profile after the stray light removal.}\n\\end{figure}\n\n\\subsection{Correction for the Zeeman splitting of the Ni~{\\scriptsize {\\rm \\it I}} line} \nIn MDI, the continuum is computed by measuring intensities at five filter \npositions (designated as F$_{0}$ through F$_{4}$, cf. Scherrer et al. \\cite{scherrer}) \non a spectral band which includes the Ni~{\\scriptsize I} absorption line. \nThe filter F$_{0}$ (whose profile is shown in Fig. 6) gives the main contribution to the\nmeasured continuum, while the intensities recorded through the other filters are used to\ncorrect for intrusions of the Ni~{\\scriptsize I} line into the continuum filter, \nmainly introduced by Doppler shifts. The claimed accuracy for such corrections is 0.2\\% \nof the continuum intensity (Scherrer et al. \\cite{scherrer}), which is perfectly \nadequate for our analysis. \n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig6.eps}} \n\\caption{Synthesised Ni~{\\scriptsize I} line profiles (solid lines). \nFilled circles represent the FTS quiet Sun (QS) spectrum for the same line. The dashed curve \nshows the position of the MDI-F$_{0}$ filter transmission profile.} \n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig7.eps}} \n\\caption{Brightness correction for the contribution of the Ni~{\\scriptsize I} \nline to the MDI-continuum measurements. The solid line shows the normalised continuum \nintensity computed including the influence of the spectral line plotted versus the \ncomputed true continuum for the same set of model atmospheres. Dashed lines are \nfor magnetic field strengths around 10\\% below or above the values listed in Table 1, \nfor a given value of temperature.} \n\\end{figure}\n\nIt is, however, not clear to what extent the Zeeman splitting of the line, as known to be\npresent in sunspots, affects the continuum intensity measurement. In order to quantify \nthis effect in sunspots, we carried out a series of calculations taking various values \nfor magnetic field strength and temperature, which approximately simulate the \nrelationship between the magnetic field and corresponding temperatures found in \nsunspots following Kopp \\& Rabin (\\cite {kopp}), Solanki et al. (\\cite{sol1993}), \nand Mathew et al. (\\cite{mathew}). The exact choice of the field \nstrength-temperature relation is not very critical, as we have found by considering \nalso other combinations (e.g. with lower or with higher field strength for a \ngiven temperature). The magnetic field strength and the corresponding temperature \nalong with other parameters used in the synthesis of Ni~{\\scriptsize I} line profiles \nare given in Table 1. The abundance is given on a logarithmic scale on which the \nhydrogen abundance is 12 and the oscillator strength implies the $\\log (gf)$ value. \n\\begin{table}\\caption{Parameters used for producing Ni~{\\scriptsize I} line profiles}\n\\begin{center}\n\\begin{tabular}{ll}\\hline \\hline\nCentre wavelength \t\t\t& \t6767.768 \\AA \\\\\nAbundance (Ni)\t\t\t\t&\t6.25\\\\\nOscillator strength \t\t\t&\t$-$1.84\\\\\nMacro turbulence \t\t\t&\t1.04 km s$^{-1}$\\\\\nMicro turbulence\t\t\t& 0.13 km s$^{-1}$\\\\\n\\hline\nTemperature (K) & Field strength (G) \\\\\n\\hline\n5750 \t \t&\t0 \\\\\n5500 \t\t&\t1100 \\\\\n5250 \t\t&\t1700 \\\\\n5000 \t\t&\t2250 \\\\\n4750 \t\t&\t2250 \\\\\n4500 \t\t&\t2500 \\\\\n4250 \t\t&\t3000 \\\\\n4000\t\t&\t3500 \\\\\n3750 \t\t&\t4000 \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig8.eps}} \n\\caption{The change in full width (solid line; left scale), equivalent width (dotted line; left scale) and \nline depth (dashed line; right scale) \nwith effective temperature.} \n\\end{figure}\n\nFigure 6 shows the computed Ni~{\\scriptsize I} line profiles along with the position \nof the MDI-F$_{0}$ filter transmission profile. The solid circles are the quiet Sun \nFTS (Fourier Transform Spectrometer) spectrum for the same line \n(Kurucz et al. \\cite{kurucz_at}). Each plotted line profile was computed using a \nmodel atmosphere from Kurucz (\\cite{kurucz}) with effective temperature and \nheight independent vertical magnetic field of the strength listed in Table 1. \nThe Ni abundance is taken from Grevesse \\& Sauval (\\cite {grevesse}) and the atomic \nparameters are obtained from the Kurucz\/NIST\/VALD \\footnote {Kurucz data base - http:\/\/www.pmp.uni-hannover.de \\\\\nNIST - http:\/\/physics.nist.gov\/PhysRefData\\\\\nVALD - http:\/\/www.atro.uu.se\/ $\\tilde{}$~vald } atomic data bases. \nAs a first step, the Ni~{\\scriptsize I} line profile taken from the quiet Sun FTS spectrum \nis fitted using the Kurucz quiet Sun model atmosphere (T$_{\\rm eff} = $ 5750 K) keeping the \nmicro- and macro-turbulence and oscillator strength as free parameters. The values \nobtained for the oscillator strength and micro- and macro-turbulence from the fit \nare maintained when computing all the remaining line profiles with various magnetic \nfield strengths. MDI theoretical filter transmission profiles are created for the \nfive filter positions across the spectral line. The transmitted intensity through \neach filter is computed and combined following Scherrer et al. (\\cite{scherrer}) to \nderive the continuum intensities. In Fig. 7 we plot the `true continuum' intensities \n(i.e., the intensity which would have been measured through the filter if the line \nwere not present in the vicinity of the MDI filter) versus the intensities resulting \nfrom the MDI continuum measurements for different effective temperatures. \nClearly, the MDI continuum measurements in the presence of the Ni~{\\scriptsize I} absorption \nline provide a lower intensity than the real continuum in the sunspots, and the difference \nvaries with the changing field strength and temperature. Naively one would expect this\ndifference to increase for increasing magnetic field strengths, whereas the \nactually found behaviour is more complex. An explanation for this is given \nin Fig. 8. The decreasing temperature reduces the line depth (which is given in units \nrelative to the continuum intensity for the relevant line profile), while the \nequivalent width initially increases before decreasing again with decreasing temperature \nif the field is left unchanged. Increasing field strength leads to enhanced line \nbroadening and a slight increase in equivalent width. The combined influence of both \neffects is plotted in Fig. 8. Note that the width is measured as the wavelength \ndifference between the two outer parts of the line profile at which it drops to $1-d\/2$, \nwhere $d$ is the line depth in units of the continuum intensity. The behaviour seen in Fig. 7 therefore \npartly reflects the dependence of the equivalent width on temperature, but quite significantly also the \nfact that the total intensity absorbed by the line in terms of the {\\it continuum intensity of the quiet Sun}\ndecreases rapidly with decreasing temperature. This is the quality more relevant for our needs, rather than \ntotal absorbed intensity relative to sunspot continuum intensity, which corresponds to the equivalent width.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8.5cm]{6356fi9a.eps} \\includegraphics[width=8.5cm]{6356fi9b.eps}\n\\caption{Cut through two simple sunspots with different effective umbral radii of around \n{\\bf (a)} 15 arc-sec and \n{\\bf (b)} 6 arc-sec.}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5.6666cm]{6356f10a.eps} \n\\includegraphics[width=5.6666cm]{6356f10b.eps}\n\\includegraphics[width=5.6666cm]{6356f10c.eps}\n\\caption{Umbral core intensity versus umbral radius, {\\bf (a)} observed, {\\bf (b)} \ncorrected for stray light, and {\\bf (c)} corrected for stray light and the influence of \nthe Ni~{\\scriptsize I} \\\/line. Here sunspots with umbral radius less \nthan 5 arc-sec and greater than 15 arc-sec are also included.}\n\\end{figure*}\n\nThe two dashed lines in Fig. 7, which are hardly distinguishable from the solid line correspond to using \ndifferent field strength-temperature relations. The upper one is found if we decrease the field strength by around \n10\\% (i.e. the correction is minutely smaller), the lower one for around 10\\% higher field. We use the solid line in \nFig. 7 to correct the observed continuum intensities in sunspots. During the data reduction process \nall the resulting intensities after the stray light removal are replaced by reading \nout the corresponding value from the computed true continuum. \n\n\\section{Brightness-radius relationships} \nBefore discussing the brightness-radius relationship,in Fig. 9 we show cuts through two different sunspots. \nThose allow us to point out various intensity values used in our study. The big sunspot (Fig. 9(a)) has an \neffective umbral radius of around 15\\arcsec\\\/. The horizontal dashed lines indicate the umbral and penumbral \nboundaries, whereas the dotted lines represent mean and minimum umbral intensities as well as mean penumbral \nintensity in this particular sunspot. Similarly, Fig. 9(b) shows a cut through a small sunspot (umbral \nradius $\\approx 6$\\arcsec). \n\n\\subsection{Umbral core and mean intensity versus umbral radius}\nFigure 10 shows the relation between umbral core intensity and umbral \nradius. The umbral radius is computed as the radius of a circle with the \nsame area as the (irregularly shaped) umbra under study. \nThe umbral core intensity is the lowest intensity value found in the particular \numbra (see Fig. 9). \n\nAll intensities are normalised to the average local quiet Sun intensity. \nFigure 10(a) shows this relation for the observed intensity and (b) for the stray light \ncorrected intensities. The influence of the Ni~{\\scriptsize I} \nline on the continuum measurement is still present in this panel. Figure 10(c) is corrected for \nboth the stray light and the effect of the Ni~{\\scriptsize I} line \non the continuum measurement. In all \nthe figures the trend remains the same. It is clear from the figure \nthat the core intensity decreases very strongly with increasing umbral radius. \n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356f11a.eps}\\includegraphics{6356f11b.eps}}\n\\caption{Mean umbral intensity versus umbral radius, {\\bf (a)} observed and, {\\bf (b)} \ncorrected for stray light and the influence of the Ni~{\\scriptsize I} \\\/line. \nHere sunspots with umbral radius less than 5 arc-sec and greater than 15 arc-sec are \nalso included.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356f12a.eps}\\includegraphics{6356f12b.eps}} \n\\caption{Power law fit (solid line) and double linear fit (dash lines) to the {\\bf (a)} \numbral core intensity and, {\\bf (b)} mean umbral intensity. \nHere the filled circles represent bins of 10 spots each.} \n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356f13a.eps}\\includegraphics{6356f13b.eps}}\n\\caption{{\\bf (a)} Mean penumbral intensity versus spot radius, corrected for stray light. \n{\\bf (b)} Linear fit to the binned mean penumbral intensity.}\n\\end{figure}\n\\begin{table*}\n\\caption{Fit parameters for radius-brightness relation}\n\\begin{tabular}{lcccccc} \n\\hline \n\\hline\n Dependence on umbral radius of,& Umbral radius\t& Constant & Exponent & Gradient & $\\sigma $\t& $\\chi^{2}$ \\\\\n\\hline\n\\bf{Power law fit} \t&\t\t&\t\t&\t &\t & & \\\\\numbral core intensity \t&(all) \t& 1.8598\t& $-1.0679$ &\t&0.063 &$3.5 \\times 10^{-3}$\\\\\numbral mean intensity \t&(all) \t\t& 0.8297 & $-0.3052$ & &0.013\t &$1.5 \\times 10^{-4}$ \\\\ \n\\bf {Double linear fit} \t&\t\t&\t\t&\t\t&\t\t& \\\\\numbral core intensity\t &$<$10$''$ \t&0.6515 &\t&$-0.0552$ \t&0.0029 \t&$6.7 \\times 10^{-4}$\\\\\n\t\t\t &$>$10$''$\t&0.2299 &\t&$-0.0094$ \t&0.0027 \t&$4.7 \\times 10^{-5}$\\\\\numbral mean intensity\t &$<$10$''$ \t&0.6536 &\t&$-0.0266$ \t&0.0013 \t&$1.2 \\times 10^{-4}$ \\\\\n\t\t\t &$>$10$''$\t&0.4858 &\t&$-0.0087$ \t&0.0026 \t&$4.3 \\times 10^{-5}$\\\\ \n\\hline \nDependence on spot radius of, \t&\t\t&\t\t&\t\t&\t\t&\\\\\n\\hline\n\\bf {Linear fit} \t\t&\t\t&\t\t&\t\t&\t\t&\\\\\npenumbral mean intensity \t&$>$10$''$\t&0.8561 &\t&$-0.0016$ \t&0.0001 \t&$1.1\\times 10^{-6}$\\\\ \n\\hline \n\\hline\n\\end{tabular}\n\\end{table*}\n\nA steeper decrease \nis found for spots with smaller umbral radius, while for the bigger \nspots a more gentle decrease in umbral core intensity with radius is observed \n(dictated by the fact that umbral intensity has to be positive). \nThis plot emphasises the need to take into account the dependence of the \numbral brightness on the size of the spot when looking for solar cycle variations.\n\nThe mean umbral intensity, plotted in Fig. 11, also shows a similar decrease \nwith increase in umbral radius. The difference between the umbral core \nand mean intensities is smallest for the smallest umbrae and increases with umbral size. \n\nIn order to obtain a relation between the umbral core and mean intensities with umbral radius,\nwe carried out two different fits to the respective corrected umbral intensities after binning \ntogether points with similar sunspot radius, such that each bin contains 10 samples. The dashed lines \nin Figs. 12(a) and (b) show the double linear \nfits to the umbral core and mean intensities, respectively. The individual linear fits are made \nto spots with umbral radii less than 10\\arcsec\\\/ and to those with radii above this limit, respectively.\nThe fit parameters along with errors and normalised $\\chi^{2}$ values are included in Table 2. \nThe solid lines in these figures show the power law fit. The power law fit to the umbral \ncore intensity seems to be a comparatively poor approximation, while the mean umbral brightness \nseems to obey the power law (i.e. it gives a very low $\\chi^2$ for half the number of free\nparameters as the double linear fit). The parameters for the power law fit are also included \nin Table 2. Figure 12 again demonstrates that the difference between the core and mean intensity \nincreases rapidly with umbral radius. It is equally clear that since the umbral core intensity \nvaries by a factor of nearly 6, the mean umbral intensity by a factor of nearly 2 between the smallest \nand the largest umbrae, employing a single value for umbral brightness of all spots is a very poor \napproximation. Such an approximation is often made e.g. for the reconstruction of solar irradiance \n(cf. Unruh et al. \\cite{unruh}, Krivova et al. \\cite{krivova}).\n\n\\subsection{Penumbral mean intensity and spot radius}\nFigure 13(a) shows the relation between mean penumbral intensity and spot radius. \nAn approximate linear relationship is evident for the spots with outer penumbral radius between \n10\\arcsec\\\/ and 30\\arcsec. The outer penumbral radius is the equivalent radius of the whole sunspot \n(including the umbra). The large scatter in mean penumbral intensities for spot \nsizes below 10\\arcsec\\\/ might result from the insufficient resolution of the \nfull disk images or may be due to the fact that the parameters for distinguishing \nbetween umbra and penumbra are possibly not appropriate for small spots. Note the \norder of magnitude smaller range of variation of penumbral contrast than of \numbral contrast. Figure 13(b) shows the linear fit to the mean penumbral brightness, \nafter binning 10 adjacent spots (taking only spots with radius greater than 10$''$). \nThe fit parameters are listed in Table 2. \n\\begin{figure*}\n\\sidecaption\n\\includegraphics[width=12cm]{6356fi14.eps}\n\\caption{Umbral core intensity versus solar cycle, {\\bf (a)} observed, {\\bf (b)} corrected for \nstray light, and {\\bf (c)} corrected for stray light and the influence of Ni~{\\scriptsize I} \\\/line. \nThe intensities are plotted for all spots with umbral radii between \n5 arc-sec and 15 arc-sec. The solid line shows the linear regression \nand the dashed lines represent the $\\pm1 \\sigma$ deviation, due to the uncertainty in the regression \ngradient. The best linear fit is given in the upper left corner.}\n\\end{figure*}\n\n\\section{Solar cycle dependence of the brightness}\nIn Fig. 14 we plot the sunspot umbral core intensity versus time elapsed since the solar cycle \nminimum. September 1996 is taken as the minimum month, the upper axis shows the corresponding \nyear. This plot includes spots with umbral radius between 5\\arcsec\\\/ and 15\\arcsec\\\/ only, in order to be \nconsistent with the work of Albregtsen \\& Maltby (\\cite {albregtsen1}). Figure 14(a) shows the observed \nintensity, (b) the stray light corrected intensity, while Fig. 14(c) shows the intensity corrected for both \nstray light and the influence of the Ni~{\\scriptsize I} \\\/line on the continuum measurements. In all the figures \nthe trend remains the same. As the umbral radius and core brightness are related, the scatter in one quantity \nalso reflects the scatter in the other.\n\nThe solid line shows the linear regression to the brightness, whereas the dotted \nlines indicate the 1$\\sigma$ error in the gradient. A feeble trend of increasing umbral \nbrightness towards the later phase of the solar cycle is observed. This increase is \nwell within the 1$\\sigma$ error bars and statistically insignificant. Table 3 lists the fit \nparameters, including the errors and normalised $\\chi^{2}$ values. All the fit parameters \nare listed for the corrected intensity values. Also, in all the remaining figures we plot \nthe corrected intensities alone. \n\\begin{figure*}\n\\includegraphics[width=12cm]{6356f15a.eps}\\\\\n\\sidecaption\n\\includegraphics[width=12cm]{6356f15b.eps}\n\\caption{Umbral core intensity versus solar cycle plotted for {\\bf (a)} umbral radii \nranging from 5 to 10 arc-sec and, {\\bf (b)} from 10 to 15 arc-sec. \nThe solid lines show the regression fits and the dashed lines the $\\pm1 \\sigma$ deviations \ndue to the uncertainty in the regression gradient.}\n\\end{figure*}\n\\begin{table*}\\caption{Fit parameters for solar cycle dependence}\n\\begin{tabular}{lcccccc}\\hline \\hline\nSolar cycle dependence of, \t&Umbral radius\t\t&Constant \t& Gradient\t& $\\sigma $ \t& Gradient$\/ \\sigma $\t& $\\chi^{2} $ \\\\ \\hline\numbral core intensity \t\t& (all)\t\t\t&0.33605\t&$-0.00785$\t&0.00649\t&$-1.2095$\t\t&0.01901\\\\\n\t\t\t\t&5$''$ - 15$''$ \t&0.22188 \t&$+0.00358$ \t&0.00577 \t&$+0.6205$\t\t&0.01015\\\\ \n\t\t\t\t&5$''$ - 10$''$ \t&0.25001\t&$+0.00227$ \t&0.00582 \t&$+0.3900$\t\t&0.00841\\\\\n\t\t\t\t&10$''$ - 15$''$\t&0.12560 \t&$-0.00269$ \t&0.00547 \t&$-0.4918$\t\t&0.00158\\\\\n\t\t\t\t&(all, northern hemisphere)&0.33957\t&$-0.01071$\t&0.00622\t&$-1.7218$\t\t&0.00085\\\\\n\t\t\t\t&(all, southern hemisphere)&0.33062\t&$-0.00310$\t&0.00874\t&$-0.3547$\t\t&0.00164\\\\\\hline\numbral mean intensity \t\t&5$''$ - 15$''$ \t&0.44346 \t&$+0.00206$ \t&0.00312 \t&$+0.6603$\t\t&0.00296\\\\ \n\t\t\t\t&5$''$ - 10$''$ \t&0.45782\t&$+0.00168$ \t&0.00294 \t&$+0.5714$\t\t&0.00215\\\\\n\t\t\t\t&10$''$ - 15$''$\t&0.39609 \t&$-0.00333$ \t&0.00513 \t&$-0.6491$\t\t&0.00139\\\\\\hline\npenumbral mean intensity \t&5$''$ - 15$''$ \t&0.82974\t&$-0.00009$ \t&0.00049 \t&$-0.1837$\t\t&0.00007\\\\ \n\t\t\t\t&5$''$ - 10$''$ \t&0.83324\t&$-0.00034$ \t&0.00042 \t&$-0.8095$\t\t&0.00004\\\\\n\t\t\t\t&10$''$ - 15$''$\t&0.81723\t&$-0.00024$ \t&0.00074 \t&$-0.3243$\t\t&0.00003\\\\\\hline\numbral radius \t\t & (all)\t\t\t&5.79088\t&$+0.19029$\t&0.13549\t&$+1.4045$\t\t&8.29028\\\\\n\t\t\t\t&5$''$ - 15$''$ \t&8.06675\t&$-0.07457$ \t&0.12525 \t&$-0.5954$\t\t&4.77502\\\\\n\t\t\t\t&5$''$ - 10$''$\t \t&7.06682\t&$-0.01697$ \t&0.08311 \t&$-0.2042$\t\t&1.71632\\\\\n\t\t\t\t&10$''$ - 15$''$ \t&11.5536\t&$+0.06823$ \t&0.20443 \t&$+0.3338$\t\t&2.20379\\\\\n\t\t\t\t&(all, northern hemisphere)&5.75164\t&$+0.21357$\t&0.11716\t&$+1.8229$\t\t&0.30162\\\\\n\t\t\t\t&(all, southern hemisphere)&5.94171\t&$+0.12221$\t&0.19719\t&$+0.6198$\t\t&0.83724\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table*}\n\\begin{figure*}\n\\includegraphics[width=12cm]{6356f16a.eps}\\\\\n\\sidecaption\n\\includegraphics[width=12cm]{6356f16b.eps}\n\\caption{Solar cycle dependence of umbral radius for {\\bf (a)} sunspots with radii between \n5 and 10 arc-sec and {\\bf (b)} with radii between 10 and 15 arc-sec.}\n\\end{figure*} \n\\begin{figure*}\n\\sidecaption\n\\includegraphics[width=12cm]{6356fi17.eps}\n\\caption{$\\mu$ versus time since activity minimum for all spots with umbral radii between \n5 arc-sec and 15 arc-sec. The solid line shows the linear regression \nand the dashed lines the $\\pm1 \\sigma$ deviation.}\n\\end{figure*}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fi18.eps}}\n\\caption{Umbral core intensity versus $\\mu$ for all spots with umbral radii between \n5 arc-sec and 15 arc-sec. \n}\n\\end{figure} \n\\begin{figure*}\n\\sidecaption\n\\includegraphics[width=12cm]{6356fi19.eps}\n\\caption{Umbral core intensity {\\bf (a)} and umbral radius {\\bf(b)} versus time since solar cycle minimum \nfor Northern (filled circles) and Southern (asterisks) hemispheres, for all observed sunspots. \nEach plotted symbol represents an average over 10 sunspots. The solid and dashed lines show the linear regression \nfits for Northern and Southern hemispheres, respectively. The best linear fits are given in the lower \nleft corner.}\n\\end{figure*}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fi20.eps}}\n\\caption{Umbral core intensity versus umbral radius. Different symbols represent spots observed \nin ascending, maximum and descending phases of solar cycle. The dates indicate the period of observation.}\n\\end{figure} \nIn Figs. 15(a) and (b) we display the umbral \ncore intensity for two different umbral size ranges, i.e. for spots with umbral \nradii in the range 5\\arcsec\\\/ - 10\\arcsec\\\/ and in 10\\arcsec\\\/ - 15\\arcsec, respectively. \nThe trend seen here is opposite for small and large spots, but is insignificant \nin both cases (see column Gradient\/$\\sigma$ in Table 3). If we plot all the sunspots, \nirrespective of radius, the results are not significantly different.\n \nIn Figs. 16(a) and (b) we plot the dependence on the time elapsed from \nactivity minimum, of the analysed sunspot umbral radius, separately for the \nsmall (5\\arcsec\\\/ - 10\\arcsec) and the large (10\\arcsec\\\/ - 15\\arcsec) spots, respectively. \nThe linear regression are overplotted. \nThe mean umbral radius of the analysed spots \nbetween 5\\arcsec\\\/ - 10\\arcsec\\\/ slightly decreases with time, whereas spots \nwith radius between 10\\arcsec\\\/ - 15\\arcsec\\\/ show the opposite trend. Similarly in \nFig. 1 the umbral radii of all studied spots are plotted. The regression parameters\nfor the full sample of spots (5\\arcsec\\\/ - 10\\arcsec) is given in Table 3. Although none of these\ntrends is statistically significant, they are opposite to the trends (also not \nstatistically significant) shown by the umbral brightness of these spots. This \nis completely consistent with the dependence of brightness on umbral radius shown \nin Figs. 10 - 12.\n\nAnother bias can be introduced by the fact that the sunspot latitude systematically \ndecrease over a solar cycle. We have to a certain extent reduced this effect by \nconsidering only sunspots at $\\mu > 0.94$. Figure 17 shows the average $\\mu$ of the \nanalysed sunspots, as expected, this displays an increase over the cycle. \nIn order to judge whether this introduces a bias into the cycle phase dependence of sunspot \nbrightness we plot in Fig. 18 umbral core brightness (i.e. contrast to local quiet Sun) \nversus $\\mu$ (for $0.94 < \\mu < 1$). We did not find any significant variation in the umbral \ncore brightness with $\\mu$.\n\nIn order to check for an asymmetry in umbral brightness between the northern and southern \nhemispheres as reported by Norton \\& Gilman (\\cite{norton}), in Fig. 19 we plot umbral core \nbrightness separately for northern and southern hemispheres. In this plot we used corrected \nintensities for all the observed spots. The intensities are binned and each bin contains 10 samples. \nThe linear regression fit provides a slightly higher gradient in the northern hemisphere. \nBut this can be well explained by the increase in umbral \nradius of the spots with the cycle phase (Fig. 19(b)). It should be noted that Norton \\& Gilman observed a significant \numbral brightness difference between the northern and southern hemisphere during the onset of cycle 23. \nDue to the restriction of $0.94 < \\mu $ in our selection criteria, we have analysed only few spots during \nthis period and hence cannot comment on that result.\n \n\nIn Table 3 we also list the parameters of the linear regressions to mean umbral and \npenumbral intensities versus time. None of the gradients is significant at even the \n1$\\sigma$ level. Also, the signs of the gradients of all umbral core and mean intensity \nsamples are opposite to those of the umbral radius of the corresponding sample, \nsuggesting that even any small gradient in the umbral brightness is due to a small \nbias in the umbral size with time. Hence we find no evidence at all for a change \nin sunspot brightness over the solar cycle. \n\nIn order to test whether the dependence of umbral core brightness on umbral radius in \nSect. 4.1 is itself dependent on solar cycle phase we plot in Fig. 20 the umbral core brightness \nversus radius but now for three different phases of the cycle. Asterisks, filled circles and \ndiamond symbols represent the spots observed in ascending, maximum and descending phases of solar \ncycle, respectively. As can be clearly seen there is no difference between the different phases. \nThis demonstrates that there is no cross-talk between cycle phase dependence and umbral radius \ndependence of umbral brightness. \n \n\\section{Discussion}\nWith a large sample of sunspots, we have tested \nif the umbral core brightness, the umbral average brightness, or the penumbral \nbrightness depend on solar cycle phase. In addition to this, we \nstudied the dependence of brightness on sunspot size.\n\nEarlier continuum observations suggested that large sunspots are \ndarker than smaller sunspots (Bray \\& Loughhead \\cite{bray}). But most \nof such observations were barely corrected for stray light (Zwaan \\cite{zwaan}). Subsequent \nobservations which were corrected for stray light showed no significant \ndependence of umbral core brightness on spot size (Albregtsen \\& Maltby \n\\cite{albregtsen2}). More recent observations however, reveal that even after stray-light \ncorrection a size dependence remains. Thus, Kopp \\& Rabin (\\cite{kopp}) \npresent observations at 1.56 ${\\rm {\\mu m}}$ that show clear evidence for the size dependence \nof umbral brightness. Also, results from two sunspots observed at the same \nwavelength combined with the Kopp \\& Rabin data confirm and strengthen the \nlinear dependence of brightness on sunspot umbral size (Solanki \\cite{solanki1997}, \nSolanki et al. \\cite{solanki1}, Kopp \\& Rabin \\cite{kopp}, R$\\mathrm{\\ddot{u}}$edi et al. \\cite{ruedi}).\n\nMart\\' \\i nez Pillet \\& V$\\acute{\\rm a}$zquez (\\cite{martinez}) confirmed the Kopp \\& Rabin result \nbased on the analysis of 7 sunspots. Collados et al. (\\cite{collados}) found from the inversion \nof Stokes profiles obtained in 3 sunspot umbrae that small umbrae are distinctly \nhotter than large umbrae. One disadvantage of all these studies is that each is restricted to a\nrelatively small number of sunspots. Another is that each sunspot was observed under \ndifferent seeing conditions, so that the level of stray light varied in an unsystematic\nmanner. Both shortcomings are addressed in the present paper. \n\nIn our analysis we found a clear dependence of umbral core brightness on \numbral size. Since we correct for the very slowly varying stray light and the MDI specific \nproblem of cross-talk of the spectral line into the continuum, as described \nin Sect. 3 and the appendix, our results are basically free from stray light contamination. \nThis is particularly true for umbral core intensities as we could show using the Mercury transit data.\nFor mean intensities some residual remains (see Appendix A).\nAlso, we have a relatively large sample of sunspots to support our results. \nWe carried out and compared two different fits to the umbral brightness-size \ndependence for the analysed spots, a double linear fit for two different umbral \nsize ranges and a power law fit to the entire data set. A similar analysis of \numbral brightness and diameter carried out at a wavelength of 1.56 $\\rm \\mu m$ \n(Solanki \\cite{solanki1997}) shows a smaller gradient than what we obtained in our \nwork. Umbral brightness for 9 spots, with umbral diameter ranging from 5\\arcsec\\\/ \nto 35\\arcsec\\\/ were included in the above study. From a linear fit, \na gradient of around $-0.012\/\\arcsec$ is obtained for the umbral brightness, \nwhereas in our case a much higher gradient of $-0.04\/\\arcsec$ is found. \nThis is not surprising since the brightness at 1.56 $\\rm \\mu m$ reacts much more weakly\nto a given temperature change than at 677 $\\rm {nm}$.\n \nAlbregtsen \\& Maltby (\\cite{albregtsen1}) reported a variation of umbral \ncore brightness with solar cycle by analysing 13 sunspots. The results \nwere mainly presented for the observations done at a wavelength of 1.67$\\rm {\\mu m}$. \nThey found an increase in umbral core intensity by as much as 0.15 from early to \nlate phase of the solar cycle. They also found no dependence of umbral brightness \non other sunspot parameters, such as size and type of the spot. In a later \npaper Maltby et al. (\\cite{maltby}) detailed this variation for a range of wavelengths \nstarting from 0.38 - 2.35 $\\rm {\\mu m}$. The nearest wavelength to our observations \n(i.e. 0.669 $\\rm {\\mu m}$) shows a variation of around 0.072 in umbral core intensity from \nearly to late phase, this corresponds to 0.0065 umbral intensity variation per year. \nBased on these findings they presented three different umbral \ncore model atmospheres for sunspots present in the early, mid and late phase of \nthe solar cycle. In a recent study carried out using MDI data Norton \\& Gilman \n(\\cite {norton}) find a relatively smooth decrease in the umbral brightness from activity minimum \nto maximum for Northern hemisphere and no distinct trend for the southern hemisphere. \nThe decrease in umbral brightness with solar cycle they found is opposite to the results \nof Maltby et al. (\\cite{maltby}). \n\nIn our analysis we found a very feeble, statistically insignificant dependence of \numbral brightness on solar cycle (i.e. any change remains well within the error bars).\nThe linear fit to umbral core brightness is given by the following \nequation,\n\\begin{eqnarray}\nI_{uc}=0.222+(0.004\\pm 0.006)\\times t\n\\end{eqnarray}\nwhere $t$ is the time elapsed from the minimum in units of years. \nAll sunspots within 5\\arcsec\\\/ - 15\\arcsec\\\/ umbral radius are included in the regression. \nIt is striking that the 1$\\sigma$ uncertainty in the gradient obtained in our \nanalysis is approximately equal to the trend found by Maltby et al. (\\cite{maltby}).\nTherefore, either the MDI wavelength is less suitable than the 1.56 $\\rm \\mu m$ wavelength \nband employed by Maltby and co-workers, or there is a selection bias affecting their results. \nIndeed, the change in umbral core intensity over the solar cycle reported by \nMaltby et al. (\\cite{maltby}), 0.072, is only $1\/3$ the umbral core intensity difference \nbetween spots with umbral radii of 5\\arcsec\\\/ and 15\\arcsec\\\/ found here and is 0.6 times the \nintensity difference between such spots at 1.56 $\\rm \\mu m$. Consequently, selection biases, \nwhich often afflict small samples, can introduce an artificial \ntrend of the correct magnitude over an activity cycle. \n\nIn order to reduce the effect of size dependence on the above relation, we grouped the sunspots \ninto two umbral radii bins. The linear regressions to these groups are given by the following \nequations,\n\\begin{eqnarray}\nI_{uc}= \\{\n\\begin{array}{ll}\n0.250+(0.002\\pm0.006)\\times t &{\\rm for} ~~5'' - 10''\\\\\n0.126-(0.003\\pm0.005)\\times t &{\\rm for} ~~10''- 15'' \n\\end{array}\n\\end{eqnarray}\nFor the group of spots with umbral radii between 5\\arcsec\\\/ - 10\\arcsec\\\/ the linear regression \nfit gives an increase in umbral brightness with increasing phase of the solar cycle, \nbut again the change is within the error bars. For the spots with umbral radii \n10\\arcsec\\\/ - 15\\arcsec\\\/ the opposite trend is found. We compared these results with the \nvariation of the sizes of the \nanalysed sunspots over the solar cycle. It turns out that for all three samples \n(5\\arcsec\\\/- 10\\arcsec\\\/, 10\\arcsec\\\/-15\\arcsec\\\/, 5\\arcsec\\\/-15\\arcsec\\\/) the average radii \nshow the opposite trend to the intensity. This suggests that at least part of any trend in \nbrightness over the solar cycle is due to a corresponding (opposite) trend in umbral radii.\n\nFrom these results it is evident that the \ndependence of the brightness of the spot on size is an important parameter to \nbe considered when the umbral brightness variation with solar cycle is studied. We believe that \nwithout showing the time dependence of the average area or size of the sunspot umbrae in the employed \nsamples, any studies of sunspot brightness (or even field strength) evolution over time are of limited \nvalue. Although there is no systematic variation in the relative distribution of \numbral areas with solar cycle (Bogdan et al. \\cite {bogdan}) in a smaller sunspot sample\na trend may be introduced by limited statistics. It would therefore be of great value to determine\nthe areas of the umbrae studied by Penn \\& Livingston (\\cite{penn}), who find a steady decrease of the \nmaximum umbral field strength over the last 7 years, since the field strength is related to brightness \n(Maltby \\cite{maltby2}, Kopp \\& Rabin \\cite{kopp}, Mart\\' \\i nez Pillet \\& V$\\acute{\\rm a}$zquez \\cite{martinez}, \nSolanki et al. \\cite{sol1993}, Mathew et al. \\cite{mathew2004}, Livingston \\cite{livingston}), \nwhich depends on size (Sect. 4.1). Also, our results imply that models of the umbral \natmosphere (e.g. Avrett \\cite{avrett}, Maltby et al. \\cite{maltby}, Caccin et al. \\cite{caccin}, \nSeverino et al. \\cite{severino}, Fontenla et al. \\cite{fontenla}) should always indicate the spot \nsize to which they refer.\n\n\nChapman et al. (\\cite{chapman})\nreported photometric observations of sunspot groups, that show considerable variation in their \nmean contrast. This could be due to umbra\/penumbra area ratio change, or due to intrinsic \nbrightness change. Our results suggest that, at least in part the later is the reason. This has \nimplications for irradiance reconstructions, particularly those using separate atmospheric \ncomponents for umbrae and penumbrae (e.g. Fligge et al. \\cite{fligge}, Krivova et al. \\cite{krivova}).\n\nThe result is also of importance for the physics of sunspots, since an explanation \nmust be found why a smaller heat flux is transported through the umbrae of larger sunspots. \nE. g. in Parker's (\\cite{parker}) spaghetti model it would imply that either the filamentation \nof the subsurface field is less efficient under larger umbrae, or that the energy flux transported \nbetween the filaments is smaller for larger filaments. One factor which probably plays a role is that\ndarker sunspots have higher field strengths (Maltby \\cite{maltby2}, Kopp \\& Rabin \\cite{kopp}, \nLivingston \\cite{livingston}) which are more efficient at blocking magnetoconvection. \nThis also implies that larger sunspots have stronger fields. \n \nThe only clear dependence we have found in our sample of sunspots is between \numbral intensity and radius. However, even this relationship shows considerable \nscatter, whose origin is not clear. We list some possibilities below:\n\\begin{enumerate}\n\\item dependence of the scattered light correction on the shape of the umbra \n(e.g. very elongated versus circular);\n\\item dependence of intrinsic brightness of umbrae on shape and complexity;\n\\item dependence of brightness on age;\n\\item (small) dependence of brightness on phase of the solar cycle;\n\\item CLV of umbral contrast (which is small according to Fig. 18).\n\\end{enumerate}\nSeparating clearly between these possibilities is beyond the scope of \nthe current paper. Our analysis is restricted to regular sunspots with single umbrae. \nComplex spots may in principle show a different behaviour. \n\\section{Concluding remarks}\nIn this paper we present the analysis of MDI continuum sunspot images aimed at \ndetecting umbral core brightness variation with solar cycle. We analysed a total of \n234 sunspots of which 164 sunspots have an umbral radius lying between 5\\arcsec\\\/ - 15\\arcsec. \nCareful corrections for stray light and the Zeeman splitting of the nearby \nNi~{\\scriptsize I} \\\/line on measured continuum intensities have been made.\nWe derive the following conclusions from our analysis. \n\\begin{itemize}\n\\item The umbral core and mean brightness decreases substantially with increasing umbral radius. \n\\item The mean penumbral intensity is also reduced with increased spot size, but by a small amount.\n\\item No significant variation in umbral core, umbral mean and mean penumbral intensities is \nfound with solar cycle.\n\\item The insignificant variation with solar cycle of the umbral intensity could be at least \npartly be explained by the dependence of the analysed spot size on \nsolar cycle.\n\\end{itemize}\n\\begin{acknowledgements}\nWe wish to express our thanks to SOHO\/MDI team for providing the full-disk continuum \nintensity images. Thanks are also due to Dr. Andreas Lagg for providing the updated \ncode for computing the line profiles and the referee Aimee Norton for her useful suggestions. \nThis work was partly supported by the Deutsche Forschungsgemeinschaft, DFG project number \nSO~711\/1-1. Funding by the Spanish National Space Program (PNE) under project ESP2003-07735 \nis gratefully acknowledged. \n\\end{acknowledgements}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\nState complexity is one of the fundamental topics in automata\ntheory. It is important from both theoretical aspect and\nimplications in automata applications, because the state complexity\nof an operation gives an upper bound of both time and space\ncomplexity of the operation. For example, programmers should know\nthe largest possible number of states that would be generated before\nthey perform an operation in an application, since they need to\nallocate enough space for the computation and make an estimate of\nthe time it takes.\n\nThe research on state complexity can be recalled to\n1950's~\\cite{RaSc59}. However, most results on state complexity came\nout after\n1990~\\cite{CCSY99,CaSaYu02,DaDoSa08,Domaratzki02,HoKu02,JiJiSz05,Jriaskova05,JiOk05,PiSh2002,SaWoYu04,Yu01,YuZhSa94}.\nTheir research focused on individual operations, e.g. union,\nintersection, star, catenation, reversal, etc, until A. Salomaa, K.\nSalomaa and S. Yu initiated the study of state complexities of\ncombined operations in 2007~\\cite{SaSaYu07}. In the following three\nyears, many papers were published on this\ntopic~\\cite{CGKY10-cat-sr,CGKY10-cat-ui,DoOk09,EsGaLiYu09,GaSaYu08,GaYu09,JiOk07,LiMaSaYu08}.\n\nPeople are interested in state complexities of combined operations\nnot only because it is a relatively new research direction but also\nbecause its importance in practice. For example, several operations\nare often applied in a certain order on languages in searching and\nlanguage processing. If we simply use the mathematical composition\nof the state complexities of individual participating operations, we\nmay get a very huge value which is far greater than the exact state\ncomplexity of the combined operation, because the resulting\nlanguages of the worst case of one operation may not be among the\nworst case input languages of the next\noperation~\\cite{GaSaYu08,JiOk07,LiMaSaYu08,SaSaYu07}. Although\ncomputer technology is developing fast, time and space should still\nbe used efficiently. Thus, state complexities of combined operations\nare at least as important as those of individual operations.\n\n\nIn~\\cite{SaSaYu07}, two combined operations were investigated:\n$(L(M)\\cup L(N))^*$ and $(L(M)\\cap L(N))^*$, where $M$ and $N$ are\n$m$-state and $n$-state DFAs, respectively. In~\\cite{LiMaSaYu08},\nBoolean operations combined with reversal were studied, including:\n$(L(M)\\cup L(N))^R$ and $(L(M)\\cap L(N))^R$. One natural question is\nwhat are the state complexities of these combined operations if we\nexchanged the orders of the composed individual operations. For\nexample, we perform star or reversal first and then perform union or\nintersection. Thus, in this paper, we investigate four particular\ncombined operations: $L(M)^*\\cup L(N)$, $L(M)^*\\cap L(N)$,\n$L(M)^R\\cup L(N)$ and $L(M)^R\\cap L(N)$.\n\nIt has been shown in~\\cite{YuZhSa94} that, (1) the state\ncomplexities of the union and intersection of an $m$-state DFA\nlanguage and an $n$-state DFA language are both $mn$, (2) the state\ncomplexity of star of a $k$-state DFA language is $\\frac{3}{4}2^k$,\nand (3), the state complexity of reversal of an $l$-state DFA\nlanguage is $2^l$. In this paper, we obtain the state complexities\nof $L(M)^*\\cup L(N)$, $L(M)^*\\cap L(N)$, $L(M)^R\\cup L(N)$ and\n$L(M)^R\\cap L(N)$ and show that they are all less than the\nmathematical compositions of individual state complexities for\n$m,n\\ge 2$.\n\nWe prove that the state complexity of $L(M)^*\\cup L(N)$ is\n$\\frac{3}{4}2^m\\cdot n-n+1$ for $m$, $n\\ge 2$ which is much less\nthan the known state complexity of $(L(M)\\cup L(N))^*$\n(\\cite{SaSaYu07}). We obtain that the state complexity of\n$L(M)^*\\cap L(N)$ is also $\\frac{3}{4}2^m\\cdot n-n+1$ for $m$, $n\\ge\n2$ whereas the state complexity of $(L(M)\\cap L(N))^*$ has been\nproved to be $\\frac{3}{4}2^{mn}$, the mathematical compositions of\nindividual state complexities (\\cite{SaSaYu07}). For $L(M)^R\\cup\nL(N)$ and $L(M)^R\\cap L(N)$, we prove both of their state\ncomplexities to be $2^m\\cdot n-n+1$ for $m$, $n\\ge 2$ while the\nstate complexities of $(L(M)\\cup L(N))^R$ and $(L(M)\\cap L(N))^R$\nare both $2^{m+n}-2^m-2^n+2$ (\\cite{LiMaSaYu08}).\n\nIn the next section, we introduce the basic notations and\ndefinitions used in this paper. In\nSections~\\ref{star-union},~\\ref{star-intersection},~\\ref{reversal-union}\nand~\\ref{reversal-intersection}, we investigate the state\ncomplexities of $L(M)^*\\cup L(N)$, $L(M)^*\\cap L(N)$, $L(M)^R\\cup\nL(N)$ and $L(M)^R\\cap L(N)$, respectively. In\nSection~\\ref{sec:conclusion}, we conclude the paper .\n\n\n\n\\section{Preliminaries}\nAn alphabet $\\Sigma$ is a finite set of letters. A word $w \\in\n\\Sigma^*$ is a sequence of letters in $\\Sigma$, and the empty word,\ndenoted by $\\varepsilon$, is the word of length 0.\n\nA {\\it deterministic finite automaton} (DFA) is usually denoted by a\n5-tuple $A = (Q, \\Sigma, \\delta, s, F)$, where $Q$ is the finite and\nnonempty set of states, $\\Sigma$ is the finite and nonempty set of\ninput symbols, $\\delta: Q\\times\\Sigma \\rightarrow Q$ is the state\ntransition function, $s\\in Q$ is the initial state, and $F\\subseteq\nQ$ is the set of final states. A DFA is said to be {\\it complete} if\n$\\delta$ is a total function. Complete DFAs are the basic model for\nconsidering state complexity. Without specific mentioning, all DFAs\nare assumed to be complete in this paper. We extend $\\delta$ to $Q\n\\times \\Sigma^* \\rightarrow Q$ in the usual way. Then this automaton\naccepts a word $w \\in \\Sigma^*$ if $\\delta(s,w) \\cap F \\neq\n\\emptyset$. Two states in a DFA are said to be {\\it equivalent} if\nand only if for every word $w \\in \\Sigma^*$, if $A$ is started in\neither state with $w$ as input, it either accepts in both cases or\nrejects in both cases. The language accepted by a DFA $A$ is denoted\nby $L(A)$. A language is accepted by many DFAs but there is only one\nessentially unique {\\it minimal} DFA for the language which has the\nminimum number of states.\n\nA {\\it non-deterministic finite automaton} (NFA) is also denoted by\na 5-tuple $B = (Q, \\Sigma, \\delta, s, F)$, where $Q$, $\\Sigma$, $s$,\nand $F$ are defined the same way as in a DFA and $\\delta:\nQ\\times\\Sigma\\rightarrow 2^Q$ maps a pair consisting of a state and\nan input symbol into a set of states rather than a single state. An\nNFA may have multiple initial states, in which case an NFA is\ndenoted $(Q, \\Sigma, \\delta, S, F)$ where $S$ is the set of initial\nstates. A language $L$ is accepted by an NFA if and only if $L$ is\naccepted by a DFA, and such a language is called a {\\it regular\nlanguage}. Two finite automata are said to be equivalent if they\naccepts the same regular language. An NFA can always be transformed\ninto an equivalent DFA by performing subset construction. The reader\nmay refer to~\\cite{HoMoUl01,Yu97} for more details about regular\nlanguages and automata theory.\n\nThe {\\it state complexity} of a regular language $L$ is the number\nof states of the minimal, complete DFA accepting $L$. The state\ncomplexity of a class of regular languages is the worst among the\nstate complexities of all the languages in the class. The state\ncomplexity of an operation on regular languages is the state\ncomplexity of the resulting languages from the operation. For\nexample, we say that the state complexity of union of an $m$-state\nDFA language and an $n$-state DFA language is $mn$. This implies\nthat the largest number of states of all the minimal, complete DFAs\nthat accept the union of an $m$-state DFA language and an $n$-state\nDFA language,\nis $mn$, and such languages exist. Thus, state complexity is a\nworst-case complexity.\n\n\n\n\\section{State complexity of $L_1^*\\cup L_2$}\\label{star-union}\nWe first consider the state complexity of $L_1^*\\cup L_2$, where\n$L_1$ and $L_2$ are regular languages accepted by $m$-state and\n$n$-state DFAs, respectively. It has been proved that the state\ncomplexity of $L_1^*$ is $\\frac{3}{4}2^m$ and the state complexity\nof $L_1\\cup L_2$ is $mn$~\\cite{Maslov70,YuZhSa94}. The mathematical\ncomposition of them is $\\frac{3}{4}2^m\\cdot n$. In the following, we\nshow that this upper bound can be lower.\n\n\\begin{theorem}\n\\label{star union upper bound}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$ such that\n$|F_M-\\{ s_M \\}|=k\\geq 1$, $m\\geq 2$, $n\\geq 1$, there exists a DFA\nof at most $(2^{m-1}+2^{m-k-1})\\cdot n-n+1$ states that accepts\n$L(M)^*\\cup L(N)$.\n\\end{theorem}\n\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ be a\ncomplete DFA of $m$ states. Denote $|F_M-\\{ s_M \\}|$ by $F_0$. Then\n$F_0=k\\geq 1$ Let $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$ be another\ncomplete DFA of $n$ states. Let DFA $M'=(Q_{M'},\\Sigma , \\delta_{M'}\n, s_{M'}, F_{M'})$ where\n\\begin{eqnarray*}\n& & s_{M'} \\notin Q_M\\mbox{ is a new start state,}\\\\\n& & Q_{M'} = \\{s_{M'}\\}\\cup \\{P\\mid P\\subseteq (Q_M-F_0)\\mbox{ \\& } P\\neq \\emptyset \\} \\\\\n& & \\qquad \\cup \\{R\\mid R\\subseteq Q_M \\mbox{ \\& } s_M\\in R \\mbox{ \\& }R\\cap F_0\\neq \\emptyset \\},\\\\\n& & \\delta_{M'}(s_{M'}, a)= \\{\\delta_M(s_M, a)\\mbox{ for any $a\\in \\Sigma$} \\},\\\\\n& & \\delta_{M'}(R, a)= \\{\\delta_M(R, a)\\}\\mbox{ for $R\\subseteq Q_M$ and $a\\in \\Sigma$ if $\\delta_M(R, a)\\cap F_0=\\emptyset$},\\\\\n& & \\delta_{M'}(R, a)= \\{ \\delta_M(R, a)\\}\\cup \\{s_M\\}\\mbox{ otherwise}, \\\\\n& & F_{M'}= \\{s_{M'}\\}\\cup\\{R\\mid R\\subseteq Q_M \\mbox{ \\& } R\\cap\nF_M\\neq \\emptyset \\}.\n\\end{eqnarray*}\n\nIt is clear that $M'$ accepts $L(M)^*$. In the second term of the\nunion for $Q_{M'}$ there are $2^{m-k}-1$ states. And in the third\nterm, there are $(2^k-1)2^{m-k-1}$ states. So $M'$ has\n$2^{m-1}+2^{m-k-1}$ states in total. Now we construct another DFA\n$A=(Q,\\Sigma , \\delta , s, F)$ where\n\\begin{eqnarray*}\n& & s=\\langle s_{M'},s_N \\rangle,\\\\\n& & Q = \\{\\langle i,j \\rangle \\mid i\\in Q_{M'}-\\{s_{M'}\\},j\\in Q_N\\}\\cup \\{s \\}, \\\\\n& & \\delta(\\langle i,j \\rangle, a)= \\langle \\delta_{M'}(i, a),\\delta_N(j, a) \\rangle \\mbox{, $\\langle i,j \\rangle \\in Q$, $a\\in \\Sigma$},\\\\\n& & F= \\{\\langle i,j \\rangle \\mid i\\in F_{M'}\\mbox{ or }j\\in F_N \\}.\n\\end{eqnarray*}\nWe can see that $$L(A)=L(M')\\cup L(N)=L(M)^*\\cup L(N).$$ Note\n$\\langle s_{M'},j \\rangle \\notin Q$, for $j\\in Q_N-\\{s_N\\}$, because\nthere is no transition going into $s_{M'}$ in DFA $M'$. So there are\nat least $n-1$ states in $Q$ are not reachable. Thus, the number of states of minimal DFA accepting $L(M)^*\\cup L(N)$ is no more than\\\\\n$$|Q|=(2^{m-1}+2^{m-k-1})\\cdot n-n+1. \\ \\ \\Box$$\n\nIf $s_M$ is the only final state of $M$($k=0$), then $L(M)^*=L(M)$.\n\n\\begin{corollary}\n\\label{star union upper bound corollary}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$, $m>1$, $n>0$,\nthere exists a DFA $A$ of at most $\\frac{3}{4}2^m\\cdot n-n+1$ states\nsuch that $L(A)=L(M)^*\\cup L(N)$.\n\\end{corollary}\n{\\bf Proof.\\ \\ } Let $k$ be defined as in the above proof. There are\ntwo cases in the following.\n\\begin{itemize}\n\\item[{\\rm (I)}]$k=0$. In this case, $L(M)^*=L(M)$. Then $A$ simply needs at most $m\\cdot\nn$ states, which is less than $\\frac{3}{4}2^m\\cdot n-n+1$ when\n$m>1$.\n\\item[{\\rm (II)}]$k\\geq 1$. The claim is clearly true by Theorem~\\ref{star union upper bound}.\n$\\ \\ \\Box$\n\\end{itemize}\nNext, we show that the upper bound $\\frac{3}{4}2^m\\cdot n-n+1$ is\nreachable.\n\\begin{theorem}\n\\label{star union lower bound}\n\nGiven two integers $m\\geq 2$, $n\\geq 2$, there exists a DFA $M$ of\n$m$ states and a DFA $N$ of $n$ states such that any DFA accepting\n$L(M)^*\\cup L(N)$ needs at least $\\frac{3}{4}2^m\\cdot n-n+1$ states.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , 0, \\{ m-1 \\})$ be a\nDFA, where $Q_M = \\{0,1,\\ldots ,m-1\\}$, $\\Sigma = \\{a,b,c\\}$ and the\ntransitions of $M$ are\n\\begin{eqnarray*}\n& & \\delta_M(i, a) = i+1 \\mbox{ mod $m$, } i=0,1, \\ldots , m-1,\\\\\n& & \\delta_M(0, b) = 0 \\mbox{, }\\delta_M(i, b) = i+1 \\mbox{ mod $m$, } i=1, \\ldots , m-1,\\\\\n& & \\delta_M(i, c) = i \\mbox{, } i=0,1, \\ldots , m-1.\n\\end{eqnarray*}\nThe transition diagram of $M$ is shown in\nFigure~\\ref{star-union-first}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{star-union-first.eps}\n \\caption{The transition diagram of the witness DFA $M$ of Theorems~\\ref{star union lower bound} and~\\ref{star intersection lower bound}}\n\\label{star-union-first}\n\\end{figure}\\\\\nLet $N=(Q_N,\\Sigma , \\delta_N , 0, \\{n-1\\})$ be another DFA, where\n$Q_N = \\{0,1,\\ldots ,n-1\\}$ and\n\\begin{eqnarray*}\n& & \\delta_N(i, a) = i \\mbox{, } i=0,1, \\ldots , n-1,\\\\\n& & \\delta_N(i, b) = i \\mbox{, } i=0,1, \\ldots , n-1,\\\\\n& & \\delta_N(i, c) = i+1 \\mbox{ mod $n$, } i=0,1, \\ldots , n-1.\n\\end{eqnarray*}\nThe transition diagram of $N$ is shown in\nFigure~\\ref{star-union-second}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{star-union-second.eps}\n \\caption{The transition diagram of the witness DFA $N$ of Theorems~\\ref{star union lower bound} and~\\ref{star intersection lower bound}}\n\\label{star-union-second}\n\\end{figure}\n\n\n\nIt has been proved in~\\cite{YuZhSa94} that the minimal DFA accepting\nthe star of an $m$-state DFA language has $\\frac{3}{4}2^m$ states in\nthe worst case. $M$ is a modification of worst case example given in\n~\\cite{YuZhSa94} by adding a $c$-loop to every state. So we design a\n$\\frac{3}{4}2^m$-state, minimal DFA $M'=(Q_{M'},\\Sigma , \\delta_{M'}\n, s_{M'}, F_{M'})$ that accepts $L(M)^*$, where\n\\begin{eqnarray*}\n& & s_{M'} \\notin Q_M\\mbox{ is a new start state,}\\\\\n& & Q_{M'} = \\{s_{M'}\\}\\cup \\{P\\mid P\\subseteq \\{0,1,\\ldots ,m-2\\}\\mbox{ \\& } P\\neq \\emptyset \\} \\\\\n& & \\qquad \\cup \\{R\\mid R\\subseteq \\{0,1,\\ldots ,m-1\\} \\mbox{ \\& } 0\\in R \\mbox{ \\& }m-1\\in R \\},\\\\\n& & \\delta_{M'}(s_{M'}, a)= \\{\\delta_M(0, a)\\mbox{ for any $a\\in \\Sigma$} \\},\\\\\n& & \\delta_{M'}(R, a)= \\{\\delta_M(R, a)\\}\\mbox{ for $R\\subseteq Q_M$ and $a\\in \\Sigma$ if $m-1\\notin \\delta_M(R, a)$},\\\\\n& & \\delta_{M'}(R, a)= \\{ \\delta_M(R, a)\\}\\cup \\{0\\}\\mbox{ otherwise}, \\\\\n& & F_{M'}= \\{s_{M'}\\}\\cup\\{R\\mid R\\subseteq \\{0,1,\\ldots ,m-1\\}\n\\mbox{ \\& } m-1\\in R\\}.\n\\end{eqnarray*}\n\nThen we construct a DFA $A=(Q,\\Sigma , \\delta , s, F)$ accepting\n$L(M)^*\\cup L(N)$ exactly as described in the proof of\nTheorem~\\ref{star union upper bound}, where\n\\begin{eqnarray*}\n& & s=\\langle s_{M'},0 \\rangle,\\\\\n& & Q = \\{\\langle i,j \\rangle \\mid i\\in Q_{M'}-\\{s_{M'}\\},j\\in Q_N\\}\\cup \\{s \\}, \\\\\n& & \\delta(\\langle i,j \\rangle, a)= \\langle \\delta_{M'}(i, a),\\delta_N(j, a) \\rangle \\mbox{, $\\langle i,j \\rangle \\in Q$, $a\\in \\Sigma$},\\\\\n& & F= \\{\\langle i,j \\rangle \\mid i\\in F_{M'}\\mbox{ or }j=n-1 \\}.\n\\end{eqnarray*}\n\nNow we need to show that $A$ is a minimal DFA.\n\\begin{itemize}\n\\item[{\\rm (I)}]All the states in $Q$ are reachable.\\\\\nFor an arbitrary state $\\langle i,j\\rangle$ in $Q$, there always\nexists a string $w_1w_2$ such that $\\delta(\\langle s_M',0\\rangle,\nw_1w_2) = \\langle i,j\\rangle$, where\n\\begin{eqnarray*}\n& & \\delta_{M'}(s_{M'}, w_1)=i\\mbox{, }w_1\\in \\{a,b\\}^*,\\\\\n& & \\delta_N (0, w_2)=j\\mbox{, }w_2\\in \\{c\\}^*.\n\\end{eqnarray*}\n\\item[{\\rm (II)}]Any two different states $\\langle i_1,j_1\\rangle$ and $\\langle i_2,j_2\\rangle$ in $Q$ are\ndistinguishable.\\\\\n\\begin{itemize}\n\\item[{\\rm 1.}]$i_1\\neq i_2$, $j_2\\neq n-1$. We can find a string $w_1$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_1)\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_1) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$, $\\delta_{M'}(i_1, w_1)\\in F_{M'}$ and\n$\\delta_M'(i_2, w_1) \\notin F_M'$.\n\n\\item[{\\rm 2.}]$i_1\\neq i_2$, $j_2= n-1$. There exists a string $w_1$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_1c)\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_1c) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$, $\\delta_{M'}(i_1, w_1)\\in F_{M'}$ and\n$\\delta_{M'}(i_2, w_1) \\notin F_{M'}$.\n\\item[{\\rm 3.}]$i_1= i_2\\notin F_{M'}$, $j_1\\neq j_2$. For this case, a string $c^{n-1-j_1}$ can distinguish the two states, since $\\delta(\\langle i_1,j_1\\rangle, c^{n-1-j_1})\\in\nF$ and $\\delta(\\langle i_2,j_2\\rangle, c^{n-1-j_1}) \\notin F$.\n\n\\item[{\\rm 4.}]$i_1= i_2\\in F_{M'}$, $j_1\\neq j_2$. A string $b^mc^{n-1-j_1}$ can distinguish them, because $\\delta(\\langle i_1,j_1\\rangle, b^mc^{n-1-j_1})\\in\nF$ and $\\delta(\\langle i_2,j_2\\rangle, b^mc^{n-1-j_1}) \\notin F$.\n\n\\end{itemize}\n\\end{itemize}\nSince all the states in $A$ are reachable and distinguishable, DFA\n$A$ is minimal. Thus, any DFA accepting $L(M)^*\\cup L(N)$ needs at\nleast $\\frac{3}{4}2^m\\cdot n-n+1$ states. $\\ \\ \\Box $\n\nThis result gives a lower bound for the state complexity of\n$L(M)^*\\cup L(N)$. It coincides with the upper bound in\nCorollary~\\ref{star union upper bound corollary}. So we have the\nfollowing Theorem~\\ref{Tight bound of star union}.\n\\begin{theorem}\n\\label{Tight bound of star union}\n\nFor any integer $m\\geq 2$, $n\\geq 2$, $\\frac{3}{4}2^m\\cdot n-n+1$\nstates are both sufficient and necessary in the worst case for a DFA\nto accept $L(M)^*\\cup L(N)$, where $M$ is an $m$-state DFA and $N$\nis an $n$-state DFA.\n\\end{theorem}\n\n\n\n\n\n\n\n\\section{State complexity of $L(M)^*\\cap L(N)$}\\label{star-intersection}\n\nSince the state complexity of intersection on regular languages is\nthe same as that of union~\\cite{YuZhSa94}, the mathematical\ncomposition of the state complexities of star and intersection is\nalso $\\frac{3}{4}2^m$. In this section, we show that the state\ncomplexity of $L(M)^*\\cap L(N)$ is $\\frac{3}{4}2^m\\cdot n-n+1$ which\nis the same as the state complexity of $L(M)^*\\cup L(N)$.\n\n\\begin{theorem}\n\\label{star intersection upper bound}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$ such that\n$|F_M-\\{ s_M \\}|=k\\geq 1$, $m>1$, $n>0$, there exists a DFA of at\nmost $(2^{m-1}+2^{m-k-1})\\cdot n-n+1$ states that accepts\n$L(M)^*\\cap L(N)$.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } We construct a DFA $A$ accepting $L(M)^*\\cap L(N)$\nthe same as in the proof of Theorem~\\ref{star union upper bound}\nexcept that its set of final states is\n\\[\nF= \\{\\langle i,j \\rangle \\mid i\\in F_{M'}\\mbox{, }j\\in F_N \\}.\n\\]\nThus, after reducing the $n-1$ unreachable states $\\langle s_{M'},j\n\\rangle \\notin Q$, for $j\\in Q_N-\\{s_N\\}$, the number of states of\n$A$ is sill no more than $(2^{m-1}+2^{m-k-1})\\cdot n-n+1. \\ \\ \\Box$\n\nSimilarly to the proof of Corollary~\\ref{star union upper bound\ncorollary}, we consider both the case that $M$ has no other final\nstate except $s_M$ ($L(M)^*=L(M)$) and the case that $M$ has some\nother final states (Theorem~\\ref{star intersection upper bound}).\nThen we obtain the following corollary. Detailed proof may be\nomitted.\n\n\\begin{corollary}\n\\label{star intersection upper bound corollary}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$, $m>1$, $n>0$,\nthere exists a DFA $A$ of at most $\\frac{3}{4}2^m\\cdot n-n+1$ states\nsuch that $L(A)=L(M)^*\\cap L(N)$.\n\\end{corollary}\nNext, we show that this general upper bound of state complexity of\n$L(M)^*\\cap L(N)$ can be reached by some witness DFAs.\n\\begin{theorem} \\label{star intersection lower bound}\n\nGiven two integers $m\\geq 2$, $n\\geq 2$, there exists a DFA $M$ of\n$m$ states and a DFA $N$ of $n$ states such that any DFA accepting\n$L(M)^*\\cap L(N)$ needs at least $\\frac{3}{4}2^m\\cdot n-n+1$ states.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } We use the same DFAs $M$ and $N$ as in the proof of\nTheorem~\\ref{star union lower bound}. Their transition diagrams are\nshown in Figure~\\ref{star-union-first} and\nFigure~\\ref{star-union-second}, respectively. Construct DFA\n$M'=(Q_{M'},\\Sigma , \\delta_{M'} , s_{M'}, F_{M'})$ that accepts\n$L(M)^*$ in the same way.\n\nThen we construct a DFA $A=(Q,\\Sigma , \\delta , s, F)$ accepting\n$L(M)^*\\cap L(N)$ exactly as described in the proof of\nTheorem~\\ref{star union lower bound} except that\n\\[\nF= \\{\\langle i,n-1 \\rangle \\mid i\\in F_{M'} \\}.\n\\]\n\nNow we prove that $A$ is minimal.\n\\begin{itemize}\n\\item[{\\rm (I)}]Every state of $A$ is reachable.\\\\\nLet $\\langle i,j\\rangle$ be an arbitrary state of $A$. Then there\nalways exists a string $w_1w_2$ such that $\\delta(\\langle\ns_{M'},0\\rangle, w_1w_2) = \\langle i,j\\rangle$, where\n\\begin{eqnarray*}\n& & \\delta_{M'}(s_{M'}, w_1)=i\\mbox{, }w_1\\in \\{a,b\\}^*,\\\\\n& & \\delta_N (0, w_2)=j\\mbox{, }w_2\\in \\{c\\}^*.\n\\end{eqnarray*}\n\\item[{\\rm (II)}]Any two different states $\\langle i_1,j_1\\rangle$ and $\\langle i_2,j_2\\rangle$ of $A$ are\ndistinguishable.\\\\\n\\begin{itemize}\n\\item[{\\rm 1.}]$i_1\\neq i_2$.\n\nWe can find a string $w_1$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_1c^{n-1-j_1})\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_1c^{n-1-j_1}) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$, $\\delta_{M'}(i_1, w_1)\\in F_{M'}$ and\n$\\delta_{M'}(i_2, w_1) \\notin F_{M'}$.\n\n\\item[{\\rm 2.}]$i_1= i_2\\notin F_{M'}$, $j_1\\neq j_2$.\n\nThere exists a string $w_2$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_2c^{n-1-j_1})\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_2c^{n-1-j_1}) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$ and $\\delta_{M'}(i_1, w_2)\\in F_{M'}$.\n\\item[{\\rm 3.}]$i_1= i_2\\in F_{M'}$, $j_1\\neq j_2$.\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, c^{n-1-j_1})\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, c^{n-1-j_1}) \\notin F.\n\\end{eqnarray*}\n\\end{itemize}\n\\end{itemize}\nDue to (I) and (II), $A$ is a minimal DFA with $\\frac{3}{4}2^m\\cdot\nn-n+1$ states which accepts $L(M)^*\\cap L(N)$. $\\ \\ \\Box $\n\nThis lower bound coincides with the upper bound in\nCorollary~\\ref{star intersection upper bound corollary}. Thus, the\nbounds are tight.\n\\begin{theorem}\n\\label{Tight bound of star intersection}\n\nFor any integer $m\\geq 2$, $n\\geq 2$, $\\frac{3}{4}2^m\\cdot n-n+1$\nstates are both sufficient and necessary in the worst case for a DFA\nto accept $L(M)^*\\cap L(N)$, where $M$ is an $m$-state DFA and $N$\nis an $n$-state DFA.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\\section{State complexity of $L_1^R\\cup L_2$}\\label{reversal-union}\nIn this section, we study the state complexity of $L_1^R\\cup L_2$,\nwhere $L_1$ and $L_2$ are regular languages. It has been proved that\nthe state complexity of $L_1^R$ is $2^m$ and the state complexity of\n$L_1\\cup L_2$ is $mn$~\\cite{Maslov70,YuZhSa94}. Thus, the\nmathematical composition of them is $2^m\\cdot n$. In this section we\nwill prove that this upper bound of state complexity of $L_1^R\\cup\nL_2$ can not be reached in any case. We will first try to lower the\nupper bound in the following.\n\n\n\\begin{theorem}\n\\label{reversal uion upper bound}\n\nLet $L_1$ and $L_2$ be two regular language accepted by an $m$-state\nand $n$-state DFAs, respectively. Then there exists a DFA of at most\n$2^m\\cdot n-n+1$ states that accepts $L_1^R\\cup L_2$.\n\\end{theorem}\n\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ be a\ncomplete DFA of $m$ states and $L_1=L(M)$. Let $N=(Q_N,\\Sigma ,\n\\delta_N , s_N, F_N)$ be another complete DFA of $n$ states and\n$L_2=L(N)$. Let $M'=(Q_M,\\Sigma , \\delta_{M'} , F_M, \\{s_M\\})$ be an\nNFA with multiple initial states. $\\delta_{M'}(p,a)=q$ if\n$\\delta_M(q,a)=p$ where $a\\in \\Sigma$ and $p,q\\in Q_M$. Clearly,\n$L(M')=L(M)^R=L_1^R$. After performing subset construction, we can\nget a $2^m$-state DFA $A=(Q_A,\\Sigma , \\delta_A , s_A, F_A)$ that is\nequivalent to $M'$. Since $A$ has $2^m$ states, one of its final\nstate must be $Q_M$. Now we construct a DFA $B=(Q_B,\\Sigma ,\n\\delta_B , s_B, F_B)$, where\n\\begin{eqnarray*}\n& & Q_B = \\{\\langle i,j \\rangle \\mid i\\in Q_A\\mbox{, } j\\in Q_N\\},\\\\\n& & s_B = \\langle s_A,s_N \\rangle,\\\\\n& & F_B = \\{\\langle i,j \\rangle\\in Q_B\\mid i\\in F_A\\mbox{ or } j\\in F_N\\},\\\\\n& & \\delta_B(\\langle i,j \\rangle, a) = \\langle i',j' \\rangle \\mbox{,\nif } \\delta_A(i,a)=i'\\mbox{ and }\\delta_N(j,a)=j'\\mbox{, }a\\in\n\\Sigma.\n\\end{eqnarray*}\nIt is easy to see that $\\delta_B(\\langle Q_M,j \\rangle, a) \\in F_B$\nfor any $j\\in Q_N$ and $a\\in \\Sigma$. This means all the states\n(two-tuples) starting with $Q_1$ are equivalent. There are $n$ such\nstates in total. Thus, the minimal DFA accepting $L_1^R\\cup L_2$ has\nno more than $2^m\\cdot n-n+1$ states.$\\ \\ \\Box $\n\n\n\nThis result gives an upper bound of state complexity of $L_1^R\\cup\nL_2$. Now let's see if this bound is reachable.\n\n\\begin{theorem}\n\\label{reversal uion lower bound}\n\nGiven two integers $m\\geq 2$, $n\\geq 2$, there exists a DFA $M$ of\n$m$ states and a DFA $N$ of $n$ states such that any DFA accepting\n$L(M)^R\\cup L(N)$ needs at least $2^m\\cdot n-n+1$ states.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , 0, \\{0\\})$ be a\nDFA, where $Q_M = \\{0,1,\\ldots ,m-1\\}$, $\\Sigma = \\{a,b,c,d\\}$ and\nthe transitions are\n\\begin{eqnarray*}\n& & \\delta_M(0, a) = m-1 \\mbox{, }\\delta_M(i, a) = i-1 \\mbox{, } i=1, \\ldots , m-1,\\\\\n& & \\delta_M(0, b) = 1 \\mbox{, } \\delta_M(i, b) = i \\mbox{, } i=1, \\ldots , m-1,\\\\\n& & \\delta_M(0, c) = 1 \\mbox{, } \\delta_M(1, c) = 0 \\mbox{, }\\delta_M(j, c) = i \\mbox{, } j=2, \\ldots , m-1,\\\\\n& & \\delta_M(k, d) = k \\mbox{, } k=0, \\ldots , m-1.\n\\end{eqnarray*}\nThe transition diagram of $M$ is shown in\nFigure~\\ref{reversal-union-first}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{reversal.eps}\n \\caption{The transition diagram of the witness DFA $M$ of Theorems~\\ref{reversal uion lower bound} and~\\ref{reversal intersection lower bound}}\n\\label{reversal-union-first}\n\\end{figure}\nLet $N=(Q_N,\\Sigma , \\delta_N , 0, \\{0\\})$ be another DFA, where\n$Q_N = \\{0,1,\\ldots ,n-1\\}$, $\\Sigma = \\{a,b,c,d\\}$ and the\ntransitions are\n\\begin{eqnarray*}\n& & \\delta_N(i, a) = i \\mbox{, } i=0, \\ldots , n-1,\\\\\n& & \\delta_N(i, b) = i \\mbox{, } i=0, \\ldots , n-1,\\\\\n& & \\delta_N(i, c) = i \\mbox{, } i=0, \\ldots , n-1,\\\\\n& & \\delta_N(i, d) = i+1 \\mbox{ mod }n \\mbox{, } i=0, \\ldots , n-1.\n\\end{eqnarray*}\nThe transition diagram of $N$ is shown in\nFigure~\\ref{reversal-union-second}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{onecircle.eps}\n \\caption{The transition diagram of the witness DFA $N$ of Theorems~\\ref{reversal uion lower bound} and~\\ref{reversal intersection lower bound}}\n\\label{reversal-union-second}\n\\end{figure}\n\nNote that $M$ is a modification of worst case example given\nin~\\cite{YuZhSa94} for reversal, by adding a $d$-loop to every\nstate. Intuitively, the minimal DFA accepting $L(M)^R$ should also\nhave $2^m$ states. Before using this result, we will prove it first.\nLet $A=(Q_A,\\Sigma , \\delta_A , \\{0\\}, F_A)$ be a DFA, where\n\\begin{eqnarray*}\n& & Q_A = \\{q\\mid q\\subseteq Q_M\\},\\\\\n& & \\Sigma = \\{a,b,c,d\\},\\\\\n& & \\delta_A(p, e) = \\{j\\mid \\delta_M(i, e)=j\\mbox{, }i\\in p\\} \\mbox{, } p\\in Q_A\\mbox{, } e\\in \\Sigma,\\\\\n& & F_A = \\{q\\mid \\{0\\}\\in q \\mbox{, }q\\in Q_A\\}.\n\\end{eqnarray*}\nClearly, $A$ has $2^m$ states and it accepts $L(M)^R$. Now let's\nprove it is minimal.\n\\begin{itemize}\n\\item[{\\rm (i)}]Every state $i \\in Q_A$ is\nreachable.\\\\\n\\begin{itemize}\n\\item[{\\rm 1.}]$i=\\emptyset$.\\\\\n$|i|=0$ if and only if $i=\\emptyset$. $\\delta_A(\\{ 0 \\}, b) =\ni=\\emptyset .$\n\\item[{\\rm 2.}]$|i|=1$.\\\\\nAssume that $i=\\{ p \\}$, $0\\leq p\\leq m-1$. $\\delta_A(\\{ 0 \\}, a^p)\n=i.$\n\\item[{\\rm 3.}]$2\\leq |i|\\leq m$.\\\\\nAssume that $i=\\{ i_1, i_2, \\ldots ,i_k \\}$, $0\\leq i_10, \n\\eeq\nand showed that there is a maximum charge and size.\nTo construct large Q-balls, Anagnostopoulos {\\it et al.} \\cite{AAFT} introduced fermions with charge of the opposite sign.\nLi {\\it et al.} \\cite{LHL} assumed a different potential, a piecewise parabolic function, and Deshaies-Jacques and MacKenzie \\cite{DM} supposed the Maxwell-Chern-Simons theory with the $V_4$ potential (\\ref{V4}) in the 2+1 dimensional spacetime; it was shown that there is a maximum charge and size of Q-balls in both models.\n\nArod\\'z and Lis \\cite{Arodz} considered gauged Q-balls with the V-shaped potential,\n\\beq\\label{VV}\nV_{\\rm V}(\\phi):=\\lambda\\frac{|\\phi |}{\\sqrt{2}}\n~~~{\\rm with} ~~~ \\lambda >0, \n\\eeq\nBecause its three-dimensional plot has the form of a cone, it would be more appropriate to call it the cone-shaped potential.\nIn addition to normal Q-balls, which have a maximum charge, they found a new type of solutions, Q-shells.\nQ-shell solutions are obtained in such a way that the scalar field and the gauge field are assumed to be constant within a certain sphere $rr_0$. \nBecause the electric charge is concentrated on the shell, large Q-balls with any amount of charge can exist without additional fermions. \nThus this model overcomes the difficulty of the $V_4$ model.\nHowever, there is another drawback that it is so simplified and singular at $\\phi=0$. \n\nIn this paper we address the question whether such large gauged Q-balls can be formed in realistic or cosmologically-motivated theories without additional fermions nor a singular potential.\nOne of the physically-motivated theories is the AD mechanism \\cite{AD}, which includes two types of potentials,\ngravity-mediation type and gauge-mediation type. \nThe former is described by \n\\bea\\label{gravity}\n&&V_{\\rm grav.}(\\phi):=\\frac{m_{\\rm grav.}^2}{2}\\phi^2\\left[\n1+K\\ln \\left(\\frac{\\phi}{M}\\right)^2\n\\right]~~ \\nonumber \\\\\n&&{\\rm with} ~~ m_{\\rm grav.}^2,~M>0,\n\\eea\nwhile the latter by\n\\beq\\label{gauge}\nV_{\\rm gauge}(\\phi):=m_{\\rm gauge}^4 \\ln\\left(1+\\frac{\\phi^2}{m_{\\rm gauge}^2}\\right)~~~\n{\\rm with} ~~~ m_{\\rm gauge}^2>0\\ .\n\\eeq\nIf we take Maclaurin expansion of the two potentials in the vicinity of $\\phi=0$, the latter can be regarded as $V_{4}$ model, and is inappropriate for our purpose.\nThus, we concentrate on investigating gauged Q-balls in the former potential. \n\nThis paper is organized as follows.\nIn Sec. II, we show the basic equations of gauged Q-balls. \nIn Sec. III, we discuss general properties of ordinary and gauged Q-balls in words of Newtonian mechanics.\nIn Sec. IV, we review previous results of $V_{4}$ and $V_{\\rm V}$ models. \nIn Sec. V, we investigate equilibrium solutions in the $V_{\\rm grav.}$ model numerically.\nSection VI is devoted to concluding remarks.\n\n\\section{basic equations}\n\nConsider an SO(2) symmetric scalar field $\\bp=(\\phi_1,\\phi_2)$ coupled to a gauged field $A_\\mu$,\n\\beq\\label{S}\n{\\cal S}=\\int d^4x\\left[\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n-\\frac{1}{2}\\eta^{\\mu\\nu}D_{\\mu}\\phi_{a} D_{\\nu}\\phi_{a}-V(\\phi) \\right],\n\\eeq\nwhere \n\\bea\n&&\\phi:=\\sqrt{\\phi_{a}\\phi_{a}},~~~\nF_{\\mu\\nu}:=\\pa_\\mu A_\\nu-\\pa_\\nu A_\\mu,\\\\\n&&D_{\\mu}\\phi_{a}:=\\pa_{\\mu}\\phi_{a}+A_{\\mu}\\epsilon_{ab}\\phi_{b}~(a,b=1,2). \n\\eea\nTo find spherically symmetric and equilibrium solutions with vanishing magnetic fields, we assume\n\\beq\\label{qball-phase}\n\\bp=\\phi(r)(\\cos\\omega t,\\sin\\omega t),~~~\nA_{0}=A_{0}(r),~~~ A_i=0,\n\\eeq\nwhere the subscript $i$ denotes spatial components and runs 1 to 3.\nIntroducing a variable,\n\\beq\n\\Omega(r):=\\omega+qA_{0}(r),\n\\eeq\nwe obtain field equations, \n\\bea\\label{FEqball}\n&&\\frac{d^2\\phi}{dr^2}+\\frac{2}{r}\\frac{d\\phi}{dr}+\\Omega^2 \\phi=\\frac{dV}{d\\phi}, \\\\\n&&\\frac{d^2\\Omega}{dr^2}+\\frac{2}{r}\\frac{d\\Omega}{dr}=\\Omega (q\\phi)^{2}. \\label{FEqball2}\n\\eea\n\nThe boundary condition we assume is \n\\bea\n&&{d\\phi\\over dr}(r=0)=0,~~{d\\Omega\\over dr}(r=0)=0, \\label{BCqball} \\\\\n&&\\phi(r\\ra\\infty)=0,~~\\Omega(r\\ra\\infty)=\\omega+\\frac{C}{r}, \\label{BCqball2}\n\\eea\nwhere $C$ is a constant.\nIn numerical calculation we must choose $\\Omega$ and $\\phi$ at $\\tilde{r}=0$ to satisfy the asymptotic conditions \n(\\ref{BCqball2}). \nIn concrete, we seek for appropriate $\\phi (0)$ for a fixed $\\Omega (0)$. \n\nWe define the energy and the charge, respectively, as\n\\bea\\label{Edef}\nE&=&\\int d^3xT_{00}\\nn\n&=&2\\pi \\int_0^{\\infty}r^2 dr\n\\left\\{\\Omega^2\\phi^2+\\left({d\\phi\\over dr}\\right)^2+\\left({d\\Omega\\over dr}\\right)^2+2V\\right\\},\\nn\nQ&=&\\int d^3x(\\phi_1D_0\\phi_2-\\phi_2D_0\\phi_1)\\nn\n&=&4\\pi \\int_0^{\\infty}r^2\\Omega\\phi^2dr,\n\\label{Qdef}\\eea\nwhere $T_{00}$ is the time-time component of the energy momentum tensor, which is defined by\n\\bea\nT_{\\mu\\nu}&=&D_\\mu\\phi_aD_\\nu\\phi_a-\\eta_{\\mu\\nu}\\left[\\frac12(D_\\lambda\\phi_a)^2+V\\right]\\nn\n&&+F_{\\mu\\lambda}F_\\nu^\\lambda-\\frac14\\eta_{\\mu\\nu}(F_{\\lambda\\sigma})^2.\n\\eea\nEquations (\\ref{FEqball}), (\\ref{FEqball2}) and (\\ref{Qdef}) indicate that the sign transformation\n$\\Omega \\to -\\Omega$ changes nothing but $Q\\to -Q$ with keeping $E$ and $\\phi(r)$ unchanged.\nThus, we choose $\\Omega >0$ in this paper.\n\n\\section{General Properties of Ordinary and Gauged Q-ball Solutions. }\n\nTo begin with, to understand the effect of gauge fields on Q-balls, we review properties of ordinary Q-ball solutions.\nThe field equations are obtained by putting $\\Omega=\\omega$=constant in Eq.(\\ref{FEqball}),\n\\beq\\label{FEOQ}\n\\frac{d^2\\phi}{dr^2}+\\frac{2}{r}\\frac{d\\phi}{dr}=\\frac{dV_\\omega}{d\\phi},~~~\nV_\\omega :=V-\\frac12\\omega^2\\phi^2.\n\\eeq\nIf one regards the radius $r$ as \\lq time\\rq\\ and the scalar amplitude $\\phi(r)$ as \\lq the position of a particle\\rq,\none can understand solutions in words of Newtonian mechanics, as shown in Fig.\\ \\ref{f1}.\nEquation (\\ref{FEOQ}) describes a one-dimensional motion of a particle under the nonconserved force \ndue to the effective potential $-V_{\\omega}(\\phi)$ and the \\lq time\\rq-dependent friction $-(2\/r)d\\phi\/dr$.\nIf one chooses the \\lq initial position\\rq\\ $\\phi(0)$ appropriately, the static particle begins to roll down the potential slope, climbs up and approaches the origin over infinite time.\n\n\\begin{figure}[htbp]\n\\psfig{file=f1,width=3.2in}\n\\caption{\\label{f1}\nInterpretation of ordinary Q-balls by analogy with a particle motion in Newtonian mechanics.}\n\\end{figure}\n\nFrom the above picture, one can derive the existing conditions of equilibrium solutions of ordinary Q-balls as follows.\nThe first condition is that the \\lq initial altitude of the particle\\rq\\ $-V_\\omega(\\phi(0))$ is larger than the \\lq final altitude\\rq\\\n$-V_\\omega(\\phi(\\infty))=0$, which leads to\n\\beq\\label{cond1}\n{\\rm max}[-V_\\omega(\\phi)]>0,~~i.e.,~~\n{\\rm min}\\left[{2V\\over\\phi^2}\\right]<\\omega^2.\n\\eeq\nThe second condition is that the \\lq particle climbs up\\rq\\ at $r\\to\\infty$, which leads to\n\\beq\\label{cond2}\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(-{dV_\\omega\\over d\\phi}\\right)=\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(\\omega^2\\phi-{dV\\over d\\phi}\\right)<0\\ .\n\\eeq\nIf the lowest-order term of $V$ is quadratic, i.e., $\\ds V=\\frac12m^2\\phi^2+O(\\phi^3)$, the second condition (\\ref{cond2}) reduces to\n\\beq\\label{cond22}\n\\omega^20$, the condition (\\ref{cond2}) is satisfied regardless of $\\omega$. Similarly, in the case of $V_{\\rm grav.}$, if we take $K<0$, the condition (\\ref{cond2}) is satisfied regardless of $\\omega$. \n\nNow let us move on to gauged Q-balls. Without specifying a potential $V$, we can show that $\\Omega^2$ is a \nmonotonically increasing function of $r$~\\cite{Arodz}.\nUsing a variable $\\ds f:=r^2\\frac{d\\Omega}{dr}$, we can rewrite Eq. (\\ref{FEqball2}) as\n\\bea\\label{FEqball2-2}\n&&\\frac{df}{dr}=\\Omega (qr\\phi)^{2},~~~\n\\frac{d\\Omega}{dr}={f\\over r^2}\\ .\n\\eea\nThe Taylor expansion of $\\Omega$ and $f$ up to the first order is expressed as\n\\bea\nf(r+\\Delta r)&=&f(r_0)+(qr_0\\phi(r))^2\\Omega(r)\\Delta r+O(\\Delta r^2),\\nn\n\\Omega(r+\\Delta r)&=&\\Omega(r)+{f(r)\\over r^2}\\Delta r+O(\\Delta r^2).\n\\label{Taylor}\\eea\nBy definition $f(0)=0$. If $\\Omega (0)>0$, then $f(\\Delta r)>0$. Equation (\\ref{Taylor}) indicates that at every step $r\\to r+\\Delta r$ both $f$ and $\\Omega$ increases. \nSimilarly, if $\\Omega (0)<0$, then $f$ and $\\Omega$ decreases at every step.\nThus we can conclude that $\\Omega^2$ is a monotonically increasing function of $r$.\n\n\\begin{figure}[htbp]\n\\psfig{file=f2a,width=3.2in}\n\\psfig{file=f2b,width=3.2in}\n\\caption{\\label{Newton}\nInterpretation of gauged Q-balls by analogy with a particle motion in Newtonian mechanics. Examples of\n(a) monotonic solutions in $V_{4}$ model and (b) nonmonotonic solutions in $V_{\\rm V}$ model.}\n\\end{figure}\n\nWe can interpret their equilibrium solutions in words of Newtonian mechanics in the same fashion, except that the potential of a particle is \\lq time\\rq-dependent,\n\\beq\nV_{\\Omega}=V-\\frac12\\Omega^2\\phi^2.\n\\eeq\nBecause the \\lq potential energy of the particle\\rq\\ $-V_\\Omega$ increases as the \\lq time\\rq\\ $r$ increases,\nthe \\lq initial altitude\\rq\\ $-V_\\Omega(0)$ is not necessarily larger than the \\lq final altitude\\rq\\ $-V_\\Omega(\\infty)=0$, that is,\nthere is no condition which corresponds to (\\ref{cond1}).\nHowever, the condition that the \\lq particle climbs up\\rq\\ at $r\\to\\infty$ should hold, we find an existing condition, which corresponds to (\\ref{cond2}),\n\\beq\\label{gaugedQexist}\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(-{dV_\\Omega\\over d\\phi}\\right)=\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(\\Omega^2\\phi-{dV\\over d\\phi}\\right)<0\\ .\n\\eeq\n\nFigure \\ref{Newton} illustrates the \\lq time-dependent potential of a fictitious particle \\rq $-V_\\Omega$.\nAs $r$ increases, $\\Omega^2$ also increases; then $-V_\\Omega$ goes up as shown in the figure.\nThere are two types of solutions.\nOne is monotonic solutions as shown in (a): $\\phi$ decreases monotonically as $r$ increases.\nThe other is nonmonotonic solutions as shown in (b): $\\phi$ increase initially, but after the sign of $dV_\\Omega\/d\\phi$ changes, $\\phi$ turns to decreases.\nThe latter type exposes a characteristic of gauged Q-balls, which appears in the $V_{\\rm V}$ and $V_{\\rm grav.}$ models.\n\n\\begin{figure}[htbp]\n\\psfig{file=f3,width=3.2in}\n\\caption{\\label{V4m02Q09r-phi}\nThe field configurations of $\\tilde{\\phi}$ and $\\tilde{\\Omega}$ for the $V_{4}$ model with $\\tilde{m}^{2}=0.2$ \nand $\\tilde{Q}=9$. The dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively.}\n\\end{figure}\n\\section{Review of previous results}\n\nIn this section we review gauged Q-ball solutions in the $V_{4}$ model \\cite{Lee} and in the $V_{\\rm V}$ model \\cite{Arodz}.\n\n\\subsection{$V_{4}$ model}\n\nFor the $V_4$ model (\\ref{V4}), the necessary condition of existing equilibrium solutions (\\ref{gaugedQexist}) is expressed as \n\\beq\\label{gaugedQexist2}\n\\lim_{r\\to\\infty}\\Omega^2 0$.\nContrary to the case of the $V_4$ model, this condition does not put any restriction on $\\Omega$. \nTherefore, large gauged Q-balls are expected in this model.\n\nUsing the normalized coupling $\\kappa:={q\\lambda}\/{\\sqrt{2}}$, we rescale the quantities as \n\\bea\n&&\\tp:=\\frac{q\\phi}{\\sqrt{\\kappa}},~~\\tilde{\\Omega}:=\\frac{\\Omega}{\\sqrt{\\kappa}},~~\n\\tilde{r}:= \\sqrt{\\kappa}r, \\nonumber \\\\\n&&\\tilde{Q}:= q^2 Q, ~~\\tilde{E}:= \\frac{q^2 E}{\\sqrt{\\kappa}}.\n\\label{rescale-VV}\n\\eea\nIn Fig.~\\ref{VVQ120r-phi}, we show the field configurations of $\\tilde{\\phi}$ and $\\tilde{\\Omega}$ \nwith $\\tilde{Q}=120$. \nThe dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively. \nIn the case of gauged Q-balls, $\\tilde{\\phi}$ initially increases as a function of $\\tilde{r}$ and \ntakes a maximum value at $\\tilde{r}=\\tilde{r}_{\\rm max}\\neq 0$; then it decreases due to the increase of $\\tilde{\\Omega}$.\nThis behavior can be understood by the effective potential shown in Fig.~\\ref{Newton} (b). \nHere we have defined $\\tilde{r}_{\\rm max}$ as the value of $\\tilde{r}$ where $\\tilde{\\phi}$ takes a maximum value.\nIn the case of ordinary Q-balls, by contrast, $\\tilde{r}_{\\rm max}$ is always zero.\n\n\\begin{figure}[htbp]\n\\psfig{file=f4,width=3.2in}\n\\caption{\\label{VVQ120r-phi}\nThe field configurations of $\\tilde{\\phi}$ and $\\tilde{\\Omega}$ for the $V_{\\rm V}$ model with $\\tilde{Q}=120$. \nThe dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively.}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f5a,width=3.2in}\n\\psfig{file=f5b,width=3.2in}\n\\caption{\\label{Omega-phiVV}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for the $V_{\\rm V}$ model. \nThe dashed line corresponds to the ordinary Q-balls. \nThe dotted and black solid lines correspond to the gauged Q-balls with $\\tilde{r}_{\\rm max}= 0$ and those with \n$\\tilde{r}_{\\rm max}\\neq 0$, respectively.\nBlue solid line corresponds to the Q-shell solutions. }\n\\end{figure}\n\nWe show the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and $\\tilde{Q}$-$\\tilde{E}$ relations in Fig.~\\ref{Omega-phiVV} (a) and (b), respectively.\nThe dashed line corresponds to the ordinary Q-balls. \nThe dotted and black solid lines correspond to the gauged case with \n$\\tilde{r}_{\\rm max}= 0$ and that with $\\tilde{r}_{\\rm max}\\neq 0$, respectively. \nBlue solid line corresponds to the Q-shell solutions that will be explained below. \n\nIn the case of ordinary Q-balls ($\\tilde{\\Omega}=\\tilde{\\omega}$), the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ relation, \nwhich was represented by the dashed line in (a), can be understood as follows.\nIn the picture of a particle motion in Newtonian mechanics, which was shown in Fig.\\ \\ref{f1}, if we ignore the ``nonconserved force\" term, $(2\/r)d\\phi\/dr$, the maximum of $\\tilde{\\phi}$, $\\tilde{\\phi}_{\\rm max}=\\tilde{\\phi}(0)$ is determined by the nontrivial solution of $V_{\\Omega}=0$. Then we obtain\n\\bea\n\\tilde{\\phi}(0)=\\frac{2}{\\tilde{\\Omega}^2}, \n\\label{phimax-VV}\n\\eea\nwhich approximates the dashed line in (a).\n\nIn the case of gauged Q-balls, the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ relation for large \n$\\tilde{\\Omega}(0)$ (small $\\tilde{Q}$), which is represented by the dotted line in (a), almost coincides with that for ordinary Q-balls.\nFor small $\\tilde{\\Omega}(0)$ (large $\\tilde{Q}$), however, the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ relation for ordinary Q-balls and that for gauged Q-balls are qualitatively different.\nNevertheless, it is surprising that there is no qualitative difference in $\\tilde{Q}$-$\\tilde{E}$ relation between\nsolutions with $\\tilde{r}_{\\rm max}= 0$ and those with $\\tilde{r}_{\\rm max}\\neq 0$.\nBoth solutions are on the same quasi-linear relation across the point $A$. \n\n$Q$ reaches a maximum at the point $B$ where cusp structure appears in the $\\tilde{Q}$-$\\tilde{E}$ plane.\nQ-ball solutions with the boundary conditions (\\ref{BCqball}) disappear at the point $C$ where $\\tilde{\\phi}(0)\\to 0$. \nHowever, Arod\\'z and Lis \\cite{Arodz} found a new type of solutions with boundary conditions (\\ref{BCqball2}) and\n\\bea\n&&\\phi (r)={d\\phi\\over dr}(r)={d\\Omega\\over dr}(r)=0,~~{\\rm for}~00.\n\\eeq\nBecause the AD gravity mediation model (\\ref{gravity}) with $K<0$ satisfies this condition, we can expect that \nit allows for large $Q$ solutions. This special property is in common with the V-shaped model.\n\nWe rescale the quantities in (\\ref{gravity}) as \n\\bea\n&&\\tp:=\\frac{q\\phi}{M},~~\\tilde{\\Omega}:=\\frac{\\Omega}{M},~~ \\nonumber \\\\\n&&\\tilde{r}:= Mr, ~~\\tilde{m}_{\\rm grav.}:= \\frac{m_{\\rm grav.}}{M},\\nonumber \\\\\n&&\\tilde{Q}:= q^2 Q, ~~\\tilde{E}:= \\frac{q^2 E}{M}.\n\\label{rescale-gravity}\n\\eea\nWe fix $\\tilde{m}_{\\rm grav.}=q=1$ below. \n\n\\begin{figure}[htbp]\n\\psfig{file=f6,width=3.2in}\n\\caption{\\label{K-1fields}\nThe field configurations of $\\tilde{\\phi}$ for gauged Q-balls with $K =-1$ and $\\tilde{Q}\\simeq 1.7$, $11$ and $103$.\n}\n\\end{figure}\n\nWe show some solutions of gauged Q-balls in Fig.~\\ref{K-1fields}; we choose $K =-1$ and obtain \nsolutions with $\\tilde{Q}=1.7$ and $11$, in which case $r_{\\rm max}=0$, and that \nwith $\\tilde{Q}=103$, in which case $r_{\\rm max}\\ne0$. \nAs $\\tilde{Q}$ increases, the field configuration becomes shell-like and the location of the shell becomes farther from the center.\nThis behavior is explained by repulsive Coulomb force of electric charge.\nThese configurations are just like ``Q-shells,\" which were obtained by Arod\\'z and Lis for the V-shaped model \\cite{Arodz}.\nThe difference is that we use the boundary condition (\\ref{BCqball}) and (\\ref{BCqball2}) consistently and \ngive tiny but nonzero value for $\\tilde{\\phi}(0)$, while they adopted the special boundary condition (\\ref{BCqshell}).\n\n\\begin{figure}[htbp]\n\\psfig{file=f7a,width=3.2in}\n\\psfig{file=f7b,width=3.2in}\n\\caption{\\label{K-1}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-1$.\nThe dashed lines correspond to ordinary Q-balls.\nThe dotted and solid lines correspond to gauged Q-balls with $\\tilde{r}_{\\rm max}= 0$ and those with \n$\\tilde{r}_{\\rm max}\\neq 0$, respectively. \n}\n\\end{figure}\n\nWe show the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-1$ \nin Fig.~\\ref{K-1}. For reference, we also plot the relations for ordinary Q-balls ($\\Omega=\\omega$), which \nare represented by the dashed lines.\nTheir extreme behavior in the thin-wall limit ($\\omega\\ra\\infty$) and in the thick-wall limit ($\\omega\\ra0$) can be discussed analytically as follows \\cite{TS2}.\nThe maximum of $\\phi$, $\\tilde{\\phi}_{\\rm max}=\\tilde{\\phi}(0)$, can be estimated by the nontrivial solution of $V_{\\Omega}=0$:\n\\bea\n\\tilde{\\phi}_{\\rm max}=e^{\\frac{1-\\tilde{\\omega}^2}{-2K}}. \n\\label{phimax-AD}\n\\eea\nBecause the energy and the charge are roughly estimated as\n\\beq\nE\\sim V(\\phi_{\\rm max})R^3,~~~\nQ\\sim\\omega\\phi_{\\rm max}^{~~~2}R^3,\n\\eeq\nwhere $R$ is the typical radius, we find\n\\bea\n\\omega\\ra0&:&\\phi_{\\rm max}\\ra{\\rm nzf},~~~E\\ra{\\rm nzf},~~~Q\\ra0,\\nn\n\\omega\\ra\\infty&:&\\phi_{\\rm max}\\ra0,~~~E\\ra0,~~~Q\\ra0,\n\\eea\nwhere nzf denotes nonzero finite. Therefore, there is an upper limit $Q_{\\rm max}$. \nThis analytic estimate agrees with the numerical results in Fig.\\ \\ref{K-1}. \nThere are two sequences of solutions which merge at the cusp. \nWe suppose by energetics that the sequences with high energy are unstable (unstable branch) while \nthose with low energy stable (stable branch). \n\n\n\\begin{figure}[htbp]\n\\psfig{file=f8a,width=3.2in}\n\\psfig{file=f8b,width=3.2in}\n\\caption{\\label{K-06}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-0.6$.\n}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f9a,width=3.2in}\n\\psfig{file=f9b,width=3.2in}\n\\caption{\\label{K-04}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-0.4$. \n}\n\\end{figure}\n\nThe results for gauge Q-balls are represented by the dashed lines ($\\tilde{r}_{\\rm max}= 0$) and \nthe solid lines ($\\tilde{r}_{\\rm max}\\neq 0$). \nThe solutions denoted by red lines correspond to those with small $\\omega$ and unstable branch, \nwhile those by black lines large $\\omega$ and stable branch. \nFor dotted lines, the gauged Q-balls are similar to the ordinary Q-balls (dashed lines). In contrast, \ndue to the nonmonotonic behavior of $\\tilde{\\phi}(\\tilde{r})$ (i.e., $\\tilde{r}_{\\rm max}\\neq 0$), the \nproperties of gauged Q-balls with solid lines and ordinary Q-balls are quite different. \n\nAs for the stable solutions denoted by the black lines, both $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ \nand $\\tilde{Q}$-$\\tilde{E}$ relations of solutions are similar to those of the $V_{\\rm V}$ model, except that cusp structure does not appear in the $\\tilde{Q}$-$\\tilde{E}$ plane in \nFig.~\\ref{K-1}(b).\nBecause $\\tilde{E}$ is a monotonically increasing function of $\\tilde{Q}$ we judge that all equilibrium solutions by black lines are stable. We also suppose by energetics that the solutions denoted by red lines are unstable. \n\n\\begin{figure}[htbp]\n\\psfig{file=f10a,width=3.2in}\n\\psfig{file=f10b,width=3.2in}\n\\caption{\\label{K-106}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-1.06$. \nThe two sequences in red lines and in black lines are about to touch.\n}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f11a,width=3.2in}\n\\psfig{file=f11b,width=3.2in}\n\\caption{\\label{K-107Omega-phi}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations with $K =-1.07$. \nThe ``recombination\" of the two sequences happens.\n}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f12a,width=3.2in}\n\\psfig{file=f12b,width=3.2in}\n\\caption{\\label{k-107Q-E}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K=-1.07$ and $\\tilde{Q}>500$.\nThe dotted lines extend from Fig.\\ \\ref{K-107Omega-phi}.}\n\\end{figure}\n\nFigures \\ref{K-06} and \\ref{K-04} show the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and $\\tilde{Q}$-$\\tilde{E}$ \nrelations for $K =-0.6$ and $-0.4$, respectively. \nWe find that, as $|K|$ decreases, the existing domain of the unstable solutions becomes small in the $\\tilde\\Omega(0)$-$\\tilde\\phi(0)$ plane and the two sequences leave away from each other.\n\nA drastic change occurs between $K=-1.06$ and $K=-1.07$, as shown in Figs.\\ \\ref{K-106} and \\ref{K-107Omega-phi}. \nAs $|K|$ increases, the two sequences approach further; eventually at some point in $-1.07500$) region for $K=-1.07$ in Fig.~\\ref{k-107Q-E}.\nComplicated structure appears along the sequence $C$ to $G$; there are several cusps about $C$-$D$-$E$-$F$. \nAs shown in Fig~\\ref{K-107r-phi}, field distributions in this region also have complicated structures. \nBeyond the point $F$, both $\\tilde{\\phi}_{\\rm max}$ and $\\tilde{r}_{\\rm max}$ monotonically increase. \nIt is interesting that small differences of boundary values $\\tilde{\\Omega}(0)$ and \n$\\tilde{\\phi}(0)$ result in such large differences in $\\tilde{Q}$ and $\\tilde{E}$. \n\n\n\\begin{figure}[htbp]\n\\psfig{file=f13a,width=3.2in}\n\\psfig{file=f13b,width=3.2in}\n\\caption{\\label{K-107r-phi}\nField distributions of $\\tilde{\\phi}$ with $K=-1.07$ for (a)solutions $C$-$D$-$E$ and (b)solutions $E$-$F$-$G$. }\n\\end{figure}\n\n\n\\section{Summary and Discussions}\n\nIn many models of gauged Q-balls, which were studied in the literature, there are upper limits for charge and size of Q-balls due to repulsive Coulomb force.\nAs a cosmologically-motivated model which could allow for gauged Q-balls with large charge and size, we have considered the gravity-mediation-type model in the Affleck-Dine mechanism.\nWe have found that stable Q-balls with any amount of charge and size exist in this model as long as $K<0$.\nAs the electric charge $Q$ increases, the field configuration of the scalar field becomes shell-like; \nbecause the charge is concentrated on the surface, the Coulomb force does not destroy the Q-ball configuration.\nThese properties are analogous to those in the V-shaped model, which was studied by Arod\\'z and Lis \\cite{Arodz}.\nBecause the V-shaped model is rather artificial, our results for the cosmologically-motivated model would be important if we consider gauged Q-balls as realistic dark matter model. \n\nWe have also found that for each $K$ there is another sequence of unstable solutions, which is separated from the other sequence of the stable solutions.\nAs $|K|$ increases, the two sequences approach; eventually at some point in $-1.07$ and their errors $\\left<\\sigma_{\\rm \\mu}\\right>$\nwould have to be weighted by the inverse variances, as:\n\\begin{equation}\n\\left< \\mu\\right> =\n{\\left(\\sum_{i=1}^{n}{\\mu_{\\rm i}\/{\\sigma^{2}_{\\rm \\mu i}}}\\right)}\n\/\n{\\left(\\sum_{i=1}^{n}{1\/{\\sigma^{2}_{\\rm \\mu i}}}\\right)}\\ .\n\\label{eqmu}\n\\end{equation}\n\\begin{equation}\n\\left< \\sigma_{\\rm \\mu}\\right> =\n{\\left(\\sum_{i=1}^{n}{1\/{\\sigma^{2}_{\\rm \\mu i}}}\\right)}^{-1\/2}\\ .\n\\label{eqsigmu}\n\\end{equation}\nWe know, however, that not all measurements are independent of each\nother, since some of the catalogues share the same plate material.\nFurthermore, though we checked as far as possible whether the star\nfound in the catalogue by the\nautomatic search procedure using its coordinates\nis indeed the white dwarf, in some cases misidentifications have occurred.\nComparing the proper motions of one star in different\nsources permits false detections to be eliminated.\n\nTo do this, we calculated the combined average and error, plus the\nquadratic deviation $\\Delta^{2}_{\\rm \\mu i}$\nof an individual measurement from this average:\n\\begin{equation}\n \\Delta^{2}_{\\rm \\mu i} =\n\\left( \\mu_{\\rm i}-\\left< \\mu\\right> \\right) ^{2}\\ .\n\\end{equation}\nIf each $\\Delta^{2}_{\\rm \\mu i}$ is divided by the corresponding\n$\\sigma_{\\rm \\mu i}^{2}$,\nthe sum over all $i$ is taken and divided by the number of measurements $n$.\nWe\nget a quantity $\\Delta_{\\rm check}$ that allows to check if the individual\nmeasurements are\nconsistent with each other and, if not, to eliminate the measurement\nwhich differs from the others:\n\\begin{equation}\n\\Delta_{\\rm check}=\n{1\\over n}\n\\left(\\sum_{i=1}^{n} \\left( \\Delta^{2}_{\\rm \\mu i}\/\\sigma^{2}_{\\mu i}\n\\right) \\right)\\ .\n\\end{equation}\nIf $\\Delta_{\\rm check}>1$, we checked the different catalogue values\nmanually in order to decide which values to choose and which to eliminate.\nHaving thus eliminated false detections the next step was to calculate\nthe quantity $\\left< \\Delta_{\\rm \\mu}\\right>$:\n\\begin{equation}\n\\left< \\Delta_{\\rm \\mu}\\right> =\n\\sqrt\n{{1\\over n} \\left(\\sum_{i=1}^{n} \\Delta^{2}_{\\rm \\mu i}\\right)}\\ .\n\\end{equation}\nWe adopted the weighted mean $\\left< \\mu\\right>$ from Eq.~(\\ref{eqmu}) \nand the maximum of $\\left< \\sigma_{\\rm \\mu}\\right>$\nand $\\left< \\Delta_{\\rm \\mu}\\right>$ as\nthe corresponding error.\nThis enabled us to obtain a realistic error estimate,\nwhich typically lies between $5\\,{\\rm mas~{yr}^{-1}}$ and\n$10\\,{\\rm mas~{yr}^{-1}}$.\n\nThe input parameters radial velocities, spectroscopic distances,\nand\nproper motion components together with their errors, are listed\nfor all white dwarfs in Table~8.\n\n\\section{Revised population classification scheme\\label{orbit}}\nIn Paper~I we presented\na new sophisticated\npopulation classification scheme based on the $U$\\\/-$V$-velocity diagram,\nthe $J_Z$-eccentricity-diagram, and the Galactic orbit.\nFor the computation of orbits and kinematic parameters, we used the code\nby \\citet{odenkirchen92} based on a Galactic potential by \\citet{allen91}.\nThe classification scheme was based on a calibration sample of\nmain-sequence stars.\nIn the meantime, new spectroscopic analyses have become available which \nallowed us to enlarge the calibration sample and to refine our\nclassification criteria.\n\n\\subsection{The calibration sample\\label{cal}}\nUnlike for main-sequence stars, the population membership of white dwarfs\ncannot be determined from spectroscopically measured metalicities.\nTherefore we have to rely on kinematic criteria.\nThose criteria have to be calibrated using a suitable calibration sample of\nmain-sequence stars.\nIn our case this sample consists of $291$ F and G main-sequence stars from\n\\citet{edvardsson93}, \\citet{fuhrmann98},\nFuhrmann (2000\\footnote{\\tt http:\/\/www.xray.mpe.mpg.de\/fuhrmann\/.},\n2004). It is important to note that the stars were \nselected from flux limited samples and not from proper motion surveys.\nThanks to the work of \\citet{fuhrmann04},\nthe number of calibration sample stars has been doubled, which makes it \nworthwhile revisiting the classification criteria outlined in Paper I.\n\nFor both samples a detailed abundance analysis was carried out.\n\\citet{fuhrmann98} combined abundances, ages, and 3D kinematics for\npopulation classification and found that the disk and halo populations\ncan be distinguished best\nin the [Mg\/Fe] versus [Fe\/H] diagram. Halo and thick-disk stars can be\nseparated by means of their [Fe\/H] abundances,\nas they possess a higher [Mg\/Fe] ratio than thin-disk stars\n\\citep[see also][]{bensby03}.\nIn Fig.~\\ref{met} the\n${\\rm [Mg\/Fe]}$ versus ${\\rm [Fe\/H]}$ abundances\nfor the $291$ main-sequence stars are shown.\nThese stars are divided into halo, thick disk, and\nthin disk according to their position in the diagram.\nThe halo stars have $[{\\rm Fe}\/{\\rm H}]<-1.05$,\nthe thick-disk stars\n$-1.05\\le[{\\rm Fe}\/{\\rm H}]\\le-0.3$ and\n$[{\\rm Mg}\/{\\rm Fe}]\\ge 0.3$, and the thin-disk stars \n$[{\\rm Fe}\/{\\rm H}]>-0.3$ and $[{\\rm Mg}\/{\\rm Fe}] \\le 0.2$.\nStars in the overlapping area between the thin and the thick disk (open\ntriangles in Fig.~\\ref{met}) were\nneglected in order to ensure a clear distinction between the\ntwo disk populations.\n\nThere are four stars left to the halo border, which according\nto \\citet{fuhrmann04} belong to the metal-weak thick disk\n(MWTD, open boxes).\nAs their kinematics are indeed incompatible with halo\nmembership, we omitted them\nfrom further analysis.\nAlso rejected was the star HD\\,148816, which though in\nthe thick-disk region in the abundance diagram, clearly shows\nhalo kinematics (not shown in the diagram).\n\nThis demonstrates that a clear distinction between halo and\nthick-disk stars by means of abundances is difficult,\nbut as will be shown later, halo and thick-disk stars\nshow very distinct kinematic\nproperties, so that they are unlikely to be confused.\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n\\begin{psfrags}\n\\psfrag{[Fe\/H]}{${\\rm [Fe\/H]}$}\n\\psfrag{[Mg\/Fe]}{${\\rm [Mg\/Fe]}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{trans}{trans}\n\\psfrag{metweak}{metweak}\n \\includegraphics[width=17cm]{2730fig1.eps}\n\\end{psfrags}}\n \\caption{${\\rm [Mg\/Fe] vs. [Fe\/H]}$ abundance diagram for the calibration \nsample (see text). \n}\n\\label{met}\n\\end{figure}\n\\subsection{The $U$\\\/-$V$-velocity diagram\\label{uv}}\nA classical tool for kinematic investigations is the\n$U$\\\/-$V$-velocity diagram. In Fig.~\\ref{uvms}, $U$ is plotted\nversus $V$ for the main-sequence stars.\nFor the thin-disk and the thick-disk stars, the mean values and standard\ndeviations of the two velocity components were calculated.\nThe values for the thin disk are:\n$\\left=3\\,\\rm{km~s^{-1}}$,\n$\\left=215\\,\\rm{km~s^{-1}}$,\n$\\sigma_{U_{\\rm ms}}=35\\,\\rm{km~s^{-1}}$, and\n$\\sigma_{V{\\rm ms}}=24\\,\\rm{km~s^{-1}}$.\nThe corresponding values for the thick disk are:\n$\\left=-32\\,\\rm{km~s^{-1}}$,\n$\\left=160\\,\\rm{km~s^{-1}}$,\n$\\sigma_{U_{\\rm ms}}=56\\,\\rm{km~s^{-1}}$, and\n$\\sigma_{V{\\rm ms}}=45\\,\\rm{km~s^{-1}}$.\nThe negative value of $\\left$ \nis explained\nin \\citet{fuhrmann04} as an effect of the Galactic bar.\nIndeed, nearly all thin-disk stars stay inside the\n$3\\sigma_{\\rm thin}$-limit, and all halo stars lie outside\nthe $3\\sigma_{\\rm thick}$-limit, as can be seen from Fig.~\\ref{uvms}.\nIn our previous paper, we used the $2\\sigma-$limit of the thin and\nthick-disk stars for finding thick-disk stars and\n$\\sqrt{U^2+(V-195)^2}\\ge 150\\,\\rm{km\\,s^{-1}}$ for finding halo stars.\nWe replaced these by the \nmore stringent \n$3\\sigma-$limits of the thin and thick-disk\nstars to obtain a clear-cut separation. \n\\begin{figure*}\n \\centering\n\\begin{psfrags}\n\\psfrag{V(kms^-1)}{$V\/{\\rm km~s^{-1}}$}\n\\psfrag{U(kms^-1)}{$U\/{\\rm km~s^{-1}}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{3 sig thin}{$3\\sigma_{\\rm thin}$-limit}\n\\psfrag{3 sig thick}{$3\\sigma-{\\rm thick}$-limit}\n \\includegraphics[width=17cm]{2730fig2.eps}\n\\end{psfrags}\n \\caption{$U$\\\/-$V$-velocity diagram for the calibration sample of \n main-sequence stars with $3\\sigma_{\\rm thin}$-, $3\\sigma_{\\rm\nthick}$-contours.}\n\\label{uvms}\n \\centering\n\\begin{psfrags}\n\\psfrag{e}{$e$}\n\\psfrag{Jz(kpc km s^-1)}{$J_Z\/{\\rm kpc\\,km~s^{-1}}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{RegionA}{Region~A}\n\\psfrag{RegionB}{Region~B}\n\\psfrag{RegionC}{Region~C}\n \\includegraphics[width=17cm]{2730fig3.eps}\n\\end{psfrags}\n \\caption{$J_Z$-$e$-diagram for the calibration sample of main-sequence stars}\n \\label{ecc1}\n\\end{figure*}\n\\subsection{The $Jz$-$e$-diagram \\label{jze}}\nThe $U$\\\/-$V$-plot is not the only source of information about population\nmembership.\nTwo important orbital parameters\nare the $z$-component of the angular momentum $J_Z$ and the\neccentricity of the orbit $e$. Both are plotted against each other\nfor the main-sequence stars in Fig.~\\ref{ecc1}.\nThe different populations can be distinguished well in this diagram.\nThe thin-disk stars cluster in a V-shaped area of\nlow eccentricity and $J_Z$ around $1\\,800\\,\\rm{kpc \\, km~s^{-1}}$,\nwhich we denote as region~A.\n\nIn general, the thick-disk stars possess higher\neccentricities $e>0.27$ and lower angular momenta.\nThey can be found in region~B.\nThere is also a clump of thick-disk\nstars with lower eccentricity around $0.2$ and higher $J_Z$.\nRegion~B is defined such that it excludes as many thin-disk\nstars as possible. The price that has to be paid for this is the\nloss of some thick-disk stars. But this way there is a high probability\nof identifying only those stars as thick-disk members that really\nbelong to the thick disk.\nIt should be noted that region~3 in our previous paper, which seemed\nto be different from the thin-disk and the thick-disk regions A and B, has\nproven to be just an extension of the thin-disk region to higher\neccentricities. Therefore it does not appear as an additional region in\nthis revised classification scheme.\n\nThe halo stars with very high eccentricity and smaller\n$J_Z$ can be found in Region~C, separated well from all other stars.\n\\subsection{Galactic orbits\\label{orb}}\nThe eccentricity was extracted from the Galactic orbit of the stars.\nThe classification can be confirmed by checking\nthe orbits themselves.\nTypical orbits for thin-disk, thick-disk and halo main-sequence stars\ncan be found \nin Paper~I and will not be repeated here.\n\\subsection{Population classification scheme \\label{class}}\n\nOur classification scheme (developed in Paper~I) combines three\ndifferent classification criteria:\ni) the position in $U$\\\/-$V$ diagram,\nii) the position in $J_Z$-$e$ diagram, and iii) the Galactic orbit.\n\nWe repeat some details here of the population classification scheme presented \nin Paper~I and then describe the new refinements and changes.\nWe classified white dwarfs as halo members if they had a value of\n$\\sqrt{U^2+(V-195)^2}\\ge 150\\,\\rm{km\\,s^{-1}}$ and \nlay in region~4 in the $J_Z$-$e$-diagram (see Paper~I).\n\nTo detect thick-disk white dwarfs, first all stars either\nsituated outside the $2\\sigma$-limit\nin the $U$\\\/-$V$-diagram or in region~2 or 3 in the $J_Z$-$e$-diagram\nwere selected as thick-disk candidates.\nIn a second step, each candidate was assigned a classification value\n$c$. $c$ was defined as the sum of the individual values $c_{\\rm UV}$,\n$c_{\\rm J_{Z}e}$ and $c_{\\rm orb}$ corresponding to the three\ndifferent criteria: position in $U$\\\/-$V$-diagram,\nposition in $J_Z$-$e$-diagram, and Galactic orbit.\n\nWe assigned $c_{\\rm UV}=+1$ to a star outside the $2\\sigma$-limit\nin the $U$\\\/-$V$-diagram, whereas one inside the $2\\sigma$-limit\ngot $c_{\\rm UV}=-1$.\nThe different regions in the $J_Z$-$e$-diagram are characterised by\n$c_{\\rm J_{Z}e}=-1$ for region~1, $0$ for region~3, and $+1$ for region~2.\nThe third classification value $c_{\\rm orb}$ described the orbits:\n$c=-1$ for orbits of thin-disk type and $c=+1$ for orbits of\nthick-disk type.\nThen the sum $c=c_{\\rm UV}+c_{\\rm J_{Z}e}+c_{\\rm orb}$ was computed.\nStars with $c=+3$ or $c=+2$ were considered as bona fide\nthick-disk members, and those with $c=+1$ as probable thick-disk members.\nIf $c \\le 0$, the star was classified as belonging to the thin disk.\n\nThe new classification scheme is more concise due to the elimination of\nregion~3. \nAs described in Sect.~\\ref{uv}, we also sharpened the selection criterion \nfor the $U$\\\/-$V$ plane by replacing the $2\\sigma$ by a $3\\sigma$ limit.\nA star is classified as a halo candidate if it lies either\noutside the $3\\sigma_{\\rm thick}$-limit in the $U$\\\/-$V$ diagram\nor in region~C in the $J_Z$-$e$ diagram.\nThen classification values $c_{\\rm UV}$, $c_{\\rm J_{Z}e}$, and $c_{\\rm orb}$\nare assigned to all halo candidates\nwhich take the value of $+1$ if the criterion favors a halo\nmembership and $-1$ if not.\nMore precisely:\n$c_{\\rm UV}=+1$ if the star lies outside the $3\\sigma_{\\rm thick}$-limit,\n$c_{\\rm J_{Z}e}=+1$ if the star lies in region~C,\nand $c_{\\rm orb}=+1$ if the star has a halo orbit.\nThen the sum $c=c_{\\rm UV}+c_{\\rm J_{Z}e}+c_{\\rm orb}$ is calculated.\nAll of the halo candidates with $c \\ge +1$\nare classified as halo members, the rest as thick-disk members.\n\nAll the remaining stars (not found to belong to the halo), \neither outside the $3\\sigma_{\\rm thin}$-limit in the $U$\\\/-$V$ diagram\nor in region~B in the $J_Z$-$e$ diagram, are classified\nas thick-disk candidates.\nThen the analogous procedure to the halo classification is applied:\n$c_{\\rm UV}=+1$ if the star lies outside the $3\\sigma_{\\rm thin}$-limit,\n$c_{\\rm J_{Z}e}=+1$ if the star lies in region~B,\nand $c_{\\rm orb}=+1$ if the star has a thick-disk orbit.\nIn contrast to Paper~I due to the elimination of region~3,\nthere is no longer a value 0 to be assigned\nto $c_{\\rm J_{Z}e}$; hence, we expect the number of thick-disk\ncandidates to decrease.\nAll of the thick-disk candidates with $c \\ge +1$\nare assigned to the thick-disk population, the rest to the thin-disk\npopulation.\n\\subsection{Consistency check for the kinematical classification \ncriteria \\label{consist}}\nIn this section a consistency check of our classification\nscheme is performed.\nThis is done by applying our kinematic classification criteria to\nour calibration main-sequence sample.\n\nThirty-three main-sequence stars are known to belong to the thick disk because \nof their abundance patterns\n(For reasons mentioned above we have excluded here the metal-weak thick-disk\nstars), and\n22 of them have a kinematical classification value $c \\geq +1$ and\nare classified as thick-disk stars.\nOnly one of them has $c=0$ and is thus misclassified as a thin-disk star.\nThis corresponds to a detection efficiency of about $67\\%$\nfor thick-disk members.\nIn addition to those 22 stars, six thin-disk main-sequence stars\nwith $c \\geq +1$ are misclassified as thick-disk stars, so that\nthe total number of stars classified as thick disk is 28 \nindicating a contamination with thin-disk stars of about $21\\%$.\n\n\n\\subsection{Application to the white dwarf sample of Paper~I\n\\label{application}}\n\n\nFurthermore, in order to be able to compare the results of Paper~I with\nthis paper, we applied the new classification scheme to the\n$107$ white dwarfs analysed in Paper~I.\nThe fraction of halo stars is not changed by this new scheme.\nDue to the elimination of region~3 in the $J_Z$-$e$ diagram, four stars\nlose their thick-disk candidate status, and we end up with a\ntotal number of eight thick-disk stars compared to twelve previously.\nThis reduces the local fraction of thick-disk white dwarfs from\n$11\\%$ to $7.5\\%$, and\ndemonstrates the uncertainty of kinematic population classification.\nEven higher errors are to be expected when the population separation\nis based on a single criterion such as the position in $U$\\\/-$V$ diagram alone,\nwhich is the case for most other kinematical studies of white dwarfs in the\nliterature.\n\n\\section{Kinematic population classification of the SPY white \ndwarfs\\label{popuclasswd}}\n\nWe calculated orbits and kinematic parameters for all $398$ white\ndwarfs (see Table~9\n).\nThe errors of $e$, $J_Z$, $U$, $V$, $W$ were\ncomputed with the Monte Carlo error propagation code\ndescribed in Paper~I.\nThey can be found in Table~9\nas well.\n\\begin{figure*}\n \\centering\n\\begin{psfrags}\n\\psfrag{V(kms^-1)}{$V\/{\\rm km~s^{-1}}$}\n\\psfrag{U(kms^-1)}{$U\/{\\rm km~s^{-1}}$}\n\\psfrag{HE0201}{\\footnotesize HE\\,0201}\n\\psfrag{HS1527}{\\footnotesize HS\\,1527}\n\\psfrag{WD0252}{\\footnotesize WD\\,0252}\n\\psfrag{WD1448}{\\footnotesize WD\\,1448}\n\\psfrag{WD1524}{\\footnotesize WD\\,1524}\n\\psfrag{WD2351}{\\footnotesize WD\\,2351}\n\\psfrag{WD2359}{\\footnotesize WD\\,2359}\n\\psfrag{WD2029}{\\footnotesize WD\\,2029}\n\\psfrag{3 sig thin}{$3\\sigma-{\\rm thin}$-limit}\n\\psfrag{3 sig thick}{$3\\sigma-{\\rm thick}$-limit}\n \\includegraphics[width=17cm]{2730fig4.eps}\n\\end{psfrags}\n \\caption{$U$\\\/-$V$-velocity diagram for the white dwarfs with\n$3\\sigma-{\\rm thin}$ and $3\\sigma-{\\rm thick}$ -- contours from \nFig.~\\ref{uvms}, Symbols with numbers are the white dwarfs \nmentioned in the text}\n \\label{uvwd}\n \\centering\n\\begin{psfrags}\n\\psfrag{e}{$e$}\n\\psfrag{Jz(kpc km s^-1)}{$J_Z\/{\\rm kpc\\,km~s^{-1}}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{RegionA}{Region~A}\n\\psfrag{RegionB}{Region~B}\n\\psfrag{RegionC}{Region~C}\n\\psfrag{HE0201}{\\footnotesize HE\\,0201}\n\\psfrag{HS1527}{\\footnotesize HS\\,1527}\n\\psfrag{WD0252}{\\footnotesize WD\\,0252}\n\\psfrag{WD1448}{\\footnotesize WD\\,1448}\n\\psfrag{WD1524}{\\footnotesize WD\\,1524}\n\\psfrag{WD2351}{\\footnotesize WD\\,2351}\n\\psfrag{WD2359}{\\footnotesize WD\\,2359}\n\\psfrag{WD2029}{\\footnotesize WD\\,2029}\n \\includegraphics[width=17cm]{2730fig5.eps}\n\\end{psfrags}\n \\caption{$Jz$-$e$-diagram of the white dwarfs}\n \\label{ecc2}\n\\end{figure*}\n\\subsection{The $U$\\\/-$V$-velocity diagram\\label{uv_wd}}\nIn Fig.~\\ref{uvwd}, the $U$\\\/-$V$-velocity diagram for the\nwhite dwarfs is shown together with the $3\\sigma$-limits\nof the thin and thick-disk stars from the calibration sample.\nThe white dwarfs can be divided into two main groups that appear\nto be separated from each other:\none group that is clustered mainly within the $3\\sigma_{\\rm thin}$-limit\nwith some stars just outside the $3\\sigma_{\\rm thin}$-border\nand another second group with smaller $V$ that lies outside or just\ninside the $3\\sigma_{\\rm thick}$-border.\nAll the white dwarfs belonging to the second group are marked with\nthe first letters of their names in Fig.~\\ref{uvwd}.\n\nThe second group comprises five stars outside the\n$3\\sigma_{\\rm thick}$-limit (which qualify as halo candidates according to\nSect.~\\ref{class}) HS\\,1527+0614, WD\\,0252$-$350,\nWD\\,1448+077, WD\\,1524$-$749, and WD\\,2351$-$365.\n Exceptional are WD\\,1448+077 and WD\\,1524$-$749, which have\na negative value of $V$; i.e. they move on retrograde orbits.\nThis behaviour is incompatible with disk membership and\nstrongly suggests that they belong to the halo.\n\nThe other three white dwarfs of the second group are HE\\,0201$-$0513,\nWD\\,2029+183 and WD\\,2359$-$324. Situated inside the\n$3\\sigma_{\\rm thick}$, they do not qualify as halo candidates but\nwe must check if they belong to the halo or to the thick disk by means of \nthe $J_Z$-eccentricity diagram and the orbits .\n\\subsection{The $J_z$-$e$-diagram \\label{jze_wd}}\nWe now move on to the $J_Z$-eccentricity diagram of the SPY\nwhite dwarfs (Fig.~\\ref{ecc2}).\nAgain, two groups of stars can be detected:\none first group starting in Region~A with a high-eccentricity tail in\nRegion~B, which represents the disk population, and a second\ngroup in the right part of Region~B and in Region~C.\nContrary to the main-sequence stars there is a gap in Region~B\nthat is not populated at all by white dwarfs.\nIf this is real or just due to selection effects cannot be said at this point.\n\nThe second group contains all the stars discussed individually in the previous\nsection and labeled by name in Fig.~\\ref{ecc2}.\nHE\\,0201$-$0513, since situated in Region~C, is\nadded to the list of halo candidates.\nThe two retrograde stars, WD\\,1448+077 and WD\\,1524$-$749,\ncan be distinguished easily by their negative value of $J_{\\rm Z}$.\n\\subsection{Galactic orbits}\n\nNext we inspect the Galactic orbits of the SPY white dwarfs.\nWe display some meridional plots of white dwarfs with thin-disk, thick-disk, \nor halo like orbits, respectively, in Figs. \\ref{wd0310} to \\ref{hs1527}.\n\n\nMost white dwarfs have thin-disk-like orbits, an example is\nWD\\,0310$-$688 (Figure~\\ref{wd0310}).\nSome orbits, like the one of WD\\,1013$-$010 (Fig.~\\ref{wd1013}),\nshow thick-disk characteristics.\nThe star WD\\,2029+183 mentioned earlier has a thick-disk orbit.\nFive stars (HS\\,1527+0614,\nHE\\,0201$-$0513, WD\\,0252$-$350, WD\\,2351$-$365, and WD\\,2359$-$324)\nhave chaotic halo orbits, as can be seen from \nFig.~\\ref{hs1527} in the case of \nHS\\,1527+0614.\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n \\begin{psfrags}\n \\psfrag{rho\/kpc}{$\\rho\/{\\rm kpc}$}\n \\psfrag{Z\/kpc}{$Z\/{\\rm kpc}$}\n \\includegraphics{2730fig6.eps}\n \\end{psfrags}\n }\n \\caption{WD\\,0310$-$688: a white dwarf with a thin-disk orbit}\n \\label{wd0310}\n\\end{figure}\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n \\begin{psfrags}\n \\psfrag{rho\/kpc}{$\\rho\/{\\rm kpc}$}\n \\psfrag{Z\/kpc}{$Z\/{\\rm kpc}$}\n \\includegraphics{2730fig7.eps}\n \\end{psfrags}\n }\n \\caption{WD\\,1013$-$010: a white dwarf with a thick-disk orbit}\n \\label{wd1013}\n\\end{figure}\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n \\begin{psfrags}\n \\psfrag{rho\/kpc}{$\\rho\/{\\rm kpc}$}\n \\psfrag{Z\/kpc}{$Z\/{\\rm kpc}$}\n \\includegraphics{2730fig8.eps}\n \\end{psfrags}\n }\n \\caption{HS\\,1527+0614: a white dwarf with a (chaotic) halo orbit.} \n \\label{hs1527}\n\\end{figure}\n\n\n\n\\subsection{Classification}\n\nWe used the population classification scheme presented\nin Sect.~\\ref{class} to divide the SPY white dwarfs into\nthe three different populations. We start with the halo candidates, \ne.g. with all white dwarfs that are either\nsituated outside the $3\\sigma$-limit of the thick disk in the\n$U$\\\/-$V$-velocity diagram or that lie in Region~C in\nthe $J_Z$-eccentricity diagram.\nSix white dwarfs fulfill these conditions:\nall but one lie outside the $3\\sigma_{\\rm thick}$-limit, \nand all lie in Region~C.\nTwo white dwarfs, WD\\,1448+077 and WD\\,1524$-$749,\nare on retrograde orbits characterised by a negative value\nof $V$ and $J_{\\rm Z}$.\nWhen the classification values of the halo white\ndwarf candidates are added, it is found that all of them have $c>1$ and\ntherefore belong to the halo population.\nWe have mentioned before that the star WD\\,2359$-$324, though it\ndoes not fulfill the criteria for a halo candidate, has an\norbit typical for a halo object.\nAs its error-bar places it near Region~C in the\n$J_Z$-$e$ diagram, we therefore decided to classify it as a halo object.\nThis leaves us with seven halo white dwarfs.\nDetails can be found in Table~\\ref{ha_class}.\n\n\nWe now move on to the remaining $32$ white dwarfs that lie either\noutside the $3\\sigma$-limit of the thin disk in the\n$U$\\\/-$V$-velocity diagram or that lie in Region~B in\nthe $J_Z$-eccentricity diagram.\nTwenty-seven of them have a classification value of $c>1$\nand are classified as thick-disk members, the remaining\nfive are assigned a thin-disk membership (see Table~\\ref{di_class}).\nAll the remaining white dwarfs are assumed to belong to the thin disk,\nleaving us with seven halo, $27$ thick-disk, and\n$364$ thin-disk out of the $398$ SPY white dwarfs.\n\n\\begin{table}\n\\caption[]\n{Classification values for the halo candidates. Note that WD2359$-$324 is \nclassified as a halo star despite having $c=-1$; see text \\label{ha_class}}\n\\begin{tabular}{lrrrrl}\n\\\\\n\\hline\nstar & $c_{\\rm UV}$ & $c_{\\rm J_Z-e}$ & $c_{\\rm orb}$ & $c$ & classification\\\\ \n\\hline\nHE\\,0201$-$0513 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.45cm} \nhalo \\\\\nHS\\,1527+0614 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.45cm} halo \\\\\nWD\\,0252$-$350 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.45cm} halo \\\\\nWD\\,1448+077 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.45cm} halo \\\\\nWD\\,1524$-$749 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.45cm} halo \\\\\nWD\\,2351$-$368 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.45cm} halo \\\\\nWD\\,2359$-$324 & \\hspace*{.3cm} $-$1 & $-$1 & +1 & $-$1 & \\hspace*{.45cm} \nhalo \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption[]\n{Classification values for the thick-disk candidates. \n\\label{di_class}}\n\\begin{tabular}{lrrrrl}\n\\\\\n\\hline\nstar & $c_{\\rm UV}$ & $c_{\\rm J_Z-e}$ & $c_{\\rm orb}$ & $c$ & class.\\\\ \n\\hline\nHE\\,0409$-$5154 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,0416$-$1034 & \\hspace*{.3cm} +1 & $-$1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,0452$-$3444 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,0508$-$2343 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,1124+0144 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick disk \\\\\nHS\\,0820+2503 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick disk \\\\\nHS\\,1338+0807 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHS\\,1432+1441 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0204$-$233 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0255$-$705 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0352+052 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0548+000 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,0732$-$427 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0956+045 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,1013$-$010 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1152$-$287 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1323$-$514 & \\hspace*{.3cm} +1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1327$-$083 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1334$-$678 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1410+168 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1426$-$276 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1507+021 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1531+184 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,1614$-$128 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} \nthick disk \\\\\nWD\\,1716+020 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1834$-$781 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1952$-$206 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,2029+183 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,2136+229 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,2253$-$081 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,2322$-$181 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,2350$-$083 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\section{Age estimates \\label{chap_ages}}\n\nThe seven halo and $27$ thick-disk white dwarfs were assigned\nto the respective populations by means of purely kinematic\ncriteria. Accordingly they must be old stars; therefore, we attempted to \nestimate their ages.\nA check to see whether their physical parameters, mass and effective\ntemperature, are compatible with their belonging to an old population\nmust now be made. Masses $M$ for the white dwarfs were derived\nfrom ${\\rm log}~g$ and the mass-radius relation by \\citet{wood95}.\n\nThe halo is older than $10\\,{\\rm Gyr}$.\n\\cite{bensby03} determined a mean age for the thick disk as\n$11.2 \\pm 4.3\\,{\\rm Gyr}$.\nIt is very probable that stars that are younger than $7\\,{\\rm Gyr}$\ndo not belong to the thick disk.\nThus the main-sequence life-time (plus about $20\\%$ for time spent \nduring the giant phases and the horizontal branch),\nplus the time the white dwarf has cooled down\nuntil it reaches its actual $T_{\\rm eff}$, has to be greater than\nthe age of the youngest stars of the respective populations.\nThe main-sequence life-time $\\tau_{\\rm ms}$ depends on the mass of the\nwhite dwarf progenitor and is approximately proportional to\n$\\tau_{\\rm ms} \\propto M^{-2.5}$ \\citep{kippenhahn94}.\nThe main-sequence life-time is $\\tau_{\\rm ms}=10\\,{\\rm Gyr}$ for the Sun,\n$7.9\\,{\\rm Gyr}$ for a $1.1\\,\\Msolar$ mass star,\n$4.3\\,{\\rm Gyr}$ for a $1.4\\,\\Msolar$ mass star, and\n$1.8\\,{\\rm Gyr}$ for a $2\\,\\Msolar$ mass star.\nAdding the $20\\%$ horizontal branch plus giant phase lifetime, the total\npre-white dwarf lifetimes would be\n$12\\,{\\rm Gyr}$, $9.5\\,{\\rm Gyr}$, $5.2\\,{\\rm Gyr}$,\nand $2.2\\,{\\rm Gyr}$, respectively.\n\nThe mass of the white dwarf is related to the mass of its\nprogenitor by the initial-to-final mass relation.\nUntil now, no definitive initial-to-final mass relation\nhas been established;\nhowever, different estimates exist from\ndifferent groups derived from theoretical considerations\nand from observational investigations of open clusters; \nsee e.g. \\citet{weidemann00} and \\citet{schroeder01}.\nUnfortunately no initial-to-final mass relation for the\nhalo and the thick disk has been derived yet, so we have to work with what \nis available for the\nthin disk and keep in mind that our age \nestimates are crude.\nAccording to \\citet{weidemann00}, stars with initial masses\nof $1\\,\\Msolar$, $1.1\\,\\Msolar$, $1.4\\,\\Msolar$, and $2\\,\\Msolar$ would\nevolve into white dwarfs with masses of\n$0.55\\,\\Msolar$, $0.555\\,\\Msolar$, $0.57\\,\\Msolar$, and $0.6\\,\\Msolar$, \nrespectively.\nThe initial-to-final mass relation of \\citet{schroeder01}, on\nthe other hand, yields white dwarf masses of\n$0.55\\,\\Msolar$, $0.565\\,\\Msolar$, $0.605\\,\\Msolar$, and $0.67\\,\\Msolar$.\n\nWe now estimate how long it takes for a C\/O core white dwarf to cool down to\n$20\\,000\\,{\\rm K}$, $10\\,000\\,{\\rm K}$, $8\\,000\\,{\\rm K}$, and\n$5\\,000\\,{\\rm K}$ using the cooling tracks of \\citet{wood95}.\nFor a $0.5\\,\\Msolar$ mass white dwarf, the respective cooling times\nwould be $0.05\\,{\\rm Gyr}$, $0.5\\,{\\rm Gyr}$, $0.9\\,{\\rm Gyr}$,\nand $4\\,{\\rm Gyr}$.\nFor a $0.6\\,\\Msolar$ mass white dwarf, the corresponding values are\n$0.08\\,{\\rm Gyr}$, $0.6\\,{\\rm Gyr}$, $1.1\\,{\\rm Gyr}$, and\n$6\\,{\\rm Gyr}$.\nHence, only for white dwarfs cooler than \n$8\\,000\\,{\\rm K}$ does\nthe cooling time contribute significantly to the total age.\n\n\nAll the halo white dwarfs we found have masses less than $0.55\\,\\Msolar$;\ni.e. their\nprogenitors had a pre-white dwarf life-time of more than $12\\,{\\rm Gyr}$.\nThey are all hotter than $14\\,000\\,{\\rm K}$, meaning\nthey have all cooled less than $0.5\\,{\\rm Gyr}$.\nDue to the large pre-white dwarf lifetime, their total age\nis perfectly compatible with halo membership.\nIt should be noted that the low mass of WD\\,0252$-$350 of\nonly $0.35\\,\\Msolar$ indicates that it probably does not possess a\nCO core but instead a He one.\n\n\nNow the masses and effective temperatures of the\nthick-disk white dwarfs detected in the SPY sample were likewise \nchecked. We found that four white dwarfs WD\\,0255$-$705,\nWD\\,0352+052, WD\\,1013$-$010, and WD\\,1334$-$678 have masses \nwhich imply ages of less than $7\\,{\\rm Gyr}$,\nwhich would \nmake them too young to belong to the thick disk. \n\nThese four stars are the coolest in our sample of thick-disk\ncandidates (see Table~\\ref{di_par}), with $T_{\\rm eff}$ ranging from 8800~K\nto 10600~K. \\cite{liebert05}\nderived the mass\ndistribution of 348 DA white dwarfs from the PG survey and found that the\naverage gravities and masses increase with decreasing effective \ntemperature for\n$T_{\\rm eff} < 12000$~K. A similar trend is found in the analysis of more\nthan 600 DA white dwarfs from the SPY survey (Voss et al., in prep.). \nThe physical reason is unknown, but two conjectures have been\npublished. The high masses inferred from\nspectroscopy below $\\approx$ 12000~K may actually be due to helium being\nbrought to the surface by the hydrogen convection zone \\citep{bergeron92,\nliebert05}. On the other hand, \n\\citet{koester05} suggest\nthat the treatment of non-ideal effects for the level population with the \nHummer-Mihalas \\citep{hummer88} occupation probability mechanism may \nbe insufficient for neutral perturbers that become important at lower \n$T_{\\rm eff}$. Since these effects are unaccounted for in the model \natmospheres, \nwe may have overestimated the masses\nof cool DA white dwarfs ($T_{\\rm eff} < 12000$~K).\n\nAs a result, the four cool \nwhite dwarf stars with thick-disk-like kinematics may have a \nlower mass and, therefore, a significantly larger age, one that is perhaps \neven \nconsistent with that of the thick disk. Therefore we regard them\nas very likely belonging to the thick disk as \nwell.\n\n\n\n\n\n\n\n\\begin{table}\n\\caption[]\n{Effective temperatures, surface gravities, and masses of the halo white\n dwarfs \\label{ha_par}}\n\\begin{tabular}{llll}\n\\\\\n\\hline\nstar & $T_{\\rm eff}$ & ${\\rm log}~g$ & $M$ \\\\ \n &${\\rm K}$ & ${\\rm cm \\,s^{-2}} $ & \\Msolar \\\\\n\\hline\n HS1527+0614 & 14015 & 7.80 & 0.50 \\\\\n WD1448+077 & 14459 & 7.66 & 0.44 \\\\\n WD2351$-$368 & 14567 & 7.81 & 0.51 \\\\\n WD0252$-$350 & 17056 & 7.42 & 0.35 \\\\\n WD2359$-$324 & 23267 & 7.65 & 0.47 \\\\\n WD1524$-$749 & 23414 & 7.61 & 0.45 \\\\\n HE0201$-$0513 & 24604 & 7.67 & 0.48 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption[]\n{Effective temperatures, surface gravities, and masses of the thick-disk white \ndwarfs. The four coolest stars have higher masses than the rest;\nhowever, the masses of the former may have been overestimated (see text).\n\\label{di_par}}\n\\begin{tabular}{llll}\n\\\\\n\\hline\nstar & $T_{\\rm eff}$ & ${\\rm log}~g$ & $M$ \\\\ \n &${\\rm K}$ & ${\\rm cm \\,s^{-2}} $ & \\Msolar \\\\\n\\hline\n WD1013$-$010 & 8786 & 8.19 & 0.71 \\\\\n WD1334$-$678 & 8958 & 8.11 & 0.66 \\\\\n WD0352+052 & 10234 & 8.00 & 0.60 \\\\\n WD0255$-$705 & 10574 & 8.09 & 0.65 \\\\\n\\hline\nWD1716+020 & 12795 & 7.66 & 0.43 \\\\\n WD2029+183 & 12976 & 7.73 & 0.47 \\\\\n WD0204$-$233 & 13176 & 7.75 & 0.47 \\\\\n WD1952$-$206 & 13742 & 7.78 & 0.49 \\\\\n WD0732$-$427 & 14070 & 7.96 & 0.58 \\\\\n WD1327$-$083 & 14141 & 7.79 & 0.50 \\\\\n WD1614$-$128 & 15313 & 7.74 & 0.48 \\\\\n HS1432+1441 & 15414 & 7.77 & 0.49 \\\\\n HE0508$-$2343 & 15835 & 7.71 & 0.47 \\\\\n HE1124+0144 & 15876 & 7.68 & 0.45 \\\\\n WD1426$-$276 & 17526 & 7.67 & 0.45 \\\\\n WD1834$-$781 & 17564 & 7.76 & 0.49 \\\\\n WD2350$-$083 & 17966 & 7.76 & 0.49 \\\\\n WD1323$-$514 & 18604 & 7.71 & 0.47 \\\\\n WD1507+021 & 19384 & 7.79 & 0.51 \\\\\n HE0452$-$3444 & 20035 & 7.82 & 0.53 \\\\\n WD1152$-$287 & 20185 & 7.64 & 0.45 \\\\\n WD1410+168 & 20757 & 7.74 & 0.49 \\\\\n WD2322$-$181 & 21478 & 7.88 & 0.56 \\\\\n HE0416$-$1034 & 23809 & 7.88 & 0.57 \\\\\n HS1338+0807 & 25057 & 7.73 & 0.50 \\\\\n HE0409$-$5154 & 26439 & 7.75 & 0.52 \\\\\n HS0820+2503 & 33330 & 7.69 & 0.51 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\nAn alternative explanation for the four cool DA stars discussed\nabove having \ngained thick-disk-like\norbits could be that\nthey might be run-away stars that were born in a binary system in the\nthin disk and were thereafter ejected from it.\nTwo ejection mechanisms have been suggested. The first one implies a close\nbinary system in which the primary undergoes a supernova explosion and\nreleases the secondary at high velocity \\citep{davies02}.\nThis study showed that, indeed, a large fraction of such\nbinaries are broken up when the primary explodes as a supernova.\nA large number of the secondaries receive kick\nvelocities of $100 \\mbox{-}200\\,{\\rm km s^{-1}}$ and travel on Galactic\norbits similar to those of thick-disk stars.\nThus a population of white dwarfs originating in\nthe thin disk may contribute significantly to the observed\npopulation of high-velocity white dwarfs.\n\nAnother possibility for explaining young white dwarfs with thick-disk-like \nkinematics was proposed by \\citet{kroupa02}, who suggests\na scenario for the thickening of galactic disks\nthrough clustered star formation. Massive star clusters may add kinematically\nhot components to galactic field populations.\n\n\n\nAs their masses may be overestimated, we think it is not required to\ninvoke such run-away scenarios to explain the origin of the four cool white \ndwarfs discussed above. A more natural explanation would be that we have simply\nunderestimated their ages. \n \nWe therefore classify \nthose $23$ white dwarfs where age and\nkinematics both indicate a thick-disk membership as bona fide \nthick-disk members. \nIn addition, the four cool \nwhite dwarfs are classified as probable thick-disk stars, i.e. \nall 27 stars are retained as thick-disk members. \nThis leaves us with a fraction of $2\\%$ halo and\n$7\\%$ thick-disk white dwarfs.\n\n\n\n\\section{Discussion\\label{dis}}\nWe have refined \nand sharpened \nthe population classification scheme\ndeveloped in Paper I and applied it to a kinematical\nanalysis of a sample of $398$ DA white dwarfs from the SPY project.\nCombining three kinematic criteria, i.e. the position in the $U$\\\/-$V$-diagram,\nthe position in the $J_Z$-$e$-diagram, and the Galactic orbit with age \nestimates, we found seven halo and $23$ thick-disk members.\n\nTo be able to discuss the kinematic parameters of the three different\npopulations white dwarfs, we calculated the mean value and standard\ndeviation of the three velocity components. Of interest are the\nasymmetric drift ($V_{\\rm lag}=220{\\rm km~s^{-1}}-$) for the thick-disk\nwhite dwarfs and the velocity dispersions of the white dwarfs of all\nthree populations (Tables~5--7).\nFor comparison, the corresponding values derived by \\citet{chiba00}\nand \\citet{soubiran03} for main-sequence stars are also shown.\n\nThe velocity dispersions that were found for the thin-disk\nwhite dwarfs are compatible with the ones of \\citet{soubiran03}.\nThe same is the case for the asymmetric drift and the velocity\ndispersions of the thick disk.\nHere agreement with the results of \\citet{soubiran03} is much\nbetter than with the earlier results of \\citet{chiba00}.\nThere, $\\sigma_{\\rm U}$ and $\\sigma_{\\rm V}$ of the halo white dwarfs\nare similar to the values of \\citet{chiba00}, while\nour $\\sigma_{\\rm W}$ is much smaller.\nThis is probably due to the fact that our local sample\ndoes not extend as far in the $Z$-direction as the\nsample of \\citet{chiba00} does.\nAlso with only seven halo white dwarfs, we have to account for\nsmall number statistics.\nIn general, the kinematic parameters\nof the white dwarfs of the three different populations\ndo not differ much from those of the main-sequence samples.\n\\begin{table}\n\\caption[]\n{Standard deviation of $U$, $V$,\n$W$ for the $361$ SPY thin-disk\nwhite dwarfs, $\\sigma_{\\rm U}$, $\\sigma_{\\rm V}$, and $\\sigma_{\\rm W}$\nfrom \\citet{soubiran03} are shown for comparison \\label{uvw_duetab}}\n\\begin{tabular}{lccc}\n\\hline\n &$\\sigma_{\\rm U}$ & $\\sigma_{\\rm V}$ & $\\sigma_{\\rm W}$\\\\\n &${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$\\\\\n\\hline\nThin-disk WDs & & & \\\\\n(our sample) & $34$ & $24$ & $18$\\\\\n\\hline\nThin-disk stars & & &\\\\\n(Soubiran et al.)& $39$ & $20$ & $20$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption[]\n{Asymmetric drift $V_{\\rm lag}$ and standard deviation of $U$, $V$,\n$W$ for the $27$ SPY thick-disk\nwhite dwarfs, $V_{\\rm lag}$, $\\sigma_{\\rm U}$, $\\sigma_{\\rm V}$, \nand $\\sigma_{\\rm W}$\nfrom \\citet{soubiran03}, and \\citet{chiba00} are shown for \ncomparison \\label{uvw_ditab}}\n\\begin{tabular}{lcccc}\n\\hline\n &$V_{\\rm lag}$ & $\\sigma_{\\rm U}$ & $\\sigma_{\\rm V}$ & $\\sigma_{\\rm W}$\\\\\n &${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$\\\\\n\\hline\nThick-disk WDs & & & \\\\\n(our sample)& $-51$ & $79$ & $36$ & $46$\\\\\n\\hline\nThick-disk stars & & & &\\\\\n(Soubiran et al.) & $-51$ & $63$ & $39$ & $39$\\\\\n\\hline\nThick-disk stars & & & &\\\\\n(Chiba \\& Beers) & $-20$ & $46$ & $50$ & $35$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption[]\n{Standard deviation of $U$, $V$,\n$W$ for the seven SPY halo\nwhite dwarfs, $V_{\\rm lag}$, $\\sigma_{\\rm U}$, $\\sigma_{\\rm V}$, and \n$\\sigma_{\\rm W}$\nfrom \\citet{chiba00} shown for comparison \\label{uvw_hatab}}\n\\begin{tabular}{lccc}\n\\hline\n & $\\sigma_{\\rm U}$ & $\\sigma_{\\rm V}$ & $\\sigma_{\\rm W}$\\\\\n & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$\\\\\n\\hline\nHalo white dwarfs & & &\\\\ \n(our sample)& $138$ & $95$ & $47$\\\\\n\\hline\nHalo stars & & &\\\\\n(Chiba \\& Beers) & $141$ & $106$ & $94$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe found seven halo white dwarfs in our sample, which\ncorresponds to a fraction of $2\\%$.\nIn Paper I we found $4\\%$ halo white dwarfs,\na deviation possibly due to small number statistics or to\na target selection effect.\nIn our first paper, we analysed stars from the early phase of\nSPY. This sample contained a relatively large fraction of white dwarfs\ndetected in proper motion surveys \n\\citep[][and references therein]{Luyten79, giclas78}. Therefore an\nover-representation of white dwarfs with high tangential velocities\nis not unexpected.\n\nOur value is lower than the one derived by\n\\citet{sion88},\nwho identified about $5$\\% of their sample as halo white dwarfs.\n\\citet{liebert89}, on the other hand, obtained a percentage of\n14\\% halo white dwarfs\nby classifying all stars that exceed a certain value of tangential velocity\nas halo members.\nWhen comparing those samples with ours, it has to be kept in mind that\nour selection criteria are \nsharper and allow us to separate thick-disk\nfrom halo stars. It is likely that a fraction of the white dwarfs classified \nas halo stars by\n\\citet{sion88} and \\citet{liebert89} actually belong to the thick disk.\nFurthermore, both samples suffer from the lack of radial velocity \ninformation.\n\nIt is difficult to compare our sample to the one of \\citet{oppenheimer01},\nbecause the inhomogeneous sky coverage of SPY does not allow us to\ncalculate a space density for halo white dwarfs.\nIt has to be taken into account that our sample is a magnitude\nlimited sample and thus biased towards\nhigh temperatures \\citep[mean temperature of $21\\,000\\,\\mathrm{K}$; see\nalso discussion in][]{schroeder04}, \nwhereas \\citet{oppenheimer01} analyse much cooler white\ndwarfs.\n\nClassically, halo white dwarfs are supposed to be cool stars\nthat originated from high mass progenitors.\nThe main contribution to the total ages of these white dwarfs is\nthe cooling time.\nThis work demonstrates that another\nclass of hot, low-mass halo white dwarfs exists\nwith low-mass progenitors that only recently have become white dwarfs so\nhave not had much time to cool down.\nThis makes this SPY sample\ncomplement to samples that focus on cool halo white dwarfs.\n\n\nThere are $27$ SPY white dwarfs classified as thick-disk members \nout of which \nfour are too cool to allow reliable ages to be derived.\nThis corresponds to a local fraction of thick-disk white dwarfs of\n$7\\%$ or $6\\%$, if we reject the four cool stars. \nThese values are somewhat lower than the $11\\%$ found \nby \\citet{silvestri02} but are much smaller than that of \nFuhrmann (2000)\\footnote{http:\/\/www.xray.mpe.mpg.de\/fuhrmann\/.},\nwho predicted a fraction of $17$\\% thick-disk white dwarfs.\nThe differences are possibly caused by the temperature bias mentioned above.\nAn over-representation of white dwarfs compared to\nlow mass main-sequence stars, which would require a truncated initial\nmass function as suggested by \\citet{favata97}, has not been found.\n\nThe question of whether thick-disk white dwarfs contribute\nsignificantly to the total mass of the Galaxy is very important for clarifying \nthe dark matter problem.\nThis contribution can be estimated from the results derived above.\nTo derive the densities of thin-disk and thick-disk white dwarfs,\nwe used the $1\/V_{\\rm max}$ method \\citep{schmidt68}.\nThe mass density of thick-disk over thin-disk white dwarfs\n${M_{\\rm thick}\\over M_{\\rm thin}}$ was calculated as described\nin Paper I.\nFor the thick disk we adopted the values of\n\\citet{ojha01}, scale length $l_{\\rm 0,thick}=3.7\\,{\\rm kpc}$, and\ntried two extreme values of the\nscale height, $h_{\\rm 0,thick}=0.8\\,{\\rm kpc}$ \\citep{ojha99} and\n$h_{\\rm 0,thick}=1.3\\,{\\rm kpc}$ \\citep{chen97}.\nFor the thin disk, we assumed\n$l_{\\rm 0,thin}=2.8\\,{\\rm kpc}$ \\citep{ojha01} and\n$h_{\\rm 0,thin}=0.25\\,{\\rm kpc}$, in between the values of\n\\citet{kroupa92} and \\citet{haywood97}.\nWe found ${M_{\\rm thick}\\over M_{\\rm thin}}=0.12 \\pm 0.36$ and\n${M_{\\rm thick}\\over M_{\\rm thin}}=0.19 \\pm 0.57$ for thick-disk scale \nheights of $0.8\\,{\\rm kpc}$\nand $1.3\\,{\\rm kpc}$, respectively.\nAccordingly, upper limits for ${M_{\\rm thick}\\over M_{\\rm thin}}$ are \n$0.48$ and $0.76$, respectively.\nOf course the errors are huge because of the poor statistics of the\nrelatively small thick-disk sample.\nNevertheless, it can be concluded\nthat the total mass of thick-disk white dwarfs is less than $48\\%$\n($76\\%$) of the total mass of thin-disk white dwarfs.\nTherefore the mass contribution of the thick-disk white dwarfs\nmust not be neglected, but it is not sufficient to account for the\nmissing dark matter.\n\n\\section{Conclusions\\label{con}}\nWe have demonstrated how a combination of sophisticated kinematic analysis\ntools can distinguish halo, thick-disk, and thin-disk white\ndwarfs. We identified a fraction of 2\\% halo and 7\\% thick-disk white\ndwarfs.\nMost of our thick-disk and halo white dwarfs\nare hot and possess low masses.\nOur results suggest that the mass present in halo and thick-disk\nwhite dwarfs is not sufficient for explaining the missing mass of the Galaxy.\nBut to draw definite conclusions, more data are needed.\nOur goal is to extend this kinematic analysis to all\n$1\\,000$ \ndegenerate stars\nfrom the SPY project,\nin order to have a large data base for\ndeciding on the population membership of white dwarfs\nand their implications for the mass and evolution of the Galaxy.\n\\acknowledgements { We thank D. Koester for providing us with the\nresults of his spectral analysis prior to publication and B. Voss for prolific\ndiscussions. \nE.-M. P. acknowledges support by the Deutsche\nForschungsgemeinschaft (grant Na\\,365\/2-1) and is grateful to the\nStudienstiftung des Deutschen Volkes for a grant. M. Altmann\nacknowledges support from the DLR~50~QD~0102 and from FONDAP~1501~0003. \nR.N.\\ is supported by a\nPPARC Advanced Fellowship. Thanks go to\nJ.~Pauli for interesting and fruitful discussions. \nThis research has made use of the SIMBAD database,\noperated at the CDS, Strasbourg, France \nand of DSS images based on photographic data obtained with the UK Schmidt\nTelescope. \n}\n\\bibliographystyle{aa}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSequential decison making is a common problem in many practical applications, broadly encompassing situations in which an agent must choose the best action to perform at each iteration while maximizing cumulative reward over some period of time \\cite{BouneffoufBG13,ChoromanskaCKLR19,RiemerKBF19,LinC0RR20,lin2020online,lin2020unified,NoothigattuBMCM19,lin2019split,lin2020story}.\nOne of the key challenges in sequential decision making is to achieve a good trade-off between the exploration of new actions and the exploitation of known actions. This exploration vs exploitation trade-off in sequential decision making is often formulated as the {\\em multi-armed bandit (MAB)} problem. In the MAB problem setting, given a set of bandit ``arms'' (actions), each associated with a fixed but unknown reward probability distribution ~\\cite {surveyDB,LR85,UCB,Bouneffouf0SW19,LinBCR18,DB2019,BalakrishnanBMR19ibm,BouneffoufLUFA14,RLbd2018,balakrishnan2020constrained,BouneffoufRCF17}, the agent selects an arm to play at each iteration, and receives a reward, drawn according to the selected arm's distribution, independently from the previous actions. \n\nA particularly useful version of MAB is the {\\em contextual multi-armed bandit (CMAB)}, or simply the {\\em contextual bandit} problem, where at each iteration, the agent observes a $N$-dimensional {\\em context}, or {\\em feature vector} prior to choosing an arm \\cite{AuerC98,AuerCFS02,BalakrishnanBMR18,BouneffoufBG12}.\nOver time, the goal is to learn the relationship between the context vectors and rewards, in order to make better action choices given the context \\cite{AgrawalG13}. Common sequential decision making problems with side information (context) that utilize the contextual bandit approach range from clinical trials \\cite{villar2015multi} to recommender systems \\cite{MaryGP15,Bouneffouf16,aaai0G20}, where the patient's information (medical history, etc.) or an online user profile provide a context for making better decisions about which treatment to propose or ad to show. The reward reflects the outcome of the selected action, such as success or failure of a particular treatment option, or whether an ad is clicked or not.\n\nIn this paper we consider a new problem setting referred to as {\\em contextual bandit with missing rewards}, where the agent can always observe the context but may not always observe the reward. \nThis setting is motivated by several real-life applications where the reward associated with a selected action can be missing, or unobservable by the agent, for various reasons. For instance, in medical decision making settings, a doctor can decide on a specific treatment option for a patient, but the patient may not come back for follow-up appointments; though the reward feedback regarding the treatment success is missing, the context, in this case the patient's medical record, is still available and can be potentially used to learn more about the patient's population. Missing rewards can also occur in information retrieval or online search settings where a user enters a search request, but, for various reasons, may not click on any of the suggested website links, and thus the reward feedback about those choices is missing. Yet another example is in online advertisement, where a user clicking on a proposed ad represents a positive reward, but the absence of a click can be negative reward (the user did not like the ad), or can be a consequence of a bug or connection loss.\n\nThe contextual bandit with missing rewards framework proposed here aims to capture the situations described above, and provide an approach to exploit all context information for future decision making, even if some rewards are missing. More specifically, we will combine unsupervised online clustering with the standard contextual bandit. Online clustering allows us to learn representations of all the context vectors, with or without the observed rewards. Utilizing the contextual bandit on top of clustering makes use of the reward information when it is available. We demonstrate on several real-life datasets that this approach consistently outperforms the standard contextual bandit approach when rewards are missing. \n \n\n\\section{Related Work}\n\\label{sec:related}\nThe multi-armed bandit problem provides a solution to the exploration versus exploitation trade-off \\cite {AllesiardoFB14,dj2020,Sohini2019}. This problem has been extensively studied. Optimal solutions have been provided using a stochastic formulation ~\\cite {LR85,UCB,BouneffoufF16}, a Bayesian formulation ~\\cite {T33}, and an adversarial formulation ~\\cite{AuerC98,AuerCFS02}. However, these approaches do not take into account the relationship between context and reward, potentially inhibiting overall performance.\nIn LINUCB ~\\cite{Li2010,ChuLRS11} and in Contextual Thompson Sampling (CTS)~\\cite{AgrawalG13}, the authors assume a linear dependency between the expected reward of an action and its context; the representation space is modeled using a set of linear predictors. However, these algorithms assume that the bandit can observe the reward at each iteration, which is not the case in many practical applications, including those discussed earlier in this paper. Authors in \\cite{bartok2014partial} considered a kind of incomplete feedback called \"Partial Monitoring (PM)\", developing a general framework for sequential decision making problems with incomplete feedback. The framework allows the learner to retrieve the expected value of actions through an analysis of the feedback matrix when possible, assuming both are known to the learner.\n\nIn \\cite{bouneffouf2020online}, authors study a variant of the stochastic multi-armed bandit (MAB) problem in which the context are corrupted. The new problem is motivated by certain online settings including clinical trial and ad recommendation applications. In order to address the corrupted-context setting, the author propose to combine the standard contextual bandit approach with a classical multi-armed bandit mechanism. Unlike standard contextual bandit methods, they were able to learn from all iteration, even those with corrupted context, by improving the computing of the expectation for each arm. Promising empirical results are obtained on several real-life datasets. \n\nIn this paper we focus on handling incomplete feedback in the bandit problem setting more generally, without assuming the existence of a systematic corruption process. Our work is somewhat comparable to online semi-supervised learning \\cite{Yver2009, ororbia2015online}, a field of machine learning that studies learning from both labeled and unlabeled examples in an online setting. However, in online semi-supervised learning, the true label is available at each iteration, whereas in the contextual bandit with missing rewards, only bandit feedback is available, and the true label, or best action, is unknown.\n\n\n\\section{Problem Setting} \n\n{ Algorithm \\ref{alg:CBP1} presents at a high-level the contextual bandit setting, where $x_t\\in C$ (we will assume here $C = \\mathbf{R}^N$) is a vector describing the context at time $t$, $r_{t,i} \\in [0,1]$ is the reward of the action $i$ at time $t$, and $r_t \\in [0,1]^K$ denotes a vector of rewards for all arms at time $t$. Also, $D_{c,r}$ denotes a joint probability distribution over $(x,r)$, $A$ denotes a set of $K$ actions, $A = \\{1,...,K\\}$, and $\\pi: C \\rightarrow A$ denotes a policy.\nWe operate under the linear realizability assumption; that is, there exists an unknown weight vector $\\theta^* \\in R$ with $ ||\\theta^*||\\leq 1$ so that,\n\n\\begin{equation*}\n\\forall k, t: \\; \\mathbb{E}[r_k(t) \\vert x_t] = \\theta_k^\\top x_t \n+ n_t .\\end{equation*}\n\nwhere $\\theta_k \\in \\mathbb{R}^d$ is an unknown coefficient vector associated with the arm $k$ which needs to be learned from data. Hence, we assume that the $r_{t,k}$ are independent random variables with expectation $x^\\top \\theta^*+ n_t$. with $n_t$ some measurement noise.\nWe also assume here that, the measurement noise $n_t$ is independent of everything and is $\\sigma$-sub-Gaussian for some $\\sigma >0$, i.e., $E[e^{\\phi\\, n_t} ] \\leq exp(\\frac{\\phi^2 \\sigma^2}{2})$ for all $ \\phi \\in R$.\n\\begin{definition}[Cumulative regret]\n{The regret of an algorithm accumulated during $T$ iterations is given as:\n\\begin{equation*}\nR(T) =\\sum ^{T}_{t=1} r_{t,k^*(t)} - \\sum^{T}_{t=1} r_{t,k(t)}.\n\\end{equation*}}\n\\end{definition}\n where $k^*(t)= \\text{argmax}_k x_{t}^\\top \\theta^*$is the best action at step $t$ according to $\\theta^*$.}\n\n\\begin{algorithm}[H]\n\t\\caption{ Contextual Bandit }\n\n\t\\label{alg:CBP1}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {\\bfseries }\\textbf{Repeat}\n\t\t\\STATE {\\bfseries } $(x_t,r_t)$ is drawn according to $D_{x,r}$\n\t\t\\STATE {\\bfseries }$x_t$ is revealed to the player\n\t\t\\STATE {\\bfseries } The player chooses an action $k =\\pi_t(x_t)$\n\t\t\t\t\\STATE {\\bfseries } The reward $r_t$\n\t\t\\STATE {\\bfseries } The player updates its policy $\\pi_t$\n\t\t\t\\STATE {\\bfseries } $t=t+1$\n\t\t\\STATE {\\bfseries }\\textbf{Until} t=T\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{LINUCB with Missing Rewards (MLINUCB)}\n\\label{sec:banditDL}\nOne solution for the contextual bandit is the LINUCB algorithm~\\cite{13} where the key idea is to apply online ridge regression to incoming data to obtain an estimate of the coefficients $\\theta_k$. \nIn order to make use of the context even in the absence of the corresponding reward, we propose to use an unsupervised learning approach; specifically, we use an online clustering step to retrieve missing rewards from available rewards with similar contexts. At each time step, the context vectors $x(t)$ are clustered into $N$ clusters where $N$ is selected \\emph{a-priori}.\n\nWe adapt the LINUCB algorithm for our setting, proposing to use a clustering step for imputing the reward data when missing. At each time step, we perform a clustering step on the context vectors where the total number of clusters $N$ is a hyperparameter. For each cluster $j$ we define the average reward for each arm as below:\n\\begin{equation}\n\\overline{r}_j=\\frac{\\sum_{\\tau=1}^{n_j} r_{\\tau}}{n_j}\n\\label{waverage}\n\\end{equation}\nAssuming $d_j=dist(x_t,\\gamma_j)$ is the metric used for clustering where $\\gamma_j$ is the $j^{th}$ cluster centroid and $n_j$ is the number of data points in cluster $j$, we choose the $m$ smallest $d_j$ as the closest clusters to $x_t$ and compute a weighted average of the average cluster rewards as formulated below: \\begin{equation}g(x_t)=\\frac{\\sum_{j=1}^{m}\\frac{\\overline{r}_j}{d_j}}{\\sum_{j=1}^{m}\\frac{1}{d_j}}\\end{equation}\nWhen $r_t$ is missing we assign\n$r_t=g(x_t)$. Note that if $m=1$, $g(x_t)$ is simply the average rewards of all the points within the cluster that $x_t$ belongs to. \n \n\\begin{algorithm}[H]\n\\caption{MLINUCB}\n \\label{alg:LINUCB}\n \\begin{algorithmic}[1]\n \\STATE {\\bfseries Input:} value for $\\alpha$, $b_0$, $\\textbf{A}_0$, $N$, $m$\n \\FOR{t=1 {\\bfseries to} T} \n \\STATE cluster \\{$x_1$, ... , $x_t$\\} into $N$ clusters\n \\FOR{all $k \\in K$}\n \\STATE $\\theta_k \\leftarrow$ \\textbf{A}$_{k_t}^{-1}*b_{k_t}$ \n \\STATE $p_{t,k} \\leftarrow \\theta^{\\top}_k x_{t} +\n \\alpha \\sqrt{x^{\\top}_{t}\n \\textbf{A}_{k_t}^{-1} x_{t}}$\n \\ENDFOR\n \\STATE Choose arm $k_t = \\text{argmax}_{k\\in K} p_{t,k}$,\n and observe real-valued payoff $r_t$\n \\IF{ $r_t$ available}\n \\STATE retrieve $r_t$ from data\n \\ELSE \n \n \\STATE $r_t \\leftarrow g(x_t)$.\n \n \n \n \\ENDIF \n \\STATE \\textbf{A}$_{k_t} \\leftarrow$ \\textbf{A}$_{k_t} + x_{t,k_t} x^{\\top}_{t,k_t}$\n \\STATE $b_{k_t} \\leftarrow b_{k_t} + r_t x_{t,k_t}$ \n \\ENDFOR\n \\end{algorithmic}\n\\end{algorithm}\n\nWe now upper bound the regret of MLINUCB. Note that the general CBP setting \\cite{abbasi2011improved} takes one context per arm instead for our setting of the one context share by actions. To upper bound our algorithm for the general CBP setting, we simply cast our setting as theirs by the following steps.\nWe simply choose a global vector $\\theta$ as the concatenation of the $K$ vectors, so $\\theta =[\\theta_{1},...,\\theta_{K}]$. We define a context $x_{t,k}$ per action with $x_t$, where $x_{t,k} =[...0,x_{t}^{\\top},0,... ]^{\\top}$ and $x_t$ being the $k$-th vector within the concatenation. All $A_t$,$r_t$, $b_t$ can be similarly defined from $A_{k(t)}$, $r_{k(t)}$, $b_{k(t)}$.\n\\begin{thm} \\label{thm:hlinucb}\nWith probability $1-\\delta$, where $0 < \\delta < 1$, the upper bound on the R(T) for the MLINUCB in the contextual bandit problem, $K$ arms and $d$ features (context size) is given as follows:\n\\begin{eqnarray}\nR(T)\\leq \\sigma (\\sqrt{d \\;log (\\frac{det(A_{T})^{1\/2} }{\\delta\\;det(\\mathbf{S})^{1\/2}} )}+\\nonumber \\\\ \\frac{||\\theta||}{\\sqrt{\\phi}})\\sqrt{18\\; T log(\\frac{det(A_{T})}{det(\\mathbf{S})})} \n\\end{eqnarray}\nwith $||x_t||_2 \\leq L$, with $\\mathbf{S}=\\mathbf{I}+ \\sum_{t \\in s} x_tx_t^\\top $ with $s \\subset T$ contains the contexts with missing rewards and $\\phi \\in R$\n\\end{thm}\nTheorem \\ref{thm:hlinucb} shows that MLINUCB has better upper bound compared to the LINUCB \\cite{abbasi2011improved}, where in LINUCB upper bound has $log(det(A_{T}))$ under the square root where we have $log(\\frac{det(A_{T})}{det(\\mathbf{H})})$. We can see that the upper bound depends on $H$, so more context with missing rewards better is the bound.\n\n\n\\section{Regret analysis of BILINUCB}\nWe now upper bound the regret of BILINUCB. General Contextual Bandit Problem (CBP) setting \\cite{abbasi2011improved} assumes one context per arm instead for BILINUCB setting with same context shared across arms. To upper bound regret of BILINUCB we cast our setting as general CBP setting in the following way.\nWe choose a global vector $\\theta$ as the concatenation of the $K$ vectors, so $\\theta =[\\theta_{1},...,\\theta_{K}]$. Next define a context $x_{t,k}$ per arm as $x_{t,k} =[...,0,x_{t}^{\\top},0,... ]^{\\top}$ with $x_t$ being the $k$-th vector within the concatenation. Let $\\mathbf{S_T} = \\mathcal{I}_D + \\sum_{t \\in s} x_t x_t^\\top $, where $s \\subset \\{1,\\ldots,T\\}$ contains the contexts with missing rewards up to step $T$, and let $\\mathbf{A}_T = \\mathbf{S_T} + \\sum_{t \\not\\in s} x_t x_t^\\top $. We have the following theorem regarding the regret bound up to step $T$.\n\nTheorem 1 of the main text shows that BILINUCB has better upper bound compared to the LINUCB \\cite{abbasi2011improved}, where in LINUCB upper bound has $\\log(\\det(A_{t}))$ under the square root where we have $\\log(\\frac{\\det(A_{t})}{\\det(\\mathbf{S}_T)})$. \nThe matrix $\\mathbf{S}_T$ is the sum of identity matrix $\\mathcal{I}_D$ and covariance matrix $\\Sigma_s= \\sum_{t \\in s} x_t x_t^\\top $ constructed using the contexts with missing reward. Both $\\mathcal{I}_D$ and $\\Sigma_s$ are real symmetric and hence Hermitian matrices. Further, $\\Sigma_s$ is positive semi-definite as a covariance matrix.\nSince all the eigenvalues of $\\mathcal{I}_D$ equal $1$ and since all the eigenvalues of $\\Sigma_s$ are non-negative, by Weyl's inequality in matrix theory for perturbation of Hermitian matrices, the eigenvalues of $\\mathbf{S}_T$ are lower bounded by $1$. Hence $\\det(\\mathbf{S}_T)$ which is the product of the eigenvalues of $\\mathbf{S}_T$ is lower bounded by $1$. Hence, BILINUCB which involves the term $\\frac{\\det (\\mathbf{A}_T)}{\\det (\\mathbf{S}_T)}$ has a provably better guarantee than LINUCB which involves only the term $\\det(\\mathbf{A}_T)$ (without $\\det(\\mathbf{S}_T)$).\n\\subsection{Proof of Theorem 1}\n\\begin{proof}\n\nWe need the following assumption:\nwe assume that the noise introduced by the imputed reward is heteroscedastic. \n Formally, let $\\rho : X \\rightarrow R$\n be a continuous, positive function, such that $\\epsilon_t$ is conditionally\n $\\rho(x_t)$-subgaussian, that is for all $t \\geq 1$ and $\\rho_t = \\rho(x_t)$,\n\\begin{equation}\n\\forall\\; \\lambda \\in R, \\quad E[e^{\\lambda n_t} | F_{t-1}, x_t] \\leq \\exp\\left(\\frac{\\lambda^2 \\rho_t^2}{2}\\right) \\text{ .}\\label{eq: noise assumption}\n\\end{equation}\nNote that this condition implies that the noise has zero mean, and common examples include Gaussian, Rademacher, and uniform random variables\nWe need the following lemma, \n\n\\begin{lem} \\label{lem:ct} \nAssuming that, the measurement noise $\\epsilon_t$ issatisfies assumption \\eqref{eq: noise assumption}.\nWith probability $1-\\delta$, where $0 < \\delta < 1$ and $\\theta^*$ lies in the confidence ellipsoid.\n\\begin{eqnarray*}\nC_{t}=\\{ \\theta: \\|\\theta-\\hat{\\theta}_{t}\\|_{A_{t}} \\leq c_{t} :=\\nonumber \\\\ \\rho_t \\sqrt{D \\log \\frac{\\det(\\mathbf{A}_{t})^{1\/2} \\det(\\mathbf{S}_t)^{-1\/2}}{\\delta}}+ \\|\\theta^*\\|_2\\}\n\\end{eqnarray*}\n\\end{lem}\n\nThe lemma is adopted from theorem 2 in \\cite{abbasi2011improved} using the noise being heteroscedastic. We follow the same step of proof, the main difference is that they have $\\mathbf{A}_T=\\lambda \\mathcal{I}_D+ \\sum_{t=1}^T x_tx_t^\\top$ and we have $\\mathbf{A}_T=\\mathbf{S}_T+ \\sum_{t \\not\\in s} x_tx_t^\\top $ with $\\mathbf{S}_T=\\mathcal{I}_D+ \\sum_{t \\in s} x_tx_t^\\top $ with $s \\subset T$ contains the contexts with missing rewards .\n \n$ R(t) = [ x_t^{*\\top} \\theta^*-x_t^\\top \\theta_t] = [ x_t^{*\\top} \\theta^*- x_t^\\top \\theta^l_t ]+[x_t^\\top \\theta_t^l -x_t^\\top \\theta_t]$ \n\nwhere $ \\theta^l$ is the parameter of the classical LINUCB, and then\n\n$ R(t) \\leq \\| x_t^{*\\top} \\theta^*- x_t^\\top \\theta^l_t \\|_2+\\|x_t^\\top \\theta^l_t -x_t^\\top \\theta_t\\|_2$ \n \nNow we investigate $\\| x_t^{*\\top} \\theta^*- x_t^\\top \\theta_t^l \\|_2$ and $\\|x_t^\\top \\theta_t^l -x_t^\\top \\theta_t\\|_2$ separately. \n \n \nFollowing the same step as the proof of theorem 2 in \\cite{abbasi2011improved} we also have the following,\n\n \n\n\n\n\n\n$\\| x_t^{*\\top} \\theta^*- x_t^\\top \\theta_t^l \\|_2\\leq 2 c_t \\|x_t\\|_{\\mathbf{A}_{t}^{-1}}$, and using Cauchy-Schwarz with $\\|\\theta_t^l -\\theta_t\\|_2\\leq \\epsilon_t $, we get\n\n$\\|x_t^\\top \\theta^l_t -x_t^\\top \\theta_t\\|_2 \\leq \\epsilon_t \\|x_t\\|_{\\mathbf{A}_{t}^{-1}} $ and then,\n\n$R(t) \\leq (2 c_t+\\epsilon_t) \\|x_t\\|_{\\mathbf{A}_{t}^{-1}}$\n\nSince $x^{\\top}\\theta_{t}^* \\in [-1,1]$ for all $x \\in X_t $ then we have $R(t) \\leq 2$. Therefore,\n\n$R(t) \\leq \\text{min}\\{(2 c_t+\\epsilon_t)\\|x\\|_{\\mathbf{A}^{-1}_{t}},2\\} \\leq 2( c_t+\\epsilon_t\/2) \\; \\text{min}\\{\\|x\\|_{\\mathbf{A}^{-1}_{t}},1\\}$\n\nOur bound on the imputed reward assures $\\epsilon_t \\leq c_t$. Therefore,\n\n$[R(t)]^2 \\leq 9 c_t^2 \\text{min}\\{\\|x\\|^2_{\\mathbf{A}^{-1}_{t}},1\\}$\n\nwe have,\n\n$R(T) \\leq \\sqrt{T\\sum_{t=1}^{T}[R(t)]^2}= \\sqrt{ \\sum_{t=1}^T 9 c_t^2 \\text{min}\\{\\|x\\|^{2}_{\\mathbf{A}^{-1}_{t}},1\\}}$ \n\n$R(T)\\leq 3 c_T \\sqrt{ T} \\sqrt{ \\sum_{t=1}^T \\text{min}\\{\\|x\\|^{2}_{\\mathbf{A}^{-1}_{t}},1\\}}$, with $c_{T}$ monotonically increasing\n\nsince $x \\leq 2\\,\\log(1+x)$ for $x \\in [0,1]$, \n\nwe have $\\sum_{t=1}^{T} \\text{min}\\{\\|x_t\\|^2_{\\mathbf{A}_{t}^{-1}}, 1\\} \\leq 2 \\sum_{t=1}^{T} \\log(1+\\|x_t\\|^2_{\\mathbf{A}^{-1}_t})\\leq 2 (\\log\\det(\\mathbf{A}_{T})-\\log\\det(\\mathbf{S}_T))$, \n\nhere we also use the fact that we have $\\mathbf{A}_T=\\mathbf{S}_T+ \\sum_{s=1}^T x_sx_s^\\top $ to get the last inequality. \n\n$R(T)\\leq 3 c_T \\sqrt{2(\\log\\det(\\mathbf{A}_{T})-\\log\\det(\\mathbf{S}_T))}$\n \nby upper bounding $c_{T}$ using lemma \\ref{lem:ct} we get our result.\n\\end{proof}\n\n\n\n\n\\section{Experiments}\n\nIn order to verify the proposed MLINUCB methodology, we ran the LINUCB and MLINUCB algorithms on four different datasets, three derived from the UCI Machine Learning Repository \\footnote{https:\/\/archive.ics.uci.edu\/ml\/datasets.html}: Covertype, CNAE-9, and Internet Advertisements, and one external dataset : Warfarin. The Warfarin dataset concerns the dosage of the drug Warfarin, where each record consists of a context of patient information and the corresponding appropriate dosage or action. The reward is then defined as 1 if the correct action is chosen and 0 otherwise. The details for each of these datasets are summarized in the Table \\ref{table:Synthetic}. \n\n\\begin{table}[ht]\n\t\\centering\n\t\\caption{Datasets}\n\t\\resizebox{0.6\\columnwidth}{!}{\n\t\t\\begin{tabular}{ l | c | r | r }\n\t\t\tDatasets & Instances & Features & Classes \\\\ \\hline\n Covertype & 500 000 & 95 & 7\\\\\n CNAE-9 & 1080 & 856 & 9\\\\\n Internet Advertisements & 3279 & 1558 & 2\\\\\n Warfarin\t\t\t\t & 5528 & 93 & 3 \\\\\n\t\t\t\n\t\t\\end{tabular}\n\t}\n\t\\label{table:Synthetic}\n\\end{table}\nTo evaluate the performance of MLINUCB and LINUCB we utilize an accuracy metric that checks the equality of the selected action and the best action, which is revealed for the purposes of evaluation. Defined as such, accuracy is inversely proportional to regret. In the following experiments we fix $m=1, \\alpha=0.25$ and utilize the mini batch K-means algorithm for clustering. In Table \\ref{accuracy}, we report the total average accuracies of running LINUCB and MLINUCB with 2, 5, 10, 15, and 20 clusters on each dataset.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Total average accuracy}\n\\label{accuracy}\n\\begin{tabular}{l|l|l|l|l}\n \\multicolumn{5}{c}{10\\% Missing Rewards} \\\\ \\hline\n & Covertype & CNAE-9 & Internet Ads & Warfarin \\\\ \\hline\nLINUCB & \\textbf{0.884} & 0.644 & 0.866 & 0.643 \\\\ \\hline\nMLINUCB - $N=2$ & 0.869 & 0.643 & 0.898 & 0.643 \\\\ \nMLINUCB - $N=5$ & 0.874 & 0.626 & 0.895 & \\textbf{0.656} \\\\ \nMLINUCB - $N=10$ & 0.880 & 0.664 & 0.894 & 0.650 \\\\ \nMLINUCB - $N=15$ & 0.877 & \\textbf{0.678} & \\textbf{0.902} & 0.647 \\\\ \nMLINUCB - $N=20$ & 0.878 & 0.675 & 0.898 & 0.653 \\\\ \\hline\n\\end{tabular}\n\\\\\\vspace{\\baselineskip}\n\\begin{tabular}{l|l|l|l|l}\n \\multicolumn{5}{c}{50\\% Missing Rewards} \\\\ \\hline\n & Covertype & CNAE-9 & Internet Ads & Warfarin \\\\ \\hline\nLINUCB & \\textbf{0.884} & 0.566 & 0.824 & 0.615 \\\\ \\hline\nMLINUCB - $N=2$ & 0.838 & 0.578 & 0.888 & 0.630 \\\\ \nMLINUCB - $N=5$ & 0.847 \t\t & 0.546 & 0.896 & \\textbf{0.641} \\\\ \nMLINUCB - $N=10$ & 0.863 & 0.592 & 0.897 & 0.640 \\\\ \nMLINUCB - $N=15$ & 0.854 & \\textbf{0.608} & \\textbf{0.903} & 0.638 \\\\ \nMLINUCB - $N=20$ & 0.853 & 0.592 & 0.901 & 0.639 \\\\ \\hline\n\\end{tabular}\n\\\\\\vspace{\\baselineskip} \n\\begin{tabular}{l|l|l|l|l}\n\n \\multicolumn{5}{c}{75\\% Missing Rewards} \\\\ \\hline\n & Covertype & CNAE-9 & Internet Ads & Warfarin \\\\ \\hline\nLINUCB & \\textbf{0.880} & 0.483 & 0.786 & 0.610 \\\\ \\hline\nMLINUCB - $N=2$ & 0.784 & 0.461 & 0.881 & 0.594 \\\\ \nMLINUCB - $N=5$ & 0.797 & 0.494 & 0.890 & 0.612 \\\\ \nMLINUCB - $N=10$ & 0.837 & \\textbf{0.521} & 0.887 & \\textbf{0.624} \\\\ \nMLINUCB - $N=15$ & 0.824 & 0.500 & 0.891 & 0.600 \\\\ \nMLINUCB - $N=20$ & 0.819 & 0.493 & \\textbf{0.896} & 0.611 \\\\ \\hline \n\\end{tabular}\n\\vspace{\\baselineskip}\n\\end{table}\n\n\nAs the MLINUCB regret upper bound is lower than the LINUCB regret upper bound when $\\epsilon$ is small, minimizing clustering error is critical to performance. Accordingly, successful MLINUCB operates on the assumption that the context vectors live in a manifold that can be described by a set of clusters. Thus MLINUCB has the potential to outperform LINUCB when this manifold assumption holds, specifically when the number of clusters chosen adequately describes the structure of the context vector space. Visualizing the context vectors suggests that some of our test datasets violate this assumption, some respect this assumption, and when an appropriate number of clusters is chosen, MLINUCB performance aligns as expected. \n\nConsider the Internet Advertisements and Warfarin datasets, where 2D projections of the context vectors capture the majority of the variance in the context vector space, 100.0\\% and 98.2\\% respectively. In Figures \\ref{Advertisement_Acc} and \\ref{warfarin_acc} the projected context vector spaces appear clustered, not randomly scattered, and MLINUCB outperforms LINUCB for most choices of $N$, the number of clusters. The Internet Advertisements dataset yields the best results - when switching from LINUCB to MLINUCB algorithms, accuracy jumps from $86.6\\%$ to $90.2\\%$ when $25\\%$ of the reward data is missing, from $82.4\\%$ to $90.3\\%$ when $50\\%$ of the reward data is missing, and from $78.6\\%$ to $89.6\\%$ when $75\\%$ of the reward data is missing.\n\nAlthough the 2D projections of the Covertype and CNAE-9 context vectors in Figures \\ref{covtree1_acc} and \\ref{CNAE_acc} appear well clustered, both projections only capture a small amount of the variance in the context vector space, $29.7\\%$ in the Covertype dataset and $13.9\\%$ in the CNAE-9 dataset. MLINUCB results do not show improvement for the cases tried for Covertype dataset suggesting that the Covertype dataset violates the manifold assumption for the context space. However in the CNAE-9 dataset, we see that MLINUCB outperforms LINUCB for most choices of $N$, which supports the observation that the context space is clustered. \n\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 29.7\\% of the variance in the Covertype dataset.]{\\includegraphics[scale=0.2]{covtree1nc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{covtree1_avg_acc.png}}\n\\caption{Covertype}\n\\label{covtree1_acc}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 13.9\\% of the variance in the CNAE-9 dataset.]{\\includegraphics[scale=0.2]{CNAEnc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{CNAE_avg_acc.png}}\n\\caption{CNAE-9}\n\\label{CNAE_acc}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 13.9\\% of the variance in the CNAE-9 dataset.]{\\includegraphics[scale=0.19]{Advertisementnc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{Advertisement_avg_acc.png}}\n\\caption{Internet Advertisements}\n\\label{Advertisement_Acc}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 98.2\\% of the variance in the Warfarin dataset.]{\\includegraphics[scale=0.2]{warfarin_datanc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{warfarin_avg_acc.png}}\n\\caption{Warfarin}\n\\label{warfarin_acc}\n\\end{figure}\n\nTaking a more in depth look at the CNAE-9 dataset, in Figure \\ref{CNAE9_alpha}, we vary LINUCB and MLINUCB's common hyperparameter $\\alpha$, which controls the ratio of exploration to exploitation, and see that MLINUCB continues to result in higher accuracies than LINUCB for most $\\alpha$. \n\nNote that $N$, the number of clusters, is a hyperparameter of the algorithm and while initialized \\emph{a-priori}, it could be changed and optimized online as more context vectors are revealed. Alternatively, we could leverage clustering algorithms that do not initialize $N$ \\emph{a-priori} and learn the best $N$ from the available data.\n\n\\begin{figure}[h]\n\\label{CNAE9_alpha}\n\\centering\n{\\includegraphics[scale=0.11]{CNAE_alpha.png}}\n\\caption{LINUCB and MLINUCB cumulative accuracies on CNAE-9 across various $\\alpha$}\n\\end{figure}\n\n\n\\section{Conclusions and Future Work}\nIn this paper we studied the effect of data imputation in the case of missing rewards for multi-arm bandit problems. We prove an upper bound for the total regret in our algorithm following the CBP upper bound. Our MLINUCB algorithm shows improvements over LINUCB in terms of total average accuracy for most cases. The main observation here is that when the context vector space lives in a clustered manifold, we can take advantage of this structure and impute the missing reward at each step given similar context in previous events. \nA very obvious next step is to try using the weighted average introduced in equation \\ref{waverage} with $m$ greater than $1$. This would use more topological information from the context feature space and wouldn't rely on a single cluster. Additionally, the algorithm doesn't rely on a fixed value for $N$ so we could optimize the value of $N$ at each event using some clustering metric to find the best $N$ at each time. This work can also be extended by replacing the simple clustering step with more complex methodologies to learn a representation of the context vector space, for example sparse dictionary learning.\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzftrv b/data_all_eng_slimpj/shuffled/split2/finalzzftrv new file mode 100644 index 0000000000000000000000000000000000000000..f485f8c0e62773ce1f20c4ac251c0738fc8f654f --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzftrv @@ -0,0 +1,5 @@ +{"text":"\\section*{Abstract}\nThe ability to register image data to a common coordinate system is a\ncritical feature of virtually all imaging studies that\nrequire multiple subject analysis,\ncombining single subject data from multiple\nmodalities, or both. However, in spite of the\nabundance of literature on the subject and the existence of\nseveral variants of registration algorithms, their practical\nutility remains\nproblematic, as commonly acknowledged even by developers of these\nmethods because the complexity of the problem\nhas resisted a general, flexible, and robust theoretical and\ncomputational framework.\n\nTo address this issue, we present\na new registration method that is\nsimilar in spirit to the current state-of-the-art technique\nof diffeomorphic mapping, but is more general and flexible. The\nmethod utilizes a Hamiltonian formalism and constructs registration as\na sequence of symplectomorphic maps in\nconjunction with a novel phase space regularization based on\nthe powerful entropy spectrum pathways (ESP)\nframework.\n\nThe main advantage of the ESP\nregularized symplectomorphic approach versus the standard\napproach of coordinates-only diffeomorphic mapping lies in use of\na common metric that remains valid even\nwith image dependent regularization.\nMoreover, the fusion of the\nHamiltonian framework with the ESP theory goes beyond just providing\nan alternative spatially varying smoothing strategy - it provides an\nefficient and straightforward way to combine multiple\nmodalities.\n\nThe method is demonstrated on the three different magnetic resonance\nimaging (MRI) modalities routinely used for human neuroimaging\napplications by mapping between high resolution anatomical\n(HRA) volumes, medium resolution diffusion weighted MRI\n(DW-MRI) and HRA volumes, and low\nresolution functional MRI (fMRI) and HRA volumes.\nThe typical processing time for high\nquality mapping ranges from less than a minute to several minutes on a\nmodern multi core CPU for typical high resolution\nanatomical ($\\sim 256^3$ voxels) MRI volumes.\n\nFor validation of the framework we developed a panel of deformations\nexpressed in analytical form that includes deformations based on known\nphysical processes in MRI that\nreproduces various distortions and artifacts typically present in\nimages collected using these different MRI modalities. Use\nof this panel allows us to quantify repeatability and reproducibility\nof our method in comparison to several available alternative\napproaches. The panel can be used in future studies especially for\nquantitative clinical validation of\ndifferent registration approaches.\n\nThe registration tool will be available as a part of the\nQUEST suite from the UCSD Center for Scientific Computation in Imaging\n(CSCI).\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nModern imaging systems are increasingly capable of acquiring\ndata sensitive to a wide range of physical parameters at multiple\nresolutions, thus offering greater sensitivity to structural and\ndynamical information in complex biological systems. However, these\ntechnological advancements present the increasingly important\ntheoretical and computational challenge of how to rigorously and\nefficiently combine, or \\textit{register}, such data in order to be\nable to accurately detect and quantify subtle and complex system\ncharacteristics.\n\n\nThe ability to register image data to a common coordinate system\nis a critical feature of virtually all imaging studies that require\nquantitative statistical analysis of group populations, as well as for\ncombining single subject modalities. Consequently, this subject has\nbeen the focus of a great deal of research. This has been a focus in\ncomputational neuroanatomy which has motivated the developed of\n\\textit{diffeomorphic} registrations methods\n\\citep{pmid15551602,pmid17761438,pmid17354694,pmid18979814,pmid22194239}\nfor which faster and more efficient algorithms continue to be\ndeveloped\\citep{pmid26221678, pmid24968094, pmid19709963,\n pmid18979813, pmid23685032}, as well as various regularizations\n\\citep{pmid24409140, pmid20879371} and additional enhancements such as\nlocal-global mixture, contrast changes, multichannel mapping, etc\n\\citep{pmid24217008, pmid21197460, pmid22972747}, and the use of\nprobabilistic diffeomorphic registration methods \\citep{pmid25320790,\n pmid20879365}. These registration advancements are important to\ngroup analyses and the development of standard atlases\n\\citep{pmid20347998,pmid24579121, pmid15501084, pmid23769915,\n pmid21995026, pmid21276861, pmid17354780} which serve a critical\nrole in the standardization of studies. The emergence of diffusion\ntensor imaging (DTI) methods and their variants for connectivity\nstudies required the extension of diffeomorphic registration methods\nto accommodate tensor data \\citep{pmid23880040, pmid22941943,\n pmid20382233, pmid19694253, pmid21134814, pmid19398016,\n pmid21316463, pmid25433212, pmid25333121, pmid24579120,\n pmid23286046, pmid22156979, pmid21761677, pmid18390342}. These\nmethods have had a profound effect on the success of numerous\nscientific studies on important clinical issues such as Alzheimer's\nand traumatic brain injury \\citep{pmid24936424, pmid23333372,\n pmid23322456, pmid20879457, pmid20211269, pmid17999940}, as well as\nstudies in other organs (cardiac, lungs, etc) \\citep{pmid24505703,\n pmid22481815, pmid16093505, pmid15508155, pmid20363173}. Another\nimportant and even more challenging task is a multi-modal registration\n(i.e. registering T1 and T2 images, or T1 and DTI, etc), as the\noptimal choice of an appropriate objective function is unknown.\nDesigning and evaluating a universal algorithm that can fit various\napplications (among subjects, multi-modal within-subject, multi-modal\nacross subjects) is an important problem that needs to be addressed,\nas existing approaches do not currently posses such universality (see,\ne.g., \\citet{pmid23739795} for a comprehensive review).\n\nIn spite of the abundance of literature and the existence of several\nvariants of diffeomorphic algorithms their practical appeal are still\nrather limited (possibly due to an interplay of a variety of reasons\n-- speed, accuracy, robustness, complexity, repeatability, etc), as\ncommonly acknowledged even by developers of these registration\nmethods. For example citing the developer of one of the relatively\nbroadly used approaches -- Large Deformation Diffeomorphic Metric\nMapping \\citep{pmid19398016, pmid24579120, pmid23286046, pmid21761677,\n pmid17999940, pmid21521665} -- ``applications of the LDDMM framework\non volumetric 3D medical images still remain limited for practical\nreasons'' \\citep{LDDMM}. Two large and thorough comparison studies\n(i.e. \\citet{pmid19195496, Ribeiro2015}) also confirm that although\ncurrently available methods are in general able to perform the\nregistration task with varying degrees of success (although some are\nexceedingly slow and some are not particular accurate), the practical\nuse limitations seem to drive an interest in improvements at least in\nterms of speed and accuracy.\n\nThe recent review paper \\citep{pmid27427472} \nconducted a retrospective analysis of the past two decades\nof the field of medical image registration since publication of the\noriginal review \\citep{pmid10638851}. It is alarming again that the\nmain conclusion of this twenty years \nretrospective is that in spite of all the progress in the\nfield of registration ``the two major problems mentioned in\n\\citep{pmid10638851} -- validation of registration methods and\ntranslation of these to the clinic -- are major problems still, which\nhave even been aggravated by the elaboration of registration\nmethods.''\n\nTo address these issues we present in this paper a new method that is\nsimilar in spirit to diffeomorphic mapping, but is more general and\nflexible. The transformation is developed within a Hamiltonian\nformalism \\citep{vialard:tel-00400379,pmid26643025,pmid19059343} in\nwhich not just the spatial coordinates are considered, but the\nentirety of phase spac\n, which is a called a\n\\textit{symplectomorphism}. \nThis theoretical construct enables a novel\nflexible, accurate, and robust computational method based on a\nsequence of energy shell transformations. The incorporation\nof phase space constraints allows us to use the same simple metric\non the space of diffeomorphisms that remains valid even with image dependent\nregularization, something that is missing in currently available\nmethods.\n\nThe generality of the Hamiltonian framework facilitates the inclusion\nof powerful prior information for spatially varying regularization in\nphase space using our recently developed method of entropy spectrum\npathways (ESP) \\citep{Frank:2014pre}. This is in contrast with the\ncurrent state-of-the-art approaches that introduce\nregularization as a differential form (almost always with constant\ncoefficients) acting on the map itself (see\ne.g.~\\citet{Beg2005,doi:10.1137\/140984002,5204344}), that effectively\napply regularization as an additional post processing step, thus\ncreating additional problems, especially for validation and comparison\nbetween different approaches and even different regularization\ntechniques. Even the existence of several techniques for spatially\nvarying smoothing strategies that have been recently proposed\n\\citep{pmid25485406,pmid25333122} do not remediate this validation\nissue (and this is in addition to being of rather limited practical\nutility, possibly adding even more speed--accuracy--complexity issues\nthan providing solutions). Generally speaking, the Hamiltonian\napproach facilitates validation of different regularizations without\ndestroying or modifying the metric on the space of diffeomorphisms.\n\nThe importance and the main advantage of symplectomorphic approach\nversus coordinates only diffeomorphic mapping can be understood from\nthe fact that use of the same common metric allows quantitative\nassessment of differences between registrations as well as evaluation\nof performance for different regularization schemes. An existence of\nvolumetric\/surface\/line measures allows accurate comparison of\nfeatures between subvolumes, surface areas or linear curves.\n\nWhat is even more important is that this fusion of the Hamiltonian\nframework with the ESP theory goes beyond just providing an\nalternative spatially varying smoothing strategy. It provides an\nefficient and straightforward way to combine multiple modalities, for\nuse in tractography, structural and functional connectivity,\netc. (although the details of implementation go beyond the subject of\nthis paper and will be reported elsewhere). \n\nOur method also incorporates fast, accurate, and flexible spatial\npreconditioning using our spherical wave decomposition (SWD)\n\\citep{swd}. The SWD approach uses fast FFT--based algorithms to\nexpand images in spherical wave modes and therefore allows to do image\nresampling, scaling, rotating and filtering with the highest possible\norder of polynomial accuracy, but at a fraction of a time.\n\nThe method is validated on a well characterized numerical phantom and\nthen demonstrated on a set of the ``standard'' neuro-MRI data\nacquisitions (HRA, DTI, rsFMRI) routinely collected at our UCSD Center\nfor FMRI (CFMRI). We demonstrate the ability to accurately\nco-register the data volumes in computational times significantly\nfaster and more accurately than current state-of-the-art methods. \nThe resulting image volumes also demonstrate previously unobserved image\ncontrasts that suggest the ability of our method to uncover more\nsubtle and important structural features in the data.\n\nIt is well known that different MRI acquisition schemes and protocols\nmay include a variety of incompatible distortions and artifacts due\nnot only to variations of scanner hardware and\npulse sequence designs but also due to intrinsic\nvariations in individual subject morphology, as well as just due to\nsimple motions. Thus validation of a registration method's\nability to disentangle the complex interplay of the acquisition\ndetails with the physical effects producing distortions within any\nparticular individuals brain is an exceedingly non-trivial problem.\nTherefore, in order to facilitate a more\nquantitative validation of all these different\nconditions we developed a panel of deformations defined analytically\nand based on well-known physical effects present in the\ndifferent MRI modalities. The deformations from the panel can be\napplied to images of different modalities and acquisition condition\nand potentially can be appropriate for quick and robust validation in\nclinical settings as well. This validation approach is somewhat\nsimilar to Gaussian deformations used in \\citet{pmid15896998}, but our\npanel includes deformations that can be attributed to a variety of\nreal physical processes present in different acquisition protocols and\nmodalities (i.e.~twist, whirl, stretch, etc).\n\nTo evaluate the practical aspects of our implementation and to\ndemonstrate the competitiveness of our approach we compared the\naccuracy and speed of phantom registration with several commonly used\nregistration methods that are often reported as top performers\n\\citep{pmid19195496} in either speed or accuracy (ANTs Diffeomorphic\nDemons, ANTs SyN, FSL FNIRT and AFNI 3dQWarp). While a variety of\nsimilarity metrics are available, for this paper we used a simple\nRoot-Mean-Square Deviation (RMSD) as a metric to evaluate\nthe accuracy of numerical phantom registration and\nwall--clock time (that characterizes the human perception of the\npassage of time from the start to the completion of a task, referred\nto as \\textit{time} afterwards) as a practical and intuitive\nmeasure of the algorithms efficiency.\n\nIn summary, this paper utilizes a Hamiltonian formalism to develop a\nnew approach to non-linear flexible image registration. The method\nbuilds a diffeomorphic mapping as a sequence of symplectomorphic maps\nwith each map embedded in a separate energy shell. The approach adds a\nnovel phase space regularization based on the\npowerful entropy spectrum pathways framework. The framework provides a\nunique opportunity to tailor image details into the\nregularization scheme by choosing an image derived regularization\nkernel. A spherical wave decomposition is applied as a\npowerful preconditioning tool in the position\ndomain to allow accurate and fast\ninterpolation, resampling and estimation of fixed shape rotation and\nscale. The result is an efficient and versatile method capable of\nfast and accurate registration of a variety of volumetric images of\ndifferent modalities and resolutions.\n\n\\section{Symplectomorphic mapping}\n\\label{sec:theory}\nWe introduce the Hamiltonian function\n$\\mathcal{H}(\\mvec{q},\\mvec{p})$ on a fixed Cartesian grid $\\mvec{x}$\nas\n\\begin{equation}\\label{eq::hamiltonian}\n\\mathcal{H}(\\mvec{q},\\mvec{p}) = \\frac{1}{2V}\\int\\l[\\mvec{p}^2 +\n \\l(I_0(\\mvec{x})-I_1(\\mvec{q}))\\r)^2\\r]d\\mvec{x}.\n\\end{equation}\nHere $I_0$ and $I_1$ are two multidimensional images defined on the\nsame fixed Cartesian grid $\\mvec{x}$, $V$ is the measure (volume) of\nthe reference $I_0$ image domain ($V\\equiv \\int d\\mvec{x}$), and\n$(\\mvec{q}(\\mvec{x},t),\\mvec{p}(\\mvec{x},t))$ is a set of canonical\ncoordinates, that define a time dependent mapping from Cartesian grid\n$\\mvec{x}$ to a new curvilinear grid\n$\\mvec{y}\\equiv\\mvec{q}(\\mvec{x},t)$, such that initially at $t=0$ the\ngrids are identical, i.e. ($\\mvec{q}(\\mvec{x},0),\\mvec{p}(\\mvec{x},0))\n\\equiv (\\mvec{x},0$).\n\nThe Hamiltonian \\cref{eq::hamiltonian} defines a flow at each location\non a fixed grid through a system of Hamilton's\nequations\n\\begin{align}\n\\label{eq::flow:q}\n\\dd{\\mvec{q}}{t} &= \\Dv{\\mathcal{H}}{\\mvec{p}} \\equiv\n\\mvec{p}\\\\\n\\label{eq::flow:p}\n\\dd{\\mvec{p}}{t} &=-\\Dv{\\mathcal{H}}{\\mvec{q}} \\equiv\n\\l(I_0-I_1\\r)\\D{I_1}{\\mvec{q}}\n\\end{align}\nwhere $\\delta\\mathcal{H}\/\\delta ...$ denotes variational (or\nfunctional) derivative.\n\nThe flow defined by \\cref{eq::flow:q,eq::flow:p}\nis called a \\textit{Hamiltonian flow} and takes place in the space\nof the coordinates $(\\mvec{q},\\mvec{p})$, which is called\n\\textit{phase space}. Diffeomorphisms in this phase space are\ncalled \\textit{Hamiltonian diffeomorphisms} or\n\\textit{symplectomorphisms} since a phase space is a symplectic\nmanifold. Thus symplectomorphisms preserve the symplectic structure\n(including the volume) of phase space. This is a very important\nfeature that will allow the generation of a shell-like sequence of\ntransformations suitable for volumetric measurements and\nquantifications.\n\nBecause the Hamiltonian function \\cref{eq::hamiltonian} and the\nreference image $I_0$ are defined on a Cartesian grid $\\mvec{x}$ we do\nnot calculate the curvilinear gradient $\\Ds{I_1}{\\mvec{q}}$\ndirectly. Instead we express $I_1(\\mvec{q})$ as a function on\na Cartesian grid $I_1(\\mvec{q}(\\mvec{x},t))$ and use\nthe chain rule to evaluate the curvilinear gradient through\na gradient on Cartesian grid $\\Ds{I_1}{\\mvec{x}}$ and Jacobian\n$J\\equiv \\Ds{\\mvec{q}}{\\mvec{x}}$ as\n$\\Ds{I_1}{\\mvec{x}}(\\Ds{\\mvec{q}}{\\mvec{x}})^{-1}$.\n\nAn evolution of the Jacobian with time can be obtained by differentiating\nthe position equation (\\cref{eq::flow:q}) on a fixed grid, giving a\nclosed set of equations\n\\begin{align}\n\\label{eq::flow1:q}\n\\dd{\\mvec{q}}{t} &= \\mvec{p}\\\\\n\\label{eq::flow1:p}\n\\dd{\\mvec{p}}{t} &=\\l(I_0-I_1\\r)\\D{I_1}{\\mvec{x}}J^{-1}\\\\\n\\label{eq::flow1:J}\n\\dd{J}{t} &=\\D{\\mvec{p}}{\\mvec{x}}\n\\end{align}\nIntegrating these equations with initial conditions\n$\\mvec{q}(\\mvec{x},0)=\\mvec{x}$, $\\mvec{p}(\\mvec{x},0)=0$, and\n$J(\\mvec{x},0) = \\mathds{1}$ generates a symplectomorphic\ntransformation $\\mvec{x} \\rightarrow \\mvec{q}(\\mvec{x},t)$. A new\nmetric can be defined for the position part $\\mvec{q}$ of the\ncanonical coordinates by introducing the metric tensor $G \\equiv\n\\{g_{ij}\\} = {(J^{-1})}^{T} J^{-1}$, where indices $i$ and $j$\ncorrespond to derivatives over $q_i$ and $q_j$ components of the\ncurvilinear coordinates $\\mvec{q}$ such that in Euclidean space\n$g_{ij}=\\delta_{ij}$ where $\\delta_{ij}$ is the Kronecker delta. The\nmetric tensor is important for providing accurate measures of line and\nsurface properties using the curvilinear coordinate system\n$\\mvec{q}$. For example, a length of a curve parameterized by\n${\\mvec{x}}(s)$ with a parameter $s$ between zero and one in Cartesian\nspace can be expressed using the metric tensor and curvilinear mapping\nas\n\\begin{equation}\n\\int\\limits^{1}_{0}\\left|\\dd{\\mvec{x}}{s}\\right|ds=\n\\int\\limits^{1}_{0}\\sqrt{g_{ij}\\dd{q^i}{s}\\dd{q^j}{s}}ds,\n\\end{equation}\nwhere repeated indices $i$ and $j$ represent summation.\n\nTo ensure that the transformation is symplectomorphic at every\nlocation on a fixed grid $\\mvec{x}$ during numerical integration we set\na small constant $\\epsilon$ and impose a requirement that both the\nJacobian and the inverse Jacobian are bounded by this constant, i.e.\n\\begin{equation}\n\\label{eq::jacobian:bound}\n\\epsilon < |J(x,t)| < \\epsilon^{-1},\n\\end{equation}\nFor the majority of the results presented in the paper a\nvalue of $\\epsilon=0.01$ was used.\nWhen the Jacobian becomes sufficiently close to zero the further\nintegration does not make sense as it will\nnot be able to guarantee either the symplectomorphic or\ndiffeomorphic properties of the flow (even\nnumerical stability of the solution can be compromised). Therefore,\nwhen the condition of \\cref{eq::jacobian:bound} is violated we stop\nnumerical integration, freeze the flow, and restart the integration\n(i.e., setting $t=0$) beginning at a new set of phase space\ncoordinate $\\{\\mvec{q}^{(n)}(\\mvec{x},0),\n\\mvec{p}^{(n)}(\\mvec{x},0)\\}$ where $n$ is the number of restart\ntimes. Since the Hamiltonian is an operator that describes the\n``energy'' of a system, we refer to these $n$ different sets of\ninitial conditions as \\textit{energy shells}. Each restart of the\nintegration therefore represents the initiation of a new energy\nshell.\n\nThe new initial conditions that define the energy shells are\nrelated to the stopping point of the coordinates in the previous\nenergy shell by the following conditions:\n\\begin{align}\n\\label{eq::embedded:ic}\n\\mvec{q}^{(n)}(\\mvec{x},0) &= \\mvec{q}^{(n-1)}(\\mvec{x},t^{(n)}-t^{(n-1)}),\\\\\n\\mvec{p}^{(n)}(\\mvec{x},0) &= 0,\\\\\nJ^{(n)}(\\mvec{x},0) &= \\mathds{1}\n\\end{align}\nRepeating this sequence of initial conditions therefore\ngenerates a set of shell-embedded symplectomorphic transformations\nsuch that the total transformation is diffeomorphic with the Jacobian\ndefined as a product of $J^{(n)}$\n\\begin{align}\n\\label{eq::jacobian:shell}\nJ\\l(\\mvec{x},t\\r) = &J^{(n)}\\l(\\mvec{x},t-t^{(n)}\\r)\\cdot\n J^{(n-1)}\\l(\\mvec{x},t^{(n)}-t^{(n-1)}\\r) \\cdot \n\\ldots \\cdot\n J^{0}\\l(\\mvec{x},t^{(1)}\\r)\n\\end{align}\nIt is worth noting that this updating\nequation for the Jacobian effectively results in an updating of the\nmetric tensor $G = {(J^{-1})}^{T} J^{-1}$ that\ncharacterizes the local geometry and assures volume preservation.\n\nWe would like to emphasize that our use of Hamiltonian framework\nprovides a major advantage over conventional approaches in both\nefficiency and accuracy. For example, similar considerations for\nlimiting the Jacobian were employed in \\citet{pmid18290061} where\nEuler equations of viscous flow were used to describe the displacement\nfield on a fixed grid. The introduction of fixed Eulerian reference\nframe resulted in frequent use of costly and inaccurate template\nregridding procedure that is completely avoided\nby our formulation.\n\nAn important practical implementation issue is that the\nnumber of shells $n$ does not have to be introduced in advance and can\nbe determined based on overall convergence (or even devised from\nrunning time constraints). In our numerical implementation the shells\nwere terminated as soon as $I_1 \\rightarrow I_0$ convergence condition\n\\begin{align}\\label{eq::convergence}\n\\int\\l[ \n \\vphantom{\n \\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n)})\\r)^2 \n \\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n-1)})\\r)^2}\n\\r. & \\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n)})\\r)^2\n-\\l.\n\\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n-1)})\\r)^2\\r]d\\mvec{x} <0\n\\end{align}\nwas not satisfied.\n\n\\section{Entropy spectrum pathways as a phase space regularization}\n\\label{sec:esp}\nThe form of Hamiltonian function used in \\cref{eq::hamiltonian}\nassumes only local input from difference between $I_0$ and $I_1$\nimages to the flow momentum $\\mvec{p}$ at every point on the fixed\ngrid $\\mvec{x}$. A more reasonable assumption would be an inclusion of\nsome information relevant to the structure of $I_0$ and $I_1$\nimages. One possible (and by far the most straightforward) way to\nprovide this structure based preconditioning\nis the entropy spectrum pathways (ESP) approach\n\\citep{Frank:2014pre} that takes into account nearest neighbor\ncoupling between adjacent grid locations.\n\nThe ESP approach starts with generating the coupling density\n$Q(\\mvec{x},\\mvec{x}^\\prime)$ which can be as simple and trivial as just the\nadjacency matrix \n\\begin{eqnarray}\nQ(\\mvec{x},\\mvec{x}^\\prime) = \n\\begin{cases}\n1 & \\mbox{if $\\mvec{x}$ and $\\mvec{x}^\\prime$ are connected}\\\\\n0 & \\mbox{if $\\mvec{x}$ and $\\mvec{x}^\\prime$ are not connected}\n\\end{cases}\n\\end{eqnarray}\nor may in general include a strength of coupling\nthrough some kind of coupling potentials that may depend on the grid\npositions. The ESP approach solves the generalized eigenvalue problem\n\\begin{equation}\\label{eq::esp:eigenproblem}\n\\lambda\\psi(\\mvec{x}) = \\int Q(\\mvec{x},\\mvec{x}^\\prime) \\psi(\\mvec{x}^\\prime)\nd\\mvec{x}^\\prime,\n\\end{equation}\nfinding the largest eigenvalue $\\lambda$ and corresponding eigenvector\n$\\psi(\\mvec{x})$ and then constructs the quantity\n\\begin{equation}\\label{eq::esp:tp}\n\\rho(\\mvec{x}^\\prime,\\mvec{x}) = \\frac{Q(\\mvec{x},\\mvec{x}^\\prime)\n \\psi(\\mvec{x}^\\prime)}{\\lambda\\psi(\\mvec{x})}\n\\end{equation}\ncalling it the transition probability density for transition between\ngrid locations $\\mvec{x}$ and $\\mvec{x}^\\prime$. The square of the\neigenvector $\\psi(\\mvec{x})$ is called the equilibrium probability\n$\\mu(\\mvec{x})$ in the sense that it represents\nthe stationary solution that satisfies the stationary point condition\n\\begin{equation}\\label{eq::esp:ep}\n\\mu(\\mvec{x}^\\prime) = \\int\n\\rho(\\mvec{x}^\\prime,\\mvec{x})\n\\mu(\\mvec{x})\nd \\mvec{x}\n\\end{equation}\n\n\\cref{eq::esp:tp} can be included in \\cref{eq::hamiltonian} to take\ninto account nonlocal effects and provide a way of regularization\nby defining a non-local Hamiltonian\n\\begin{align}\n\\label{eq::hamiltonian:non:local}\n\\mathcal{H}^{nl}(\\mvec{q},\\mvec{p}) =& \\frac{1}{2V}\\int \\int\n\\l[\n\\delta(\\mvec{x},\\mvec{x}^\\prime) \\mvec{p}^2 \\r.\n+ \\l.\n\\rho(\\mvec{x},\\mvec{x}^\\prime)\n\\l(I_0(\\mvec{x}^\\prime)-I_1(\\mvec{q}))\\r)^2\\r]d\\mvec{x}d\\mvec{x}^\\prime,\n\\end{align}\nhere $\\delta(\\mvec{x},\\mvec{x}^\\prime)$ is Dirac delta function,\n$\\mvec{q}\\equiv\\mvec{q}(\\mvec{x}^\\prime,t)$ and\n$\\mvec{p}\\equiv\\mvec{p}(\\mvec{x}^\\prime,t)$. This\nnonlocal expression for the Hamiltonian function produces non-local\nHamilton's equations \n\\begin{align}\n\\label{eq::flowR:q}\n\\dd{\\mvec{q}}{t} &= \\mvec{p}\\\\\n\\label{eq::flowR:p}\n\\dd{\\mvec{p}}{t} &=\\int\\l[\\rho(\\mvec{x},\\mvec{x}^\\prime)\n\\l(I_0-I_1\\r)\\D{I_1}{\\mvec{x}}J^{-1}\n\\r]d\\mvec{x}^\\prime\n\\\\\n\\label{eq::flowR:J}\n\\dd{J}{t} &=\\D{\\mvec{p}}{\\mvec{x}}\n\\end{align}\nwhere the momentum equation (\\cref{eq::flowR:p}) is the\nnon-local version of \\cref{eq::flow1:p} that now includes the\nconvolution of a local potential (gradient of squared image difference\nin our case) with a kernel $\\rho(\\mvec{x},\\mvec{x}^\\prime)$ that\ndepends on the coupling between grid locations.\n\nAlternatively the non-local Hamiltonian function can be specified as\n\\begin{align}\n\\label{eq::hamiltonian:non:local1}\n\\mathcal{H}^{nl}(\\mvec{q},\\mvec{p}) =& \\frac{1}{2V}\\int \\int\n\\rho(\\mvec{x},\\mvec{x}^\\prime) \n\\l[\n\\mvec{p}^2 \n+\n\\l(I_0(\\mvec{x}^\\prime)-I_1(\\mvec{q}))\\r)^2\\r]d\\mvec{x}d\\mvec{x}^\\prime,\n\\end{align}\nproviding alternative non-local form for the coordinate equation (\\cref{eq::flow1:q}) as well\n\\begin{align}\n\\label{eq::flowR:qn}\n\\dd{\\mvec{q}}{t} &= \n\\int\\rho(\\mvec{x},\\mvec{x}^\\prime)\\,\\mvec{p}\\,\nd\\mvec{x}^\\prime\n\\end{align}\n\n\nAssuming that the coupling density $Q(\\mvec{x},\\mvec{x}^\\prime)$ does\nnot depend on position $\\mvec{x}$ but depends only on a difference\nbetween them (i.e. $Q(\\mvec{x},\\mvec{x}^\\prime) \\equiv\nQ(\\mvec{x}-\\mvec{x}^\\prime)$), the ESP scheme can provide a variety of\nposition independent regularization kernels often used as convolution\nfilters in image registration \\citep{pmid17761438}. As a trivial\nexample, an eigenvalue problem (\\cref{eq::esp:eigenproblem}) for\nposition independent Gaussian coupling density\n$Q(\\mvec{x}-\\mvec{x}^\\prime)=\\exp(-(\\mvec{x}-\\mvec{x}^\\prime)^T S\n(\\mvec{x}-\\mvec{x}^\\prime)))$ in infinite $n$-dimensional domain has\nmaximum eigenvalue $\\lambda = \\sqrt{\\pi^n\/\\det{S}}$ and a trivial\neigenvector $\\psi(\\mvec{x})=\\mathrm{const}$, resulting in the commonly\nused Gaussian regularization kernel. This simple\nillustration is merely meant to demonstrate that the commonly used\nGaussian kernel is naturally derived from our very general\nprocedure. In practice, more complex coupling schemes can provide\nmore informative prior information, resulting in more robust warping\nschemes.\n\nWe would like to emphasize the significant\nadvantages that ESP regularization provides. Its general\nformulation \\citep{Frank:2014pre} is probabilistic in nature and\nprovides a framework for the incorporation of available information.\nIn the present context of image registration it naturally provides a\nmechanism to incorporate information from either or both of the\n$I_0$ and $I_1$ images. The position dependent coupling naturally\ncreates image dependent regularization. Moreover, the ESP\napproach can also include any information that is not present in\nthe images themselves but known \\textit{a priori} and related to\nimages in some quantitative way can be easily included into the\ncoupling scheme with some sort of linear or nonlinear\nparameterization. We have recently demonstrated this ability\nto incorporate multiple priors in ESP coupling in the related\nproblem of multi-modal parameter estimation \\citet{quna}, where the\nsymplectomorphic registration method of this paper was used for \nregistration of multiple modalities.\nAdditionally, incorporation of the ESP method into the\nHamiltonian formalism provides a\nsimple and efficient way for introduction of different image matching\nterms by modification of the position--based part of either\nlocal or nonlocal Hamiltonian function. This provides great\nflexibility for tailoring the method to specific applications.\n\n\\section{Spherical waves decomposition as a position domain preconditioning}\n\\label{sec:swd}\n\nThe set of Hamilton's equations\n(\\cref{eq::flowR:q,eq::flowR:p,eq::flowR:J}) used in the previous\nsections to generate a sequence of energy shell-embedded\nsymplectomorphic transformations (\\cref{eq::jacobian:shell}) requires\nequal dimensionality of images $I_0$ and $I_1$. However, in many cases\nthe images to be registered are of different spatial resolutions so\nthat some form of interpolation is required. To provide an\neffective way to do position domain resampling, interpolation,\nfiltering and estimation of best orthogonal transform in a single step\nwe used the spherical waves decomposition (SWD) approach \\citep{swd}.\n\nThe SWD approach uses fast algorithms to expand both $I_0$ and $I_1$\nimages in spherical wave modes\n\\begin{align}\\label{eq::swd}\nf_{lmn}^{\\{0,1\\}} =& \\int_0^a\\int_0^\\pi\\int_0^{2\\pi}\nI_{\\{0,1\\}}(r,\\theta,\\phi)R_{nl}(r)\nY_{l}^{m\\star}(\\theta,\\phi) r^2 dr\n\\sin\\theta\nd\\theta d\\phi,\n\\end{align}\nwhere $Y_{l}^{m\\star}(\\theta,\\phi)$ are the spherical harmonics, and\n$R_{nl}(r)$ can be expressed through the spherical Bessel function\n\\begin{equation}\\label{eq::bessels}\nR_{ln}(r)=\\frac{1}{\\sqrt{\\mathcal{N}_{ln}}}j_l(k_{ln}r),\n\\end{equation}\nwith an appropriate choice of normalization constants\n$\\mathcal{N}_{ln}$ and the discrete spectrum wave numbers $k_{ln}$\ndetermined by the boundary conditions. The number of modes\n($l,m=0\\dots L_{max}$ and $n=1\\dots N_{max}$) are determined by the\nhighest image resolution. The details of definitions of the spherical\nharmonics $Y_{l}^{m}(\\theta,\\phi)$ and spherical Bessel Functions\n$j_l(r)$ can be found in \\citet{swd}. The interpolation and\nresampling are then implemented as fast inverse spherical wave\ntransform\n\\begin{equation}\\label{eq::swd:inv}\nI_{\\{0,1\\}}^{NL}(r,\\theta,\\phi) =\n\\sum_{n=1}^{N}\\sum_{l=0}^{L}\\sum_{m=-l}^{l}\n\\mathcal{F}_{lmn}f_{lmn}^{\\{0,1\\}}R_{ln}(r)Y_{l}^{m}(\\theta,\\phi),\n\\end{equation}\nusing appropriate grid locations $(r,\\theta,\\phi)$ and assigning\n$f_{lmn}$ to zeros for modes with $n>N_{max}$ or $l,m>L_{max}$. A\nvariety of low\/band\/high pass filters can be used for frequency domain\nfilter $\\mathcal{F}$ following the standard image processing\ntechniques.\n\nThe scale and the amount of rigid rotation between images can be\neasily and effectively estimated using the decomposition of\nthe radial and spherical parts using the partial transforms\n\\begin{align}\\label{eq::swd:inv:rtp}\nI_{\\{0,1\\}}^{N}(r) &=\\frac{1}{2\\sqrt{\\pi}}\n\\sum_{n=1}^{N}\n\\frac{1}{\\sqrt{\\mathcal{N}_{0n}}}\n\\mathcal{F}_{00n}f_{00n}^{\\{0,1\\}}j_{0}(k_{0n}r),\\\\\nI_{\\{0,1\\}}^{L}(\\theta,\\phi) &=\n\\sum_{l=0}^{L}\n\\frac{1}{\\sqrt{\\mathcal{N}_{l1}}}\n\\sum_{m=-l}^{l}\\mathcal{F}_{lm1}f_{lm1}^{\\{0,1\\}}\nY_{l}^{m}(\\theta,\\phi),\n\\end{align}\nand finding the parameters of the similarity transformation (scale $s_r$ and\nrotation angles $\\theta_r$ and $\\phi_r$) by solving the two (one and two\ndimensional) minimization problems\n\\begin{align}\\label{eq::rigid}\ns_r &= \\arg \\min_{s_r}\n\\int\\limits_{0}^{R_{max}}\n\\l[\\l(I_{0}^{N}(r)\\r)^2-\\l(I_{1}^{N}(s_r r)\\r)^2\\r] dr,\\\\\n(\\theta_r,\\phi_r) &= \\arg\n\\min_{\\theta_r \\phi_r}\n\\int\\limits_{0}^{2\\pi}\n\\int\\limits_{0}^{\\pi}\n \\l[\\l(I_{0}^{L}(\\theta,\\phi)\\r)^2 - \n\\l(I_{1}^{L}(\\theta -\n \\theta_r,\\phi-\\phi_r)\\r)^2\\r] d\\theta d\\phi,\n\\end{align}\nusing small number of modes ($L a \\ge 0$, \n\\[\n \\frac{1}{x}\n \\#\\big\\{ n \\le x : a < d_n\/\\log p_n \\le b \\big\\} \n \\sim\n \\int_a^b\n \\e^{-t}\n \\dd t\n \\quad \n (x \\to \\infty).\n\\]\nHowever, we do not even know of any specific limit point of the \nsequence $(d_n\/\\log p_n)$, except for $0$ and $\\infty$, \nthe former having been known for just a decade, thanks to the \ngroundbreaking work of Goldston--Pintz--Y{\\i}ld{\\i}r{\\i}m \n\\cite{GPY}.\n(The latter follows from a 1931 result of Westzynthius \n\\cite{WES}.) \n\nThis limit point lacuna notwithstanding, Hildebrand and Maier \n\\cite{HM} showed in 1988 that a positive (but unspecified) \nproportion of nonnegative real numbers are limit points of \n$(d_n\/\\log p_n)$.\nMore recently, the second author, Banks and Maynard \\cite{BFM} \nhave shown that in fact at least $12.5\\%$ of nonnegative real \nnumbers are limit points of $(d_n\/\\log p_n)$.\nThe proof strategy in \\cite{BFM} incorporates an \n``Erd{\\H o}s--Rankin'' type construction for producing long gaps \nbetween consecutive primes into the celebrated Maynard--Tao sieve, \nwhich was originally developed to produce short gaps between \nprimes.\nMore recently still, Ford, Green, Konyagin, and Tao \\cite{FGKT}, \nand (independently) Maynard \\cite{MAY2}, have settled the \nnotorious ``Erd{\\H o}s--Rankin problem'' by showing that $\\infty$ \nis a limit point of $(d_n\/R(p_n))$, where%\n\\footnote{%\nWe define $\\log_2 T \\defeq \\log\\log T$, \n$\\log_3 T \\defeq \\log\\log\\log T$ and so on. \n}\n$\n R(T)\n \\defeq \n \\log T \\log_2 T \\log_4 T\/(\\log_3 T)^2\n$.\n\nWe are therefore motivated to study limit points of \n$(d_n\/R(p_n))$.\nUsing basically the same strategy as in \\cite{BFM}, and the work \nof Ford, Green, Konyagin, and Tao \\cite{FGKT}, \nPintz \\cite{PIN2,PIN3} has shown that at least $25\\%$ of \nnonnegative real numbers are limit points of $(d_n\/R(p_n))$.\nIn fact, Pintz's result is that the same statement holds if the \nnormalizing function $R(T)$ is replaced by any function --- \nsubject to certain technical conditions --- that tends to infinity \nno faster than $R(T)$, for example $\\log T\\log_2 T\/(\\log_3 T)^2$.\n\nFord, Green, Konyagin, Maynard and Tao \\cite{FGKMT} have actually \nshown that, for infinitely many $n$, $d_n \\gg R(p_n)\\log_3 p_n$.\nThe purpose of this paper is to fully integrate the work of the \nfive-author paper \\cite{FGKMT} into the study of limit points of \nnormalized prime gaps initiated in \\cite{BFM, PIN2, PIN3}.\nIn so doing, we extend the aforementioned result of Pintz in three \nways. \n\nFirst, we show that the normalizing function $R(T)$ may be \nreplaced by any ``reasonable'' function that tends to infinity \nmore slowly than $R(T)\\log_3 T$, for example\n$R_1(T) = \\log T\\log_2 T\/\\log_3 T$.\nSecond, we show that the $25\\%$ may conditionally be improved to \n$33\\frac{1}{3}\\%$ or even $50\\%$ on a certain conjecture \nconcerning the level of distribution of the primes.\nThird, we also consider ``chains'' of normalized, consecutive gaps \nbetween primes (cf.\\ Theorem \\ref{thm:chains}).\n\nPrecisely what we mean by a ``reasonable'' function is best \nexplained in context, so we defer the statement of our main result \nto \\S\\ref{sec:BFM} (cf.\\ Theorem \\ref{thm:general}).\nExamples of ``reasonable'' functions are $\\log_6 T$, \n$\\sqrt{\\log T}$, $\\log_2 T\/\\sqrt{\\log_3 T}$, $(\\log T)^{7\/9}$, \n$\\log T$, $R(T)$, $R_1(T)$ and $R_1(T)\\log_5 T$.\nAny one of these could replace $R_1(T)$ in the following special \ncase of Theorem \\ref{thm:general}, which will serve as a \nplaceholder.\n\n\\begin{theorem}\n \\label{thm:main}\nLet $d_n \\defeq p_{n+1} - p_n$, where $p_n$ denotes the $n$th \nsmallest prime, and let $\\LP[R_1]$ denote the set of limit points \nin $[0,\\infty]$ of the sequence $(d_n\/R_1(p_n))_{p_n \\ge T_0}$, \nwhere \n\\[\n R_1(T)\n \\defeq \n \\log T \\log_2 T\/\\log_3 T\n\\]\nand $T_0$ is large enough so that $\\log_3 T_0 \\ge 1$.\nGiven any five nonnegative real numbers $\\alpha_1,\\ldots,\\alpha_5$ \nwith $\\alpha_1 \\le \\cdots \\le \\alpha_5$, we have \n$\n \\{\\alpha_j - \\alpha_i : 1 \\le i < j \\le 5\\} \n \\cap \n \\LP[R_1]\n \\ne \n \\emptyset.\n$\n\\end{theorem}\n\nAs in \\cite[Corollary 1.2]{BFM}, one may deduce from Theorem 1.1 \nthat, with $\\lambda$ denoting the Lebesgue measure on $\\RR$, \n\\begin{equation}\n\\label{eq:BFM1.4}\n \\lambda([0,X] \\cap \\LP[R_1])\n \\ge \n X\/(4(1 + 1\/2 + 1\/3 + 1\/4))\n \\quad (X \\ge 0), \n\\end{equation} \nand%\n\\footnote{%\nHere, by $o(1)$ we mean a positive quantity that tends to zero as \n$X$ tends to infinity.\n} \n(with an ineffective $o(1)$),\n\\begin{equation}\n\\label{eq:BFM1.3}\n \\lambda([0,X] \\cap \\LP[R_1])\n \\ge \n (1 - o(1))\n X\/4\n \\quad (X \\to \\infty).\n\\end{equation}\nAs we will see, assuming a certain variant of the \nElliott--Halberstam conjecture (cf.\\ Hypothesis \\ref{hyp:EH} \nbelow), one has\n$\n\\{\\alpha_2 - \\alpha_1, \\alpha_3 - \\alpha_1, \\alpha_3 - \\alpha_2\\} \n \\cap \\LP[R_1] \n \\ne \n \\emptyset\n$\nfor any {\\em three} nonnegative real numbers \n$\\alpha_1 \\le \\alpha_2 \\le \\alpha_3$, with corresponding \nimprovements to \\eqref{eq:BFM1.4} and \\eqref{eq:BFM1.3}\n(viz.\\ $2 = 3 - 1$ replaces $4 = 5 - 1$).\n\n\\subsection*{Acknowledgments}\n\nThe second author gratefully acknowledges the hospitality of \nBrigham Young University, where the work on this paper commenced.\n\n\n\\section{Notation and terminology}\n \\label{sec:notation}\n\nWe rely heavily on the paper \\cite{FGKMT} of Ford, Green, \nKonyagin, Maynard and Tao, and we follow their notation and \nconventions.\nWe explain these conventions here, among others, for completeness' \nsake.\n\n\\begin{enumerate}[label=---]\n \\item The set of all primes is denoted by $\\bP$; $p,q,s$ stand \n for primes; $p_n$ denotes the $n$th smallest prime.\n \\item For $a,b \\in \\ZZ$, we define \n $a \\pod{b} \\defeq \\{a + bc : c \\in \\ZZ\\}$.\n %\n Thus, $a_1 \\equiv a_2 \\pod{b}$ if and only if \n $a_1 \\pod{b} = a_2 \\pod{b}$. \n \\item A finite set $\\cH$ of integers is {\\em admissible} if and \n only if $\\cH$ is not a complete set of residues modulo $p$, \n for any prime $p$.\n \\item We say an integer is {\\em $x$-smooth} ($x \\in \\RR$) if and \n only if its prime divisors are all less than or equal to \n $x$. \n \\item For $n \\in \\ZZ$ and $\\cH \\subseteq \\ZZ$, we define \n $n + \\cH \\defeq \\{n + q : q \\in \\cH\\}$. \n \\item For statements $S$, $\\ind{S} \\defeq 1$ if $S$ is true and \n $\\ind{S} \\defeq 0$ if $S$ is false.\n \\item The cardinality of a set $\\cS$ is denoted by $\\#\\cS$ or \n $\\#(\\cS)$.\n %\n The indicator function for $\\cS \\subseteq \\cT$ (with $\\cT$ \n clear in context) is denoted $\\ind{\\cS}$.\n %\n That is, for $t \\in \\cT$, \n $\\ind{\\cS}(t) \\defeq \\ind{t \\in \\cS}$. \n \\item We write $\\PP$ for probability and $\\EE$ for expectation.\n \\item Boldface symbols such as $\\bX$ or $\\ba$ denote random \n variables, while non-boldface symbols such as $X$ or $a$ \n denote their deterministic counterparts.\n %\n Vector-valued random variables are indicated in arrowed \n boldface, for instance \n $\\vec{\\ba} = (\\vec{\\ba}_s)_{s \\in \\cS}$ denotes a random \n tuple of random variables indexed by the set $\\cS$.\n \\item If $\\bX$ takes at most countably many values, we define the \n {\\em essential range} of $\\bX$ to be the set of all $X$ \n such that $\\PP(\\bX = X) \\ne 0$.\n \\item If $E$ is an event of nonzero probability, \n \\[\n \\PP(F \\mid E)\n \\defeq \n \\frac{\\PP(F \\land E)}{\\PP(E)}\n \\]\n for any event $F$, and \n \\[\n \\EE(\\bX \\mid E)\n \\defeq \n \\frac{\\EE(\\bX \\ind{E})}{\\PP(E)}\n \\]\n for any absolutely integrable real-valued random variable \n $\\bX$.\n %\n If $\\bY$ is another random variable taking at most \n countably many values, we define the conditional \n probability $\\PP(F \\mid \\bY)$ to be the random variable \n that equals $\\PP(F \\mid \\bY = Y)$ on the event $\\bY = Y$ \n for each $Y$ in the essential range of $\\bY$, and similarly \n define the conditional expectation $\\EE(\\bX \\mid \\bY)$ to \n be the random variable that equals $\\EE(\\bX \\mid \\bY = Y)$ \n on the event $\\bY = Y$.\n \\item Throughout, $x$ denotes a parameter to be thought of as \n tending to infinity.\n %\n \\item Thus, $o(1)$ signifies a quantity that tends to zero as \n $x \\to \\infty$ and $X \\sim Y$ denotes that \n $X = (1 + o(1))Y$.\n \\item Expressions of the form $X = O(Y)$, $X \\ll Y$ and $Y \\gg X$ \n all denote that $|X| \\le c|Y|$ throughout the domain of \n $X$, for some constant $c > 0$.\n \\item The constant $c$ is to be taken as independent of any \n parameter unless indicated otherwise, as in \n $X \\ll_{\\delta,A} Y$ for instance, in which $c$ depends on \n $\\delta$ and $A$.\n \\item We write $X = O_{\\le}(Y)$ to denote that one can take \n $c = 1$.\n \\item We write $X \\asymp Y$ to denote that $X \\ll Y \\ll X$.\n\\end{enumerate}\n\n\n\\section{Proof strategy}\n \\label{sec:outline}\n \nLet $x$ be a large number and set \n$\n y \\defeq cx\\log x\\log_2 x\/\\log_3 x\n$, \nwhere $c > 0$ is a certain small constant.\nFord, Green, Konyagin, Maynard and Tao \\cite{FGKMT} show that if \n$C$ is large enough, then there exists a vector \n$(c_p \\pod{p})_{p \\le Cx}$ of residue classes for which \n\\[\n \\big((x,y] \\cap \\ZZ \\big) \n \\setminus \\, {\\textstyle \\bigcup_{p \\le Cx} c_p \\pod{p}}\n =\n \\emptyset.\n\\]\nThus, if $b \\pod{W}$ is the residue class modulo \n$W \\defeq \\prod_{p \\le Cx} p = \\e^{(1 + o(1))Cx}$ for which \n$b \\equiv -c_p \\pod{p}$ for each $p \\le Cx$, then for \n$n \\equiv b \\pod{W}$ with $n + x > Cx$, \n\\[\n \\bP \\cap (n + x,n + y] = \\emptyset.\n\\]\n\nWe generalize this slightly by proving that if $\\cH$ is any set of \n$K$ primes in $(x,y]$ with $K \\le \\log x$ (say), the residue \nclasses may be chosen so that \n\\[\n \\big((x,y] \\cap \\ZZ \\big) \n \\setminus \\, {\\textstyle \\bigcup_{p \\le Cx} c_p \\pod{p}}\n =\n \\cH\n\\]\nand hence \n\\[\n \\bP \\cap (n + x,n + y] = \\bP \\cap n + \\cH.\n\\]\nNote that $\\cH$, being a set of $K \\le \\log x$ primes greater than \n$p_K = O(K\\log K)$, is admissible. \n\nNow let $M \\ge 2$ be an integer with $M \\mid K$ and \nlet $\\cH = \\cH_1 \\cup \\cdots \\cup \\cH_M$ be a partition of $\\cH$ \ninto $M$ subsets of equal size.\nAs was shown in \\cite{BFM}, with $M = 9$, a smaller choice of $y$ \nand a minor technical condition on $\\cH$, the Maynard--Tao sieve \nmethod establishes that for large $N$, there exists \n$n \\in (N,2N] \\cap b \\pod{W}$ and a pair $i < j$ for which \n\\[\n \\#(\\bP \\cap n + \\cH_{i}), \\#(\\bP \\cap n + \\cH_{j}) \\ge 1,\n\\] \nprovided $K$ is sufficiently large and $W \\le N^{\\eta}$ for some \nsmall $\\eta$.\n\nChoosing $j - i$ to be minimal, we obtain a pair of consecutive \nprimes in $n + \\cH$.\nWe may carefully choose our primes in $\\cH$ so that the spacings \nbetween them grow faster than $x\/\\log x$ but slower than $y$.\n\nActually, for reasons related to level of distribution and \n``Siegel'' zeros, we require that $W$ not be a multiple of a \ncertain putative ``exceptional'' modulus less than\n$N^{O(\\eta)}$.\nThe largest prime divisor $p'$ of this exceptional modulus, if it \nexists, satisfies $p' \\gg \\log_2 N^{\\eta} \\gg \\log x$.\nFor this reason, we introduce a set $\\cZ$ of ``unusable'' primes, \nwhich has the properties of $\\{p'\\}$.\nTheir effect is negligible.\n\nPintz \\cite{PIN3} has very recently given an elegant \nsimplification of part of this argument, which we take advantage \nof in this paper, and which shows that one can take $M = 5$.\n\n\n\\section{A modification of Ford--Green--Konyagin--Maynard--Tao}\n \\label{sec:FGKMT}\n \n\\subsection{Main results}\n \\label{subsec:fgkmtmain}\n\nGiven a large number $x$ we define \n\\begin{equation}\n \\label{eq:fgkmt3.1}\n y \\defeq cx\\frac{\\log x \\log_3 x}{\\log_2 x},\n\\end{equation}\nwhere $c$ is a certain (small) fixed positive constant, and \n\\begin{equation}\n \\label{eq:fgkmt3.2}\n z \\defeq x^{\\log_3 x\/(4\\log_2 x)}.\n\\end{equation}\nWe then define\n\\begin{align}\n \\cS & \\defeq \\{\\text{$s$ prime} : (\\log x)^{20} < s \\le z\\}, \\label{eq:fgkmt3.3} \\\\\n \\cP & \\defeq \\{\\text{$p$ prime} : x\/2 < p \\le x\\}, \\label{eq:fgkmt3.4} \\\\\n \\cQ & \\defeq \\{\\text{$q$ prime} : x < q \\le y\\}. \\label{eq:fgkmt3.5} \n\\end{align}\nFor vectors of residue classes \n$\\vec{a} \\defeq (a_s \\pod{s})_{s \\, \\in \\, \\cS}$ \nand \n$\\vec{b} \\defeq (b_p \\pod{p})_{p \\, \\in \\, \\cP}$, \nwe define sifted sets\n\\[\n S(\\vec{a})\n \\defeq \n \\ZZ \n \\, \\setminus \\, \n \\textstyle{\\bigcup_{s \\, \\in \\, \\cS}} \\,\n a_s \\pod{s}\n\\quad \n \\text{and}\n \\quad \n S(\\vec{b})\n \\defeq \n \\ZZ \n \\, \\setminus \\, \n \\textstyle{\\bigcup_{p \\, \\in \\, \\cP}} \\,\n b_p \\pod{p}.\n\\] \nWe note that in view of the prime number theorem (with suitably \nstrong error term) and \\eqref{eq:fgkmt3.1}, \n\\begin{equation}\n \\label{eq:Qsize}\n \\#\\cQ \n = \n \\frac{y}{\\log x}\n \\bigg(\n 1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\n \\bigg).\n\\end{equation}\nFinally, let $K$ be any natural number satisfying \n\\begin{equation}\n \\label{eq:Kbnd}\n K \\le \\log x.\n\\end{equation}\nBy \\eqref{eq:Qsize}, we may suppose $x$ is large enough so that \n$\\cQ$ contains at least $K$ primes.\nSince $p_K \\ll K\\log K$, we may also suppose that $p_K \\le x$.\nWe fix any \n\\begin{equation}\n \\label{eq:cHdef}\n \\cH \n \\defeq \\{q_1,\\ldots,q_K\\}\n \\subseteq \\cQ \n \\quad \n \\text{with}\n \\quad \n \\#\\cH = K.\n\\end{equation}\nNote that $\\cH$, being a set of $K$ primes larger than $p_K$, is \nan admissible set.\n\n\\begin{theorem}[Sieving for primes]\n \\label{thm:fgkmt2}\nFor all sufficiently large $x$, there exist vectors of residue \nclasses \n$\\vec{a} = (a_s \\pod{s})_{s \\, \\in \\, \\cS}$ \nand \n$\\vec{b} = (b_p \\pod{p})_{p \\, \\in \\, \\cP}$ \nsuch that $\\cH \\subseteq S(\\vec{a}) \\cap S(\\vec{b})$ and \n\\begin{equation}\n \\label{eq:fgkmt3.6}\n \\#(\\cQ \\cap S(\\vec{a}) \\cap S(\\vec{b}))\n \\ll\n \\frac{x}{\\log x}, \n\\end{equation}\nwhere the implied constant is absolute.\n\\end{theorem}\n\nThe only difference between Theorem \\ref{thm:fgkmt2} and \n\\cite[Theorem 2]{FGKMT} is our additional requirement that \n$\\cH \\subseteq S(\\vec{a}) \\cap S(\\vec{b})$.\nUnsurprisingly, the proof of Theorem \\ref{thm:fgkmt2} follows that \nof \\cite[Theorem 2]{FGKMT} very closely, even verbatim in many \nparts.\nNevertheless, the details must be checked, and by including them \nhere we are also able to point out the minor differences between \nthe two proofs.\n\nWe now introduce a set $\\cZ$ of ``unusable'' primes with the \nproperty that for any $p' \\in \\cZ$,\n\\begin{equation}\n \\label{eq:Zsparse}\n \\sums[p \\ge p'][p \\in \\cZ]\n \\frac{1}{p}\n \\ll\n \\frac{1}{p'}\n \\ll\n \\frac{1}{\\log x}.\n\\end{equation}\n\n\\begin{corollary}\n \\label{cor:thm2}\nLet $C$ be a sufficiently large but fixed positive constant.\nFor all sufficiently large $x$, there exists a vector of residue \nclasses \n$(c_p \\pod{p})_{p \\le Cx, \\, p \\, \\not\\in \\, \\cZ}$ such that \n$\n \\cH\n =\n \\big(\\ZZ \\cap (x,y]\\big)\n \\setminus \\, \n {\\textstyle \\bigcup_{p \\le Cx, \\, p \\, \\not\\in \\, \\cZ}} \n \\, c_p \\pod{p}.\n$ \n\\end{corollary}\n\nWe deduce Corollary \\ref{cor:thm2} from Theorem \\ref{thm:fgkmt2} \nwith the aid of Lemma 5.1 of \\cite{BFM}, which is as follows.\n\n\\begin{lemma}\n \\label{lem:BFM5.1}\nLet $\\sH,\\sT$ be sets of integers, $\\sP$ a set of \nprimes, such that for some $x \\ge 2$, \n$\\sH \\subseteq \\sT \\subseteq [0,x^2]$ and \n$\n \\#\\{p \\in \\sP : p > x\\} \n > \n \\#\\sH + \\#\\sT\n$.\nIf $\\sH$ is admissible then there exists a vector of residue \nclasses $(\\gamma_p \\pod{p})_{p \\, \\in \\, \\sP}$ such that \n$\n \\sH\n = \n \\sT \n \\, \\setminus \\,\n {\\textstyle \\bigcup_{p \\, \\in \\, \\sP}}\\, \\gamma_p \\pod{p}.\n$\n\\end{lemma}\n\n\\begin{proof}[Deduction of Corollary \\ref{cor:thm2}]\nWe choose $x$, $\\vec{a}$ and $\\vec{b}$ so that the conclusions \nof Theorem \\ref{thm:fgkmt2} hold, and work with the enlarged \nsifted sets\n\\[\n S_{\\cZ}(\\vec{a})\n \\defeq \n \\ZZ \n \\, \\setminus \\, \n \\textstyle{\\bigcup_{s \\, \\in \\, \\cS \\, \\setminus \\, \\cZ}} \\,\n a_s \\pod{s}\n\\quad \n \\text{and}\n \\quad \n S_{\\cZ}(\\vec{b})\n \\defeq \n \\ZZ \n \\, \\setminus \\, \n \\textstyle{\\bigcup_{p \\, \\in \\, \\cP \\, \\setminus \\, \\cZ}} \\,\n b_p \\pod{p}.\n\\] \nNote that if $n \\in S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b})$, then \neither $n \\in S(\\vec{a}) \\cap S(\\vec{b})$, \n$n \\equiv a_s \\pod{s}$ for some $s \\in \\cS \\cap \\cZ$ \nor \n$n \\equiv b_p \\pod{p}$ for some $p \\in \\cP \\cap \\cZ$.\nNow, \n\\[\n \\sum_{s \\, \\in \\, \\cS \\cap \\cZ}\n \\sums[n \\le y][n \\equiv a_{s} \\pod{s}]\n 1\n \\, +\n \\sum_{p \\, \\in \\, \\cP \\cap \\cZ}\n \\sums[n \\le y][n \\equiv b_{p} \\pod{p}]\n 1\n \\le \n y\n \\sums[p \\, \\in \\, \\cZ][p > (\\log x)^{20}]\n \\frac{1}{p'}\n \\ll\n \\frac{y}{(\\log x)^{20}}\n\\]\nby \\eqref{eq:fgkmt3.3}, \\eqref{eq:fgkmt3.4} and \n\\eqref{eq:Zsparse}.\nThus, the elements of \n$\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b})$\nthat are not in \n$\\cQ \\cap S(\\vec{a}) \\cap S(\\vec{b})$ \nnumber at most $y\/(\\log x)^{20} \\ll x\/(\\log x)^{19}$ \nby \\eqref{eq:fgkmt3.1}.\nWe conclude from \\eqref{eq:fgkmt3.6} that \n\\begin{equation}\n \\label{eq:QZbnd}\n \\#(\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b}))\n \\ll\n \\frac{x}{\\log x}.\n\\end{equation}\n\nLet \n\\[\n \\cL \n \\defeq \n \\{\n \\text{$\\ell$ prime} : \n \\ell \\in [2,(\\log x)^{20}] \\cup (z,x\/2] \n \\}\n\\]\nso that $\\cS \\cup \\cL \\cup \\cP$ is a partition of the primes less \nthan or equal to $x$.\nWe define a vector of residue classes \n$(c_p \\pod{p})_{p \\le x, \\, p \\not\\in \\cZ}$ \nby setting \n\\begin{align*}\n c_p \n \\defeq \n \\begin{cases}\n a_p & p \\in \\cS \\, \\setminus \\, \\cZ \\\\\n b_p & p \\in \\cP \\, \\setminus \\, \\cZ \\\\\n 0 & p \\in \\cL \\, \\setminus \\, \\cZ. \n \\end{cases}\n\\end{align*}\nRecalling that Theorem \\ref{thm:fgkmt2} gives \n$\\cH \\subseteq S(\\vec{a}) \\cap S(\\vec{b})$, and noting that $\\cH$ \nconsists of primes larger than $x$ by definition, we see that\n\\[\n \\cH\n \\subseteq\n \\cT\n \\defeq \n \\big(\\ZZ \\cap (x,y]\\big)\n \\setminus \\, \n {\\textstyle \\bigcup_{p \\le x, \\, p \\not\\in \\cZ}} \\,\n c_p \\pod{p}.\n\\]\n\nNow, if $n \\in \\cT$ then $x < n \\le y$ and either \n\\begin{enumerate}[label=(\\arabic*)]\n \\item $n$ is divisible by a prime $p' > x\/2$, \n \\item $n$ is divisible by a prime $p' \\in (z,x\/2]$, or\n \\item $n$ is $z$-smooth.\n\\end{enumerate}\n\nIn case (1), $n = mp'$ for some \n$m \\le y\/p' < 2y\/x = o(\\log x)$ by \\eqref{eq:fgkmt3.1} and so, by \n\\eqref{eq:Zsparse}, $m$ is not divisible by any prime in $\\cZ$ \n(provided $x$ is sufficiently large, as we assume).\nNor is $m$ divisible by any other prime $p < 2y\/x$, since \nsuch primes are in $\\cL \\, \\setminus \\, \\cZ$ and $c_{p} = 0$ for \nsuch primes.\nHence $m = 1$, and $n$ must belong to \n$\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b})$.\nThus,\n\\begin{equation}\n \\label{eq:(1)bnd}\n \\#\\{n \\in \\cT : \\text{(1) holds} \\} \n \\le \n \\#(\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b}))\n \\ll\n \\frac{x}{\\log x}\n\\end{equation}\nby \\eqref{eq:QZbnd}.\nIn case (2), the prime $p'$ must belong to $\\cZ$, for otherwise \n$c_{p'} = 0$. \nThus, \n\\begin{equation}\n \\label{eq:(2)bnd}\n \\#\\{n \\in \\cT : \\text{(2) holds} \\}\n \\ll\n y\n \\sums[p > z, \\, p \\in \\cZ] \n \\frac{1}{p}\n \\ll\n \\frac{y}{z}\n = \n o\\Big(\\frac{x}{\\log x}\\Big)\n\\end{equation}\nby \\eqref{eq:fgkmt3.2} and \\eqref{eq:fgkmt3.1}.\nAs shown in \\cite[Theorem 2 et seq.]{FGKMT}, smooth number \nestimates give \n\\begin{equation} \n \\label{eq:(3)bnd}\n \\#\\{n \\in \\cT : \\text{(3) holds} \\}\n = \n o\\Big(\\frac{x}{\\log x}\\Big).\n\\end{equation}\n\nCombining \\eqref{eq:(1)bnd}, \\eqref{eq:(2)bnd} and \n\\eqref{eq:(3)bnd}, we obtain $K + \\#\\cT \\ll x\/\\log x$ in view of \n\\eqref{eq:Kbnd}.\nWe may therefore choose our constant $C$ to be large enough so \nthat\n\\[\n \\#\\{\\text{$p$ prime}: p \\in (x,Cx], \\, p \\not\\in \\cZ \\} \n >\n K + \\#\\cT.\n\\] \n(The number of primes in $\\cZ$ that belong to $(x,Cx]$ is \nnegligible, for \n\\[\n \\#\\{p \\in \\cZ : p \\le Cx\\} \\ll \\log x,\n\\]\nas can be seen from \\eqref{eq:Zsparse} [write $1 = p\/p$ and sum \ndyadically].)\nAs $\\cH$ is an admissible subset of $\\cT \\subseteq (x,y]$, we \nmust conclude, in view of Lemma \\ref{lem:BFM5.1}, that for \nsufficiently large $x$ there exist residue classes $c_p \\pod{p}$ \nfor $p \\in (x,Cx]$, $p \\not\\in \\cZ$, such that \n\\[\n \\cH\n =\n \\cT\n \\, \\setminus \\,\n {\\textstyle \\bigcup_{p \\, \\in \\, (x,Cx], \\, p \\, \\not\\in \\, \\cZ}} \n \\, c_p \\pod{p}\n =\n \\big(\\ZZ \\cap (x,y]\\big)\n \\setminus \\, \n {\\textstyle \\bigcup_{p \\le Cx, \\, p \\, \\not\\in \\, \\cZ}} \n \\, c_p \\pod{p}.\n\\] \n\\end{proof}\n\nIn order to prove Theorem \\ref{thm:fgkmt2} we must first establish \nthe following result, which is analogous to \n\\cite[Theorem 4]{FGKMT}. \n\n\\begin{theorem}[Random construction]\n \\label{thm:fgkmt4}\nLet $x$ be sufficiently large.\nThere exists a positive number $C$ with \n\\begin{equation}\n \\label{eq:fgkmt4.29}\n C \\asymp \\frac{1}{c},\n\\end{equation}\nthe implied constants being independent of $c$, a set of \npositive integers $\\{h_1,\\ldots,h_r\\}$ with $r \\le \\sqrt{\\log x}$, \nand random vectors \n$\\vec{\\ba} = (\\ba_s \\pod{s})_{s \\, \\in \\, \\cS}$ \nand \n$\\vec{\\bn} = (\\bn_p)_{p \\, \\in \\, \\cP}$\nof residue classes $\\ba_s \\pod{s}$ and integers $\\bn_p$ \nrespectively, satisfying the following.\n\\begin{enumerate}\n \\item For every $\\vec{a} = (a_s \\pod{s})_{s \\in \\cS}$ in the \n essential range of $\\vec{\\ba}$, we have \n \\[\n \\cH \\cap a_s \\pod{s} = \\emptyset \\quad (s \\in \\cS).\n \\]\n \\item For every $\\vec{n} = (n_p \\pod{p})_{p \\in \\cP}$ in the \n essential range of $\\vec{\\bn}$, we have \n \\[\n \\cH \\cap n_p \\pod{p} = \\emptyset \\quad (p \\in \\cP).\n \\]\n \\item For every $\\vec{a}$ in the essential range of $\\vec{\\ba}$, \n we have\n \\[\n \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n \\le \n x^{-3\/5}\n \\quad \n (p \\in \\cP),\n \\]\n where \n $\n \\be_p(\\vec{a}) \\defeq \\{\\bn_p + h_ip : i \\le r\\}\n \\cap \n \\cQ \n \\cap \n S(\\vec{a})\n $.\n \\item With probability $1 - o(1)$, we have \n \\begin{equation}\n \\label{eq:fgkmt4.30}\n \\#(\\cQ \\cap S(\\vec{\\ba}))\n \\sim \n 80cx\\frac{\\log_2 x}{\\log x}.\n \\end{equation}\n \\item Call an element $\\vec{a}$ in the essential range of \n $\\vec{\\ba}$ ``good'' if, for all but at most \n $\\frac{x}{\\log x \\log_2 x}$ elements \n $q \\in \\cQ \\cap S(\\vec{a})$, one has \n \\begin{equation}\n \\label{eq:fgkmt4.31}\n \\sum_{p \\in \\cP}\n \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n =\n C + O_{\\le}\\bigg(\\frac{1}{(\\log_2 x)^{2}}\\bigg).\n \\end{equation}\n %\n Then $\\vec{\\ba}$ is good with probability $1 - o(1)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{Proof of Theorem \\ref{thm:fgkmt4}}\n \\label{subsec:thm4pf}\n\nLet $x,c,y,z,\\cS,\\cP,\\cQ,K$ and $\\cH$ be as in \nTheorem \\ref{thm:fgkmt4}.\nWe set\n\\begin{equation}\n \\label{eq:fgkmt6.8}\n r \\defeq \\lfloor (\\log x)^{1\/5} \\rfloor\n\\end{equation}\nand let $\\{h_1,\\ldots,h_r\\}$ be the admissible set with \n$h_i \\defeq (2i-1)^2$ for $i \\le r$.\nOur first lemma is a special case of \\cite[Theorem 5]{FGKMT}.\n\\begin{lemma}[Existence of a good sieve weight]\n \\label{thm:fgkmt5}\nThere exist positive quantities\n\\begin{equation}\n \\label{eq:fgkmt6.2}\n \\tau \\ge x^{o(1)} \n \\quad \n \\text{and}\n \\quad \n u \\asymp \\log_2 x,\n\\end{equation}\nand a function $w : \\cP \\times \\ZZ \\to \\RR^+$ supported on \n$\\cP \\times [-y,y]$, satisfying the following.\n\\begin{enumerate} \n \\item Uniformly for $p \\in \\cP$, \n\\begin{equation}\n \\label{eq:fgkmt6.4}\n \\sum_{n \\in \\ZZ} w(p,n)\n =\n \\bigg(\n 1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n \\bigg)\n \\tau\n \\frac{y}{(\\log x)^r}.\n\\end{equation}\n \\item Uniformly for $q \\in \\cQ$ and $i \\le r$, \n\\begin{equation}\n \\label{eq:fgkmt6.5}\n \\sum_{p \\in \\cP}\n w(p,q - h_ip)\n =\n \\bigg(\n 1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n \\bigg)\n \\tau\n \\frac{u}{r} \n \\frac{x}{2(\\log x)^r}.\n\\end{equation} \n \\item Uniformly for $(p,n) \\in \\cP\\times \\ZZ$, \n\\begin{equation}\n \\label{eq:fgkmt6.7}\n w(p,n) = O\\big(x^{1\/3 + o(1)}\\big).\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\nWe choose $\\tau$, $u$ and $w : \\cP \\times \\ZZ \\to \\RR^+$ \naccording to Lemma \\ref{thm:fgkmt5}, and define \n$w_{\\cH} : \\cP \\times \\ZZ \\to \\RR^+$ by setting \n\\[\n w_{\\cH}(p,n) \n \\defeq \n \\begin{cases}\n w(p,n) & \\text{if $\\cH \\cap n \\pod{p} = \\emptyset$} \\\\\n 0 & \\text{otherwise.} \n \\end{cases}\n\\]\n\n\\begin{lemma}\n \\label{lem:rcb2}\nStatements \\textup{(}i\\textup{)}, \\textup{(}ii\\textup{)} and \n\\textup{(}iii\\textup{)} of Lemma \\ref{thm:fgkmt5} all hold with \n$w_{\\cH}$ in place of $w$, provided the hypothesis $q \\in \\cQ$ in \n\\textup{(}ii\\textup{)} is replaced by the hypothesis \n$q \\in \\cQ\\setminus \\cH$.\n\\end{lemma}\n\n\\begin{proof}\nWe only need to consider (i) and (ii).\nFor every $p \\in \\cP$ we have \n\\begin{align*}\n 0 \\le \\sum_{n \\in \\ZZ} (w(p,n) - w_{\\cH}(p,n))\n \\le \\sum_{j=1}^K \\sums[|n| \\le y][n \\equiv q_j \\pod{p}] w(p,n)\n \\ll x^{1\/3 + o(1)}y\/p\n \\ll x^{-2\/3 + o(1)}y,\n\\end{align*}\nwhich gives the analog of \\eqref{eq:fgkmt6.4} for $w_{\\cH}$ in \nview of \\eqref{eq:fgkmt6.2}.\nFor every $q \\in \\cQ\\setminus \\cH$ and $i \\le r$, we see \nsimilarly that \n\\begin{align*}\n 0 \n & \\le \n \\sum_{p \\in \\cP} (w(p,q - h_ip) - w_{\\cH}(p,q - h_ip)) \n \\\\\n & \\hspace{45pt}\n \\le\n x^{1\/3 + o(1)}\n \\sum_{j=1}^K\n \\sum_{q - h_ip \\equiv q_j \\pod{p}} 1 \n \\le \n x^{1\/3 + o(1)}\n \\sum_{j=1}^K\n \\sum_{p \\mid q - q_j} 1 \n \\ll\n x^{1\/3 + o(1)},\n\\end{align*}\nwhich gives the analog of \\eqref{eq:fgkmt6.5}.\n\\end{proof}\n\nFor each $p \\in \\cP$, let $\\tilde{\\bn}_p$ denote the random \ninteger with probability density \n\\[\n \\PP(\\tilde{\\bn}_p = n) \n \\defeq \n \\frac{w_{\\cH}(p,n)}{\\sum_{n' \\in \\ZZ} w_{\\cH}(p,n')}\n\\]\nfor all $n \\in \\ZZ$.\nUsing Lemma \\ref{lem:rcb2}, we verify that \n\\begin{equation}\n \\label{eq:fgkmt6.9}\n \\sum_{p \\in \\cP}\n \\PP(q = \\tilde{\\bn}_p + h_ip)\n =\n \\ind{\\cQ\\setminus\\cH}(q)\n \\bigg(\n 1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n \\bigg)\n \\frac{u}{r}\n \\frac{x}{2y}\n \\quad \n (q \\in \\cQ, i \\le r),\n\\end{equation}\n\\begin{equation}\n \\label{eq:fgkmt6.10}\n \\PP(\\tilde{\\bn}_p = n) \\ll x^{-2\/3 + o(1)}\n \\quad (p \\in \\cP, n \\in \\ZZ)\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:probnph0}\n \\PP(\\cH \\cap \\tilde{\\bn}_p \\pod{p} \\ne \\emptyset) = 0\n \\quad (p \\in \\cP).\n\\end{equation}\nBy \\eqref{eq:fgkmt6.10}, the analog of \\eqref{eq:fgkmt6.9} holds \nwith a single prime deleted from $\\cP$.\n\nWe choose the random vector \n$\\vec{\\ba} \\defeq (\\ba_s \\pod{s})_{s \\in \\cS}$ by selecting each \n$\\ba_s \\pod{s}$ uniformly at random from \n\\begin{equation}\n \\label{eq:defOmega}\n \\Omega_{\\cH}(s)\n \\defeq \n (\\ZZ\/s\\ZZ)\\setminus\\{q \\pod{s} : q \\in \\cH\\},\n\\end{equation}\nindependently in $s$ and independently of the $\\tilde{\\bn}_p$.\nNote, then, that for any random vector $\\vec{\\ba}$, \n$\\cH \\subseteq S(\\vec{\\ba})$.\n\nThe sifted set $S(\\vec{\\ba})$ is a random periodic subset of $\\ZZ$ \nwith density \n\\[\n \\sigma_{\\cH}\n \\defeq \n \\prod_{s \\in \\cS}\n \\bigg(\n 1 - \\frac{1}{\\#\\Omega_{\\cH}(s)}\n \\bigg).\n\\]\nLet us compare $\\sigma_{\\cH}$ with the quantity \n\\[\n \\sigma \n \\defeq \n \\prod_{s \\in \\cS}\n \\bigg(\n 1 - \\frac{1}{s}\n \\bigg)\n\\]\ndefined in \\cite{FGKMT}. \nAs $(\\log x)^{20} - \\log x \\le s - K \\le \\#\\Omega_{\\cH}(s) \\le s$ \n(cf.\\ \\eqref{eq:fgkmt3.3} and \\eqref{eq:Kbnd}), one may verify, in \na straightforward manner, that \n\\begin{equation}\n \\label{eq:sigmahatsigma}\n \\sigma_{\\cH} \n = \n \\sigma\\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{19}}\\bigg)\\bigg). \n\\end{equation}\nConsequently, in the estimates that follow, $\\sigma_{\\cH}$ and \n$\\sigma$ are interchangeable. \n\nAs noted in \\cite{FGKMT}, by the prime number theorem (with \nsuitably strong error term), \\eqref{eq:fgkmt3.2} and \n\\eqref{eq:fgkmt3.3}, \n\\[\n \\sigma \n = \n \\bigg(\n 1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n \\bigg)\n \\frac{80\\log_2 x}{\\log x \\log_3 x\/\\log_2 x},\n\\]\nso by \\eqref{eq:fgkmt3.1} we have \n\\begin{equation}\n \\label{eq:fgkmt6.11}\n \\sigma y\n =\n \\bigg(\n 1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n \\bigg) \n 80c x\\log_2 x.\n\\end{equation}\nAlso, by \\eqref{eq:fgkmt6.8} we have \n\\begin{equation}\n \\label{eq:fgkmt6.12}\n \\sigma^r = x^{o(1)}.\n\\end{equation}\n\nLet\n\\begin{equation}\n \\label{eq:fgkmt6.14}\n X_p(\\vec{a})\n \\defeq \n \\PP(\\tilde{\\bn}_p + h_ip \\in S(\\vec{a}) \\, \\, \\hbox{for all} \\, \\, i \\le r)\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:rcb11a}\n \\cP(\\vec{a})\n \\defeq \n \\bigg\\{\n p \\in \\cP : \n X_p(\\vec{a})\n =\n \\bigg(\n 1 + O_{\\le}\\bigg(\\frac{1}{(\\log x)^{6}}\\bigg)\n \\bigg)\n \\sigma^r\n \\bigg\\}.\n\\end{equation}\nIt will transpire that with probability $1 - o(1)$, most primes in \n$\\cP$ lie in $\\cP(\\vec{\\ba})$.\n\nWe now define $\\bn_p$ in a slightly complicated way.\nLet \n\\[\n Z_p(\\vec{a};n)\n \\defeq \n \\ind{\n (n + h_j p \\, \\in \\, S(\\vec{a}) \\, \\, \\forall j \\le r)\n }\n \\PP(\\tilde{\\bn}_p = n).\n\\]\nSuppose we are in the event that $\\vec{\\ba} = \\vec{a}$.\nIf $p \\in \\cP \\, \\setminus \\, \\cP(\\vec{a})$, we set \n$\\bn_p = 0$.\nOtherwise, let $\\bn_p$ be the random integer with \n\\begin{equation}\n \\label{eq:rcb11}\n \\PP(\\bn_p = n \\mid \\vec{\\ba} = \\vec{a})\n =\n \\frac{Z_p(\\vec{a};n)}{X_p(\\vec{a})},\n\\end{equation}\nwith the $\\bn_p$ jointly conditionally independent on the \nevent $\\vec{\\ba} = \\vec{a}$. \n(We easily verify that \n$\n \\sum_{n \\in \\ZZ} Z_p(\\vec{a};n) = X_p(\\vec{a})\n$,\nso that \\eqref{eq:rcb11} makes sense.)\n\n\n\\begin{proof}%\n[Deduction of Theorem \\ref{thm:fgkmt4} \n \\textup{(}i\\textup{)} -- \\textup{(}iii\\textup{)}]\nLet $p \\in \\cP$.\nWe claim that \n\\[\n \\PP(\\cH \\cap \\bn_p \\pod{p} \\ne \\emptyset) = 0.\n\\]\nTo prove the claim it suffices to show \n$\n \\PP(\\cH \\cap \\bn_p \\pod{p} \\ne \\emptyset \\mid \\vec{\\ba} = \\vec{a}) \n = 0\n$\nfor every $\\vec{a}$.\nThis is easily checked if $p \\not\\in \\cP(\\vec{a})$, since \n$\\cH \\cap 0 \\pod{p} \\subseteq \\cQ \\cap 0 \\pod{p} = \\emptyset$; \notherwise \n\\[\n \\PP(\\cH \\cap \\bn_p \\pod{p} \\ne \\emptyset \\mid \\vec{\\ba} = \\vec{a})\n \\le \n \\frac{\\PP(\\cH \\cap \\tilde{\\bn}_p \\pod{p} \\ne \\emptyset)}\n {X_p(\\vec{a})}\n =\n 0.\n\\]\n\nWe see that Theorem \\ref{thm:fgkmt4} (i) and (ii) hold and we can \nnow prove Theorem \\ref{thm:fgkmt4} (iii).\nGiven $\\vec{a}$ in the essential range of $\\vec{\\ba}$, and \n$q \\in \\cQ \\cap S(\\vec{a})$, we have \n\\[\n \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n \\le \n \\PP( \\bn_p + h_ip = q \\, \\, \\hbox{for some} \\, \\, i \\le r\\mid \\vec{\\ba} = \\vec{a}).\n\\]\nThe right-hand side is $0$ if $p \\not\\in \\cP(\\vec{a})$.\nOtherwise,\n\\begin{align*}\n \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n & \\le \n r \\max_{n \\in \\ZZ} \\PP(\\bn_p = n \\mid \\vec{\\ba} = \\vec{a}) \n \\ll\n r\\sigma^{-r} \\max_{n \\in \\ZZ} \\PP(\\tilde{\\bn}_p = n) \\\\\n & \\ll\n x^{-2\/3 + o(1)}.\n\\end{align*}\n(Here, we have used \\eqref{eq:fgkmt6.8}, \\eqref{eq:fgkmt6.10} and \n\\eqref{eq:fgkmt6.12}.)\n\\end{proof}\n\nThe following lemma is analogous to \\cite[Lemma 6.1]{FGKMT}.\n\n\\begin{lemma}\n \\label{lem:rcb4}\nLet $n_1,\\ldots,n_t$ be distinct integers of magnitude $x^{O(1)}$, \n$t \\le \\log x$, such that \n$\\cH \\cap \\{n_1,\\ldots,n_t\\} = \\emptyset$.\nThen for all sufficiently large $x$, \n\\[\n \\PP(n_1,\\ldots,n_t \\in S(\\vec{\\ba}))\n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n \\sigma^t,\n\\]\nwhere the implied constant is absolute.\n\\end{lemma}\n\\begin{proof}\nFor $s \\in \\cS$, let \n$\n t_{s}\n \\defeq \n \\#(\\Omega_{\\cH}(s) \\cap \\{n_1 \\pod{s},\\ldots,n_t \\pod{s}\\}) \n$.\nWe have \n\\[\n \\PP(n_1,\\ldots,n_t \\in S(\\vec{\\ba}))\n =\n \\prod_{s \\in \\cS}\n \\bigg(1 - \\frac{t_{s}}{\\#\\Omega_{\\cH}(s)}\\bigg).\n\\]\nNote that for $s \\in \\cS$, \n$\n 1 - t_{s}\/\\#\\Omega_{\\cH}(s)\n = \n 1 + O\\br{t\/s}\n =\n 1 + O\\br{1\/(\\log x)^{19}}\n$.\n\nLet $\\cS'$ be the set of primes $s \\in \\cS$ such that either \n$n_i \\equiv n_j \\pod{s}$ for some $i \\ne j$ or \n$\\cH \\cap n_i \\pod{s} \\ne \\emptyset$ for some $i$.\nFor $i \\ne j$ we have $1 \\le |n_i - n_j| \\ll x^{O(1)}$, so \n$n_i - n_j$ has at most $O(\\log x)$ prime divisors.\nSimilarly, for each $i$ and $j$, $n_i - q_j$ has at most \n$O(\\log x)$ prime divisors.\nWe see that $|\\cS'| \\ll (t^2 + tK)\\log x \\ll (\\log x)^3$ \n(cf.\\ \\eqref{eq:Kbnd}), and \n\\begin{align*}\n & \n \\prod_{s \\in \\cS'}\n \\bigg(1 - \\frac{t}{\\#\\Omega_{\\cH}(s)}\\bigg)^{-1}\n \\bigg(1 - \\frac{t_{s}}{\\#\\Omega_{\\cH}(s)}\\bigg)\n \\\\\n & \\hspace{30pt} \n =\n \\prod_{s \\in \\cS'}\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{19}}\\bigg)\\bigg)^{|\\cS'|}\n =\n 1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg).\n\\end{align*}\nFor $s \\in \\cS \\, \\setminus \\, \\cS'$ we have $t_{s} = t$.\nThus, \n\\begin{align*}\n \\prod_{s \\in \\cS}\n \\bigg(1 - \\frac{t_{s}}{\\#\\Omega_{\\cH}(s)}\\bigg)\n & = \n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n \\prod_{s \\in \\cS}\n \\bigg(1 - \\frac{t}{\\#\\Omega_{\\cH}(s)}\\bigg)\n \\\\\n & =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg) \n \\sigma^t\n \\prod_{s \\in \\cS}\n \\bigg(1 + O\\bigg(\\frac{t^2}{s^2}\\bigg)\\bigg) \n \\\\\n & = \n \\sigma^t\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\n\\end{proof}\n\n\\begin{proof}%\n[Deduction of Theorem \\ref{thm:fgkmt4} \\textup{(}iv\\textup{)}]\nLet $\\cR \\defeq \\cQ\\setminus \\cH$.\nRecalling \\eqref{eq:cHdef} and \\eqref{eq:Kbnd}, we have \n\\begin{equation}\n \\label{eq:rcb13}\n \\#(\\cQ \\cap S(\\vec{\\ba})) \n = \n \\#(\\cR \\cap S(\\vec{\\ba})) + K \n =\n \\#(\\cR \\cap S(\\vec{\\ba})) + O_{\\le}(\\log x).\n\\end{equation}\nLet \n$\n X \n \\defeq \n \\sum_{q \\in \\cR} \\ind{q \\in S(\\vec{\\ba})}. \n$\nWe have\n\\[\n \\EE X \n =\n \\sum_{q \\in \\cR}\n \\PP(q \\in S(\\vec{\\ba}))\n =\n (\\#\\cR)\\sigma\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n\\]\nfrom Lemma \\ref{lem:rcb4}.\nNote that by \\eqref{eq:Qsize} we have \n\\[\n (\\#\\cQ)\\sigma\n = \n \\frac{\\sigma y}{\\log x}\n \\bigg(1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\\bigg).\n\\]\nBy \\eqref{eq:rcb13}, the same estimate holds for $(\\#\\cR)\\sigma$,\nand\n\\begin{equation}\n \\label{eq:rcb14}\n \\EE \\#(\\cQ \\cap S(\\vec{\\ba}))\n =\n \\frac{\\sigma y}{\\log x}\n \\bigg(1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\\bigg).\n\\end{equation}\nWe similarly have \n\\begin{align*}\n \\EE X^2\n & =\n \\sum_{q_1 \\in \\cR}\n \\sum_{q_2 \\in \\cR}\n \\ind{q_1,q_2 \\in S(\\vec{\\ba})}\n \\\\\n & = \n \\sums[(q_1,q_2) \\in \\cR^2][q_1 \\ne q_2]\n \\sigma^2\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n + \n \\sum_{q \\in \\cR}\n \\sigma\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n \\\\\n & = \n \\sigma^2(\\#\\cR)^2\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\nThus, \n\\begin{equation}\n \\label{eq:rcb15}\n \\EE (X - \\EE X)^2 \n =\n \\EE X^2 - (\\EE X)^2\n \\ll\n \\frac{\\sigma^2(\\#\\cR)^2}{(\\log x)^{16}}.\n\\end{equation}\n\nNow we use Chebyshev's inequality: \n\\begin{align}\n \\begin{split}\n \\label{eq:rcb16}\n \\PP\\big(|X - \\EE X| > (\\#\\cR)\\sigma(\\log x)^{-3}\\big)\n & \n \\le \n (\\#\\cR)^{-2}\\sigma^{-2}(\\log x)^6 \\, \\EE (X - \\EE X)^2\n \\\\\n & \n \\ll\n \\frac{1}{(\\log x)^{10}}.\n \\end{split}\n\\end{align}\nRecalling \\eqref{eq:rcb13} again we get \n\\[\n \\#(\\cQ \\cap S(\\vec{\\ba}))\n =\n \\frac{\\sigma y}{\\log x}\n \\bigg(1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\\bigg)\n\\]\nwith probability $1 - O((\\log x)^{-10})$, and \nTheorem \\ref{thm:fgkmt4} (iv) follows on recalling that, by \n\\eqref{eq:fgkmt6.11}, \n$\n \\sigma y\/\\log x \\sim 80cx \\log_2 x \/\\log x.\n$\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:rcb5}\n\\textup{(}i\\textup{)}\nWith probability \n$1 - O\\big(\\frac{1}{\\log x}\\big)$, \n$\\cP(\\vec{\\ba})$ contains all but \n$O\\big(\\frac{\\#\\cP}{(\\log x)^3}\\big)$ \nof the primes in $\\cP$.\n\\textup{(}ii\\textup{)}\nWe have \n\\[\n \\EE \\, \\#\\cP(\\vec{\\ba}) \n = (\\#\\cP)\n \\bigg(\n 1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\n \\bigg).\n\\]\n\\end{lemma}\n\\begin{proof}\nWe have \n\\begin{align*}\n \\EE \\, X_p(\\vec{\\ba})\n & = \n \\sum_{n \\in \\ZZ}\n \\PP(\\tilde{\\bn}_p = n)\n \\PP(n + h_i p \\in S(\\vec{\\ba}) \\,\\, \\forall i \\le r)\n \\\\\n & = \n \\sum_{n \\in \\ZZ} \n \\PP(\\tilde{\\bn}_p = p)\n \\sigma^r\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg) \n \\\\\n & = \n \\sigma^r\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\n(For the second step, we supplement Lemma \\ref{lem:rcb4} with \nthe observation that $\\PP(\\bn_p = n) = 0$ whenever \n$n + h_ip \\in \\cH$ for some $i \\le r$.) \n\nLet $\\tilde{\\bn}_p^{(1)}$ and $\\tilde{\\bn}_p^{(2)}$ be independent \nrandom variables having the same probability distribution as \n$\\tilde{\\bn}_p$.\nThen \n\\begin{align*}\n X_p(\\vec{\\ba})^2\n & = \n \\PP\\big(\\tilde{\\bn}_p^{(1)} \\in S(\\vec{\\ba}) \\,\\, \\forall i \\le r\\big)\n \\PP\\big(\\tilde{\\bn}_p^{(2)} \\in S(\\vec{\\ba}) \\,\\, \\forall i \\le r\\big)\n \\\\\n & = \n \\PP\\big(\\tilde{\\bn}_p^{(l)} \\in S(\\vec{\\ba}) \\,\\, \\forall l \\le 2, i \\le r\\big).\n\\end{align*}\nArguing as above, \n\\[\n \\EE X_p(\\vec{\\ba})^2\n = \n \\sum_{n_1 \\in \\ZZ}\n \\sum_{n_2 \\in \\ZZ} \n \\PP\\big(\\tilde{\\bn}_p^{(1)} = n_1\\big)\n \\PP\\big(\\tilde{\\bn}_p^{(2)} = n_2\\big)\n \\sigma^{t(n_1,n_2)}\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg),\n\\]\nwhere $t(n_1,n_2)$ is the number of distinct integers \n$n_l + h_ip$ ($l \\le 2$, $i \\le r$).\n\nNow fix $n_1$.\nThere are less than $r^2$ values of $n_2$ for which \n$t(n_1,n_2) \\ne 2r$.\nSince $\\PP(\\bn_p^{(2)} = n_2) \\ll x^{-2\/3 + o(1)}$ (cf.\\ \n\\eqref{eq:fgkmt6.10}), we obtain\n\\begin{align*}\n &\n \\EE X_p(\\vec{\\ba})^2\n \\\\\n & \\hspace{15pt}\n =\n \\sum_{n_1 \\in \\ZZ}\n \\PP(\\tilde{\\bn}_p^{(1)} = n_1)\n \\bigg\\{\n \\sigma^r \n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n \\bigg(1 - O\\bigg(\\frac{1}{x^{1\/2}}\\bigg)\\bigg)\n + O\\bigg(\\frac{1}{x^{1\/2}}\\bigg)\n \\bigg\\}\n \\\\\n & \\hspace{15pt}\n = \n \\sigma^{2r}\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\nArguing as in \\eqref{eq:rcb15}, \\eqref{eq:rcb16}, \n\\[\n \\PP\n \\Big(\n |\n X_p(\\vec{\\ba}) \n - \\big(\n 1 + O\\big({\\textstyle \\frac{1}{(\\log x)^{16}}}\\big)\n \\big) \n \\sigma^r\n |\n > \n {\\textstyle \n \\frac{1}{2} \n \\frac{{\\displaystyle \\sigma^r}}{(\\log x)^6}}\n \\Big)\n \\ll\n \\frac{1}{(\\log x)^4}.\n\\]\nThus, with probability \n$1 - O\\big(1\/(\\log x)^4\\big)$, \nwe have \n\\[\n X_p(\\vec{\\ba})\n =\n \\bigg(\n 1 + O_{\\le}\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\n \\bigg) \n \\sigma^r,\n\\]\nthat is, $p \\in \\cP(\\vec{\\ba})$.\nMoreover, \n\\begin{align*}\n & \n \\frac{\\#\\cP}{(\\log x)^3}\n \\PP\n \\Big(\n {\\textstyle \\sum_{p \\in \\cP} } \n \\ind{p \\not\\in \\cP(\\vec{\\ba})}\n >\n {\\textstyle\\frac{\\#\\cP}{(\\log x)^3}}\n \\Big)\n \\\\\n & \\hspace{60pt}\n \\le\n \\EE \\, \n \\Big(\n \\sum_{p \\in \\cP} \n \\ind{p \\not\\in \\cP(\\vec{\\ba})}\n \\Big)\n = \n \\sum_{p \\in \\cP}\n \\PP(p \\not\\in \\cP(\\vec{\\ba}))\n \\ll\n \\frac{\\#\\cP}{(\\log x)^4}.\n\\end{align*}\nSo with probability $1 - \\big(1\/\\log x\\big)$, \n$\\cP(\\vec{\\ba})$ contains all but \n$O\\big(\\frac{\\#\\cP}{(\\log x)^3}\\big)$ of the primes $p \\in \\cP$.\nFinally, \n\\[\n \\EE \\, \\#\\cP(\\vec{\\ba})\n = \n \\sum_{p \\in \\cP}\n \\PP(p \\in \\cP(\\vec{\\ba}))\n =\n (\\#\\cP)\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\\bigg).\n\\]\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:rcb6}\nLet $q \\in \\cQ$ and let $\\vec{a}$ be in the essential range of \n$\\vec{\\ba}$.\nThen\n\\begin{equation}\n \\label{eq:rcb17}\n \\sigma^{-r}\n \\sum_{i=1}^r \n \\sum_{p \\, \\in \\, \\cP(\\vec{a})}\n Z_p(\\vec{a};q - h_ip)\n =\n \\big(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n \\sum_{p \\, \\in \\, \\cP}\n \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a}).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nRecalling \\eqref{eq:rcb11}, the left-hand side of \n\\eqref{eq:rcb17} is \n\\begin{align*}\n & \n \\sigma^{-r}\n \\sum_{i=1}^r \n \\sum_{p \\, \\in \\, \\cP(\\vec{a})}\n X_p(\\vec{a}) \n \\PP(\\bn_p = q - h_ip \\mid \\vec{\\ba} = \\vec{a})\n \\\\\n & \\hspace{60pt}\n = \n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n \\sigma^{-r}\n \\sum_{i=1}^r \n \\sum_{p \\, \\in \\, \\cP(\\vec{a})}\n \\PP(\\bn_p = q - h_ip \\mid \\vec{\\ba} = \\vec{a}). \n\\end{align*}\nSince $q - h_ip \\ne \\bn_p$ if $p \\not\\in \\cP(\\vec{a})$, we may \nrewrite this as \n\\[\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n \\sigma^{-r}\n \\sum_{i=1}^r \n \\sum_{p \\, \\in \\, \\cP}\n \\PP(\\bn_p = q - h_ip \\mid \\vec{\\ba} = \\vec{a}), \n\\]\nand the lemma follows.\n\\end{proof}\n\nIt is convenient to write \n\\[\n U(q,\\vec{a})\n \\defeq \n \\sigma^{-r}\n \\sum_{i=1}^r \n \\sum_{p \\, \\in \\, \\cP}\n Z_p(\\vec{a};q - h_ip).\n\\]\n\n\\begin{lemma}\n \\label{lem:rcb7}\nWe have\n\\begin{equation}\n \\label{eq:rcb18}\n \\EE \\,\n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n \\sigma^r U(q,\\vec{\\ba})\n \\Big)\n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n \\frac{\\sigma y}{\\log x}\n \\frac{\\sigma^{r-1}ux}{2y}\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:rcb19}\n \\EE \\,\n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n \\sigma^r U(q,\\vec{\\ba})^2\n \\Big)\n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n \\frac{\\sigma y}{\\log x}\n \\bigg(\\frac{\\sigma^{r-1}ux}{2y}\\bigg)^2.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe begin with \\eqref{eq:rcb19}.\nLet $\\tilde{\\bn}_p^{(1)}$ and $\\tilde{\\bn}_p^{(2)}$ be \nindependent copies of $\\tilde{\\bn}_p$ that are also independent of \n$\\vec{\\ba}$.\nWe observe that for any $n_1,n_2$, and $p_1,p_2 \\in \\cP$, \n\\[\n Z_{p_1}(\\vec{\\ba};n_1)\n Z_{p_2}(\\vec{\\ba};n_2)\n =\n \\ind{\n (n_l + h_jp_l \\, \\in \\, S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, \\, j \\le r)\n }\n \\,\n \\PP(\\tilde{\\bn}_p^{(1)} = n_1)\n \\PP(\\tilde{\\bn}_p^{(2)} = n_2).\n\\]\nThe left-hand side of \\eqref{eq:rcb19} is thus\n\\begin{align}\n \\label{eq:rcb20}\n \\begin{split}\n & \n \\EE \\,\n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n \\,\n \\sum_{i_1,i_2 = 1}^r\n \\,\n \\sum_{p_1,p_2 \\in \\cP}\n Z_{p_1}(\\vec{\\ba};q - h_{i_1}p_1)\n Z_{p_2}(\\vec{\\ba};q - h_{i_2}p_2)\n \\Big)\n \\\\\n & \n =\n \\sum_{q \\in \\cQ}\n \\,\n \\sum_{i_1,i_2 = 1}^r\n \\big( \n \\PP(q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r)\n \\\\\n & \\hspace{90pt} \n \\times \n \\PP(\\tilde{\\bn}_{p_1}^{(1)} = q - h_{i_1}p_1)\n \\PP(\\tilde{\\bn}_{p_2}^{(2)} = q - h_{i_2}p_2)\n \\big).\n \\end{split}\n\\end{align}\n(For $q \\in \\cQ$, the event \n$q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r$ \nis identical to the event \n$q \\in \\cQ \\cap S(\\vec{\\ba})$, \n$q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r$.) \n\nLet $\\Sigma_1,\\Sigma_2$ be the contributions to the right-hand \nside of \\eqref{eq:rcb20} from $p_1 \\ne p_2$, respectively \n$p_1 = p_2$.\n\nFix $p_1,p_2$ in $\\cP$ with $p_1 \\ne p_2$, and fix \n$i_1,i_2 \\le r$.\nThe number of distinct integers $q + (h_j - h_{i_l})p_l$ \n($l \\le 2, j \\le r$) is $2r - 1$ since \n$\n (h_j - h_{i_1})p_1 \\ne (h_j - h_{i_2})p_2\n$\nfor $i_1 \\ne j$.\nHence \n\\begin{equation}\n \\label{eq:rcb21}\n \\begin{split}\n & \n \\PP(q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r)\n \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)\n \\PP(\\tilde{\\bn}_p^{(2)} = q - h_{i_2}p_2)\n \\\\ \n & \\hspace{30pt}\n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg) \n \\sigma^{2r-1}\n \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)\n \\PP(\\tilde{\\bn}_p^{(2)} = q - h_{i_2}p_2). \n \\end{split}\n\\end{equation}\n(Both sides are $0$ if there are $j,i_l$ with \n$q + (h_j - h_{i_l})p_l \\in \\cH$; otherwise \nLemma \\ref{lem:rcb4} applies.)\n\nWe combine \\eqref{eq:rcb21} with the remark after \n\\eqref{eq:probnph0} to obtain \n\\begin{align}\n \\label{eq:rcb22}\n \\begin{split}\n \\sumsstxt[][][1]\n & \n =\n \\sums[q \\in \\cQ \\setminus \\cH][i_1,i_2 \\le r]\n \\sigma^{2r-1}\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n \\bigg(\\frac{ux}{2ry}\\bigg)^2\n \\\\\n & = \n \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n \\frac{\\sigma y}{\\log x}\n \\bigg(\\frac{\\sigma^{r-1}ux}{2y}\\bigg)^2.\n \\end{split}\n\\end{align}\nSimilarly, \n\\begin{align*}\n \\sumsstxt[][][2]\n & \n =\n \\sum_{q \\in \\cQ}\n \\sum_{i_1 = 1}^r\n \\sum_{p_1 \\in \\cP}\n \\PP(q + (h_j - h_{i_1})p_1 \\in S(\\vec{\\ba}) \\, \\, \\forall j \\le r)\n \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)^2\n \\\\\n & \n \\ll\n x^{-3\/5}\n \\sum_{q \\in \\cQ}\n \\sum_{i_1 = 1}^r \n \\sum_{p_1 \\in \\cP}\n \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)\n \\\\\n & \n \\ll\n x^{-3\/5}(\\#\\cQ),\n\\end{align*}\nwhich together with \\eqref{eq:rcb22} yields \\eqref{eq:rcb19}.\n\nMuch the same argument gives for the left-hand side of \n\\eqref{eq:rcb18} the expression \n\\begin{align*}\n & \n \\sum_{q \\in \\cQ}\n \\sum_{i = 1}^r\n \\sum_{p \\in \\cP}\n \\PP(q + (h_j - h_i)p \\in S(\\vec{\\ba}) \\, \\, \\forall j \\le r)\n \\PP(\\tilde{\\bn}_p = q - h_ip)\n \\\\\n & \\hspace{30pt}\n = \n \\sum_{q \\in \\cQ}\n \\sum_{i=1}^r\n \\sum_{p \\in \\cP}\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n \\sigma^r\n \\PP(\\tilde{\\bn}_p = q - h_ip)\n \\\\\n & \\hspace{30pt}\n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n \\frac{\\sigma y}{\\log x}\n \\frac{\\sigma^{r-1}ux}{2y}.\n\\end{align*}\n\\end{proof}\n\nWe now specify that the quantity $C$ in Theorem \\ref{thm:fgkmt4} \nis \n\\[\n C \\defeq \\frac{ux}{2\\sigma y},\n\\]\nso that $C \\asymp 1\/c$.\n\n\\begin{lemma}\n \\label{lem:rcb8}\nWith probability $1 - o(1)$, we have \n\\[\n U(q,\\vec{\\ba}) \n = \n \\bigg(\n 1 + O_{\\le}\\bigg(\\frac{1}{(\\log_2 x)^3}\\bigg)\n \\bigg)\n C\n\\]\nfor all but at most $\\frac{x}{2\\log x\\log_2 x}$ of the primes \n$q \\in \\cQ \\cap S(\\vec{\\ba})$.\n\\end{lemma}\n\n\\begin{proof}\nUsing \\eqref{eq:rcb14} and Lemma \\ref{lem:rcb7}, we find that \n\\begin{align} \n \\label{eq:rcb23}\n \\begin{split}\n & \n \\EE \\,\n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n (U(q,\\vec{\\ba}) - C)^2\n \\Big)\n \\\\\n & \n =\n \\EE \\,\n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})} U(q,\\vec{\\ba})^2 \n \\Big)\n -\n 2C \\,\n \\EE \\, \n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})} U(q,\\vec{\\ba})\n \\Big)\n + \n C^2 \\,\n \\EE \\,\n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})} 1\n \\Big) \n \\\\\n & \n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n \\frac{u^2x^2}{4\\sigma y\\log x}\n \\\\\n & \\hspace{45pt} \n -\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n \\frac{2ux}{2\\sigma y}\n \\frac{ux}{2\\log x}\n +\n \\big(1 + O\\big({\\textstyle \\frac{1}{\\log x}}\\big)\\big)\n \\frac{u^2x^2}{4\\sigma^2y^2}\n \\frac{\\sigma y}{\\log x}\n \\\\\n & \n \\ll\n \\frac{u^2x^2}{\\sigma y(\\log x)(\\log_2 x)^{10}}\n \\\\\n & \n \\ll\n \\frac{C^2\\sigma y}{(\\log x)(\\log_2 x)^{10}}.\n \\end{split}\n\\end{align}\nLet $V$ be the event \n\\[\n \\#\\big\\{\n q \\in \\cQ \\cap S(\\vec{\\ba}) : \n |U(q,\\vec{\\ba}) - C| \n >\n {\\textstyle \\frac{C}{(\\log_2 x)^3}}\n \\big\\}\n >\n \\frac{x}{2\\log x \\log_2 x}.\n\\]\nEvidently \n\\[\n \\EE \\,\n \\Big(\n \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n (U(q,\\vec{\\ba}) - C)^2\n \\Big)\n \\ge \n \\PP(V)\n \\frac{x}{2\\log x\\log_2 x}\n \\frac{C^2}{(\\log_2 x)^6}.\n\\]\nCombining this with \\eqref{eq:rcb23}, and recalling that \n$\\sigma y \\asymp x\\log_2 x$ (cf.\\ \\eqref{eq:fgkmt6.11}), we obtain \n\\[\n \\PP(V) \\ll \\frac{1}{(\\log_2 x)^2}.\n\\]\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:rcb9}\nWe have \n\\begin{equation}\n \\label{eq:rcb24}\n \\EE \\, \n \\Big(\n \\sum_{n \\in \\ZZ}\n \\sigma^{-r}\n \\sum_{p \\in \\cP(\\vec{\\ba})} \n Z_p(\\vec{\\ba};n)\n \\Big)\n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\\bigg)\n (\\#\\cP)\n\\end{equation}\nand \n\\begin{equation}\n \\label{eq:rcb25}\n \\EE \\, \n \\Big(\n \\sum_{n \\in \\ZZ}\n \\sigma^{-r}\n \\sum_{p \\in \\cP \\setminus \\cP(\\vec{\\ba})} \n Z_p(\\vec{\\ba};n)\n \\Big)\n \\ll\n \\frac{\\#\\cP}{(\\log x)^4}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nThe left-hand side of \\eqref{eq:rcb24} is \n\\begin{align*}\n & \n \\sigma^{-r}\n \\sum_{\\vec{\\ba}}\n \\PP(\\vec{\\ba} = \\vec{a})\n \\sum_{p \\in \\cP(\\vec{a})}\n \\sum_{n \\in \\ZZ}\n \\PP(n + h_ip \\in S(\\vec{a}) \\, \\, \\forall i \\le r)\n \\PP(\\tilde{\\bn}_p = n)\n \\\\\n & \\hspace{30pt} = \n \\sigma^{-r}\n \\sum_{\\vec{\\ba}}\n \\PP(\\vec{\\ba} = \\vec{a})\n \\sum_{p \\in \\cP(\\vec{a})}\n X_p(\\vec{a})\n \\\\\n & \\hspace{30pt} = \n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n \\sum_{\\vec{\\ba}}\n \\PP(\\vec{\\ba} = \\vec{a})\n \\sum_{p \\in \\cP(\\vec{a})} 1\n \\\\\n & \\hspace{30pt} = \n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n \\EE \\, \\#\\cP(\\vec{a})\n \\\\\n & \\hspace{30pt} = \n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\\bigg)\n (\\# \\cP)\n\\end{align*}\nby Lemma \\ref{lem:rcb5}.\nThis proves \\eqref{eq:rcb24}.\n\nNow, \n\\begin{align}\n \\label{eq:rcb26}\n \\begin{split}\n \\EE \\,\n \\Big(\n \\sum_{n \\in \\ZZ} \\sigma^{-r}\n \\sum_{p \\in \\cP} Z_p(\\vec{\\ba};n)\n \\Big)\n & \n =\n \\sigma^{-r}\n \\sum_{p \\in \\cP}\n \\sum_{n \\in \\ZZ}\n \\PP(\\tilde{\\bn}_p = n)\n \\PP(n + h_jp \\in S(\\vec{a}) \\,\\, \\forall j \\le r)\n \\\\\n & \n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n \\sum_{p \\in \\cP}\n \\sum_{n \\in \\ZZ} \n \\PP(\\tilde{\\bn}_p = n)\n \\\\\n & \n =\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n (\\# \\cP).\n \\end{split}\n\\end{align}\n(Here we have used Lemma \\ref{lem:rcb4} with a familiar argument.)\nWe obtain \\eqref{eq:rcb25} on subtracting \\eqref{eq:rcb24} from \n\\eqref{eq:rcb26}.\n\\end{proof}\n\n\\begin{proof}%\n[Deduction of Theorem \\ref{thm:fgkmt4} \\textup{(}v\\textup{)}]\nIn view of Lemmas \\ref{lem:rcb6} and \\ref{lem:rcb8} it suffices to \nshow that with probability $1 - O\\big(1\/(\\log x)^3\\big)$, the \nnumber of $q$ in $\\cQ \\cap S(\\vec{\\ba})$ with \n\\begin{equation}\n \\label{eq:rcb27}\n \\sum_{i = 1}^r\n \\sum_{p \\in \\cP \\setminus \\cP(\\vec{\\ba})}\n \\sigma^{-r} Z_p(a; q - h_ip)\n >\n \\frac{ux}{\\sigma y(\\log_2 x)^3}\n\\end{equation}\nis at most $\\frac{x}{2\\log x\\log_2 x}$.\n\nLet $W$ be the event that \\eqref{eq:rcb27} holds for more than \n$\\frac{x}{2\\log x\\log_2 x}$ primes in \n\\linebreak \n$\\cQ \\cap S(\\vec{\\ba})$.\nThen \n\\[\n \\PP(W)\n \\le \n \\PP\n \\Big(\n \\sum_{q \\in \\cQ}\n \\sum_{i=1}^r\n \\sum_{p \\in \\cP\\setminus \\cP(\\vec{\\ba})}\n \\sigma^{-r}\n Z_p(\\vec{\\ba};q - h_ip) > v\n \\Big),\n\\]\nwhere \n\\[\n v \n \\defeq \n \\frac{ux}{\\sigma y(\\log_2 x)^3}\n \\cdot \n \\frac{x}{2\\log x \\log_2 x}\n =\n \\frac{x}{(\\log x)^{2 - o(1)}}.\n\\]\nThus, \n\\begin{multline*}\n \\PP(W) \n \\le \n \\frac{1}{v} \n \\EE \\, \n \\Big(\n \\sum_{q \\in \\cQ}\n \\sum_{i=1}^r \n \\sum_{p \\in \\cP\\setminus \\cP(\\vec{\\ba})}\n \\sigma^{-r}\n Z_p(q - h_ip)\n \\Big)\n \\\\\n \\le \n \\frac{r}{v}\n \\EE \\,\n \\Big(\n \\sum_{n \\in \\ZZ}\n \\sigma^{-r} \n \\sum_{p \\in \\cP\\setminus \\cP(\\vec{\\ba})}\n Z_p(\\vec{\\ba};n)\n \\Big)\n \\ll\n \\frac{r}{v}\n \\frac{x}{(\\log x)^5}\n \\ll\n \\frac{1}{(\\log x)^2}.\n\\end{multline*}\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:fgkmt2}}\n \\label{subsec:thm2pf}\nWe require one further lemma for the proof of \nTheorem \\ref{thm:fgkmt2}, viz.\\ the following, which is a special \ncase of \\cite[Corollary 3]{FGKMT}.\n\n\\begin{lemma}\n \\label{lem:rcb10}\nLet $\\cQ'$ be a set of primes with $\\#\\cQ' > (\\log_2 x)^3$.\nFor each $p \\in \\cP$, let $\\be_p$ be a random subset of $\\cQ'$ \nwith \n\\[\n \\# \\be_p \\le r, \n \\quad \n \\PP(q \\in \\be_p) \\le x^{-3\/5} \\quad (q \\in \\cQ').\n\\]\nSuppose that for all but at most $\\frac{\\#\\cQ'}{(\\log_2 x)^2}$ \nelements $q \\in \\cQ'$, we have \n\\[\n \\sum_{p \\in \\cP} \\PP(q \\in \\be_p)\n =\n C + O_{\\le}\\bigg(\\frac{1}{(\\log_2 x)^2}\\bigg),\n\\]\nwhere $C$ is independent of $q$ and \n\\begin{equation}\n \\label{eq:rcb28}\n {\\textstyle \\frac{5}{4}}\\log 5 \\le C \\ll 1.\n\\end{equation}\nSuppose that for any distinct $q_1,q_2 \\in \\cQ'$, \n\\begin{equation}\n \\label{eq:rcb29}\n \\sum_{p \\in \\cP'} \\PP(q_1,q_2 \\in \\be_p)\n \\le \n x^{-1\/20}.\n\\end{equation}\nThen for any positive integer $m$ with \n\\[\n m \\le \\frac{\\log_3 x}{\\log 5},\n\\]\nwe can find random sets $\\be_p' \\subseteq \\cQ'$ for each \n$p \\in \\cP$ such that $\\be_p'$ is either empty or is in the \nessential range of $\\be_p$, and \n\\begin{equation}\n \\label{eq:rcb30}\n \\#\\{q \\in \\cQ' : q \\not\\in \\be_p' \\,\\, \\textup{for all} \\,\\, p \\in \\cP \\}\n \\sim \n 5^{-m}(\\#\\cQ'),\n\\end{equation}\nwith probability $1 - o(1)$.\n\\end{lemma}\n\n\\begin{proof}[Deduction of Theorem \\ref{thm:fgkmt2}]\nBy \\eqref{eq:fgkmt4.29}, we may choose $c$ small enough so that \n\\eqref{eq:rcb28} holds.\nTake \n\\[\n m = \\Big\\lfloor \\frac{\\log_3 x}{\\log 5} \\Big\\rfloor.\n\\]\nLet $\\vec{\\ba}$ and $\\vec{\\bn}$ be as in Theorem \\ref{thm:fgkmt4}.\nSuppose that we are in the probability $1 - o(1)$ event that \n$\\vec{\\ba}$ takes a value $\\vec{a}$ for which \\eqref{eq:fgkmt4.31} \nholds.\nFix some $\\vec{a}$ within this event.\nWe apply Lemma \\ref{lem:rcb10} with $\\cQ' = \\cQ \\cap S(\\vec{a})$, \n$\\be_p = \\be_p(\\vec{a})$.\nWe need only check the hypothesis \\eqref{eq:rcb29}.\nWe have \n\\[\n \\sum_{p \\in \\cP}\n \\PP(q_1,q_2 \\in e_p(\\vec{a})\n \\le \n \\sums[p \\mid q_1 - q_2][p \\in \\cP]\n \\PP(q_1 \\in e_p(\\vec{a}))\n \\le \n x^{-3\/5}\n\\]\n(the sum has at most one term).\n\nLet $\\be_p'(\\vec{a})$ be the random variables provided by Lemma \n\\ref{lem:rcb10}.\nRecalling \\eqref{eq:fgkmt4.30}, \n\\[\n \\#\\{q \\in \\cQ' : q \\not\\in \\be_p' \\,\\, \\text{for all} \\,\\, p \\in \\cP \\}\n \\sim \n 5^{-m}\\#(\\cQ \\cap S(\\vec{a}))\n \\ll\n \\frac{x}{\\log x}\n\\]\nwith probability $1 - o(1)$.\nSince $e_p'(\\vec{a})$ is either empty or \n\\[\n e_p'(\\vec{a})\n =\n \\{\\tilde{\\bn}_p' + h_ip : i \\le r\\}\n \\cap \n \\cQ \\cap S(\\vec{a})\n\\]\nfor some random integer $\\tilde{\\bn}_p'$, it follows that \n\\[\n \\#\\{\n q \\in \\cQ \\cap S(\\vec{a}) : \n q \\not\\equiv \\tilde{\\bn}_p' \\pod{p} \n \\,\\, \\text{for all} \\,\\, p \\in \\cP\n \\}\n \\ll\n \\frac{x}{\\log x}\n\\]\nwith probability $1 - o(1)$.\nThe bound \\eqref{eq:fgkmt3.6} follows on setting $b_p = n_p'$ for \na specific $\\vec{n}' = (\\tilde{\\bn}_p')$ for which this bound \nholds.\nThat $\\cH$ is contained in $S(\\vec{a}) \\cap S(\\vec{b})$ follows \nfrom parts (i) and (ii) of Theorem \\ref{thm:fgkmt4}.\n\\end{proof}\n\n\n\\section{A modification of Maynard--Tao}\n \\label{sec:MT}\n \n\\begin{definition} \n \\label{def:w}\nWe consider functions of the form \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$, with $T_1,x_1 \\ge 1$.\nLet us say that such a function $f_1$ is ``of the first kind'' if \nand only if \n(i) it is a strictly increasing bijection, \n(ii) $f_1(T) \\le \\log T$ for $T \\ge T_1$,\n(iii) $f_1(2T)\/f_1(T) \\to 1$ as $T \\to \\infty$\nand\n(iv) for $0 < \\eta \\le 1$, there exists $L_{\\eta} \\ge 1$ such that \n$f_1(T)\/f_1(T^{\\eta}) \\to L_{\\eta}$ as \n$T \\to \\infty$.\n\\end{definition}\n\n\\begin{definition}\n \\label{def:sparse}\nWe consider (possibly empty) sets $\\cZ(T)$, $T \\ge 2$, of primes \nless than or equal to $T$.\nLet us that such a set is ``repulsive'' if and only if for any \n$p' \\in \\cZ(T)$, \n$\\sum_{p \\in \\cZ(T), \\, p \\ge p'} 1\/p \\ll 1\/p' \\ll 1\/\\log_2 T$.\n\\end{definition}\n\nGiven a function $\\upsilon : \\NN \\to \\RR$ with finite support and \nany arithmetic progression $a \\pod{D}$ with $(a,D) = 1$, we \ndefine \n\\[\n \\Delta(\\upsilon;a \\pod{D})\n \\defeq \n \\sum_{n \\equiv a \\pod{D}}\n \\upsilon(n)\n -\n \\frac{1}{\\phi(D)}\n \\sum_{(n,D) = 1}\n \\upsilon(n),\n\\]\nwhere $\\phi$ is Euler's totient function.\n\n\\begin{hypothesis} \n \\label{hyp:EH}\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\nFor any given $A > 0$ and $\\delta \\in (0,\\theta)$, if \n$\\eta = \\eta(A,\\delta) \\in (0,\\theta - \\delta)$ is a \nsufficiently small, fixed number then, for $N \\ge T_1^{1\/\\eta}$, \nthere is a repulsive subset $\\cZ \\defeq \\cZ(N^{4\\eta})$ of the \nprimes less than or equal to $N^{4\\eta}$ such that, with \n$W \\defeq \\prod_{p \\le N^{\\eta}, \\, p \\not\\in \\cZ} p$ and \n$Z \\defeq \\prod_{p \\in \\cZ} p$, we have\n\\[\n \\sums[r \\le N^{\\theta}\/(N^{\\delta}W)]\n [(r,WZ) = 1]\n [\\text{$r$ squarefree}]\n \\max_{N \\le M \\le 2N}\n \\max_{(rW,a) = 1} \n |\\Delta(\\ind{\\bP}\\ind{(M,M + N]}; a \\pod{rW})|\n \\ll_{\\delta,A}\n \\frac{N}{\\phi(W)(\\log N)^A}.\n\\]\n\\end{hypothesis}\n\n\\begin{theorem}\n \\label{thm:BFM4.3}\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\nSuppose that Hypothesis \\ref{hyp:EH} holds.\nFix a positive integer $a$.\nIn the notation of Hypothesis \\ref{hyp:EH}, if \n$K = K_{\\theta,a}$ is a sufficiently large integer multiple of \n$\\lceil (2\/\\theta) a \\rceil + 1$, if $A = A_K$ is sufficiently \nlarge and if $\\delta = \\delta_{\\theta,a}$ is sufficiently small, \nthen the following holds for $N \\ge N(T_1,K,\\eta)$.\nLet $\\cH \\defeq \\{H_1,\\ldots,H_K\\} \\subseteq [0,N]$ be an \nadmissible set of $K$ distinct integers for which \n$\\prod_{1 \\le i < j \\le K}(H_j - H_i)$ is \n$f_1(N^{\\eta})$-smooth, and let $b$ be an integer such that \n\\[\n \\textstyle (\\prod_{i = 1}^K (b + H_i),W) = 1.\n\\]\nThen for any partition \n\\[\n \\cH \n = \n \\cH_1 \\cup \\cdots \\cup \\cH_{\\lceil (2\/\\theta) a \\rceil + 1} \n\\]\nof $\\cH$ into $\\lceil (2\/\\theta) a \\rceil + 1$ sets of equal \nsize, there exists some $n \\in (N,2N] \\cap b \\pod{W}$, and $a + 1$ \ndistinct indices \n$\n i_1,\\ldots,i_{a+1} \n \\in \n \\{1,\\ldots,\\lceil (2\/\\theta) a \\rceil + 1\\}\n$, \nsuch that \n\\[\n \\#(\\bP \\cap n + \\cH_{i_1}),\n \\ldots,\n \\#(\\bP \\cap n + \\cH_{i_{a + 1}}) \\ge 1.\n\\]\n\\end{theorem}\n\n\n\\begin{theorem}\n \\label{thm:BFM4.2}\nHypothesis \\ref{hyp:EH}, and therefore the statement of \nTheorem \\ref{thm:BFM4.3}, holds with $\\theta = 1\/2$ and any \nfunction $f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\n\\end{theorem}\n\n\\begin{proof}%\n[Proof of Theorems \\ref{thm:BFM4.2} and \\ref{thm:BFM4.3}]\nThat Hypothesis \\ref{hyp:EH} holds with $\\theta = 1\/2$ and any \nfunction $f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind \nis a consequence of Lemma 4.1 and Theorem 4.2 of \\cite{BFM}.\n\nWe prove Theorem \\ref{thm:BFM4.3} by following Pintz's \\cite{PIN3} \nmodification to the proof of Theorem 4.3 (i) in \\cite{BFM}.\nThere are many parameters involved and it is important to keep \ntrack of their interdependencies.\nIt is also important to note that the implicit constants in all \n$O$-terms are absolute, that is, independent of all \nparameters.\n\nOnly the unconditional case $\\theta = 1\/2$ is considered in \n\\cite{BFM,PIN3}, whereas here we are considering $\\theta \\le 1$.\nTo do this, we need to note that on Hypothesis \\ref{hyp:EH}, the \nterm $4 + O(\\delta)$ may be replaced by $(2\/\\theta) + O(\\delta)$ \non the right-hand side of the inequality in \n\\cite[Lemma 4.5 (iii)]{BFM}.\nIn the proof of this lemma in \\cite[\\S 4.2]{BFM}, the support of \nthe smooth function $G : [0,\\infty) \\to \\RR$, which is \n$[0,1\/4 - 2\\delta]$, may be replaced by \n$[0,(\\theta\/2) - 2\\delta]$, and the rest of the proof may be \ncarried out, mutatis mutandis.\n\nAs in \\cite{PIN3}, we begin with the following observation.\nSuppose $K$ and $M$ are positive integers with $M \\mid K$, and \nlet $\\cH = \\cH_1 \\cup \\cdots \\cup \\cH_M$ be a partition of a set \n$\\cH$ of integers into $M$ subsets of equal size.\nSuppose also that $\\mu'$ and $\\mu$ are positive real numbers \nwith \n\\[\n \\mu' \n \\defeq\n \\max_{v \\in \\NN}\n \\bigg(v - \\mu \\binom{v}{2}\\bigg).\n\\]\n\nGiven an integer $n$, consider the expression \n\\[\n \\sum_{j=1}^M\n \\Big\\{ \n \\sum_{H \\in \\cH_j} \\ind{\\bP}(n + H)\n -\n \\mu \n \\sums[H,H' \\in \\cH_j][H \\ne H']\n \\ind{\\bP}(n + H)\\ind{\\bP}(n + H') \n \\Big\\},\n\\]\nwhere in the double sum each unordered pair \n$\\{H,H'\\} \\subseteq \\cH_j$ with $H \\ne H'$ is counted once only. \nSuppose $\\#(\\bP \\cap n + \\cH_j) = 0$ for all but at most $a$ of \nthe subsets $\\cH_j$.\nThen the above expression is at most $\\mu' a$.\nConsequently, if \n\\[\n \\sum_{H \\in \\cH}\n \\ind{\\bP}(n + H)\n -\n \\mu' a\n - \n \\mu\n \\sum_{j=1}^M\n \\sums[H,H' \\in \\cH_j][H \\ne H']\n \\ind{\\bP}(n + H)\n \\ind{\\bP}(n + H')\n\\]\nis positive then $\\#(\\bP \\cap n + \\cH_j) \\ge 1$ for at least \n$a + 1$ of the subsets $\\cH_j$.\n\nNote that when $\\mu$ is the reciprocal of a positive integer, \nwe have \n\\[\n \\mu' \n = \n \\textstyle \n \\frac{1}{2}\\big(1 + \\frac{1}{\\mu}\\big),\n\\]\nthe maximum being attained when $v = 1\/\\mu$ and \n$v = 1 + 1\/\\mu$.\n\nNow, fix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\nSuppose that Hypothesis \\ref{hyp:EH} holds.\nFix any positive integer $a$ and let $M = M_{\\theta,a}$ be \nthe integer satisfying \n\\[\n M - 2 < (2\/\\theta)a \\le M - 1.\n\\]\nLet $\\iota = \\iota_{\\theta,a}$ be a small, fixed quantity to \nbe specified.\nSet \n\\[\n \\delta = \\delta_{\\theta,a} \\defeq \\iota^2\/(aM).\n\\]\nLet $K = K_{\\theta,a}$ be the integer satisfying\n\\[\n \\e^{aM^2\/(\\delta(M-1))} < K \\le \\e^{aM^2\/(\\delta(M-1))} + M\n \\quad \n \\text{and}\n \\quad \n M \\mid K.\n\\]\nFinally, let\n\\[\n \\rho \\defeq \\frac{aM^2\/(M-1)}{\\delta\\log K} < 1.\n\\]\n\nNow let $A = A(K)$, $\\eta = \\eta(A,\\delta)$, \n$N \\ge N(T_1,K,\\eta)$, $\\cH = \\{H_1,\\ldots,H_K\\}$ and $b \\pod{W}$ \nbe as in the statement of the theorem.\nLet $\\cH = \\cH_1 \\cup \\cdots \\cup \\cH_M$ be any partition of $\\cH$ \ninto $M$ subsets of equal size.\nConsider the expression\n\\[\n S \n \\defeq \n \\hspace{-7pt} \n \\sums[N < n \\le 2N][n \\equiv b \\pod{W}]\n \\hspace{-3pt}\n \\Big\\{\n \\sum_{H \\in \\cH} \\ind{\\bP}(n + H)\n -\n \\frac{1 + M}{2}a\n -\n \\frac{1}{M}\n \\sum_{j=1}^M\n \\sums[H,H' \\in \\cH_j][H \\ne H']\n \\hspace{-5pt}\n \\ind{\\bP}(n + H)\\ind{\\bP}(n + H')\n \\Big\\}\n \\nu_{\\cH}(n),\n\\]\nwhere $\\nu_{\\cH} : \\NN \\to [0,\\infty)$ is the nonnegative weight \ngiven by \n\\[\n \\nu_{\\cH}(n)\n \\defeq \n \\Big( \n \\sums[d_1,\\ldots,d_K][d_i \\mid n + H_i \\,\\, \\forall i \\le K]\n \\lambda_{d_1,\\ldots,d_K}\n \\Big)^2,\n\\]\nand where $(\\lambda_{d_1,\\ldots,d_K})$ is the Maynard--Tao sieve \nas used in \\cite[\\S 4]{BFM}.\nThe aim is to show that $S > 0$, for in that case, by the \nobservation made at the beginning of the proof, there must exist \nsome $n \\in (N,2N] \\cap b \\pod{W}$ and $a + 1$ subsets $\\cH_j$ \nfor which $\\#(\\bP \\cap n + \\cH_{j}) \\ge 1$.\n\nAt this point we invoke Lemmas 4.5 and 4.6 in \\cite{BFM} (with \n$4 + O(\\delta)$ in the latter replaced by \n$2\/\\theta + O(\\delta) \\le (M - 1)\/a + O(\\delta)$).\nTo ease notation define $\\mathfrak{S}$ by the relation \n$\n S = \\mathfrak{S}NW^{-1}B^{-K}I_K(F),\n$\nwith $B$ and $I_K(F)$ as defined in \\cite[\\S 4.2]{BFM}.\nAlso let $\\xi = (\\log K)^{-1\/2}$.\n\nAs in \\cite[\\S 4.2]{BFM} and \\cite[(3.13)]{PIN3}, we find that the \nrelevant estimates yield \n\\begin{align*}\n \\mathfrak{S}\n & \n \\ge\n \\sum_{H \\in \\cH} \\frac{aM^2\/(M-1)}{K}(1 + O(\\xi))\n - \\frac{1 + M}{2}a \n \\\\\n & \\hspace{30pt} \n - \\frac{1}{M}\n \\sum_{j=1}^M \n \\sums[H,H' \\in \\cH_j][H \\ne H']\n \\frac{M-1}{a}\\cdot \n \\frac{(aM^2\/(M-1))^2}{K^2}\n (1 + O(\\delta + \\xi))\n \\\\\n & \n =\n \\frac{aM^2}{M - 1}(1 + O(\\xi))\n - \n \\frac{1 + M}{2}a \n -\n \\binom{K\/M}{2}\n \\frac{aM^4(1 + O(\\delta + \\xi))}{K^2(M-1)}\n .\n\\end{align*}\nRecalling that $\\e^{aM^2\/(\\delta(M-1))} < K$, we see that \n$M\/K < \\delta$ and hence \n\\[\n \\binom{K\/M}{2}\n = \n \\frac{K^2}{2M^2}\\Big(1 - \\frac{M}{K}\\Big)\n =\n \\frac{K^2}{2M^2}(1 + O(\\delta)).\n\\]\nWe also have $\\xi^2 < \\delta\/(aM) = \\iota^2\/(a^2M^2)$, so \n$\\delta + \\xi = O(\\iota\/(aM))$.\nWe therefore have\n\\begin{align*}\n \\mathfrak{S}\n & \\ge \n \\frac{aM^2}{M-1}\\Big(1 + O\\Big(\\frac{\\iota}{aM}\\Big)\\Big) \n - \\frac{1 + M}{2}a \n - \\frac{aM^2}{2(M-1)}\\Big(1 + O\\Big(\\frac{\\iota}{aM}\\Big)\\Big)\n \\\\\n & =\n \\frac{a(1 + O(\\iota))}{2(M - 1)}.\n\\end{align*}\nTaking $\\iota$ sufficiently small gives $\\mathfrak{S} > 0$, and \nhence $S > 0$, as desired.\n\\end{proof}\n\n\n\n\n\\section{Main Theorem and Deduction of Theorem \\ref{thm:main}}\n \\label{sec:BFM}\n\nRecall Definition \\ref{def:w}, in which functions ``of the first \nkind'' are introduced.\nWe now define a second kind of function.\n\n\\begin{definition} \n \\label{def:2ndkind}\nWe consider functions $f_2 : [x_2,\\infty) \\to [z_2,\\infty)$, with \n$x_2,z_2 \\ge 1$. \\linebreak\nLet us say that such a function $f_2$ is ``of the second kind'' \nif and only if\n(i) it is a strictly increasing bijection, \n(ii) \n\\[\n (x\/\\log x)\/f_2(x) \\to 0\n \\quad \n \\text{and} \n \\quad \n f_2(x)\/(x\\log x \\log_3 x\/\\log_2 x) \\to 0\n\\]\nas $x \\to \\infty$ and \n(iii) for any $C > 0$, $f_2(Cx)\/(Cf(x)) \\to 1$ as $x \\to \\infty$.\n\\end{definition}\n\n\\begin{theorem}\n \\label{thm:general}\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ \nof the first kind, and suppose Hypothesis \\ref{hyp:EH} holds.\nFix a function \n$f_2 : [x_1,\\infty) \\to [z_2,\\infty)$ \nof the second kind and let \n$f \\defeq f_2 \\circ f_1$.\nLet $d_n \\defeq p_{n+1} - p_n$, where $p_n$ denotes the $n$th \nsmallest prime, and let $\\LP[f]$ denote the set of limit \npoints in $[0,\\infty]$ of the sequence \n$(d_n\/f(p_n))_{p_n \\ge T_1}$.\nThen given any $\\lceil (2\/\\theta) \\rceil + 1$ nonnegative real numbers \n$\n\\alpha_1,\\ldots,\\alpha_{\\lceil (2\/\\theta) \\rceil + 1}\n$ \nwith \n\\[\n \\alpha_1 \\le \\cdots \\le \\alpha_{\\lceil (2\/\\theta) \\rceil + 1},\n\\] \nwe have \n\\begin{equation}\n \\label{eq:genthm1}\n \\{\\alpha_j - \\alpha_i : 1 \\le i < j \\le \\lceil (2\/\\theta) \\rceil + 1\\} \n \\cap \n \\LP[f]\n \\ne \n \\emptyset.\n\\end{equation}\nConsequently, letting $\\lambda$ denote the Lebesgue measure on \n$\\RR$, we have \n\\begin{equation}\n \\label{eq:genthm2}\n \\lambda([0,X] \\cap \\LP[f])\n \\ge \n c_1(\\theta)X\n \\quad (X \\ge 0)\n\\end{equation}\nand \n\\begin{equation}\n \\label{eq:genthm3}\n \\lambda([0,X] \\cap \\LP[f])\n \\ge \n (1 - o(1))\n c_2(\\theta)X\n \\quad (X \\to \\infty),\n\\end{equation}\nwhere \n\\begin{equation}\n \\label{eq:genthm4}\n c_1(\\theta)\n \\defeq\n (\\lceil (2\/\\theta) \\rceil(1 + 1\/2 + \\cdots + 1\/\\lceil (2\/\\theta) \\rceil))^{-1}\n \\quad\n \\text{and}\n \\quad \n c_2(\\theta) \\defeq 1\/\\lceil (2\/\\theta) \\rceil.\n\\end{equation}\n\\end{theorem}\n\n\\vspace*{1em}\n\n\\begin{center}\n \\label{tab:1}\n\\begin{tabular}{|c|c|c|c|} \n \\hline \n $\\theta$ & $\\lceil (2\/\\theta) \\rceil + 1$ & $c_1(\\theta)$ & $c_2(\\theta)$ \\\\ \\hline \\hline\n $1\/2 \\le \\theta < 2\/3$ & $5$ & $3\/25$ & $1\/4$ \\\\ \\hline \n $2\/3 \\le \\theta < 1\\phantom{\/1}$ & $4$ & $2\/11$ & $1\/3$ \\\\ \\hline \n $ \\theta = 1$ & $3$ & $1\/3\\phantom{0}$ & $1\/2$ \\\\ \\hline \n\\end{tabular}\n \\vspace*{1em}\n \\captionof{table}{Possible values of $\\lceil (2\/\\theta) \\rceil + 1$, $c_1(\\theta)$ and $c_2(\\theta)$.}\n\\end{center}\n\n\\begin{proof}[Deduction of Theorem \\ref{thm:main}]\nIn view of Theorem \\ref{thm:BFM4.2}, we may unconditionally apply \nTheorem \\ref{thm:general} with $\\theta = 1\/2$ and any function \n$\nf_1 : [T_1,\\infty) \\to [x_1,\\infty)\n$ \nof the first kind.\nLet \n$\nf_1 : [\\e^{\\e^{\\e}},\\infty) \\to [\\e^{\\e},\\infty)\n$\nbe given by $f_1(T) = \\log T$, and let \n$\nf_2 : [\\e^{\\e},\\infty) \\to [\\e^{\\e + 1},\\infty)\n$\nbe given by $f_2(x) = x\\log x\/\\log_2 x$.\nThen $f_1$ is of the first kind, $f_2$ is of the \nsecond kind and \n$\n f_2 \\circ f_1(T) \n =\n R_1(T)\n =\n \\log T \\log_2 T\/\\log_3 T\n$\nfor $T \\ge \\e^{\\e^{\\e}}$.\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:goodktuple}\nLet $K$ be a natural number and let $K = K_1 + \\cdots + K_M$ be a \npartition of $K$. \nLet $x$ and $y$ be real numbers such that $K \\le y\/x \\le \\log x$.\nIf $x$ is sufficiently large, then for any $M$ \n\\textup{(}possibly overlapping\\textup{)} subintervals \n$(v_i,v_i + x\/\\log x] \\subseteq (x,y]$, $i \\le M$, there exist $M$ \npairwise disjoint sets of primes \n$\\cH_i \\subseteq (v_i,v_i + x\/\\log x]$ with $|\\cH_i| = K_i$, such \nthat if $\\cH_1 \\cup \\cdots \\cup \\cH_M = \\{q_1,\\ldots,q_K\\}$, then \n$\\prod_{1 \\le i < j \\le K}(q_j - q_i)$ is $x$-smooth.\n\\end{lemma}\n\n\\begin{proof}\nFor any $M$ sets $\\cJ_i$ with $|\\cJ_i| \\ge K$, \n$i \\le M$, there exist $M$ pairwise disjoint sets \n$\\cH_i$ such that $\\cH_i \\subseteq \\cJ_i$ and $|\\cH_i| = K_i$, \n$i \\le M$.\nFor the sets $\\cJ_i$, let $D$ be the integer satisfying \n$y\/x \\le D < 1 + y\/x$.\nAs $D < 1 + \\log x$ and $y \\le x\\log x$, a suitably strong \nversion of the prime number theorem for arithmetic progressions \n(cf.\\ \\cite[\\S22 (4)]{DAV}) yields, for \n$(v_i,v_i + x\/\\log x] \\subseteq (x,y]$, \n\\[\n \\sums[v < p \\le v + x\/\\log x][p \\equiv 1 \\pod{D}] 1\n =\n \\frac{x}{\\phi(D)(\\log x)^2} + O\\bigg(\\frac{y}{(\\log y)^5}\\bigg)\n \\ge \n \\frac{x}{(\\log x)^3} + O\\bigg(\\frac{x}{(\\log x)^4}\\bigg).\n\\]\nAs $K \\le \\log x$, we see that if $x$ is sufficiently large, \nthen there are at least $K$ primes $p \\equiv 1 \\pod{D}$ in \n$(v_i,v_i + x\/\\log x]$.\nIf $q < q'$ are any two such primes, as $q' - q \\le y$ \nand $q' \\equiv q \\pod{D}$, any prime divisor $p$ of $q' - q$ \nmust either divide $D < 1 + \\log x$ or be less than or equal to \n$y\/D \\le x$.\nHence $q' - q$ is $x$-smooth.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:general}]\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ \nof the first kind, and suppose Hypothesis \\ref{hyp:EH} holds.\nIn accordance with Theorem \\ref{thm:BFM4.3} (in which we take \n$a = 1$), let $K = K_{\\theta}$ be a sufficiently large integer \nmultiple of $\\lceil (2\/\\theta) \\rceil + 1$, let $A = A_K$ be \nsufficiently large and $\\delta = \\delta_{\\theta}$ and \n$\\eta = \\eta(A,K)$ be sufficiently small.\n\nIn accordance with Corollary \\ref{cor:thm2}, let $C$ be a \nsufficiently large but fixed positive constant and let $x$ be a \nsufficiently large number.\nSuppose, as we may, that $Cx \\ge x_1$, and \n(cf.\\ Definition \\ref{def:w} (i)) set \n\\[\n N \\defeq (f_1^{-1}(Cx))^{1\/\\eta}.\n\\]\nThus, $Cx = f_1(N^{\\eta})$ and $N$ tends to infinity with $x$.\nSuppose $x$ is large enough so that in accordance with \nTheorem \\ref{thm:BFM4.3}, $N \\ge N(T_1,K,\\eta)$.\n\nBy Definition \\ref{def:w} (iv) there exists $L_{\\eta} \\ge 1$ such \nthat $f_1(N)\/x \\to CL_{\\eta}$ as $x \\to \\infty$.\nFix nonnegative real numbers \n$\\alpha_1,\\ldots,\\alpha_{\\lceil (2\/\\theta) \\rceil + 1}$ with \n$\\alpha_1 \\le \\cdots \\le \\alpha_{\\lceil (2\/\\theta) \\rceil + 1}$ \nand set \n\\begin{equation}\n \\label{eq:betaeta}\n \\beta_i \\defeq \\alpha_iC L_{\\eta}, \n \\quad i \\le \\lceil (2\/\\theta) \\rceil + 1.\n\\end{equation}\n\nFix a function $f_2 : [x_2,\\infty) \\to [z_2,\\infty)$ of the second \nkind and consider the intervals\n\\begin{equation}\n \\label{eq:intervals}\n (x + \\beta_i f_2(x),x + \\beta_i f_2(x) + x\/\\log x],\n \\quad \n i \\le \\lceil (2\/\\theta) \\rceil + 1.\n\\end{equation}\nRecall that $y \\defeq cx\\log x \\log_3 x\/\\log_2 x$, where $c > 0$ \nis a certain constant (cf.\\ \\eqref{eq:fgkmt3.1}). \nBy Definition \\ref{def:2ndkind} (ii) we have \n$x\/\\log x = o(f_2(x))$ and $f_2(x) = o(y)$.\nSuppose, then, that $x$ is large enough (in terms of \n$\\beta_{\\lceil (2\/\\theta) \\rceil+1}$) so that the intervals in \n\\eqref{eq:intervals} are all contained in $(x,y]$. \n\nIn accordance with Lemma \\ref{lem:goodktuple}, choose \n$\\lceil (2\/\\theta) \\rceil + 1$ pairwise disjoint sets of primes \n$\\cH_i$ of equal size, with \n\\begin{equation}\n \\label{eq:rightsize}\n \\cH_i \n \\subseteq \n (x + \\beta_i f_2(x),x + \\beta_i f_2(x) + x\/\\log x],\n \\quad \n i \\le \\lceil (2\/\\theta) \\rceil + 1. \n\\end{equation}\nThus, letting \n\\[\n \\cH \n \\defeq \n \\cH_1 \\cup \\cdots \\cup \\cH_{\\lceil (2\/\\theta) \\rceil + 1}\n \\eqdef \\{q_1,\\ldots,q_K\\},\n\\]\nwe have that $\\prod_{1 \\le i < j \\le K}(q_j - q_i)$ is $x$-smooth, \nand hence $f_1(N^{\\eta})$-smooth (we may suppose that $C \\ge 1$).\n\nAs $K$ is fixed we may of course suppose that $K \\le \\log x$ and \n$p_K \\le x$, so that \\eqref{eq:Kbnd} is satisfied and $\\cH$, being \na set of $K$ primes larger than $p_K$, is admissible.\nWe may of course also suppose that $y \\le N$, so that \n$\\cH \\subseteq [0,N]$.\nThus, $\\cH$ satisfies each of the hypotheses of \nTheorem \\ref{thm:BFM4.3}.\n\nLet $\\cZ(N^{\\eta})$ be as in Hypothesis \\ref{hyp:EH}, so that \n$\\cZ(N^{\\eta})$ is repulsive (cf.\\ Definition \\ref{def:sparse}), \nand note that since $x \\le f_1(N^{\\eta}) \\le \\log N^{\\eta}$ (cf.\\ \nDefinition \\ref{def:w} (ii)), \n\\[\n \\sums[p \\in \\cZ(N^{\\eta})][p \\ge p']\n \\frac{1}{p}\n \\ll\n \\frac{1}{p'}\n \\ll\n \\frac{1}{\\log_2 N^{\\eta}}\n \\le \n \\frac{1}{\\log f_1(N^{\\eta})}\n \\le \n \\frac{1}{\\log x}.\n\\]\nThus, \\eqref{eq:Zsparse} is satisfied with \n$\\cZ = \\cZ(N^{\\eta})$.\nTherefore, by Corollary \\ref{cor:thm2} there exists a vector of \nresidue classes \n$\n(c_p \\pod{p})_{p \\le f_1(N^{\\eta}), \\, p \\, \\not\\in \\, \\cZ(N^{\\eta})}\n$ \nsuch that \n\\begin{equation}\n \\label{eq:Hsurvives}\n \\cH\n =\n \\big(\\ZZ \\cap (x,y]\\big)\n \\setminus \\, \n {\\textstyle \\bigcup_{p \\le f_1(N^{\\eta}), \\, p \\, \\not\\in \\, \\cZ(N^{\\eta})}} \n \\, c_p \\pod{p}.\n\\end{equation} \n\nLet $b \\pod{W}$ be the arithmetic progression modulo \n\\[\n W \\defeq \\prods[p \\le f_1(N^{\\eta})][p \\not\\in \\cZ(N^{\\eta})] p\n\\]\nsuch that $b \\equiv -c_p \\pod{p}$ for all primes \n$p \\le f_1(N^{\\eta})$ with $p \\not\\in \\cZ(N^{\\eta})$.\nBy \\eqref{eq:Hsurvives} we have $(\\prod_{i=1}^K(b + q_i),W) = 1$.\n\nEach hypothesis of Theorem \\ref{thm:BFM4.3} now accounted for, we \nconclude that there is some $n \\in (N,2N] \\cap b \\pod{W}$, and a \npair of indices \n$i_1,i_2 \\in \\{1,\\ldots,\\lceil (2\/\\theta) \\rceil + 1\\}$, \n$i_1 < i_2$, such that \n\\[\n \\#(\\bP \\cap n + \\cH_{i_1}) \\ge 1\n \\quad\n \\text{and}\n \\quad \n \\#(\\bP \\cap n + \\cH_{i_2})\n \\ge \n 1.\n\\] \nIf there are more than two such indices, we take $i_2 - i_1$ to \nbe minimal.\n\nThus, if $p$ is the largest prime in $\\bP \\cap n + \\cH_{i_1}$ and \n$p'$ is the smallest prime in $\\bP \\cap n + \\cH_{i_2}$, then $p$ \nand $p'$ are {\\em consecutive}, that is, $p = p_t$ and \n$p' = p_{t+1}$ for some $t$.\nIndeed, by \\eqref{eq:Hsurvives} and the definition of \n$b \\pod{W}$, for any $n \\equiv b \\pod{W}$ with \n$n + x \\ge f_1(N^{\\eta})$ we have\n\\[\n \\bP \\cap (n + x,n + y] = \\bP \\cap n + \\cH. \n\\]\n\nBy \\eqref{eq:rightsize} and \\eqref{eq:betaeta}, and since \n$x\/\\log x = o(f_2(x))$, we have \n\\[\n p_{t+1} - p_t\n =\n (\\beta_{i_2} - \\beta_{i_1})f_2(x) + O\\Big(\\frac{x}{\\log x}\\Big)\n =\n (\\alpha_{i_2} - \\alpha_{i_1} + o(1))CL_{\\eta}f_2(x). \n\\]\nSince there are only $O(1\/\\theta^2)$ distinct pairs of indices \nfrom which $i_1$ and $i_2$ may be chosen, we deduce that there \nexists a single pair $i_1 < i_2$ such that, for arbitrarily large \n$N$, we have \n\\[\n p_{t+1} - p_t \n = (\\alpha_{i_2} - \\alpha_{i_1} + o(1))CL_{\\eta}f_2(x),\n\\]\nfor some pair of consecutive primes \n$p_t,p_{t+1} \\in (N,N + y] \\subseteq (N,3N]$.\n\nFinally, using Definition \\ref{def:w} (i), (iii) and (iv) and \nDefinition \\ref{def:2ndkind} (i) and (iii), we find that \n$\n CL_{\\eta} f_2(x)\\sim f_2(f_1(N)) \\sim f_2(f_1(3N)).\n$ \nWe conclude that \n\\[\n \\frac{p_{t+1} - p_t}{f_2(f_1(p_t))}\n = \n (1 + o(1))(\\alpha_j - \\alpha_i).\n\\]\nWe deduce \\eqref{eq:genthm2} and \\eqref{eq:genthm3} by using the \nargument of \\cite[Corollary 1.2]{BFM}.\n\\end{proof}\n\nAs in \\cite[Theorem 1.3]{BFM}, we may also consider ``chains'' of \nnormalized, consecutive gaps between primes. \nUsing essentially the same argument as above, but using (the \nunconditional) Theorem 4.3 (ii) of \\cite{BFM} in place of \nTheorem \\ref{thm:BFM4.3}, one may verify the \nfollowing result. \n\n\\begin{theorem}\n \\label{thm:chains}\nFix any integer $a$ with $a \\ge 2$.\nFix functions $f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ and \n$f_2 : [x_1,\\infty) \\to [z_2,\\infty)$ of the first and second \nkinds respectively, and let $f \\defeq f_2 \\circ f_1$.\nLet $d_n \\defeq p_{n+1} - p_n$, where $p_n$ denotes the $n$th \nsmallest prime, and let $\\LP[a,f]$ denote the set of limit \npoints in $[0,\\infty]^a$ of the sequence of ``chains'' \n\\[\n \\textstyle \n \\Big( \n \\frac{d_n}{f(p_n)},\\ldots,\\frac{d_{n + a - 1}}{f(p_{n + a -1})}\n \\Big)\n\\]\nfor $p_n \\ge T_1$.\nGiven \n$\\boldsymbol{\\alpha} = (\\alpha_1,\\ldots,\\alpha_K) \\in \\RR^K$, let \n$S_a(\\boldsymbol{\\alpha})$ be the set \n\\[\n \\big\\{ \n \\big( \n \\alpha_{J(2)} - \\alpha_{J(1)},\n \\ldots,\n \\alpha_{J(a+1)} - \\alpha_{J(a)}\n \\big)\n :\n 1 \\le J(1) < \\cdots < J(a + 1) \\le K\n \\big)\n \\big\\}.\n\\]\nFor any $8a^2 + 16a$ nonnegative real numbers \n$\\alpha_1 \\le \\cdots \\le \\alpha_{8a^2 + 16a}$, we have \n\\[\n S_a(\\boldsymbol{\\alpha}) \n \\cap \n \\LP[f]^a\n \\ne \n \\emptyset.\n\\]\n\\end{theorem}\n\nLet us call a function ``reasonable'' if it is of the form \n$f_2 \\circ f_1$, where $f_1$ is a function of the first kind and \n$f_2$ is a function of the second kind. \nTheorem \\ref{thm:chains} shows that for any $a$ there are \ninfinitely many chains of consecutive prime gaps with \n$d_n,\\ldots,d_{n + a - 1} > f(p_n)$ for any reasonable function \n$f$.\nThere are reasonable functions $f$ for which $f(T)\/(R(T)\\log_3 T)$ \ntends to $0$ arbitrarily slowly (recall that \n$R(T) = \\log T\\log_2 T\\log_4 T\/(\\log_3 T)^2$ is the \nErd{\\H o}s--Rankin function).\nWe believe that in a forthcoming paper \\cite{FMT}, Ford, Maynard \nand Tao show that for any $a$ there are infinitely many chains of \nconsecutive prime gaps with \n$d_n,\\ldots,d_{n + a - 1} \\gg R(p_n)\\log_3 p_n$.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nTwo knots $K$ and $K'$ are \\emph{equivalent} if there is a homeomorphism of \n${\\mathbb S}^{3}$ sending $K$ to $K'$. Given a knot $K \\subset {\\mathbb S}^{3}$ and an integer \n$p \\geq 2$ one can construct the (total space of the) $p$-fold cyclic cover \n$M_{p}(K)$ of ${\\mathbb S}^{3}$ branched along $K$: it is a fundamental object in knot \ntheory. There are non-prime knots all of whose cyclic branched covers are \nhomeomorphic. This is no longer true for prime knots: S. Kojima \\cite{Ko} \nproved that for each prime knot $K \\subset {\\mathbb S}^{3}$ there is an integer $n_{K} \n\\geq 2$ such that two prime knots $K$ and $K'$ are equivalent if their $p$-fold \ncyclic branched covers are homeomorphic for some $p > \\max (n_{K}, n_{K'})$.\n\n\\bigskip\n\nThere are many examples of prime knots in ${\\mathbb S}^{3}$ which are not equivalent \nbut share homeomorphic $p$-fold cyclic branched covers due to C. Giller \n\\cite{Gi}, C. Livingston \\cite{Li}, Y. Nakanishi \\cite{Na}, M. Sakuma \n\\cite{Sa1}. Moreover there is no universal bound for $n_{K}$.\n\nThe main goal of this article is to study the relationship between prime knots \nand their cyclic branched covers when the number of sheets is an odd prime \nnumber. \n\n\\smallskip\n\n\\begin{Definition} \nLet $K \\subset {\\mathbb S}^3$ be a prime knot. A knot $K' \\subset {\\mathbb S}^3$ which is not \nequivalent to $K$ and which has the same $p$-fold cyclic branched cover as $K$ \nis called a \\emph{$p$-twin} of $K$.\n\\end{Definition}\n\n\\medskip\n\nThere are examples of prime knots, even hyperbolic knots (e.g. Montesinos \nknots) with an arbitrarily large number of non-equivalent $2$-twins. In \ncontrast, for an odd prime number $p$, the number of $p$-twins is very \nrestricted, according to our main result:\n\n\\smallskip\n\n\\begin{Theorem}\\label{thm:twins} \nLet $K\\subset {\\mathbb S}^3$ be a prime knot. Then:\n\n\\item{(i)} There are at most two odd prime numbers $p$ for which $K$ admits a \n$p$-twin.\n\n\\item{(ii)} For a given odd prime number $p$, $K$ admits at most one $p$-twin.\n\n\\item{(iii)} Suppose that a prime knot $K$ admits the same knot $K'$ as a \n$p$-twin and a $q$-twin for two distinct odd prime numbers $p$ and $q$. Then \n$K$ has two commuting rotational symmetries of order $p$ and $q$ with trivial \nquotients.\n\\end{Theorem}\n\n\\medskip\n\nA \\emph{rotational symmetry of order $p$} of a knot $K \\subset {\\mathbb S}^3$ is an \norientation preserving periodic diffeomorphism $\\psi$ of the pair $({\\mathbb S}^3, K)$ \nwith period $p$ and non-empty fixed-point set disjoint from $K$. We say that \nthe rotational symmetry $\\psi$ has \\emph{trivial quotient} if $K\/\\psi$ is the \ntrivial knot.\n\nFor hyperbolic knots Theorem \\ref{thm:twins} is in fact a consequence of B. \nZimmermann's result in \\cite{Zim1} whose proof uses the orbifold theorem and \nthe Sylow theory for finite groups. \n\nThe result in Theorem \\ref{thm:twins} is sharp: for any pair of coprime \nintegers $p> q >2$ B. Zimmermann has constructed examples of prime hyperbolic \nknots with the same $p$-fold and $q$-fold branched coverings \\cite{Zim2}. \n\nThe second named author \\cite{Pao2} has proved that a hyperbolic knot is \ndetermined by three cyclic branched covers of pairwise distinct orders. The \nfollowing, straightforward corollary of Theorem \\ref{thm:twins}, shows that a \nstronger conclusion holds for arbitrary prime knots when we focus on branched \ncoverings with odd prime orders.\n\n\\smallskip\n\n\\begin{Corollary}\\label{cor: three covers} A prime knot is determined by three \ncyclic branched covers of pairwise distinct odd prime orders. More \nspecifically, for every knot $K$ there is at least one integer $p_K \\in \n\\{3, 5, 7 \\}$ such that $K$ is determined by its $p_K$-cyclic branched cover. \n\\end{Corollary}\n\n\\medskip\n\nAnother straightforwards consequence of Theorem \\ref{thm:twins} is:\n\n\\smallskip\n\n\\begin{Corollary}\\label{cor:composite}\nLet $K=K_1\\sharp...\\sharp K_t$ and $K'=K'_1\\sharp...\\sharp K'_t$ be two \ncomposite knots with the same cyclic branched covers of orders $p_j$, \n$j=1,2,3$, for three fixed, pairwise distinct, odd prime numbers. Then, after\na reordering, the (non oriented) knots $K_i$ and $K'_i$ are equivalent for all\n$i=1,...,t$.\n\\end{Corollary}\n\n\\medskip\n\nPart (ii) of Theorem \\ref{thm:twins} states that for a given odd prime number \n$p$ a closed, orientable $3$-manifold can be the $p$-fold cyclic branched cover \nof at most two non-equivalent knots in ${\\mathbb S}^3$. In \\cite{BPZ} it has been shown \nthat an integer homology sphere which is a $n$-fold cyclic branched cover of \n${\\mathbb S}^3$ for four distinct odd prime numbers $n$ is in fact ${\\mathbb S}^3$. By putting \ntogether these two results we get the following corollary:\n\n\\begin{Corollary}\\label{cor:homologysphere} \nLet $M$ be an irreducible integer homology $3$-sphere. Then: there are at most \nthree distinct knots in ${\\mathbb S}^3$ having $M$ as cyclic branched cover of odd prime \norder.\n\\end{Corollary}\n\nOur main task will be to prove Theorem \\ref{thm:twins} for a satellite knot: \nthat is a knot whose exterior ${\\mathbb S}^3\\setminus{\\mathcal U}(K)$ has a non trivial \nJaco-Shalen-Johannson decomposition \\cite{JS}, \\cite{Jo} (in the sequel we use \n$JSJ$-decomposition for short). Otherwise the knot is called simple: in this \ncase, due to Thurston's hyperbolization theorem \\cite{Th2}, its exterior is \neither hyperbolic, and the proof follows already from the works in \\cite{Pao2} \nand \\cite{Zim1}, or it is a torus knot and a simple combinatorial argument \napplies.\n \nThe proof of Theorem \\ref{thm:twins} for satellite knots relies on the study \nof the \\emph{partial symmetries} of the exterior $E(K)$ of $K$ induced by the \ncovering transformations associated to the twins of $K$ and on the localization \nof their axes of fixed points in the components of the $JSJ$-decomposition of \n$E(K)$. In particular the proof uses the following result about rotational \nsymmetries of prime knots which is of interest in its own right.\n\n\\smallskip\n\n\\begin{Theorem}\\label{thm:three rotations} \nLet $K$ be a knot in ${\\mathbb S}^3$ admitting three rotational symmetries with trivial \nquotients and whose orders are three pairwise distinct numbers $>2$. Then $K$ \nis the trivial knot.\n\\end{Theorem}\n\n\\medskip\n\nSince the trivial knot admits a rotational symmetry with trivial quotient of\norder $p$ for each integer $p \\ge 2$, the above Theorem \\ref{thm:three\nrotations} can be interpreted as a characterisation of the trivial knot, i.e. a\nknot is trivial if and only if it admits three rotational symmetries of\npairwise distinct orders $>2$ and trivial quotients. \n\n\n\\section{Rotational symmetries of knots}\n\nA \\emph{rotational symmetry} of order $p$ of a knot $K \\subset {\\mathbb S}^3$ is an \norientation preserving, periodic diffeomorphism $\\psi$ of the pair $({\\mathbb S}^3, K)$ \nof order $p$ and non-empty fixed-point set disjoint from $K$. We say that the \nrotational symmetry $\\psi$ has \\emph{trivial quotient} if $K\/\\psi$ is the \ntrivial knot.\n\n\\smallskip\n\n\\begin{Remark}\\label{rem:lift} \nLet $K$ be a knot and let $\\psi$ be a rotational symmetry of $K$ of order $p$. \nThe symmetry $\\psi$ lifts to a periodic diffeomorphism $\\tilde\\psi$ of the \n$p$-fold branched cover $M_p(K)$ with order $p$ and non-empty fixed-point set, \nwhich commutes with the covering transformation $h$ of $K$ acting on $M_p(K)$. \nThen the symmetry $\\psi$ has trivial quotient if and only if \n$(M,Fix(\\tilde\\psi))\/<\\tilde\\psi> \\cong ({\\mathbb S}^3,K')$. Moreover in this case $K$ \nand $K'$ have a common quotient link with two trivial components (see \n\\cite{Zim1}).\n\nIn particular a symmetry of a knot $K$ induced by the covering transformation \nassociated to a $p$-twin $K'$ of $K$ is a $p$-rotational symmetry with trivial \nquotient. This follows from the fact that the two commuting deck \ntransformations associated to the two twins induce on $M_p(K)$ a \n${\\mathbb Z}\/p{\\mathbb Z} \\oplus {\\mathbb Z}\/p{\\mathbb Z}$-cover of ${\\mathbb S}^3$ branched over a link with two unknotted \ncomponents.\n\\end{Remark}\n\n\\medskip\n\nThe main result of this section is the following theorem whose assertion (i) is \nTheorem \\ref{thm:three rotations}:\n\n\\smallskip\n\n\\begin{Theorem}\\label{thm:rotations} \nLet $K$ be a knot in ${\\mathbb S}^3$.\n\n\\item{(i)} Assume that $K$ admits three rotational symmetries with trivial \nquotients and whose orders are three pairwise distinct numbers $>2$. Then $K$ \nis the trivial knot.\n\n\\item{(ii)} Assume that $K$ admits two rotational symmetries $\\psi$ and \n$\\varphi$ with trivial quotients and of distinct orders $>2$. Then the \nfixed-point sets $Fix(\\psi)$ and $Fix(\\varphi)$ sit in the $JSJ$-component of \n$E(K)$ which contains $\\partial E(K)$.\n\n\\end{Theorem}\n\n\\medskip\n\nWe prove first a weaker version of Theorem \\ref{thm:three rotations} that we \nshall use in the remaining of this section (see also \\cite[Scholium]{Pao2}).\n\n\\smallskip\n\n\\begin{Proposition}\\label{prop:commuting rotations} \nLet $K$ be a knot in ${\\mathbb S}^3$ admitting three commuting rotational symmetries of\norders $p>q>r\\ge2$. If the symmetries of order $q$ and $r$ have trivial\nquotients, then $K$ is the trivial knot.\n\\end{Proposition}\n\n {\\bf Proof.} \nDenote by $\\varphi$, $\\psi$ and $\\rho$ the three symmetries. If two of them \n-say $\\varphi$, $\\psi$- have the same axis, then by hypothesis the one with \nsmaller order -say $\\psi$- must have trivial quotient, i.e. $K\/\\psi$ is the \ntrivial knot. Since the three symmetries commute, $\\varphi$ induces a \nrotational symmetry of $K\/\\psi$ which is non trivial for the order of \n$\\varphi$ is larger than that of $\\psi$. The axis ${\\mathcal A}$ of this induced symmetry \nis the image of $Fix(\\psi)$ in the quotient by the action of $\\psi$. In \nparticular $K\/\\psi$ and ${\\mathcal A}$ form a Hopf link and $K$ is the trivial knot: this \nfollows from the equivariant Dehn lemma, see \\cite{Hil}. We can thus assume \nthat the axes are pairwise disjoint. Note that even if $r=2$, since the \nsymmetries commute, the symmetry of order $2$ cannot act as a strong inversion \non the axes of the other two symmetries. In this case we would have that the \naxis of $\\rho$, which is a trivial knot, admits two commuting rotational \nsymmetries, $\\varphi$ and $\\psi$, with distinct axes, which is impossible: this \nfollows, for instance, from the fact (see \\cite[Thm 5.2]{EL}) that one can find \na fibration of the complement of the trivial knot which is equivariant with \nrespect to the two symmetries.\n\\qed\n\n\\bigskip\n\nThe proof of Theorem \\ref{thm:rotations} is based on a series of Lemmata. \n\nThe first result concerns the structure of the $JSJ$-decomposition of the \n$p$-fold cyclic branched cover $M$ of a prime knot $K \\subset {\\mathbb S}^3$. Let $h$ be \nthe covering transformation, then the quotient space $M\/$ has a natural \norbifold structure, denoted by ${\\mathcal O}_p(K)$, with underlying space ${\\mathbb S}^3$ and \nsingular locus $K$ with local group a cyclic group of order $p$ (cf.\n\\cite[Chap. 2]{BMP}). According to Bonahon-Siebenmann \\cite{BS} and the \norbifold theorem \\cite{BoP}, \\cite{CHK}, such an orbifold admits a \ncharacteristic collection of toric $2$-suborbifolds, which split ${\\mathcal O}_p(K)$ \ninto geometric suborbifolds. Moreover this characteristic collection of toric \n$2$-suborbifolds lifts to the $JSJ$-collection of tori for $M$. It follows that \nfor $p > 2$ the Bonahon-Siebenmann characteristic collection of toric \n$2$-suborbifolds coincides with the $JSJ$-collection of tori for the exterior \n$E(K) = {\\mathbb S}^3\\setminus{\\mathcal U}(K)$ of $K$.\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:JSJ}\nLet $p >2$ be an integer and let $M$ be the $p$-fold cyclic branched cover of a \nprime knot $K$ in the $3$-sphere. Then:\n\n\\item{(a)} The dual graph associated to the $JSJ$-decomposition of $M$ is a \ntree.\n\n\\item{(b)} The fixed-point set of the group of deck transformations is entirely \ncontained in one geometric piece of the decomposition. \n\\end{Lemma}\n\n {\\bf Proof.} \n\n\\noindent{\\bf (a)} Note, first of all, that $M$ is irreducible since $K$ is \nprime. Hence the Bonahon-Siebenmann decomposition of the orbifold ${\\mathcal O}_p(K)$ \nlifts to the $JSJ$-collection for $M$ since $p>2$. Moreover, the graph dual to \nthe Bonahon-Siebenmann decomposition of the orbifold ${\\mathcal O}_p(K)$, which lifts to \nthe $JSJ$-decomposition for $M$, is a tree. Cutting along a torus of former \ndecomposition and considering the component $C$ which does not contain \n$K$ one gets the complement of a knot in ${\\mathbb S}^3$. The lemma follows now from the \nfact that each connected component of a cyclic branched cover of $C$ has a\nunique boundary component.\n\n\\noindent{\\bf (b)} Note that the group of deck transformations preserves the \n$JSJ$-collection of tori. If $p>2$, the fixed-point set of this group does not \nmeet any torus of the $JSJ$-decomposition, because each $JSJ$-torus is\nseparating and the fixed point set is connected. Since the fixed point set is \nconnected, it is entirely contained in one geometric piece of the \n$JSJ$-decomposition.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\nNote that the conclusion of the first part of the lemma holds also\nfor covers of order $2$. For covers of prime order this\nproperty follows also from the fact that $M_p(K)$ is a ${\\mathbb Z}\/p{\\mathbb Z}$-homology sphere (see\n\\cite{Go}). \n\\end{Remark}\n\n\\medskip\n\n\\begin{Lemma}\\label{lem:prime} \nIf a knot $K \\subset {\\mathbb S}^3$ has a rotational symmetry with trivial quotient, \nthen $K$ is prime.\n\\end{Lemma}\n\n {\\bf Proof.} \nM. Sakuma \\cite[Thm 4]{Sa2} showed that the only possible rotational symmetries \nof a composite knot must either permute cyclically its prime summands, or act \nas a symmetry of one prime summand while permuting the remaining ones. In \nparticular the quotient knot cannot be trivial.\n\\qed\n\n\\bigskip\n\nThe following is a key lemma for the proofs of Theorems \\ref{thm:twins} and \n\\ref{thm:rotations}. \n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:companion} \nLet $K$ be a knot admitting a rotational symmetry $\\psi$ of order $p>2$ and \nconsider the $JSJ$-decomposition of its exterior $E(K) = {\\mathbb S}^3 \\setminus{\\mathcal U}(K)$.\n\n\\item{(i)} $T$ is a torus of the decomposition which does not separate\n$\\partial E(K)$ from $Fix(\\psi)$ if and only if the orbit $\\psi T$ has $p$ \nelements.\n\n\\item{(ii)} Under the assumption that $\\psi$ has trivial quotient, each torus \nwhich separates $\\partial E(K)$ from $Fix(\\psi)$ corresponds to a prime \ncompanion of $K$ on which $\\psi$ acts with trivial quotient.\n\\end{Lemma}\n\n {\\bf Proof.} \nLet $T$ be a torus of the $JSJ$-decomposition of $E(K)$ considered as a torus \ninside $S^3$: $T$ separates the $3$-sphere into a solid torus containing $K$ \nand the exterior of a non trivial knot $K_T$ which is a companion of $K$. Note \nthat, since the order of the symmetry $\\psi$ is $>2$, its axis cannot meet $T$. \nAssume that the axis $Fix(\\psi)$ of the symmetry is contained in the solid \ntorus. \n\nIf the orbit of $T$ under $\\psi$ does not contain $p$ elements, then a \nnon-trivial power of $\\psi$ leaves $T$ invariant, and thus it also leaves the \nsolid torus and the knot exterior invariant. The restriction of this power of \n$\\psi$ to the solid torus acts as a rotation of order $m >1$ around its core \nand leaves invariant each meridian. This non-trivial power of $\\psi$ would then \nbe a rotational symmetry about the non trivial knot $K_T$ which is absurd \nbecause of the proof of the Smith's conjecture (see \\cite{MB}). \n\nFor the reverse implication, it suffices to observe that the geometric pieces \nof the decomposition containing $\\partial E(K)$ and $Fix(\\psi)$ must be \ninvariant by $\\psi$, and so must be the unique geodesic segment joining the \ncorresponding vertices in the tree dual to the decomposition.\n\nFor the second part of the Lemma, note that $K_T\/\\psi$ is a companion of \n$K\/\\psi$, which is trivial by hypothesis. In particular $K_T\/\\psi$ is also \ntrivial and thus, by Lemma \\ref{lem:prime}, must be prime.\n\\qed\n\n\\bigskip\n\nThe following lemma gives a weaker version of assertion (ii) of Theorem \n\\ref{thm:rotations} under a commutativity hypothesis:\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:two rotations}\nLet $K$ be a prime knot admitting two commuting rotational symmetries $\\psi$ \nand $\\varphi$ of orders $p,q>2$. Then:\n\n\\item{(i)} The fixed-point sets of $\\psi$ and $\\varphi$ are contained in the\nsame geometric component of the $JSJ$-decomposition for $E(K)$;\n\n\\item{(ii)} If $\\psi$ has trivial quotient and $p \\not = q$, the fixed-point \nsets of $\\psi$ and $\\varphi$ sit in the component which contains \n$\\partial E(K)$.\n\\end{Lemma}\n\n {\\bf Proof.} \n\n\\noindent{\\bf Part (i)} Let $v_{\\psi}$ (respectively $v_{\\varphi}$) the\nvertex of the graph $\\Gamma_K$ dual to the $JSJ$-decomposition of $E(K)$ \ncorresponding to the geometric component containing $Fix(\\psi)$ (respectively \n$Fix(\\varphi)$). Since the two rotational symmetries commute, $\\psi$ \n(respectively $\\varphi$) must leave $Fix(\\varphi)$ (respectively $Fix(\\psi)$) \ninvariant, and so the geodesic segment of $\\Gamma_K$ joining $v_{\\psi}$ to \n$v_{\\varphi}$ must be fixed by the induced actions of $\\psi$ and $\\varphi$ on \n$\\Gamma_K$. If this segment contains an edge $e$, the corresponding $JSJ$-torus\n$T$ in $E(K)$ cannot separate both $Fix(\\varphi)$ and $Fix(\\psi)$ from \n$\\partial E(K)$. This would contradict part (i) of Lemma \\ref{lem:companion}.\n\n\\medskip\n\n\\noindent{\\bf Part (ii)} Let $M$ be the $p$-fold cyclic branched cover of $K$ \nand let $h$ be the associated covering transformation. According to Remark \n\\ref{rem:lift} the lift $\\tilde \\psi$ of $\\psi$ to $M$ is the deck \ntransformation of a cyclic cover of ${{\\mathbb S}}^3$ branched along a knot $K'$. Note \nthat both $\\tilde \\psi$ and $\\tilde \\varphi$ (the lift of $\\varphi$ to $M$) \ncommute on $M$ with the covering transformation $h$. In particular \n$\\tilde \\varphi$ and $h$ induce commuting rotational symmetries of $K'$ with \norder $q$ and $p$ respectively. According to part $(i)$, $Fix(\\varphi)$ and \n$Fix(h)$ belong to the same piece of the $JSJ$-decomposition of $M$. Since \n$Fix(h)$ maps to $K$ and $p \\not = q$, $Fix(\\varphi)$ sits in the $JSJ$-piece \nof $E(K)$ which contains $\\partial E(K)$ and the conclusion follows since \n$Fix(\\psi)$ belongs to the same $JSJ$-piece as $Fix(\\varphi)$.\n\\qed\n\n\\bigskip\n\n\\begin{Lemma}\\label{lem:torus} \nLet $K$ be a knot admitting a rotational symmetry $\\psi$ with trivial quotient \nand of order $p>2$. Let $M$ be the $p$-fold cyclic branched cover of $K$ and \ndenote by $\\pi:M \\longrightarrow({\\mathbb S}^3,K)$ the associated branched cover. Let \n$T$ be a torus in the $JSJ$-collection of tori of $E(K)$.\n\n\\item{(i)} The torus $T$ is left invariant by $\\psi$ if and only if \n$\\pi^{-1}(T)$ is connected.\n\n\\item{(ii)} If $\\pi^{-1}(T)$ is connected, then the companion $K_T$ of $K$ \ncorresponding to $T$ is prime and the winding number of $T$ with respect to $K$ \nis prime with $p$, so in particular it is not zero.\n\n\\item{(iii)} The torus $T$ is not left invariant by $\\psi$ if and only if \n$\\pi^{-1}(T)$ has $p$ components.\n\\end{Lemma}\n\n {\\bf Proof.} \n\n\\noindent{\\bf Part (i)}. According to Remark \\ref{rem:lift}, the $p$-fold \ncyclic branched cover $M$ of $K$ admits two commuting diffeomorphisms of order \n$p$, $h$ and $h'=\\tilde\\psi $, such that: $(M,Fix(h))\/ \\cong ({\\mathbb S}^3,K)$ on \nwhich $h'$ induces the $p$-rotational symmetry $\\psi$ with trivial quotient, \nand $(M,Fix(h'))\/ \\cong ({\\mathbb S}^3,K')$ on which $h$ induces a $p$-rotational \nsymmetry $\\psi'$ with trivial quotient. The preimage $\\pi^{-1}(T) = \\tilde T$ \nis connected if and only if it corresponds to a torus $\\tilde T$ of the \n$JSJ$-decomposition of $M$ which is left invariant by $h$. If, by \ncontradiction, $\\psi$ does not leave $T$ invariant, then the $h'$-orbit of \n$\\tilde T$ consists of $m>1$ elements. Cutting $M$ along these $m$ separating \ntori, one gets $m+1$ connected components. \n\n\\smallskip\n\n\\begin{Claim}\\label{claim:component} \nBoth $Fix(h)$ and $Fix(h')$ must be contained in the same connected component.\n\\end{Claim}\n\n {\\bf Proof.} \nThe diffeomorphism $h'$ cyclically permutes the $m$ connected components which \ndo not contain $Fix(h')$. Since $h$ and $h'$ commute, $h$ leaves invariant each \nof these $m$ components and it acts in the same way on each of them (that is, \nthe restrictions of $h$ to each component are conjugate). Since the set \n$Fix(h)$ is connected, the claim follows.\n\\qed\n\n\\bigskip\n\nThe $m$ components permuted by $h'$ project to a connected submanifold of the \nexterior $E(K')$ of the knot $K'$ with connected boundary the image $T'$ of $\\tilde T$. This submanifold is invariant by the action of $\\psi'$ \nbut does not contain $Fix(\\psi')$. This contradicts Lemma \n\\ref{lem:companion}(i). To conclude the proof of Lemma \\ref{lem:torus} (i), it \nsuffices to observe that $h$ and $h'$ play symmetric roles.\n\n\\medskip\n\n\\noindent{\\bf Part (ii)} The first part of assertion (ii) is a straightforward \nconsequence of assertion (i) and of Lemma \\ref{lem:companion}. The second part \nfollows from the fact that for $\\pi^{-1}(T)$ to be connected, the winding \nnumber of $T$ and $p$ must be coprime.\n\n\\medskip\n\n\\noindent{\\bf Part (iii)} is a consequence of the proof of part (i) of Lemma \n\\ref{lem:companion} and of the fact that $h$ and $h'$ play symmetric roles.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of Theorem \\ref{thm:rotations}.} The proof is achieved in three \nsteps.\n\n\\medskip\n\n\\noindent {\\bf Step 1.} Theorem \\ref{thm:rotations} is true under the \nassumption that the rotational symmetries commute pairwise.\n\nIn this case, assertion (i) is the statement of Proposition \\ref{prop:commuting \nrotations}. Assertion (ii) follows from Lemma \\ref{lem:two rotations}. \n\n\\bigskip\n\n\\noindent{\\bf Step 2.} Theorem \\ref{thm:rotations} is true under the assumption \nthat every companion of $K$ is prime (i.e. $K$ is \\emph{totally prime}) and has \nnon vanishing winding number (i.e. $K$ is \\emph{pedigreed}).\n\nAssume that we are in the hypotheses of Theorem \\ref{thm:rotations}. Then Lemma \n\\ref{lem:prime} assures that $K$ is a prime knot. If $K$ is also totally prime \nand pedigreed then M. Sakuma \\cite[Thm4 and Lemma 2.3]{Sa2} proved that, up to \nconjugacy, the rotational symmetries belong either to a finite cyclic subgroup \nor to an $S^1$-action in $Diff^{+,+}(S^3,K)$. Thus after conjugacy, step 1 \napplies. For part (ii) note that the distances of the fixed point set of the \nsymmetries to the vertex containing $\\partial E(K)$ in the $JSJ$-graph \n$\\Gamma_ K$ do not change by conjugacy.\n\n\\bigskip\n\n\\noindent {\\bf Step 3.} Reduction of the proof to step 2.\n\nIf $K$ is not totally prime or pedigreed, then it is non-trivial. We shall\nconstruct a non trivial, totally prime and pedigreed knot verifying the \nhypothesis of Theorem \\ref{thm:rotations}. Assertion (i) then follows by \ncontradiction. For Assertion (ii) we need to verify that the construction does \nnot change the distance of the pieces containing the axes of rotations to the \nroot containing $\\partial E(K)$. Roughly speaking we consider the $JSJ$-tori \nclosest to $\\partial E(K)$ and corresponding either to non-prime or to winding \nnumber zero companions. Then we cut $E(K)$ along these tori and keep the \ncomponent $W$ containing $\\partial E(K)$ and suitably Dehn-fill $W$ along these \ntori to get the exterior of a non-trivial knot $\\hat K$ in ${\\mathbb S}^3$, which \nverifies Sakuma's property. \n\nMore precisely, let $\\Gamma_K$ be the tree dual to the $JSJ$-decomposition of \n$E(K)$ and let $\\Gamma_0$ be its maximal (connected) subtree with the following \nproperties: \n\\begin{itemize}\n\n\\item $\\Gamma_0$ contains the vertex $v_\\partial$ corresponding to the \ngeometric piece whose boundary contains $\\partial E(K)$. Note that the\ngeometric piece of the decomposition corresponding to $v_\\partial$ cannot be a\ncomposing space for $K$ is prime;\n\n\\item no vertex of $\\Gamma_0$ corresponds to a composing space (i.e. a space \nhomeomorphic to a product $S^1 \\times B$ where $B$ is an $n$-punctured disc \nwith $n \\geq 2$);\n\n\\item no edge of $\\Gamma_0$ corresponds to a torus whose meridian has linking \nnumber $0$ with $K$.\n\n\\end{itemize}\nDenote by $X(\\Gamma_0)$ the submanifold of $E(K)$ corresponding to $\\Gamma_0$.\n\nThe following claim describes certain properties of $X(\\Gamma_0)$ with respect \nto a rotational symmetry $\\psi$ of $({\\mathbb S}^3,K)$.\n\n\\begin{Claim}\\label{claim:sym} \nLet $\\psi$ be a rotational symmetry of $({\\mathbb S}^3,K)$ with order $p >2$ and trivial\nquotient. Then:\n\n\\item{(i)} The fixed-point set of $\\psi$ is contained in $X(\\Gamma_0)$.\n\n\\item{(ii)} The tree $\\Gamma_0$ is invariant by the automorphism of $\\Gamma_K$ \ninduced by $\\psi$ and the submanifold $X(\\Gamma_0)$ is invariant by $\\psi$.\n\\end{Claim}\n\n {\\bf Proof.} \n\n\\noindent{\\bf Assertion {(i)}.} Let $\\gamma$ be the unique geodesic segment in \n$\\Gamma_K$ which joins the vertex $v_\\partial$ to the vertex corresponding to \nthe geometric piece containing $Fix(\\psi)$ (see Lemma \\ref{lem:JSJ}; note that \nhere we use $p>2$). According to assertion (ii) of Lemma \\ref{lem:companion}, \nno vertex along $\\gamma_i$ can be a composing space. Since the linking number \nof $K$ and $Fix(\\psi)$ must be coprime with $p$, no torus corresponding to an \nedge of $\\gamma$ can have winding number $0$ (see Lemma \\ref{lem:torus}). \n\n\\medskip\n\n\\noindent{\\bf Assertion {(ii)}.} This is just a consequence of the maximality \nof $\\Gamma_0$ and the fact that elements of the group $\\langle\\psi \\rangle$ \ngenerated by $\\psi$ must preserve the $JSJ$-decomposition of $E(K)$ and the \nwinding numbers of the $JSJ$-tori, as well as send composing spaces to \ncomposing spaces.\n\\qed\n\n\\bigskip\n\nLet $\\pi :M_{p}(K) \\longrightarrow({\\mathbb S}^3,K)$ be the $p$-fold cyclic \nbranched cover. Let $T$ be a torus of the $JSJ$-collection of tori for $E(K)$. \nDenote by $E_T$ the manifold obtained as follows: cut $E(K)$ along $T$ and \nchoose the connected component whose boundary consists only of $T$. Note that \n$E_T$ is the exterior of the companion $K_T$ of $K$ corresponding to $T$.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:meridian-longitude}\nLet $T$ be a torus of $\\partial X(\\Gamma_0)\\setminus\\partial E(K)$. The \npreimage $\\pi^{-1}(T)$ consists of $p$ components, each bounding a copy of \n$E_T$ in $M_{p}(K)$. In particular, there is a well-defined\nmeridian-longitude system $(\\mu_T,\\lambda_T)$ on each boundary component of\n$X(\\Gamma_0)$, different from $\\partial E(K)$, which is preserved by taking the\n$p$-fold cyclic branched covers.\n\\end{Claim}\n {\\bf Proof.} \nAccording to Lemma \\ref{lem:torus}, the preimage of $T$ is either connected or \nconsists of $p$ components. If the preimage of $T$ were connected, the tree \n$\\Gamma_0$ would not be maximal according to Lemma \\ref{lem:torus}(ii). The \nremaining part of the Claim is then easy.\n\\qed\n\n\\bigskip\n\nWe wish now to perform Dehn fillings on the boundary of $X(\\Gamma_0)$ in order \nto obtain a totally prime and pedigreed knot admitting pairwise distinct \nrotational symmetries with trivial quotients. On each component $T$ of\n$\\partial X(\\Gamma_0)\\setminus\\partial E(K)$ we fix the curve $\\alpha_n = \n\\lambda_T+n\\mu_T$.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:surgery} \nFor all but finitely many $n \\in {\\mathbb Z}$ the Dehn filling of each component $T$ of \n$\\partial X(\\Gamma_0)\\setminus\\partial E(K)$ along the curve $\\alpha_n$\nproduces the exterior of a non-trivial, prime and pedigreed knot $\\hat K$ in \n${\\mathbb S}^3$.\n\\end{Claim}\n\n {\\bf Proof.} \nNote that by the choice of surgery curves the resulting manifold \n$\\hat X(\\Gamma_0)$ is the exterior of a knot $\\hat K$ in the $3$-sphere, i.e. \n$\\hat X(\\Gamma_0)\\subset {\\mathbb S}^3$, and thus is irreducible. We distinguish two \ncases:\n\n\\medskip\n\n\\noindent{\\bf {(1)}} The $JSJ$-component $X_T$ of $X(\\Gamma_0)$ adjacent to $T$ \nis Seifert fibred. Then, by the choice of $\\Gamma_0$, $X_T$ is a cable space \n(i.e. the exterior of a $(a,b)$-torus knot in the solid torus bounded by $T$ in\n${\\mathbb S}^3$). Moreover the fiber $f$ of the Seifert fibration of $X(\\Gamma_0)$ is \nhomologous to $a\\mu_{T} + b\\lambda_{T}$ on $T$ and the intersection number\n$\\vert \\Delta(f,\\mu_T) \\vert = b > 1$. The intersection number of the filling \ncurve $\\alpha_n$ with the fiber $f$ is then $\\vert \\Delta(f,\\alpha_n) \\vert =\n\\vert na -b \\vert$ and is $> 1$ for all but finitely many $n \\in {\\mathbb Z}$. In this \ncase the resulting manifold $X_T(\\alpha_n)$ is the exterior of a non trivial \ntorus knot which is prime and pedigreed \\cite{CGLS}.\n\n\\medskip\n\n\\noindent {\\bf{( 2)}} The $JSJ$-component $X_T$ of $X(\\Gamma_0)$ adjacent to \n$T$ is hyperbolic. By Thurston's hyperbolic Dehn filling theorem \n\\cite[Chap. 5]{Th1} (see also \\cite[Appendix B]{BoP}) for all but finitely many \n$n \\in {\\mathbb Z}$ the Dehn filling of each component $T \\subset \\partial X_T \\cap \n(\\partial X(\\Gamma_0)\\setminus\\partial E(K))$ along the curve $\\alpha_n$ \nproduces a hyperbolic manifold $X_T(\\alpha_n)$ with finite volume.\n\n\\medskip\n\nTherefore for all but finitely many $n$'s $\\in {\\mathbb Z}$ the Dehn filling of each \ncomponent $T \\subset \\partial X(\\Gamma_0)\\setminus\\partial E(K)$ along the \ncurve $\\alpha_n$ produces a $\\partial$-irreducible $3$-manifold $\\hat \nX(\\Gamma_0) \\subset {\\mathbb S}^3$ such that each Seifert piece of its \n$JSJ$-decomposition is either a Seifert piece of $X(\\Gamma_0)$ or a non-trivial \ntorus knot exterior. Hence it corresponds to the exterior of a non-trivial knot \n$\\hat{K} \\subset {\\mathbb S}^3$ which is totally prime. It is also pedigreed by the \nchoice of $\\Gamma_0$.\n\\qed\n\n\\bigskip\n\nLet $\\psi$ a rotational symmetry of $({\\mathbb S}^3,K)$ with order $p >2$. Then the \nrestriction ${\\psi}_{\\vert_{X(\\Gamma_0)}}$, given by Claim \\ref{claim:sym} \nextends to $\\hat X(\\Gamma_0)$, giving a $p$-rotational symmetry $\\hat\\psi$ of \nthe non-trivial, totally prime and pedigreed knot $({\\mathbb S}^3,\\hat K)$. In order to \nbe able to apply step 2 to the knot $\\hat K$ and the induced rotational \nsymmetries, we still need to check that the rotational symmetry $\\hat \\psi$ has \ntrivial quotient when $\\psi$ has trivial quotient. This is the aim of the \nfollowing:\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:quotient} \nIf the knot $K\/\\psi$ is trivial, then the knot $\\hat K\/\\hat\\psi$ is trivial.\n\\end{Claim}\n\n {\\bf Proof.} \nLet $\\pi:M_{p}(K) \\longrightarrow({\\mathbb S}^3,K)$ be the $p$-fold cyclic \nbranched cover. Let $h$ be the deck transformation of this cover and $h'$ \nthe lift of $\\psi$. According to Remark \\ref{rem:lift}, $h'$ is the deck \ntransformation for the $p$-fold cyclic cover of the $3$-sphere branched along \na knot $K'$. Note that, by Claim \\ref{claim:meridian-longitude},\n$M_{p}(K)\\setminus\\pi^{-1}(X(\\Gamma_0) \\cup {\\mathcal U}(K))$ is a disjoint union of $p$ \ncopies of $E(K) \\setminus X(\\Gamma_0)$. It follows that the $p$-fold cyclic \nbranched cover $M_{p}(\\hat K)$ of $\\hat K$ is the manifold obtained by a \n$(\\lambda_T+n\\mu_T)$-Dehn filling on all the boundary components of \n$\\pi^{-1}(X(\\Gamma_0) \\cup {\\mathcal U}(K))$. The choice of the surgery shows that both \n$h$ and $h'$ extend to diffeomorphisms $\\hat h$ and $\\hat h'$ of order \n$p$ of $M_{p}(\\hat K)$. By construction we have that $M_{p}(\\hat K)\/<\\hat\nh> \\cong {\\mathbb S}^3$. In the same way $M_{p}(\\hat K)\/<\\hat h'>$ is obtained\nfrom $M_{p}(K)\/ \\cong{\\mathbb S}^3$ by cutting off a copy of $E(K) \\setminus\nX(\\Gamma_0)$ and Dehn filling along $\\partial X(\\Gamma_0)$. The choice of\nthe surgery curve assures that the resulting manifold is again ${\\mathbb S}^3$ and the \nconclusion follows from Remark \\ref{rem:lift}.\n\\qed\n\n\\bigskip\n\nFrom the non-trivial prime knot $K$, we have thus constructed a non-trivial, \ntotally prime and pedigreed knot $\\hat K$ which has the property that every \nrotational symmetry $\\psi$ of $K$ with trivial quotient and order $>2$ \ninduces a rotational symmetry $\\hat \\psi$ of $\\hat K$ with trivial quotient and \nthe same order. Moreover by the choice of the Dehn filling curve in the \nconstruction of $\\hat K$, the vertex containing $Fix(\\hat \\psi)$ remains at the \nsame distance from the vertex containing $\\partial E(\\hat K)$ in the \n$JSJ$-tree $\\Gamma_ {\\hat K}$ as the vertex containing $Fix(\\psi)$ from the \nvertex containing $\\partial E(K)$ in the $JSJ$-tree $\\Gamma_ K$. Then the \nconclusion is a consequence of step 2.\\qed \n\n\n\\section{Twins of a prime knot}\n\nIn this section we prove Theorem \\ref{thm:twins}. If $K$ is trivial, the\ntheorem is a consequence of the proof of Smith's conjecture (see \n\\cite{MB}). We shall thus assume in the remaining of this section that $K$ is \nnon trivial and $p$ is an odd prime number.\n\nLet $M$ be the common $p$-fold cyclic branched cover of two prime knots $K$ and \n$K'$ in ${\\mathbb S}^3$. Let $h$ and $h'$ be the deck transformations for the coverings \nof $K$ and $K'$ respectively. By the orbifold theorem \\cite{BoP}, see also \n\\cite{CHK} one can assume that $h$ and $h'$ act \\emph{geometrically} on the \ngeometric pieces of the $JSJ$-decomposition of $M$, i.e. by isometries on the \nhyperbolic pieces and respecting the fibration on the Seifert fibred ones.\n\nThe following lemma describes the Seifert fibred pieces of the \n$JSJ$-decomposition of the $p$-fold branched cyclic cover $M$ (see also\n\\cite{Ja} and \\cite[Lemma 2]{Ko}).\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:seifert}\nLet $p$ be an odd prime integer and let $M$ be the $p$-fold cyclic branched \ncover of ${\\mathbb S}^3$ branched along a prime, satellite knot $K$. If $V$ is a Seifert \npiece in the $JSJ$-decomposition for $M$. Then the base $B$ of $V$ can be:\n\n\\begin{enumerate}\n\n\\item A disc with $2$, $p$ or $p+1$ singular fibres;\n\n\\item A disc with $1$ hole, i.e. an annulus, with $1$ or $p$ singular fibres;\n\n\\item A disc with $p-1$ holes and $1$ singular fibre;\n\n\\item A disc with $p$ holes and $1$ singular fibre;\n\n\\item A disc with $n$ holes, $n\\ge2$.\n\n\\end{enumerate}\n\\end{Lemma}\n\n {\\bf Proof.} \nIt suffices to observe that $V$ projects to a Seifert fibred piece $V'$ of the \nBonahon-Siebenmann decomposition for the orbifold ${\\mathcal O}_p(K)$. There are four \npossible cases:\n\n\\noindent{\\bf (a)} $V'$ contains $K$: $V'$ is topologically a non trivially \nfibred solid torus and $K$ is a regular fibre of the fibration, i.e. a torus \nknot $K(a,b)$, since it cannot be the core of the fibred solid torus. The knot\n$K$ lifts to a singular fibre of order $p$ if $p$ does not divide $ab$ and to \na regular fibre otherwise. The core of the solid torus is a singular fibre of \norder -say- $a$. It lifts to a regular fibre if $a=p$, a singular fibre of \norder $a\/(a,p)$ if $p$ does not divide $b$, or to $p$ singular fibres of order \n$a$ if $p$ divides $b$. Thus $V$ has $p$ boundary components if $p$ divides $a$ \nand $1$ otherwise. An Euler characteristic calculation shows that $B$ is either \na disc with $2$ or $p$ singular fibres, or a disc with $p-1$ holes and with at \nmost $1$ singular fibre.\n\n\\noindent{\\bf (b)} $V'$ is the complement of a torus knot $K(a,b)$ in ${\\mathbb S}^3$. \nIn this case, $V$ is either a copy of $V'$, and $B$ is a disc with $2$ singular\nfibres or $V$ is a true $p$-fold cover of $V'$. In this case $V$ has exactly\none boundary component. Reasoning as in case (a), we see that the two singular\nfibres of $V'$ must lift to either $2$ singular fibres, or $1$ regular fibre\nand $p$ singular fibres or $1$ singular fibre and $p$ singular fibres. In\nparticular $B$ is a disc with $2$, $p$ or $p+1$ singular fibres.\n\n\\noindent{\\bf (c)} $V'$ is the complement of a torus knot $K(a,b)$ in a solid \ntorus, i.e. a cable space, and its base is an annulus with $1$ singular fibre. \nReasoning as in (b) we find that $B$ can be a disc with $1$ hole and $1$ or $p$ \nsingular fibres or a disc with $p$ holes and at most $1$ singular fibre.\n\n\\noindent{\\bf (d)} $V'$ is a composing space with at least $3$ boundary\ncomponents and thus so is $V$. More precisely, note that either $V'$ lifts to\n$p$ disjoint copies of itself, or $V$ and $V'$ are homeomorphic and $V'$ is \nobtained by quotienting $V$ via the $p$-translation along the ${\\mathbb S}^1$ fibre. In \nthis case $B$ is a disc with at least $2$ holes.\n\nThis analysis ends the proof of Lemma \\ref{lem:seifert}.\n\\qed\n\n\\bigskip\n\n\\begin{Proposition}\\label{prop:subtree}\nLet $M$ be the common $p$-fold cyclic branched cover of two prime knots $K$ and \n$K'$ in ${\\mathbb S}^3$, $p$ an odd prime number, and let $h$ be the deck transformation \nfor the covering of $K$. Let $\\Gamma$ be the tree dual to the\n$JSJ$-decomposition of $M$. The deck transformation $h'$ for the covering of\n$K'$ can be chosen (up to conjugacy) in such a way that:\n\n\\item{(i)} There exists a subtree $\\Gamma_f$ of $\\Gamma$ on which the actions\ninduced by $h$ and $h'$ are trivial;\n\n\\item{(ii)} The vertices of $\\Gamma$ corresponding to the geometric pieces of\nthe decomposition which contain $Fix(h)$ and $Fix(h')$ belong to $\\Gamma_f$;\n\n\\item{(iii)} Let $M_f$ the submanifold of $M$ corresponding to $\\Gamma_f$. The\nrestrictions of $h$ and $h'$ to $M_f$ commute.\n\\end{Proposition}\n\n {\\bf Proof.} \nThe proof relies on the study of the actions of the two covering \ntransformations $h$ and $h'$ on the $JSJ$-decomposition of the common $p$-fold \ncyclic branched covering $M$. Since $\\Gamma$ is finite, the group generated by \nthe tree automorphisms induced by $h$ and $h'$ is finite as well. Standard \ntheory of group actions on trees assures that a finite group acting on a tree \nwithout inversion must have a global fixed point and that its fixed-point set \nis connected. Thus part (i) of the proposition follows, using the fact that $h$ \nand $h'$ have odd orders.\n\n\\medskip\n\nChoose now $h'$, up to conjugacy in $Diff^+(M)$, in such a way that\n$\\Gamma_f$ is maximal. We want to show that, in this case, $M_f$ contains \n$Fix(h)$ and $Fix(h')$. Assume by contradiction that the vertex $v_h$ of \n$\\Gamma$ corresponding to the geometric piece containing $Fix(h)$, whose\nexistence is ensured by Lemma \\ref{lem:JSJ}, does not belong to $\\Gamma_f$. Let \n$\\gamma_h$ the unique geodesic path in $\\Gamma$ connecting $v_h$ to $\\Gamma_f$. \nLet $e_h$ the edge in $\\gamma_h$ adjacent to $\\Gamma_f$ and denote by $T$ the \ncorresponding torus of the $JSJ$-collection of tori for $M$. Let $U$ be the \nconnected component of $M\\setminus T$ which contains $Fix(h)$. Consider the \n$\\langle h,h'\\rangle$-orbit of $U$. This orbit is the disjoint union of $h$ \n(and $h'$) orbits of $U$. Remark that the $h$-orbit of $U$ is $\\{U\\}$.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:orbit}\nThe orbit $\\langle h,h'\\rangle U$ must contain an $h$-orbit, different from \n$\\{U\\}$ and containing a unique element.\n\\end{Claim}\n\n {\\bf Proof.} \nOtherwise all the $h$-orbits in $\\langle h,h'\\rangle U$ different from $\\{U\\}$ \nwould have $p$ elements, since $p$ is prime. In particular, the cardinality of \n$\\langle h,h'\\rangle U$ would be of the form $kp+1$. This implies that at least \none of the $h'$-orbits in $\\langle h,h'\\rangle U$ must contain one single \nelement $U'$. Up to conjugacy with an element of $\\langle h,h'\\rangle$ (whose \ninduced action on $\\Gamma_f$ is trivial), we can assume that $U=U'$, \ncontradicting the hypothesis that $h'$ was chosen up to conjugacy in such a \nway that $\\Gamma_f$ is maximal.\n\\qed\n\n\\bigskip\n\nLet $U'\\neq U$ the element of $\\langle h,h'\\rangle U$ such that $h(U')=U'$. \nNote that $U$ and $U'$ are homeomorphic since they belong to the same \n$\\langle h,h'\\rangle$-orbit.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:knot complement}\n$U$ is homeomorphic to the exterior $E({{\\mathcal K}})$ of a knot ${\\mathcal K} \\subset {\\mathbb S}^3$\nadmitting a free symmetry of order $p$.\n\\end{Claim}\n\n {\\bf Proof.} \nThe first part of the Claim follows from the fact that, by maximality of \n$\\Gamma_f$, $h'$ cannot leave $U$ invariant, so must freely permute $p$ copies\nof $U$ belonging to $\\langle h,h'\\rangle U$. Thus $U$ must appear as a union of\ngeometric pieces of the $JSJ$-splitting of $E(K')$. The second part follows \nfrom the fact that $h$ must act freely on $U'$ which is homeomorphic to $U$.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:free quotient} \nNote that the quotient of $U$ by the action of its free symmetry of order $p$ \nis also a knot exterior because $h$ acts freely on $U'$ and $U'$ must project \nto a union of geometric pieces of the $JSJ$-splitting of $E(K)$.\n\\end{Remark}\n\n\\medskip\n\n\\begin{Claim}\\label{claim:non-free quotient}\n$U$ admits a rotational symmetry of order $p$ whose quotient $U\/\\langle \nh\\rangle$ is topologically a solid torus.\n\\end{Claim}\n\n {\\bf Proof.} \nThe quotient $U\/\\langle h\\rangle$ is obtained by cutting ${\\mathbb S}^3$ along an \nessential torus in $E(K)$. Since $K \\subset U\/\\langle h\\rangle$, it must be a \nsolid torus. \n\\qed\n\n\\bigskip\n\nIt follows from Claim \\ref{claim:non-free quotient} and Lemma \\ref{lem:prime} \nthat the knot ${\\mathcal K}$ is prime. Moreover, according to Claims \\ref{claim:knot \ncomplement} and \\ref{claim:non-free quotient}, ${\\mathcal K}$ admits a rotational \nsymmetry and a free symmetry, both of order $p$. This is however impossible \nbecause M. Sakuma \\cite[Thm. 3]{Sa2} showed that a prime knot can only have one \nsymmetry of odd order up to conjugacy. This contradiction proves part (ii) of \nProposition \\ref{prop:subtree}.\n\n\\medskip\n\nTo prove part (iii) we shall consider two cases, according to the structure\nof $\\Gamma_f$.\n\n\\medskip\n\n\\noindent {\\bf Case (a)}: {\\it $\\Gamma_f$ contains an edge.}\nChoose an edge in $\\Gamma_f$ and let $T$ be the corresponding torus in the\n$JSJ$-collection of tori for $M$. Let $V$ be a geometric piece of the \n$JSJ$-decomposition of $M$ adjacent to $T$. Then Lemma \\ref{lem:commutation} \nbelow together with a simple induction argument show that $h'$ can be chosen \n(up to conjugacy) in such a way that its restriction to $M_f$ commutes with the \nrestriction of $h$.\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:commutation}\nIf the covering transformations $h$ and $h'$ preserve a $JSJ$-torus $T$ of $M$ \nthen, up to conjugacy in $Diff^{+}(M)$, $h$ and $h'$ commute on the union of \nthe geometric components of the $JSJ$-decomposition adjacent to $T$.\n\\end{Lemma}\n\n {\\bf Proof.} \nFirst we show that $h$ and $h'$ commute on each geometric component adjacent to \n$T$. Since $h$ and $h'$ preserve the orientation of $M$, we deduce that \n$h(V)=V$ and $h'(V)=V$, and that $h$ and $h'$ act geometrically on the \ngeometric piece $V$. A product structure on $T$ can always be induced by the \ngeometric structure on $V$: either by considering the induced Seifert fibration \non $T$ if $V$ is Seifert fibred, or by identifying $T$ with a section of a cusp \nin the complete hyperbolic manifold $V$. Since $h$ and $h'$ are isometries of \norder $p$, for such a product structure on $T$ they act as (rational) \ntranslations, i.e. their action on $T={{\\mathbb S}}^1\\times{{\\mathbb S}}^1$ is of the form \n$(\\zeta_1,\\zeta_2) \\mapsto (e^{2i\\pi r_1\/p}\\zeta_1,e^{2i\\pi r_2\/p}\\zeta_2)$, \nwhere $p$ and at least one between $r_1$ and $r_2$ are coprime. Thus $h$ and \n$h'$ commute on $T$.\n\n\\medskip\n\nIf $V$ is hyperbolic, we have just seen that $h$ and $h'$ are two isometries of \n$V$ which commute on the cusp corresponding to $T$. Thus they must commute on \n$V$.\n\n\\medskip\n\nIf $V$ is Seifert fibred, then the Seifert fibration is unique up to isotopy, \nand $h$ and $h'$ preserve this fibration. \n\n\\smallskip\n\n\\begin{Remark}\nNote that the quotient of $V$ by a fiber-preserving diffeomorphism of\nfinite order $h$ only depends on the combinatorial behaviour of $h$, i.e. its\ntranslation action along the fibre and the induced permutation on cone points \nand boundary components of the base. In particular, the conjugacy class of $h$ \nonly depends on these combinatorial data. Note moreover that two geometric\nsymmetries having the same combinatorial data are conjugate via a\ndiffeomorphism isotopic to the identity.\n\\end{Remark}\n\n\\medskip\n\nSince the translation along the fibres commutes with every fiber-preserving \ndiffeomorphism of $V$, it suffices to see whether $h$ and $h'$ commute, up to a \nconjugation of $h'$, on the base $B$ of $V$. It is enough then to consider the \npossible actions of order $p$ on the possible bases. According to Lemma \n\\ref{lem:seifert} the possible actions of $h$ and $h'$ are described below:\n\n\\begin{enumerate}\n\n\\item If $B$ is a disc with $2$ singular fibres, or an annulus with $1$ \nsingular fibre, or a disc with $n$ holes, $n\\neq p$, or a disc with $p-1$ holes\nand $1$ singular fibre, then the action on $B$ is necessarily trivial and there \nis nothing to prove. Note that, according to the proof of Lemma \n\\ref{lem:seifert}, if $B$ is a disc with $p-1$ holes with one singular fibre, \nno boundary torus is left invariant, so this possibility in fact does not \noccur.\n\n\\item If $B$ is a disc with $p$ holes and $1$ singular fibre or a disc with \n$p+1$ singular fibres, then the only possible action is a rotation about a \nsingular fibre cyclically permuting the holes or the remaining singular fibres. \n\n\\item If $B$ is a disc with $p$ singular fibres then the action must be a\nrotation about a regular fibre which cyclically exchanges the singular fibres.\n\n\\item If $B$ is an annulus with $p$ singular fibres the action must be a free\nrotation cyclically exchanging the singular fibres. Note that in the three\nlatter cases the action can never be trivial on the base.\n\n\\item If $B$ is a disc with $n$ holes then two situations can arise: either the\naction is trivial on the base (case (d) in the proof of Lemma \n\\ref{lem:seifert}; note that in case (a), when $n=p-1$, all boundary components \nmust be cyclically permuted), or $n=p$ and the action is a rotation about a \nregular fibre which cyclically permutes the $p$ holes (see part (c) of Lemma \n\\ref{lem:seifert}).\n\n\\end{enumerate}\n\nWe shall now show that, if both $h$ and $h'$ induce non trivial actions on the\nbase of $V$, then, up to conjugacy, $h$ and $h'$ can be chosen so that their \nactions on $B$ coincide. Note that for $h$ and $h'$ to commute it suffices that \nthe action of $h'$ on $B$ coincides with the action of some power of $h$, \nhowever this stronger version will be needed in the proof of Corollary\n\\ref{cor:extension}.\n\nFirst of all remark that, if $B$ is a disc with $p+1$ singular fibres (case 2) \nand $h$ and $h'$ leave invariant distinct singular fibres, then all the\nsingular fibres must have the same order (in fact, must have the same\ninvariants). This means that, after conjugating $h'$ by a homeomorphism of $V$\nwhich is either an isotopy exchanging two regular fibres or a Dehn twist along\nan incompressible torus exchanging two singular fibres, one can assume that, in\ncases 2 and 3, $h$ and $h'$ leave set-wise invariant the same fibre. Note that\nthis homeomorphism is isotopic to the identity on $\\partial V$ and thus extends\nto $M$. In fact, using Lemma \\ref{lem:seifert} one can show that the fibres\ncannot all have the same order.\n\nSince the actions of $h$ and $h'$ consist in permuting exactly $p$ holes or \nsingular fibres, it suffices to conjugate $h'$ via a homeomorphism of $V$ \n(which is a composition of Dehn twists along incompressible tori) in such a way \nas to exchange the order of the holes or singular fibres so that $h'$ and $h$\ncyclically permute them in the same order. Note that in the case of singular \nfibres this product of Dehn twists is isotopic to the identity on $\\partial V$ \nand thus extends to $M$. In the case of holes, the product of Dehn twists \nextends to $M$ since it induces the identity on the fundamental groups of the \ntori of $\\partial V$ and the connected components of $M\\setminus V$ adjacent to \nboundary tori different from $T$ are necessarily homeomorphic. \n\nOnce the two diffeomorphisms $h$ and $h'$ commute on the two geometric pieces \nadjacent to $T$, the commutation can be extended on a product neighborhood of \n$T$, since the two finite abelian groups generated by the restrictions of $h$ \nand $h'$ on each side of $T$ have the same action on $T$. Indeed, the slope of \nthe translation induced by $h'$ on $T$ has been left unchanged by the \nconjugation.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:same action}\nNote that in case 1 of the proof of the above Lemma, the actions of $h$ and\n$h'$ must coincide after taking a power, i.e. $h$ and $h'$ generate the same\ncyclic group. This is not necessarily true in the remaining cases, even if $h$ \nand $h'$ induce the same action on $B$. Indeed, they can induce different\ntranslations along the fibres. Nevertheless, in both cases, to assure that the\nactions of $h$ and $h'$ coincide on $V$, it suffices to check that they \ncoincide on $T$.\n\\end{Remark}\n\n\\medskip\n\n\\noindent {\\bf Case (b)}: {\\it $\\Gamma_f$ is a single vertex.}\nLet $V=M_f$ be the geometric piece corresponding to the unique vertex of \n$\\Gamma_f$. If $V=M$, then the result is already known. We can thus assume that\n$V\\neq M$. According to part (ii) of Proposition \\ref{prop:subtree}, we \ncan assume that the fixed-point sets of $h$ and $h'$ are contained in $V$.\nIf $V$ is Seifert fibred then, case (a) of the proof of Lemma \\ref{lem:seifert} \nshows that the base $B$ of $V$ is either a disc with $2$ or $p+1$ singular \nfibres, or a disc with $p-1$ holes and with $1$ or $2$ singular fibres. In the \nfirst case the boundary torus of $V$ is preserved by $h$ and $h'$ and the \nassertion follows from Lemma \\ref{lem:commutation}. In the second case the \naction on the base is necessarily a rotation fixing two points (either the \nunique singular fibre and a regular one, or the two singular fibres) and \ncyclically permuting the $p$ boundary components. Then conjugating $h'$ by a \nproduct of Dehn twists along incompressible tori, which extends to $M$ as in \nthe proof of Lemma \\ref{lem:commutation}, leads to the desired conclusion.\n\n\\medskip\n\nThe case where $V$ is hyperbolic is due to B. Zimmermann \\cite{Zim1}. We give \nthe argument for completeness. Since $V$ is hyperbolic, we consider the group \n${\\mathcal I}_V$ of isometries of $V$ induced by diffeomorphisms of $M$ which leave $V$ \ninvariant. Let ${\\mathcal S}$ be the $p$-Sylow subgroup of ${\\mathcal I}_V$. Up to conjugacy, we \ncan assume that both $h=h_{\\vert_V}$ and $h'=h'_{\\vert_V}$ belong to ${\\mathcal S}$. If \nthe groups $\\langle h\\rangle$ and $\\langle h'\\rangle$ generated by $h$ and $h'$ \nare conjugate, we can assume that $h=h'$ and we are done. So we assume that \n$\\langle h\\rangle$ and $\\langle h'\\rangle$ are not conjugate. Then it suffices \nto prove that $h'$ normalises $\\langle h\\rangle$ because each element \nnormalising $\\langle h\\rangle$ must leave invariant $Fix(h)$ and the subgroup \nof ${\\mathcal I}_V$ which leaves invariant a simple closed geodesic, like $Fix(h)$, must \nbe a finite subgroup of ${\\mathbb Z}\/2{\\mathbb Z}\\ltimes({\\mathbb Q}\/{\\mathbb Z}\\oplus{\\mathbb Q}\/{\\mathbb Z})$. In particular, \nelements of odd order must commute. Assuming that $\\langle h\\rangle$ and \n$\\langle h'\\rangle$ are not conjugate, we have that $\\langle h\\rangle \n\\subsetneq {\\mathcal S}$ and, by \\cite[Ch 2, 1.5]{Su}, either $\\langle h\\rangle$ is \nnormal in ${\\mathcal S}$ and we have reached the desired conclusion, or there exist an \nelement $\\hat{h}=ghg^{-1}$, conjugate to $h$ in ${\\mathcal S}$, which normalises \n$\\langle h\\rangle$ and such that \n$\\langle h\\rangle \\cap\\langle{\\hat{h}}\\rangle=\\{1\\}$.\n\nWe want to show that $h'$ normalises $\\langle h\\rangle$. Assume, by \ncontradiction that $h'$ is not contained in $\\langle h,\\hat{h}\\rangle =\n{\\mathbb Z}\/p{\\mathbb Z}\\oplus{\\mathbb Z}\/p{\\mathbb Z}$. Then this group is smaller than ${\\mathcal S}$ and again we are \nable to find a new cyclic group $H$ of order $p$ whose intersection with \n$\\langle h,\\hat{h}\\rangle$ is reduced to the identity and which normalises \n$\\langle h,\\hat{h}\\rangle$. Since the order of $H$ is an odd prime number and \nsince $\\langle h\\rangle$ and $\\langle\\hat{h}\\rangle$ are the only subgroups of \n$\\langle h,\\hat{h}\\rangle$ which fix point-wise a geodesic by \\cite[Proposition \n4]{MZ}, $H$ would commute with $\\langle h,\\hat{h}\\rangle$ which is a \ncontradiction to the structure of a group leaving a geodesic invariant. This \nfinal contradiction shows that, up to conjugacy, the subgroups $\\langle \nh\\rangle$ and $\\langle h'\\rangle$ either commute or coincide on $V$. This \nfinishes the proof of Proposition \\ref{prop:subtree}.\n\\qed\n\n\\bigskip\n\nThe following proposition shows that a prime knot $K$ having a $p$-twin either \nadmits a rotational symmetry of order $p$, or a well-specified submanifold \n$E_p(K)$ built up of geometric pieces of the $JSJ$-decomposition of $E(K)$ \nadmits a symmetry of order $p$ with non-empty fixed-point set.\n\n\\smallskip\n\n\\begin{Definition} \nLet $K$ be a prime knot in ${\\mathbb S}^3$. For each odd prime number $p$ we define \n$E_p(K)$ to be the connected submanifold of $E(K)$ containing $\\partial E(K)$ \nand such that $\\partial E_p(K) \\setminus \\partial E(K)$ is the union of the \n$JSJ$-tori of $E(K)$ with winding number $p$ which are closest to $\\partial \nE(K)$.\n\\end{Definition}\n\n\\medskip\n\n\\begin{Proposition}\\label{prop:orbifold}\nLet $K$ be a prime knot and let $p$ be an odd prime number. Then for any \n$p$-twin $K'$, the deck transformation of the branched cover\n$M\\longrightarrow({{\\mathbb S}}^3,K')$ induces on $E_p(K)$ a symmetry of order $p$, with\nnon-empty fixed-point set and which extends to ${\\mathcal U}(K)$.\n\\end{Proposition}\n\n {\\bf Proof.} \nFirst we show that the deck transformation of the branched cover\n$M\\longrightarrow({{\\mathbb S}}^3,K')$ associated to a $p$-twin of $K$ induces on \n$E_p(K)$ a symmetry of order $p$.\n \nLet $K'$ be a $p$-twin of $K$. Let $h$ and $h'$ be the deck transformations on \n$M$ for the $p$-fold cyclic branched covers of $K$ and $K'$. We shall start by \nunderstanding the behaviour of $h$ and $h'$ on $M$. We have seen in Proposition\n\\ref{prop:subtree} that $h$ and $h'$ can be chosen to commute on the \nsubmanifold $M_f$ of $M$ corresponding to the maximal subtree of $\\Gamma$ on \nwhich both $h$ and $h'$ induce a trivial action. Let $\\Gamma_c$ the maximal \n$\\langle h,h'\\rangle$-invariant subtree of $\\Gamma$ containing $\\Gamma_f$, such \nthat, up to conjugacy, $h$ and $h'$ can be chosen to commute on the \ncorresponding submanifold $M_c$ of $M$.\n\nIf $M_c = M$ then after conjugation $h'$ commutes with $h$ on $M$, but is\ndistinct from $h$ because the knots $K$ and $K'$ are not equivalent. Hence it \ninduces a rotational symmetry of order $p$ of the pair $(S^3,K)$ and we are \ndone.\n\nSo we consider now the case where $\\partial M_c$ is not empty. It is sufficient \nto show that $E_p(K) \\subset M_c\/$: then the symmetry of order $p$ induced \nby $h'$ on $M_c\/$ must preserve $E_p(K)$ since each $JSJ$-torus of $E(K)$ \ncan only be mapped to another torus of the family with the same winding number \nand the same distance from $\\partial E(K)$. First we show:\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:permutation}\nLet $T$ be a connected component of $\\partial M_c$. The $h$-orbit of $T$ \nconsists of $p$ elements which are permuted in the same way by $h$ and $h'$.\n\\end{Claim}\n\n {\\bf Proof.} \nLet $T$ be a torus in $\\partial M_c$ and let $U$ be the connected component of \n$M\\setminus M_c$ adjacent to $T$. Because of Lemma \\ref{lem:commutation}, $T$ \ncannot be preserved by both $h$ and $h'$ for else $M_c$ would not be maximal.\nWithout loss of generality, we can assume that either:\n\\medskip\n\n\\noindent{\\bf (a)} $h(T) \\neq T$ and $h'(T) \\neq T$;\n\n\\medskip\n\n\\noindent or\n\n\\medskip\n\n\\noindent{\\bf (b)} $h(T)=T$ but $h'(T)\\neq T$; in this case since $h$ and\n$h'$ commute on $M_c$, we have that $h(h{'}^{\\alpha}(U)) = h{'}^{\\alpha}(U)$. \nThen part (ii) of Proposition \\ref{prop:subtree} implies that $h$ acts freely \non $h{'}^{\\alpha}(U)$ for each $\\alpha=0,...,p-1$.\n\n\\medskip\n\nIn case (a), the orbit of $T$ by the action of the group $\\langle h,h'\\rangle$ \nconsists of $p$ or $p^2$ elements which bound on one side $M_c$ and on the \nother side a manifold homeomorphic to $U$. If the orbit consist of $p$ \nelements, since $h$ and $h'$ commute on $M_c$, up to choosing a different\ngenerator in $\\langle h'\\rangle$ we can assume that $h$ and $h'$ permute the\nelements of the orbit in the same way. Indeed, we have \n$h'h(T)=hh'(T)=h(h^{\\alpha}(T))=h^{\\alpha}(h(T))$.\n\nIf the orbit consist of $p^2$ elements, $U$ is a is a knot exterior and there \nis a well-defined longitude-meridian system on each component of the $\\langle \nh,h'\\rangle$-orbit of $T$. In particular, there is a unique way to glue a copy \nof $U$ along the projection of $T$ in $M_c\/\\langle h,h'\\rangle$. This implies \nthat $h$ and $h'$ commute up to conjugacy on $M_c\\cup\\langle h,h'\\rangle U$, \ncontradicting the maximality of $M_c$. Note also that in this latter case the \nstabiliser of each component of $\\langle h,h'\\rangle U$ is reduced to the \nidentity which clearly extends to $\\langle h,h'\\rangle U$.\n\n\\medskip\n\nAssume we are in case (b). Consider the restriction of $h$ and\n$h_\\alpha=h{'}^{-\\alpha}hh{'}^\\alpha$ to $U$. Since $h$ and $h'$ commute on\n$M_c$, $h$ and $h_\\alpha$ coincide on $T$. Let $V$ be the geometric piece of \nthe $JSJ$-decomposition for $M$ adjacent to $T$ and contained in $U$. Using \nLemma \\ref{lem:commutation}, we see that $h$ and $h_\\alpha$ commute on $V$ and \nthus coincide on it, because they coincide on $T$. Thus $h$ and $h'$ commute on\n$M_c\\cup_{\\alpha=0}^{p-1}h{'}^\\alpha(V)$, and again we reach a contradiction to\nthe maximality of $M_c$.\n\\qed\n\n\\bigskip\n\nWe can thus assume to be in case (a) and that the $\\langle h,h'\\rangle$-orbit\nof $T$ has $p$ elements.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:winding number}\nEach torus in the boundary of $M_c\/$ has winding number $p$ with respect\nto $K$.\n\\end{Claim}\n\n {\\bf Proof.} \nSince a boundary component $T$ of $M_c\/$ lifts to $p$ boundary components of \n$M_c$, the winding number of $T$ with respect to $K$ must be a multiple of $p$. \nWe shall now reason by induction on the number $n$ of boundary components of \n$M_c\/$. If $n=0$ there is nothing to prove.\n\nIf $n = 1$ the quotient spaces $M_c\/$ and $M_c\/$ are solid tori, i.e. \nthe exterior of a trivial knot which can be identified with a meridian of each \nsolid torus. Note that the winding number of $T$ is precisely the linking \nnumber of $K$ with such a meridian. Note, moreover, that the spaces $M_c\/$ \nand $M_c\/$ have a common quotient ${\\mathcal O}$ which is obtained by quotienting \n$M_c\/$ (respectively $M_c\/$) via the the symmetry $\\psi$ (respectively \n$\\psi'$) of order $p$ and with non-empty fixed-point set, induced by $h'$ \n(respectively $h$). Since $\\psi'$ preserves $\\partial{(M_c\/)}$ and has \nnon-empty fixed-point set, $Fix(\\psi')$ and the meridian of \n$\\partial{(M_c\/)}$ must form a Hopf link, in particular, their linking \nnumber is $1$. The image of $Fix(\\psi')$ and of the meridian of \n$\\partial{(M_c\/)}$ form again a Hopf link in ${\\mathcal O} =(M_c\/)\/\\psi$. By \nlifting them up to $M_c\/$ we see that the meridian lifts to a meridian and \nthe image of $Fix(\\psi')$ lifts to $K$ which thus have linking number $p$. \nHence the property is proved in this case.\n\nIf $n>1$, we shall perform trivial Dehn surgery on $n-1$ boundary components of\n$M_c\/$. Note that such a surgery does not change the winding number of the \nremaining boundary components (for the boundary components are unlinked), that \nthe symmetry of order $p$ of $M_c\/$ extends to the resulting solid torus, \nand that the surgery can be lifted on $M_c$ in such a way that the quotient of \nthe resulting manifold by the action of the diffeomorphism induced by $h'$ is \nagain a solid torus. This last property follows from the fact that each \nconnected component of $(E(K)\\setminus(M_c\/\\langle h\\rangle)$ is the exterior \nof a knot which lifts in $M$ to $p$ diffeomorphic copies. These $p$ copies of \nthe knot exterior are permuted by $h'$ and a copy appears in the \n$JSJ$-decomposition of $E(K')$. This means that on each boundary component \nthere is a well-defined meridian-longitude system which is preserved by $h$ and \n$h'$ and by passing to the quotient. The claim follows now from case $n=1$.\n\\qed\n\n\\bigskip\n\nNow Claims \\ref{claim:permutation} and \\ref{claim:winding number} imply that \n$E_p(K)$ is a submanifold of $M_c\/\\langle h\\rangle \\cap E(K)$. \n\nNote, moreover, that the fixed-point set of the induced symmetry is contained \nin $M_f\/\\langle h\\rangle\\subset M_c\/\\langle h\\rangle$. In particular, each \ntorus of the $JSJ$-family separating such fixed-point set from $K$ lifts to a \nsingle torus of the $JSJ$-family for $M$ and its winding number cannot be a \nmultiple of $p$. We can thus conclude that the fixed-point set of the symmetry \ninduced by $h'$ is contained in $E_p(K)$. This finishes the proof of\nProposition \\ref{prop:orbifold}.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:orbifold} \nNote that $M_c\/h \\cap E(K)$ can be larger than $E_p(K)$ for there might be tori \nof the $JSJ$-collection for $M$ which have an $\\langle h,h'\\rangle$-orbit \ncontaining $p^2$ elements and which project to tori with winding number $p$. \nNote also that $E_p(K)$ coincides with $E(K)$ if there are no \n$JSJ$-tori in $E(K)$ with winding number $p$.\n\\end{Remark}\n\n\\medskip\n\n\\begin{Remark}\\label{rem:commutation} \nThe deck transformations $h$ and $h'$ cannot commute on the submanifolds $U$ of \n$M$ corresponding to branches of $\\Gamma$ whose $h$- and $h'$-orbits coincide \nand consist of $p$ elements, if $h$ and $h'$ are different; that is, the \nstabiliser $h'h^{-1}$ is a finite order diffeomorphism of $U$ if and only if it \nis trivial. To see this, assume that there is a unique orbit of this type and \nassume by contradiction that $h$ and $h'$ commute on $M$ and are distinct. The \ndiffeomorphism $h'$ would induce a non trivial symmetry of $E(K)$ of order $p$ \nand non-empty fixed-point set which fixes set-wise the projection of $U$ and \nacts freely on it. This contradicts the first part of Lemma \n\\ref{lem:companion}. If there are $n>1$ such orbits an equivariant Dehn surgery \nargument on $n-1$ components leads again to a contradiction\n\\end{Remark}\n\n\\medskip\n\nHere is a straightforward corollary of Proposition \\ref{prop:orbifold} which \ngeneralises a result proved by B. Zimmermann \\cite{Zim1} for hyperbolic knots.\n\n\\smallskip\n\n\\begin{Corollary}\\label{cor:p-symmetry}\nLet $K$ be a prime knot and let $p$ be an odd prime number. If $K$ has no\ncompanion of winding number $p$ and has a $p$-twin, then $K$ admits a \nrotational symmetry of order $p$ with trivial quotient.\n\\qed\n\\end{Corollary}\n\n\\bigskip\n\nSo far we have proved that if a prime knot $K$ has a $p$-twin either $E(K)$ \nadmits a $p$-rotational symmetry or a well-specified submanifold $E_p(K)$ of \n$E(K)$ admits a symmetry of order $p$ with non-empty fixed-point set. We shall \nsay that the $p$-twin induces a \\emph{symmetry}, respectively a \\emph{partial \nsymmetry}, of $K$.\n\n\\smallskip\n\n\\begin{Proposition}\\label{prop:twin}\nLet $K$ be a prime knot. Assume that $K$ has a $p$-twin and a $q$-twin for two \ndistinct odd prime numbers.\n\n\\item{(i)} At least one twin, say the $q$-twin, induces a $q$-rotational \nsymmetry $\\psi_q$ of $K$. Moreover:\n\n\\item{(ii)} If the $p$-twin induces a partial $p$-symmetry of $K$, then \n$\\partial E_p(K) \\setminus \\partial E(K)$ is a $JSJ$-torus which separates \nthe fixed point set $Fix(\\psi_q)$ from $\\partial E(K)$.\n\\end{Proposition}\n\n\\medskip\n\nFirst we study some properties of partial symmetries induced by $p$-twins for \nan odd prime number $p$.\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:partial companion}\nLet $K$ be a prime knot and let $\\psi$ be the partial symmetry of order $p$ \ninduced on $E_p(K)$ by a $p$-twin. Let $T$ be a torus of the $JSJ$-collection \nof $E_p(K)$ which is not in the boundary. Then $T$ does not separate $\\partial \nE(K)$ from $Fix(\\psi)$ if and only if its $\\psi$-orbit has $p$ elements. \nMoreover, this is the case if and only if the lift of $T$ to the $p$-fold \ncyclic branched cover of $K$ has $p$ elements.\n\\end{Lemma}\n\n {\\bf Proof.} \nIt suffices to perform $\\psi$-equivariant Dehn fillings on the boundary\ncomponents $\\partial E_p(K) \\setminus \\partial E(K)$ of $E_p(K)$ in such a way \nthat the resulting manifold is a knot exterior $E(\\hat K)$ and that the graph \ndual to the $JSJ$-decomposition of $E(\\hat K)$ remains unchanged after filling \n(see the proof of Theorem \\ref{thm:rotations}). Part (i) of Lemma \n\\ref{lem:companion} then applies to the resulting knot $\\hat K$ and the induced \nrotational symmetry. To apply Lemma \\ref{lem:torus} it suffices to note that, \nas in the proof of Claim \\ref{claim:winding number}, the fillings can be chosen \nin such a way that the induced fillings on the quotient $E_p(K)\/\\langle \\psi \n\\rangle$ give also a solid torus (see Remark \\ref{rem:lift}).\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:case b} \nIn particular, case (b) of the proof of Claim \\ref{claim:permutation} cannot\nhappen for a torus $T$ in the situation of Lemma \\ref{lem:partial companion}.\n\\end{Remark}\n\n\\medskip\n\n\\begin{Lemma}\\label{lem:vertex}\nLet $K$ be a prime knot and let $\\psi$ be the partial symmetry of order $p$ \ninduced on $E_p(K)$ by a $p$-twin. Let $T \\subset \\partial E_p(K) \\setminus \n\\partial E(K)$ be a torus which is $\\psi$-invariant. Let $e_{T}$ be the \ncorresponding edge in the tree dual to the $JSJ$-decomposition of $E_p(K)$. Let \n$v_K$ and $v_\\psi$ be the vertices corresponding to the geometric pieces \ncontaining $\\partial E(K)$ and $Fix(\\psi)$ respectively. Then $v_\\psi$ belongs \nto the unique geodesic joining $v_K$ to $e_{T}$ in this $JSJ$-tree.\n\\end{Lemma}\n\n {\\bf Proof.} \nIf we cut ${\\mathbb S}^3$ along a torus of the $JSJ$-collection of $E_p(K)$, the \nconnected component which does not contain $K$ is a knot exterior and is thus \ncontained in a ball in ${\\mathbb S}^3$. If the conclusion of the Lemma were false, then \nwe could find two tori of the $JSJ$-decomposition of $E_p(K)$ contained in two \ndisjoint balls, one torus separating $Fix(\\psi)$ from $K$ and the other \ncoinciding with $T$ or separating it from $K$. In particular the linking number \nof $Fix(\\psi)$ and a meridian of the solid torus bounded by $T$ (i.e. the \nwinding number of $T$ with respect to $Fix(\\psi)$) would be zero. This is \nimpossible since $\\psi$ leaves set-wise invariant $T$.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:adjacency}\nLemma \\ref{lem:vertex} has two interesting consequences. Since $h$ and $h'$\nplay symmetric roles, we deduce that $Fix(\\psi)$ and $\\partial E(K)$ must \nbelong to the same geometric piece of the $JSJ$-decomposition of $E_p(K)$. This \nfollows from the fact that, in $E_p(K') \\cup {\\mathcal U}(K')$, $Fix(\\psi)$ maps to \n$K'$, $K$ maps to $Fix(\\psi')$, and $T$ maps to a $\\psi'$-invariant torus. \nMoreover, each invariant boundary torus $T$ is adjacent to the geometric \ncomponent containing $Fix(\\psi)$ and $K$, else, we would get a contradiction to \nLemma \\ref{lem:partial companion}.\n\\end{Remark}\n\n\\medskip\n\n{\\bf Proof of Proposition \\ref{prop:twin}(i).}\nWe argue by contradiction, assuming that there are a $p$-twin and a $q$-twin of \n$K$ which induce only partial symmetries of $E(K)$ for two distinct odd \nprime numbers $p$ and $q$. Then $\\partial E_p(K)$ and $\\partial E_q(K)$ are not \nempty. Moreover, we must have $E(K)\\setminus E_p(K) \\subset E_q(K)$ since the \nwinding number along nested tori is multiplicative and thus the winding number \nof any $JSJ$-torus contained in $E(K)\\setminus E_p(K)$ must be of the form \n$kp$ and cannot be $q$. In particular $\\partial E_p(K) \\setminus \\partial E(K) \n\\subset \\text{int}(E_q(K))$.\n\nLet $T \\in \\partial E_p(K) \\setminus \\partial E(K)$ be a torus and let $\\psi$ \nbe the $q$-symmetry with non-empty fixed-point set induced on $E_q(K)$ by the \n$q$-twin. Since the winding number of $T$ is $p$, its lift to the $q$-fold \ncyclic branched cover of $K$ is connected. According to part (i) of Lemma \n\\ref{lem:companion} and to Lemmata \\ref{lem:torus} and \\ref{lem:partial \ncompanion}, $T$ must separate $\\partial E(K)$ from $Fix(\\psi)$. Since \n$Fix(\\psi)$ is connected, we see that so must be $\\partial E_p(K) \\setminus \n\\partial E(K) = T$. The final contradiction is then reached by applying Remark \n\\ref{rem:adjacency}.\n\\qed \n\n\\bigskip\n\n{\\bf Proof of Proposition \\ref{prop:twin}(ii).} \nThis is a consequence of the proof of part (i): note that in the proof $\\psi$ \nmay be a global or partial symmetry.\n\\qed\n\n\\bigskip\n\nWe are now in a position to prove Theorem \\ref{thm:twins}.\n\n\\medskip\n\n{\\bf Proof of part (i) of Theorem \\ref{thm:twins}.}\nWe argue by contradiction, assuming that $K$ admits twins for three distinct, \nodd prime numbers $p, q, r$. Under this assumption, it follows that $K$ is a \nnon-trivial knot. \n\nIf the three twins induce rotational symmetries of the knot $K$, then part (i) \nof Theorem \\ref{thm:rotations} gives a contradiction.\n\nTherefore part (i) of Proposition \\ref{prop:twin} implies that twins of orders, \nsay $q$ and $r$, induce rotational symmetries $\\psi_q$ and $\\psi_r$ of $K$ \nhaving order $q$ and $r$ respectively, while a $p$-twin induces only a partial \nrotational symmetry of $E(K)$ of order $p$. \n\nThen part (ii) of Proposition \\ref{prop:twin} shows that $\\partial E_p(K) \n\\setminus \\partial E(K)$ is a $JSJ$-torus in $E(K)$ which separates $\\partial \nE(K)$ from both $Fix(\\psi_q)$ and $Fix(\\psi_r)$. This contradicts part (ii) of \nTheorem \\ref{thm:rotations} which states that $Fix(\\psi_q)$ and \n$Fix(\\psi_r)$ must sit in the $JSJ$-component containing $\\partial E(K)$.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of part (ii) of Theorem \\ref{thm:twins}.}\nLet $K$ be a prime knot and let $p$ be an odd prime number. We assume that $K$ \nhas at least two non-equivalent $p$-twins $K_1$ and $K_2$ and look for a \ncontradiction. \n\n\\medskip\n\nIf both $\\psi_{1}$ and $\\psi_{2}$ are rotational symmetries of order $p$ of\n$K$, then by M. Sakuma \\cite[Thm. 3]{Sa2} they are conjugate since $K$ is \nprime. This would contradict the hypothesis that the knots $K_1$ and $K_2$ \nare not equivalent.\n\n\\medskip\n\nAssume now that at least one symmetry, say $\\psi_{1}$ is partial. Then \n$\\psi_{1}$ and $\\psi_{2}$ are rotational symmetries of order $p$ of the \nsubmanifold $E_p(K) \\subset E(K)$. Let $X_0$ be the geometric piece of \nthe $JSJ$-decomposition of $E(K)$ containing $\\partial E(K)$. Then $\\psi_1$ \n(respectively $\\psi_2$) generates a finite cyclic subgroup $G_1$ (respectively\n$G_2$) of the group $Diff^{+,+}(X_0, \\partial E(K))$ of diffeomorphisms of the \npair $(X_0, \\partial E(K))$ which preserve the orientations of $X_0$ and of \n$\\partial E(K)$. Moreover, one can assume that $G_1$ and $G_2$ act\ngeometrically on $X_0$.\n\nIf $X_0$ admits a hyperbolic structure, it is a consequence of the proof of the \nSmith conjecture (see for example \\cite[Lemma 2.2]{Sa2}) that the subgroup of \n$Diff^{+,+}(X_0, \\partial E(K))$ consisting of restrictions of isometries of \n$X_0$ is finite cyclic. Hence $G_1 = G_2$ and up to taking a power \n$\\psi_1 = \\psi_2$ on $X_0$.\n\nIf $X_0$ is Seifert fibred, then it must be a cable space, since $K$ is prime. \nThe uniqueness of the Seifert fibration and the fact that the basis of the \nSeifert fibration has no symmetry of finite order imply that the cyclic groups \n$G_1$ and $G_2$ belong to the circle action $S^1 \\subset Diff^{+,+}(X_0, \n\\partial E(K))$ inducing the Seifert fibration of $X_0$, see \n\\cite[Lemma 2.3]{Sa2}. Since $G_1$ and $G_2$ have the same prime order, up to \ntaking a power $\\psi_1 = \\psi_2$ on $X_0$.\n\nLet $h_1$ and $h_2$ be the deck transformations on $M$ associated to the\n$p$-fold cyclic coverings branched along $K_1$ and $K_2$, and which induce \n$\\psi_1$ and $\\psi_2$. Then by taking a suitable powers, $h_1$ and $h_2$ \ncoincide up to conjugacy on the geometric piece $\\widetilde X_0$ of the \n$JSJ$-decomposition of $M$ containing the preimage of $K$. The following lemma \nshows that they will coincide on $M$, contradicting our hypothesis.\n\\qed\n\n\\bigskip\n\n\\begin{Lemma}\\label{cor:extension}\nIf the covering transformations $h$ and $h'$ preserve a $JSJ$-piece or a\n$JSJ$-torus of $M$ and coincide on it, then they can be chosen, up to\nconjugacy, to coincide everywhere.\n\\end{Lemma}\n\n {\\bf Proof.} \nThis is a consequence of the proofs of Propositions \\ref{prop:subtree} and\n\\ref{prop:orbifold}. We shall start by showing that we can always assume that \nthere is a piece $V$ of the $JSJ$-decomposition on which $h$ and $h'$ coincide. \nTo this purpose, assume that $h$ and $h'$ coincide only on a $JSJ$-torus $T$. \nAccording to Lemma \\ref{lem:commutation} and Remark \\ref{rem:same action}, $h$ \nand $h'$ coincide on the geometric pieces of the decomposition adjacent to $T$, \nwhich are also invariant. Consider now the maximal subtree $\\Gamma_1$ of \n$\\Gamma$ such that the restrictions of $h$ and $h'$ to the corresponding \nsubmanifold $M_1$ of $M$ coincide, up to conjugacy, and such that \n$V\\subset M_1$. Let $S$ be a $JSJ$-torus for $M$ in the boundary of $M_1$. \nSince $h$ and $h'$ coincide on $M_1$, the $h$-orbit and the $h'$-orbit of $S$ \ncoincide as well and consist of either one single element $\\{S\\}$ or $p$ \nelements $\\{S,h(S)=h'(S),...,h^{p-1}(S)={h'}^{p-1}(S)\\}$. In the former case, \naccording to Lemma \\ref{lem:commutation}, $\\Gamma_1$ would not be maximal. In \nthe latter case, we are precisely in the situation described in part (a) of \nClaim \\ref{claim:permutation}. Once more, $\\Gamma_1$ is not maximal because one \ncan impose that $h$ and $h'$ act in the same way on the $p$ connected \ncomponents with connected boundary obtained by cutting $M$ along the $\\langle \nh,h'\\rangle$-orbit of $S$ (see Remark \\ref{rem:commutation}). This \ncontradiction shows that $M=M_1$ and the lemma is proved.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of part (iii) of Theorem \\ref{thm:twins}.}\nFirst we analyse the case of a knot admitting two twins, one of which induces a \npartial symmetry. \n\n\\smallskip\n\n\\begin{Proposition}\\label{prop:partial}\nLet $K$ be a prime knot admitting a $p$-twin $K'$ and a $q$-twin $K''$ for two distinct\nodd prime numbers $p$ and $q$. If $K'$ induces a partial symmetry of $K$ then\n$K'$ and $K''$ are not equivalent.\n\\end{Proposition}\n\n {\\bf Proof.} \nBy part (ii) of Proposition \\ref{prop:twin}, $E_p(K)$ has a unique boundary \ncomponent which separates $\\partial E(K)$ from the fixed-point set of the\n$q$-rotational symmetry $\\psi$ induced by $K''$. By cutting ${\\mathbb S}^3$ along \n$T = \\partial E_p(K)$ we obtain a solid torus $V=E_p(K)\\cup {\\mathcal U}(K)$ containing \n$K$, and a knot exterior $E_T$. $K$ admits a $q$-rotational symmetry $\\psi$ \ninduced by $K''$ which preserves this decomposition and induces a \n$q$-rotational symmetry with trivial quotient (see Lemma \\ref{lem:companion}) \non $E_T$ and a free $q$-symmetry $\\tilde\\psi$ on $V$. The covering \ntransformation for the knot $K'$ induces a $p$-symmetry $\\varphi$ of $V$ with \nnon-empty fixed-point set.\n\nAssume now by contradiction that $K'=K''$. Since $K'$ induces a partial \nsymmetry of $K$ and vice versa, $S^3$ admits a decomposition into two \npieces: $V'=E_p(K')\\cup {\\mathcal U}(K')$ and $E_T $. On the other hand, since $K''$ \ninduces a genuine $q$-rotational symmetry of $K$, $K''$ admits a \n$q$-rotational symmetry $\\psi''$ induced by $K$ which preserves the \naforementioned decomposition and induces a $q$-rotational symmetry with trivial \nquotient on $E_T$. Using the fact that $E_T$ is the exterior of a prime knot \n(see Lemma \\ref{lem:prime}) and M. Sakuma's result \\cite[Thm. 3]{Sa2}, we see \nthat the two $q$-rotational symmetries with trivial quotient induced by $\\psi$ \nand $\\psi''$ on $E_T$ act in the same way. Let now $E_0$ be the smallest knot \nexterior of the $JSJ$-decomposition of $E_T$ on which $\\psi=\\psi''$ induces a \n$q$-rotational symmetry with trivial quotient (this is obtained by cutting \n$E_T$ along the torus of the $JSJ$-decomposition closest to $Fix(\\psi)$ \n-respectively $Fix(\\psi'')$- and separating it from $T$. Consider now the lift,\ndenoted by $(X,{\\mathcal K})$, to $(S^3,K'')$ of $(E_0, Fix(\\psi))\/\\psi$. We claim that \n$(X,{\\mathcal K})=(V',K')$. Indeed, $X$ contains $K''=K'$ by construction, and its \nboundary is the unique torus of the $JSJ$-decomposition which is left invariant \nby the $q$-rotational symmetry of $K''$ -by construction again- and which is \nclosest to $K''$ (compare Remark \\ref{rem:adjacency}). Since \n$E_0\/\\psi=E_0\/\\psi''$, and a solid torus has a unique $q$-fold cyclic cover, we \ndeduce that $(V',K')=(X,{\\mathcal K})=(V,K)$. In particular, the deck transformations \nfor $K$ and $K'$ on their common $p$-fold cyclic branched cover can be chosen \nto coincide on the lift of $V=V'$. Lemma \\ref{cor:extension} implies that \n$K=K'$ contradicting the fact that $K'$ is a $p$-twin.\n\\qed\n\n\\bigskip\n\nLet $K'$ be a $p$-twin and a $q$-twin of $K$ for two distinct odd prime numbers \n$p$ and $q$. Proposition \\ref{prop:partial} implies that $K'$ induces two \nrotational symmetries $\\psi_p$ and $\\psi_q$ of $K$ with trivial quotients and \norders $p$ and $q$. Part (ii) of Theorem \\ref{thm:rotations} shows that the \nfixed-point sets $Fix(\\psi_p)$ and $Fix(\\psi_q)$ lie in the $JSJ$-component of \n$E(K)$ which contains $\\partial E(K)$. Then the proof of part (iii) of Theorem \n\\ref{thm:twins} follows from the following:\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:commuting symmetries}\nLet $K$ be a prime knot admitting two rotational symmetries $\\psi$ and\n$\\varphi$ of odd prime orders $p > q$. If the fixed-point sets of $\\psi$ and \n$\\varphi$ lie in the component which contains $\\partial E(K)$, then the two \nsymmetries commute up to conjugacy.\n\\end{Lemma}\n\n {\\bf Proof.} \nReasoning as in the proof of part (ii) of Theorem \\ref{thm:twins}, one can show\nthat $\\psi$ and $\\varphi$ commute on the component which contains $\\partial\nE(K)$. Since all other components are freely permuted according to part (i) of\nLemma \\ref{lem:companion}, the conclusion follows as in the proof of part (a)\nof Claim \\ref{claim:permutation}.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of Corollary \\ref{cor:composite}.}\nFirst of all note that, because of the uniqueness of the Milnor-Kneser \ndecomposition of the covers of $K$ and $K'$, the number of prime summands of $K$ \nand $K'$ is the same. After ditching components of $K$ and $K'$ that appear in \nboth decompositions in equal number, we can assume that $K_i$ is not equivalent \nto $K'_\\ell$, for all $i,\\ell=1,...,t$. If $K$ and $K'$ have three common \ncyclic branched covers of odd prime orders, we deduce that for each \n$i=1,...,t$, $K_i$ is not determined by its $p_j$-fold cyclic branched cover, \n$j=1,2,3$, for it is also the $p_j$-fold cyclic branched cover of some \n$K'_{i_j}$ not equivalent to $K_i$. Hence $K_i$ would have twins for three\ndistinct odd prime orders which is impossible by Theorem \\ref{thm:twins}. \n\\qed\n\n\n\\section{Examples}\n\nExamples of prime knots admitting a $p$-twin which induces a global rotational\nsymmetry of order $p$ were first constructed by Y. Nakanishi \\cite{Na} and M.\nSakuma \\cite{Sa1}. They considered a prime link with two trivial components \nwhose linking number is $1$. By taking the $p$-fold cyclic cover of ${\\mathbb S}^3$ \nbranched along the first (respectively the second) component of the link one\ngets again ${\\mathbb S}^3$ and the second (respectively first) component lifts to a\nprime knot. The two knots thus constructed have the same $p$-fold cyclic\nbranched cover by construction (see also Remark \\ref{rem:lift}), moreover, by \ncomputing their Alexander polynomial they were shown to be distinct.\n\nIn \\cite[Thm 3 and Cor. 1]{Zim1} B. Zimmerman showed that if a hyperbolic knot \nhas a $p$-twin, for $p\\ge 3$, then the $p$-twin induces a global symmetry and \nthe two knots are thus obtained by Y. Nakanishi and M. Sakuma's construction \nwhere the quotient link is hyperbolic and admits no symmetry which exchanges \nits two components.\n\nAs a matter of fact, the links considered by Y. Nakanishi and M. Sakuma are in \nfact hyperbolic and so are the resulting twins if $p$ is at least $3$, \naccording to the orbifold theorem \\cite{BoP}, see also \\cite{CHK}. Note that, \nwhen $p=2$, the situation, even in the case of hyperbolic knots, is much more \ncomplex and there are several ways to construct $2$-twins of a given knot. In \nthis section we shall see how one can construct, for each given odd prime $p$, \ntwo prime, non simple, knots which are $p$-twins, and such that the symmetries \nthey induce are not global.\n\nThe first construction shows that the number $\\nu$ of components of $\\partial \nE_p(K)\\setminus\\partial E(K)$ can be arbitrarily large. This means that the \nsituation encountered in Proposition \\ref{prop:twin}(ii) is extremely special. \nThe second construction shows that our result is indeed best possible even for\nprime knots with $p$-twins inducing partial symmetries: we shall construct \nprime knots admitting a $p$-twin inducing a partial symmetry and a $q$-twin \ninducing a global rotational symmetry. \n\n\\bigskip\n\n\\subsection{Knots admitting a $p$-twin inducing only a partial symmetry} \n\n\\medskip\n\nAssume we are given a hyperbolic link $L=L_1\\cup...\\cup L_{\\nu+2}$, with $\\nu+2\n\\ge 3$ components, satisfying the following requirements:\n\n\\medskip\n\n{\\bf Property $*$}\n\n\\begin{enumerate}\n\n\\item The sublink $L_3\\cup...\\cup L_{\\nu+2}$ is the trivial link;\n\n\\item For each $i=1,2$ and $j=3,...,\\nu+2$, the sublink $L_i\\cup L_j$ is a Hopf\nlink;\n\n\\item ${\\rm lk}(L_1,L_2)$ is prime with $p$;\n\n\\item No symmetry of $L$ exchanges $L_1$ and $L_2$.\n\n\\end{enumerate}\n\n\\medskip\nWe shall consider the orbifold ${\\mathcal O}=({\\mathbb S}^3,(L_1\\cup L_2)_p)\\setminus\n{\\mathcal U}(L_3\\cup...\\cup L_{\\nu+2})$ which is the $3$-sphere with singular set of\norder $p$ the (sub)link $L_1\\cup L_2$ and an open tubular neighbourhood of the\n(sub)link $L_3\\cup...\\cup L_{\\nu+2}$ removed. ${\\mathcal O}$ is hyperbolic if $p\\ge3$, \nand will represent the quotient of ${\\mathcal O}_p=E_p(K)\\cup{\\mathcal U}(K)$ and \n${\\mathcal O}_p'=E_p(K')\\cup{\\mathcal U}(K')$ via the action of the partial $p$-symmetries. \nIndeed, to obtain ${\\mathcal O}_p$ (respectively ${\\mathcal O}_p'$) take the $p$-fold cyclic \norbifold cover of $({\\mathbb S}^3,(L_1\\cup L_2)_p)\\setminus\n{\\mathcal U}(L_3\\cup...\\cup L_{\\nu+2})$ which desingularises $L_2$ (respectively $L_1$).\nObserve that one can fix a longitude-meridian system on each boundary \ncomponent of ${\\mathcal O}$, induced by those of $L_i$, $i=3,\\dots,\\nu+2$. Note that, \nbecause of condition 4 of Property $*$, the two orbifolds ${\\mathcal O}_p$ and ${\\mathcal O}_p'$ \nwith the fixed peripheral systems are distinct. \n\nRemark that ${\\mathcal O}_p$ and ${\\mathcal O}'_p$ can be obtained by the orbifold covers,\nanalogous to those described above, of $({\\mathbb S}^3,(L_1\\cup L_2)_p)$ (which are \ntopologically ${\\mathbb S}^3$) by removing open regular neighbourhoods of the lifts of \nthe components $L_3\\cup...\\cup L_{\\nu+2}$. Note that these components \nlift to trivial components whose linking number with the lift of $L_i$, \n$i=1,2$, is precisely $p$, because of condition 2, and which form again a \ntrivial link.\n\nFor each $j=3,...,\\nu+2$, choose a knot exterior $E({\\mathcal K}_j)$ to be glued along \nthe $j$-th boundary component of ${\\mathcal O}_p$ and ${\\mathcal O}'_p$ in such a way that a \nfixed longitude-meridian system on $E({\\mathcal K}_j)$ is identified with the lift of \nthe longitude-meridian system on the $j$-th boundary component of ${\\mathcal O}$. The underlying spaces of the\norbifolds ${\\mathcal O}_p\\cup_{j=3}^{\\nu+2} E({\\mathcal K}_j)$ and ${\\mathcal O}'_p\\cup_{j=3}^{\\nu+2} \nE({\\mathcal K}_j)$ are topologically ${\\mathbb S}^3$ and it is easy to see that their singular \nsets are connected (see condition 3). The resulting knots have the same \n$p$-fold cyclic branched cover, however, since ${\\mathcal O}_p$ and ${\\mathcal O}'_p$ are \ndistinct, they are not equivalent.\n\n\\bigskip\n\n\\begin{Remark}\nObserve that we have just shown that the number of connected components of\n$\\partial E_p(K)\\setminus\\partial E(K)$, which is precisely $\\nu$, can be \narbitrarily large. Note also that if $\\nu\\ge2$, according to Proposition\n\\ref{prop:twin}, the knot $K$ has no $q$-twins for $q\\neq p$ odd prime.\n\\end{Remark}\n\n\\bigskip\n\nWe shall now prove that links with Property $*$ exist. Notice that for $\\nu=1$\nlinks satisfying all the requirements where constructed by Zimmermann in\n\\cite{Zim2}, see also \\cite{Pao1}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{partial.eps}\n\\end{center}\n\\caption{The link $L$ and its Bonahon-Siebenmann decomposition.}\n\\label{fig:one}\n\\end{figure}\n\nConsider the link given in Figure \\ref{fig:one} for $\\nu=3$ (the generalization \nfor arbitrary $\\nu\\ge1$ is obvious). Most conditions are readily checked just \nby looking at the figure, and we only need to show that $L$ is hyperbolic and \nhas no symmetries which exchange $L_1$ and $L_2$. To this purpose, we shall \ndescribe the Bonahon-Siebenmann decomposition of the orbifold $({\\mathbb S}^3,(L)_2)$, \nwhere all components have ${\\mathbb Z}\/2{\\mathbb Z}$ as local group. The decomposition consists \nof one single hyperbolic piece (see Figure \\ref{fig:one}) and $\\nu+1$ \n(respectively $1$) Seifert fibred pieces if $\\nu\\ge2$ (respectively $\\nu=1$). \nSince the Seifert fibred pieces contain no incompressible torus, the \nhyperbolicity of $L$ follows.\n\nNote now that every symmetry of $L$ must leave invariant the unique hyperbolic\npiece of the decomposition. This piece is obtained by quotienting the\nhyperbolic knot $10_{155}$ via its full symmetry group ${\\mathbb Z}\/2{\\mathbb Z}\\oplus{\\mathbb Z}\/2{\\mathbb Z}$ and \nthus has no symmetries (for more details see \\cite{Pao1}), so we conclude that \nthe components $L_1$ and $L_2$ are non exchangeable.\n\n\\bigskip\n\n\\subsection{Knots admitting a $p$-twin inducing a partial symmetry and a\n$q$-twin inducing a global symmetry}\n\n\\medskip\n\nLet ${\\mathcal K}$ be a hyperbolic knot admitting a $p$-twin and a $q$-twin; the twins \nof ${\\mathcal K}$ induce global symmetries, so that ${\\mathcal K}$ admits a $p$- and a\n$q$-rotational symmetry with trivial quotient (see \\cite{Zim2}, where a method\nto construct hyperbolic knots with two twins is described). Remove a tubular\nneighbourhood of the axis of the symmetry of order $q$ (note that the two\nsymmetries have disjoint axes), and use the resulting solid torus $V$ to \nperform Dehn surgery on the exterior $E$ of the $(2,q)$-torus knot. Denote by \n$K$ the image of ${\\mathcal K}$ after surgery. We require that:\n\n\\begin{enumerate}\n\n\\item The resulting manifold is ${\\mathbb S}^3$;\n\n\\item The $q$-rotational symmetry of $E$ and the restriction of the\n$q$-rotational symmetry of ${\\mathcal K}$ to $V$ give a global $q$-rotational symmetry \nof $K$;\n\n\\item The $q$-rotational symmetry of $K$ has trivial quotient.\n\n\\end{enumerate}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{trivial.eps}\n\\end{center}\n\\caption{Satellising so that the induced rotation has trivial quotient.}\n\\label{fig:two}\n\\end{figure}\n\nNote that the last requirement can be met by choosing appropriately the\nlongitude when satellising, as illustrated in Figure \\ref{fig:two}. We claim \nthat $K$ admits a $q$-twin, $K''$, and a $p$-twin, $K'$. $K''$ is obtained by \nthe standard method described in Remark \\ref{rem:lift}. Note that $K\\neq K''$, \nfor the roots of the $JSJ$-decompositions of the exteriors of $K$ and $K''$ are \nhyperbolic and Seifert fibred respectively. To construct $K'$, consider the \n$p$-twin ${\\mathcal K}'$ of ${\\mathcal K}$ and let $V'$ be the solid torus obtained by removing \nthe axis of the $q$-rotational symmetry of ${\\mathcal K}'$. Note that $V$ and $V'$ have \na common quotient obtained by taking the space of orbits of the $p$-rotational \nsymmetries, however $V$ and $V'$ are different orbifolds by construction. Fix a \nlongitude-meridian system on $V$ (the one used for the surgery): by first \nquotienting and then lifting it, get a longitude-meridian system on $V'$ that \nmust be used to perform surgery along a copy of $E$. The image of ${\\mathcal K}'$ after \nthe surgery will be $K'$. Note that, when taking the $p$-fold cyclic branched \ncovers of $K$ and $K'$, the hyperbolic orbifolds $V$ and $V'$ lift to the same \nmanifold by construction, while the Seifert fibred part lifts, in both cases, \nto $p$ copies of $E$. Again by construction, the gluings are compatible and the \ntwo covers coincide. It is also evident that $K'$ can only induce a partial \nsymmetry of $K$, and the claim is proved.\n\n\\bigskip\n\n\\begin{Remark}\nNote that according to Proposition \\ref{prop:partial} the $p$-twins and \n$q$-twins obtained in this construction cannot be equivalent.\n\\end{Remark}\n\n\n\\section{Homology spheres as cyclic branched covers}\n\nBy the proof of the Smith conjecture Corollary \\ref{cor:homologysphere} is true \nfor the $3$-sphere $S^3$. So from now on we assume that the integral homology \nsphere $M$ is not homeomorphic to $S^3$. Then by \\cite[Thm1]{BPZ}, $M$ can be a \n$p_i$-fold cyclic branched cover of ${\\mathbb S}^3$ for at most three pairwise distinct \nodd prime numbers $p_i$. Moreover if $M$ is irreducible and is the $p_i$-fold \ncyclic branched cover of ${\\mathbb S}^3$ for three pairwise distinct odd prime numbers \n$p_i$, then the proof of \\cite[Corollary 1.(i)]{BPZ} shows that for each prime \n$p_i$, $M$ is the $p_i$-fold cyclic branched cover of precisely one knot. Since \na knot admits at most one $p$-twin for an odd prime integer $p$, we need only \nto consider the case when the irreducible integral homology sphere $M$ is the \nbranched cover of ${\\mathbb S}^3$ for precisely two distinct odd primes, say $p$ and \n$q$. Moreover \\cite[Corollary 1.(ii)]{BPZ} shows that $M$ has a non trivial \n$JSJ$-decomposition. \n\nLooking for a contradiction, we can assume that, for each prime, $M$ is the \nbranched covering of two distinct knots with covering transformations $\\psi$, \n$\\psi'$ of order $p$ and $\\varphi$, $\\varphi'$ of order $q$. \n\nIf each rotation of order $p$ commutes with each rotation of order $q$ up to \nconjugacy, then the contradiction follows from the following claim which is an \neasy consequence of Sakuma's result \\cite[Thm. 3]{Sa2} (see \\cite[Claim \n8]{BPZ}). \n\n\\begin{Claim}\\label{claim:unique symmetry} \nLet $n\\ge3$ be a fixed odd integer. Let $\\rho$ be a rotation with trivial \nquotient of an irreducible manifold $M$. All the rotations of $M$ of order $n$ \nwhich commute with $\\rho$ are conjugate in $Diff(M)$ into the same cyclic group \nof order $n$. \n\\qed\n\\end{Claim}\n\nOtherwise, consider the subgroup $G=\\langle \\psi, \\psi', \\varphi, \\varphi' \n\\rangle$ of diffeomorphisms of $M$. According to the proof of \\cite[Proposition \n4]{BPZ}, each rotation of order $p$ commutes with each rotation of order $q$ up \nto conjugacy, unless the induced action of $G$ on the dual tree of the \n$JSJ$-decomposition for $M$ fixes precisely one vertex corresponding to a \nhyperbolic piece $V$ of the decomposition and $\\{p,q\\}=\\{3,5\\}$. In this case, \none deduces as in the proof of \\cite[Corollary 1.(ii)]{BPZ} that the \nrestrictions of $\\psi$ and $\\psi'$ (respectively $\\varphi$ and $\\varphi'$) \ncoincide up to conjugacy on $V$. Then the desired contradiction follows from \nLemma \\ref{cor:extension} which implies that $\\psi$ and $\\psi'$ (respectively \n$\\varphi$ and $\\varphi'$) coincide up to conjugacy on $M$.\n\\qed\n\n\n\\begin{footnotesize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{\\large \\textbf{Supplemental Material}\\\\Stochastic Thermodynamics of\n Learning}\n\n\\begin{center}\n Sebastian Goldt and Udo Seifert\\\\\n \\emph{II. Institut f\u00fcr Theoretische Physik, Universit\u00e4t Stuttgart, 70550\n Stuttgart, Germany}\\\\\n (Dated: \\today)\n\\end{center}\n\n In this supplemental material, we discuss the stochastic thermodynamics of\n neural networks in detail in section~\\ref{sec:stochastic-thermodyanmics} and\n derive our main result, eq.~\\eqref{eq:inequality} of the main text, in\n section~\\ref{sec:inequalities}. Furthermore, we complement our discussion Hebbian\n learning in the thermodynamic limit with additional analytical calculations in\n section~\\ref{sec:hebbian-calculations}.\n\n\\section{Stochastic thermodynamics of neural networks}\n\\label{sec:stochastic-thermodyanmics}\n\nWe now give a detailed account of the stochastic thermodynamics of neural\nnetworks. For simplicity, here we will focus on batch learning; the\ngeneralisation to online learning is straightforward. For a network with $N$\nweights $\\wn\\in\\mathbb{R}^N$ learning $P$ samples $\\vsample[\\mu]\\in\\{\\pm1\\}^N$\nwith their labels $\\tlab[\\mu]=\\pm1$, $\\mu=1,2,\\dots,P$, we have $N$ Langevin\nequations~\\cite{s_vankampen1992}\n\\begin{equation}\n \\label{eq:langevin}\n \\dot{\\w}_n(t) = - \\wn(t) + f(\\wn(t), \\{\\sample[\\mu]_n, \\tlab[\\mu]\\}, t) + \\zeta_n(t).\n\\end{equation}\nThe Gaussian noise $\\zeta_n(t)$ has correlations\n$\\avg{\\zeta_n(t)\\zeta_m(t')}=2T\\delta_{nm}\\delta(t-t')$ for $n,m=1,\\dots,N$\nwhere $T$ is the temperature of the surrounding medium and we have set\nBoltzmann's constant to unity to render entropy dimensionless. The\nweights $\\vw$ determine the transition rates of the $P$ independent two-state\nprocesses for the predicted labels $\\lab[\\mu]$ via\n\\begin{equation}\n k_\\mu^+\/k_\\mu^- = \\exp \\left(\\act^\\mu\/T\\right)\n\\end{equation}\nwhere $\\act^\\mu$ is the input-dependent activation\n\\begin{equation}\n \\act^\\mu\\equiv\\frac{1}{\\sqrt{N}}\\vw\\cdot\\vsample[\\mu]\n\\end{equation}\nFor the remainder of this supplemental material, we set $T=1$, rendering energy\ndimensionless. We assume that the thermal noise in each subsystem, like $\\wn$ or\n$\\lab[\\mu]$, is independent of all the others. This multipartite\nassumption~\\cite{s_horowitz2015} allows us to write the master equation for the\ndistribution $p(\\tlabs,\\vw,\\labs, t)$ with\n$\\tlabs\\equiv(\\tlab[1],\\dots,\\tlab[P])$ and $\\labs\\equiv(\\lab[1],\\dots,\\lab[P])$\nas\n\\begin{equation}\n \\label{eq:master}\n \\partial_t p(\\tlabs,\\vw,\\labs, t)=-\\sum_{n=1}^N \\partial_n j_n(t) + \\sum_{\\mu=1}^Pj_\\mu(t),\n\\end{equation}\nwhere $\\partial_t\\equiv\\partial\/\\partial t$,\n$\\partial_n\\equiv \\partial\/\\partial\\, \\wn$ and the probability currents for the\n$n$-th weight $\\wn$ and the $\\mu$-th predicted label $\\lab[\\mu]$ are given by\n\\begin{subequations}\n \\label{eq:currents}\n \\begin{align}\n j_n(t)=& \\left[-\\wn+f(\\wn, \\vsample[\\mu(t)], \\tlab[\\mu(t)], t) - \\partial_n\\right]\n p(\\tlabs,\\vw,\\labs, t), \\\\\n j_\\mu(t) =& k^+ p(\\tlabs,\\vw,\\lab[1],\\dots,-\\lab[\\mu],\\dots,\\lab[P], t)\n - k^- p(\\tlabs, \\vw, \\labs, t).\n \\end{align}\n\\end{subequations}\nWe choose symmetric rates $k^\\pm_\\mu=\\gamma\\exp(\\pm\\act^\\mu\/2)$ with\n$\\gamma\\gg1$. Initially, the true labels $\\tlabs$, weights $\\vw$ and predicted\nlabels are all uncorrelated with\n\\begin{align}\n p_0(\\tlab[\\mu])=&1\/2, \\\\\n p_0(\\lab[\\mu])=&1\/2, \\quad \\text{and} \\\\\n p_0(\\vw) =& \\frac{1}{(2\\pi)^{N\/2}} \\exp(-\\vw\\cdot\\vw\/2).\n\\end{align}\nSince the following discussion applies to the time-dependent\ndynamics~\\eqref{eq:master}, we understand that all quantities that will be\nintroduced in the remainder of this section have an implicit\ntime-dependence via the distribution $p(\\tlabs, \\vw, \\labs, t)$ or the\ncurrents~\\eqref{eq:currents}.\n\nOur starting point for the stochastic thermodynamics of this system is the\nwell-known total entropy production $\\dot{S}^\\tot$ of the network which obeys\nthe following second-law like inequality~\\cite{s_Seifert2012}\n\\begin{equation}\n \\dot{S}^\\tot = \\partial_t S(\\tlabs, \\vw, \\labs) + \\dot{S}^\\m \\ge 0\n\\end{equation}\nwith equality in equilibrium only. Here, we have the Shannon\nentropy~\\cite{s_cover2006} of the system,\n\\begin{equation}\n \\label{eq:shannon_all}\n S(\\tlabs, \\vw, \\labs) = - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n p(\\tlabs, \\vw, \\labs) \\ln p(\\tlabs, \\vw, \\labs).\n\\end{equation}\nHere, we include the variables $\\tlabs$, $\\vw$ and $\\labs$ as arguments of the\nfunction $S$ in a slight abuse of notation to emphasise that we consider the\nShannon entropy of the full distribution $p(\\tlabs, \\vw, \\labs)$. $\\dot{S}^\\m$\ngives the rate of entropy production in the medium. For a system at constant\ntemperature $T=1$, $\\dot{S}^\\m\\equiv\\dot{Q}$, the rate of heat dissipation into\nthe medium~\\cite{s_Seifert2012}. Let us first focus on the change in Shannon\nentropy by differentiating~\\eqref{eq:shannon_all} with respect to time,\n\\begin{equation}\n \\partial_t S(\\tlabs, \\vw, \\labs) = - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n \\dot{p}(\\tlabs, \\vw, \\labs) \\ln p(\\tlabs, \\vw, \\labs),\n\\end{equation}\nwhere we have used that $p(\\tlabs, \\vw, \\labs)$ is, of course, normalised. Using\nthe master equation~\\eqref{eq:master}, we find that\n\\begin{equation}\n \\partial_t S(\\tlabs, \\vw, \\labs) = \\sum_{n=1}^N \\dot{S}_n + \\sum_{\\mu=1}^P \\dot{S}_\\mu\n\\end{equation}\nwhere \n\\begin{align}\n \\dot{S}_n \\equiv& \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n \\partial_n j_n(t) \\ln p(\\tlabs, \\vw, \\labs),\\\\\n \\dot{S}_\\mu \\equiv& - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n j_\\mu \\ln p(\\tlabs, \\vw, \\labs),\n\\end{align}\nare the rate of change of the Shannon entropy $S(\\tlabs, \\vw, \\labs)$ due to the\ndynamics of $\\w_n$ and $\\lab[\\mu]$, respectively. The key point here is that\nmultipartite dynamics, a consequence of the uncorrelated noise across\nsubsystems, lead to a linear splitting of the probability currents and hence to\na linear splitting of all quantities which are functions of the total\nprobability current. Similarly, for the rate of heat dissipation $\\dot{Q}$,\nwe can write\n\\begin{equation}\n \\dot{Q} = \\sum_{n=1}^N \\dot{Q}_n + \\sum_{\\mu=1}^P\\dot{Q}_\\mu\n\\end{equation}\nwhere\n\\begin{equation}\n \\dot{Q}_n = \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n j_n(t) F_n(\\tlabs, \\vw, \\labs)\n\\end{equation}\nwith the total force on the $n$-th weight\n$F_n=- \\wn(t) + f(\\wn(t), \\{\\sample[\\mu]_n, \\tlab[\\mu]\\}, t)$, while\n\\begin{equation}\n\\dot{Q}_\\mu = \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n j_\\mu(t) \\lab[\\mu] \\vw\\cdot\\vsample[\\mu]\/2.\n\\end{equation} \nFinally, total entropy production $\\dot{S}^\\tot$ can also be split,\n\\begin{equation}\n \\dot{S}^\\tot = \\sum_{n=1}^N \\dot{S}^\\tot_n + \\sum_{\\mu=1}^P \\dot{S}^\\tot_\\mu.\n\\end{equation}\nIt can easily be shown that each of these total entropy productions of a\nsubsystem obeys a separate second-law like inequality, \\emph{e.g.}\n\\begin{equation}\n \\label{eq:2nd-law-short-n}\n \\dot{S}^\\tot_n =\\dot{S}_n(\\tlabs, \\vw, \\labs) + \\dot{Q}_n \\geq 0\n\\end{equation}\nfor the $n$-th weight. \n\nWriting\n\\begin{equation}\np(\\tlabs, \\vw, \\labs) = p(\\wn)p(\\tlabs, \\vwOthers, \\labs |\\wn)\n\\end{equation}\nwith $\\vwOthers\\equiv(\\cdots, \\w_{n-1}, \\w_{n+1},\\cdots)$, we can split\n$\\dot{S}_n(\\tlabs,\\vw,\\labs)$ into two parts: first, the change of Shannon\nentropy of the marginalized distribution $p(\\wn)$,\n\\begin{equation}\n \\dot{S}_n(\\wn) = \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n \\partial_n j_n(t) \\ln p(\\wn) = \\partial_t S(\\wn),\n\\end{equation}\nwhere the last equality follows from the fact that an entropy change of the\nmarginalized distribution $p(\\wn)$ can only come from the dynamics of $\\wn$. The\nsecond part is called the learning rate \\cite{s_hartich2014}\n\\begin{equation}\n \\label{eq:lw}\n l_n(\\w_n; \\tlabs, \\labs, \\vwOthers) = \n - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\; \\partial_n j_n(t) \\ln p(\\tlabs,\\labs, \\vwOthers |\\w_n)\n\\end{equation}\nor information flow \\cite{s_allahverdyan2009,s_horowitz2014}. We emphasise that this\nlearning rate $l_n$ is thermodynamic and has nothing to do with the learning\nrate $\\lr$ that goes into the definition of the learning algorithms, see for\nexample eq.~\\eqref{eq:hebbian-force} of the main text. To avoid confusion, we\nwill refer to $l_n$ as the thermodynamic learning rate for the remainder of this\nsupplemental material. The second law \\eqref{eq:2nd-law-short-n} for the $n$-th\nweight hence becomes\n\\begin{equation}\n \\label{eq:2nd-law-long-n}\n \\dot{S}^\\tot_n = \\partial_t S(\\wn) + \\dot{Q}_n - l_n(\\w_n; \\tlabs, \\labs, \\vwOthers)\\geq 0\n\\end{equation}\nThe thermodynamic learning rate is a thermodynamically consistent measure of how\nmuch the dynamics of $\\wn$ change the mutual information\n$\\mutual{\\wn}{\\tlabs, \\vwOthers, \\labs}$, in particular for a system that\ncontinuously rewrites a single memory~\\cite{s_horowitz2014a}.\n\nWe can further refine the second law~\\eqref{eq:2nd-law-long-n} by exploiting the\ncausal structure of the dynamics, as was recently suggested by\nHorowitz~\\cite{s_horowitz2015}. The subsystem $\\wn$ directly interacts only with\nthose degrees of freedom that appear in its probability current $j_n(t)$\n\\eqref{eq:currents}. From inspection of the current $j_n(t)$, we see that $\\wn$\nis directly influenced only by itself and the given labels $\\tlabs$. Keeping\nthis in mind, we use the chain rule for mutual information~\\cite{s_cover2006} to\nwrite\n\\begin{equation}\n \\mutual{\\wn}{\\tlabs, \\vwOthers, \\labs}= \\mutual{\\wn}{\\tlabs} +\n \\mutualGiven{\\wn}{\\vwOthers, \\labs}{\\tlabs},\n\\end{equation}\nwhere we use the conditional mutual information\n\\begin{align}\n \\label{eq:mutual-info-conditional}\n \\mutualGiven{\\wn}{\\vwOthers, \\labs}{\\tlabs} =& S(\\wn|\\tlabs) -\n S(\\wn|\\vwOthers, \\labs, \\tlabs) \\\\\n =&-\n \\sum_{\\labs,\\tlabs}\\int_{-\\infty}^\\infty \\diff{\\vw} \\; p(\\tlabs, \\vw, \\labs) \\ln\n \\frac{p(\\tlabs, \\vw, \\labs)p(\\tlabs)}{p(\\wn, \\tlabs)p(\\vwOthers, \\labs, \\tlabs)}.\n\\end{align}\nAccordingly, we split the thermodynamic learning rate~\\eqref{eq:lw} into a\nthermodynamic learning rate of the $n$-th weight with the degrees of freedom\nthat it directly interacts with, \\emph{i.e.} the true labels $\\tlabs$,\n\\begin{equation}\n \\label{eq:lw-refined}\n l_n(\\wn; \\tlabs) = - \\sum_{\\labs,\\tlabs}\\int_{-\\infty}^\\infty \\diff{\\vw}\n \\; \\partial_n j_n(t)\\ln p(\\tlabs|\\wn),\n\\end{equation}\nand a thermodynamic learning rate with the other subsystems given the true labels,\n\\begin{equation}\n \\label{eq:lw-conditional}\n l_n(\\wn; \\vwOthers, \\labs|\\tlabs) = - \\sum_{\\labs,\\tlabs}\\int_{-\\infty}^\\infty\n \\diff{\\vw} \\; \\partial_n j_n(t)\\ln \\left(\\frac{p(\\wn, \\vwOthers, \\labs\n |\\tlabs)}{p(\\wn|\\tlabs)p(\\vwOthers, \\labs|\\tlabs)}\\right).\n\\end{equation}\nHorowitz proved \\cite{s_horowitz2015} the following second-law like inequality\nincluding the refined thermodynamic learning rate~\\eqref{eq:lw-refined},\n\\begin{equation}\n \\label{eq:2nd-law-refined}\n \\partial_t S(\\wn) + \\dot{Q}_n - l_n(\\wn; \\tlabs) \\geq 0.\n\\end{equation}\nwhich is the basis for our proof of the main inequality,\nequation~\\eqref{eq:inequality} of the main text.\n\n\\section{Derivation of inequality~$\\eqref{eq:inequality}$ of the main text}\n\\label{sec:inequalities}\n\nThe stochastic thermodynamics of neural networks yields $N$ inequalities of the\nform~\\eqref{eq:2nd-law-refined}. Integrating over time and summing over all the\nweights, we find\n\\begin{equation}\n \\sum_{n=1}^N \\left[ \\Delta S(\\wn) + \\Delta Q_n\\right] \\ge \\sum_{n=1}^N \\int_0^\\infty \\diff{t} \\; l_n(\\wn; \\tlabs) = \\sum_{n=1}^N \\Delta\n \\mutual{\\wn}{\\tlabs}\n\\end{equation}\nThe precise definition of all the terms are discussed in the main text and in\nsection \\ref{sec:stochastic-thermodyanmics} of this supplemental material. The\ncrucial point for the last equality is that the labels $\\tlabs$ are static, so\nthat the mutual information $\\mutual{\\wn}{\\tlabs}$ changes only due to the\ndynamics of $\\wn$ and hence $\\partial_t \\mutual{\\wn}{\\tlabs}=l_n(\\wn;\\tlabs)$\n\\cite{Note6}. To make progress towards our main result,\ninequality~\\eqref{eq:inequality} of the main text, we need to show that\n\\begin{equation}\n \\label{eq:1}\n \\sum_{n=1}^N \\Delta \\mutual{\\wn}{\\tlabs} \\ge \\sum_{\\mu=1}^P \\Delta \\mutual{\\tlab[\\mu]}{\\lab[\\mu]}.\n\\end{equation}\n\nFirst, we note that from the chain rule of mutual information \\cite{s_cover2006},\nwe have\n\\begin{equation}\n \\Delta \\mutual{\\vw}{\\tlabs}=\\Delta \\mutual{\\w_1, \\dots, \\w_n}{\\tlabs}=\\sum_{n=1}^N\\Delta \\mutualGiven{\\wn}{\\tlabs}{\\w_{n-1},\\dots,\\w_1}\n\\end{equation}\nwith the conditional mutual information \\cite{s_cover2006}\n\\begin{equation}\n \\mutualGiven{\\wn}{\\tlabs}{\\w_{n-1},\\dots,\\w_1} \\equiv S(\\wn|\\w_{n-1},\\dots,\\w_1) -\n S(\\wn|\\tlabs, \\w_{n-1},\\dots,\\w_1).\n\\end{equation}\nDue to the form of the Langevin equation for the single weight,\neq.~\\eqref{eq:langevin}, individual weights are uncorrelated, and hence the\nconditional mutual information simplifies to\n\\begin{align}\n \\Delta \\mutualGiven{\\wn}{\\tlabs}{\\w_{n-1},\\dots,\\w_1} &=\n \\Delta S(\\wn|\\w_{n-1},\\dots,\\w_1)-\\Delta S(\\wn|\\tlabs,\\w_{n-1},\\dots,\\w_1)\\\\\n &= \\Delta S(\\wn) - \\Delta S(\\wn|\\tlabs)\\\\\n &= \\Delta \\mutual{\\wn}{\\tlabs}\n\\end{align}\nsuch that\n\\begin{equation}\n \\sum_{n=1}^N\\Delta \\mutual{\\wn}{\\tlabs} = \\Delta \\mutual{\\vw}{\\tlabs}.\n\\end{equation}\n\nNext, we show that\n\\begin{equation}\n \\label{eq:first-inequality}\n \\Delta \\mutual{\\vw}{\\tlabs} = \\sum_{\\mu=1}^P\n \\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}\\stackrel{!}{\\ge} \\sum_{\\mu=1}^P\\Delta \\mutual{\\vw}{\\tlab[\\mu]}.\n\\end{equation}\nusing the independence of the given labels $\\tlabs$. We first note that\n\\begin{align}\n \\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}= & \\Delta S(\\tlab[\\mu]|\\tlab[\\mu-1],\\dots,\\tlab[1])\n - \\Delta S(\\tlab[\\mu]|\\vw, \\tlab[\\mu-1],\\dots,\\tlab[1]) \\\\\n =& \\Delta S(\\tlab[\\mu]) - \\Delta S(\\tlab[\\mu]|\\vw, \\tlab[\\mu-1],\\dots,\\tlab[1]) \n\\end{align}\nwhile\n\\begin{equation}\n \\Delta \\mutual{\\vw}{\\tlab[\\mu]}=\\Delta S(\\tlab[\\mu]) - \\Delta S(\\tlab[\\mu]|\\vw)\n\\end{equation}\nHence for\n$\\Delta\n\\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}\\stackrel{!}{\\ge}\\Delta\n\\mutual{\\vw}{\\tlab[\\mu]}$, we need\n\\begin{align}\n \\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}-\n \\Delta \\mutual{\\vw}{\\tlab[\\mu]} & \\\\ \n = \\quad & \\Delta S(\\tlab[\\mu]|\\vw) - \\Delta S(\\tlab[\\mu]|\\vw,\n \\tlab[\\mu-1],\\dots,\\tlab[1]) \\label{eq:first-step} \\\\\n = \\quad & \\Delta \\mutualGiven{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}{\\vw} \\\\\n \\ge \\quad & 0\n\\end{align}\nwhere we first used that the $\\tlab[\\mu]$ are independent and identically\ndistributed. The last inequality follows since any mutual information,\nconditional or not, is always greater than or equal to zero \\cite{s_cover2006}. We\nhave thus shown that\n$\\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}\\ge\\Delta\n\\mutual{\\vw}{\\tlab[\\mu]}$ and hence~\\eqref{eq:first-inequality} is true.\n\nFinally, to prove that\n$\\Delta \\mutual{\\vw}{\\tlab[\\mu]}>\\Delta \\mutual{\\tlab[\\mu]}{\\lab[\\mu]}$, we\nconsider the full probability distribution $p(\\tlabs,\\vw,\\labs)$. From the\nmaster equation, eq.~\\eqref{eq:master}, we can write this distribution as\n\\begin{equation}\n p(\\tlabs, \\vw, \\labs) = p(\\tlabs)p(\\vw|\\tlabs)\\left[p^{(0)}(\\labs|\\vw) +\n \\frac{1}{\\gamma}p^{(1)}(\\labs|\\vw) + \\mathcal{O}(1\/\\gamma^2)\\right]\n\\end{equation}\nwith $\\gamma\\gg1$ for physiological reasons as described in the text -- it takes\nthe neuron longer to learn than to generate an action potential. Hence to first\norder, $\\tlabs\\rightarrow\\vw\\rightarrow\\labs$ is by definition a Markov chain~\\cite{s_cover2006}. Integrating out all the labels, true and predicted, except\nfor the $\\mu$-th one, we have the Markov chain\n$\\tlab[\\mu]\\rightarrow\\vw\\rightarrow\\lab[\\mu]$. For such a Markov chain, it is\neasy to show the following data processing inequality \\cite{s_cover2006},\n\\begin{equation}\n \\Delta \\mutual{\\tlab[\\mu]}{\\vw} \\ge \\Delta \\mutual{\\tlab[\\mu]}{\\lab[\\mu]},\n\\end{equation}\nwhich completes our derivation.\n\n\\section{Hebbian learning in the thermodynamic limit}\n\\label{sec:hebbian-calculations}\n\nIn this section, we provide additional analytical calculations for Hebbian\nlearning in the thermodynamic limit for long times $t\\rightarrow\\infty$.\n\n\n\n\\subsection{Direct integration of the full distribution $p(\\tlabs, \\vw, \\labs)$ }\n\nTo compute the mutual information between the true and predicted label of a\ngiven sample, $\\mutual{\\tlab[\\mu]}{\\lab[\\mu]}$, we need the distribution\n$p(\\tlab[\\mu], \\lab[\\mu])$ or, since both $\\tlab[\\mu]$ and $\\lab[\\mu]$ are\nsymmetric binary random variables, the probability that\n$\\tlab[\\mu] = \\lab[\\mu]$. Our aim in this section is to obtain this probability\nfor Hebbian learning in the thermodynamic limit with $t\\rightarrow\\infty$ by\ndirect integration of the full distribution over the true labels, weights and\npredicted labels for a given set of samples $\\{\\vsample[\\mu]\\}$, which will also\ngive additional motivation for introducing the stability $\\Delta^\\mu$ of a\nsample.\n\nWe start with the full probability distribution\n\\begin{equation}\n p(\\tlabs, \\vw, \\labs) = \\left(\\frac{1}{2}\\right)^P \\left(\\prod_{n=1}^N\n \\frac{e^{-(\\wn-\\lr \\mathcal{F}_n)^2\/2}}{\\sqrt{2\\pi}}\\right)\n \\left(\\prod_{\\mu=1}^P \\frac{e^{\\lab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\\right),\n\\end{equation}\nwhere $\\lr$ is the learning rate and $\\mathcal{F}_n$ is a suitably scaled\naverage over the samples and labels,\n\\begin{equation}\n \\mathcal{F}_n=\\frac{1}{\\sqrt{N}}\\sum_{\\rho=1}^P \\tlab[\\rho]\\sample[\\rho]_n\n\\end{equation}\nWhile the sum over the predicted labels $\\lab[\\rho\\neq\\mu]=\\pm1$ is trivial, we\ncan integrate over the true labels by noting that we can rewrite the exponent as\n\\begin{equation}\n p(\\tlabs, \\vw, \\lab[\\mu]) = \\left(\\frac{1}{2}\\right)^P \\left(\\prod_{n=1}^N\n \\frac{e^{-(\\wn-\\lr \\tlab[\\mu] \\sample[\\mu]_n\/\\sqrt{N}- \\lr \\mathcal{F}^{\\overline{\\mu}}_n)^2\/2}}{\\sqrt{2\\pi}}\\right)\n \\frac{e^{\\lab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\n\\end{equation}\nwhere the only dependence of the weight distribution on the true labels\n$\\tlab[\\rho\\neq\\mu]$ is now confined to the sum\n\\begin{equation}\n \\mathcal{F}^{\\overline{\\mu}}_n\\equiv\\frac{1}{\\sqrt{N}}\\sum_{\\rho\\neq\\mu}^P \\tlab[\\rho]\\sample[\\rho]_n.\n\\end{equation}\nIn the thermodynamic limit, this allows us to replace the sum over all\n$\\tlab[\\mu\\neq\\rho]$ by an integral over the stochastic variable\n$\\mathcal{F}^{\\overline{\\mu}}_n$, which is normally distributed by the central\nlimit theorem and has mean 0 and variance $\\alpha$. Carrying out the integral,\nwe find\n\\begin{equation}\n \\label{eq:3}\n p(\\tlab[\\mu], \\vw, \\lab[\\mu]) = \\left(\\prod_{n=1}^N\n \\frac{e^{-(\\wn-\\lr \\tlab[\\mu] \\sample[\\mu]_n\/\\sqrt{N})^2\/2(1+\\alpha\\lr^2)}}{\\sqrt{2\\pi(1+\\alpha\\lr^2)}}\\right)\n \\frac{e^{\\lab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\n\\end{equation}\nSince both $\\tlab[\\mu]$ and $\\lab[\\mu]$ are binary random variables and\n$\\tlab[\\mu]=\\pm1$ with equal probabilities, the mutual information between the\ntrue and predicted label can be written as\n\\begin{equation}\n \\mutual{\\tlab[\\mu]}{\\lab[\\mu]} = \\ln 2 - S[p(\\tlab[\\mu]=\\lab[\\mu])]\n\\end{equation}\nwith the shorthand for the binary entropy\n$S[p]=-p \\ln p - (1-p)\\ln(1-p)$~\\cite{s_cover2006}. With $\\lab[\\mu]=\\tlab[\\mu]$ in\nthe exponential term of eq.~\\eqref{eq:3} and noting that\n$(\\tlab[\\mu]\\sample[\\mu]_n)^2=1$ for all $\\tlab[\\mu]$, $\\sample[\\mu]_n$, we then\nhave\n\\begin{equation}\n \\label{eq:4}\n p(\\tlab[\\mu] =\\lab[\\mu], \\vw) = \\left(\\prod_{n=1}^N\n \\frac{e^{-(\\wn \\tlab[\\mu] \\sample[\\mu]_n -\\lr\/\\sqrt{N})^2\/2(1+\\alpha\\lr^2)}}{\\sqrt{2\\pi(1+\\alpha\\lr^2)}}\\right)\n \\frac{e^{\\tlab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\n\\end{equation}\nIt thus becomes clear that $\\vw\\cdot\\vsample[\\mu]\\tlab[\\mu]$ is the sum of $N$\nrandom variables with mean $\\lr\/\\sqrt{N}$ and variance $1+\\alpha\\lr^2$. We are\nthen motivated to introduce the stability of a sample,\n\\begin{equation}\n \\Delta^\\mu \\equiv \\frac{1}{\\sqrt{N}} \\vw\\cdot\\vsample[\\mu]\\tlab[\\mu] = \\act^\\mu \\tlab[\\mu].\n\\end{equation}\nwhich, from eq.~\\eqref{eq:4}, is normally distributed with mean $\\lr$ and\nvariance $1+\\alpha\\lr^2$. Introducing the stability allows us to replace the\nintegral over all the weights by an integral over the stability,\n\\begin{equation}\n p(\\tlab[\\mu] = \\lab[\\mu]) = \\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\n \\frac{e^{-(\\Delta^\\mu -\\lr)^2\/2(1+\\alpha\\lr^2)}}{\\sqrt{2\\pi(1+\\alpha\\lr^2)}}\n \\frac{e^{\\Delta^\\mu}}{1+e^{\\Delta^\\mu}} = \\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\n p(\\Delta^\\mu) \\frac{e^{\\Delta^\\mu}}{1+e^{\\Delta^\\mu}}\n\\end{equation}\nwhich is the distribution obtained as eq.~\\eqref{eq:pR} of the main text.\n\n\\subsection{Direct derivation of the distribution of stabilities}\n\n\\label{sec:stabilities}\n\nLet us quickly show how the distribution of stabilities\n\\begin{equation}\n \\label{eq:supp_stability}\n \\Delta^\\mu \\equiv \\frac{1}{\\sqrt{N}} \\vw\\cdot\\vsample[\\mu]\\tlab[\\mu],\n\\end{equation}\n$\\mu=1,\\dots,P$, is obtained directly from its definition. The weights are given\nby\n\\begin{equation}\n \\label{eq:supp_mean-weight}\n \\vw = \\frac{1}{\\sqrt{N}}\\lr \\sum_{\\rho=1}^P \\vsample[\\rho]\\tlab[\\rho] + \\mathbf{y}\n\\end{equation}\nwith $\\mathbf{y}=(y_1,y_2,\\dots,y_N)$ where $y_n$ are normally distributed\nrandom variables with mean 0 and variance 1 arising from the thermal\nfluctuations in equilibrium. Substituting eq. \\eqref{eq:supp_mean-weight} into\n\\eqref{eq:supp_stability}, we have\n\\begin{align}\n \\Delta^\\mu = & \\frac{1}{N} \\lr \\sum_{\\rho=1}^P \\tlab[\\rho]\\tlab[\\mu]\n \\vsample[\\rho]\\cdot\\vsample[\\mu] +\n \\frac{1}{\\sqrt{N}}\\tlab[\\mu]\\vsample[\\mu]\\cdot\\mathbf{y} \\\\\n = & \\lr + \\frac{1}{N}\\lr\\sum_{\\rho\\neq\\mu}^P\\tlab[\\rho]\\tlab[\\mu]\n \\vsample[\\rho]\\cdot\\vsample[\\mu] + \\frac{1}{\\sqrt{N}}\\tlab[\\mu]\\vsample[\\mu]\\cdot\\mathbf{y}\n\\end{align}\nwhere going to the last line we have used the fact that\n$\\vsample[\\mu]\\cdot\\vsample[\\mu]=N$. By inspection, we see that the second term\nis the sum of $N(P-1)\\approx NP$ random numbers $\\pm \\lr \/ N$ and the last term is\nthe sum of $N$ random numbers $y_n\/\\sqrt{N}$. By the central limit theorem,\n$\\Delta^\\mu$ is hence normally distributed with mean $\\overline{\\avg{\\Delta^\\mu}}=\\lr$ and\nvariance\n\\begin{equation}\n \\overline{\\avg{(\\Delta^\\mu)^2}}-\\overline{\\avg{\\Delta^\\mu}}^2 = \\lr^2 + NP \\frac{\\lr^2}{N^2} +\n N\\frac{1}{N}-\\lr^2 = 1+\\alpha\\lr^2.\n\\end{equation}\n\n\\subsection{Analytical approximation for $\\mutual{\\tlab}{\\lab}$}\n\n\\label{sec:approximation}\n\nWe quantify the success of learning using the mutual information per sample,\n\\begin{equation}\n \\mutual{\\tlab[\\mu]}{\\lab[\\mu]} = \\ln 2 - S(p^\\mu_\\textrm{C})\n\\end{equation}\nwhere $S(p)=-[p \\ln p + (1-p)\\ln(1-p)]$ is the binary Shannon entropy and\n$p^\\mu_\\textrm{C}$ is defined as\n\\begin{equation}\n \\label{eq:pC}\n p^\\mu_\\textrm{C}\\equiv p(\\lab[\\mu]=\\tlab[\\mu]) = \\int_{-\\infty}^\\infty \\dd \\! \\Delta^\\mu \\; p(\\Delta^\\mu)\\frac{e^{\\Delta^\\mu}}{e^{\\Delta^\\mu}+1}\n\\end{equation}\nThe stabilities $\\Delta^\\mu$ are normally distributed with mean $\\lr$ and variance\n$1+\\alpha \\lr^2$ (see section \\ref{sec:stabilities}). This integral does not\nhave a closed-form analytical solution, but here we will demonstrate a very good\nanalytical approximation.\n\nTo that end, we first rewrite the sigmoid function in the integrand in terms of\nthe hyperbolic tangent and exploit the similarity of the latter to the error\nfunction:\n\\begin{align}\n \\pC^\\mu = & \\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\\; p(\\Delta^\\mu)\n \\frac{e^{\\Delta^\\mu\/2}}{e^{\\Delta^\\mu\/2}+e^{-\\Delta^\\mu\/2}} \\\\\n = & \\frac{1}{2} + \\frac{1}{2}\\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\\; p(\\Delta^\\mu)\n \\tanh(\\Delta^\\mu \/ 2) \\\\\n \\simeq & \\frac{1}{2} + \\frac{1}{2}\\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\\; p(\\Delta^\\mu)\n \\erf(\\gamma \\Delta^\\mu \/ 2)\n\\end{align}\nwhere we choose $\\gamma=4\/5$ by inspection of the graphs of the two\nfunctions. Now the convolution of a normal distribution and an error function\nhas an exact solution,\n\\begin{equation}\n \\frac{1}{\\sqrt{2\\pi d^2}}\\int_{-\\infty}^\\infty \\dd x \\erf(ax+b)\n \\exp\\left(-\\frac{(x-c)^2}{2d^2}\\right)\n = \\erf\\left(\\frac{b+ac}{\\sqrt{1+2a^2d^2}}\\right).\n\\end{equation}\nSetting $a=\\gamma\/2$, $b=0$, $c=\\lr$ and $d^2=1+\\alpha\\lr^2$, we find that\n\\begin{align}\n \\pC^\\mu(\\alpha, \\lr) \\simeq & \\frac{1}{2} + \\frac{1}{2}\\erf\\frac{\\gamma\\lr\/2}{\\sqrt{1+\\gamma^2(1+\\alpha\n \\lr^2)\/2}}\\\\\n = & \\frac{1}{2} + \\frac{1}{2}\\erf\\frac{\\lr\/2}{\\sqrt{25\/16+1\/2+\\alpha\n \\lr^2\/2}}\\\\ \n \\simeq & \\frac{1}{2} + \\frac{1}{2}\\erf\\frac{\\lr\/2}{\\sqrt{2(1+\\alpha\n \\lr^2\/4)}}\\\\\n = & p(\\Delta^\\mu > 0 | \\alpha, \\lr\/2) \\label{eq:approximation}\n\\end{align}\nwhere in the last line we recognise by inspection that our result is nothing but\nthe integral over the distribution of stabilities $p(\\Delta^\\mu|\\alpha, \\lr\/2)$ from\n0 to $\\infty$. The probability that the neuron predicts the correct label is\nhence given by the probability that the neuron learned the label correctly,\n$\\Delta^\\mu>0$, with \\emph{half the learning rate}.\n\n\\bibliographystyle{aip}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Right Tail}\n\nLet us first describe the essentially inviscid \ninstantons producing the right tails of the PDFs \nfor gradients and differences \\cite{Pol95,95GM,GK96}. \nAt $t=0$, the \nfield $p$ is localized near the origin. At moving backwards in time the \nviscosity will spread the field $p$. Nevertheless, \na positive velocity slope\n``compresses'' the field $p$ so that one can expect that the width of $p$ remains\nmuch smaller than $L$. Then, it is possible to formulate the closed \nsystem of equations for the quantities $a(t)$ and\n$c(t)=-i \\int dx\\,x\\,p(t,x)$ since for narrow $p$ and small $x$ we can put \n$\\int dx'\\chi(x-x')p(t,x')\\to-i\\partial_x\\chi(x)c(t)\\approx2i\\omega^3xc(t)$:\n\\begin{equation}\n\\partial_tc=2ac, \\quad\n\\partial_ta=-a^2+2\\omega^3c.\n\\label{vca} \\end{equation}\nThe instanton is a separatrix \nsolution of (\\ref{vca}). \nThe initial condition \n$a(0)c(0)=n$ by virtue of the energy conservation \ngives $a(0)=\\omega^3c^2(0)\/n=\\omega n^{1\/3}$. \nFor differences, $w=2a(0)\\rho$.\nOne can check that \n$ {\\cal I}_{\\rm extr}=i\\int dt\\, c\\partial_t a\\sim a(0)c(0)=n $ which \nis negligible in comparison with $n\\ln[a(0)]$ so that\n$\\langle(u')^n\\rangle\\sim[a(0)]^n \\sim \\omega^n n^{n\/3}$ \nwhich gives the right cubic tails of the PDFs \n$\\ln{\\cal P}(u')\\sim-(u'\/\\omega)^3$ \\cite{GK96} and\n$\\ln{\\cal P}(w)\\sim-[w\/(\\rho\\omega)]^3$ \n\\cite{Pol95,95GM}. \nOne can show that the width of $p$ is much less than $L$ through the \ntime of evolution $T\\sim n^{-1\/3}\\omega^{-1}$ \ngiving the main contribution into the action \\cite{95GM}. \nThe right tails of ${\\cal P}(u')$ and ${\\cal P}(w)$ are thus\nuniversal i.e. independent of the large-scale properties of the pumping. \nAbove consideration does not imply that the instanton is completely inviscid,\nit may well have viscous shock at $x\\sim L$, this has no influence\non the instanton answer (since $p$ is narrow) while may influence the fluctuation\ncontribution i.e. predexponent in the PDF.\n\n\nThe main subject of this paper is the analysis of the instantons that give\nthe tails of ${\\cal P}(u)$ and the left tails of ${\\cal P}(u')$ \nand ${\\cal P}(w)$ corresponding to negative $a$, $w$. \nEven though the field $p$ is narrow at $t=0$, we cannot use the simple \nsystem (\\ref{vca}) to describe those instantons. \nThe reason is that sweeping by a negative velocity slope provides for \nstretching (rather than compression) of the field $p$ at moving backwards \nin time. As a result, the support of $p(x)$ stretches up to $L$ so that\none has to account for the given form of the pumping correlation function\n$\\chi(x)$ at $x\\simeq L$. This leads to a nonuniversality of ${\\cal P}(u)$ and of \nthe left tails of ${\\cal P}(u')$ and ${\\cal P}(w)$ which depend\non the large-scale properties of the pumping. As we shall see, the form of\nthe tails is universal, nonuniversality is related to a single constant in PDF.\nAdditional complication in analytical description is due to\nthe shock forming from negative slope near the\norigin. The shock cannot be described in terms of the inviscid \nequations so that we should use the complete system\n(\\ref{va2},\\ref{vam}) to describe what can be called\nviscous instantons.\n\nApart from a narrow front near $x=0$, the velocity field \nhas $L$ as the only characteristic scale of change. The life time $T$ of\nthe instanton is then determined by the moment when the position of $p$ \nmaximum reaches $L$ due to sweeping by the velocity \n$u_0$: $T\\sim L\/u_0$. Such a velocity $u_0$ itself has been created during \nthe time $T$ by the forcing so that $u_0\\sim|c|_{max}TL\\omega^3$. \nTo estimate the maximal value of $|c(t)|$, let us consider the backward \nevolution from $t=0$. We first notice that the width of $p$ (which was zero \nat $t=0$) is getting larger than the width of the velocity front $\\simeq u_0\/a$\nalready after the short time $\\simeq a^{-1}$. After that time, the \nvalues of $c$ and $a$ are of order of their values at $t=0$. \nThen, one may consider that $p(t,x)$ propagates (backwards in time) \nin the almost homogeneous velocity field $u_0$ so that \n$$\\partial_t c=-i\\int_{-\\infty}^{\\infty} dx\\, xup_x\\approx 2iu_0\\int_0^\\infty dx\\, p\n\\ .$$ The (approximate) integral\nof motion $i\\int dx\\, p$ can be estimated by it's value at $t=0$ which \nis $n\/2u_0$. Therefore, we get $c_{max}\\simeq nT$ so that\n$T\\simeq n^{-1\/3}\\omega^{-1}$ and $u_0\\simeq L\\omega n^{1\/3}$. \nAt the viscosity-balanced shock, the velocity $u_0$ and the gradient $a$ \nare related by $u_0^2\\simeq\\nu a$ so that $a(0)\\simeq \\omega{\\rm Re}\\,n^{2\/3}$.\n\nLet us briefly describe now the consistent analytic procedure of the derivation of\nthe function $c(t)$ that confirms above estimates. We use the\nCole-Hopf substitution \\cite{Burg} for the velocity $\\partial_x\\Psi=-{u}\\Psi\/{2\\nu}$\nand introduce $P=2\\nu\\partial_xp\/\\Psi$.\nThe saddle-point equations for $\\Psi$ and $P$ \n\\begin{eqnarray} &&\n\\partial_t\\Psi-\\nu\\partial_x^2\\Psi+\\nu F\\Psi=0,\n\\label{ha7} \\\\ &&\n\\partial_t P+\\nu\\partial_x^2P-\\nu FP\n-{2\\nu}\\lambda'(x)\\delta(t)\\Psi^{-1}=0\n\\label{ha3} \\end{eqnarray}\ncontain $F$ determined by $\\partial_xF(t,x)=-{i}\n\\int dx'\\chi(x-x')p(t,x')\/{2\\nu^2}$ and fixed by the condition $F(t,0)=0$. \nWe introduce the evolution operator $\\hat U(t)$\nwhich satisfies the equation $\\partial_t\\hat U=\\hat H\\hat U$ with\n$\\hat H(t)=\\nu(\\partial_x^2-F)$. It is remarkable that one \ncan develop the closed description\nin terms of two operators $\\hat A=\\hat U^{-1} x\\hat U$ and $\\hat B=\n\\hat U^{-1}\\partial_x\\hat U$:\n$$\\partial_t\\hat A=-2\\nu\\hat B\\,,\\quad \\partial_t\\hat B=-\\nu F_x(t,\\hat A)\\ .$$\nSince we are \ninteresting in the time interval when $p(t,x)$ is narrow, it is enough for our\npurpose to consider $x\\ll L$ where $F(t,x)=c(t)x^2\\omega^3\/2\\nu^2$. Further\nsimplification can be achieved in this case and the closed ODE for $c(t)$ can\nbe derived after some manipulations:\n$$(\\partial_t c)^2=4\\omega^3c^3+16\\xi_2^2+4\\omega^3\\xi_1^3\\ ,$$\nwhere $\\xi_1\\!=i\\int\\! dx\\lambda(x) x$ and $4\\xi_2\\!=-{i}\\int\\! dx\\lambda(x)\n\\partial_x[xu(0,x)]$. Integrating we get\n\\begin{eqnarray}&&\nt=\\frac{1}{2}\n\\int_{c(0)}^{c}\\frac{dx}{\\sqrt{\\omega^3x^3+4\\xi_2^2+\\xi_1^3}}\\ ,\n\\label{b7}\\end{eqnarray}\nwhich describes $c(t)$ in an implicit form. Further analysis depends on the\ncase considered. For the gradients, we substitute $\\xi_1=n\/a_0$ and $\\xi_2=-n\/2$\nand see that, as time goes backwards, negative $c(t)$ initially decreases by \nthe law $c(t)=c(0)+2nt$ until $T= \\omega^{-1}(n\/2)^{-1\/3}$ then it grows\nand the approximation looses validity when $c(t)$ approaches zero and the account\nof the pumping form $\\chi(x)$ at $x\\simeq L$ is necessary. Requiring the width\nof $p(x)$ at this time to be of order $L$ we \nget the estimate $a(0)\\simeq \\omega{\\rm Re}\\,n^{2\/3}$ and thus confirm the above\npicture.\nThe main contribution to the saddle-point value (\\ref{vaa})\nis again provided by the term $[\\partial_xu(0,0)]^n$ \nand we find $\\langle(u')^n\\rangle\\simeq[a(0)]^n\\simeq\n(\\omega{\\rm Re})^n n^{2n\/3}$, which corresponds to the following left \ntail of PDF at $u'\\gg \\omega{\\rm Re}$\n\\begin{equation} \n{\\cal P}(u')\\propto \n\\exp[-C(-u'\/\\omega{\\rm Re})^{3\/2}]\\ .\n\\label{an2} \\end{equation}\nFor higher derivatives $u^{(k)}$, by using (\\ref{b7}) we get initial growth \n$c(t)=c(0)+n(k+1)t$ which gives\n$u^{(k)}(0,0)\\sim N^{k+1}L^{1-k}\\omega {\\rm Re}^{k}$ leading to\n$\\langle[u^{(k)}]^n\\rangle\\sim\\omega {\\rm Re}^{k} \nL^{1-k} n^{(k+1)\/3}$ which can be rewritten in terms of PDF: \n\\begin{eqnarray} &&\n{\\cal P}\\left(|u^{(k)}|\\right)\\!\\propto\\!\n\\exp\\left[-C_k\\left({|u^{(k)}|L^{k-1}}\/\n{\\omega{\\rm Re}^k}\\right)^{3\/(k+1)}\\right].\n\\label{hd4}\\end{eqnarray}\nNote that the non-Gaussianity increases with increasing $k$. On the other hand,\nthe higher $k$ the more distant is the validity region of (\\ref{hd4}): \n$u^{(k)}\\gg u^{(k)}_{\\rm rms}\\sim L^{1-k}\\omega{\\rm Re}^k$.\n\nFor the differences, $\\xi_1={2n\\rho_0}\/{w}$ and\n$4\\xi_2=-{n}[1+{2\\rho_0u_x(0,\\rho_0)}\/{w}]$ and we get \n$\\langle w^n\\rangle\\simeq (L\\omega)^nn^{n\/3}$\nwhich corresponds to the cubic left tail \n\\begin{equation} {\\cal P}(w)\\propto \\exp\\{-B[w\/(L\\omega)]^3\\}\n\\label{an3} \\end{equation}\nvalid at $w\\gg L\\omega$. In the intermediate region $L\\omega\\gg w\\gg\\rho\\omega$,\nthere should be a power asymptotics which is the subject of current debate\n\\cite{Pol95,GK96,KS96}.\nIt is natural that $\\rho$-dependence of ${\\cal P}(w)$ cannot be found in a \nsaddle-point approximation; as a predexponent, it can be obtained only at the\nnext step by calculating the contribution of fluctuations around the instanton\nsolution. This is consistent with the known fact that the scaling exponent\nis $n$-independent for $n>1$: $\\langle w^n(\\rho)\\rangle\\propto\\rho$.\n\nFor the velocity, $\\lambda(x)=-{in}\\delta(x)\/u(0,0)$ is an even function\nso that $F$ is a linear (rather than quadratic)\nfunction of $x$ for narrow $p$: $F(x)={\\chi(0)bx}\/{2\\nu^2}$ with\n$b=-{i}\\int dx p(x)$. Direct calculation shows that energy and momentum\nconservation makes $b$ time independent: $b=n\/u(0,0)$.\nIt is easy then to get the $n$-dependence of $u(0,0)$: \nVelocity stretches the field $p$ so that the width of $p$\nreaches $L$ at $T\\simeq L\/u(0,0)$ while the velocity itself is produced by\nthe pumping during the same time: $u(0,0)\\simeq \\chi(0)bT=\\chi(0)nT\/2u(0,0)\n\\simeq n\\chi(0)L\/u(0,0)$. That gives $u(0,0)\\simeq L\\omega n^{1\/3}$ and\n$$ {\\cal P}(u)\\propto \\exp\\{-D[u\/(L\\omega)]^3\\}\\ .$$ \nThe product $L\\omega$ plays the role of the root-mean square velocity\n$u_{\\rm rms}$. The numerical factors $C$, $B$ and $D$ \nare determined by the evolution at $t\\simeq T$ i.e. by the behavior of pumping\ncorrelation function $\\chi(x)$ at $x\\simeq L$.\n\n\nWe thus found the main exponential factors in the PDF tails. \nComplete description of the tails requires the analysis\nof the fluctuations around the instanton which will be the subject of\nfuture detailed publications. Here, we briefly outline some important\nsteps of this analysis. The account of the fluctuations in the Gaussian\napproximation is straightforward and leads to the shift of ${\\cal I}_{\\rm extr}$\ninsignificant at $n\\gg1$. However, the terms of the perturbation theory with \nrespect to the interaction of fluctuations are infrared divergent (proportional\nto the total observation time). That means that there is a soft mode which\nis to be taken into account exactly. Such an approach has been already developed\nin \\cite{95FKLM} for the simpler problem of the PDF tails for a passive scalar\nadvected by a large-scale velocity where the comparison with the exactly\nsolvable limits was possible. \nA soft mode usually corresponds to a global symmetry\nwith a continuous group: if one allows the slow spatio-temporal\nvariations of the parameters of the transformation then small variations of\nthe action appears. \nOur instantons break Galilean invariance so that the respective\nGoldstone mode has to be taken into account. Namely, under the transformation\n\\begin{equation}x\\rightarrow x-r,\\ u(x)\\rightarrow u(x-r)+v,\\ \nr=\\int_t^0v(\\tau)d\\tau\\ ,\\label{sym}\\end{equation}\nthe action is transformed as ${\\cal I}\\rightarrow{\\cal I}-i\\int dxdtp\\partial_tv$.\nThe source term $\\int dxdt\\lambda u$ is invariant with respect to (\\ref{sym}) for\nantisymmetric $\\lambda(x)$. To integrate exactly along the direction specified by\n(\\ref{sym}) in the functional space we use Faddeev-Popov trick \ninserting \nthe additional factor \n\\begin{equation}\n1=\\int{\\cal D}v(t)\\delta\\left[u\\biggl(t,\\int_t^0v(\\tau)d\\tau\\biggr)-\nv(t)\\right] {\\cal J}\\ .\n\\label{unity}\\end{equation}\ninto the integrand in (\\ref{si1},\\ref{sio}).\nJacobian ${\\cal J}$ is determined by a regularization of (\\ref{sym})\naccording to our choice of the retarded regularization for the initial\nintegral: at \ndiscretizing time we put $\\partial_tu+u\\partial_xu\n\\rightarrow ({u_n-u_{n-1}})\/{\\epsilon}+u_{n-1} u'_{n-1}$ (otherwise, \nsome additional $u$-dependent term appears \\cite{DP78}). \nThe discrete version of (\\ref{sym}),\n$p_n(x)\\rightarrow p_n(x-\\epsilon\\sum_{j=n}^{N-1}v_j)$,\n$u_n(x)\\rightarrow u_n(x-\\epsilon\\sum_{j=n}^{N-1}v_j)-v_n$, $u_N(x)\\rightarrow\nu_N(x)-v_N$ gives \n$${\\cal J}=\\exp\\left[\\int_{-T}^0dtu'\\biggl(t,\\int_t^0 v(\\tau)d\\tau\\biggr)\\right]\\ .$$\nSubstituting (\\ref{unity}) into (\\ref{si1},\\ref{sio})\nand making (\\ref{sym}) we calculate $\\int{\\cal D}v$ as a Fourier integral \n(the saddle-point method is evidently\ninapplicable to such an integration) and conclude that after the integration\nover the mode (\\ref{sym}) the measure ${\\cal D}u{\\cal D}pe^{i{\\cal I}}$\nacquires the additional factor\n$$\\prod\\limits_t\\!\\delta\\biggl[\\int\\partial_t^2 p(t,x)dx\\biggr]\n\\delta[u(t,0)]\\exp\\left[\n\\int_{-T}^0\\!u'(t,0)dt\\right].$$\nThe last (jacobian) term here exactly corresponds\nto the term $u'{\\cal P}(u')$ in the equation for ${\\cal P}(u')$ \nderived in \\cite{GK93,GK96}. This term \nmakes the perturbation theory for the fluctuations around the instanton to be\nfree from infrared \ndivergences, the details will be published elsewhere. \n\n\nLet us summarize.\nAt smooth almost inviscid ramps, velocity differences and gradients are \npositive and linearly related $w(\\rho)\\approx \n2\\rho u'$ so that the right tails of PDFs have the same cubic form \n\\cite{Pol95,95GM,GK96}.\nThose tails are universal i.e. they are determined by a single characteristics\nof the pumping correlation function $\\chi(r)$, namely, by it's second\nderivative at zero $\\omega=[-(1\/2)\\chi''(0)]^{1\/3}$. \nContrary, the left tails found here contain nonuniversal constant which \ndepends on a large-scale behavior of the pumping. The left tails\ncome from shock fronts where $w^2\\simeq -\\nu u'$ so that cubic tail \nfor velocity differences (\\ref{an3}) corresponds to semi-cubic tail for gradients\n(\\ref{an2}).\nThe formula (\\ref{an3}) is valid for $w\\gg u_{\\rm rms}\\simeq L\\omega$\nwhere ${\\cal P}(w)$ should coincide with \na single-point ${\\cal P}(u)$ since the \nprobability is small for both $u(\\rho)$ and $u(-\\rho)$ being large \nsimultaneously. Indeed, we saw that the tails of\n$\\ln{\\cal P}(u)$ at $u\\gg u_{\\rm rms}$ are cubic as well.\nNote that \n(\\ref{hd4}) is the same as obtained for decaying turbulence with white (in space)\ninitial conditions\nby a similar method employing the saddle-point approximation in the \npath integral with time as large parameter \\cite{Avel}. \nThat, probably, means that white-in-time forcing\ncorresponds to white-in-space initial conditions. Note that\nif the pumping has a finite correlation time $\\tau$ then our\nresults, strictly speaking, are valid for \n$u,w\\ll L\/\\tau$ and $u'\\ll1\/\\tau$.\n\nWe are grateful to M. Chertkov, V. Gurarie, D. Khmelnitskii, R. Kraichnan \nand A. Polyakov for useful discussions. \nThis work was supported \nby the Minerva Center for Nonlinear Physics (I. K. and V. L.), \nby the Minerva Einstein Center (V. L.), by the Israel Science Foundation\nfounded by the Israel Academy (E.B.) and by \nthe Cemach and Anna Oiserman Research Fund (G.F.).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhlgh b/data_all_eng_slimpj/shuffled/split2/finalzzhlgh new file mode 100644 index 0000000000000000000000000000000000000000..34544fde92f13259b37627bfea9086c78c29c5db --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhlgh @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn the era of information explosion, the recommender system has\nbecome one of the most effective ways to help users to discover\nwhat they are interested in enormous data. Generally speaking, the recommender systems usually follow two steps: learn vectorized representations (aka. embeddings) of users and items and then model interactions among them (e.g., whether a user buy an item). Collaborative filtering (CF) learns node embedding based on the historical interactions on user-item bipartite graph and performs item recommendation based on the parameters.\n\nAs a matter of fact, there exist diverse relations among various types of nodes (e.g., buy relation and social relation) in real-world recommendation scenario, also widely known as heterogeneous graph \\cite{17tkde_shi}. Taking the dataset Movielens as an example, it contains three types\nof nodes include movie, user and genre. \nMeta-path \\cite{11vldb_pathsim}, a composite relation connecting two objects, is a\nwidely used structure to capture the semantics. \nThe semantics revealed by different meta-paths are able to describe the characteristics of nodes from different aspects. For example, meta-path User-Movie (U-M) describes the preference of user, while meta-path User-User (U-U) describes social influence among users. \nBesides basic meta-path, multi-hop meta-path (e.g., U-U-U) which captures high-order semantics and enrich the connections among users is able to improve the node embedding and alleviate the cold-start problem.\n\nBased on the above analysis, when designing heterogeneous graph neural network for recommendation, we need to address the following requirements.\n\\begin{itemize}\n\\item \\textbf{Heterogeneity of graph.} The heterogeneity is an intrinsic property of heterogeneous graph, i.e., various types of nodes and edges. How to handle such complex structural information for recommendation is an urgent problem that needs to be solved.\n\\item \\textbf{High-order semantic preservation. } High-order semantic information which captures diverse long-term dependencies among nodes plays the key role in improving node embedding and alleviating the cold-start problem in recommender system. How to inject high-order semantic into node embedding is a fundamental problem in recommender system.\n\\item \\textbf{Rich semantics fusion.} Different meaningful and complex semantic information are involved in heterogeneous graph, which are usually reflected by diverse meta-paths. For example, meta-path U-M and U-U can describe the preference and social influence of user and then comprehensively describe the characteristics of user from different aspects. How to select the most meaningful meta-paths and fuse rich semantics to improve node embedding is an open problem.\n\\end{itemize}\nIn this paper, we propose \\textbf{H}eterogeneous\n\\textbf{G}raph neural network for \\textbf{Rec}ommendation, named HGRec, which mainly considers high-order semantic preservation and rich semantics fusion. Specifically, \nsemantic aggregation layer injects high-order semantic into node embedding via multi-hop meta-path and semantic fusion layer fuse rich semantics revealed by multiple meta-paths. \nAfter that, the overall model can be optimized via back propagation in an end-to-end manner.\n\nThe contributions of our work are summarized as follows:\n\\begin{itemize}\n\\item We highlight the critical importance of rich high-order semantics in improving node embedding for recommendation system.\n\\item We propose a heterogeneous graph neural network based recommendation system, which explicitly injects high-order semantic into node embedding via multi-hops meta-path and fuses rich semantics via multiple meta-paths for comprehensive node embedding.\n\\item Empirical studies on real-world heterogeneous graphs \n demonstrate the state-of-the-art performance of HGRec and potentially\ngood interpretability for the recommendation results.\n\n\\end{itemize}\n\\section{METHODOLOGY}\nIn this section, we present the proposed model \\textbf{H}eterogeneous\n\\textbf{G}raph neural network for \\textbf{Rec}ommendation (HGRec). The basic idea of HGRec is to learn representative node\nembedding of users and items by injecting and fusing high-order semantics. The proposed HGRec first adopts embedding layer to initialize node embedding. Then, semantic aggregation layer and semantic fusion layer will inject high-order semantic into node embedding via multi-hops meta-path and fuses rich semantics via multiple meta-paths, respectively. Lastly, we leverage the fused embedding of user and item for recommendation.\n\\subsection{Embedding Initialization}\nFollowing the previous works \\cite{he2017neural,wang2019neural}, we random initialize node embedding matrix and use look-up to get the initial embedding of user $u$ and item $i$, denoted as $\\mathbf{e}_u\\in \\mathbb{R}^d$ and $\\mathbf{e}_i \\in \\mathbb{R}^d $, respectively. Here $d$ is the dimension of node embedding.\n\n\\subsection{Semantic Aggregation Layer}\nAfter obtain initial node embedding, we propose semantic aggregating layer to aggregate multi-hops meta-path based neighbors and update node embedding, so the high-order semantic information is well preserved. For clearly, we first introduce the first-order aggregation in semantic aggregation layer and then generalize it to multiple\nsuccessive layers (aka. high-order semantic aggregation).\n\n\\textbf{First-order Semantic Aggregation}\nTaking one user $u$ and one user-related meta-path $\\Phi^U$ as an example, \nwe propose semantic aggregation layer $\\mathcal{A}$ to aggregate meta-path based neighbors $\\mathcal{N}^{\\Phi^U}_u$ and get the first-order user embedding $\\mathbf{e}_{u }^{\\Phi^U, 1}$, shown as follows:\n\\begin{equation}\n \\mathbf{e}_{u }^{\\Phi^U, 1}=\\mathcal{A}(u, \\Phi^U).\n\\end{equation}\nRather than simple neighbor combination, we consider the complex interaction between node and its neighbors in aggregating process. Specifically, we encode the interaction between node $u$ and its neighbor $k$ into aggregating process via $\\mathbf{e}_{k} \\odot \\mathbf{e}_{u}$, where $\\odot$ denotes the element-wise product. The overall aggregating process is shown as follows:\n\n\\begin{equation}\n\\mathbf{e}_{u }^{\\Phi^U, 1}= \\mathbf{W}_{1}^{\\Phi^U} \\mathbf{e}_{u} + \\sum_{k\\in \\mathcal{N}^{\\Phi^U}_u}\\left(\\mathbf{W}_{1}^{\\Phi^U} \\mathbf{e}_{k}+\\mathbf{W}_{2}^{\\Phi^U}\\left(\\mathbf{e}_{k} \\odot \\mathbf{e}_{u}\\right)\\right) ,\n\\end{equation}\nwhere \n$\\mathbf{W}_{1}^{\\Phi^U},\\mathbf{W}_{2}^{\\Phi^U}$ are weight matrixes.\nThe first-order semantic aggregation only aggregates one-hop meta-path based neighbors into node embedding, while high-order semantic revealed by multi-hops meta-path plays a crucial role in improving node embedding.\n\n\\textbf{High-order Semantic Aggregation} Considering the high-order semantic revealed by multi-hops meta-path, we stack first-order semantic aggregation for multiple layers and recurrently aggregate corresponding meta-path based neighbors, \nso the high-order semantic is injected into node embedding, shown as follows:\n\\begin{equation}\n\\mathbf{e}_{u }^{\\Phi^U, L}=\\mathcal{A}^L(\\cdots \\mathcal{A}^2(\\mathcal{A}^1(u, \\Phi^U))),\n\\end{equation}\nwhere $\\mathbf{e}_{u }^{\\Phi^U, L}$ denotes the $L$-order user embedding. \nThen, we concatenate different order user embedding and get the semantic-specific embedding of user $u$, shown as follows:\n\\begin{equation}\n \\mathbf{e}_{u }^{\\Phi^U}=\\mathbf{e}_{u }^{\\Phi^U, 1}||\\mathbf{e}_{u }^{\\Phi^U, 2}||,\\cdots, ||\\mathbf{e}_{u }^{\\Phi^U, L},\n\\end{equation}\nwhere $||$ is the concatenation operation. However, one meta-path cannot comprehensively describe the characteristics of node from different aspects. Considering a set of user-related meta-paths $\\{ \\Phi_1^U, \\Phi_2^U, \\cdots, \\Phi_{K_1}^U\\}$, we can get $K_1$ groups of user embeddings $ \\{\\mathbf{E}_{u }^{\\Phi_1^U},\\mathbf{E}_{u }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{u }^{\\Phi_{K_1}^U} \\}$.\n\nSimilar to user embedding, given a set of item-related meta-paths $\\{ \\Phi_1^I, \\Phi_2^I, \\cdots, \\Phi_{K_2}^I\\}$, we can get $K_2$ groups of item embeddings $\\{\\mathbf{E}_{i }^{\\Phi_1^I},\\mathbf{E}_{i }^{\\Phi_2^I},\\cdots,\\mathbf{E}_{i }^{\\Phi_{K_2}^I} \\}$.\n\n\\subsection{Semantic Fusion Layer}\nAfter obtaining multiple higher-order node embedding, we need to learn the importance of different meta-paths and fuse them properly for better recommendation. Given $K_1$ groups of user embeddings $ \\{\\mathbf{E}_{u }^{\\Phi_1^U},\\mathbf{E}_{u }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{u }^{\\Phi_{K_1}^U} \\}$, we propose semantic fusion layer $\\mathcal{F}$ to learn the weights of different meta-paths (e.g., $w^{\\Phi_1^U}, w^{\\Phi_2^U}, \\cdots, w^{\\Phi_{K_1}^U}$), shown as follows:\n\n\\begin{equation}\n (w^{\\Phi_1^U}, w^{\\Phi_2^U}, \\cdots, w^{\\Phi_{K_1}^U}) = \\mathcal{F}(\\mathbf{E}_{u }^{\\Phi_1^U},\\mathbf{E}_{u }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{u }^{\\Phi_{K_1}^U}).\n\\end{equation}\nTo learn the importance of each meta-path (e.g., $\\alpha_{\\Phi _{k}^U}$), we first project node embedding into the attention space and then use a semantic attention vector $\\mathbf{q}_{U}$ to measure the importance of meta-path specific embedding, \n\\begin{equation}\n \\alpha_{\\Phi _{k}^U}=\\frac{1}{\\left |V \\right |}\\sum_{i\\in V }\\mathbf{q}_U^{\\top}\\cdot \\tanh\\left (\\mathbf{W}_U\\cdot \\mathbf{e}_{u}^{\\Phi^U_k }+\\mathbf{b}_U\\right ),\n\\end{equation}\nwhere $\\mathbf{W}_U$ and $\\mathbf{b}_U$ are weight and bias, respectively.\nThen, we normalize them via softmax function and get meta-path weights $w_{\\Phi _{k}^U}$, shown as follows:\n\\begin{equation}\n w_{\\Phi _{k}^U}=\\frac{\\exp\\left ( w_{\\Phi _{k}^U} \\right )}{\\sum_{k=1}^{K_1}\\exp\\left ( w_{\\Phi _{k}^U} \\right )}.\n\\end{equation}\nWith the learned weights as coefficients, we\ncan fuse multiple user embeddings to obtain the final\nembedding $\\mathbf{E}_u$ as follows:\n\n\\begin{equation}\n \\mathbf{E}_u=\\sum_{k=1}^{K_1} w_{\\Phi _{k}^U}\\cdot \\mathbf{E}_u^{\\Phi _{k}^U}.\n\\end{equation}\n\nSimilar to user embedding, we can fuse $K_2$ groups of item embeddings $\\{\\mathbf{E}_{i }^{\\Phi_1^U},\\mathbf{E}_{i }^{\\Phi_2^U},\\cdots,\\mathbf{E}_{i }^{\\Phi_{K_2}^U} \\}$ and obtain the final\nembedding of item $\\mathbf{E}_i$.\n\n\\subsection{Model Prediction}\nThe final part of the model is to recommend items for users based on their embedding. Here we calculate the inner product of user and item for recommendation, as follows:\n\n\\begin{equation}\n \\hat{y}_{ui}=(\\mathbf{E}_{u })^\\top \\mathbf{E}_{i}.\n\\end{equation}\nThen, we calculate BPR loss \\cite{wang2019neural} and optimize the parameters, as follows:\n\\begin{equation}\n L=\\sum_{\\left ( u,i,j \\right )\\in \\mathcal{O}}-\\ln\\sigma \\left ( \\hat{y}_{ui}- \\hat{y}_{uj} \\right )+\\lambda \\left \\| \\Theta \\right \\|_{2}^{2},\n\\end{equation}\nwhere $\\mathcal{O}=\\left\\{(u, i, j) |(u, i) \\in \\mathcal{R}^{+},(u, j) \\in \\mathcal{R}^{-}\\right\\}$ denotes the pairwise\ntraining data,$ \\mathcal{R}^{+}$ indicates the observed interactions, $\\mathcal{R}^{-}$ is\nthe unobserved interactions, $\\Theta $ denotes all trainable model parameters, and $\\lambda $\ncontrols the L2 regularization strength to prevent overfitting.\n\n\\section{EXPERIMENTS}\n\n\nWe conduct experiments on three heterogeneous graphs: Amazon, Yelp and Movielens (details are shown in Table \\ref{dataset}).\n\\begin{table}[]\n\\caption{Statistics of the datasets.} \n\\label{dataset}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\nDatasets & Relation(A-B) & \\#A & \\#B & \\#A-B \\\\ \\hline\n\\multirow{3}{*}{Movielens} & User - Movie & 943 & 1,682 & 100,000 \\\\ \\cline{2-5} \n & User - User & 943 & 943 & 47,150 \\\\ \\cline{2-5} \n & Movie - Genre & 1,682 & 18 & 2,861 \\\\ \\hline\n\\multirow{4}{*}{Amazon} & User - Item & 6,170 & 2,753 & 195,791 \\\\ \\cline{2-5} \n & Item - Cate. & 2,753 & 22 & 5,508 \\\\ \\cline{2-5} \n & Item - Brand & 2,753 & 334 & 2,753 \\\\ \\hline\n\\multirow{4}{*}{Yelp} & User - Item & 16,239 & 14,284 & 198,397 \\\\ \\cline{2-5} \n &User - User & 16,239 & 16,239 & 158,590 \\\\ \\cline{2-5} \n & Item - City & 14,284 & 47 & 14,267 \\\\ \\hline \n\\end{tabular}\n\\end{table}\nWe compare with some state-of-art baselines, include BPRMF \\cite{bpr}, NMF \\cite{he2017neural}, GAT \\cite{18iclr_gat}, MCRec \\cite{hu2018leveraging}, NGCF \\cite{wang2019neural}, to verify the effectiveness of the proposed model. We also test a variant of HGRec, denotes as HGRec-, which assigns the same importance to each meta-path.\n\n\nFor evaluation, we split datasets into training set and test set with 8:2 ratio and employ Pre@10, Recall@10, HR@10 and NDCG@10 as evaluation metrics. \n\nWe randomly initialize parameters and optimize models with Adam. For the proposed HGRec, we set the L2 regularization to 1e-2, the dimension of the semantic attention vector $\\mathbf{q}$ to 64, the dropout to 0.8, and \nthe learning rate to 5e-4, 1e-3 and 5e-3 on Movielens Amazon and Yelp, respectively.\nWe also use early stopping with a patience of 100 to aviod overfitting.\n\n\\subsection{Overall Performance Analysis}\nThe experiment results are shown in Table \\ref{perform} and we have the following observations:.\n\n\n\\begin{table}[]\n\\caption{Overall Performance Comparison }\n\\label{perform}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{Movielens} \\\\ \\cline{2-5} \n & Pre@10 & Rec@10 & NDCG@10 & HR@10 \\\\ \\hline\nBMF & 0.3251 & 0.2096 & 0.4081 & 0.8928 \\\\ \\hline\nNMF & 0.1704 & 0.1163 & 0.2336 & 0.7739 \\\\ \\hline\nGAT & 0.2068 & 0.1210 & 0.2556 & 0.7548 \\\\ \\hline\nMCRec & 0.3310 & 0.2129 & 0.2624 & 0.9025 \\\\ \\hline\nNGCF & 0.3369 & 0.2179 & 0.4178 & 0.9045 \\\\ \\hline\nHGRec- & \\textbf{0.3670} & \\textbf{0.2412} & \\textbf{0.4551} & 0.9172 \\\\ \\hline\nHGRec & 0.3667 & 0.2405 & 0.4547 & \\textbf{0.9193} \\\\ \\hline\nImprov. & 6.70$\\%$ &6.80$\\%$ & 8.19$\\%$ & 1.36$\\%$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{Amazon} \\\\ \\cline{2-5} \n & Pre@10 & Rec@10 & NDCG@10 & HR@10 \\\\ \\hline\nBMF & 0.0490 & 0.0881 & 0.1176 & 0.3232 \\\\ \\hline\nNMF & 0.0168 & 0.0264 & 0.0463 & 0.1371 \\\\ \\hline\nGAT & 0.0410 & 0.0810 & 0.1096 & 0.2998 \\\\ \\hline\nMCRec & 0.0309 & 0.0697 & 0.1131 & 0.3027 \\\\ \\hline\nNGCF & 0.0495 & 0.0870 & 0.1150 &0.3224 \\\\ \\hline\nHGRec- & 0.0553 & 0.0988 & 0.1313 & 0.3503 \\\\ \\hline\nHGRec & \\textbf{0.0588} & \\textbf{0.1054} & \\textbf{0.1384} & \\textbf{0.3746} \\\\ \\hline\nImprov. &5.95$\\%$ &6.26$\\%$ &5.13$\\%$ & 6.48$\\%$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Models} & \\multicolumn{4}{c|}{Yelp} \\\\ \\cline{2-5} \n & Pre@10 & Rec@10 & NDCG@10 & HR@10 \\\\ \\hline\nBMF & 0.0039 & 0.0287 & 0.0150 & 0.0291 \\\\ \\hline\nNMF &0.0012 & 0.0265 & 0.0233 & 0.0398 \\\\ \\hline\nGAT &0.0038 & 0.0240 & 0.0171 & 0.0363 \\\\ \\hline\nMCRec & 0.0031 & 0.0531 & 0.0201 & 0.0432 \\\\ \\hline\nNGCF & 0.0073 & 0.0410 & 0.0271 &0.0667 \\\\ \\hline\nHGRec- & 0.0076 & 0.0433 & 0.0237 & 0.0506 \\\\ \\hline\nHGRec & \\textbf{0.0078} & \\textbf{0.0447} & \\textbf{0.0310} & \\textbf{0.0671} \\\\ \\hline\nImprov. &6.41$\\%$ &3.13$\\%$ &12.6$\\%$ &1.03$\\%$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\textbullet ~The proposed HGRec consistently performances better than baselines with significant gap on all the datasets. In particular, HGRec improves over the strongest baseline NGCF w.r.t.\nRecall@10 by 6.80\\%, 17.49\\%,3.13\\% in Movielens\nAmazon and Yelp, respectively. The results demonstrate that injecting rich high-order semantics into the node embedding indeed improves the recommendation performance.\n\n\\textbullet ~Compare HGRec with HGRec- , we can observe that HGRec outperforms HGRec- on all datasets. This proves that the semantic fusion layer is able to identify the importance of meta-paths and then enhance the performance of HGRec.\n\n\\textbullet ~Graph neural network based recommendation models show their superiorities over traditional MF based models, demonstrating the importance of nonlinear structural\ninteractions among nodes.\n\\begin{table}[]\n\\caption{Effectiveness of Layer Number}\n\\label{movie_depth}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{L} & \\multicolumn{2}{c|}{Movielens} & \\multicolumn{2}{c|}{Amazon} \\\\\\cline{2-5} \n & Rec@10 & NDCG@10 & Rec@10 & NDCG@10 \\\\\\hline\n1 & 0.2390 & 0.4506 & 0.0947 & 0.1251 \\\\\\hline\n2 & 0.2391 & 0.4526 & 0.0864 & 0.1151 \\\\\\hline\n3 & \\textbf{0.2405} & \\textbf{0.4547} & \\textbf{0.1054} & \\textbf{0.1384} \\\\\\hline\n 4 & 0.2391 & 0.4513 & 0.0743 & 0.1064 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\useunder{\\uline}{\\ul}{}\n\n\n\\textbf{Effect of Layer Numbers.} \nTo investigate the whether high-order semantic improves node embedding, we vary the model depth (e.g., $L=1,2,3,4$) and show the results on Table \\ref{movie_depth}. We can find that with the growth of model depth, the performance of HGRec are sustainable growth and achieves the best performance\nwhen $L$ is set to 3, indicating the effectiveness of high-order semantic. After that, the performance of\nHGRec starts to degenerate which may because of overfitting.\n\n\n\n\n\n\n\n\\section{Conclusion and Future Work}\nIn this work, we highlight the critical importance of rich high-order semantics in improving node embedding for better recommendation. Specifically, we design a semantic aggregation layer which aggregates multi-hop meta-path neighbors so as to inject high-order semantic into node embedding. To describe the characteristics of node comprehensively, we leverage a semantic fusion layer to fuse rich semantic revealed by multiple meta-paths. Experimental results demonstrates the superiority of the proposed model and show the potentially good interpretability for the recommendation results.\n\n\n\n\n\n\\nocite{langley00}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\n\nThermally driven turbulent flows widely occur in geophysical flows and industrial processes. Examples are thermal convection in the atmospheric and mantle convection \\cite{mck1974,Wyngaard1992}, in the ocean \\cite{cheng2019fast}, and in many industrial processes \\cite{Bejan}. Rayleigh--B\\'enard convection (RBC), a fluid layer heated from below and cooled from above, is an ideal model for the study of thermally driven turbulent flows \\cite{ahl09,loh10,chi12}. \nThe main challenge for thermal turbulence studies is to explore the flow dynamics and heat transfer of the system in a wide range of the control parameters. The main attention has been on how the heat transfer depends on the Rayleigh number, which is the dimensionless temperature difference and measures the intensity of the thermal driving of the system. The Rayleigh number is defined as $\\text{Ra}=\\frac{\\beta g \\Delta L^3 }{\\nu \\kappa}$, where $g$ is the gravitational acceleration, $\\beta$ is the isobaric thermal expansion coefficient, $\\nu$ the kinematic viscosity, $\\kappa$ is the thermal diffusivity, $\\Delta$ is the temperature difference between the hot plate and the cold plate, and $L$ is the thickness of the fluid layer between the aforementioned plates. \n\nTo achieve high Rayleigh numbers, numerous strategies have been proposed in the past decades. Looking at the definition of $\\text{Ra}$, one can see that large $\\text{Ra}$ can be reached in several ways. \nOne approach is to use a working fluid for which the parameter $\\frac{\\beta}{{\\nu}\\kappa}$ has a large value. This approach has been widely used in the community, such as using helium gas at cryogenic temperatures \\cite{cas89,cha97,nie00,Urban2010} or mercury \\cite{san89}. Recently, it was found that pure gas, particularly gasses with high molecular weight and under high pressure (up to 19-bar), is also an effective way to push $\\text{Ra}$ to high values at a constant $\\text{Pr}$ number \\cite{fun09,ahl09b}. Another commonly used approach in the field of turbulent thermal convection is to increase the system thickness $L$ \\cite{nie00,fun05,nik05,sun05e,pui07,fun09,ahl09b,Urban2010}. \n\nThe fact that hitherto not much attention was on the changing the gravitational acceleration provides the motivation of the present work. In this study, we propose a novel system, annular centrifugal RRC (ACRBC), which is a cylindrical annulus with cooled inner and heated outer walls under a solid body rotation. \nIn this system, the buoyancy force can be efficiently enhanced by replacing the gravitational acceleration ($g$) by the centrifugal acceleration, and consequently, $\\text{Ra}$ can be increased by increasing the rotation rate for a given fluid and temperature difference. \nAnother advantage of this system is that the thermal convection in rapidly rotating cylindrical annulus has been recognized as a good model for the study of the flows in planetary cores and stellar interiors \\cite{hid58,bus74,hid75,bus76,aue95,kan19} and also in engineering applications under strong rotation, \\textit{i.e.}~flow in gas turbines \\cite{mar05,mic15,pit17}. The flow dynamics in this system can give insights into flows in geophysical context and flows in industrial processes. In this work, we aim to study the heat transfer and the turbulent structures in ACRBC.\n\n\n\n\n\\section{Governing equations, simulations, and experiments}\n\\subsection{Governing equations and numerical simulations}\n\nUsing the parameter definitions as shown in Fig.~\\ref{fig:setup}(b), the governing dimensionless Boussinesq equations in a rotating reference frame \\cite{lop13} can be expressed as\n\\begin{small}\n\\begin{align}\n \\nabla \\cdot {\\bf u} &= 0 \\\\\n\\frac{\\partial \\theta}{\\partial t} +{\\bf u}\\cdot \\nabla \\theta &= \\frac{1}{\\sqrt{RaPr}}\\nabla^2 \\theta \\\\\n\\frac{\\partial \\bf{u}}{\\partial t} +{\\bf u}\\cdot \\nabla {\\bf u} &= -\\nabla p+\\text{Ro}^{-1} \\hat{\\bm{\\omega}} \\times {\\bf u} \\nonumber \\\\ \n&\\phantom{=} +\\sqrt{\\frac{Pr}{Ra}}\\nabla^2 {\\bf u} -\\theta \\frac{2(1-\\eta)}{(1+\\eta)}\\bf r\n\\end{align}\n\\end{small}\n\\noindent where $\\bm{\\hat{\\omega}}$ is the unit vector pointing in the direction of the angular velocity, $\\bf u$ is the velocity vector normalized by the free fall velocity $\\sqrt{\\omega ^2\\frac{R_o+R_i}{2}\\beta \\Delta L}$, $t$ is the dimensionless time normalized by $\\sqrt{L\/(\\omega ^2\\frac{R_o+R_i}{2}\\beta\\Delta)}$, and $\\theta$ is the temperature normalized by $\\Delta = T_h - T_c$. Here, $\\omega$, $R_i$, $R_o$, and $\\Delta$ are the rotational speed, the thickness of the fluid layer between the two cylinders, the outer radius of the inner cylinder, the inner radius of the outer cylinder, and the temperature difference between the hot ($T_h$) and cold ($T_c$) cylinders, respectively.\n\n\nFrom the above governing equations, the relevant control parameters in ACRBC are the Rayleigh number (characterizing the thermal driving strength) \n$Ra=\\frac{1}{2}\\omega^2(R_o+R_i)\\beta \\Delta L^3\/(\\nu \\kappa)$,\nthe inverse Rossby number (measuring Coriolis effects)\n$Ro^{-1}=2(\\beta \\Delta (R_o+R_i)\/(2L))^{-1\/2}$,\nthe Prandtl number (fluid property) $\\text{Pr}=\\nu\/\\kappa$, and the radius and aspect ratio (geometric properties) $\\eta=R_i\/R_o$ and $\\Gamma=H\/L$. The key response parameter of the system is the Nusselt number (measuring the ratio of the total heat transport over the conductive one) $\\text{Nu}=-Q\\ln(\\eta)\/(\\alpha \\Delta 2 \\pi H)$. Here, $Q$ is the measured heat input through the outer cylinder into the system per unit of time; $\\alpha=\\kappa\\rho c_p$ is the thermal conductivity of the working fluid with $\\rho$ and $c_p$ being the density and the specific heat capacity of the fluid, respectively; and $H$ is the height of the gap between two cylinders. It should be noted that the definition of $\\text{Nu}$ in ACRBC is slightly different from that in classical RBC because of the cylindrical geometry with a heat flux in the radial direction, and the detailed derivations of the $\\text{Nu}$ for ACRBC are documented in the Supplementary Materials.\n\n\nDirect numerical simulations (DNS) are performed using an energy conserving second-order finite-difference code \\cite{ver96,poe15cf,zhu18afid}. The code has been extensively validated and used in previous studies in both the Cartesian and cylindrical coordinates for convective systems \\cite{poe15cf,zhu18afid,zhu18prl,zhu18np}. In all numerical simulations and experiments, the radius ratio is fixed at $\\eta=0.5$. No-slip boundary conditions were used for the velocity and constant temperature boundary conditions for the inner and outer cylinders. Periodic boundary conditions were used for the top and bottom surfaces. For three-dimensional simulations, the aspect ratio was chosen the same as in the experiments $\\Gamma=H\/L=1$ (the only two exceptions are for the cases at $\\text{Ra}=1.16\\times10^9$ and $\\text{Ra}=2.2\\times10^9$, where the height of the domain is $L\/4$ and $L\/8$, respectively, as the flows here are quasi two-dimensional). For the cases of which the flows are quasi two-dimensional at large $\\text{Ra}$, we use two-dimensional simulations. Pr was fixed at 4.3 for all the simulations. The adequate resolution was ensured for all cases, for example, at $\\text{Ra}=4.7 \\times 10^8$, $4608\\times384\\times384$ grid points were used and at $\\text{Ra}=4.7 \\times 10^{10}$ with grid points $18432\\times1536$. To obtain sufficient statistics, each simulation was run at least 200 free fall time units. As reported in \\citep{kun10,kun11}, the boundary layers in rotation system are expected to be thiner compared with classical RBC. Hence, the verification of boundary layer grid resolution is necessary. A posteriori check of grid resolution shows that at $\\text{Ra}=4.7\\times10^{10}$, there are 48 grid points inside thermal boundary layers and 64 grid points inside viscous boundary layers, which guarantees to resolve the boundary layer adequately. Besides, we have conducted a set of grid independence studies and checked the spatial resolution in bulk region to resolve all relevant scales (Kolmogorov scale and Batchelor scale). The parameter space that was explored can be found in Fig. \\ref{fig:setup}(c). More details about the DNS are documented in the Supplementary Materials.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{fig1_mod.pdf}\n\\caption{{\\bf System configuration and parameter space.} (a) Three-dimensional render of the experimental set-up. A fluid is confined in a rotating cylindrical annulus having its inner(blue) and outer(red) cylindrical surfaces made out of copper, which is known for its excellent thermal conductivity. The bottom plate is made of teflon for thermal insulation, and top plate is made of plexiglass allowing flow visualization. The coolant pipes and electric cables go through the hollow stainless steel shaft and connect to a slip ring and a rotary union. The toothed pulley is driven by a servo motor. (b) Schematical diagram of the set-up, which defines the geometric parameters. (c) Explored parameter space of $\\text{Ra}$ and $\\text{Ro}^{-1}$ for our study of supergravitational thermal convection. The red discs, the blue circles and the blue triangles correspond to the parameters used in the experiments (EXP), in the three-dimensional (3D) numerical simulations and in the two-dimensional (2D) numerical simulations, respectively. The parameter space is divided into three regimes according to the influence of Coriolis force. In addition, for comparison we have also performed the simulations for $\\text{Ro}^{-1}=10^{-5}$ (not shown here), for which the Coriolis force is negligible. For more details about the experimental apparatus, we refer to the Supplementary Materials and movie S1.}\n\\label{fig:setup}\n\\end{figure*}\n\n\n\\bigskip\n\n\\subsection{Experiments}\nExperiments are performed in a cylindrical annulus with solid-body rotation as sketched in Fig.~\\ref{fig:setup} (a). The two concentric cylinders are machined from a solid piece of copper to control the symmetry of the system, and their surfaces are electroplated with a thin layer of nickel to prevent oxidation. The inner cylinder has an outer radius of $R_i=\\unit{120}{\\milli\\meter}$, and the outer cylinder has an inner radius of $R_o=\\unit{240}{\\milli\\meter}$, resulting in a gap of L=$\\unit{120}{\\milli\\meter}$, and a radius ratio of $\\eta = R_i\/R_o = 0.5$. The gap with a height $H=\\unit{120}{\\milli\\meter}$ is sandwiched by a top plexiglass plate and a bottom teflon plate, resulting in an aspect ratio of $\\Gamma = H\/L = 1$. Plexiglass with an excellent transparency is used as the material of the top lid, allowing us to visualize the flow field, whereas teflon with an excellent corrosion resistance and a good strength is machined as the bottom base. These end plates and cylinders are fixed together and leveled on a rotating aluminum frame with a rotation rate up to \\unit{546}{rpm}. Four silicone rubber film heaters are attached to the outside of the outer cylinder, and are supplied by a DC power supply (Ametek, \\texttt{XG 300-5}) with 0.005\\% long-term stability. We use water as our working fluid, which has $\\text{Pr} \\approx 4.3$. \n\n\nInside the cold inner cylinder, 16 channels are machined for the coolant fluid to pass through, and the inner cylinder is regulated at constant temperature using a temperature-controlled circulating bath (PolyScience, \\texttt{AP45R-20-A12E}). The coolant pipes and electric cables go through the hollow stainless steel shaft and connect with a slip ring and a rotary union (Moflon, \\texttt{MEPH200}), which has 2 channels for liquids, 6 channels for powers (\\unit{2220}{\\watt}), and 48 channels for electrical signals. The shaft is driven by a toothed belt and the pulley has a gear ratio of $60:32\\approx1.88:1$. The whole system is powered by a servomotor (Yaskawa, \\texttt{SGM7G-1AA}), which has a rated power of \\unit{11}{\\kilo\\watt}.\n\n\nFor high-precision temperature and heat flux measurements, it is essential to minimize the heat leakage from the experimental apparatus to the surroundings. Various thermal shields that are regulated at appropriate temperatures are installed in the system to prevent heat losses. Furthermore, the whole system is placed in a big toughened glass box where the temperature is controlled by a Proportional-Integral-Derivative (PID) controller to match the mean temperature of the bulk fluid. For more detailed descriptions on the experimental set-up, we refer to Supplementary Materials. The measurement techniques are documented in Methods. \n\n\n\n\n\\begin{figure*} [ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig2_mod.pdf}\n\\caption{{\\bf Effects of the Coriolis force on the heat transfer and flow structures.} (a) Nusselt number ($\\text{Nu}$) as a function of $\\text{Ro}^{-1}$ for $\\text{Ra}=10^6$, $2.2\\times10^6$, $4.7\\times10^6$, $10^7$, $2.2\\times10^7$, $4.7\\times10^7$, $10^8$, $2.2\\times10^8$ and $4.7\\times10^8$ for $\\text{Pr} = 4.3$. (d) Root mean square axial velocity fluctuation $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ versus $Ro^{-1}$ for $\\text{Ra}=10^7$ and $\\text{Ra}=10^8$. (b,c,e,f) Instantaneous temperature fields from DNS at $\\text{Ra}=10^8$ for $\\text{Ro}^{-1}=0.1$, $0.5$, $1$, and $25$ ($\\text{Pr} = 4.3$). The inner and outer surfaces locate 0.02 $L$ away from the cold and hot cylinder correspondingly.}\n\\label{fig:Ro}\n\\end{figure*}\n\n\n\\section {Results} \n\n\n\n\\subsection{Effects of the Coriolis force on the heat transport and flow structures}\nA series of numerical simulations are performed to explore the influence of the Coriolis force on the heat transfer and flow structures. In numerical experiments, $\\text{Ro}^{-1}$ varies from $10^{-2}$ to $4\\times 10^2$ with several fixed Rayleigh numbers $\\text{Ra}=10^{6}$, $2.2\\times10^6$, $4.7\\times10^6$, $10^7$, $2.2\\times10^7$, $4.7\\times10^7$, $10^8$, $2.2\\times10^8$, and $4.7\\times10^8$. As shown in Fig.~\\ref{fig:Ro} (a), for these $\\text{Ra}$ studied, the influence of Coriolis force can be divided into three regimes. In regime \\uppercase\\expandafter{\\romannumeral1} ($\\text{Ro}^{-1}<\\text{Ro}^{-1}_{c1}$), the Coriolis force is small and negligible as compared to the buoyancy so that the $\\text{Nu}$ does not depend on $\\text{Ro}^{-1}$. In regime \\uppercase\\expandafter{\\romannumeral3} ($\\text{Ro}^{-1}>\\text{Ro}^{-1}_{c2}$), the rotation is so strong that the flow is nearly constrained by the effect of the Taylor-Proudman theorem~\\cite{pro16,tay22}, resulting in a quasi-two-dimensional flow state and thus a reduction of $\\text{Nu}$ compared with regime \\uppercase\\expandafter{\\romannumeral1}. For regime \\uppercase\\expandafter{\\romannumeral2} ($\\text{Ro}^{-1}_{c1}<\\text{Ro}^{-1}<\\text{Ro}^{-1}_{c2}$), the flow is governed by the combination of $\\text{Ra}$ and $\\text{Ro}^{-1}$ with rich flow states. In this regime, the general trend is that Nu decreases with $\\text{Ro}^{-1}$. The instantaneous temperature fields for several $\\text{Ro}^{-1}$ at $\\text{Ra}=10^8$ (see Fig.~\\ref{fig:Ro}(b,c,e,f)) from DNS support the explanation of this $\\text{Ro}$ dependence on the heat transport. As illustrated in Fig.~\\ref{fig:Ro}(b), the flow is three-dimensional at $\\text{Ro}^{-1}=0.1$ because the Coriolis force is too small to influence the flow effectively. However, with $\\text{Ro}^{-1}$ increasing, the enhanced Coriolis force tends to suppress vertical variation of the convection flow, and the flow gradually becomes two-dimensional, which is a manifestation of the Taylor-Proudman theorem. The two-dimensionalization of the flow field should be responsible for the reduction of heat transport at $\\text{Ro}^{-1} > \\text{Ro}^{-1}_{c1}$, which was also reported in \\cite{poe13}. As is evident in Fig.~\\ref{fig:Ro} (f), the flow is nearly two-dimensional at $\\text{Ro}^{-1} \\gtrsim \\text{Ro}^{-1}_{c2}$. So, further increased $\\text{Ro}^{-1}$ saturates the influence of the Coriolis force, which explains the nearly constant $\\text{Nu}$ when $\\text{Ro}^{-1} \\gtrsim \\text{Ro}^{-1}_{c2}$. Moreover, we use root mean square axial velocity fluctuation $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ to measure the influence of the Taylor-Proudman theorem, as shown in Fig.~\\ref{fig:Ro}(d). It shows that $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ has nearly a constant value when $\\text{Ro}^{-1}<\\text{Ro}^{-1}_{c1}$, then gradually decreases with $\\text{Ro}^{-1}$, and finally approaches to around zero after $\\text{Ro}^{-1}>\\text{Ro}^{-1}_{c2}$. The overall trend of $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$ versus $\\text{Ro}^{-1}$ is consistent with the dependence of $\\text{Nu}$ on $\\text{Ro}^{-1}$ in general.\n\nWe have noticed that the critical $\\text{Ro}^{-1}_{c1}$ and $\\text{Ro}^{-1}_{c2}$ depend on the $\\text{Ra}$. As illustrated in Fig.~\\ref{fig:Ro} (d), for $\\text{Ra}=10^7$, we have $\\text{Ro}^{-1}_{c1}\\approx0.1$ and $\\text{Ro}^{-1}_{c2}\\approx10$, while for $\\text{Ra}=10^8$, we have $\\text{Ro}^{-1}_{c1}\\approx0.15$ and $\\text{Ro}^{-1}_{c2}\\approx15$. According to the numerical data explored, we can roughly determine the boundaries of the three regimes, which is plotted in Fig.~\\ref{fig:Ro} (a) with green lines. In addition, we have also included the boundaries in Fig.~\\ref{fig:setup} (c), and extrapolated the boundaries to the parameter space of experiments. (More detailed discussions of the flow structures, $\\langle (u_z)_{rms}\\rangle _{r,\\varphi, z}$, and $\\text{Ro}^{-1}_{c}$ for these $\\text{Ra}$ cases are available in Supplementary Materials.) \n\n\n\n\n\\begin{figure*} [ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig3_mod.pdf}\n\\caption{{\\bf Global heat transport.} (a) Nusselt number ($\\text{Nu}$) as a function of $\\text{Ra}$ from experiments (the solid symbols), DNS (the open symbols) in ACRBC and the prediction from Grossmann-Lohse (G--L) theory\\cite{gro00} in classical RBC (dashed line). (b) The same plots as (a), but the vertical axis is compensated. Inset: an enlarged portion of the compensated plot at the large $\\text{Ra}$ regime, which shows the transition of the effective scaling exponent ($\\text{Nu} \\propto \\text{Ta}^{\\gamma}$) to $\\gamma > 1\/3$.}\n\\label{fig:heat}\n\\end{figure*}\n\n\n\\subsection{Global turbulent heat transport}\nAs shown in the section above, $\\text{Nu}$ does not depend much on $\\text{Ro}^{-1}$ for high $Ro^{-1}$ regime, in which we will study how heat transfer depends on $\\text{Ra}$ in this section. We have performed 48 experiments and more than 130 numerical simulations. Fig.~\\ref{fig:setup}(c) shows the parameter space of the experimental and numerical studies. In the experimental studies, we note that the existence of Earth's gravity and lids is unavoidable. Several numerical simulations have been performed to study the influences of Earth's gravity and lids, which show that their effects on $\\text{Nu}$ are small. \n In addition, we note that the centrifugal force increases linearly with radial location. To study the influence of nonuniform driving force, we have performed two sets of simulations with the radial-dependent centrifugal acceleration $\\omega^2 r$ and with a constant artificial acceleration $\\omega^2 (R_o+R_i)\/2$. The results show that the effect of the radial-dependent gravity has a similar role as the non-Oberbeck-Boussinesq effect \\citep{gro00,ahl06,sug09}, which does not have much influence on the heat transport and flow structures in the current parameter regime (see the Supplementary Materials for details). \n\n\n\n\nFig.~\\ref{fig:heat} (a) shows the measured $\\text{Nu}$ as a function of $\\text{Ra}$ for different $\\text{Ro}$. Each measurement lasts at least 4 hours after the system has reached a thermally steady situation to ensure a statistically stationary state (for the detailed measurement procedure, see the Supplementary Materials). As being evident in Fig.~\\ref{fig:heat}(a), the experiments and numerical simulations are in excellent agreement. Combining experiments and simulations, this study covers more than four decades of $\\text{Ra}$, \\textit{i.e.} from $10^6$ to $6.5\\times 10^{10}$. The range of $\\text{Ro}^{-1}$ is from 18 to 58 in experiments, and the data series show a consistent dependence of $\\text{Nu}$ on $\\text{Ra}$. For comparison, we also plot the data calculated from Grossmann--Lohse (G--L) theory \\cite{gro00} in the corresponding $\\text{Ra}$ range. \nTo better reveal the local exponent, we plot the compensated data in Fig.~\\ref{fig:heat}(b). The experimental and numerical data have a lower amplitude as compared to the classical RBC (G--L line) due to the different flow geometry. However, the scaling dependence of $\\text{Nu}$ versus $\\text{Ra}$ shows a good agreement between the data and G-L theory at $\\text{Ra} \\lessapprox 10^{10}$ with a scaling exponent $\\gamma=0.27\\pm0.01$, which is close to the typical value found in two-dimensional RBC. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{fig4_mod.pdf}\n\\caption{{\\bf Revolution of convective rolls.} (a, b) Experiments at $\\text{Ra}=1.8\\times 10^9$ and $\\text{Ro}^{-1}=13.8$. (a) Snapshots of streak images revealing the flow patterns. Seeing from the top, the whole system rotates clockwise in experiments. The corresponding movie is available in movie S2. (b) Time series of local temperature fluctuations. The measured point locates \\unit{30}{\\milli\\meter} away from the cold inner cylinder and at mid-height. (c, d) Simulations at $\\text{Ra}=10^7$, and $\\text{Ro}^{-1}=1$: (c) Snapshots (Top view) of instantaneous temperature fields. (d) The averaged azimuthal velocity profile along the radial direction. The reference of the frame is on the clockwise direction, and the corresponding movie is available in the Supplementary Movie.}\n\\label{fig:flowpattern}\n\\end{figure*}\n\nIt is very unexpected that the local effective exponent $\\gamma$ of $\\text{Nu} \\propto \\text{Ra}^{\\gamma}$ even exceeds one-third at $\\text{Ra} \\gtrsim 4\\times 10^{10} $ in experiments. \nIs this scaling regime connected to the appearance of ultimate regime of Rayleigh--B\\'enard turbulence \\cite{kra62,gro11,he12a}? If it is not in the ultimate regime, what causes this steep scaling exponent? \nAs our current $\\text{Ra}$ is much lower than the transition $\\text{Ra}$ for ultimate turbulence observed in the classical RBC, we cannot provide a concrete answer on this question. The possible answers are that (i) because of the different way of the driving and the geometry, the transition Ra to the ultimate regime at the current system is lower than that in the classical RBC. (ii) Another possibility is that the enhanced local slope is due to the emergence of new flow states. As demonstrated in Fig.~\\ref{fig:setup} (c), with $\\text{Ra}$ increasing, the location of ($\\text{Ra},\\, \\text{Ro}^{-1}$) where the enhanced local effective scaling appears seems to move from regime III to regime II. Thus the flow may change from a two-dimensional flow state to a three-dimensional flow state due to the strong thermal driving, resulting in a higher $\\text{Nu}$ in this $\\text{Ra}$ range. \nThe two-dimensional simulations for high $\\text{Ra}$ coincide well with the experiments, indicating that within the $\\text{Ra}$ range explored in two-dimensional simulations the flow is still quasi-two-dimensional. Unfortunately, we cannot push the simulations to the Ra regime with the effective exponent larger than one-third, in which the flow may not be in the quasi-two-dimensional state. We incline to anticipate that the changing from a two-dimensional flow state to a three-dimensional flow state may be responsible for the enhancement of local effective exponent at $Ra \\gtrsim 4\\times10^{10}$. \nWe notice that the scaling range with $\\gamma > 1\/3$ is very narrow in the current work, therefore further work at even higher Ra is needed to clarify this issue. \n\n\n\\subsection{Zonal flow}\n\n\nWe now study the dynamics of zonal flow experimentally and numerically. The \"zonal flow\" phenomenon has been investigated in experiments of geophysical and astrophysical flows \\cite{hei05,har15}. Fig.~\\ref{fig:flowpattern}(b) shows the time series of local temperature fluctuations in water at $\\text{Ra}=6.6\\times 10^9$, $\\text{Pr}=4.3$, and $\\text{Ro}^{-1}=18$, which unexpectedly shows a noticeable periodicity. To prove that this novel phenomenon is connected to the azimuthal movement of the coherent flow, we perform flow visualization with aqueous glycerol solution as mentioned in Methods to demonstrate the flow pattern.\n\n\nFig.~\\ref{fig:flowpattern}(a) shows some typical streak images from flow visualization at three different time (for video, see movie S2), where there are four pairs of rolls parallel to the rotating axis. For reference, we highlight one of the convective rolls using a yellow ellipse. From Fig.~\\ref{fig:flowpattern} (aI--aIII), unexpectedly the convection rolls move clockwise around the center with a faster rotation rate than the background rotation of the experimental system, which we name \"zonal flow\". The flow visualization further justifies that the revolution of the cold plume arms triggers the periodic temperature signals measured by the thermistors. \n\n\nNext to the experiments, we also find this phenomenon in the simulations (see Fig.~\\ref{fig:flowpattern}(c) and movie S3).\nWe use a black ellipse to highlight a selected cold plume. As shown in Fig.~\\ref{fig:flowpattern}(c),\non the reference of the (clockwise) rotating frame, the plume arms still evolve in a clockwise direction, indicating a net rotation of the coherent flow (zonal flow).\nThis zonal flow is further quantified through the averaged (axially and azimuthally in space and over time) azimuthal velocity profile versus the radial distance from the inner cylinder wall $r$, as shown in Fig.~\\ref{fig:flowpattern}(d).\n\n\nWhat causes this net rotation of the convection rolls? \nThe dashed lines in Fig. \\ref{fig:flowpattern}(c\\uppercase\\expandafter{\\romannumeral1}) mark the direction of hot\/cold plumes without Coriolis force. But because of the effects of Coriolis force, the plumes deflect to their left when the system rotates clockwise. As is evident in the Fig. \\ref{fig:flowpattern}(c\\uppercase\\expandafter{\\romannumeral1}) where the plumes are just formed in the beginning stage, the deflected angle of hot plumes ($\\angle a$) is approximately equal to the deflected angle of cold plumes ($\\angle b$). However, because of the different curvature of the cylinders, the similar deflected angles of the hot and cold plumes induce the different effects. The hot plumes directly affect the position (A) where the cold plumes are ejected, resulting in the clockwise rotation of the cold plumes and flow, whereas the impact of the cold plumes does not directly affect the motion of the hot plumes due to a relatively large distance the impact position of the cold plumes (B) and the ejecting position of the hot plumes. Thus, the hot plumes win and push the overall flow to move in the same direction of the background rotation. \nWe also perform the experiments and simulations with the opposite direction of the background rotation, and the results are consistent. In addition, several numerical simulations have been performed to test the dependence of zonal flow on the radius ratio $\\eta$, which shows that the zonal flows become weaker and weaker with $\\eta$ increasing from 0.4 to 0.9. The difference in the curvature of the two cylinders plays the key role for the net rotation of convection rolls. One may expect to observe different types of zonal flows at different $\\eta$, such as two large scale winds with the opposite directions near the plates \\cite{har15}. This remarkable net rotation of the coherent flow structures along the same direction of the background rotation could be connected to many relevant flow phenomena in nature, which deserves systematical studies in the future.\n\n \n\n\\section{Summary}\nIn summary, we have experimentally and numerically studied the global heat transport and turbulent flow structures in a rotating annulus with hot outer cylinder and cold inner cylinder, \\textit{i.e.}, ACRBC. In experiments, the mean centrifugal acceleration covers the range from 5 to 60 times gravitational acceleration. We show that the effective scaling exponent of global heat transport transitions to $\\gamma=1\/3$ at $\\text{Ra} \\approx 10^{10}$ and finally exceeds $\\gamma > 1\/3$ for $\\text{Ra} \\gtrapprox 4\\times 10^{10}$. \nUnexpectedly, we observed the faster azimuthal motion of the coherent overall flow structure faster than the background rotation of the system, which we call zonal flow, and provided a physical understanding on this remarkable net rotation motion of the coherent structures. This novel experimental approach sheds new lights on how to efficiently extend the parameter regimes for the study of buoyancy driven turbulent flows. Furthermore, the findings in the current system driven by the centrifugal acceleration can help us understand phenomena in geophysical and astrophysical flows. \n\n\n\\section{Methods}\n{\\it Measurement techniques.} In our experiments, all the temperature measurements are based on negative temperature coefficient thermistors. We use $6\\frac{1}{2}$-digit multimeter (Keithley, \\texttt{2701}) to measure the resistances of the thermistors, and the resistance can be converted to temperature using the Steinhart-Hart equation\\cite{lav97}. Two types of thermistors are used: One with a head diameter of \\unit{2.5}{\\milli\\meter} (Omega, \\texttt{44131}) is used to measure the temperature of the inner and outer cylinders, and the other with a head diameter of \\unit{300}{\\micro\\meter} (Measurement Specialties, \\texttt{GAG22K7MCD419}) is inserted into the convection cell to resolve fast temperature fluctuations, which is located \\unit{30}{\\milli\\meter} away from the cold cylinder and at mid-height in the vertical direction. \n\nTo visualize the flow field, we use nylon fibers with length $l = \\unit{3}{\\milli\\meter}$, and diameter $d = \\unit{0.5}{\\milli\\meter}$. Nylon has a density of $\\rho_{p}\\approx \\unit{1.14}{\\gram\\per\\centi\\meter\\cubed}$ that is a little heavier than water. In the flow visualization experiments, we use aqueous glycerol solution with 54\\% glycerol by volume to match the density of nylon. The properties of the aqueous glycerol solution are listed in the Supplementary Materials for reference. Four light-emitting diods are used as light source, and a charge-coupled device camera (Ximea, \\texttt{MD028MU-SY}) is put on the top of the cell to record images. As the system rotates rapidly, it is difficult to make the camera rotate synchronously. So we keep the camera fixed, and then set the frame rate of the camera the same as the rotational speed. Processing the images taken by camera through a MATLAB code, we can get streak images to visualize the flow field. More processing details can be found in the Supplementary Materials.\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsubsection{Irreducible algebraic representations}\n\nThe simplest irreducible algebraic representations are the weighted Specht modules $S^{\\lambda,k}$ where as an $S_n$ representation this is just the usual Specht module $S^\\lambda$, and each copy of $\\mathbb{C}^\\times$ just acts by scalars via the character $z \\to z^k$.\n\nIf $\\lambda^1, \\lambda^2, \\dots, \\lambda^m$ are partitions of total size $n$, and $k_1 < k_2 < \\dots < k_m$ are distinct integers then the induced representation \n\\begin{equation}\\label{algirr}\n\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})\n\\end{equation}\nis irreducible and moreover every irreducible algebraic representation of $T \\rtimes S_n$ is obtained this way. Hence we see the irreducible algebraic representations of $T \\rtimes S_n$ can be naturally labeled by collections of partitions of total size $n$ indexed by the integers .\n\n\\subsubsection{Induction of representations}\n\nMoreover, it is easy to describe the induction of two weighted Specht modules of the same $\\mathbb{C}^\\times$ weight. We just get a direct sum of weighted Specht modules of same $\\mathbb{C}^\\times$ weight with multiplicities coming from the usual Littlewood-Richardson rule. More precisely: \n$$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\mu|})} ^ {\\mathbb{C}^\\times \\wr S_{|\\lambda| + |\\mu|} }(S^{\\lambda,k} \\otimes S^{\\mu,k}) = \\bigoplus_\\nu c_{\\lambda, \\mu}^\\nu S^{\\nu, k}$$\nIt's then clear how to decompose the induction of two arbitrary irreducibles, you just run the Littlewood-Richardson rule separately on each $\\mathbb{C}^\\times$-weight. In other words the Grothendieck ring of $\\bigoplus_n \\text{Rep}(\\mathbb{C}^\\times \\wr S_{n})$ with monoidal structure coming from induction is just a direct sum indexed by $\\mathbb{Z}$ of copies of the ring of symmetric functions $\\Lambda$.\n\n\\subsubsection{Restriction to $S_n$}\n\nFinally, we will close out this section by noting that because irreducible representations of $T \\rtimes S_n$ are just induced up from products of weighted Specht modules, we can easily describe the restriction of arbitrary irreducible $T \\rtimes S_n$ representations to $S_n$. The irreducible representation \n\n$$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})$$\njust becomes\n$$\\text{Ind}_{S_{|\\lambda^1|} \\times S_{|\\lambda^2|} \\times \\dots S_{|\\lambda^m|}}^{S_{n} }(S^{\\lambda^1} \\otimes S^{\\lambda^2} \\otimes \\dots \\otimes S^{\\lambda^m})$$\nwhich of course can be decomposed using the Littlewood-Richardson rule. Explicitly in terms of symmetric functions we have the following corollary.\n\n\\begin{corollary}\\label{restmult}\nThe multiplicity of the Specht module $S^\\mu$ inside the restriction to $S_n$ of $$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})$$ is equal to the multiplicity of the Schur function $s_\\mu$ in the product $s_{\\lambda^1}s_{\\lambda^2}\\dots s_{\\lambda^m}$\n\n\\end{corollary}\n\nAs suggested in the introduction we see that the difficulty in understanding the $S_n$ action on $\\overline{W(\\lambda)_\\nu}$ lies in understanding it as a $T \\rtimes S_n$ representation, and that then restricting to $S_n$ is straightforward.\n\n\\end{subsection}\n\n\\begin{subsection}{Polynomial representations of $T \\rtimes S_n$}\n\nWe say that an algebraic representation of $T \\rtimes S_n$ is \\emph{weakly polynomial} if the $T$-weights that occur are all polynomial weights. The weighted Specht module $S^{\\lambda, k}$ is weakly polynomial if and only if $k \\ge 0$, and more generally an irreducible representation of the form (\\ref{algirr}) is weakly polynomial if and only if $k_i \\ge 0$ for all $i$. The class of weakly polynomial representations is preserved by induction and the corresponding Grothendieck ring of $\\bigoplus_n \\text{Rep}^{wp}(\\mathbb{C}^\\times \\wr S_{n})$ is again a direct sum of copies of $\\Lambda$, this time indexed by $\\mathbb{Z}_{\\ge 0}$.\n\nLet $V$ denote the standard $n$-dimensional representation of $T \\rtimes S_n$ obtained by restricting the defining representation of $GL_n$ to $T \\rtimes S^n$. As a representation of $S_n$ this is just the standard $n$-dimensional permutation representation, and $T$ acts on the permutation basis with weights $(1,0,0,\\dots, 0)$, $(0,1,0,0,\\dots 0)$, $\\dots$, and $(0,0,\\dots, 0, 1)$. In the notation of (\\ref{algirr}) we have that $$V \\cong \\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})} ^ {\\mathbb{C}^\\times \\wr S_{n} }(S^{(n-1),0} \\otimes S^{(1),1}).$$\n\nWe say that a representation of $T \\rtimes S_n$ is \\emph{polynomial of degree $d$} if it is a direct summand of copies of $V^{\\otimes d}$, and say that a representation is \\emph{polynomial} if it is a direct sum of polynomial representations of various degrees. In particular, a representation of $T \\rtimes S_n$ is polynomial if and only if it is a direct summand of the restriction of a polynomial representation of $GL_n$. \n\n\\medskip\n\n\\noindent \\textbf{Warning:} We'll note that unlike the classes of algebraic and weakly polynomial representations, induction from $(\\mathbb{C}^\\times \\wr S_{n}) \\times (\\mathbb{C}^\\times \\wr S_{m})$ to $\\mathbb{C}^\\times \\wr S_{n+m}$ does not preserve the class of polynomial representations. However it is obvious by the definition that polynomial representations are preserved by the internal tensor product of $T \\rtimes S_n$ representations.\n\n\n\\subsubsection{Irreducible polynomial representations}\n\nThe following proposition characterizes which irreducible weakly polynomial representations are polynomial.\n\n\\begin{proposition}\\label{polirr}\nGiven partitions $\\lambda^1, \\lambda^2, \\dots, \\lambda^m$ of total size $n$, and distinct nonnegative integers $0 \\le k_1 < k_2 < \\dots < k_m$ then the irreducible weakly polynomial representation\n\n$$\\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^m|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{\\lambda^1,k_1} \\otimes S^{\\lambda^2,k_2} \\otimes \\dots \\otimes S^{\\lambda^m,k_m})$$\nis polynomial (of degree $k_1+k_2+\\dots k_m$) if and only if either $k_1 \\ne 0$ or $k_1 = 0$ and $\\lambda^1$ is a one-row partition.\n\n\\end{proposition}\n\nBefore proving Proposition \\ref{polirr} it will be convenient to introduce a new labeling for these representations. If $\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell$ is a sequence of (possibly empty) partitions of total size $m \\le n$ then define $$V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} := \\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{n-m}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^\\ell|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{(n-m),0} \\otimes S^{\\lambda^1,1} \\otimes \\dots \\otimes S^{\\lambda^\\ell,\\ell})$$\n\nFor example in this notation the standard $n$-dimensional representation $V$ from the previous section is now denoted by $V^{(1)}$, and if instead we took the $n$-dimensional permutation representation for $S_n$ and had $T$ act on the permutation basis with weights $(2,0,0,\\dots, 0)$, $(0,2,0,0,\\dots 0)$, $\\dots$, and $(0,0,\\dots, 0, 2)$ this representation would be labeled $V^{\\emptyset, (1)}$. Under this labeling the degree of $V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}$ is given by $|\\lambda^1| + 2|\\lambda^2| + 3|\\lambda^3| + \\dots + \\ell | \\lambda^\\ell|.$\n\nWe'll note that adding empty partitions to the end of the sequence $\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell$ doesn't change the corresponding representation. It will occasionally be useful to think of this as being a labeling by infinite sequences of partitions where all but finitely many are the empty partition. In particular, the trivial representation just corresponds to the sequence where all the partitions are empty.\n\nAt times we will want to use this notation to describe representations of $\\mathbb{C}^\\times \\wr S_{n}$ for different values of $n$, in which case we will add subscripts $V_{n}^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}$ to avoid ambiguity.\n\n\\medskip\n\nSince by definition all irreducible polynomial representations are direct summands of tensor powers of $V^{(1)}$ it will be useful to see how tensoring with $V^{(1)}$ looks under this labeling. The following lemma gives such a description:\n\n\\begin{lemma}\\label{tensorV} \\hspace{1cm} \\\\\n\n \\noindent If $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| = n$ then $$V^{(1)} \\otimes V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} = \\bigoplus_{\\substack{{\\lambda^i}' = \\lambda^i - \\square \\\\ {\\lambda^{i+1}}' = \\lambda^{i+1} + \\square}} V^{\\lambda^1, \\dots ,{\\lambda^i}' , {\\lambda^{i+1}}', \\dots, \\lambda^\\ell}$$\nand if $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| < n$\n\n$$V^{(1)} \\otimes V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} = \\bigoplus_{\\substack{{\\lambda^i}' = \\lambda^i - \\square \\\\ {\\lambda^{i+1}}' = \\lambda^{i+1} + \\square}} V^{\\lambda^1, \\dots ,{\\lambda^i}' , {\\lambda^{i+1}}', \\dots, \\lambda^\\ell} \\oplus \\bigoplus_{{\\lambda^1}' = \\lambda^1 + \\square} V^{{\\lambda^1}', \\lambda^2,\\dots \\lambda^\\ell}$$\nwhere the sums are over all ways of removing\/adding a box to the corresponding Young diagrams.\n\\end{lemma}\n\nIn terms of our labeling by sequences of partitions this just says that tensoring with $V^{(1)}$ corresponds to the following process: If we start with one sequence of partitions we create new ones in all possible ways by removing one box from one partition and adding it to the next partition in the sequence, or if there aren't already $n$ total boxes we can add one box to the first partition.\n\nWe'll mostly be interested in the case where $n$ is strictly larger than the degree so we are always allowed to add a box to the first partition, but we've included the general case here for completion. In terms of the Young diagrams involved here is a picture of the first few rows of the Bratteli diagram for tensoring with $V^{(1)}$ when $n$ is at least $3$ (when $n$ is less than $3$ just delete all sequences of Young diagrams with more than $n$ boxes):\n\n\\medskip\n\n\\begin{tikzpicture}[scale=.7]\n \\node (empty) at (0,0) {\\scalebox{1.5}{$\\emptyset$}};\n \\node (1) at (0,-2) {$\\ydiagram{1}$};\n \\node (a) at (0,-5) {$\\ydiagram{1,1}$};\n \\node (b) at (3,-5) {$\\ydiagram{2}$};\n \\node (c) at (6,-5) {$\\Big{(} \\scalebox{1.5}{$\\emptyset$}, \\ydiagram{1}$ \\ \\Big{ )}};\n \\node (d) at (0,-9) {$\\ydiagram{1,1,1}$};\n \\node (e) at (3,-9) {$\\ydiagram{2,1}$};\n \\node (f) at (6,-9) {$\\ydiagram{3}$};\n \\node (g) at (10,-9) {$\\Big{(} \\ \\ydiagram{1}, \\ydiagram{1}$ \\ \\Big{ )}};\n \\node (h) at (14,-9) {$\\Big{(}$ \\scalebox{1.5}{$\\emptyset$}, \\scalebox{1.5}{$\\emptyset$}, $\\ydiagram{1}$ \\ \\Big{ )}};\n \n \\draw (empty)--(1)--(a)--(d)\n (1)--(b)--(e)\n (1) -- (c)--(g)\n (a)--(e)\n (b)--(f)\n (a)--(g)--(b) \n (c)--(h);\n\\end{tikzpicture}\n\n\\medskip\n\n\\noindent \\textbf{Proof of Lemma \\ref{tensorV}:} $V^{(1)}$ is induced from $(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})$ so in order to tensor with it we may use the push-pull formula:\n\n$$\\text{Ind}(U) \\otimes W \\cong \\text{Ind}(U \\otimes \\text{Res}(W))$$\n \nSo to calulate this first we need to restrict $V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}$ to $(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})$. We already saw that induction between algebraic representations was controlled by the Littlewood-Richardson rule, hence by Frobenius reciprocity it follows that if $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| = n$ then\n\n$$\\text{Res} (V_n^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}) = \\bigoplus_{{\\lambda^i}' = \\lambda^i - \\square} V_{n-1}^{\\lambda^1, \\dots {\\lambda^i}' \\dots, \\lambda^\\ell} \\otimes S^{(1),i}$$\n\nand if $|\\lambda^1| + |\\lambda^2| + \\dots + |\\lambda^\\ell| < n$ then \n\n$$\\text{Res} (V_n^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell}) = \\bigoplus_{{\\lambda^i}' = \\lambda^i - \\square} V_{n-1}^{\\lambda^1, \\dots {\\lambda^i}' \\dots, \\lambda^\\ell} \\otimes S^{(1),i} \\hspace{.4cm} \\oplus \\hspace{.4cm} V_{n-1}^{\\lambda^1, \\lambda^2, \\dots, \\lambda^\\ell} \\otimes S^{(1),0}$$\n\nNext we need to tensor these factors with the $(\\mathbb{C}^\\times \\wr S_{n-1}) \\times (\\mathbb{C}^\\times \\wr S_{1})$ representation that is the trivial representation of $(\\mathbb{C}^\\times \\wr S_{n-1})$ tensored with $S^{(1),1}$. Of course tensoring with the trivial representation does nothing, and the second factor is just a character of $\\mathbb{C}^\\times$ so we see this just sends $V_{n-1}^{\\mu^1, \\dots, \\mu^\\ell} \\otimes S^{(1),i}$ to $V_{n-1}^{\\mu^1, \\dots, \\mu^\\ell} \\otimes S^{(1),i+1}$.\n\nFinally, we need to induce back up to $(\\mathbb{C}^\\times \\wr S_{n-1})$. But we know by the Littlewood-Richardson rule that $$\\text{Ind} V_{n-1}^{\\mu^1, \\dots, \\mu^\\ell} \\otimes S^{(1),i+1} = \\bigoplus_{{\\mu^{i+1}}' = \\mu^{i+1} + \\square} V_n^{\\mu^1, \\dots {\\mu^{i+1}}', \\dots, \\mu^\\ell}$$\nPutting this all together gives the desired formula. \\hfill $\\square$\n\n\\medskip\n\nFrom here Proposition \\ref{polirr} follows immediately: By induction every irreducible subrepresentation of $(V^{(1)})^{\\otimes d}$ is of the desired form, and conversely it's clear that any sequence of partitions can be obtained from the empty sequence by the process described in Lemma \\ref{tensorV} of adding boxes to the first partition and moving them down the sequence arbitrarily (moreover the multiplicity is given by the number of downward walks on the Bratteli diagram from the empty sequence to the desired sequence of partitions).\n\n\\medskip\n\n\\noindent \\textbf{Example:} If $n$ is at least $2$ Lemma \\ref{tensorV} gives us the decomposition of the tensor square of the defining representation:\n\n$$V^{(1)} \\otimes V^{(1)} \\cong V^{(1,1)} \\oplus V^{(2)} \\oplus V^{\\emptyset, (1)}$$\nIn terms of $GL_n$ representations this is familiar, if we take the usual decomposition of $GL_n$ representations\n$$V \\otimes V = \\Lambda^2(V) \\oplus \\text{Sym}^2(V)$$\nthen $\\Lambda^2(V)$ restricts to $V^{(1,1)}$, and if we think of $\\text{Sym}^2(V)$ as degree $2$ polynomials of standard basis vectors $x_1, x_2, \\dots x_n$ then $V^{\\emptyset, (1)}$ corresponds to the span of the polynomials $x_i^2$ and $V^{(2)}$ corresponds to the span of the polynomials $x_ix_j$ with $i \\ne j$.\n\n\n\\end{subsection}\n\n\\end{section}\n\n\\begin{section}{Permutation modules, symmetric powers, and multiset combinatorics}\\label{combo}\n\n\\begin{subsection}{Weighted permutation modules}\n\nRecall for the symmetric group $S_n$ and a partition (or composition) $\\lambda =(\\lambda_1,\\lambda_2, \\dots, \\lambda_\\ell)$, the permutation module $M(\\lambda)$ is defined as $$M(\\lambda) := \\text{Ind}_{S_{\\lambda_1} \\times S_{\\lambda_2} \\times \\dots S_{\\lambda_\\ell}}^{S_n}(\\mathbf{1}) = \\text{Ind}_{S_{\\lambda_1} \\times S_{\\lambda_2} \\times \\dots S_{\\lambda_\\ell}}^{S_n}(S^{(\\lambda_1)} \\otimes S^{(\\lambda_2)}\\otimes \\dots \\otimes S^{(\\lambda_\\ell)} )$$\nwhich combinatorially just means $M(\\lambda)$ is the linearized representation of the action of $S_n$ on ordered set partitions of $\\{1,2,\\dots,n\\}$ into sets $A_1, A_2, \\dots A_\\ell$ with $|A_i| = \\lambda_i$ for each $i$.\n\nFor algebraic representations of $T \\rtimes S_n$ there are natural analogs of these modules where we allow the copies of $\\mathbb{C}^\\times$ to act by characters and then induce up. Explicitly we have the \\emph{weighted permutation modules} $M(\\lambda, \\mathbf{k})$ where $\\lambda = (\\lambda_1,\\lambda_2, \\dots, \\lambda_\\ell)$ is a composition of $n$ and $\\mathbf{k} = (k_1, k_2, \\dots, k_\\ell)$ is a list of $\\mathbb{C}^\\times$ weights of the same length\n$$M(\\lambda, \\mathbf{k}) := \\text{Ind}_{(\\mathbb{C}^\\times \\wr S_{|\\lambda_1|}) \\times (\\mathbb{C}^\\times \\wr S_{|\\lambda_2|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda_\\ell|})}^{ \\mathbb{C}^\\times \\wr S_{n} }(S^{(\\lambda_1),k_1} \\otimes S^{(\\lambda_2),k_2} \\otimes \\dots \\otimes S^{(\\lambda_\\ell),k_\\ell})$$\nNote that we can rearrange the parts of the composition or the order of the $k_i$'s, so long as we permute the other accordingly. In particular, at times it will be convenient to reorder things so that $\\lambda$ is a partition and other times it will be convenient to order the $k_i$'s in increasing order. Just note you can't in general do both simultaneously.\n\nIt's then easy to see that a weighted permutation module is polynomial if and only if each $k_i$ is non-negative and at most one of them is equal to zero. In this case it will be convenient to reindex as before by lists of partitions $\\lambda^1, \\lambda^2, \\dots \\lambda^j$ of total size at most $n$. If we denote $\\lambda^i = (\\lambda^i_1, \\lambda^i_2,\\dots, \\lambda^i_{\\ell_i})$ and $|\\lambda^1|+ \\dots + |\\lambda^j| = m$ then define\n$$\\tilde{M}(\\lambda^1, \\lambda^2,\\dots, \\lambda^k) := M((n-m, \\lambda^1_1, \\lambda^1_2, \\dots, \\lambda^j_{\\ell_j}), (0, 1,1,\\dots, 2,2,\\dots, j,j))$$\nwhere there are $\\ell_1$ $1$'s, $\\ell_2$ $2$'s, and so on. \n\n In other words, $\\tilde{M}(\\lambda^1, \\lambda^2,\\dots, \\lambda^k)$ is just the representation induced from the representation of $(\\mathbb{C}^\\times \\wr S_{n-m})\\times (\\mathbb{C}^\\times \\wr S_{|\\lambda^1|}) \\times \\dots (\\mathbb{C}^\\times \\wr S_{|\\lambda^j|})$ where the first factor acts trivially and in the $\\mathbb{C}^\\times \\wr S_{|\\lambda^i|}$ factor we take a copy of the $S_n$ permutation module $M(\\lambda^i)$ with the copies of $\\mathbb{C}^\\times$ scaling by the character $z \\to z^i$.\n \n\\medskip\n\n\\noindent \\textbf{Examples:} The defining representation $V^{(1)}$ is isomorphic to the permutation module $\\tilde{M}((1))$. If $n \\ge 2$ the tensor square $V^{(1)} \\otimes V^{(1)}$ decomposes as $\\tilde{M}((1,1)) \\oplus \\tilde{M}(\\emptyset, (1))$. In terms of the standard basis vectors $x_1, x_2, \\dots x_n$ of $V^{(1)}$, $\\tilde{M}((1,1))$ is the span of all vectors of the form $x_i \\otimes x_j$ with $i \\ne j$, and $\\tilde{M}(\\emptyset, (1))$ is spanned by the vectors $x_i \\otimes x_i$.\n\n\\medskip\n\nFor the rest of the section we will recall some facts about permutation modules for $S_n$ and describe appropriate analogs for weighted permutation modules.\n\n\\subsubsection{Decomposition into irreducibles}\n\nIn the unweighted case, the decomposition of a permutation module into irreducibles given by the Kostka numbers which combinatorially count semistandard Young tableaux. Explicitly we have $$M(\\lambda) \\cong \\bigoplus_\\mu K_{\\lambda,\\mu} S^\\mu$$\nwe'll note however that this is just a special case of the Littlewood-Richardson rule (or even the Pieri rule) applied to the induction of the trivial representation of the Young subgroup corresponding to $\\lambda$.\n\nSo unsurprisingly, the decomposition of weighted permutation modules will also be governed by Kostka numbers. Our notation for polynomial representations will be more convenient for stating this result (and ultimately will be the case we care about), but we'll note that it holds for algebraic representations as well.\n\n\\begin{lemma}\\label{multikostka}\n$$\\tilde{M}(\\lambda^1, \\lambda^2, \\dots, \\lambda^m) = \\bigoplus_{\\mu^1, \\mu^2, \\dots, \\mu^m} K_{\\mu^1,\\lambda^1}K_{\\mu^2,\\lambda^2}\\dots K_{\\mu^m,\\lambda^m} V^{\\mu^1,\\mu^2,\\dots, \\mu^m}$$\n\\end{lemma}\n\n\\noindent \\textbf{Proof:} We've already seen that for algebraic $T \\rtimes S_n$ modules induction is governed by the Littewood-Richardson rule on each $\\mathbb{C}^\\times$-weight separately. As such, the decomposition of a weighted permutation modules is given by a product of Kostka numbers for each $\\mathbb{C}^\\times$ weight. \\hfill $\\square$\n\n\n\\subsubsection{$S_n$-invariants in weighted permutation modules}\n\nThe usual permutation module $M(\\lambda)$ contains a one dimensional space of $S_n$-invariants. Indeed this is true anytime a representation of a finite group is constructed as the linearization of a transitive group action, and the space of invariants is spanned by the sum of the elements of the set being acted upon (or alternatively one can see this using Frobenius reciprocity as we are inducing up the trivial representation from a subgroup).\n\nFor weighted permutation modules, asking for a $T \\rtimes S_n$ invariant vector is asking too much. Indeed this only occurs when $T$ acts trivially (which in the $\\tilde{M}(\\lambda, \\mathbf{k})$ notation is when $\\mathbf{k}$ is the zero vector), which just factors through the unweighted case. Nevertheless, we still have the following:\n\n\\begin{lemma}\\label{perminv}\nThe permutation module $\\tilde{M}(\\lambda, \\mathbf{k})$ has a one dimensional space of $S_n$-invariants.\n\\end{lemma}\n\n\\noindent \\textbf{Proof:} By construction $M(\\lambda, \\mathbf{k})$ restricted to $S_n$ is isomorphic to $M(\\lambda)$, which as we just said has a one dimensional space of invariants. \\hfill $\\square$\n\n\\subsubsection{Tensor products of weighted permutation modules}\n\nAnother key fact about permutation modules for $S_n$ is that the tensor product of two permutation modules is isomorphic to a direct sum of permutation modules. More precisely, if $\\lambda$ and $\\mu$ are partitions of $n$ then $M(\\lambda) \\otimes M(\\mu)$ decomposes into permutation modules as follows:\n\nDefine a \\emph{tabloid} of type $(\\lambda, \\mu)$ to be a matrix of non-negative integers such that the $i$th row sums to $\\lambda_i$ and the $j$th column sums to $\\mu_j$ for all $i$ and $j$, and let $T(\\lambda, \\mu)$ be the set of all tabloids of type $(\\lambda, \\mu)$. Note that tabloids are just a convenient way of indexing the double cosets of the corresponding Young subgroups. For every tabloid $T \\in T(\\lambda, \\mu)$, let $M(T)$ denote the corresponding permutation module where we think of the nonzero entries of $T$ as a composition of $n$. Then we have that\n$$M(\\lambda) \\otimes M(\\mu) = \\bigoplus_{T(\\lambda,\\mu)} M(T)$$\nand in fact this holds over arbitrary rings, although we'll stick to complex representations here (see \\cite{Stanley} chapter 7). In particular we'll note that Lemma \\ref{perminv} then says the space of $S_n$-invariants in $M(\\lambda) \\otimes M(\\mu)$ is therefore equal to $|T(\\lambda,\\mu)|$, which is equal to the number of double cosets of the corresponding Young subgroups.\n\nFor weighted permutation modules essentially nothing changes, we just need to keep track of the weights. If we are decomposing the tensor product $M(\\lambda, \\mathbf{k}) \\otimes M(\\lambda, \\mathbf{k'})$, then define $T(\\lambda, \\mu)$ as before but this time define $M(T, \\mathbf{k + k'})$ to be the weighted permutation module where if there is a nonzero entry in position $(i,j)$ we weight it by $k_i + k_j'$. We have the following lemma:\n\n\\begin{lemma}\\label{permtensor}\n$$M(\\lambda, \\mathbf{k}) \\otimes M(\\lambda, \\mathbf{k'}) = \\bigoplus_{T(\\lambda,\\mu)} M(T, \\mathbf{k + k'})$$\n\\end{lemma}\n\nThe proof is essentially identical to the unweighted case which is well known and a simple application of Mackey's theorem, so we'll omit it.\n\n\\medskip\n\n\\noindent \\textbf{Example}: Suppose $n = 5$ and we want to decompose $M(\\lambda,\\mathbf{k}) \\otimes M(\\mu, \\mathbf{k'})$ where $\\lambda = \\mu = (3,2)$, $\\mathbf{k} = (0,2)$ and $\\mathbf{k'} = (1,4)$. The relevant tabloids in $T(\\lambda, \\mu)$ are:\n\n\n$$\\begin{pmatrix}\n3 & 0 \\\\\n0 & 2\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n2 & 1 \\\\\n1 & 1\n\\end{pmatrix} \n\\hspace{1cm}\n\\begin{pmatrix}\n1 & 2 \\\\\n2 & 0\n\\end{pmatrix}\n$$\nwhich give us the decomposition $$M(\\lambda,\\mathbf{k}) \\otimes M(\\mu, \\mathbf{k'}) = M((3,2), (1,6)) \\oplus M((2,1,1,1), (1,3,4,6)) \\oplus M((1,2,2), (1,3,4))$$\n\n\\medskip\n\nNext, suppose we want to decompose a tensor product of three or more permutation modules \n$$M(\\lambda^1) \\otimes M(\\lambda^2) \\otimes \\dots \\otimes M(\\lambda^m)$$\nas a direct sum of permutation modules. Of course one could just repeatedly use the rule from above for a product of two permutation modules in terms of tabloids, however it turns out that it's often simpler to just do it in one step.\n\nDefine a \\emph{multitabloid} of type $\\lambda^1, \\lambda^2, \\dots, \\lambda^m$ to be a $m$-dimensional array of nonnegative integers $b_{i_1,i_2,\\dots, i_m}$ such that the $\\ell$th generalized row sum in the $j$th direction is $\\lambda^j_\\ell$. That is,\n$$\\sum_{\\substack{(i_1,i_2,\\dots, i_m) \\\\ i_j = \\ell}} b_{i_1,i_2,\\dots, i_m} = \\lambda^j_\\ell$$\nif we then let $T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)$ denote the set of all such multitabloids. Then by an easy induction on the number of terms $m$ we get the following corollary:\n\n\\begin{corollary} \\ \\\\\n\\begin{enumerate}\n\\item For unweighted permutation modules $M(\\lambda^1), M(\\lambda^2), \\dots , M(\\lambda^m)$ we have: $$M(\\lambda^1) \\otimes M(\\lambda^2) \\otimes \\dots \\otimes M(\\lambda^m) = \\bigoplus_{B \\in T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)} M(B)$$\n\n\\item For weighted permutation modules $M(\\lambda^1, \\mathbf{k}^1), M(\\lambda^2, \\mathbf{k}^2), \\dots , M(\\lambda^m,\\mathbf{k}^m)$ we have $$M(\\lambda^1, \\mathbf{k}^1) \\otimes M(\\lambda^2, \\mathbf{k}^2) \\otimes \\dots \\otimes M(\\lambda^m, \\mathbf{k}^m) = \\bigoplus_{B \\in T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)} M(B, \\mathbf{k}^1+ \\mathbf{k}^2 + \\dots + \\mathbf{k}^m)$$\n\nWhere we view $T \\in T(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)$ as a composition by taking the nonzero entries, and $\\mathbf{k}^1+ \\mathbf{k}^2 + \\dots + \\mathbf{k}^m$ assigns the entry $b_{i_1,i_2,\\dots, i_m}$ of $B$ the weight $k^1_{i_1} + k^2_{i_2}+ \\dots +k^m_{i_m}$.\n\n\\end{enumerate}\n\\end{corollary}\n \n\n\n\\subsubsection{Stable tensor products for polynomial permutation modules}\n\nWe'll note our description of the decomposition of a tensor product of weighted permutation modules immediately gives us a compatible description for the polynomial permutation representations. Indeed one just needs to recall that $\\tilde{M}(\\lambda^1, \\lambda^2, \\dots, \\lambda^k)$ has an extra factor of size $(n-m)$ in degree zero when we write it as $M(\\lambda, \\mathbf{k})$. \n\n The benefit of this notation for polynomial permutation representations is that it is independent of $n$ provided $n$ is large enough, and it is often easier to work with. For example, when we look at the matrices involved in decomposing a tensor product of permutation modules only thing that changes when we vary $n$ is the upper left entry. \n \n For example recall from a previous example the decomposition\n \n $$V^{(1)} \\otimes V^{(1)} = \\tilde{M}(1) \\otimes \\tilde{M}(1)= \\tilde{M}((1,1)) \\oplus \\tilde{M}(\\emptyset, (1))$$\n in our other notation this is $$M((n-1,1), (0,1)) \\otimes M((n-1,1), (0,1))$$\n and if $n$ is at least $2$ the corresponding tabloids are\n \n $$\\begin{pmatrix}\nn-2 & 1 \\\\\n1 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\nn-1 & 0 \\\\\n0 & 1\n\\end{pmatrix} $$\nand we see that as $n$ changes the only thing that changes is the entry in the upper left corner. Moreover we'll note that since the sum of the entries of these matrices sum to $n$, one can always recover that entry from knowing the rest of the matrix and $n$.\n\nThis motivates the notion of \\emph{stable tabloids}, introduced in \\cite{Harman} to study periodicity phenomena in the modular representation theory of symmetric groups, and independently in \\cite{OZ1} to study certain symmetric functions related to $S_n$-representations. If $\\lambda = (\\lambda_1, \\lambda_2,\\dots \\lambda_m)$ and $\\mu = (\\mu_1, \\mu_2, \\dots ,\\mu_\\ell)$ are compositions of arbitrary sizes a stable tabloid of type $(\\lambda,\\mu)$ is a $(m+1)\\times(\\ell+1)$ matrix (with rows and columns indexed from zero to $m$ and $\\ell$ respectively) such that:\n\n\\begin{enumerate}\n\\item The $(0,0)$ entry is empty and all other entries are non-negative integers.\n\n\\item For $i \\ge 1$ the $i$th row sums to $\\lambda_i$ and the $i$th column sums to $\\mu_i$.\n\n\\end{enumerate}\n\nThe point being that for $n$ sufficiently large these are just obtained from deleting the upper left entries from the matrices in $T(\\lambda[n],\\mu[n])$ where $\\lambda[n] = (n-|\\lambda|, \\lambda_1, \\lambda_2,\\dots \\lambda_m)$ and $\\mu[n] = (n-|\\mu|, \\mu_1, \\mu_2, \\dots ,\\mu_\\ell)$ are the padded partitions.\n\nIf we let $\\tilde{T}(\\lambda,\\mu)$ denote the set of stable tabloids of type $(\\lambda, \\mu)$ then it follows (See \\cite{Harman} or \\cite{OZ1}) that the decomposition of the tensor product of (unweighted) permutation modules $M(\\lambda[n]) \\otimes M(\\mu[n])$ for $n$ sufficiently large is just:\n\n$$M(\\lambda[n]) \\otimes M(\\mu[n]) = \\bigoplus_{T \\in \\tilde{T}(\\lambda,\\mu)} M(T[n])$$\nWhere $T[n]$ is the the padded composition of $n$ obtained by taking the nonzero entries of $T$ along with an extra part of size $(n-|T|)$.\n\nIn the weighted case, Lemma \\ref{permtensor} says that if we decompose a tensor product of two polynomial weighted permutation modules of the form $\\tilde{M}(\\lambda^1, \\lambda^2, \\dots, \\lambda^m)$, then the decomposition into weighted permutation modules will again be indexed by weighted tabloids. To get the weights for a stable tabloid one just adds the corresponding weights on the rows and columns, with the caveat that the zeroth row and column have weight zero, in particular the ``extra\" part of size $n-|T|$ always gets weight zero.\n\n\\medskip\n\n\\noindent \\textbf{Remark:} We'll note that if $n$ is below the stable range we can still recover $T(\\lambda[n],\\mu[n])$ from $\\tilde{T}(\\lambda,\\mu)$ just by throwing out those stable tabloids where the sum of the entries is larger than $n$ and then filling in the upper left corner appropriately. Hence in this setting it will often be easy to deduce non-stable decompositions from the calculations in the stable range.\n\n\\medskip\n\nAs in the non-stable case, if we want to take a tensor product of three or more permutation modules\n$$M(\\lambda^1[n]) \\otimes M(\\lambda^2[n]) \\otimes \\dots \\otimes M(\\lambda^m[n])$$\n we can combine things into a single combinatorial object, a \\emph{stable multitabloid}. If $\\lambda_i$ has length $\\ell_i$ these will be $m$-dimensional arrays of dimensions $(\\ell_1 +1) \\times (\\ell_2 +1) \\times \\dots \\times (\\ell_m +1)$ such that:\n \n \\begin{enumerate}\n \\item The $(0,0,\\dots, 0)$ position is left empty, and all other entries $b_{i_1,i_2, \\dots i_m}$ are non-negative integers.\n \n \\item The $r$th generalized row sum in the $j$th direction is $\\lambda^j_r$. That is, $$\\sum_ {\\substack{(i_1,i_2, \\dots i_m) \\\\ i_j = r}} b_{i_1,i_2, \\dots i_m} = \\lambda^j_r$$\n \n \\end{enumerate}\n As before this can be seen by a simple inductive argument on the number of terms $m$, iterating the $m=2$ case at every step. Similarly to the case with two factors we can extend this to tensor products of weighted permutation modules by keeping track of the weights accordingly.\n\n\\end{subsection}\n\n\\begin{subsection}{Symmetric powers}\nWe are now ready to start talking about symmetric powers of the defining representation $V$ and tensor products thereof. The key observation is that as a representation of $T \\rtimes S_n$ the symmetric power $Sym^k(V)$ decomposes as a direct sum of weighted permutation modules.\n\nMore precisely, let $A_k$ be the set of all $k$-tuples of non-negative integers $(a_1,a_2, \\dots, a_k)$ such that \n$$a_1 + 2a_2 + 3a_3+\\dots+ka_k = k$$\nand let $A_k^n$ denote the subset of $A_k$ such that\n$$a_1 + a_2 + a_3+\\dots+a_k \\le n$$\nin particular note that if $n \\ge k$ then $A_k^n = A_k$. The following proposition describes $Sym^k(V)$ as a sum of weighted permutation modules.\n\n\\begin{proposition}\\label{symdec}\n $$Sym^k(V) \\cong \\bigoplus_{A_k^n} \\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_k))$$\n\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} If $x_1,x_2,\\dots x_n$ is the standard basis of $V$ then $Sym^k(V)$ is the space of homogeneous degree $k$ polynomials in the $x_i$'s. Then for $(a_1,a_2, \\dots, a_k) \\in A_k^n$ consider the space spanned by all monomials $x_1^{b_1}x_2^{b_2} \\dots x_n^{b_n}$ where $b_i = 1$ for $a_1$ values of $i$, $b_i = 2$ for $a_2$ values of $i$, and so on. This space is isomorphic to the weighted permutation module $\\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_k))$, with the monomials forming the permutation basis. $Sym^k(V)$ is spanned by monomials and each monomial lies in some such space for a unique $(a_1,a_2, \\dots, a_k) \\in A_k^n$. \\hfill $\\square$\n\n\\medskip\n\nWe'll note that in this case the weighted permutation modules $\\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_k))$ involved are all irreducible $T \\rtimes S_n$ representations so we could also have written:\n\n$$Sym^k(V) \\cong \\bigoplus_{A_k^n} V^{(a_1),(a_2),(a_3), \\dots, (a_k)}$$\n\n\n\\subsubsection{Combinatorics of $A_k$}\n\nIn preparation for the next section we'd like to give two interpretations for $A_k$, which will generalize nicely to tensor products of symmetric powers.\n\n\\medskip\n\n\\noindent \\textbf{Polynomial interpretation:} Given a sequence $(a_1,a_2, \\dots, a_k)$ consider the polynomial $$a(x) := a_1x+a_2x^2+a_3x^3+\\dots+a_kx^k$$\nthe condition that $(a_1,a_2, \\dots, a_k) \\in A_k$ just corresponds to the condition that $a'(1) = k$, and the condition that $(a_1,a_2, \\dots, a_k) \\in A_k^n$ adds the additional constraint that $a(1) \\le n$. Hence $A_k^n$ can be naturally identified with the set of polynomials $a(x) \\in \\mathbb{Z}_{\\ge 0}[x]$ such that $a(0)= 0$, $a(1) \\le n$, and $a'(1) = k$.\n\n\\medskip\n\n\\noindent \\textbf{Multiset partition interpretation:} Suppose $M$ is a multiset. A \\emph{multiset partition} of $M$ is a collection (i.e a multiset) $\\{ M_1, M_2, \\dots, M_m\\}$ of non-empty sub-multisets of $M$ such that the multiplicity of an element of $M$ is equal to the sum of the multiplicities of it in the $M_i$'s. \n\n$A_k$ can naturally be thought of as an indexing set for multiset partitions of the multiset $M = \\{1,1,1, \\dots, 1\\} = \\{1^k\\}$ consisting of a single element with multiplicity $k$. Indeed an element $(a_1,a_2, \\dots, a_k) \\in A_k$ just corresponds to the unique multiset partition with $a_1$ parts of size $1$, $a_2$ parts of size 2, and so on. Under this identification $A_k^n$ is just indexing those multiset partitions of $M$ into at most $n$ parts.\n\n\n\\vspace{.5cm}\n\nAs mentioned before, in this case the decomposition into weighted permutation modules also gives the decomposition into irreducible $T \\rtimes S_n$ modules. If we forget about the action of $T$ this gives us a combinatorial interpretation for the decomposition of $Sym^k(V)$ as a representation of $S_n$.\n\n For $(a_1, a_2, \\dots, a_k) \\in A_k^n$ define the \\emph{associated composition of $n$} as $(n-(a_1 + a_2 + a_3+\\dots+a_k), a_1, a_2, \\dots, a_k)$, and define the associated composition of $n$ to a multiset partition of $M = \\{1,1,1, \\dots, 1\\}$ via the identification above. \n\n\\begin{corollary} If $\\lambda$ is a partition of $n$ then the multiplicity of $S^\\lambda$ in $Sym^k(V)$ is equal to the number of pairs $(P,T)$ where $P$ is a multiset partition of $\\{1^k\\}$ with at most $n$ parts and $T$ is a semistandard Young tableau of shape $\\lambda$ and content equal to the associated composition of $n$ to $P$.\n\n\n\\end{corollary}\n\n\\noindent \\textbf{Proof:} Proposition \\ref{symdec} and the above interpretation tells us that $Sym^k(V)$ decomposes into weighted permutation modules indexed by these multiset partitions. Restricting a weighted permutation modules to $S_n$ just gives an ordinary permutation module corresponding to the associated composition of $n$, and permutation modules decompose into irreducibles with multiplicities given by Kostka numbers which count semistandard Young tableaux. \\hfill $\\square$\n\n\\medskip\n\nIn particular, the case where $\\lambda = (n)$ and we are just looking at the space of symmetric group invariants is of particular interest so we'll state it as a separate corollary.\n\n\\begin{corollary}\nThe space of $S_n$-invariants in $Sym^k(V)$ is equal to the number of multiset partitions of $\\{1^k\\}$ with at most $n$ parts.\n\n\\end{corollary}\n\n\\end{subsection}\n\n\\begin{subsection}{Tensor products of symmetric powers}\n\nNow let's extend this analysis from a single symmetric power of $V$ to a product of symmetric powers $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V).$$\nWe already saw how to decompose a single symmetric power into weighted permutation modules, and we also saw how to decompose a tensor product of two weighted permutation modules, so all that remains is to put it all together and keep track of the terms to get a concise combinatorial description.\n\n\\subsubsection{The two factor case}\n\nFirst let's go through how one would do this in the case where there are just two factors $Sym^{k_1}(V) \\otimes Sym^{k_2}(V)$ just to demonstrate the idea.\n\n\\begin{enumerate}\n\\item First one would use Proposition \\ref{symdec} to write $$Sym^{k_i}(V) \\cong \\bigoplus_{A_{k_i}^n} \\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_{k_i}))$$ for $i = 1,2$.\n\n\\item Then for each pair in $(a,a') \\in A_{k_1}^n \\times A_{k_2}^n$ we would decompose $$\\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_{k_1})) \\otimes \\tilde{M}((a'_1),(a'_2),(a'_3), \\dots, (a'_{k_2}))$$\naccording to Lemma \\ref{permtensor} into weighted permutation modules indexed by stable tabloids of type $(a,a')$ (where we think of $a = (a_1,\\dots, a_{k_1})$ and $a' = (a'_1,\\dots, a'_{k_2})$ as compositions) with appropriate weights.\n\n\\end{enumerate}\n\nCombining this into a single step by summing over all stable tabloids as $(a,a')$ varies over $A_{k_1}^n \\times A_{k_2}^n$ we are summing over all $(k_1+1) \\times (k_2+1)$ matrices with rows and columns indexed from $0$ to $k_1$ and $k_2$ respectively such that:\n\n\\begin{itemize}\n\n\\item The $(0,0)$ entry is blank, and all other entries are non-negative integers $b_{ij}$.\n\n\\item The sum of the entries weighted by their row number is equal to $k_1$ and the sum of the entries weighted by their column number is equal to $k_2$. That is,\n\n$$\\sum_{i,j} i b_{ij} = k_1 \\hspace{1cm} \\sum_{i,j} j b_{ij} = k_2.$$\n\n\n\\item The sum of the entries of the matrix is at most $n$ (if we are in the stable range where $n > k_1 +k_2$ this condition is redundant).\n\n\\end{itemize}\n\nFor each such matrix we get a weighted permutation module $\\tilde{M}(\\lambda_1,\\lambda_2, \\dots, \\lambda_{k_1 + k_2})$ where $\\lambda_\\ell$ is the partition obtained by taking the entries $b_{ij}$ with $i+j = \\ell$.\n\n\\medskip\n\n For example if we want to decompose $Sym^2(V) \\otimes Sym^2(V)$ there are $8$ such stable tabloids that appear:\n \n \n $$\\begin{pmatrix}\n\\ & 2 & 0 \\\\\n2 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 1 & 0 \\\\\n1 & 1 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 0 & 0 \\\\\n0 & 2 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix} \n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 1 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 1 & 0\n\\end{pmatrix}$$\n\n $$\n\\begin{pmatrix}\n\\ & 0 & 0 \\\\\n1 & 0 & 1 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 2 & 0 \\\\\n0 & 0 & 0 \\\\\n1 & 0 & 0\n\\end{pmatrix} \n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 0 & 1 \\\\\n2 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}\n\\hspace{1cm}\n\\begin{pmatrix}\n\\ & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix} $$\n\nWe then read off the corresponding weighted permutation modules by looking at the diagonals that go from the lower left to the upper right to obtain the decomposition\n\n\\begin{equation}\n\\begin{split}\nSym^2(V) \\otimes Sym^2(V) = \\tilde{M}((2,2)) \\oplus \\tilde{M}((1,1),(1)) \\oplus \\tilde{M}(\\emptyset, (2)) \\\\ \\oplus \\ 2\\tilde{M}((1),\\emptyset, (1)) \\oplus 2\\tilde{M}((2),(1)) \\oplus \\tilde{M}(\\emptyset, \\emptyset, \\emptyset, (1))\n\\end{split}\n\\end{equation}\nwhich holds for all $n \\ge 4$. If $n$ is less than $4$ we obtain the decomposition by keeping just those terms where the sum of the sizes of the partitions involved is at most $n$.\n\n\\subsubsection{The general case}\n\nNow let's extend this to the general case of $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V).$$\nHere the program will be basically identical to the two factor case\n\n\\begin{enumerate}\n\\item First one would use Proposition \\ref{symdec} to write $$Sym^{k_i}(V) \\cong \\bigoplus_{A_{k_i}^n} \\tilde{M}((a_1),(a_2),(a_3), \\dots, (a_{k_i}))$$ for $i = 1,2,\\dots, m$.\n\n\\item Then for each $m$-tuple in $(a^1,a^2,\\dots, a^m) \\in A_{k_1}^n \\times A_{k_2}^n\\times \\dots A_{k_m}^n$ we would decompose $$ \\bigotimes_i \\tilde{M}((a^i_1),(a^i_2),(a^i_3), \\dots, (a^i_{k_i}))$$\naccording to Lemma \\ref{permtensor} into weighted permutation modules indexed by stable multitabloids of type $(a^1,a^2, \\dots, a^m)$ (where we think of $a^i = (a^i_1,\\dots, a^i_{k_i})$ as compositions) with appropriate weights.\n\n\\end{enumerate}\n\nAs before we can combine this into a single step by counting up the multitabloids as we vary our choice of $(a^1,a^2,\\dots, a^m) \\in A_{k_1}^n \\times A_{k_2}^n\\times \\dots A_{k_m}^n$. Now the objects we care about are $(k_1+1)\\times (k_2 +1) \\times \\dots \\times (k_m+1)$ arrays with rows, columns, etc. indexed from $0$ to $k_1$, $k_2$, and so on respectively such that:\n\n\n\\begin{itemize}\n\n\\item The $(0,0, \\dots , 0)$ entry is blank, and all other entries are non-negative integers $b_{i_1,i_2,\\dots, i_m}$.\n\n\\item The sum of the entries weighted by their $jth$ index value is equal to $k_j$ for all $j$. That is:\n\n$$\\sum_{i_1,i_2,\\dots, i_m} i_j b_{i_1,i_2,\\dots, i_m} = k_j$$\n\n\n\\item The sum of the entries of the array is at most $n$ (if we are in the stable range where $n > k_1 +k_2 + \\dots +k_m$ this condition is redundant).\n\n\\end{itemize}\n\n\\medskip\n\nIf we let $A^n_{k_1,k_2,\\dots,k_m}$ denote the set of such arrays then then putting everything together gives the following proposition describing the decomposition of a product of symmetric powers of $V$ into permutation modules.\n\n\\begin{proposition}\\label{symproddec}\n$$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V) = \\bigoplus_{B \\in A^n_{k_1,k_2,\\dots,k_m}} \\tilde{M}(\\lambda_1,\\lambda_2, \\dots, \\lambda_{k_1 + k_2+\\dots +k_m})$$\nwhere $\\lambda_\\ell = \\lambda_\\ell (B)$ is the partition obtained from the $B$ by taking the nonzero entries $b_{i_1,i_2,\\dots,i_m}$ with $i_1+i_2+\\dots+i_m = \\ell$.\n\n\\end{proposition}\n\n\n\\subsubsection{Combinatorics of $A^n_{k_1,k_2,\\dots,k_m}$}\n\nAs in the case of a single symmetric power, we will now give two interpretations for the set $A^n_{k_1,k_2,\\dots,k_m}$, one in terms of polynomials and one in terms of multiset partitions.\n\n\\medskip\n\n\\noindent \\textbf{Polynomial interpretation}: Given an array $B$ of numbers $b_{i_1, i_2, \\dots, i_m}$ (for $i_j \\ge 0$) with finitely many nonzero entries we can construct the polynomial\n\n$$P_B(x_1,x_2, \\dots x_m) := \\sum_{i_1, i_2, \\dots, i_m} b_{i_1, i_2, \\dots, i_m}x_1^{i_1}x_2^{i_2}\\dots x_m^{i_m}$$\nthen the condition that $B \\in A^n_{k_1,k_2,\\dots,k_m}$ translates into\n\n\\begin{enumerate}\n\\item $P_B(x_1,x_2, \\dots x_m)$ has constant term zero and all other coefficients are non-negative integers.\n\n\\item For $i =1, 2, \\dots m$ $$ \\frac{d}{dx_i}P_B(x_1,x_2, \\dots x_m) |_{(x_1,x_2, \\dots x_m) = (1,1,\\dots 1)} = k_i$$\n\n\\item$ P_B(1,1,\\dots 1) \\le n$\n\n\\end{enumerate}\n\nHence we may identify $A^n_{k_1,k_2,\\dots,k_m}$ with the set of such polynomials satisfying these three conditions.\n\n\\medskip\n\n\\noindent \\textbf{Multiset partition interpretation:} $A^n_{k_1,k_2,\\dots,k_m}$ has a natural bijection with the set of multiset partitions of $\\{1^{k_1}, 2^{k_2}, \\dots m^{k_m} \\}$. Explicitly, an array $B \\in A^n_{k_1,k_2,\\dots,k_m}$ with entries $b_{i_1, i_2, \\dots, i_m}$ corresponds to the multiset partition where $\\{1^{i_1}, 2^{i_2}, \\dots, m^{i_m}\\}$ appears $b_{i_1, i_2, \\dots, i_m}$ times.\n\n\\vspace{.5cm}\n\nWe find the multiset partition interpretation to be more conceptually satisfying and will mostly use it to state results, however we've included the polynomial interpretation as we feel it may be more amenable to computations.\n\n\\medskip\n\n The \\emph{type}, $\\text{Type}(P)$, of a multiset partition $P \\vDash M$ is the sequence $\\lambda^1, \\lambda^2, \\lambda^3, \\dots$ of partitions where $\\text{Type}(P)^i = \\lambda^i$ records the multiplicities of the parts of $P$ of size $i$. \n \n For example, if $M = \\{1,1,2\\}$ and $P = \\{\\{1\\},\\{1\\},\\{2\\}\\}$ then $\\text{Type}(P)$ is the sequence $\\lambda^1, \\lambda^2, \\lambda^3, \\dots$ where $\\lambda^1 = (2,1)$ and all other partitions are the empty set. This is since $P$ has three parts of size $1$, two of which are the same as one another (i.e. $\\{1\\}$) and one that appears with multiplicity $1$ (i.e $\\{2\\}$). By convention we'll drop off the trailing empty sets and just write $\\text{Type}(P) = ((2,1))$. If instead we took $P' = \\{1,1,2\\}$ then $\\text{Type}(P') = (\\emptyset, \\emptyset, (1))$.\n \nTranslating Proposition \\ref{symproddec} into this language we get the following corollary:\n\n\\begin{corollary}\\label{symtoperm}\n\n$$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V) = \\bigoplus_{\\substack{P \\vDash \\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\} \\\\ |P| \\le n}}\\tilde{M}(\\text{Type}(P))$$\n\n\\end{corollary}\n\n\\medskip\n\nWe are now ready to give our main result, a combinatorial interpretation for the decomposition of a tensor product of symmetric powers into irreducible $T \\rtimes S_n$ modules.\n\n\\begin{proposition}\\label{maindecomp}\nThe multiplicity of $V^{\\lambda^1, \\lambda^2, \\dots, \\lambda^j}$ inside $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$\nis equal to the number of tuples $(P,T_1,T_2, \\dots T_j)$ such that $P$ is a multiset partition of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with at most $n$ parts and $T_i$ is a semistandard Young tableau of shape $\\lambda^i$ and content $\\text{Type}(P)^i$.\n\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} Corollary \\ref{symtoperm} gives us a decomposition of this tensor product of symmetric powers into permutation modules indexed by multiset partitions. Lemma \\ref{multikostka} then gives us the decomposition of a permutation module in terms of Kostka numbers. Finally, the combinatorial interpretation of Kostka numbers as counting semistandard Young tableaux gives the result. \\hfill $\\square$\n\n\\medskip\n\n\\noindent \\textbf{Remark:} In the case where each $k_i$ is $1$ this refines Proposition $5.10$ in \\cite{BHH} to the case of weighted permutation modules, but the proof is morally very similar.\n\n\\medskip\n\nIf we forget about the action of $T$, Corollary \\ref{symtoperm} recovers an interpretation of the decomposition of $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$ into irreducible representations of $S_n$ due to Orellana and Zabrocki (\\cite{OZ1} Theorem 5). \n\n Define the \\emph{unweighted type} $\\text{UType}(P)$ of a multiset partition $P$ to be the partition recording the multiplicities of the multisets in $P$ (in other words just combine the partitions making up $\\text{Type}(P)$ into a single partition). And let $\\text{UType}(P)[n]$ be the corresponding partition of $n$ obtained by adding a part of size $(n-|P|)$ to $\\text{UType}(P)$.\n\n\\begin{proposition}\nThe multiplicity of the Specht module $S^\\lambda$ inside $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$\nis equal to the number of tuples $(P,T)$ such that $P$ is a multiset partition of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with at most $n$ parts and $T$ is a semistandard Young tableau of shape $\\lambda$ and content $\\text{UType}(P)[n]$.\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} Upon restricting to $S_n$ Corollary \\ref{symtoperm} tells us that $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V) = \\bigoplus_{\\substack{P \\vDash \\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\} \\\\ |P| \\le n}}M(\\text{UType}(P)[n])$$\nas a representation of $S_n$. Then the decomposition of $M(\\text{UType}(P)[n])$ is just given by appropriate Kostka numbers, which again count semistandard Young tableaux of fixed shape and content. \\hfill $\\square$\n\n\\medskip\n\n\\noindent \\textbf{Remark:} In \\cite{OZ1} Orellana and Zabrocki combine a pair $(P,T)$ into a single combinatorial object: a multiset tableau. However morally these are the same description, and we'll leave it as an exercise to any interested parties to make explicit the bijection between these pairs and appropriate multiset tableaux. \n\nWe'll also note that they stated their results in terms of a new basis of the ring of symmetric functions they defined corresponding to irreducible symmetric group representations of large symmetric groups. In this context we'll mention that one should not expect such a basis to exist for $T \\rtimes S_n$, as (among other reasons) it is possible for two elements of $T \\rtimes S_n$ to be $GL_n$-conjugate but not $T \\rtimes S_n$-conjugate.\n\n\\medskip\n\nFinally, we'll close out this section by stating as a corollary the important special cases of the above propositions where we just look at the space of $S_n$-invariants.\n\n\\begin{corollary}\n\nIf $\\mu = (\\mu_1, \\mu_2, \\dots, \\mu_\\ell)$ is a partition of $k_1+k_2+\\dots+k_m$ with at most $n$ parts the dimension of $S_n$-invariants in the symmetrized weight space\n\n$$(Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V))_{\\bar{\\mu}}$$\nis equal to the number of multiset partitions of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with parts of sizes $\\mu_1, \\mu_2, \\dots, \\mu_\\ell$. In particular, the dimension of the space of $S_n$-invariants inside full space $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$\nis equal to the number multiset partitions of $\\{1^{k_1},2^{k_2}, \\dots m^{k_m} \\}$ with at most $n$ parts.\n\n\\end{corollary}\n\nIn particular if all the $k_i$'s are equal to $1$ and $m > n$ this recovers the fact that the space of $S_n$-invariants in $V^{\\otimes m}$ is given by the Bell number $B_m$. If we let the $k_i$'s be arbitrary but assume $n$ is sufficiently large this gives a representation theoretic context for the generalized Bell numbers studied in \\cite{Griffiths}.\n\n\n\\end{subsection}\n\\end{section}\n\n\\begin{section}{Future directions and computations}\\label{future}\n\nThe main motivating problem of finding a combinatorial interpretation for the restriction of an irreducible polynomial representation of $GL_n$ to $T \\rtimes S_n$ remains open. In light of Proposition \\ref{maindecomp} and the fact that weighted permutations satisfy an upper triangularity property with respect to the irreducibles it is reasonable to expect that such an interpretation could be formulated in terms of multiset partitions and Young tableaux, however for the time being such an interpretation remains elusive.\n\nFor the remainder of the paper we will outline some future directions of study beyond this motivating question. In particular we will outline a version of Schur-Weyl duality for $T \\rtimes S_n$, highlight a connection between the restriction problem and the Foulkes conjecture, and include some low-degree computations of restriction from $GL_n$ to $T \\rtimes S_n$.\n\n\\begin{subsection}{Schur-Weyl duality for $T \\rtimes S_n$}\n\nNow we'll breifly describe a version of Schur-Weyl duality for $T \\rtimes S_n$. As before let $V$ denote the defining representation of $GL_n$. We have a tower of subgroups \n$$S_n \\subset T\\rtimes S_n \\subset GL_n$$\nacting on $V$ and hence on $V^{\\otimes k}$. Therefore we get a reverse inclusion of endomorphism algebras \n$$\\text{End}_{S_n}(V^{\\otimes k}) \\supset \\text{End}_{T \\rtimes S_n}(V^{\\otimes k}) \\supset \\text{End}_{GL_n}(V^{\\otimes k})$$\nwhere the outer two terms are familiar instances of Schur-Weyl duality.\n\nOn the right we have classical Schur-Weyl duality in which $S_k$ acts $GL_n$-equivariantly on $V^{\\otimes k}$ by permuting the factors. For any $n$ and $k$ these maps span $ \\text{End}_{GL_n}(V^{\\otimes k})$, and if $n \\ge k$ this provides an isomorphism $\\mathbb{C}[S_k] \\cong \\text{End}_{GL_n}(V^{\\otimes k})$.\n\nOn the left the situation is similar. The partition algebra $Par(n,k)$ acts $S_n$-equivariantly on $V^{\\otimes k}$ for all $n$ and $k$. The map $Par(n,k) \\to \\text{End}_{S_n}(V^{\\otimes k})$ is always surjective, moreover the kernel has an explicit combinatorial description. In particular, if $n \\ge 2k$ this map is an isomorphism (See \\cite{HR}, \\cite{BH1}, \\cite{BH2} for more details).\n\nAs such one may expect that we can explicitly describe the endomorphism algebra $\\text{End}_{T \\rtimes S_n}(V^{\\otimes k})$ at least for $n$ sufficiently large compared to $k$ and indeed this is the case. Recall that $Par(n,k)$ has a basis indexed by set-partitions of the set $\\{1,2,\\dots,k\\} \\cup \\{1',2',\\dots k'\\}$, we say that such a set-partition is \\emph{balanced} if each part $P$ of the partition satisfies $|P \\cap \\{1,2,\\dots,k\\}| = |P \\cap \\{1',2',\\dots,k'\\}|$. \n\n\\medskip\n\nThe following proposition describes a version of Schur-Weyl duality for $T\\rtimes S_n$:\n \n\\begin{proposition} Let $Par^{bal}(k)$ denote the subspace of $Par(n,k)$ spanned by the balanced set partitions. The following holds:\n\\begin{enumerate}\n\\item $Par^{bal}(k)$ is a subalgebra of $Par(n,k)$.\n\\item The obvious identification of these subspaces for $Par(n,k)$ and $Par(m,k)$ is an isomorphism of algebras (motivating the notation $Par^{bal}(k)$ not involving $n$).\n\\item $Par^{bal}(k)$ acts $T\\rtimes S_n$-equivariantly on $V^{\\otimes k}$ giving an algebra homomorphism $Par^{bal}(k) \\to \\text{End}_{T \\rtimes S_n}(V^{\\otimes k})$, this is surjective for all values of $n$ and $k$ and is an isomorphism whenever $n \\ge k$.\n\n\n\\end{enumerate}\n\\end{proposition}\n\n\\noindent \\textbf{Sketch of proof:} For parts $1$ and $3$ we use the description of Schur-Weyl duality for $S_n$ and check that $Par^{bal}(k)$ is exactly the subalgebra of $Par(n,k)$ that preserves $S_n$-orbits of $T$-weights. For part $2$ one just notes that in the partition algebra $Par(n,k)$, the dependence on $n$ arises when we stack two partition diagrams and there are isolated components in the middle, but for balanced partitions that can never happen. \\hfill $\\square$\n\n\\medskip\n\nWe didn't pursue this direction any further, but here are a few problems we think may be interesting:\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe $Par^{bal}(k)$ by generators and relations.\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Is there a double centralizer property? In other words, does the image of $T \\rtimes S_n$ span the algebra of $Par^{bal}(k)$-endomorphisms of $V^{\\otimes k}$? If so, use this to describe the representations of $Par^{bal}(k)$.\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe the kernel of the natural map $Par^{bal}(k) \\to \\text{End}_{T \\rtimes S_n}(V^{\\otimes k})$ for $n < k$.\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe the $T \\rtimes S_n$ endomorphism algebras of $V^{\\otimes r} \\otimes V^{* \\otimes s}$. We'll note that this should be something in between a partition algebra and a walled Brauer algebra.\n\\end{subsection}\n\n\\medskip\n\n\\noindent \\textbf{Problem:} Describe diagrammatically the endomorphism algebras of $$Sym^{k_1}(V) \\otimes Sym^{k_2}(V) \\otimes \\dots \\otimes Sym^{k_m}(V)$$ as $S_n$ and $T \\rtimes S_n$ representations. We'll go ahead and coin the terms ``multiset partition algebras\" and ``balanced multiset partition algebras\".\n\n\n\\begin{subsection}{$S_n$-invariants, plethysm, and Foulkes conjecture}\n\nA particularly important subcase of general problem of restricting an irreducible representation $W(\\lambda)$ from $GL_n$ to $T \\rtimes S_n$ is to understand the the space of $S_n$-invariants in each symmetrized weight space $W(\\lambda)_{\\bar{\\mu}}$ as $\\mu$ varies over $S_n$-orbits of weights.\n\nIf $\\mu$ has $m_1$ parts of size $1$, $m_2$ parts of size $2$, and so on. Let \n\n$$S_{\\mu} := S_1^{m_1} \\times S_2^{m_2} \\times \\dots \\times S_k^{m_k}$$\ndenote the Young subgroup of $S_{|\\mu|}$ corresponding to $\\mu$, and let $N(\\mu)$ denote its normalizer in $S_{|\\mu|}$. Note that $N(\\mu)$ is just the product of the wreath products $S_i \\wr S_{m_i}$. \n\nThe following proposition, essentially due to Gay, relates the space of $S_n$-invariants in the symmetrized weight space $W(\\lambda)_{\\bar{\\mu}}$to the representation theory of $S_{|\\lambda|}$.\n\n\\begin{proposition} \\textbf{(\\cite{Gay} Theorem 2)} The dimension of the space of $S_n$-invariants in $W(\\lambda)_{\\bar{\\mu}}$is equal to the multiplicity of the Specht module $S^\\lambda$ in $\\text{Ind}_{N(\\mu)}^{S_{|\\mu|}}(\\mathbf{1})$.\n\\end{proposition}\n\n\\noindent \\textbf{Remark:} We'll note that Gay only considered the ``zero\" weight space case where $\\mu = (a,a,a,\\dots,a)$. However his proof easily generalizes, and moreover one can reduce the general case to this by a straightforward application of the Littlewood-Richardson rule for restricting to the Levi subgroup $GL_{m_1} \\times GL_{m_2} \\times \\dots \\times GL_{m_k}$.\n\n\\medskip\n\nFinding a combinatorial interpretation of such modules induced from normalizers of Young subgroups is an important open problem in combinatorial representation theory which in the language of symmetric functions is equivalent to decomposing the plethysms $h_a[h_b]$ into Schur functions. \n\nIn particular one important conjecture in this area is Foulkes conjecture from 1950, about embedding one such induced module into another. Translated into the language of this paper Foulkes conjecture is the following:\n\n\\begin{conjecture} \\textbf{(Foulkes conjecture \\cite{Foulkes})} If $a < b$ then the space of $S_n$-invariants in the symmetrized weight space of weight $(a,a,\\dots, a, 0,0,\\dots,0)$ is at least as large as the space of $S_n$-invariants in the symmetrized weight space of weight $(b,b,\\dots, b, 0,0,\\dots,0)$ for any polynomial representation of $GL_n$ of degree $ab$.\n\n\\end{conjecture}\n\nWe'll note that the usual representation theoretic formulations of this conjecture are either entirely about general linear group representations or entirely about symmetric group representations. As far as we can tell this ``mixed\" formulation of the conjecture seems to be missing from much of the literature on Foulkes conjecture.\n\n It suggests a possible approach to the conjecture by studying the representation theory of the spherical subalgebra $eAe$, where $$A = U(\\mathfrak{gl}_n) \\rtimes \\mathbb{C}(S_n) \\hspace{.5cm} \\text{ and } \\hspace{.5cm} e = \\frac{1}{n!}\\sum_{\\sigma \\in S_n} \\sigma$$\nwhich naturally acts on the space of $S_n$-invariants inside a $GL_n$ representation. \n\n\\medskip\n\nUsing Proposition \\ref{maindecomp} we can give a purely combinatorial formulation of a weak version of Foulkes conjecture for tensor products of symmetric powers:\n\n\\begin{conjecture} \\textbf{(Weak Foulkes conjecture)} If $a < b$ and $M$ is a multiset of size $ab$ then the number of multiset partitions of $M$ into $b$ parts of size $a$ at least as large as the number of multiset partitions of $M$ into $a$ parts of size $b$.\n\\end{conjecture}\n\nNote that in terms of symmetric functions this is equivalent to the conjecture that $h_b[h_a]-h_b[h_a]$ has nonnegative coefficients when expressed in the basis of monomial symmetric functions (whereas the full Foulkes conjecture says it has non-negative coefficients in the basis of Schur functions).\n\n\n\n\\end{subsection}\n\n\n\n\\subsection{Low degree calculations}\nProposition \\ref{maindecomp} combined with the Jacobi-Trudi identity gives us a combinatorial method for computing the decomposition of an irreducible polynomial representation of $GL_n$ to $T \\rtimes S_n$ (although not a positive combinatorial formula). Here are some explicit calculations for polynomial representations of degree at most $4$, where we always assume we are in the stable range with $n$ larger than the degree.\n\n$$ W( \\emptyset) \\longrightarrow V^\\emptyset $$\n$$W(1) \\longrightarrow V^{(1)} $$\n$$W(2) \\longrightarrow V^{(2)} \\oplus V^{\\emptyset, (1)}$$\n$$ W(1,1) \\longrightarrow V^{(1,1)}$$\n$$W(3) \\longrightarrow V^{(3)} \\oplus V^{(1),(1)} \\oplus V^{\\emptyset,\\emptyset, (1)}$$\n$$W(2,1) \\longrightarrow V^{(2,1)} \\oplus V^{(1),(1)}$$\n$$W(1,1,1) \\longrightarrow V^{(1,1,1)}$$\n$$W(4) \\longrightarrow V^{(4)} \\oplus V^{(2),(1)} \\oplus V^{\\emptyset, (2)} \\oplus V^{(1), \\emptyset, (1)} \\oplus V^{\\emptyset,\\emptyset, \\emptyset, (1)}$$\n$$W(3,1) \\longrightarrow V^{(3,1)} \\oplus V^{(1,1),(1)} \\oplus V^{(2),(1)} \\oplus V^{\\emptyset, (1,1)} \\oplus V^{(1),\\emptyset,(1)} $$\n$$ W(2,2) \\longrightarrow V^{(2,2)} \\oplus V^{(2),(1)} \\oplus V^{\\emptyset, (2)}$$\n$$W(2,1,1) \\longrightarrow V^{(2,1,1)} \\oplus V^{(1,1),(1)}$$\n$$W(1,1,1,1) \\longrightarrow V^{(1,1,1,1)}$$\n\n\n\n\n\\end{section}\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\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn this paper, we study the 2D viscous\nshallow water equations with a more general diffusion\n\\begin{equation}\\label{1.1}\n\\left\\{\n\\begin{array}{ll}\nh_t+\\mbox{div}(hu)=0,\\\\\nh(u_t+u\\cdot\\nabla u)-\\nu\\nabla\\cdot(hD(u))-\\nu\\nabla(h \\textrm{div}(u))+h\\nabla h=0, \\\\\nu(0,\\cdot)=u_0,\\,\\,h(0,\\cdot)=h_0 ,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $h(t,x)$ is the height of fluid surface,\n$u(t,x)=(u_1(t,x),u_2(t,x))$ is the horizontal velocity vector\nfield, $D(u)=\\f12(\\nabla u+\\nabla u^t)$ is the deformation tensor, and\n$\\nu>0$ is the viscous coefficient. If the diffusion terms in (\\ref{1.1}) are\nreplaced by $-\\nu\\nabla\\cdot(h\\nabla u)$,\nthen (\\ref{1.1}) turns into the usual viscous shallow water equations.\n\nRecently, the viscous shallow water equations have been widely\nstudied by Mathematicians, see the review paper\\cite{BDM}. Bui\n\\cite{Bui} proved the local existence and uniqueness of classical\nsolutions to the Cauchy-Dirichlet problem for the shallow water\nequations with initial data $h_0$, $u_0$ in H\\\"{o}lder spaces as\nwell as $h_0$ bounded away from vacuum. Kloeden\\cite{Klo} and\nSundbye\\cite{Su1} independently proved global existence and\nuniqueness of classical solutions to the Cauchy-Dirichlet problem in\nSobolev spaces. Later, Sundbye\\cite{Su2} also proved global\nexistence and uniqueness of classical solutions to the Cauchy\nproblem. However, for all above results (except \\cite{Bui}), the\nauthors only consider the case when the initial data $h_0$ is a\nsmall perturbation of some positive constant $\\bar h_0$ and $u_0$ is\nsmall in some sense. Very recently, Wang and Xu\\cite{WX} proved the local\nwell-posedness of the Cauchy problem in Sobolev spaces\nfor the large data $u_0$ and $h_0$ closing to $\\bar h_0$. More\nprecisely, they obtained the following result.\n\\begin{Theorem}\\label{WX}\\cite{WX} Let $\\bar{h}_0$ be a strictly positive constant and $s>2$.\nAssume that\n\\begin{eqnarray*}&&(i)\\quad(u_0,h_0-\\bar{h}_0)\\in\nH^s(\\mathop{\\bf R\\kern 0pt}\\nolimits^2)\\otimes H^s(\\mathop{\\bf R\\kern 0pt}\\nolimits^2);\\hspace{7cm}\\\\\n&&(ii)\\quad\\|h_0-\\overline{h}_0\\|_{H^s}\\ll\\overline{h}_0.\\end{eqnarray*} Then\nthere exist a positive time $T$ and a unique solution $(u,h)$ of\n$(\\ref{1.1})$ such that\n\\begin{eqnarray}\\label{1.2} u,\\,\\, h-\\bar h_0\\in\nL^\\infty([0,T],H^s),\\quad \\nabla u\\in L^2([0,T]; H^s).\n\\end{eqnarray}\nMoreover, there exists a strictly positive constant $c$ such that if\n\\begin{eqnarray}\\label{1.3}\n\\|u_0\\|_{H^s}+ \\|h_0-\\bar{h}_0\\|_{H^s}\\le c,\n\\end{eqnarray}\nthen we can choose $T=+\\infty$.\n\\end{Theorem}\n\nOne purpose of this paper is to study the well-posedness of\n$(\\ref{1.1})$ for the initial data with the minimal regularity.\nFor the incompressible Navier-Stokes equations, such research has been initiated by\nFujita and Kato\\cite{FK}, see also \\cite{Can1, Can2, Mey2} for other relevant results. They proved local well-posedness for the incompressible Navier-Stokes equations\nin the scaling invariant space. The scaling invariance means that\nif $(u,p)$ is a solution of the incompressible Navier-Stokes equations with initial data $u_0(x)$, then\n\\begin{eqnarray}\n\\label{1.4}u_\\lambda(t,x)\\triangleq\\lambda u(\\lambda^2 t,\\lambda x),\\quad\np_\\lambda(t,x)\\triangleq \\lambda^2 p(\\lambda^2 t,\\lambda x)\n\\end{eqnarray}\n is also a solution of the incompressible Navier-Stokes equations with\n$u_{0,\\lambda}\\triangleq \\lambda u_0(\\lambda x)$.\nObviously, $\\dot H^{\\frac d 2-1}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$\nis a scaling invariant space under the scaling of (\\ref{1.4}), i.e.\n$$\\|u_\\lambda\\|_{\\dot{H}^{\\frac d 2-1}}=\\|u\\|_{\\dot{H}^{\\frac d 2-1}}.$$\nThe equations $(\\ref{1.1})$ have no scaling invariance like the incompressible Navier-Stokes equations.\nHowever, due to the similarity of the structure between (\\ref{1.1}) and the incompressible Navier-Stokes equations,\nwe still solve (\\ref{1.1}) for initial data whose regularity fits with the scaling of (\\ref{1.4}).\nIt should be pointed out that R. Danchin was the first to consider the similar problem for the compressible\nNavier-Stokes equations, and some ideas of this paper is motivated by \\cite{D1}.\n\nThe second purpose of this paper is to prove the local well-posedness of (\\ref{1.1}) under more natural assumption that\nthe initial height is bounded away from zero.\nFor the initial data with slightly higher regularity,\nthis can be easily obtained by modifying the argument of Danchin\\cite{D5}.\nHowever, for the initial data with low regularity, his method is not applicable any more, since\nthe proof of \\cite{D5} relies on the fact that some profits can be gained from the inclusion map\n$B^s\\hookrightarrow L^\\infty$ in the case of $s>\\frac d2$.\nFor this reason, we have to introduce some kind of weighted Besov space $E^s_T$(see Section 3) which is crucial to\nget rid of the condition that the initial height $h_0$ is close to $\\bar{h}_0$.\nOne important observation is that the $E^s_T$ norm of the solution is small for small time $T$.\n\n\n\nBefore stating our main result, let us first introduce some notations and definitions.\nChoose a radial function $\\varphi\\in{\\cal S}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ such that\n\n$${\\rm supp}\\,\\varphi\\subset\\{\\xi\\in \\mathop{\\bf R\\kern 0pt}\\nolimits^d;\n\\frac{5}{6}\\le|\\xi|\\le \\frac{12}{5}\\},\\quad \\sum_{k\\in{\\mathbf Z}}\\varphi(2^{-k}\\xi)=1,\n\\quad\\xi\\in\\mathop{\\bf R\\kern 0pt}\\nolimits^d\\setminus \\{0\\}.$$\nHere $\\varphi_k(\\xi)=\\varphi(2^{-k}\\xi)$,\n$k\\in {\\mathbf Z}$.\n\n\n\\begin{Def}\\label{Def1.1}\nLet $k\\in {\\mathbf Z}$, the Littlewood-Paley projection operators $\\Delta_k$ and $S_k$ are defined as follows\n$$\\Delta_kf=\\varphi(2^{-k}D)f,\\quad\nS_kf=\\sum_{j\\le k-1}\\Delta_jf, \\quad \\textrm{for}\\quad f\\in {\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d).$$\n\\end{Def}\n\n\nWe denote the space ${\\cal Z'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ by the dual space of ${\\cal\nZ }(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)=\\{f\\in {\\cal S}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d);\\,D^\\alpha \\hat{f}(0)=0;\n\\forall\\alpha\\in\\mathop{\\bf N\\kern 0pt}\\nolimits^d \\,\\mbox{multi-index}\\}$, it also can be\nidentified by the quotient space of ${\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)\/{\\cal P}$ with the\npolynomials space ${\\cal P}$. The formal equality $$f=\\sum_{k\\in{\\mathbf Z}}\\Delta_kf$$ holds true for $f\\in {\\cal Z'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$\nand is called the homogeneous Littlewood-Paley decomposition. It\nhas nice properties of quasi-orthogonality: with our choice of\n$\\varphi$,\n\\begin{eqnarray}\\label{1.5}\n\\Delta_j\\Delta_kf=0\\quad i\\!f\\quad|j-k|\\ge 2\\quad \\textrm{and}\n\\quad \\Delta_j(S_{k-1}\\Delta_k\nf)=0\\quad i\\!f\\quad|j-k|\\ge 4.\n\\end{eqnarray}\n\n\\begin{Def}\\label{Def1.2}\nLet $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$, $1\\le p, r\\le+\\infty$. The homogeneous Besov space $\\dot{B}^{s}_{p,r}$\nis defined by\n$$\\dot{B}^{s}_{p,r}=\\{f\\in {\\cal Z'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\\,\\|f\\|_{\\dot{B}^{s}_{p,r}}<+\\infty\\},$$\nwhere\n\\begin{eqnarray*}\n&&\\|f\\|_{\\dot{B}^{s}_{p,r}}=\\left\\{\\begin{array}{l}\\displaystyle\\bigg(\\sum_{k\\in{\\mathbf Z}} 2^{ksr}\n\\|\\Delta_kf\\|^r_{p}\\bigg)^{\\frac{1}{r}},\\quad\\mbox{for}\\quad r<+\\infty,\\\\\n\\displaystyle\\sup_{k\\in{\\mathbf Z}}2^{ks}\\|\\Delta_kf\\|_{p},\\quad\\mbox{for}\\quad r=+\\infty.\\end{array}\\right.\\end{eqnarray*}\n\\end{Def}\nIf\\, $p=r=2$, $\\dot{B}^s_{2,2}=\\dot{H}^s$,\nand if\\, $d=2$, we have $\\dot{B}^{1}_{2,1}\\hookrightarrow L^\\infty$ and\n$$\\|f\\|_{\\infty}\\le C\\|f\\|_{\\dot{B}^{1}_{2,1}}.$$\nWe refer to \\cite{Ch1,Tr} for more details.\n\nIn addition to the general time-space space such as $L^\\rho(0,T; \\dot{B}^{s}_{p,r})$,\nwe introduce a useful mixed time-space homogeneous Besov space\n$\\widetilde{L}^\\rho_T(\\dot{B}^{s}_{p,r})$\nwhich is initiated in \\cite{CL} and is used in the proof of the uniqueness.\n\\begin{Def}\\label{Def1.3}\nLet $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$, $1\\le p,r,\\rho\\le+\\infty$, $00}2^{k\\sigma}\\|\\Delta_k f\\|_{2}.$$\nLet $m=-[\\frac{d}{2}+1-s]$, we define\n\\begin{eqnarray*}\n&&\\widetilde{B}^{s,\\sigma}_2(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)=\\{f\\in{\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\n\\|f\\|_{\\widetilde{B}^{s,\\sigma}_2}<+\\infty\\}\\quad\n\\hbox{if}\\quad m<0,\\\\\n&&\\widetilde{B}^{s,\\sigma}_2(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)=\\{f\\in{\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)\/{\\cal P}_m:\n\\|f\\|_{\\widetilde{B}^{s,\\sigma}_2}<+\\infty\\}\\quad\n\\hbox{if}\\quad m\\ge0,\\end{eqnarray*}\nwhere ${\\cal P}_m$ denotes the set of polynomials of degree $\\le m$.\n\\end{Def}\n\nThroughout this paper, we will denote\n$\\dot{B}^{s}_{2,1}$ by $B^s$, and $\\widetilde{B}^{s,\\sigma}_2$ by\n$\\widetilde{B}^{s,\\sigma}$. The following facts can be easily verified by using the definition of $\\widetilde{B}^{s,\\sigma}$:\\vspace{0.1cm}\n\n(i) $\\widetilde{B}^{s,s}=\\dot{B}^s_{2,1}$;\n\n(ii)\\,If $s\\le \\sigma$, then $\\widetilde{B}^{s,\\sigma}=\\dot{B}^s_{2,1}\\cap\\dot{B}^\\sigma_{2,1}$.\nOtherwise, $\\widetilde{B}^{s,\\sigma}=\\dot{B}^s_{2,1}+\\dot{B}^\\sigma_{2,1}$.\n\\vspace{0.2cm}\n\n\nNow we state our main result as follows.\n\\begin{Theorem}\\label{Them1.2} Let $\\bar{h}_0$ be a positive constant.\nAssume that\n\n(i)\\, $(u_0,h_0-\\bar{h}_0)\\in B^0(\\mathop{\\bf R\\kern 0pt}\\nolimits^2)\\otimes\n\\widetilde{B}^{0,1}(\\mathop{\\bf R\\kern 0pt}\\nolimits^2);$\\vspace{0.1cm}\n\n(ii)\\,\\,$h_0\\ge \\bar{h}_0$.\\\\ \\vspace{0.1cm}\nThen there exist a positive time $T$ and a unique solution $(u,h)$ of $(\\ref{1.1})$ such that\n\\begin{eqnarray}\\label{1.6}\nu\\in C([0,T]; B^0)\n\\cap L^1(0,T; B^2),\\,\\,\nh-\\bar{h}_0\\in C([0,T]; \\widetilde{B}^{0,1})\\cap\nL^1(0,T; \\widetilde{B}^{2,1}),\\quad h\\ge \\f12 \\bar{h}_0.\n\\end{eqnarray}\nMoreover, there exists a strictly positive constant $c$ such that if\n\\begin{eqnarray}\\label{1.7}\n\\|u_0\\|_{B^0}+\n\\|h_0-\\bar{h}_0\\|_{\\widetilde{B}^{0,1}}\\le c,\\end{eqnarray}\nthen we can choose $T=+\\infty$.\n\\end{Theorem}\n\nThe structure of this paper is as follows.\n\nIn Section 2, we recall some useful multilinear estimates in the Besov spaces .\nIn Section 3, we prove the existence of solution. In Section 4,\nwe prove the uniqueness of the solution.\nFinally, in the Appendix, we prove some multilinear estimates in the weighted Besov spaces.\n\nThroughout the paper, $C$ denotes various ``harmless'' large\nfinite constants, and $c$ denotes various ``harmless\" small\nconstants. We shall sometimes use $X\\lesssim Y$ to\ndenote the estimate $X\\le CY$ for some constant $C$.\nWe denote $\\|\\cdot\\|_p$ by the $L^p$ norm of a function.\n\\vspace{.3cm}\n\n\\setcounter{equation}{0}\n\\section{Multilinear estimates in the Besov spaces}\n\nLet us first recall the Bony's paraproduct decomposition.\n\n\\begin{Def}\\label{Def2.1}\nWe shall use the following Bony's paraproduct decomposition(see \\cite{BC,B})\n\\begin{eqnarray}\\label{2.1}\nfg=T_fg+T_gf+R(f,g),\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{2.2}\nT_fg=\\sum_{k\\in{\\mathbf Z}}S_{k-1}f\\Delta_kg\\quad \\textrm{and} \\quad\nR(f,g)=\\sum_{k\\in{\\mathbf Z}}\\sum_{|k'-k|\\le 1}\\Delta_{k} f\\Delta_{k'}g.\n\\end{eqnarray}\n\\end{Def}\n\nNext, let us recall some useful lemmas and multilinear estimates in the Besov spaces.\n\n\\begin{Lemma}(Bernstein's inequality)\\label{Lem2.1}\nLet $1\\le p\\le q\\le+\\infty$. Assume that $f\\in {\\cal S'}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$,\nthen for any $\\gamma\\in{\\mathbf Z}^d$,\nthere exist\nconstants $C_1$, $C_2$ independent of $f$, $j$ such that\n\\begin{eqnarray*}\n&&{\\rm supp}\\hat f\\subseteq \\{|\\xi|\\le\nA_02^{j}\\}\\Rightarrow\n\\|\\partial^\\gamma f\\|_q\\le C_12^{j{|\\gamma|}+j d(\\frac{1}{p}-\\frac{1}{q})}\\|f\\|_{p},\n\\\\\n&&{\\rm supp}\\hat f\\subseteq \\{A_12^{j}\\le|\\xi|\\le A_22^{j}\\}\\Rightarrow\n\\|f\\|_{p}\\le C_22^{-j|\\gamma|}\\sup_{|\\beta|=|\\gamma|}\\|\\partial^\\beta f\\|_p.\n\\end{eqnarray*}\n\\end{Lemma}\nThe proof can be found in \\cite{Ch1}.\n\n\\begin{Proposition}\\label{Prop2.2}\nIf $s>0$, $f,g\\in B^s\\cap L^\\infty$.\nThen $fg\\in B^s\\cap L^\\infty$ and\n\\begin{equation}\\label{2.3}\n\\|fg\\|_{B^s}\\le C(\\|f\\|_{\\infty}\\|g\\|_{B^s}+\\|g\\|_{\\infty}\\|f\\|_{B^s}).\n\\end{equation}\nIf $s_1$, $s_2\\le\\frac{d}{2}$ such that $s_1+s_2>0$, $f\\in B^{s_1}$, and\n$g\\in B^{s_2}$. Then $fg\\in B^{s_1+s_2-\\frac{d}{2}}$ and\n\\begin{eqnarray}\\label{2.4}\n\\|fg\\|_{B^{s_1+s_2-\\frac{d}{2}}}\\le C\\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{eqnarray}\nIf $|s|<\\frac{d}{2}$, $1\\le r\\le+\\infty$, $f\\in \\dot{B}^s_{2,r}$ ,and\n$g\\in {B}^{\\frac{d}{2}}$. Then $fg\\in \\dot{B}^{s}_{2,r}$ and\n\\begin{equation}\\label{2.5}\n\\|fg\\|_{\\dot{B}^{s}_{2,r}}\\le C\\|f\\|_{\\dot{B}^{s}_{2,r}}\n\\|g\\|_{{B}^{\\frac{d}{2}}}.\n\\end{equation}\nIf $s\\in(-\\frac{d}{2},\\frac{d}{2}]$, $f\\in B^s$, and\n$g\\in \\dot{B}^{-s}_{2,\\infty}$. Then $fg\\in \\dot{B}^{-\\frac{d}{2}}_{2,\\infty}$ and\n\\begin{eqnarray}\\label{2.6}\n\\|fg\\|_{\\dot{B}^{-\\frac{d}{2}}_{2,\\infty}}\\le C\\|f\\|_{{B}^{s}}\n\\|g\\|_{\\dot{B}^{-s}_{2,\\infty}}.\\end{eqnarray}\nIf $1\\le \\rho_1, \\rho_2, \\rho\\le \\infty, s\\in(-\\frac{d}{2},\\frac{d}{2}]$,\n$f\\in \\widetilde{L}^{\\rho_1}_T(B^s)$, and\n$g\\in \\widetilde{L}^{\\rho_2}_T(\\dot{B}^{-s}_{2,\\infty})$. Then there holds\n\\begin{eqnarray}\\label{2.7}\n\\|fg\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{-\\frac{d}{2}}_{2,\\infty})}\\le\nC\\|f\\|_{\\widetilde{L}^{\\rho_1}_T({B}^{s})}\n\\|g\\|_{\\widetilde{L}^{\\rho_2}_T(\\dot{B}^{-s}_{2,\\infty})},\n\\end{eqnarray}\nwhere $\\frac{1}{\\rho_1}+\\frac{1}{\\rho_2}=\\frac{1}{\\rho}$.\n\\end{Proposition}\n{\\it Proof.}\\quad\nFor the sake of simplicity, we only present the proof of (\\ref{2.4}) below, the others can be\ndeduced in the same way (see also \\cite{D3,RS}).\nBy the Bony's paraproduct decomposition and the\nproperty of quasi-orthogonality (\\ref{1.5}), for fixed $j\\in{\\mathbf Z}$, we write\n\\begin{align}\n\\Delta_j(fg)&=\\sum_{|k-j|\\le3}\\Delta_j(S_{k-1}f\\Delta_kg)+\n\\sum_{|k-j|\\le3}\\Delta_j(S_{k-1}g\\Delta_kf)\n+\\sum_{k\\ge j-2}\\sum_{|k-k'|\\le 1}\\Delta_j(\\Delta_{k}f\\Delta_{k'}g)\\nonumber\\\\\n&\\triangleq {I+II+III}.\\nonumber\\end{align}\nThanks to the definition of Besov space $B^s$, we have\n\\begin{align}\\label{2.8}\n\\|fg\\|_{B^{s_1+s_2-\\frac{d}{2}}}&\\le\\bigg(\\sum_{j\\in\n{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}\\|I\\|_{2}\\bigg)\n+\\cdots+\\bigg(\\sum_{j\\in\n{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}\\|III\\|_{2}\\bigg)\\nonumber\\\\\n&\\triangleq {I'+II'+III'}.\\end{align}\nIt suffices to estimate the\nabove three terms separately.\nUsing the Young's inequality and lemma \\ref{Lem2.1}, we have\n\\begin{align}\n\\|\\Delta_j(S_{k-1}f\\Delta_kg)\\|_{2}&\\lesssim\n\\|S_{k-1}f\\|_{\\infty}\\|\\Delta_kg\\|_{2}\n\\lesssim\n\\sum_{k'\\le k-2}\\|\\Delta_{k'} f\\|_{\\infty}\\|\\Delta_kg\\|_{2}\\nonumber\\\\\n&\\lesssim \\sum_{k'\\le k-2}2^{k' s_1}\n\\|\\Delta_{k'} f\\|_{2}2^{k'(\\frac{d}{2}-s_1)}\\|\\Delta_kg\\|_{2}\n\\nonumber\\\\\n&\\lesssim \\|f\\|_{B^{s_1}}\n\\|\\Delta_kg\\|_{2}2^{k(\\frac{d}{2}-s_1)},\\nonumber\n\\end{align}\nwhere we have used the fact $s_1\\le \\frac{d}{2}$ in the last inequality.\nHence, we get\n\\begin{align}\\label{2.9}\n{I'}&\\lesssim \\|f\\|_{B^{s_1}}\\sum_{j\\in{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}\n\\sum_{|k-j|\\le3}2^{k(\\frac{d}{2}-s_1)}\n\\|\\Delta_kg\\|_{2}\n\\nonumber\\\\&\\lesssim \\|f\\|_{B^{s_1}}\n\\sum_{|\\ell|\\le3}2^{-(s_1+s_2-\\frac{d}{2})\\ell}\n\\sum_{j\\in{\\mathbf Z}}2^{s_2(j+\\ell)}\n\\|\\Delta_{j+\\ell}g\\|_{2}\n\\lesssim \\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{align}\nSimilarly, using the fact $s_2\\le\\frac{d}{2} $, we can obtain\n\\begin{align}\\label{2.10}\n{II'}&\\lesssim \\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{align}\nNow we turn to estimate $III'$. From Lemma \\ref{Lem2.1} and H\\\"{o}lder inequality\n, it follows that\n\\begin{eqnarray*}\n\\|\\Delta_j(\\Delta_{k}f\\Delta_{k'}g)\\|_{2}\\lesssim\n2^{j\\frac{d}{2}}\\|\\Delta_{k}f\\Delta_{k'}g\\|_{1}\n\\lesssim 2^{j\\frac{d}{2}}\\|\\Delta_{k}f\\|_{2}\\|\\Delta_{k'}g\\|_{2}.\n\\end{eqnarray*}\nSo, we get by Minkowski inequality that for $s_1+s_2>0$\n\\begin{align}\\label{2.11}\n{III'}&\\lesssim \\sum_{j\\in{\\mathbf Z}}2^{(s_1+s_2-\\frac{d}{2})j}2^{j\\frac{d}{2}}\n\\bigg(\\sum_{k\\ge j-2}\\sum_{|k-k'|\\le1}\n\\|\\Delta_{k}f\\|_{2}\n\\|\\Delta_{k'}g\\|_{2}\\bigg)\n\\nonumber\\\\&\\lesssim\n\\sum_{\\ell\\ge-2}2^{-(s_1+s_2)\\ell}\n\\sum_{j\\in{\\mathbf Z}}2^{s_1(j+\\ell)}\n\\|\\Delta_{j+\\ell}f\\|_{2}\\|g\\|_{B^{s_2}}\n\\lesssim \\|f\\|_{B^{s_1}}\\|g\\|_{B^{s_2}}.\n\\end{align}\nSumming up (\\ref{2.8})-(\\ref{2.11}), we get the desired inequality\n(\\ref{2.4}). \\hfill $ \\blacksquare $ \\vskip 3mm\n\n\\begin{Proposition}\\label{Prop2.3}\n\\textrm{(1)} Let $s>0$. Assume that $F\\in\nW^{[s]+2,\\infty}_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ such that $F(0)=0$. Then there\nexists a constant $C(s,d,F)$ such that if $u\\in B^s\\cap L^\\infty$,\nthere holds \\begin{eqnarray}\\label{2.12} &&\\|F(u)\\|_{B^s}\\le\nC(1+\\|u\\|_{\\infty})^{[s]+1}\\|u\\|_{B^s} ;\\end{eqnarray} and if $u\\in\n\\dot{B}^s_{2,\\infty}\\cap L^\\infty$, there holds \\begin{eqnarray}\\label{2.13}\n&&\\|F(u)\\|_{\\dot{B}^s_{2,\\infty}}\\le\nC(1+\\|u\\|_{\\infty})^{[s]+1}\\|u\\|_{\\dot{B}^s_{2,\\infty}} . \\end{eqnarray}\n(2)Assume that $G\\in W^{[\\frac{d}{2}]+3,\\infty}_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ such\nthat $G'(0)=0$. Then there exists a functions $C(s,d,G)$ such that\nif $-\\frac{d}{2}0$. The function space $E^{s}_T$\nis defined by\n$$E^{s}_T=\\{f\\in {\\cal Z'}((0,T)\\times\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\\,\\|f\\|_{E^{s}_T}<+\\infty\\},$$\nwhere\n\\begin{eqnarray*}\n\\|f\\|_{E^{s}_T}\\triangleq\\sum_{k\\in Z}2^{ks}\\omega_k(T)\\|\\Delta_k f\\|_{L^\\infty_T(L^2)}.\n\\end{eqnarray*}\n\\end{Def}\n\n\\begin{Def}{\\label{Def3.2}}\nLet $s_1, s_2\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$ and $T>0$. The function space $\\widetilde{E}^{s_1,s_2}_T$\nis defined by\n$$\\widetilde{E}^{s_1,s_2}_T=\\{f\\in {\\cal Z'}((0,T)\\times\\mathop{\\bf R\\kern 0pt}\\nolimits^d):\\,\\|f\\|_{\\widetilde{E}^{s_1,s_2}_T}<+\\infty\\},$$\nwhere\n$$\\|f\\|_{\\widetilde{E}^{s_1,s_2}_T}\\triangleq \\sum_{k\\le0}2^{ks_1}\\omega_k(T)\\|\\Delta_k f\\|_{L^\\infty_T(L^2)}\n+\\sum_{k\\ge 1}2^{ks_2}\\omega_k(T)\\|\\Delta_k f\\|_{L^\\infty_T(L^2)}.$$\n\\end{Def}\n\\begin{rmk}\nIf $s_1\\le s_2$, then $\\widetilde{E}^{s_1,s_2}_T=E^{s_1}_T\\cap E^{s_2}_T$.\nOtherwise, $\\widetilde{E}^{s_1,s_2}_T=E^{s_1}_T+E^{s_2}_T$.\n\\end{rmk}\n\nLet $(u,h)$ be a smooth solution of (\\ref{3.2}). We want to establish the following {\\it a-priori} estimates for $(h,u)$:\n\\begin{eqnarray}\\label{3.3}\n&&\\|u\\|_{L^1_T(B^2)}+\\|u\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)E_k(0)\n+C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|\\Delta_k{\\cal G}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla \\Delta_k{\\cal H}(t)\\|_{L^1_T(L^2)}\n+C\\sum_{k<1}\\omega_k(T)\\|\\Delta_k{\\cal H}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+C\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{3.4}\n&&\\|u\\|_{L^\\infty_T(B^0)}+\\|h\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h\\|_{L^1_T(\\widetilde{B}^{2,1})}\\nonumber\\\\\n&&\\quad\\le E_0+C\\Bigl(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0,1})}\n+\\|{\\cal G}\\|_{L^1_T(B^0)}+\\int_0^TV'(t)(\\|u(t)\\|_{B^0}+\\|h(t)\\|_{\\widetilde{B}^{0,1}})dt\\Bigr),\n\\end{eqnarray}\nwhere $V(t)=\\|v(t')\\|_{L^1_t(B^2)}$ and\n$$\nE_0=\\sum_{k\\in{\\mathbf Z}}E_k(0),\\quad E_{k}(t)=\\bigg\\{\n\\begin{array}{ll}E_{hk}(t)\n&\\quad k\\ge 1,\\\\\nE_{lk}(t)&\\quad k< 1,\n\\end{array}\\bigg.\n$$\nwith\n\\begin{eqnarray*}\n&& E_{hk}^2(t)=\\f12\\|u_k(t)\\|^2_{2}+\\|\n\\nabla h_k(t)\\|^2_{2}+(u_k(t), \\nabla h_k(t)), \\quad \\textrm{and}\\\\\n&& E_{lk}^2(t)= \\f12\\|u_k(t)\\|^2_{2}+\\f12\\|\nh_k(t)\\|^2_{2}+\\f18(u_k(t),\\nabla h_k(t)).\n\\end{eqnarray*}\n\nLet us begin with the proof of (\\ref{3.3}) and (\\ref{3.4}). Set\n$$\nu_k=\\Delta_k u,\\quad h_k=\\Delta_k h,\\quad {\\cal H}^k=\\Delta_k {\\cal\nH},\\quad {\\cal G}^k=\\Delta_k{\\cal G}.\n$$\nThen we get by applying the operator $\\Delta_k$ to (\\ref{3.2}) that\n\\begin{equation}\\label{3.5}\n\\left\\{\n\\begin{array}{ll}\n\\partial_th_k+\\Delta_k(v\\cdot\\nabla h)+\\mbox{div} u_k={\\cal H}_k,\\\\\n\\partial_tu_k-(\\nabla\\cdot D(u_k)+\\nabla \\textrm{div}\\,u_k)+\\Delta_k(v\\cdot\\nabla u)+\\nabla h_k={\\cal G}_k, \\\\\nu_k(0,\\cdot)=\\Delta_ku_0, \\quad h_k(0,\\cdot)=\\Delta_kh_0.\n\\end{array}\n\\right.\n\\end{equation}\nMultiplying the second equation of (\\ref{3.5}) by $u_k$, and\nintegrating the resulting equation over $\\mathop{\\bf R\\kern 0pt}\\nolimits^2$, we obtain\n\\begin{eqnarray}\\label{3.6} \\frac{1}{2}\\frac{d}{dt}\\|u_k\\|^2_{2} +\\f12\\|\\nabla\nu_k\\|^2_{2}+\\f32\\|\\mbox{div} u_k\\|_2^2+(\\nabla h_k,u_k)=({\\cal G}_k, u_k)\n-(\\Delta_k(v\\cdot\\nabla u), u_k). \\end{eqnarray}\n\nIn the following, we will deal with the high frequency and the low frequency of $h$ in a different manner.\n\n\n\n\\vspace{.15cm}\n\n{\\bf High frequencies}:\\, $k\\ge 1$.\\vspace{.15cm}\n\nFirstly, applying $\\nabla$ to the first equation of $(\\ref{3.5})$,\nand multiplying it by $\\nabla h_k$, then integrating the resulting\nequation over $\\mathop{\\bf R\\kern 0pt}\\nolimits^2$, we obtain \\begin{eqnarray}\\label{3.7}\n\\frac{1}{2}\\frac{d}{dt}\\|\\nabla h_k\\|^2_{2}+ (\\nabla\\mbox{div} u_k, \\nabla\nh_k)=(\\nabla {\\cal H}_{k}, \\nabla h_k)-(\\nabla\\Delta_k(v\\cdot \\nabla\nh), \\nabla h_k).\\quad \\end{eqnarray} Secondly, applying the operator $\\nabla$\nto the first equation of $(\\ref{3.5})$ and taking the $L^2$ product\nof the resulting equation with $u_k$; then taking the $L^2$ product\nof second equation of $(\\ref{3.5})$ with $\\nabla h_k$, we get by\nsumming them up that\n\\begin{eqnarray}\\label{3.8} &&\\frac{d}{dt}(u_k, \\nabla\nh_k)-\\|\\mbox{div} u_k\\|^2_{2} -2(\\nabla \\mbox{div} u_k,\\nabla h_k)+\\|\\nabla\nh_k\\|^2_{2}\\nonumber\\\\&& \\quad=(\\nabla{\\cal H}_{k}, u_k)+({\\cal\nG}_{k}, \\nabla h_k) -(\\nabla\\Delta_k(v\\cdot \\nabla h),\nu_k)-(\\Delta_k(v\\cdot\\nabla u), \\nabla h_k), \\end{eqnarray} where we used the\nfact that\n$$\n(\\nabla\\cdot D(u_k)+\\nabla \\textrm{div}\\,u_k, \\nabla h_k)=2(\\nabla \\mbox{div} u_k,\\nabla h_k).\n$$\nThen we get by summing up (\\ref{3.6}), (\\ref{3.7})$\\times 2$, and (3.8) that\n\\begin{eqnarray}\\label{3.9}\n&&\\frac{d}{dt}\\bigl[\\f12\\|u_k\\|^2_{2}+\\|\n\\nabla h_k\\|^2_{2}+(u_k, \\nabla h_k)\\bigr]\\nonumber\\\\\n&&\\qquad\\qquad+\\bigl[\\|\\nabla h_k\\|^2_{2}+\\f12\\|\\nabla u_k\\|^2_{2}+\\f12\\|\\mbox{div} u_k\\|^2_2\n+(\\nabla h_k, u_k)\\bigr]\\nonumber\\\\\n&&=\\Bigl[(\\nabla {\\cal H}_{k}, u_k)+2(\\nabla {\\cal H}_{k},\\nabla h_k)\n+({\\cal G}_{k}, u_k)+({\\cal G}_{k}, \\nabla h_k)\\Bigr]\\nonumber\\\\\n&&\\quad-(\\Delta_k(v\\cdot\\nabla u), u_k)\n-2(\\nabla\\Delta_k(v\\cdot\\nabla h), \\nabla h_k)\\nonumber\\\\\n&&\\quad-\\Bigl[(\\nabla\\Delta_k(v\\cdot\\nabla h), u_k)\n+(\\Delta_k(v\\cdot\\nabla u), \\nabla h_k)\\Bigr]\\nonumber\\\\\n&&\\triangleq I+II+III+IV.\n\\end{eqnarray}\nNote that\n\\begin{eqnarray*}\n(u_k,\\nabla h_k)\\le \\f13\\|u_k\\|^2_{2}+\n\\f34\\|\\nabla h_k\\|^2_{2},\n\\end{eqnarray*}\nhence, we get by the definition of $E_{hk}$ that\n\\begin{eqnarray}\\label{3.10}\n\\f16(\\|u_k\\|^2_{2}+\\|\\nabla h_k\\|^2_{2})\\le E_{hk}^2\\le 2(\\|u_k\\|^2_{2}+\\|\\nabla h_k\\|^2_{2}).\n\\end{eqnarray}\nSimilarly, using the fact that $\\f562^k\\ge \\f53$ and (\\ref{3.10}), we have\n\\begin{eqnarray}\\label{3.11}\n\\|\\nabla h_k\\|^2_{2}+\\f12\\|\\nabla u_k\\|^2_{2}+\\f12\\|\\mbox{div} u_k\\|^2_{2}\n+(\\nabla h_k, u_k)\\ge\\frac{1}{8}E_{hk}^2.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.9})-(\\ref{3.11}), we obtain\n\\begin{eqnarray}\\label{3.12}\n\\frac{d}{dt}E_{hk}^2+cE_{hk}^2\\le C|I+II+III+IV|.\n\\end{eqnarray}\n\nIn order to obtain (\\ref{3.3}), we use Lemma \\ref{Lem5.1} to deal with the right hand terms of (\\ref{3.12}).\nFirstly, we get by using the Cauchy-Schwartz inequality and (\\ref{3.10}) that\n\\begin{eqnarray}\\label{3.13}\n|I|\\le C(\\|\\nabla {\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2})E_{hk}.\n\\end{eqnarray}\nFrom Lemma \\ref{Lem5.1} and (\\ref{3.10}), it follows that\n\\begin{eqnarray}\\label{3.14}\n|II+III+IV|\\le C(\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2})E_{hk}.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.12}), and (\\ref{3.13})-(\\ref{3.14}), we obtain\n\\begin{eqnarray}\\label{3.15}\n\\frac{d}{dt}E_{hk}+cE_{hk}\\le C\\Bigl(\\|\\nabla {\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2}\\Bigr),\n\\end{eqnarray}\nwhich implies that\n\\begin{align}\\label{3.16}\n\\|E_{hk}(t)\\|_{L^\\infty_T}\\le & E_{hk}(0)+C\\Bigl(\\|\\nabla {\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&+\\|{\\cal F}_k^1(t)\\|_{L^1_T(L^2)}+\\|{\\cal \\widetilde{F}}_k^0(t)\\|_{L^1_T(L^2)}\\Bigr).\n\\end{align}\nFurthermore, by (\\ref{5.1}) and (\\ref{5.2}), there holds\n\\begin{eqnarray*}\n\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\Big(\\|{\\cal F}_k^1(t)\\|_{L^1_T(L^2)}+\\|{\\cal \\widetilde{F}}_k^0(t)\\|_{L^1_T(L^2)}\\Big)\\le C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}\n+\\|h\\|_{E^{1}_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{eqnarray*}\nMultiplying $\\omega_k(T)$ on both sides of \\eqref{3.16}, then\nsumming up the resulting equation over $k\\ge 1$, we obtain\n\\begin{align}\\label{3.17}\n\\sum_{k\\ge 1}\\omega_k(T)\\|E_{hk}(t)\\|_{L^\\infty_T}\\le &\\sum_{k\\ge 1}\\omega_k(T)E_{hk}(0)\\nonumber\\\\\n&+C\\sum_{k\\ge 1}\\omega_k(T)\\Big(\\|\\nabla {\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)}\\Big)\\nonumber\\\\\n&+C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{E^{1}_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{align}\n\nNext, we use the decay effect of the parabolic operators to estimate $\\|u\\|_{L^2_T(B^1)\\cap L^1_T(B^2)}$. It follows from (\\ref{3.6}) and Lemma 5.1 that\n\\begin{eqnarray*}\n\\frac{d}{dt}\\|u_k\\|_2+c2^{2k}\\|u_k\\|_2\\le C(\\|\\nabla h_k(t)\\|_2+\\|{\\cal G}_{k}(t)\\|_{L^2}\n+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{L^2}),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray*}\n\\|u_k\\|_2\\le e^{-ct2^{2k}}\\|u_k(0)\\|_{2}+Ce^{-ct2^{2k}}\\ast_t\\Bigl(\\|\\nabla h_k(t)\\|_2\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{2}\\Bigr),\n\\end{eqnarray*}\nwhere the sign $\\ast$ denotes the convolution of functions defined in $\\mathop{\\bf R\\kern 0pt}\\nolimits^+$, more precisely,\n$$e^{-ct2^{2k}}\\ast_t f\\triangleq \\int_0^t\ne^{-c(t-\\tau)2^{2k}}f(\\tau)d\\tau.$$\nTaking the $L^r$ norm for $r=1,2$ with respect to $t$, we get by using the Young's inequality that\n\\begin{eqnarray*}\n\\|u_k\\|_{L^r_T(L^2)}\\le C2^{-2k\/r}e^r_k(T)\\Big(\\|u_k(0)\\|_{2}+\n\\|\\nabla h_k\\|_{L^1_T(L^2)}+\\|{\\cal G}_{k}\\|_{L^1_T(L^2)}+\\|{\\cal \\widetilde{F}}_{k}^0\\|_{L^1_T(L^2)}\\Big),\n\\end{eqnarray*}\nwhich together with (\\ref{5.2}) implies that\n\\begin{align}\\label{3.18}\n&\\sum_{k\\ge1}\\Bigl(2^{2k}\\|u_k\\|_{L^1_T(L^2)}+2^{k}\\|u_k\\|_{L^2_T(L^2)}\\Bigr)\\le C\\sum_{k\\ge1}\\omega_k(T)\\|u_k(0)\\|_2\n\\nonumber\\\\&\\qquad+C\\sum_{k\\ge1}\\omega_k(T)\\Bigl(\\|\\nabla h_k\\|_{L^1_T(L^2)}+\\|{\\cal G}_{k}\\|_{L^1_T(L^2)}\\Bigr)\n+C\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)},\n\\end{align}\nwhere we used the fact that\n\\begin{eqnarray*}\ne_k^1(T)+e_k^2(T)\\le \\omega_k(T).\n\\end{eqnarray*}\nOn the other hand, it follows from (\\ref{3.15}) that\n\\begin{eqnarray*}\n\\|E_{hk}\\|_2\\le e^{-ct}E_{hk}(0)+Ce^{-ct}\\ast_t\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_2\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal{F}}_{k}^1(t)\\|_{2}+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{2}\\Bigr).\n\\end{eqnarray*}\nTaking the $L^1$ norm with respect to $t$, we get by using the Young's inequality that\n\\begin{align}\\label{3.19}\n\\|E_{hk}\\|_{L_T^1}&\\le C(1-e^{-cT})E_{hk}(0)+C(1-e^{-cT})\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L_T^1(L^2)}\\nonumber\\\\\n&\\quad+\\|{\\cal{F}}_{k}^1(t)\\|_{L_T^1(L^2)}+\\|{\\cal \\widetilde{F}}_{k}^0(t)\\|_{L_T^1(L^2)}\\Bigr).\n\\end{align}\nNote that for $k\\ge 1$\n$$\n1-e^{-ct}\\le 1-e^{-ct2^{2k}}\\le \\omega_k(t),\n$$\nwhich together with (\\ref{3.19}) and Lemma \\ref{Lem5.1} gives\n\\begin{align}\\label{3.20}\n\\sum_{k\\ge 1}\\|E_{hk}\\|_{L_T^1}\\le& C\\sum_{k\\ge 1}\\omega_k(T)E_{hk}(0)\n+C\\sum_{k\\ge 1}\\omega_k(T)\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L_T^1(L^2)}\\Bigr)\\nonumber\\\\\n&+C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{E^1_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{align}\nPlugging \\eqref{3.20} into (\\ref{3.18}), we obtain\n\\begin{align}\\label{3.21}\n&\\sum_{k\\ge 1}\\Bigl(2^{2k}\\|u_k\\|_{L^1_T(L^2)}+2^{k}\\|u_k\\|_{L^2_T(L^2)}\\Bigr)\\nonumber\\\\&\\quad\\le C\\sum_{k\\ge1}\n\\omega_k(T)E_{hk}(0)\n+C\\sum_{k\\ge 1}\\omega_k(T)\\Bigl(\\|\\nabla {\\cal H}_k(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L_T^1(L^2)}\\Bigr)\\nonumber\\\\\n&\\qquad+C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{E^{1}_T}\\|v\\|_{L^1_T(B^2)}\\Big).\n\\end{align}\n\nOn the other hand, in order to obtain (\\ref{3.4}), we use Proposition \\ref{Prop2.4} to deal with the right hand terms of (\\ref{3.12}).\nApplying (\\ref{2.16}) with $s_1=s_2=0$ to $II$, (\\ref{2.16}) with $s_1=0, s_2=1$ to $III$, (\\ref{2.18})\nwith $t_1=t_2=0, s_1=0, s_2=1$ to $IV$, we obtain\n\\begin{eqnarray}\\label{3.22}\n|II+III+IV|\\le CE_{hk}\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}}),\n\\end{eqnarray}\nwith $\\displaystyle\\sum_{k\\in {\\mathbf Z}}\\alpha_k\\le 1$ and $V(t)=\\|v(t')\\|_{L^1_t(B^2)}$. From (\\ref{3.13}) and (\\ref{3.22}), it follows that\n\\begin{eqnarray*}\n\\frac{d}{dt}E_{hk}+cE_{hk}\\le C\\Bigl(\\|\\nabla {\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}})\\Bigr),\n\\end{eqnarray*}\nfrom which, a similar proof of (\\ref{3.21}) ensures that\n\\begin{eqnarray}\\label{3.23}\n&&\\sum_{k\\ge 1}\\Bigl(\\|E_{hk}\\|_{L_T^1}+\\|E_{hk}\\|_{L_T^\\infty}\\Bigr)\\le C\\sum_{k\\ge 1}E_{hk}(0)\\nonumber\\\\\n&&\\qquad+C\\Bigl(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0,1})}\n+\\|{\\cal G}\\|_{L^1_T(B^0)}+\\int_0^TV'(t)(\\|u(t)\\|_{B^0}+\\|h(t)\\|_{\\widetilde{B}^{0,1}})dt\\Bigr).\n\\end{eqnarray}\n\n\\vspace{.15cm}\n\n{\\bf Low frequencies}:\\, $k<1$.\\vspace{.15cm}\n\nMultiplying the first equation of $(\\ref{3.5})$ by $h_k$, we get by integrating\nthe resulting equation over $\\mathop{\\bf R\\kern 0pt}\\nolimits^2$ that\n\\begin{eqnarray}\\label{3.24}\n\\frac{1}{2}\\frac{d}{dt}\\|h_k\\|^2_{2}+\n(\\mbox{div} u_k, h_k)=({\\cal H}_{k}, h_k)-(\\Delta_k(v\\cdot \\nabla h),h_k).\n\\end{eqnarray}\nSumming up (\\ref{3.6}), $(\\ref{3.8})\\times\\frac{1}{8}$, and (\\ref{3.24}), we obtain\n\\begin{align}\\label{3.25}\n&\\frac{d}{dt}\\Bigl[\\f12\\|u_k\\|^2_{2}+\\f12\\|\nh_k\\|^2_{2}+\\frac{1}{8}(u_k,\\nabla h_k)\\Bigr]\n\\nonumber\\\\\n&\\qquad+\\Bigr[\\frac{1}{8}\\|\\nabla h_k\\|^2_{2}\n+\\f12\\|\\nabla u_k\\|^2_{2}+\\frac{11}{8}\\|\\mbox{div} u_k\\|^2_{2}\n-\\frac{1}{4}(\\nabla\\mbox{div}\\, u_k,\\nabla h_k)\\Bigr]\\nonumber\\\\&=\\Bigl[\\f18(\\nabla {\\cal H}_{k}, u_k)+({\\cal H}_{k},h_k)\n+({\\cal G}_{k}, u_k)+\\f18({\\cal G}_{k}, \\nabla h_k)\\Bigr]\\nonumber\\\\\n&\\quad-(\\Delta_k(v\\cdot\\nabla u), u_k)\n-(\\Delta_k(v\\cdot\\nabla h),h_k)\\nonumber\\\\\n&\\quad-\\f18\\Bigl[(\\nabla\\Delta_k(v\\cdot\\nabla h), u_k)\n+(\\Delta_k(v\\cdot\\nabla u), \\nabla h_k)\\Bigr]\\nonumber\\\\\n&\\triangleq I+II+III+IV.\n\\end{align}\nNote that $2^k\\le 1$, we get by the Cauchy-Schwartz inequality that\n\\begin{eqnarray*}\n\\f18(u_k,\\nabla h_k)\\le \\f3{10}\\|u_k\\|_{2}\\|h_k\\|_{2}\\le \\f14\\|u_k\\|^2_{2}+\n\\f14\\|h_k\\|^2_{2},\n\\end{eqnarray*}\nhence, we get by the definition of $E_{lk}$ that\n\\begin{eqnarray}\\label{3.26}\n\\f14(\\|u_k\\|^2_{2}+\\|h_k\\|^2_{2})\\le E_{lk}^2\\le 2(\\|u_k\\|^2_{2}+\\|h_k\\|^2_{2}).\n\\end{eqnarray}\nSimilarly, we can prove\n\\begin{eqnarray*}\n\\frac{1}{4}(\\nabla\\mbox{div}\\, u_k,\\nabla h_k)\\le \\f3{5}\\|\\nabla u_k\\|_{2}\\|\\nabla h_k\\|_{2}\\le \\f{9} {10}\\|\\nabla u_k\\|_2^2+\\f1 {10}\\|\\nabla h_k\\|_2^2,\n\\end{eqnarray*}\nwhich together with (\\ref{3.26}) implies that\n\\begin{eqnarray}\\label{3.27}\n&&\\frac{1}{8}\\|\\nabla h_k\\|^2_{2}\n+\\f12\\|\\nabla u_k\\|^2_{2}+\\frac{11}{8}\\|\\mbox{div} u_k\\|^2_{2}\n-\\frac{1}{4}(\\nabla\\mbox{div}\\, u_k,\\nabla h_k)\\nonumber\\\\\n&&\\qquad\\ge \\frac{1}{160}2^{2k}(\\|u_k\\|^2_{2}+\\|\nh_k\\|^2_{2})\\ge \\frac{1}{320}2^{2k}E_{lk}^2.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.25})-(\\ref{3.27}), we obtain\n\\begin{eqnarray}\\label{3.28}\n\\frac{d}{dt}E_{lk}^2+c2^{2k}E_{lk}^2\\le C|I+II+III+IV|.\n\\end{eqnarray}\n\nIn order to obtain (\\ref{3.3}), we use Lemma 5.1 to estimate the right hand terms of (\\ref{3.28}).\nUsing the fact that $2^k\\le 1$, we get by the Cauchy-Schwartz inequality and (\\ref{3.26}) that\n\\begin{eqnarray}\\label{3.29}\n|I|\\le C(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2})E_{lk}.\n\\end{eqnarray}\nUsing Lemma 5.1 and (\\ref{3.26}), we have\n\\begin{eqnarray}\\label{3.30}\n|II+III+IV|\\le C(\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2})E_{lk}.\n\\end{eqnarray}\n\nBy summing up (\\ref{3.28})-(\\ref{3.30}), we obtain\n\\begin{eqnarray*}\n\\frac{d}{dt}E_{lk}+c2^{2k}E_{lk}\\le C\\Bigl(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2}\\Bigr),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray*}\nE_{lk}\\le e^{-c2^{2k}t}E_{lk}(0)+Ce^{-c2^{2k}t}\\ast_t\\Bigl(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\|{\\cal F}_k^1(t)\\|_{2}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{2}\\Bigr).\n\\end{eqnarray*}\nTaking the $L^r$ norm with respect to $t$, we get by using the Young's inequality that\n\\begin{align}\n\\|E_{lk}\\|_{L^r_T}&\\le C2^{-2k\/r}e^r_k(T)\\Bigl(E_{lk}(0)+\\|{\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\quad+\\|{\\cal F}_k^1(t)\\|_{L^1_T(L^2)}+\\|{\\cal F}^0_k(t)\\|_{2}+\\|{\\cal \\widetilde{F}}^0_k(t)\\|_{L^1_T(L^2)}\\Bigr),\\nonumber\n\\end{align}\nfrom which and Lemma 5.1, it follows that\n\\begin{eqnarray}\\label{3.31}\n&&\\sum_{k<1}\\omega_k(T)\\|E_{lk}\\|_{L^\\infty_T}\\le C\\sum_{k<1}\\omega_k(T)E_{lk}(0)+\\sum_{k<1}\\omega_k(T)(\\|{\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)})\\nonumber\\\\\n&&\\qquad\\quad +C\\Big(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)}\\Big),\n\\end{eqnarray}\nand\n\\begin{align}\\label{3.32}\n&\\sum_{k<1}(2^{2k}\\|E_{lk}\\|_{L^1_T}+2^{k}\\|E_{lk}\\|_{L^2_T})\\nonumber\\\\&\\quad\\le C\\sum_{k<1}\\omega_k(T)E_{lk}(0)\n+\\sum_{k<1}\\omega_k(T)(\\|{\\cal H}_{k}(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}(t)\\|_{L^1_T(L^2)})\\nonumber\\\\\n&\\qquad+C\\Bigl(\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)}\\Bigr).\n\\end{align}\n\nOn the other hand, in order to obtain (\\ref{3.4}),\nwe use Proposition \\ref{Prop2.4} to deal with the right hand terms of (\\ref{3.28}).\nApplying (\\ref{2.17}) with $s_1=s_2=0$ to $II$, (\\ref{2.17}) with $s_1=0, s_2=1$ to $III$, (\\ref{2.19})\nwith $t_1=t_2=0, s_1=0, s_2=1$ to $IV$, we obtain\n\\begin{align}\\label{3.33}\n|II+III+IV|\\le CE_{lk}\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}}),\n\\end{align}\nwith $\\displaystyle\\sum_{k\\in {\\mathbf Z}}\\alpha_k\\le 1$ and $V(t)=\\|v(t')\\|_{L^1_t(B^2)}$. From (\\ref{3.32}) and (\\ref{3.36}), it follows that\n\\begin{eqnarray*}\n\\frac{d}{dt}E_{lk}+c2^{2k}E_{lk}\\le C\\Bigl(\\|{\\cal H}_{k}(t)\\|_{2}\n+\\|{\\cal G}_{k}(t)\\|_{2}+\\alpha_kV'(t)(\\|u\\|_{B^0}+\\|h\\|_{\\widetilde{B}^{0,1}})\\Bigr),\n\\end{eqnarray*}\nfrom which and a similar proof of (\\ref{3.21}) ensure that\n\\begin{eqnarray}\\label{3.34}\n&&\\sum_{k<1}\\Bigl(2^{2k}\\|E_{lk}\\|_{L_T^1}+\\|E_{lk}\\|_{L_T^\\infty}\\Bigr)\\le \\sum_{k<1}E_{hk}(0)\\nonumber\\\\\n&&\\qquad+C\\Bigl(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0,1})}\n+\\|{\\cal G}\\|_{L^1_T(B^0)}+\\int_0^TV'(t)(\\|u(t)\\|_{B^0}+\\|h(t)\\|_{\\widetilde{B}^{0,1}})dt\\Bigr).\n\\end{eqnarray}\n\n\\vspace{.2cm}\n\\noindent{\\bf The completion of the {\\it a-priori} estimates}\\vspace{.2cm}\n\n\\noindent Firstly, adding up \\eqref{3.17}, \\eqref{3.21}, \\eqref{3.31}, and \\eqref{3.32} yields that\n\\begin{eqnarray}\\label{3.35}\n&&\\|u\\|_{L^1_T(B^2)}+\\|u\\|_{L^2_T(B^1)}+\\|h\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)E_k(0)\n+C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|{\\cal G}_k(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla{\\cal H}_k(t)\\|_{L^1_T(L^2)}\n+C\\sum_{k<1}\\omega_k(T)\\|{\\cal H}_k(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\|u\\|_{L^2_T(B^1)}\\|v\\|_{L^2_T(B^1)}+C\\|h\\|_{\\widetilde{E}^{0,1}_T}\\|v\\|_{L^1_T(B^2)},\n\\end{eqnarray}\nwhere we used the fact that\n$$\n\\|h\\|_{E^{1}_T}\\le C\\|h\\|_{\\widetilde{E}^{0,1}_T}.\n$$\nOn the other hand, adding up \\eqref{3.23} and \\eqref{3.34} gives rise to\n\\begin{eqnarray}\\label{3.36}\n&&\\|u\\|_{L^\\infty_T(B^0)}+\\|h\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h\\|_{L^1_T(\\widetilde{B}^{2,1})}\n\\nonumber\\\\&&\\qquad \\le E_0+C\\Big(\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0, 1})}+\\|{\\cal G}\\|_{L^1_T(B^0)}+\n\\int_0^TV'(t)(\\|u\\|_{B^{0}}+\\|h\\|_{\\widetilde{B}^{0, 1}})dt\\Big),\n\\end{eqnarray}\nwhich together with the Gronwall inequality implies that\n\\begin{eqnarray}\\label{3.37}\n&&\\|u\\|_{L^\\infty_T(B^0)}+\\|h\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h\\|_{L^1_T(\\widetilde{B}^{2,1})}\\nonumber\\\\\n&&\\qquad\\qquad\\le Ce^{C\\|v\\|_{L^1_T(B^2)}}\\Big(E_0+\\|{\\cal H}\\|_{L^1_T(\\widetilde{B}^{0, 1})}+\\|{\\cal G}\\|_{L^1_T(B^0)}\\Big).\n\\end{eqnarray}\nFinally, let us remark that\n\\begin{eqnarray*}\nE_0\\thickapprox(\\|h_0\\|_{\\widetilde{B}^{0,1}}+\\|u_0\\|_{B^0}).\n\\end{eqnarray*}\n\n\n\\subsection{The uniform estimate of the approximate sequence of solutions}\n\nIn this subsection, we will construct the approximate solutions of (\\ref{3.1})\nand present the uniform estimate of the approximate solutions.\nLet us first define the approximate sequence $(h^n, u^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits}$ of (\\ref{3.1}) by the following system:\n\\begin{align}\\label{3.38} \\left\\{\n\\begin{aligned}\n&\\partial_t h^{n+1}+u^n\\cdot\\nabla h^{n+1}+\\mbox{div} u^{n+1}=\n{\\cal H}^n,\\\\\n&\\partial_tu^{n+1}-(\\nabla\\cdot D(u^{n+1})+\\nabla \\textrm{div}\\,u^{n+1})+u^{n}\\cdot\\nabla u^{n+1}+\\nabla h^{n+1}\n={\\cal G}^{n},\\qquad\\quad\\\\\n&(h^{n+1}, u^{n+1})|_{t=0}=\\sum_{|k|\\le n+N}\\Delta_k(h_0,u_0),\n\\end{aligned}\n\\right.\n\\end{align}\nwhere\n$${\\cal H}^n\\triangleq -h^n\\mbox{div} u^n,\\qquad {\\cal G}^{n}\\triangleq \\frac{\\nabla h^n}{1+h^n}\\widetilde{\\nabla }u^n,\n\\quad \\textrm{with}\\quad \\widetilde{\\nabla} u^n=D(u^n)+\\mbox{div} u^n, $$\nand $N$ is a fixed large integer such that\n$$\n1+h^n(0)\\ge \\f34,\\qquad \\textrm{for} \\quad n\\ge 1.\n$$\nSet $(h^0, u^0)=(0,0)$ and solve the linear system, we can define $(h^n, u^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits_0}$ by the induction.\nNext, we are going to prove by the induction that there exist positive constants $\\eta$, $K$, and $T$ such that\nthe following bounds hold for all $n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits_0$:\n\\begin{align}\n&1+h^n\\ge\\f12,\\label{3.39}\\\\\n&\\|u^{n}\\|_{L^1_T(B^2)\\cap L^2_T(B^1)}+\\|h^n\\|_{\\widetilde{E}^{0,1}_T}\n\\le \\eta,\\label{3.40}\\\\\n&\\|u^n\\|_{L^\\infty_T(B^0)}+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})\\cap L^1_T(\\widetilde{B}^{2,1})}\n\\le KE_0.\\label{3.41}\n\\end{align}\nAssume that (\\ref{3.39})-(\\ref{3.41}) hold for $(h^n, u^n)$,\nwe need to prove that (\\ref{3.39})-(\\ref{3.41}) also hold for $(h^{n+1}, u^{n+1})$.\nApplying the {\\it a-priori} estimates (\\ref{3.35}) and (\\ref{3.37}) to $(h^{n+1}, u^{n+1})$, we obtain\n\\begin{eqnarray}\\label{3.42}\n&&\\|u^{n+1}\\|_{L^1_T(B^2)}+\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C{\\cal Q}_0(T)+C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|{\\cal G}_k^{n}(t)\\|_{L^1_T(L^2)}\n+C\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla {\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k<1}\\omega_k(T)\\|{\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}+C\\|u^{n+1}\\|_{L^2_T(B^1)}\\|u^n\\|_{L^2_T(B^1)}\\nonumber\\\\\n&&\\qquad+C\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\|u^n\\|_{L^1_T(B^2)},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{3.43}\n&&\\|u^{n+1}\\|_{L^\\infty_T(B^0)}+\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h^{n+1}\\|_{L^1_T(\\widetilde{B}^{2,1})}\\nonumber\\\\\n&&\\qquad\\qquad\\le Ce^{C\\|u^{n}\\|_{L^1_T(B^2)}}\\Big(E_0+\\|{\\cal H}^{n}\\|_{L^1_T(\\widetilde{B}^{0,1})}+\\|{\\cal G}^{n}\\|_{L^1_T(B^0)}\\Big),\n\\end{eqnarray}\nwith\n$${\\cal Q}_0(T)\\triangleq\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)E_{k}(0).$$\nThanks to (\\ref{2.4}), we have\n\\begin{eqnarray*}\n\\|{\\cal H}^n\\|_{{B}^{0}}\\le C\\|{h}^n\\|_{{B}^{0}}\n\\|u^n\\|_{B^2},\\quad \\hbox{and}\\quad\n\\|{\\cal H}^n\\|_{{B}^{1}}\\le C\\|{h}^n\\|_{{B}^{1}}\n\\|u^n\\|_{B^2},\n\\end{eqnarray*} which together with the fact that $\\widetilde{B}^{0,1}={B}^{0}\\cap{B}^{1}$ yields\n\\begin{eqnarray}\\label{3.44}\n\\|{\\cal H}^n\\|_{L^1_T(\\widetilde{B}^{0,1})}\n\\le C\\|{h}^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\|u^n\\|_{L^1_T(B^2)}\\le CKE_0\\eta.\n\\end{eqnarray}\nWe rewrite ${\\cal G}^n$ as\n\\begin{eqnarray*}\\label{3.45}\n\\frac{\\nabla h^n}{1+h^n}\\widetilde{\\nabla} u^n=\n(1+h^n)\\nabla\\bigg(\\frac{h^n}{1+h^n}\\bigg)\\widetilde{\\nabla} u^n.\n\\end{eqnarray*}\nUsing (\\ref{2.4}) and (\\ref{2.12}), we get\n\\begin{align}\\label{3.45}\n\\|{\\cal G}^{n}\\|_{L^1_T(B^0)}\n&\\le C\\Bigl\\|\\nabla\\bigg(\\frac{h^n}{1+h^n}\\bigg)\\Bigr\\|_{L^\\infty_T(B^0)}\\|(1+h^n)\\widetilde{\\nabla} u^n\\|_{L^1_T(B^1)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(L^\\infty)})^2\\|h^n\\|_{L^\\infty_T(B^1)}(1+\\|h^n\\|_{L^\\infty_T(B^1)})\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})})^3\n\\|h^{n}\\|_{L^\\infty_T(B^1)}\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le CKE_0(1+KE_0)^3\\eta.\n\\end{align}\nPlugging (\\ref{3.44}) and (\\ref{3.45}) into (\\ref{3.43}) yields that\n\\begin{eqnarray}\\label{3.46}\n\\|u^{n+1}\\|_{L^\\infty_T(B^0)}+\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h^{n+1}\\|_{L^1_T(\\widetilde{B}^{2,1})}\n\\le Ce^{C\\eta}\\Big(E_0+KE_0(1+KE_0)^3\\eta\\Big).\n\\end{eqnarray}\nWe take $T, \\eta>0$ small enough and $K=4C$ such that\n\\begin{align}\ne^{C\\eta}\\le 2,\\quad K(1+KE_0)^3\\eta\\le 1,\\tag{$\\Re_1$}\n\\end{align}\nfrom which and (\\ref{3.46}), it follows that\n\\begin{eqnarray*}\n\\|u^{n+1}\\|_{L^\\infty_T(B^0)}+\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}+\\|h^{n+1}\\|_{L^1_T(\\widetilde{B}^{2,1})}\\le KE_0.\n\\end{eqnarray*}\nThis proves (\\ref{3.41}) for $(u^{n+1},h^{n+1})$.\n\nNext, we prove (\\ref{3.40}) for $(u^{n+1},h^{n+1})$.\nApplying Lemma \\ref{Lem5.2} with $s_1=0$ and $s_2=1$, (\\ref{2.4}) with $s_1=s_2=1$,\nand Lemma \\ref{Lem5.4} with $s=1$, we obtain\n\\begin{align}\\label{3.47}\n\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)\\|{\\cal G}_k^{n}(t)\\|_{L^1_T(L^2)}\n&\\le C\\Bigl\\|\\nabla\\bigg(\\frac{h^n}{1+h^n}\\bigg)\\Bigr\\|_{E^0_T}\\|(1+h^n)\\widetilde{\\nabla} u^n\\|_{L^1_T(B^1)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(L^\\infty)})^3\\|h^n\\|_{E^1_T}(1+\\|h^n\\|_{L^\\infty_T(B^1)})\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le C(1+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})})^4\n\\|h^{n}\\|_{\\widetilde{E}^{0,1}_T}\\|u^n\\|_{L^1_T(B^2)}\\nonumber\\\\\n&\\le C(1+KE_0)^4\\eta^2.\n\\end{align}\nOn the other hand, we apply Lemma 5.2 with $s_1=0, s_2=1$ to get\n\\begin{eqnarray*}\n&&\\sum_{k\\ge 1}\\omega_k(T)\\|\\nabla {\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}+\\sum_{k<1}\\omega_k(T)\\|{\\cal H}_k^{n}(t)\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\quad\\le C\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)(\\|\\nabla h^n_k\\|_{L^\\infty_T(L^2)}+\n\\|h^n_k\\|_{L^\\infty_T(L^2)})\\|\\mbox{div} u^n\\|_{L^1_T(B^1)}\\nonumber\\\\\n&&\\qquad+C\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{2k}\\|u^n_k\\|_{L^1_T(L^2)}\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\nonumber\\\\\n&&\\quad\\triangleq I+II.\n\\end{eqnarray*}\nObviously, we have\n\\begin{eqnarray}\\label{3.48}\nI\\le C\\|h^n\\|_{\\widetilde{E}^{0,1}_T}\\|u^n\\|_{L^1_T(B^2)}\\le C\\eta^2.\n\\end{eqnarray}\nIn order to estimate $II$, we first fix $k_0\\ge 1$ such that\n\\begin{eqnarray}\\label{3.49}\n\\sum_{k\\ge k_0}\\|u_k(0)\\|_{2}\\le \\f {\\eta} {16CKE_0}.\n\\end{eqnarray}\nThen we write\n\\begin{align}\nII&=\\sum_{k\\ge k_0}\\omega_k(T)2^{2k}\\|u^n_k\\|_{L^1_T(L^2)}\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\n+\\sum_{k\\le k_0}\\omega_k(T)2^{2k}\\|u^n_k\\|_{L^1_T(L^2)}\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\nonumber\\\\\n&\\triangleq II_1+II_2.\\nonumber\n\\end{align}\nUsing (\\ref{3.18}), (\\ref{3.47}), and (\\ref{3.49}), we obtain\n\\begin{align}\\label{3.50}\nII_1&\\le CKE_0\\Bigl[\\sum_{k\\ge k_0}\\omega_k(T)\\|u_k(0)\\|_2+\n\\sum_{k\\ge k_0}\\omega_k(T)\\Bigl(\\|\\nabla h_k^n\\|_{L^1_T(L^2)}\n+\\|{\\cal G}_{k}^{n-1}\\|_{L^1_T(L^2)}\\Bigr)\\nonumber\\\\\n&\\quad+\\|u^n\\|_{L^2_T(B^1)}\\|u^{n-1}\\|_{L^2_T(B^1)}\\Bigr]\\nonumber\\\\\n&\\le CKE_0\\Bigl[\\f {\\eta} {16CKE_0}+\n\\sum_{k\\ge k_0}\\omega_k(T)\\|\\nabla h_k^n\\|_{L^1_T(L^2)}+(1+KE_0)^4\\eta^2\\Bigr].\n\\end{align}\nOn the other hand, thanks to (\\ref{3.19}) and Lemma \\ref{Lem5.1}, we have\n\\begin{align}\n\\sum_{k\\ge k_0}\\omega_k(T)\\|\\nabla h_k^n\\|_{L_T^1(L^2)}&\\le C(1-e^{-cT})\\sum_{k\\ge k_0}\\omega_k(T)E_{hk}(0)\\nonumber\\\\\n&\\quad+C(1-e^{-cT})\\sum_{k\\ge k_0}\\omega_k(T)\\Bigl(\\|\\nabla {\\cal H}_k^{n-1}(t)\\|_{L_T^1(L^2)}\n+\\|{\\cal G}_{k}^{n-1}(t)\\|_{L_T^1(L^2)}\\Bigr)\\nonumber\\\\\n&\\quad+C\\|u^n\\|_{L^2_T(B^1)}\\|u^{n-1}\\|_{L^2_T(B^1)}+C\\|h^n\\|_{E^1_T}\\|u^{n-1}\\|_{L^2_T(B^1)}\\nonumber\\\\\n&\\le C(1-e^{-cT})E_0+C(1-e^{-cT})KE_0\\eta+C(1+KE_0)^4\\eta^2,\\nonumber\n\\end{align}\nwhere we used (\\ref{3.44}) and (\\ref{3.47}) in the second inequality.\nPlugging the above inequality into (\\ref{3.50}) yields that\n\\begin{eqnarray}\\label{3.51}\nII_1\\le CKE_0\\Bigl[\\f {\\eta} {16CKE_0}+\n(1-e^{-cT})(E_0+KE_0\\eta)+(1+KE_0)^4\\eta^2\\Bigr].\n\\end{eqnarray}\nNote for $k\\le k_0$, we can choose $T>0$ small enough so that\n\\begin{align}\n\\omega_k(T)\\le \\f 1{16CKE_0\\eta},\\tag{$\\Re_2$}\n\\end{align}\nso we get\n\\begin{eqnarray}\\label{3.52}\n|II_2|\\le \\f \\eta {16}.\n\\end{eqnarray}\n\nPlugging (\\ref{3.47}), (\\ref{3.48}), (\\ref{3.51}), (\\ref{3.52}) into (\\ref{3.42}), we get\n\\begin{eqnarray}\\label{3.53}\n&&\\|u^{n+1}\\|_{L^1_T(B^2)}+\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\nonumber\\\\\n&&\\quad\\le C{\\cal Q}_0(T)+\\f \\eta {8}+ C(1+KE_0)^5\\eta^2+\nCKE_0(1-e^{-cT})(E_0+KE_0\\eta)\\nonumber\\\\\n&&\\qquad+C\\eta(\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}).\n\\end{eqnarray}\nNote that ${\\cal Q}_0(0)=0$, we can take $T, \\eta$ small enough such that\n\\begin{align}\n&C\\eta\\le \\f12,\\quad C{\\cal Q}_0(T)\\le \\f \\eta 8,\\quad C(1+KE_0)^5\\eta<\\f18,\n\\quad \\textrm{and}\\nonumber\\\\\n&CKE_0(1-e^{-cT})(E_0+KE_0\\eta)\\le \\f \\eta 8, \\tag{$\\Re_3$}\n\\end{align}\nwhich together with (\\ref{3.53}) gives\n\\begin{eqnarray*}\n\\|u^{n+1}\\|_{L^1_T(B^2)}+\\|u^{n+1}\\|_{L^2_T(B^1)}+\\|h^{n+1}\\|_{\\widetilde{E}^{0,1}_T}\\le\\eta.\n\\end{eqnarray*}\n\nFinally, let us prove (\\ref{3.39}) for $h^{n+1}$.\nWe rewrite the first equation of \\eqref{3.38} as\n\\begin{eqnarray*}\n\\partial_t(1+h^{n+1})+u^{n}\\cdot\\nabla(1+h^{n+1})+\\mbox{div} u^{n+1}-{\\cal H}^{n}=0.\n\\end{eqnarray*}\nThen $1+h^{n+1}$ can be represented as\n\\begin{align}\\label{3.54}\n(1+h^{n+1})(t,x)=&(1+h^{n+1}_0)((\\psi^n)^{-1}_t(x))+\\int_0^t\\mbox{div}\\,u^{n+1}(\\tau,\\psi^n_\\tau((\\psi^n)^{-1}_t(x)))d\\tau\\nonumber\\\\\n&+\\int_0^t{\\cal H}^{n}(\\tau,\\psi^n_\\tau((\\psi^n)^{-1}_t(x)))d\\tau,\n\\end{align}\nwhere the flow map $\\psi^{n}_t$ is defined by\n\\begin{eqnarray*}\\left\\{\n\\begin{aligned}\n&\\partial_t\\psi^{n}_t(x)=u^{n}(t,\\psi^{n}_t(x))\\\\\n&\\psi^{n}_t|_{t=0}=x.\n\\end{aligned}\\right.\n\\end{eqnarray*}\nThanks to the inclusion map $B^1\\hookrightarrow L^\\infty$ and (\\ref{2.4}), we get\n\\begin{eqnarray*}\n&&\\int_0^t\\|\\mbox{div}\\,u^{n+1}(\\tau,\\psi^n_\\tau((\\psi^n)^{-1}_t(x)))\\|_{\\infty} d\\tau\\le \\|u^{n+1}\\|_{L^1_t(B^2)}\\le \\eta,\\\\\n&&\\int_0^t\\|{\\cal H}^{n}(\\tau,\\psi_\\tau(\\psi^{-1}_t(x)))\\|_\\infty d\\tau \\le\\|h^{n}\\mbox{div} u^{n}\\|_{L^1_t(B^1)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\qquad\\le C\\|h^{n}\\|_{L^\\infty_t(\\widetilde{B}^{0,1})}\\|u^{n}\\|_{L^1_t(B^2)}\\le CKE_0\\eta,\n\\end{eqnarray*}\nfrom which and (\\ref{3.54}), it follows that\n\\begin{eqnarray}\\label{3.55}\n1+h^{n+1}\\ge \\f34-(1+CKE_0)\\eta.\n\\end{eqnarray}\nWe take $\\eta$ small enough such that\n\\begin{align}\n(1+CKE_0)\\eta\\le \\f14, \\tag{$\\Re_4$}\n\\end{align}\nwhich together with (\\ref{3.55}) ensures that\n\\begin{eqnarray*}\n1+h^{n+1}\\ge \\f12.\n\\end{eqnarray*}\nSo far, we have show that $T$, $\\eta$ can be chosen small enough such that\nthe assumption $(\\Re_1)-(\\Re_4)$ hold under which\nthe approximate solutions $(u^n, h^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits_0}$ is uniformly bounded in\n$${\\cal E}_T\\triangleq \\Bigl(L^\\infty_T(B^0)\\cap L^1_T(B^2)\\Bigr)\\times\n\\Bigl(L^\\infty_T(\\widetilde{B}^{0,1})\\cap L^1_T(\\widetilde{B}^{2,1})\\Bigr).$$\nIt should be pointed out that if $\\|u_0\\|_{B^0}+\n\\|h_0\\|_{\\widetilde{B}^{0,1}}$ is small enough, we can take $T=+\\infty$ such that\nthe assumption $(\\Re_1)-(\\Re_4)$ hold.\n\n\\subsection {The existence of the solution}\nNow let us turn to prove the existence of the solution, and\nthe standard compact arguments will be used.\nIn the section 3.2, we have showed that the approximate solutions $(h^n,u^n)_{n\\in\\mathop{\\bf N\\kern 0pt}\\nolimits}$\nsatisfy \\eqref{3.39}-\\eqref{3.41}, and without loss of generality, we can assume the following:\n\\begin{align}\n1&+h^n\\ge\\f12,\\label{3.56}\\\\\n\\|u^n\\|_{L^\\infty_T(B^0)\\cap L^1_T(B^2)}&+\\|h^n\\|_{L^\\infty_T(\\widetilde{B}^{0,1})\\cap L^1_T(\\widetilde{B}^{2,1})}\n\\le KE_0.\\label{3.57}\n\\end{align}\nUsing the interpolation and the fact that $B^0\\cap B^1=\\widetilde{B}^{0,1}$, we have\n\\begin{eqnarray*}\n&&\\|h^n\\|_{L^2_T(B^1)}\\lesssim\\|h^n\\|^{\\frac{1}{2}}_{L^\\infty_T(\\widetilde{B}^{0,1})}\n\\|h^n\\|^{\\frac{1}{2}}_{L^1_T(\\widetilde{B}^{2,1})},\\quad\n\\|u^n\\|_{L^2_T(B^1)}\\lesssim\n\\|u^n\\|^{\\frac{1}{2}}_{L^\\infty_T(B^0)}\\|u^n\\|^{\\frac{1}{2}}_{L^1_T(B^2)},\\\\\n&&\\|h^n\\|_{L^4_T(B^{\\frac{1}{2}})}\\lesssim\\|h^n\\|^{\\frac{1}{2}}_{L^\\infty_T(\\widetilde{B}^{0,1})}\n\\|h^n\\|^{\\frac{1}{2}}_{L^2_T(B^{1})},\\quad\\|u^n\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\\lesssim\n\\|u^n\\|^{\\frac{1}{4}}_{L^\\infty_T(B^0)}\\|u^n\\|^{\\frac{3}{4}}_{L^1_T(B^2)},\n\\end{eqnarray*} from which and (\\ref{3.56}), it follows that\n\\begin{eqnarray}\\label{3.58}\n\\|h^n\\|_{L^2_T(B^1)}+\\|u^n\\|_{L^2_T(B^1)}+\\|h^n\\|_{L^4_T(B^{\\frac{1}{2}})}+\\|u^n\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\n\\lesssim KE_0.\n\\end{eqnarray}\n\nNow, we show that $(h^n, u^n)$ is uniformly bounded in $C^{\\frac{1}{2}}_{loc}(B^0)\\times\nC^{\\frac{1}{4}}_{loc}(B^{-\\frac{1}{2}})$. Using (\\ref{2.4}), (\\ref{3.57})\nand (\\ref{3.58}), it is easy to verify that\n\\begin{eqnarray*}\n&&\\|u^{n}\\cdot\\nabla h^{n+1}\\|_{L^2_T(B^{0})}\\lesssim\n\\|u^{n}\\|_{L^2_T(B^{1})}\\|h^{n+1}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\lesssim (KE_0)^2,\\\\\n&&\\|h^{n}\\mbox{div} u^{n}\\|_{L^2_T(B^{0})}\\lesssim\n\\|u^{n}\\|_{L^2_T(B^{1})}\\|h^{n}\\|_{L^\\infty_T(\\widetilde{B}^{0,1})}\\lesssim\n(KE_0)^2, \\end{eqnarray*}\nfrom which and the first equation of (\\ref{3.38}), it follows that $\\partial_t\nh^{n}$ is uniformly bounded in $L^2_T(B^0)$ which implies $h^{n}$ is\nuniformly bounded in $C^{\\frac{1}{2}}_{loc}(B^0)$.\nOn the other hand, thanks to (\\ref{2.4}), (\\ref{3.56}) and (\\ref{2.12}), we have\n\\begin{eqnarray*}\n&&\\|u^{n}\\cdot\\nabla u^{n+1}\\|_{L^\\frac{4}{3}_T(B^{-\\frac{1}{2}})}\\lesssim\n\\|u^{n}\\|_{L^\\infty_T(B^0)}\\|u^{n+1}\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\\lesssim (KE_0)^2,\\\\\n&&\\bigg\\|\\frac{\\nabla h^{n}}{1+h^{n}}\\widetilde{\\nabla} u^{n}\\bigg\\|_{L^\\frac{4}{3}_T(B^{-\\frac{1}{2}})}\n\\lesssim C(1+\\|h^{n}\\|_{L^\\infty_T(B^1)})^3\\|u^{n}\\|_{L^\\frac{4}{3}_T(B^\\frac{3}{2})}\\lesssim C(1+KE_0)^3KE_0,\n\\end{eqnarray*}\nfrom which and the second equation of (\\ref{3.38}), it follows that\n$\\partial_t u^{n}$ is uniformly bounded in $L^\\frac{4}{3}_T(B^{-\\frac{1}{2}})$\nwhich implies $u^{n}$ is uniformly bounded\nin $C^{\\frac{1}{4}}_{loc}(B^{-\\frac{1}{2}})$.\n\nNext, we claim that the inclusions $B^0\\cap B^1\\hookrightarrow L^2$ and\n$B^{-\\frac{1}{2}}\\cap B^0\\hookrightarrow \\dot{H}^{-\\frac{1}{2}}$ are locally\ncompact. Indeed, these can be proved by noting that for $s'\\nonumber\\\\\n&=\n\\Big<(1+h^n)\\nabla\\Big(\\frac{h^n}{1+h^n}-\\frac{h}{1+h}\\Big)\\widetilde{\\nabla} u^n,\\,\\theta\\Big>\\nonumber\\\\&\\quad+\n\\Big<(h^n-h)\\nabla\\Big(\\frac{h}{1+h}\\Big)\\widetilde{\\nabla} u^n,\\,\\theta\\Big>+\n\\Big<(1+h)\\nabla\\Big(\\frac{h}{1+h}\\Big)\\widetilde{\\nabla} (u^n-u),\\,\\theta\\Big>\\nonumber\\\\&\n\\triangleq{I}_1+{I}_2+{I}_3.\\nonumber\n\\end{align}\nThanks to (\\ref{2.4}) and (\\ref{3.56}), we have\n\\begin{align}\n{I}_1&\\le\\bigg\\|\\frac{\\psi(h^n-h)}{(1+h^n)(1+h)}\\bigg\\|_2\n\\|\\nabla((1+h^n)\\widetilde{\\nabla} u^n\\theta)\\|_2\\lesssim\\|\\theta(h^n-h)\\|_2\\|(1+h^n)\\widetilde{\\nabla} u^n\\|_{B^1}\\nonumber\\\\\n&\\lesssim \\|\\theta(h^n-h)\\|_2(1+\\|h^n\\|_{\\widetilde{B}^{0,1}})\\|u^n\\|_{B^2},\\nonumber\n\\end{align}\nwhere $\\psi \\in C_0^\\infty([0, T^*)\\times\\mathop{\\bf R\\kern 0pt}\\nolimits^2)$,\\,and $\\psi=1$ on \\,$\\textrm{supp}\\,\\theta$.\nFor ${I}_2$, we have\n\\begin{align}\n{I}_2&\\le\\|\\theta(h^n-h)\\|_2\n\\bigg\\|\\nabla\\bigg(\\frac{h}{1+h}\\bigg)\\widetilde{\\nabla} u^n\\bigg\\|_2\n\\lesssim\\|\\theta(h^n-h)\\|_2\\|\\nabla h\\|_2\\|\\nabla u^n\\|_{L^\\infty}\\nonumber\\\\\n&\\lesssim \\|\\theta(h^n-h)\\|_2\\|h\\|_{\\widetilde{B}^{0,1}}\\|u^n\\|_{B^2}.\\nonumber\n\\end{align}\nUsing (\\ref{3.56}) and the interpolation, we get\n\\begin{align}\n{I}_3&\\le\\bigg\\|(1+h)\\nabla\\bigg(\\frac{h}{1+h}\\bigg)\\bigg\\|_2\n\\|\\widetilde{\\nabla} (u^n-u)\\theta\\|_2\\lesssim (1+\\|h\\|_\\infty)\\|\\nabla h\\|_2\\|(u^n-u)\\psi\\|_{\\dot{H}^1}\\nonumber\\\\\n&\\lesssim (1+\\|h\\|_{\\widetilde{B}^{0,1}})\\|h\\|_{B^1}\\|u^n-u\\|^{\\frac{3}{5}}_{\\dot{H}^2}\n\\|(u^n-u)\\theta\\|^{\\frac{2}{5}}_{\\dot{H}^{-\\frac{1}{2}}}.\\nonumber\n\\end{align}\nThus, by (\\ref{3.59}), we get as $n\\rightarrow 0$\n$$\n\\big\\longrightarrow 0.\n$$\nFollowing the argument in \\cite{D1}, we can also prove that $(u,h)$ is continuous in\ntime with values in $B^{0}\\times \\widetilde{B}^{0,1}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\n\\section{Uniqueness}\n\nIn this section, we will prove the uniqueness of the solution. Firstly, let us recall some known results.\n\\begin{Lemma}\\label{Lemma4.1}(Osgood's lemma) Let $\\rho$ be a measurable positive\nfunction and $\\gamma$ a positive locally integrable function, each defined on\nthe domain $[t_0, t_1]$. Let $\\mu: [0, \\infty)\\rightarrow [0,\\infty)$\nbe a continuous nondecreasing function, with $\\mu(0)=0$. Let $a\\ge0$,\nand assume that for all $t$ in $[t_0, t_1]$,\n$$\\rho(t)\\le a+\\int_{t_0}^{t}\\gamma(\\tau)\\mu(\\rho(\\tau))d\\tau.$$\nIf $a>0$, then\n$$-{\\cal M}(\\rho(t))+{\\cal M}(a)\\le \\int_{t_0}^{t}\\gamma(\\tau)d\\tau,\n\\quad\\mbox{where}\\quad{\\cal M}(x)=\\int_x^1\\frac{d\\tau}{\\mu(\\tau)}.$$\nIf $a=0$ and ${\\cal M}=\\infty$, then $\\rho\\equiv0.$\n\\end{Lemma}\nThis Lemma can be understood as a generalization of classical Gronwall Lemma and\ncan be found in \\cite{Ch1}.\n\\begin{Proposition}\\label{Prop4.1}\nLet $s\\in (-\\frac{d}{p}, 1+\\frac{d}{p})$, and $1\\le p,r\\le+\\infty$.\nLet $v$ be a vector field such that $\\nabla v\\in L^1_T(\\dot{B}^{\\frac{d}{p}}_{p,r}\\cap L^\\infty)$.\nAssume that $f_0\\in \\dot{B}^{s}_{p,r},$ $g\\in L^1_T(\\dot{B}^{s}_{p,r})$ and\n$f\\in L^\\infty_T(\\dot{B}^{s}_{p,r})\\cap C([0,T]; {\\cal S}')$ is the solution of\n\\begin{align}\n\\bigg\\{\\begin{aligned}\n&\\partial_t f+v\\cdot \\nabla f =g,\\\\\n&f(0,x)=f_0.\n\\end{aligned}\n\\bigg.\\nonumber\\end{align}\nThen there exists a constant $C(s,p,d)$ such that\nfor $t\\in[0,T]$\n\\begin{align}\\label{4.1}\n\\|f\\|_{\\widetilde{L}^\\infty_t(\\dot{B}^{s}_{p,r})}\\le Ce^{CV(t)}\\bigg(\n\\|f_0\\|_{\\dot{B}^{s}_{p,r}}+\\int_0^t e^{-CV(\\tau)}\\|g(\\tau)\\|_{\\dot{B}^{s}_{p,r}}d\\tau\\bigg),\n\\end{align}\nwhere $V(t)\\triangleq\\int_0^t\\|\\nabla v(\\tau)\\|_{\\dot{B}^{\\frac{d}{p}}_{p,r}\\cap L^\\infty}d\\tau.$\nIf $r<+\\infty$, then $f$ belongs to $C([0,T]; \\dot{B}^s_{p,r})$.\n\\end{Proposition}\nThe proof can be found in \\cite{D4}.\n\\begin{Proposition}\\label{Prop4.3}\nLet $T>0$, $s\\in \\mathop{\\bf R\\kern 0pt}\\nolimits$, and $1\\le q,r\\le+\\infty$. Assume that\n$u_0\\in \\dot{B}^{s}_{2,q},$ $g\\in \\widetilde{L}^1_T(\\dot{B}^{s}_{2,q})$ and\n$u$ is the solution of\n\\begin{align}\n\\,\\bigg\\{\\begin{aligned}\n&\\partial_t u-\\nu \\widetilde{\\Delta} u=g,\\nonumber\\\\\n&u(0,x)=u_0,\n\\end{aligned}\n\\bigg.\\end{align}\nwhere $\\widetilde{\\Delta} u=\\nabla\\cdot D(u)+\\nabla \\mbox{div} \\,u$.\nThen there exists a constant $C(s,d,\\nu)$ such that\n\\begin{align}\\label{4.2}\n(r\\nu)^{\\frac1r}\\|u\\|_{\\widetilde{L}^r_T(\\dot{B}^{s+\\frac2r}_{2,q})}\\le\n\\Big(&\\sum_{k\\in{\\mathbf Z}}\\big({1-e^{-r\\nu 2^{2k}T}}\\big)^{\\frac q r}2^{qks}\\|\\Delta_ku_0\\|_2^q\\Big)^{\\frac1q}\n\\nonumber\\\\&+C\\Big(\\sum_{k\\in{\\mathbf Z}}\\big({1-e^{-r\\nu 2^{2k}T}}\\big)^{\\frac qr}2^{qks}\\|\\Delta_kg\\|_{L^1_T(L^2)}^q\\Big)^{\\frac1q}.\n\\end{align}\nIf $q<+\\infty$, then $u$ belongs to $C([0,T]; \\dot{B}^{s}_{2,q})$.\n\\end{Proposition}\nThe proof is similar to the case when the diffusion term $\\widetilde{\\Delta} u$ is replaced by $\\Delta u$.\nWe can refer to \\cite{Ch2} see the details.\n\n\nNow we introduce the logarithmic interpolation inequality (see \\cite{D3})\n\\begin{Proposition}\\label{Prop4.4} For any $1\\le p,\\rho\\le+\\infty$, $s\\in\\mathop{\\bf R\\kern 0pt}\\nolimits$ and $0<\\epsilon\\le 1$,\nwe have\n\\begin{eqnarray}\\label{4.3}\n\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^s_{p,1})}\\le\nC\\frac{\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^s_{p,\\infty})}}{\\epsilon}\n\\log\\bigg(e+\\frac{\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{s-\\epsilon}_{p,\\infty})}\n+\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^{s+\\epsilon}_{p,\\infty})}}\n{\\|f\\|_{\\widetilde{L}^\\rho_T(\\dot{B}^s_{p,\\infty})}}\\bigg).\n\\end{eqnarray}\n\n\\end{Proposition}\n\nNow, let us prove the uniqueness of the solution of (\\ref{3.1}).\nLet $(u_1, h_1)$, $(u_2, h_2)$ $\\in\\big(L^\\infty_T(B^0)\\cap\nL^1_T(B^2))\\times L^\\infty_T(\\widetilde{B}^{0,1})$\nbe two solutions of (\\ref{3.1}) with the same initial data.\nThe difference $\\vartheta\\triangleq h_2-h_1$, $w\\triangleq u_2-u_1$ satisfies the following system:\n\\begin{align}\\label{4.4}\n\\left\\{\n\\begin{aligned}\n&\\partial_t\\vartheta+u_2\\cdot\\nabla\\vartheta=\n-\\mbox{div} w-w\\nabla h_1-\\vartheta\\mbox{div} u_2-h_1\\mbox{div} w,\\\\\n&\\partial_tw-\\nu\\widetilde{\\Delta} w=-\\nabla\\vartheta-u_2\\cdot\\nabla w-w\\cdot\\nabla u_1\n+\\nu(1+h_1)\\nabla\\Big(\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} w\\\\ &\\qquad\\qquad\\qquad+\\nu(1+h_1)\n\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} u_2\n+\\nu\\vartheta\\nabla\\Big(\\frac{h_2}{1+h_2}\\Big)\\widetilde{\\nabla}u_2,\\\\\n&\\vartheta(0,x)=0,\\quad w(0,x)=0.\n\\end{aligned}\\right.\n\\end{align}\nWithout loss of generality, we assume that there holds for sufficiently small $T$\n\\begin{align}\\label{4.5}\n&1+h_1\\ge \\f12, \\\\\\label{4.6}\n&\\|h_1\\|_{\\widetilde{E}^{0,1}_T}\\le \\varepsilon,\n\\end{align}\nwhere $\\varepsilon>0$ is small enough. Applying the Proposition \\ref{Prop4.1} to the first\nequation of (\\ref{4.4}) yields\n\\begin{align}\\label{4.7}\n\\|\\vartheta(t)\\|_{\\dot{B}^0_{2,\\infty}}\\lesssim\n\\int_0^t\\!e^{C(V_2(t)-V_2(\\tau))}\\|w\\cdot\\nabla\nh_1+\\vartheta\\mbox{div} u_2\n+h_1\\mbox{div} w+\\mbox{div} w\\|_{\\dot{B}^0_{2,\\infty}}d\\tau,\n\\end{align} with $V_2(t)\\triangleq\\int_0^t\\|\\nabla\nu_2\\|_{\\dot{B}^1_{2,\\infty}\\cap L^\\infty}d\\tau$. It follows from\n(\\ref{2.5}) with $s=0$ that\n\\begin{align} \\|w\\cdot\\nabla\nh_1\\|_{\\dot{B}^0_{2,\\infty}}&\\lesssim\n\\|\\nabla h_1\\|_{\\dot{B}^0_{2,\\infty}}\\|w\\|_{B^1}\n\\lesssim\\|w\\|_{{B}^1}\\|h_1\\|_{{B}^1},\\nonumber\\\\\n\\|\\vartheta\\mbox{div} u_2\\|_{\\dot{B}^0_{2,\\infty}}&\\lesssim\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\n\\|u_2\\|_{{B}^2},\\nonumber\\\\\n\\|h_1\\mbox{div} w\\|_{\\dot{B}^0_{2,\\infty}}&\\lesssim\n\\|\\mbox{div} w\\|_{\\dot{B}^0_{2,\\infty}}\n\\|h_1\\|_{{B}^1}\\lesssim\\|w\\|_{{B}^1}\n\\|h_1\\|_{{B}^1},\\nonumber\n\\end{align}\nwhere we have used $B^1\\hookrightarrow\\dot{B}^1_{2,\\infty}$.\nPlugging the above estimates into (\\ref{4.7}), we get\n\\begin{align}\\label{4.8}\n\\|\\vartheta(t)\\|_{\\dot{B}^0_{2,\\infty}}\\lesssim\n\\int_0^t\\!e^{C(V_2(t)-V_2(\\tau))}\\Bigl[\\|w\\|_{B^1}\n(1+\\|h_1\\|_{B^1})+\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\n\\|u_2\\|_{B^2}\\Bigr]d\\tau.\n\\end{align}\nRecall that $u^i\\in L^1_T(B^2)$, we can take a $T\\in (0,\\infty)$ small enough so\nthat\n$$C\\|u_2\\|_{L^1_T(B^2)}\\le \\f1 4,$$\nwhich together with (\\ref{4.8}) implies that for $t\\le T$\n\\begin{align}\\label{4.9}\n\\|\\vartheta\\|_{L^\\infty_t(\\dot{B}^0_{2,\\infty})}\\lesssim\n\\|w\\|_{L^1_t(B^1)}(1+\\|h_1\\|_{L^\\infty_t(B^1)}).\n\\end{align}\nApplying (\\ref{4.3}) to the term\n$\\|w\\|_{L^1_t(B^1)}$ yields\n\\begin{align}\\label{4.10}\n\\|\\vartheta\\|_{L^\\infty_t(\\dot{B}^0_{2,\\infty})}\\lesssim&\n\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}\n\\log\\bigg(e+\\frac{\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^0_{2,\\infty})}\n+\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^2_{2,\\infty})}}\n{\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}}\\bigg)(1+\\|h_1\\|_{L^\\infty_t(B^1)}).\n\\end{align}\nThanks to $B^s\\hookrightarrow\\dot{B}^s_{2,\\infty}$\nand $\\widetilde{B}^{0,1}\\hookrightarrow B^1$, we have\n\\begin{align}\\label{4.11}\n\\|\\vartheta\\|_{L^\\infty_t(\\dot{B}^0_{2,\\infty})}\\lesssim\n\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}\n\\log\\bigg(e+\\frac{W(t)}\n{\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}}\\bigg)\n\\end{align}\nwith $$W(t)\\triangleq\\|u^i\\|_{\\widetilde{L}^1_t(B^0)}\n+\\|u^i\\|_{\\widetilde{L}^1_t(B^2)},$$ and for finite $t$, $W(t)< +\\infty$.\n\nNext, we deal with the second equation of (\\ref{4.4}). We get by\napplying (\\ref{2.7}) with $s=1$, $s=0$ respectively that\n\\begin{align}\\label{4.12}\n\\|u_2\\cdot\\nabla w\\|_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\\lesssim\n\\|u_2\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})},\\\\\\label{5.12}\n\\|w\\cdot\\nabla u_1\\|_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\\lesssim\n\\|u_1\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}.\n\\end{align}\nWe can deduce $h_i\\in C(0,T; \\mathop{\\bf R\\kern 0pt}\\nolimits^2)$ $(i=1,2)$ from the fact $B^1\\hookrightarrow C$.\nMoreover, due to (\\ref{4.5}), we can assume $h_1(t,x)+1\\ge \\delta$\nfor all $t\\le T$, $x\\in \\mathop{\\bf R\\kern 0pt}\\nolimits^2$. Since $h_1$, $h_2$ have the same initial data,\nfrom the continuity of $h_2$, there exists a $\\widetilde{T}\\le T$ such that\n$$h_2(x,t)+1\\ge \\delta,\\quad\\hbox{for all}\\quad t\\in[0, \\widetilde{T}],\\quad x\\in\\mathop{\\bf R\\kern 0pt}\\nolimits^2.$$\nIt follows from (\\ref{2.6}) with $s=1$, (\\ref{2.13}) and\n$B^1\\hookrightarrow\\dot{B}^1_{2,\\infty}\\cap L^\\infty$ that\n\\begin{align}\n&\\Big\\|(1+h_1)\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} u_2\\Big\\|\n_{\\dot{B}^{-1}_{2,\\infty}}\\nonumber\\\\&\\qquad\\lesssim\n\\Big\\|(1+h_1)\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\Big\\|\n_{\\dot{B}^{-1}_{2,\\infty}}\\|\\widetilde{\\nabla} u_2\\|\n_{B^1}\\nonumber\\\\&\\qquad\\lesssim (1+\\|h_1\\|_{B^1})\n\\Big\\|\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big\\|\n_{\\dot{B}^0_{2,\\infty}}\\|u_2\\|_{B^2}\\nonumber\\\\&\\qquad\\lesssim (1+\\|h_1\\|_{B^1})\n(\\|h_1\\|_{B^1}+\\|h_2\\|_{B^1})\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\n\\|u_2\\|_{B^2},\\nonumber\n\\end{align}\nwhich together with\n $L^1_t(\\dot{B}^{-1}_{2,\\infty})\\subset\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})$\nyields\\begin{align}\\label{4.14}\n\\Big\\|&(1+h_1)\\nabla\\Big(\\frac{h_2}{1+h_2}-\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} u_2\\Big\\|\n_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\\nonumber\\\\&\n\\lesssim\\int_0^t(1+\\|h_1\\|_{B^1})(\\|h_1\\|_{B^1}+\\|h_2\\|_{B^1})\n\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}\\|u_2\\|_{B^2}d\\tau.\n\\end{align}\nThanks to (\\ref{2.6}),\n(\\ref{2.12}), and\n$L^1_t(\\dot{B}^{-1}_{2,\\infty})\\subset\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})$, we get\n\\begin{align}\\label{4.15}\n&\\Big\\|\\vartheta\\nabla \\Big(\\frac{h_2}{1+h_2}\\Big)\\widetilde{\\nabla} u_2\\Big\\|\n_{\\widetilde{L}^1_t(\\dot{B}^{-1}_{2,\\infty})}\n\\lesssim\\int_0^t\\|\\vartheta\\|_{\\dot{B}^{0}_{2,\\infty}}\\|h_2\\|_{B^1}\\|u_2\\|_{B^2}d\\tau.\\end{align}\nThanks to Lemma 5.3 with $s_1=s_2=0$, Lemma 5.4 with $s=1$, and (\\ref{2.5}) with $s=0$, we have\n\\begin{align} \\label{4.17}\n\\sup_{k\\in{\\mathbf Z}}&\\, \\omega_k(t)2^{-k}\\Big\\|\\Delta_k\\Big((1+h_1)\\nabla\\Big(\\frac{h_1}{1+h_1}\\Big)\\widetilde{\\nabla} w\\Big)\\Big\\|_{L^1_t(L^2)}\n\\nonumber\\\\&\\lesssim\n\\Big\\|\\nabla\\Big(\\frac{h_1}{1+h_1}\\Big)\\Big\\|_{E^{0}_t}\n\\|(1+h_1)\\widetilde{\\nabla} w\\|_{\\widetilde{L}^1_t(\\dot{B}^0_{2,\\infty})}\\nonumber\\\\&\\lesssim\n\\Big\\|\\frac{h_1}{1+h_1}\\Big\\|_{E^{1}_t}(1+\\|h_1\\|_{L^\\infty_t(B^1)})\n\\|\\nabla w\\|_{\\widetilde{L}^1_t(\\dot{B}^0_{2,\\infty})}\n\\nonumber\\\\&\\lesssim\\|h_1\\|_{\\widetilde{E}^{0,1}_t}\n(1+\\|h_1\\|_{L^\\infty_t(\\widetilde{B}^{0,1})})^4\\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^1_{2,\\infty})}.\n\\end{align}\nIn terms of Proposition 4.3, (\\ref{4.12})-(\\ref{4.17}) and\n$\\widetilde{B}^{0,1}\\hookrightarrow B^1$, we finally obtain\n\\begin{align}\\label{4.18}\n\\|w\\|&_{\\widetilde{L}^1_t(\\dot{B}^{1}_{2,\\infty})}+\n\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}\\nonumber\\\\ \\lesssim &\n\\|u_2\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}\n+\\|u_1\\|_{L^2_t(B^1)}\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}\\nonumber\\\\&+\\|h_1\\|_{\\widetilde{E}^{0,1}_t}\n\\big(1+\\|h_1\\|_{L^\\infty_t(\\widetilde{B}^{0,1})}\\big)^4\n\\|w\\|_{\\widetilde{L}^1_t(B^1_{2,\\infty})}\\nonumber\\\\\n&+\\int_0^t(1+\\|h_1\\|_{\\widetilde{B}^{0,1}})(1+\\|h_1\\|_{\\widetilde{B}^{0,1}}\n+\\|h_2\\|_{\\widetilde{B}^{0,1}})(1+\\|u_2\\|_{B^2})\n\\|\\vartheta\\|_{\\dot{B}^0_{2,\\infty}}d\\tau.\n\\end{align}\nLet us define $$Z(t)\\triangleq \\|w\\|_{\\widetilde{L}^1_t(\\dot{B}^{1}_{2,\\infty})}\n+\\|w\\|_{\\widetilde{L}^2_t(\\dot{B}^{0}_{2,\\infty})}.$$\nDue to (\\ref{4.6}), if $T$ is chosen small enough,\nthen the first four terms of the right side of (\\ref{4.18}) can be absorbed by the left\nside $Z(t)$. Noting that $r\\log(e+\\frac{W(T)}{r})$ is increasing, from\n(\\ref{4.11}) and (\\ref{4.18}), it follows that\n\\begin{align}\\label{4.19}\nZ(t)&\\lesssim\\int_0^t(1+W'(\\tau))\nZ(\\tau)\\log\\Big(e+\\frac{W(\\tau)}{Z(\\tau)}\\Big)d\\tau\\nonumber\\\\&\n\\lesssim \\int_0^t(1+W'(\\tau))\nZ(\\tau)\\log\\Big(e+\\frac{W(T)}{Z(\\tau)}\\Big)d\\tau.\n\\end{align}\nIt is easy to verify that\n$$1+W'(\\tau)\\in L^1_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^+)\\quad\\hbox{and}\n\\quad\\int_0^1\\frac{dr}{r\\log(e+\\frac{W(T)}{r})}=+\\infty.$$\nHence by Osgood Lemma, we have $Z\\equiv0$ on $[0,\\widetilde{T}]$, i.e. $w\\equiv0$, then from\n(\\ref{4.9}),\n$\\vartheta=h_2-h_1\\equiv0$. Then a standard\ncontinuous argument gives the uniqueness.\n\n\n\\setcounter{equation}{0}\n\\section{Appendix}\n\nIn this appendix, we prove some multilinear estimates in the weighted Besov space.\n\n\\begin{Lemma}\\label{Lem5.1}\nLet $A$ be a homogeneous smooth function of degree $m$. Assume that $-\\frac d2<\\rho\\le\\frac d2$. Then there hold\n\\begin{eqnarray}\n&&\\Big|(A(D)\\Delta_k(v\\cdot\\nabla h),A(D)\\Delta_kh)\\Big|\\le C\\|{\\cal\nF}_k^m(t)\\|_2\\|A(D)\\Delta_kh\\|_2,\\label{5.1a}\\\\\n&&\\Big|(A(D)\\Delta_k(v\\cdot\\nabla u),A(D)\\Delta_ku)\\Big|\\le C\\|{\\cal\n\\widetilde{F}}_k^m(t)\\|_2\\|A(D)\\Delta_ku\\|_2,\\label{5.2a}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&&\\Big|(A(D)\\Delta_k(v\\cdot\\nabla h),\\Delta_k u)+(\\Delta_k(v\\cdot\\nabla\nu),A(D)\\Delta_kh)\\Big|\\nonumber\\\\\n&&\\qquad\\qquad\\le C\\big(\\|{\\cal {F}}_k^m(t)\\|_2+\\|{\\cal \\widetilde{F}}_k^0(t)\\|_2\\big)\\big(\\|\\Delta_k\nu\\|_2+\\|A(D) \\Delta_kh\\|_2\\big),\\label{5.3a}\n\\end{eqnarray}\nwhere ${\\cal F}_k^m(t)$ and ${\\cal \\widetilde{F}}_k^m(t)$ satisfy\n\\begin{eqnarray}\\label{5.1}\n&&\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}\\|{\\cal\nF}_k^m(t)\\|_{L^1_T(L^2)}\\le C\\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\frac{d}2+1})},\\\\\n&&\\sum_{k\\in{\\mathbf Z}}2^{k(\\rho-m)}\\|{\\cal \\widetilde{F}}_k^m(t)\\|_{L^1_T(L^2)}\n\\le C\\|u\\|_{L^2_T(B^{\\rho+1})}\\|v\\|_{L^2_T(B^{\\frac{d}2})}.\\label{5.2}\n\\end{eqnarray}\n\\end{Lemma}\n\n\\noindent{\\it Proof.}\\quad Let us first prove (\\ref{5.1a}).\nUsing the Bony's paraproduct decomposition, we write\n\\begin{align}\\label{5.3}\n\\big(A(D)\\Delta_k(v\\cdot\\nabla h), A(D)\\Delta_kh\\big)&=\\big(A(D)\\Delta_k(T'_{\\pa_j h}v^j), A(D)\\Delta_kh\\big)+J_k,\n\\end{align}\nwhere\n\\begin{eqnarray*}\n&&T'_fg=T_fg+R(f,g),\\quad \\textrm{and} \\\\\n&&J_k=\\sum_{|k'-k|\\le3}([A(D)\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_jh, A(D)\\Delta_kh)\\\\\n&&\\qquad+\\sum_{|k'-k|\\le3}((S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh, A(D)\\Delta_kh)\\\\\n&&\\qquad+(S_{k-1}v^jA(D)\\Delta_{k}\\pa_jh, A(D)\\Delta_kh)\n\\end{eqnarray*}\nWe get by integration by parts that\n$$(S_{k-1}v^jA(D)\\Delta_{k}\\pa_jh, A(D)\\Delta_kh)=-\\frac12\\big(S_{k-1}\\mbox{div}\\, vA(D)\\Delta_{k}h, A(D)\\Delta_kh\\big).$$\nLet us set\n\\begin{eqnarray*}\n&&{\\cal F}_{k,0}^m(t)=A(D)\\Delta_k(T'_{\\pa_j h}v^j),\\\\\n&&{\\cal F}_{k,1}^m(t)=\\sum_{|k'-k|\\le3}[A(D)\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_jh,\\\\\n&&{\\cal F}_{k,2}^m(t)=\\sum_{|k'-k|\\le3}(S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh,\\\\\n&&{\\cal F}_{k,4}^m(t)=-\\frac12 S_{k-1}\\mbox{div}\\, vA(D)\\Delta_{k}h.\n\\end{eqnarray*}\nBy the Cauchy-Schwartz inequality, we get\n\\begin{eqnarray*}\n\\big(A(D)\\Delta_k(v\\cdot\\nabla h), A(D)\\Delta_kh\\big)\\le \\|{\\cal\nF}_k^m(t)\\|_2\\|A(D)\\Delta_kh\\|_2,\n\\end{eqnarray*}\nwith ${\\cal F}_k^m(t)=\\displaystyle\\sum_{i=0}^3{\\cal F}_{k,i}^m(t).$ So,\nit remains to prove that ${\\cal F}_k^m(t)$ satisfies (\\ref{5.1}). For the simplicity, we set\n$$\\widetilde{\\Delta}_{k}=\\displaystyle\\sum_{|k'-k|\\le 1}\\Delta_{k'},\\quad\n\\widetilde{\\widetilde{\\Delta}}_{k}=\\displaystyle\\sum_{|k'-k|\\le 3}\\Delta_{k'}.$$\nThanks to the definition of ${\\cal F}_{k,0}^m(t)$ and Lemma \\ref{Lem2.1}, we have\n\\begin{align}\n\\|{\\cal F}_{k,0}^m(t)\\|_{L^1_T(L^2)}&\\le \\sum_{|k'-k|\\le3}2^{km}\\|S_{k'-1}\\pa_jh\\|_{L^\\infty_T(L^\\infty)}\\|\\Delta_{k'}v^j\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\quad+\\sum_{k'\\ge k-2}2^{(m+\\frac d2)k}\\|\\Delta_k(\\Delta_{k'}\\pa_jh\\widetilde{\\Delta}_{k'}v^j)\\|_{L^1_T(L^1)}\\nonumber\\\\\n&\\triangleq I+II.\\nonumber\n\\end{align}\nThanks to Lemma \\ref{Lem2.1}, we have\n\\begin{align}\n2^{k(\\rho-m)}I &\\lesssim 2^{k\\rho}\n\\sum_{k'\\le k+1}2^{k'(1+\\f d2)}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\lesssim\\sum_{k'\\le k+1}2^{(k'-k)(1+\\f d2-\\rho)}2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}2^{k(1+\\f d2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)},\\nonumber\n\\end{align}\nfrom which and the definition of $\\omega_k(T)$, it follows that\n\\begin{eqnarray}\\label{5.4}\n&&\\sum_{k\\in {\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}I\\nonumber\\\\\n&&\\lesssim \\sum_{k'\\in {\\mathbf Z}}2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\ge k'-1}\\omega_k(T)2^{(k'-k)(1+\\f d2-\\rho)}2^{k(1+\\f d2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\lesssim \\sum_{k'\\in {\\mathbf Z}}\\omega_{k'}(T)2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\ge k'-1}2^{(k'-k)(\\f d2-\\rho)}2^{k(1+\\f d2)}\\|\\widetilde{\\widetilde{\\Delta}}_{k}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&&\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}, \\end{eqnarray} where we\nused the assumption $\\rho\\le \\f d 2$ in the last inequality. Set\n$e_k(T)=e^1_{k}(T)+e^2_{k}(T)$. Using Lemma \\ref{Lem2.1}, we also have\n\\begin{align}\n\\omega_k(T)2^{k(\\rho-m)}II&\\lesssim \\omega_k(T)2^{k(\\rho+\\f d2)}\n\\sum_{k'\\ge k-2}2^{k'}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\\nonumber\\\\\n&\\lesssim 2^{k(\\rho+\\f d2)} \\sum_{k'\\ge\nk-2}2^{k'}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\n\\sum_{k'\\ge \\widetilde{k}\\ge k}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\nonumber\\\\\n&\\quad+2^{k(\\rho+\\f d2)} \\sum_{k'\\ge\nk-2}2^{k'}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\n\\sum_{\\widetilde{k}\\ge k, \\widetilde{k}\\ge k'}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\nonumber\\\\\n&\\triangleq II_1+II_2.\\nonumber\n\\end{align}\nNote that for $\\widetilde{k}\\le k'$\n$$\ne_{\\widetilde{k}}(T)\\le e_{k'}(T)\\le \\omega_{k'}(T),\n$$\nfrom which and $\\rho>-\\f d 2$, we deduce that\n\\begin{align}\\label{5.5}\n\\sum_{k\\in Z}II_1&\\lesssim \\sum_{k'\\in Z}\\omega_{k'}(T)2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}2^{k'(\\f d2+1)}\n\\|\\widetilde{\\Delta}_{k'}v\\|_{L^1_T(L^2)}\n\\sum_{k\\le k'+2}2^{(k-k')(\\rho+\\f d2)}\\nonumber\\\\\n&\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{align}\nSimilarly, we can obtain\n\\begin{align}\\label{5.6}\n\\sum_{k\\in {\\mathbf Z}}II_2&\\lesssim \\sum_{k'\\in {\\mathbf Z}}2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\le k'+2}2^{(k-k')(\\f d2+\\rho)}\n\\sum_{\\widetilde{k}\\ge k'}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\|v\\|_{L^1_T(B^{\\f d2+1})}\\nonumber\\\\\n&\\lesssim \\sum_{k'\\in {\\mathbf Z}}\\omega_{k'}(T)2^{k'\\rho}\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\sum_{k\\le k'+2}2^{(k-k')(\\f d2+\\rho+1)}\n\\|v\\|_{L^1_T(B^{\\f d2+1})}\\nonumber\\\\\n&\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{align}\n\nBy summing up (\\ref{5.4})-(\\ref{5.6}), we obtain\n\\begin{eqnarray}\\label{5.7}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}\\|{\\cal F}_{k,0}^m(t)\\|_{L^1_T(L^2)}\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{eqnarray}\n\nNote that $A(D)\\Delta_k=2^{km}\\widetilde{\\varphi}(2^{-k}D)$ with $\\widetilde{\\varphi}(\\xi)=A(\\xi)\\varphi(\\xi)$.\nSet $\\tilde\\theta={\\cal F}^{-1}\\tilde\\varphi$, we get by using the Taylor's formula that\n\\begin{align}\\label{5.8}\n{\\cal F}_{k,1}^m(t)=\\sum_{|k'-k|\\le3}2^{k(m-1)}\\int_{\\mathop{\\bf R\\kern 0pt}\\nolimits^d}\\int_0^1\\tilde\\theta(y)\n(y\\cdot S_{k'-1}\\nabla v^j(x-2^{-k}\\tau y))\\Delta_{k'}\\pa_jh(x-2^{-k}y) d\\tau dy,\\nonumber\n\\end{align}\nfrom which and Lemma \\ref{Lem2.1}, it follows that\n\\begin{align}\n\\|{\\cal F}_{k,1}^m(t)\\|_{L^1_T(L^2)}&\\lesssim2^{k(m-1)}\\sum_{|k'-k|\\le3}\n\\|S_{k'-1}\\nabla v^j\\|_{L^1_T(L^\\infty)}\\|\\Delta_{k'}\\pa_jh\\|_{L^\\infty_T(L^2)}\\nonumber\\\\&\n\\lesssim2^{km}\\sum_{|k'-k|\\le3}\n\\|\\Delta_{k'}h\\|_{L^\\infty_T(L^2)}\\|v\\|_{L^1_T(B^{\\frac d 2 +1})},\\nonumber\n\\end{align}\nthus, we get\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(\\rho-m)}\\|{\\cal F}_{k,1}^m(t)\\|_{L^1_T(L^2)}\n\\lesssim \\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})}.\n\\end{align}\nThanks to the fact $|k'-k|\\le3$ and Lemma \\ref{Lem2.1}, we have\n\\begin{align}\n\\|(S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh\\|_{L^1_T(L^2)}\n\\lesssim 2^{km}\\|\\Delta_{k}h\\|_{L^\\infty_T(L^2)}\\|v\\|_{L^1_T(B^{\\f d2+1})},\\nonumber\n\\end{align}\nfrom which, it follows that\n\\begin{align}\\label{5.9}\n\\sum_{k\\in{\\mathbf Z}}&\\omega_k(T)2^{k(\\rho-m)}\\big(\\|{\\cal F}_{k,2}^m(t)\\|_{L^1_T(L^2)}\n+\\|{\\cal F}_{k,3}^m(t)\\|_{L^1_T(L^2)}\\big)\\lesssim\\|h\\|_{E^\\rho_T}\\|v\\|_{L^1_T(B^{\\f d2+1})},\n\\end{align}\nwhich together with (\\ref{5.7}) and (\\ref{5.8}) yields (\\ref{5.1}).\n\nUsing the decomposition (\\ref{5.3}) with $h$ instead of $u$ and Lemma \\ref{Lem2.1}, (\\ref{5.2a}) can be easily proved. We omit it here.\nIn order to prove (\\ref{5.3a}), we use the decomposition\n\\begin{align}\n\\big(A(D)\\Delta_k(v\\cdot\\nabla h),\\Delta_k u\\big)+\\big(\\Delta_k(v\\cdot\\nabla u),A(D)\\Delta_kh\\big)=I_k+J_k,\\nonumber\n\\end{align}\nwith\n\\begin{align}\nI_k=&\\big(A(D)\\Delta_k(T'_{\\pa_j h}v^j),\\Delta_k u\\big)+\\big(\\Delta_k(T'_{\\pa_j u}v^j),A(D)\\Delta_kh\\big)\\nonumber\\\\\n\\triangleq&\\big({\\cal F}_{k,0}^m(t),\\Delta_k u\\big)+\\big({\\cal \\widetilde{F}}_{k,0}^0(t),A(D)\\Delta_kh\\big)\\nonumber\\\\\nJ_k=&\\sum_{|k'-k|\\le3}\\Big([A(D)\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_jh, \\Delta_ku\\Big)+\n\\Big((S_{k'-1}-S_{k-1})v^jA(D)\\Delta_{k}\\Delta_{k'}\\pa_jh, \\Delta_ku\\Big)\\nonumber\\\\\n&+\\sum_{|k'-k|\\le3}\\Big([\\Delta_k, S_{k'-1}v^j]\\Delta_{k'}\\pa_ju, A(D)\\Delta_kh\\Big)+\n\\Big((S_{k'-1}-S_{k-1})v^j\\Delta_{k}\\Delta_{k'}\\pa_ju, A(D)\\Delta_kh\\Big)\\nonumber\\\\\n&-\\Big(S_{k-1}\\mbox{div} vA(D)\\Delta_{k}h, \\Delta_ku\\Big)\\nonumber\\\\\n\\triangleq&\\big({\\cal F}_{k,1}^m(t),\\Delta_k u\\big)+\\big({\\cal F}_{k,2}^m(t),\\Delta_k u\\big)+\n\\big({\\cal \\widetilde{F}}_{k,1}^0(t),A(D)\\Delta_kh\\big)\n+\\big({\\cal\\widetilde{ F}}_{k,2}^0,A(D)\\Delta_kh\\big)+\\big({\\cal F}_{k,3}^m(t), \\Delta_ku\\big),\n\\nonumber\\end{align}\nfrom which, a similar proof of (\\ref{5.1}) gives (\\ref{5.3a}). This completes the proof of Lemma \\ref{Lem5.1}. \\hfill $ \\blacksquare $ \\vskip 3mm\n\n\n\\begin{Lemma}\\label{Lem5.2}\nLet $s_1\\le\\frac d2-1$, $s_2\\le\\frac d2$, and $s_1+s_2>0$. Then there holds\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|\\Delta_k(fg)\\|_{L^1_T(L^2)}\n\\le C\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks_1}\\|\\Delta_kf\\|_{L^{r_1}_T(L^2)}\\|g\\|_{L^{r_2}_T(B^{s_2})},\\label{5.10}\n\\end{align}\nwhere $1\\le r_1,r_2\\le \\infty$ and $\\f 1 {r_1}+\\f 1 {r_2}=1$.\n\\end{Lemma}\n\n\\noindent{\\it Proof.}\\quad Using the Bony's paraproduct decomposition, we write\n\\begin{align}\n\\Delta_k(fg)&=\\sum_{|k'-k|\\le3}\\Delta_k(S_{k'-1}f\\Delta_{k'}g)+\n\\sum_{|k'-k|\\le3}\\Delta_k(S_{k'-1}g\\Delta_{k'}f)\\nonumber\\\\\n&\\quad+\\sum_{k'\\ge k-2}\\Delta_k(\\Delta_{k'}f\\widetilde{\\Delta}_{k'}g)\\triangleq I+II+III.\\nonumber\n\\end{align}\nA similar proof of (\\ref{5.4}) ensures that for $s_1\\le\\frac d2-1$\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|I\\|_{L^1_T(L^2)}\\lesssim\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks_1}\\|\\Delta_kf\\|_{L^{r_1}_T(L^2)}\\|g\\|_{L^{r_2}_T(B^{s_2})},\\nonumber\n\\end{align}\nwhile $II$ can be directly deduced for $s_2\\le\\frac d2$. On the other hand,\na similar proof of (\\ref{5.5}) and (\\ref{5.6}) gives for $s_1+s_2>0$\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|III\\|_{L^1_T(L^2)}\\lesssim \\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks_1}\\|\\Delta_kf\\|_{L^{r_1}_T(L^2)}\\|g\\|_{L^{r_2}_T(B^{s_2})}.\\nonumber\n\\end{align}\nThis completes the proof of Lemma \\ref{Lem5.2}. \\hfill $ \\blacksquare $ \\vskip 3mm\n\nSimilarly, we can also prove the following lemma.\n\n\\begin{Lemma}\\label{Lem5.3}\nLet $s_1\\le\\frac d2-1$, $s_2<\\frac d2$ and $s_1+s_2\\ge0$. Then there holds\n\\begin{align}\n\\sup_{k\\in{\\mathbf Z}}\\omega_k(T)2^{k(s_1+s_2-\\frac d2)}\\|\\Delta_k(fg)\\|_{L^1_T(L^2)}\n\\le C\\|f\\|_{E^{s_1}_T}\\|g\\|_{\\widetilde{L}^1_T(\\dot B^{s_2}_{2,\\infty})}.\\label{5.11}\n\\end{align}\n\\end{Lemma}\n\n\\begin{Lemma}\\label{Lem5.4}\nLet $s>0$. Assume that $F\\in W^{[s]+3,\\infty}_{loc}(\\mathop{\\bf R\\kern 0pt}\\nolimits^d)$ with $F(0)=0$. Then\nthere holds\n\\begin{eqnarray}\n\\|F(f)\\|_{E^s_T}\\le C(1+\\|f\\|_{L^\\infty_T(L^\\infty)})^{[s]+2}\\|f\\|_{E^s_T}.\\label{5.12}\n\\end{eqnarray}\n\\end{Lemma}\n\n\\noindent{\\it Proof.}\\quad We decompose $F(f)$ as\n\\begin{align}\nF(f)=\\sum_{k'\\in{\\mathbf Z}}F(S_{k'+1}f)-F(S_{k'}f)&=\\sum_{k'\\in{\\mathbf Z}}\\Delta_{k'}f\\int_0^1F'(S_{k'}f+\\tau\\Delta_{k'}f)d\\tau\\nonumber\\\\\n&\\triangleq\\sum_{k'\\in{\\mathbf Z}}\\Delta_{k'}f\\, m_{k'},\\nonumber\n\\end{align}\nwhere $m_{k'}=\\int_0^1F'(S_{k'}f+\\tau\\Delta_{k'}f)d\\tau$. Furthermore, we write\n\\begin{align}\n\\Delta_kF(f)=\\sum_{k'k'}2^{(k-k')(s-|\\al|+1)}(1+\\|f\\|_{L^\\infty_T(L^\\infty)})^{|\\al|}\\|F'\\|_{W^{|\\al|,\\infty}}\n\\nonumber\\\\&\\quad \\lesssim(1+\\|f\\|_{L^\\infty_T(L^\\infty)})^{[s]+2}\\|F'\\|_{W^{[s]+2,\\infty}}\\|f\\|_{E^s_T}.\\label{5.14}\n\\end{align}\n\nNext, let us turn to the proof of $II$. We get by using Lemma 2.1 that\n\\begin{align}\n\\|II\\|_{L^\\infty_T(L^2)}&\\lesssim\\sum_{k\\ge k'}\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}.\\nonumber\n\\end{align}\nThen we write\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks}\\|II\\|_{L^\\infty_T(L^2)}\n&\\lesssim \\sum_{k\\in {\\mathbf Z}}2^{ks}\\sum_{k'\\ge k}\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}\n\\sum_{k'\\ge \\widetilde{k}\\ge k}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T)\\nonumber\\\\\n&\\quad+\\sum_{k\\in {\\mathbf Z}}2^{ks}\\sum_{k'\\ge k}\\|\\Delta_{k'}f\\|_{L^\\infty_T(L^2)}\n\\sum_{\\widetilde{k}\\ge k', \\widetilde{k}\\ge k}2^{-(\\widetilde{k}-k)}e_{\\widetilde{k}}(T),\\nonumber\n\\end{align}\nfrom which, a similar proof of (\\ref{5.5}) and (\\ref{5.6}) ensures that\n\\begin{align}\n\\sum_{k\\in{\\mathbf Z}}\\omega_k(T)2^{ks}\\|II\\|_{L^\\infty_T(L^2)}\\lesssim\\|f\\|_{E^s_T}.\\label{5.15}\n\\end{align}\n\nBy summing up (\\ref{5.14}) and (\\ref{5.15}), we deduce the inequality (\\ref{5.12}).\nThis completes the proof of Lemma \\ref{Lem5.4}. \\hfill $ \\blacksquare $ \\vskip 3mm\n\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgments.} The authors thank the referees\n for their invaluable comments and suggestions which helped improve the paper greatly.\n Q. Chen and C.\nMiao were partially supported by the NSF of China (No.10571016).\nZ. Zhang is supported by the NSF of China(No.10601002).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}}\n\n\\IEEEPARstart{T}{here} has been a fundamental paradigm shift in the landscape of how we build software over the last 2 decades.\nThe compute stack was originally envisaged with the assumption that a single program would run on a single machine.\nIn this traditional model, system software abstractions subsumed the management of processor, memory, disk and physically connected peripherals within the context of the machine.\nHowever, this landscape rapidly changed with the evolution toward software being served on the backbone provided by the internet.\nNow, an `application' is realized through the cooperation of multiple distinct sub-applications (services) running collaboratively.\nFor example a single application my contain self one or more self-contained database, memcache, logging, application logic, AI model applications interfacing each other over APIs as shown in Figure~\\ref{fig:intro} (left).\nWe call these applications \\emph{diffuse applications}.\n\nThis work contends that the fundamental programming paradigms in computing has not evolved at pace.\nThe abstractions envisioned during the era of the single machine computational model is still present at the programming interface and throughout the runtime stack leading to significant and costly complexity.\n\n\nTo address this complexity, two keystone abstractions have recently emerged to facilitate the development of these diffuse applications.\nThe first of these abstractions is the introduction and rapid dissemination of containerization service platforms.\nWith what started as a key insight articulated in ``The Datacenter as a Computer,'' Google would innovate their Borg system and ultimately released it open source as Kubernetes.\nWith Kubernetes, the underlying hardware resources would be abstracted away with the introduction of \\emph{pods} (virtual machines), and other resources that can be virtually networked together and otherwise configured irrespective of the physical hardware.\nToday, Kubernetes is the most prevalent containerized service abstraction layer in cloud computing.\nThe second of keystone abstraction would be coined ``Severless Computing'' and gained prominence with the introduction of Amazons Lamda functions.\nThis FaaS abstraction would facilitate the development of diffuse applications at the level of functions and abstract away the underlying containerized service ecosystem.\nA programer can simply make function calls in their favorite language without every needing to be aware of where the function will run nor the system level resources that would be allocated or managed.\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/jaseci_stack.pdf}\n \\caption{Comparison between status quo development of production grade \\emph{diffuse} applications (left), and the Jaseci technology stack that hides and automates an expanded set of subsystems through raising the level of abstraction (right).}\n \\label{fig:intro}\n\\end{figure}\n\n\n\nThough these two abstractions have been highly impactful, these innovations in our stack architecture represent a bottom up evolution of abstractions.\nAs a result, programmers are still left with single-machine abstractions at the programming interface and must grapple with a significant amount of complexity.\nFor example, traditional languages and their runtime stacks are predominately designed with the goal of hiding and managing intra-machine resources while what is needed for diffuse applications is the hiding and management of inter-machine resources.\nAnalogous to the virtualization and management of allocated memory on the heap provided by garbage collectors in modern languages (intra-machine), the virtualization and management of resources such as microservice creation, scheduling and orchestration alongside policies for organizing distributed databases, mem caches, logging and other highly complex subsystems (inter-machine) is not only needed, but as we show in this work, possible and practical.\nWithout this raising of the level of abstraction, it has become prohibitively difficult for a single engineer to invent, build, deploy, launch, and scale modern cutting edge applications.\n\nTo the best of our knowledge, we are not aware of a thorough, wholistic, and top-down design of a serverless programming paradigm and computational stack from the language level down through the system runtime stack to hide this expanded set of resources.\n\nIn this work, we present a wholistic design approach with the goal of abstracting away and automating a new class of underlying systems, allowing a programmer to articulate solutions and diffuse applications at the problem level.\nWe present the design of a \\emph{diffuse runtime execution engine} we call \\textbf{Jaseci}, and a \\emph{data-spacial programming language} we call \\textbf{Jac}.\nThe design of Jaseci and Jac has initially been inspired to by sophisticated emerging AI applications at scale and is driven by two key insightss.\n\\begin{itemize}\n \n \n\n \\item \\emph{Higher level abstractions are needed at the language level to allow single creators to work at the problem level to build end-to-end diffuse AI products.}\n \\item \\emph{A new set of abstractions across the language runtime and system stack is needed to automate and hide the class of inter-machine resources from the programmer.}\n\n \n\\end{itemize}\n\n\\noindent\nTo this end we present techniques across two categories,\n\n\\begin{enumerate}\n \n \n\n \\item \\emph{Jac Language} - A language that introduces a new set of abstractions, namely \\textbf{data-spacial scoping} and \\textbf{agent oriented programming}. These abstractions natively facilitates the emerging need to reason about and solve problems with graph representations as well as the need for algorithmic modularity and encapsulation to hide a new class of inter-machine resources.\n \\item \\emph{Jaseci Diffuse Runtime Engine} - A runtime that raises the abstraction layer to the problem solving level where the runtime engine subsumes responsibility for not only for the optimization of program code, but the orchestration, configuration, and optimization of the full cloud compute stack and inter-machine resources (such tasks as container formation, scaling and optimization).\n\n \n\\end{enumerate}\n\nJaseci and Jac is fully functional, open-source~\\cite{jaseci-website,jaseci-github,jaseci-pypi}, and used in production for four real-world products today.\nThese commercial products were built entirely on the Jaseci staci and includes Myca~\\cite{myca-website}, HomeLendingPal~\\cite{hlp-website}, ZeroShotBot~\\cite{zsb-website} and TrueSelph~\\cite{ts-website}.\nAcross these and other projects, the Jac language has been used by dozens of programmers in the creation of production software and Jaseci deployments support tens of thousands of production queries per day currently.\nIn practice, our initial infrastructure has been leveraged in practice to achieve 10x reduction in development time and near 100\\% elimination of typical backend code needed for a complicated AI based application.\n\nThe specific contributions of this paper include:\n\\begin{itemize}\n \\item We formulate the problem of development complexity and present a top down programing paradigm and runtime stack for diffuse applications.\n \\item We describe the design and implementation Jaseci's \\textbf{diffuse runtime execution engine}.\n \\item We introduce Jac, a language that implements a \\textbf{data-spacial} programming paradigm (the first of its kind).\n \\item We describe the utility of Jaseci and Jac through real world case studies of building out a real production scale-out product.\n\\end{itemize}\n\nWe find that the wholistic design philosophy and resulting paradigm of Jaseci and Jac is a promising one.\nMultiple development teams have adopted the data spacial programming model of Jac and the diffuse runtime execution engine in Jaseci to build sophisticated AI products with significantly reduced complexity and teaming.\n\n\n\\section{The Case for a New Computational Model}\\label{sec:motivation}\n\nThough recent advancements in serverless computing has been instrumental in improving the ability of teams to more rapidly develop software, significant challenges remain in the development of cutting edge applications and products in our current compute landscape.\nAn demonstrative problem domain with this challenge are those characterized by applications that include sophisticated AI pipelines on their critical path.\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/team_sizes.pdf}\n \\caption{Comparison of typical development team required to realize production grade AI application today (left), and the ability of a single software developer to realize such an application with Jaseci (right). }\n \\label{fig:dev}\n\\end{figure}\n\n\n\\subsection{Problem Scenario}\nFigure~\\ref{fig:dev}A shows the typical set of often siloed roles needed to create software in this environment.\nThe first critical role needed is an \\emph{architect \/ tech-lead} responsible for architecting the software solution across disparate components, programming languages, frameworks, and SDKs.\nIf a microservice ecosystem is needed (which is a must for modern\nAI applications), the architect will also decide what will and won't be its own service (container) and define the interfaces between these disparate services.\nFor the AI model work, the role of a \\emph{data scientist \/ ML engineer} is needed.\nThis role typically works primarily in Jupyter notebooks selecting, creating, training and tuning ML models to support application features.\nProduction software engineering is typically outside of the scope of this expertise in practice.\nThe role of a \\emph{backend engineer} is needed for implementing the main services of the application and taking the code out of Jupyter notebooks to build the models into the backend (server-side) of the application.\nThe \\emph{backend engineer} is also responsible for supporting new features and creating their API interfaces for \\emph{frontend} engineers.\nOne of the key roles any software team needs to deploy an AI product is a \\emph{DevOps engineer}.\nThis role is solely responsible for deploying and configuring containers to run on a cloud and ensure these containers are operational and scaled to the load requirements of the software.\nThis responsibility covers configuring software instance pods, database pods, caching layers, logging services, and parameterizing replicas and auto-scaling heuristics.\n\n\n\nIn this traditional model of software engineering, many challenges and complexity emerge.\nAn example is the (quite typical) scenario of the first main server-side implementation of the application being a monoservice while DB, caching, and logging are microservices.\nAs the \\emph{ML engineer} introduces models of increasing size, the \\emph{dev-ops} person alerts the team that the cloud instances, though designated as \\emph{large}, only have 8gb of ram.\nMeanwhile new AI models being integrated exceed this limit.\nThis event leads to a re-architecture of the main monoservice to be split out AI models into microservices and interfaces being designed or adopted leading to significant backend work \/ delays.\nIn this work, we aim to create a solution that would move all of this decisioning and work under the purview of the automated runtime system.\n\n\n\n\n\nUltimately, the mission of Jaseci is to accelerate and democratize the development and deployments of end-to-end scalable AI applications as presented in Figure~\\ref{fig:dev}B.\nTo this end, we present a novel set of higher level abstractions for programming sophisticated software in a micro-service\/serverless AI and a full stack architecture and programming model that abstracts away and automates much of the complexity of building diffuse applications on a distributed compute substrate of potentially thousands of compute nodes.\n\n\\section{Data Spacial Programming Model with Jac}\\label{sec:jac}\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/data_spacial.pdf}\n \\caption{A visualization of the behavior of scopes and problem solving abstractions provided by the near ubiquitous function \/ method based languages (left) and the data spacial programming model (right).}\n \\label{fig:benefit}\n\\end{figure}\n\nTraditionally in computer science, the task of raising the level abstraction in a computational model has primarily been for the goal of increasing programmer productivity.\nThis productivity comes from allowing engineers to function at the problem level while hiding the complexity of the underlying system.\nThe Jac language introduces a set of new abstractions guided by these principles based on two key insights.\nFirst, Jac recognizes the emerging need for programmers to reason about and solve problems with graph representations of data.\nSecond, Jac further supports the need for algorithmic modularity and encapsulation to change and prototype production software in place of prior running codebases.\nBased on these insights, we introduce two new sets of abstractions.\nAs shown in Figure~\\ref{fig:benefit}b, Jac's \\textbf{data-spacial scoping} natively facilitates graph based problem solving by replacing the traditional \\emph{temporal} notion of scope with a function's activation record with scoping that is flattened and spatially laid out in graph structure.\nThis type of scoping allows for richer semantics for the organization of the data relevant to the problem being solved.\nFigure~\\ref{fig:benefit}b also depicts Jac's \\textbf{agent oriented programming} as little robots.\nEach robot carries scope with it as it walks and performs compute relevant to where it sits on the graph.\nThese `agent' abstractions capture the need for algorithmic modality and encapsulation when introducing solutions to already sophisticated codebases.\nJac can be used solely to build out complete solutions or as glue code with components built in other languages.\nBy leveraging these new language abstractions, HomeLendingPal~\\cite{hlp-website} was able to create a production grade conversational AI experience with $\\sim$300 lines of code in contrast to the tens of thousands it would take to build in a traditional programming language.\n\n\\section{Diffuse Runtime Execution with Jaseci}\\label{sec:arch}\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/jaseci_arch.pdf}\n \\caption{The architecture of the Jaseci diffuse runtime execution. The runtime stack includes and combines information from interpreter level profiling, cloud monitoring and profiling, microservice orchestrator and optimizer. Container linked libraries are also depicted. }\n \\label{fig:benefit2}\n\\end{figure}\n\nJaseci's cloud-scale runtime engine presents a higher level abstraction of the software stack.\nThe \\emph{diffuse} runtime engine subsumes responsibility not only for the optimization of program code, but also the orchestration, configuration, and optimization of constituent micro services and the full cloud compute stack.\nDuties such as container formation, microservice scaling, scheduling and optimization are automated by the runtime.\nFor example, as shown in Figure~\\ref{fig:benefit2}c Jaseci introduces the concept of \\textbf{container linked libraries} to complement traditional notions of statically and dynamically linked libraries.\nFrom the programmers perspective, they need not know whether a call to a library is fused with the running programming instance or a remote call to a microservice somewhere in a cluster.\nThe decisioning of what \\emph{should} be a microservice and what should be statically in the programs object scope is made automatically and seamlessly by the Jaseci \\textbf{microservice orchestration engine}.\nUnderlying in-cluster microservices are encapsulated and hidden with this abstraction.\nWith the runtime having full visibility and control over the diffuse application, high complexity runtime decisions and heuristics such as autoscaling is brought under the purview of the runtime software stack, relieving the need of manual configuration.\nWith this Jaseci runtime, a single frontend engineer was able to implement the full ZeroShotBot~\\cite{zsb-website} application (which uses a number of transformer neural networks) without writing a single line of traditional `backend' code.\nThis implementation currently support tens of thousands of queries a day across about $\\sim$12 business customers with tens of thousands of individual end users in a single production environment.\n\n\\section{Case Studies and Discussion}\\label{sec:casestudy}\n\nJaseci is available on Github~\\cite{jaseci-github} under MIT open source license and is composed of an ecosystem of tools spanning 3 packages.\nThese include \\textbf{Jaseci Core}, its core execution engine, \\textbf{Jaseci Serv}, its diffuse runtime cloud-scale execution engine, and \\textbf{Jaseci Kit}, a collection of cutting edge AI engines provided by the Jaseci community.\nIn addition to these main codebases, an experimental toolkit we call \\textbf{Jaseci Studio} is in development to provide visual programming and debugging tooling for developers building with Jaseci.\n\nThere are a number of notable examples of Jaseci's use in production.\nThese users include four selected start-up companies that have adopted Jac and Jaseci as their development engine and have already launched their products built using Jaseci.\n\n\\indent \\textbf{ myca.ai}~\\cite{myca-website} - a B2C personal productivity platform that uses AI to understand personal behavior trends and help users allocate their time, prioritize their tasks and achieve personal growth goals. Using Jaseci, myca.ai's back-end development only took 1 month and myca.ai was launched within 3 months' development to the public. Myca.ai is one of the fast growing personal growth tool and has received positive feedback from their users.\n\n\\indent \\textbf{ZeroShotBot}~\\cite{zsb-website} - a B2B company that develops a cutting edge conversational AI platform using Jaseci. The product development took 2 months and was done by frontend engineers. Zeroshotbot has gained significant market traction and has been in business discussions with major logos such as Volaris, Pizzahut to provide readily deployable FAQ chatbots.\n\n\n\n\\indent \\textbf{Truselph}~\\cite{ts-website} - A minority founded startup. Truselph creates an avatar of the person and builds conversational intelligence that allows the general public to interact with the avatar and ask questions, while the avatar will be able to provide personalized answers with emotions and facial expressions. Truselph is in partnership with Lenovo to co-develop Truselph powered Kiosks for retail stores and is in business discussions with chains such as Sephora.\n\n\n\\indent \\textbf{ Home Lending Pal}~\\cite{hlp-website} - an AI Powered Mortgage Advisor. Home Lending Pal is a minority founded start-up that helps people, especially under-served minority population to navigate through the mortgage and home purchase process. Home Lending Pal adopted Jaseci to provide two main product features: 1 - personalized mortgage advice and 2 - Kev, an AI-powered chatbot that will answer users questions about the process and give them a plan to improve their finances.\n\n\\balance\n\\section{Conclusion}\nJaseci is a novel computational model invented, designed and implemented to address this challenge.\nJaseci includes a novel programming model we call \\emph{data-spacial programming} and a runtime engine we call the \\emph{diffuse execution environment} to enable rapid development of large scale and nimble AI applications.\nOur initial infrastructure has been used in practice to achieve 10x reduction in development time and near 100\\% elimination of typical backend code needed for a complicated AI based application.\nJaseci~\\cite{jaseci-website} was open sourced in 2021~\\cite{jaseci-github}~\\cite{jaseci-pypi}.\nToday Jaseci is in production with 4 distinct commercial products built on the engine, including Myca~\\cite{myca-website}, HomeLendingPal~\\cite{hlp-website}, ZeroShotBot~\\cite{zsb-website} and TrueSelph~\\cite{ts-website}.\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\n\\section{Introduction}\\label{sec:introduction}}\n\\else\n\\section{Introduction}\n\\label{sec:introduction}\n\\fi\n\n\n\n\n\\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''\nfor IEEE Computer Society journal papers produced under \\LaTeX\\ using\nIEEEtran.cls version 1.8b and later.\nI wish you the best of success.\n\n\\hfill mds\n \n\\hfill August 26, 2015\n\n\\subsection{Subsection Heading Here}\nSubsection text here.\n\n\n\\subsubsection{Subsubsection Heading Here}\nSubsubsection text here.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe conclusion goes here.\n\n\n\n\n\n\n\n\n\\appendices\n\\section{Proof of the First Zonklar Equation}\nAppendix one text goes here.\n\n\\section{}\nAppendix two text goes here.\n\n\n\\ifCLASSOPTIONcompsoc\n \n \\section*{Acknowledgments}\n\\else\n \n \\section*{Acknowledgment}\n\\fi\n\n\nThe authors would like to thank...\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\n\n\n\n\\section{Introduction}\nThis demo file is intended to serve as a ``starter file''\nfor IEEE Computer Society conference papers produced under \\LaTeX\\ using\nIEEEtran.cls version 1.8b and later.\nI wish you the best of success.\n\n\\hfill mds\n \n\\hfill August 26, 2015\n\n\\subsection{Subsection Heading Here}\nSubsection text here.\n\n\n\\subsubsection{Subsubsection Heading Here}\nSubsubsection text here.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe conclusion goes here.\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcompsoc\n \n \\section*{Acknowledgments}\n\\else\n \n \\section*{Acknowledgment}\n\\fi\n\n\nThe authors would like to thank...\n\n\n\n\n\n\n\n\n\\section{Introduction}\\label{sec:introduction}}\n\n\n\n\n\\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''\nfor IEEE Computer Society journal papers produced under \\LaTeX\\ using\nIEEEtran.cls version 1.8b and later.\nI wish you the best of success.\n\n\\hfill mds\n \n\\hfill August 26, 2015\n\n\\subsection{Subsection Heading Here}\nSubsection text here.\n\n\n\\subsubsection{Subsubsection Heading Here}\nSubsubsection text here.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe conclusion goes here.\n\n\n\n\n\n\n\n\n\\appendices\n\\section{Proof of the First Zonklar Equation}\nAppendix one text goes here.\n\n\\section{}\nAppendix two text goes here.\n\n\n\\ifCLASSOPTIONcompsoc\n \n \\section*{Acknowledgments}\n\\else\n \n \\section*{Acknowledgment}\n\\fi\n\n\nThe authors would like to thank...\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhtoh b/data_all_eng_slimpj/shuffled/split2/finalzzhtoh new file mode 100644 index 0000000000000000000000000000000000000000..0ed9ef1591c7d8459b4357bd1682c76b374c562b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhtoh @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nMagnetic fields on the Sun operate at a large range of spatial and temporal scales, and all of them play a fundamental role in its structure, its dynamics, and its energy budget. \nThese fields connect the inside of the Sun with its atmosphere providing means for storing, transporting, and depositing energy among the different atmospheric layers. They also draw pathways throughout the interplanetary medium, magnetically linking the Sun to the Earth and the entire solar system. Magnetic fields are at the core of the solar cycle and they are the seeds of space weather.\n\nActive regions (ARs) are the most visible manifestation of magnetic fields on the Sun. They appear on the solar surface in the form of magnetic dipoles. It is generally believed that the magnetic field of ARs emerges in the form of $\\Omega$-loops whose rise is triggered by deep convective flows and buoyant instabilities in a toroidal field located near the base of the convection zone \\citep{parker1955}. \nAlthough this paradigm explains many of the observed aspects of ARs and the solar cycle, these flux tubes do not emerge as a whole; rather, they break through the photosphere in small bundles \\citep[see, e.g.][]{birch}, slowly piling up magnetic flux at the surface, that then aggregates into stronger concentrations and gives rise to sunspots \\citep{cheung2010}. \n\\cite{rolf} describe observational evidence of small-scale emergence of magnetic flux (associated to elongated granulation) as the main contributor to the overall flux of an AR.\nThis is the picture of {\\em resistive emergence} described by \\cite{pariat2004}, in which the $\\Omega$-loops rising from the convection zone develop spatial undulations, whose crests emerge because of the Parker instability, while their troughs remain trapped below the surface, until they are released by magnetic reconnection. In their work, these locations are often associated with Ellerman Bombs \\citep[EB,][]{ellerman} and bright points. \nThe serpentine nature of the emerging fields results in small-scale moving dipolar features \\citep[MDFs,][]{bernasconi2002} of opposite polarity to that of the AR. These are the footpoints of U-loops where the serpentine field is anchored below the surface due to trapped mass, and become the channels where the material that was carried up by the crests of the undulating field, flows back down into the surface \\citep[see][]{cheung2010, centeno2012}. \n\nWhen studying the small-scale details in AR emergence sites, \\cite{ortiz14} reported semi-circular magnetic footpoints, straddling several granules, braketing patches of horizontal field in the photosphere. Chromospheric spectral analysis showed that these footpoints always surround a bubble of cool chromospheric material that increases in size as it ascends into higher layers. Their results are in qualitative agreement with numerical simulations of flux emergence \\citep{juan2008, juan2009}.\nIn a subsequent paper, \\cite{jaime2015} show that these granular sized magnetic bubbles emerge up to the chromosphere, where the pre-exisiting field seems to hinder their further ascent. Signatures of heating around the edges of the bubble are reported, but no clear physical mechanisms are identified.\nRecent joint observations with the Swedish Solar Telescope, the Interface Region Imaging Spectrograph (IRIS), and Hinode have revealed the transition region response to these small-scale events in the emergence sites of ARs. Upward velocities predominate in the emergence sites and, when the orientation of the emerging flux differs from that of the overarching filament system, brightenings ensue. The transition region response is typically delayed by 10 minutes with respect to the photospheric emergence \\citep{ortiz16}.\n\nMagnetic reconnection is one of the most likely mechanisms responsible for heating the upper atmospheric layers. The reconnection process leads to energy conversion, from magnetic into thermal and kinetic energy \\citep{priest2000}, and it shows up in the observations in the form of brightenings and\/or surges \\citep{yokoyama}. Ellerman bombs occur in areas where intricate magnetic topologies are likely to lead to magnetic reconnection \\citep{georgoulis2002}, such as places where newly emerged magnetic flux interacts with preexisting ambient fields, the locations above MDFs and the neutral lines surrounding the boundaries of supergranular cells.\nAlthough there is some controversy in the literature of what constitutes an EB \\citep{vissers2015, rutten2016}, they can be grouped in a category of events that involve magnetic reconnection and subsequent heating of mid-chromospheric layers and tend to happen rather frequently in the emergence site of ARs.\n\n\n\n\n In this work we study, at a constant, high resolution, the detailed structure and evolution of the emergence site of a developing active region using data from the second flight of the {\\sc Sunrise} mission. The scientific payload onboard {\\sc Sunrise II} captured two small flux emergence events in meticulous detail, registering the photospheric and chromospheric responses to the incipient magnetic field that rises from below the surface to become part of the larger-scale active region.\nThe instruments and the observations used in this work are presented in Section \\ref{observations}. In Section \\ref{photosphere}, we delve into a detailed description of the photospheric signatures of the emerging magnetic flux, whilst Section \\ref{chromosphere} analyzes the chromospheric response to these events. All of the information gathered from the observations is then folded into a coherent picture discussed in Section \\ref{discussion}, followed by speculations and final remarks. \n\n\n\n\\section{Observations}\\label{observations}\n\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.55]{imax_548_15_lev2_cont.eps}\n\\includegraphics[angle=0,scale=0.55]{imaxsufi_548_15_300_397.eps}\n\\caption{Left: Reconstructed continuum image from the IMaX instrument. The blue rectangle represents the smaller FOV of the SUFI instrument, whilst the red rectangle delimits the area of interest for this work. Within it, two small flux emergence events (labeled `East EFR' and `West EFR', respectively) show the typical signatures of elongated granulation. Middle and right panels: images from SUFI in 2 wavelength bands, at 300 (UV continuum) and 397 nm (Ca {\\sc ii} H). Notice how the FOV of SuFI only captures the East EFR and the confluence point between both magnetic patches.\\label{fig:context}}\n\\end{figure}\n\n\n\nThe data presented in this paper were obtained during the first day of the second flight of the balloon-borne mission {\\sc Sunrise II} \\citep{solanki2016}. {\\sc Sunrise II} was launched from Kiruna (Sweden) on 2013 June 12, and it flew for 5 consecutive days at an average flight altitude of 35~km. This granted the possibility of a 24~hr duty cycle in a near-space environment, which minimized turbulence perturbations and allowed for ultraviolet (UV) observations. The scientific payload was similar to that of the first flight \\citep{bar11, berkefeld2011}, comprising the Sunrise Ultraviolet Filter Imager \\citep[SuFI,][]{gan11} and the Imaging Magnetograph Experiment \\citep[IMaX,][]{pillet11}.\nTwo main data reduction levels were produced for each of the instruments: Level 1.x refers to calibrated data prior to deconvolution with the instrument PSF, whereas Level 2.x data were subjected to a spatial reconstruction. Despite the higher contrast of the latter, the increased noise and the small artifacts that arise from the deconvolution process render Level 2.x data risky for the analysis of weak polarization signals. Throughout this paper we make use of Level 1.2 data (see below) in all quantitative aspects of the analysis, relegating the reconstructed data to visualization purposes only. This comes at the expense of spatial resolution.\n\n\nAfter refurbishment, the IMaX instrument on {\\sc Sunrise II} remained essentially unchanged with respect to its first flight \\citep{pillet11, solanki2016}. IMaX uses a LiNbO$_3$ solid etalon for spectral analysis and liquid crystal variable retarders for polarization modulation. \nThe data presented in this paper were acquired using the so-called \\emph{V8-4} observing mode, which produces the full Stokes vector ($I$, $Q$, $U$, $V$) at 8 wavelengths around the magnetically sensitive Fe {\\sc i} 525.0208 nm line, which has a Land\\'e factor of 3 and forms in the photosphere of the Sun.\n The spectral region was sampled at $[-12, -8, -4, 0, 4, 8, 12, 22.7]$ pm with respect to the reference wavelength, yielding 7 wavelength samples within the line and one in the red continuum. In order to improve the signal to noise ratio, 4 accumulations were taken for each polarization modulation state and each wavelength position. A whole scan operation in this observing mode takes approximately 36.5~s. The time series used in this paper is composed of 28 individual observation cycles and is $\\sim 17$ minutes long.\nThe total field of view (FOV) covers $51^{\\prime\\prime} \\times 51^{\\prime\\prime}$ with a pixel size of $0.055^{\\prime\\prime}$. \n\n\\noindent A further set of treatments was applied to the level 1 data used for this study. First, the whole time series was carefully aligned to correct for remaining jitter. Second, a temporal interpolation was applied to the images in order to compensate for the non-simultaneity of the measurements in each cycle. As noted above, an IMaX observation cycle takes approximately 36.5 s, during which the instrument tunes through the different wavelength positions and polarization modulation states. The aim of this interpolation is to put the whole array of data into a common time frame, thus reducing the artifacts introduced by solar evolution. \nLastly, a wavelength correction across the FOV was performed to compensate for the wavelength blueshift introduced by the IMaX etalon in its collimated configuration \\citep[see, e.g.,][]{puschmann06,reardon08,pillet11}. In order to correct for this effect, the data reduction pipeline includes a model of the wavelength blueshift that is obtained from a surface fit to a third-degree polynomial over the FOV of a flat-field image. \nAll of these treatments led to the production of the Level 1.2 data used in this paper.\n\n\nSimultaneous measurements from the SuFI instrument were carried out in a subset of the IMaX field of view (delimited by the blue rectangle in the left panel of Fig. \\ref{fig:context}) at a high cadence. SuFI cycled through 3 broadband wavelength channels (i.e. a 5\\,nm wide channel in the UV continuum around 300 nm and two narrower bands of 0.11 nm and 0.18 nm around the the Ca {\\sc ii} H line at 396.8 nm) in approximately 7~s \\citep{riethmueller2013}, producing brightness images of the photosphere and the low chromosphere in a portion of the IMaX FOV, with a \n$\\sim 0.02^{\\prime\\prime}$\/px spatial sampling. A sample of the images of a SuFI cycle is shown in the middle and right panels of Figure \\ref{fig:context}. \n\nThe data used for this work were obtained on 2013 June 12, starting at 23:39 UT and spanning $\\sim$17 minutes. The telescope pointed to the newly emerged active region NOAA AR 11768, which was located at $\\sim 11^{\\circ}$ S, $\\sim 16^{\\circ}$ W in solar coordinates. The larger FOV of the IMaX instrument (left panel of Fig. \\ref{fig:context}) covered the emergence site and the trailing polarity of the AR, which comprised several pores and a penumbra-like structure.\nThis paper focuses on a particular section of the FOV, delimited by the red rectangle in Figure~\\ref{fig:context}. During the time series, this area shows two separate instances of magnetic field emergence (evidenced by the elongated granulation in the continuum image) that contribute to the overall emergence of magnetic field in the AR. At the time of the snapshot in this figure, the two emerging flux regions (EFR) are already present and clearly visible. Throughout this paper we refer to them as {\\em East EFR} and {\\em West EFR}, whilst the location in between them, where the opposite polarity footpoints of the two EFRs meet, is referred to as the {\\em confluence point} (enlarged images of the FOV of the red rectangle can be seen in Figure \\ref{fig:evolution}). In this paper, we carry out a detailed study of the temporal evolution of the magnetic and thermodynamic properties in the photosphere as well as the chromospheric response to the newly emerged flux.\n \nData from the Helioseismic and Magnetic Imager \\citep[HMI,][]{hmi} on board the Solar Dynamics Observatory \\citep[SDO,][]{sdo} were used as context to understand the evolution of the AR before and after the {\\sc Sunrise II} observations. Data from the Atmospheric Imaging Assembly \\cite[AIA,][]{aia_paper} were also used to study other facets of the chromospheric response to the emerging flux.\n\n\n\\subsection{Spectral line inversion and p-mode filtering of IMaX data} \\label{inversions}\n\nIn order to obtain the magnetic properties of the emerging flux site at the photosphere we proceeded to carry out the spectral line inversion of the spectropolarimetric data from the IMaX instrument. For this purpose we used a Milne-Eddington (ME) spectral line inversion code\\footnote{See reference to HEXIC in http:\/\/www2.hao.ucar.edu\/csac\/csac-spectral-line-inversions.}, generalized from the Very Fast Inversion of the Stokes Vector \\citep[VFISV,][]{vfisv, vfisv2}. The ME approximation solves the polarized radiative transfer equation assuming that the thermodynamical and magnetic properties of the atmosphere are height independent, except for the source function, which varies linearly with optical depth. The generation of spectral line polarization is taken to be exclusively due to the Zeeman effect. The inversion algorithm itself uses the Levenberg-Marquardt technique \\citep{press} for non-linear least-squares minimization of a merit function that quantifies the difference between the synthetic and the observed spectral lines. All inversions were performed by fixing the magnetic filling factor to 1, but they were ran twice, using two different values for the stray light fraction, $0\\%$ and $25\\%$ (the latter being the approach taken by the IMaX data calibration team). This was done in order to explore the quantitative and qualitative differences in the results between these two extremes. We found that the transverse component of the magnetic field increased by $\\sim 10\\%$ when adopting the higher stray light fraction, whilst the longitudinal component was amplified, typically, by several hundred gauss. All of the magnetic field values quoted throughout the paper refer to the $25\\%$ stray light inversions, however, no qualitative differences ensue from the two approaches, which speaks to the robustness of the results.\n\nThe inversion was carried out over the FOV in the red box of Fig. \\ref{fig:context} for the 17 minute time series. This provided, for each pixel, the vector magnetic field in spherical coordinates (i.e. the pixel-averaged magnetic field strength, $B$, the inclination of the field with respect to the LOS, $\\theta$, and its azimuth in the transverse plane, $\\chi$), the line-of-sight velocity, $v_{\\rm LOS}$, and 5 other parameters that contain the thermodynamical information encoded in the spectral line (the source function and its gradient, the damping parameter, the Doppler width and the line-to-continuum opacity ratio).\nThe simplistic approximation of the Milne-Eddington atmosphere does not allow us to retrieve actual physical thermodynamical parameters, nor is it able to interpret asymmetries in the spectral line. However, it is a conservative approximation that provides a good first-order estimate of the magnetic and dynamic properties prevailing in the atmosphere \\citep[e.g.][]{deltoro2016}.\n\nIn order to determine the real LOS velocity flows associated with the magnetic flux emergence, we removed the p-mode oscillations using a subsonic filter, following the space-time filtering described by \\cite{title89}.\nThe velocities were obtained from the Level 1.2 calibrated spectra using the center-of-gravity method to determine the position of the spectral line. Then, the filter was applied over the inferred Doppler shifts. The reason to use the velocities computed with the center-of-gravity method rather than from the spectral line inversions was to avoid any spurious effect due to non-converging pixel inversions.\n\n\\section{The lower layers: photosphere}\\label{photosphere}\n\nIn this section we describe the physical properties and evolution of the emergence site at the photosphere. The information presented here is inferred from the IMaX data, through spectral line inversions in the case of the magnetic properties and through the center of gravity analysis in the case of LOS velocities (see Section \\ref{inversions}). The emerging magnetic field interacts with the surface granulation in complex ways, leading to morphology changes, altered plasma flows and signatures of heating.\n\n\n\\subsection{Vector magnetic field and LOS velocities}\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.37]{bar.ps}\n\\includegraphics[angle=0,scale=.37]{imax_sequence_3col.ps}\n\\caption{Evolution of the two emerging flux areas inside the red box of Figure \\ref{fig:context}. The left column shows the continuum intensity (at 525.043\\,nm) in the background with overlaid LOS velocity contours (red and blue correspond to redshifts and blueshifts at $\\pm 0.5$ and $\\pm 0.8$ km\/s). The middle column depicts the evolution of the magnetic field, where the background is now the LOS magnetogram and the red lines represent the transverse component of the magnetic field. In the right column, the LOS velocities after p-mode filtering are shown.\\label{fig:evolution}}\n\\end{figure}\n\n\nFigure \\ref{fig:evolution} displays the evolution of the emerging flux area, sampled every six time steps. The grayscale background of the left column depicts the continuum (at 525.043\\,nm) intensity images, which show the pattern of the surface granulation in the area of the emerging magnetic field. A close look reveals elongated granules ($\\sim 5^{\\prime\\prime}$ long) during some stages of the emergence. Overplotted velocity contours at $\\pm 0.8$ and $\\pm 0.5$~kms$^{-1}$ represent the plasma LOS velocity, where blue and red contours correspond to material moving towards and away from the observer, respectively.\nThe middle column exhibits the evolution of the magnetic field. The grayscale background now represents the line-of-sight magnetic flux density, saturated at $\\pm 500$~G (this quantity is obtained by computing $B\\cdot cos\\theta$ from the spectral line inversion results). The dashed lines show the direction of the transverse component of the magnetic field for pixels where the field strength is larger than $280$~G (this threshold was set for clarity in the figure). The color-coding of the lines changes progressively from yellow to red to black as the magnetic field inclination with respect to the surface goes from $- 40^{\\circ}$ (pointing inwards) to $40^{\\circ}$ (pointing outwards). Red corresponds to purely transverse magnetic field, i.e. parallel to the surface. Purely based on continuity arguments it is clear that the magnetic field comes out of the surface at the right footpoint of the West EFR (label \\#1 in first row), goes back into the surface at its left footpoint (\\#2), immediately threads up at the right footpoint of the East EFR (\\#3) just to pierce back into the surface at the left-most footpoint (\\#4). \nThis column clearly shows the magnetic signature of the two emerging flux patches. The {\\em confluence region} comprises labels 2 and 3 (around $x \\sim 8^{\\prime\\prime}$ and $y \\sim 4^{\\prime\\prime}$), where the right footpoint of the East EFR meets the left footpoint of the West EFR. Based on the connectivity of the magnetic field, this opposite polarity patch is an MDF like the ones described in \\cite{bernasconi2002} and \\cite{xu2010}. \n\nThe column on the right presents the evolution of the line of sight velocity, saturated between $-0.5$ and $0.5$ km\/s. Blue and red correspond to upflows and downflows, respectively. The East side emergence was already developed when IMaX started observing, while the West one was captured during its incipient stages. \nThe LOS velocity shows a strong blueshift accompanying the West EFR as it rises to the surface, revealing a tight spatial correlation between horizontal magnetic fields and upflows. \nA string of downflow patches lights up the perimeter of the both EFRs during the entire sequence. The strongest and most concentrated downflows appear at the main footpoints, and especially in the confluence area. \nThe surface granulation (left column) also shows the typical signatures of an emerging flux region. The presence of granules elongated in the direction of the field coincides with the presence of a strong transverse magnetic component (red lines in middle panels). As soon as this transverse component weakens, the elongated granules break down into normal quiet Sun granulation (see a more detailed analysis in Section \\ref{elongated}).\n\nThese stuctures resemble emergent magnetic bubbles delimited by their footpoints that, upon ascending, perturb the granulation pattern due to the strength of their magnetic field. This is in agreement with a typical flux emergence scenario in which, as the magnetic field rises through the surface, it interacts with the convection, changing the shape of the granulation and dragging material up into higher layers. The displaced material has to drain back down to the surface, mostly guided along the field lines to the footpoints \\citep{cheung2007b, solanki2003}. \nOnce the magnetic bubble has risen above the surface, the upflows subside, the transverse fields fade and the granulation returns to its usual behaviour. In the chromospheric layers, these EFRs also resemble magnetic bubbles (see, for instance, the right panel of Fig. \\ref{fig:sufi_filaments}, corresponding to SuFI's 397 nm channel).\n\n\n\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.73]{imax_azimuth_lines.ps}\n\\caption{Transverse magnetic field lines at the location of the West EFR (the spatial coordinates refer to those of Fig. \\ref{fig:evolution}), which was captured from its very early stages. The background images show the continuum intensity and the red lines represent the transverse magnetic field lines connecting the footpoints of this small emerging bubble. The sequence evolves from left to right and top to bottom. \\label{fig:azi_lines}}\n\\end{figure}\n\nFigure \\ref{fig:azi_lines} shows the evolution of the shape of the magnetic field lines at the location of the West emergence.\nIn this sequence (from left to right and top to bottom), the background images show the evolution of the continuum intensity. Each one of the red lines superposed on these images is constructed using a 2D version of the procedure introduced by \\cite{solanki2003}: we first take a pixel in the left footpoint (\\#2) and calculate the projection of the vector magnetic field onto the transverse plane (in the LOS reference frame). This is plotted over the background image with a short red dash. The direction of the vector points towards the next pixel in the sequence, which is then taken as the origin and the calculation of the transverse field is repeated. The red dashes are only drawn on the image when the magnetic field strength exceeds the threshold of 250 G. The process is iterated until the line spans the length of the emerging region. Ten ``field lines'' are represented, spanning the distance between the two footpoints (\\#1 and \\#2). \nIt is important to emphasize that these are not magnetic field lines in the mathematical sense. They are constructed from the direction of the magnetic field in the transverse plane, which is derived from the spectral line inversions. However, they give a sense of continuity and the behavior of the magnetic field topology.\n\nAt the beginning of the sequence, only a small area is covered by magnetic field above the $250$~G threshold. In the first two panels of Fig. \\ref{fig:azi_lines}, the {\\em field lines} cross a few granules with no obvious alignment between them and the convective pattern. In these early moments of the emergence, the largest values of the transverse magnetic field strength (no more than $30^{\\circ}$ from the transverse plane) are between 400-500\\,G, which lie just below the photospheric equipartition value \\citep{cheung2007b}.\nAs the sequence progresses, the emerging flux patch brings stronger transverse fields to the surface and widens sideways, adopting an overall rugby-ball shape. By 23:42, the transverse field strengths reach the $600-700$\\,G range, increasing up to $800$\\,G 2 minutes later, where they plateau and remain strong for the next 5 minutes. It is during this time that the granules adopt elongated shapes aligned with the direction of the field. Around 23:48, the transverse field strength starts to decay, as the elongated granules break up into smaller and more regular-shaped convective cells.\nBased on the inferred surface field strengths, the convective motions dominate over the magnetic field at the beginning and the end of the sequence, however, in the middle, the field is well enough above the equipartition value to alter the shape of the granulation \\citep{requerey2015, requerey2016}.\nTowards the end of the sequence, the magnetic signal weakens at the center of the emergence (hence the lack of dashed lines in that area). This is expected: as the field rises, it expands sideways becoming intrinsically weaker, and it moves upwards beyond the region of sensitivity of the IMaX spectral line, leading to vanishing polarization signals. \nThe overall behavior though, shows field lines that fan out at the center of the emergence while remaining constrained and trapped at the footpoints, where downflows prevail. The plasma that was dragged up to the surface by the buoyant transverse fields, is funneled back down through the footpoints, guided by the field lines, returning to the surface.\n\n\n\n\\subsection{Elongated granulation}\\label{elongated}\n\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.73]{breaking_granule.ps}\n\\caption{Two snaphots in the evolution of an enlongated granule in the East EFR, seen in continuum intensity (left) and line-of-sight velocity (right) saturated at $\\pm 0.8$~kms$^{-1}$. Blue indicates upflow and red downflow. The spatial coordinates refer to those of Figure \\ref{fig:evolution}. \\label{fig:granule}}\n\\end{figure}\n\nElongated granules are one of the most common signatures of magnetic flux emergence in the photosphere. Figure \\ref{fig:granule} shows in detail one of the elongated granules that appears during the emergence sequence: the continuum intensity on the left and, on the right, the LOS velocity saturated between $\\pm 0.8$~kms$^{-1}$. The top row presents the granule when it is at its longest, coinciding with a time when strong transverse fields are present at the surface ($700 - 800$\\,G). The LOS velocity for this snapshot reveals an obvious upflow pattern along the length of the bright granule while the dark elongated intergranular lane flanking it features significant downflows. The nature of the convective flow is preserved, but its morphology has changed to adapt to the presence of the magnetic field.\nA few minutes later (second row), when the transverse field at the surface has weakened, the elongated structure breaks into two smaller granules, separated by a darker channel in between them. The velocity pattern changes accordingly with a neutral velocity channel appearing at the location of the rupture.\n\n\n\n\\subsection{Confluence area}\\label{sec:bp}\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.5]{bright_point2.ps}\n\\includegraphics[angle=0,scale=.5]{mean_intensity_profiles.ps}\n\\caption{Left: snapshot of the continuum intensity when the bright point (marked with the green arrow) in the confluence area has developed. Black and white patches show two polarities of the longitudinal magnetic field saturated at $\\pm 250$~G and colored contours present the line-of-sight velocity saturated at $\\pm 0.6$~kms$^{-1}$. Blue indicates upflow and red downflow. Right: Average granular (purple) and intergranular (red) intensity profiles. An exaple of a spectral profile at the location of the bright point is shown in black. \\label{fig:brightpoint}}\n\\end{figure}\n\n\nWhen observing the evolution of the photospheric continuum intensity of the confluence area in between the two EFRs, a prominent bright point develops exactly at this position. It is flanked by the opposite polarity footpoints of both emerging patches, that come together as the West EFR develops and brings magnetic flux to the surface. \nThe left panel of Figure \\ref{fig:brightpoint} selects the instance during the lifetime of the bright point when it reaches its maximum brightness. In the background, the continuum intensity image reveals the granulation pattern as well as the bright point, whose location is indicated by the green arrow. Overplotted, the black and white patches show the longitudinal magnetic field saturated at $\\pm 250$~G, whilst the blue and red contours mark the line-of-sight velocities at $\\pm 0.6$~kms$^{-1}$. In this snapshot, the BP is no more than $\\sim 0.5''$ wide and sits in the middle of the confluence area, where the right footpoint of the East EFR meets the left footpoint of the West EFR. It is surrounded by the strong downflows of the magnetic footpoints (red contours) and the upwelling motions of the West emerging flux region (blue contours).\nThe photospheric spectral profiles at the location of the bright point are very shallow in comparison to other locations in the field of view. The right panel of Figure \\ref{fig:brightpoint} shows examples of a bright point (black) intensity profile, as well as the average granular (purple) and average intergranular (red) profiles. As expected, the intergranule profiles are shallow, have low continuum intensity values and are red-shifted with respect to the reference position of the spectral line. The granule profiles, on the other hand, appear blue-shifted with respect to the intergranular profiles, have a high continuum intensity and are rather deep \\citep{dravins}. In the bright point, the continuum is typically $2\\%-5\\%$ higher than in the brightest granules, and the spectral line is wide and shallow. The spectral line seems to be slightly red-shifted, yet shows a strong asymmetry in the blue wing of the line. The higher continuum as well as the width of the spectral line indicate high temperatures and significant thermal motions, whilst the asymmetry is most likely due to a strong gradient of velocities along the line of sight. Also, the shallowness of the line indicates high temperatures in the upper layers (see section \\ref{temperature_stratification} for a detailed study of the temperature statification in the FOV). The confluence of two opposite polarity footpoints and the associated chromospheric brightenings (described in Section \\ref{chromosphere}) lead us to interpret the spectral profiles in the bright point as a signature of heating due to magnetic reconnection.\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.5]{imax_flux_cancel.ps}\n\\caption{Flux history of the confluence area (black) and light curve of bright point (red). The flux history is calculated separately for the positive polarity (solid line), which is already present at the beginning of the series, and the negative polarity (dashed line), which appears as the West EFR breaks through the surface. The threshold to calculate the fluxes is set at 300 G. The red solid line shows the light curve of the continuum intensity at the bright point, $I_{\\rm BP}$, normalized to the average quiet Sun continuum intensity, $I_{\\rm QS}$. \\label{fig:fluxhistory}}\n\\end{figure}\n\nAt the beginning of the time series, the confluence area only has a patch of positive magnetic polarity, corresponding to the right footpoint (\\#3) of the East EFR. As the West EFR develops, it brings flux to the surface and a negative polarity footpoint (\\#2) that moves into the confluence area. Figure \\ref{fig:fluxhistory} shows the flux history of the opposite polarity footpoints that come together in this small region. This is computed within a $3.3^{\\prime\\prime} \\times 3.7^{\\prime\\prime}$ box with a $300$\\,G threshold in order to spatially isolate the footpoints. With these constraints, the positive foopoint is always contained within the box and does not change much. Nevertheless, some positive magnetic flux does appear around $500$\\,s. The negative polarity, on the other hand, enters the computation box as the West EFR brings flux to the surface. At the beginning, the only existing polarity is the positive one. But as the sequence advances, the negative flux grows as the positive flux decreases. This indicates two things: magnetic flux is being brought up to the surface by the West EFR and, as its negative footpoint moves into the confluence, the opposite polarities come together and partially cancel each other. Because the negative polarity footpoint is moving into the confluence at the same time as it cancels out with the positive polarity, its flux curve (dashed line) does not change as significantly as the positive flux curve (solid line). We can use the latter to make an estimate of the rate of flux cancellation between 50 and 650\\,s, by attributing all the decay exclusively to cancellation. The positive magnetic flux loss derived from this measurement is $\\sim -2.5 \\cdot 10^{16}$\\,Mx\/s, which is 10 times larger than that found by \\cite{reid2016} (but it is similar to that quoted by \\cite{kuckein2012} for the flux loss rate of an entire AR). However, this is still a conservative estimate, since we are not correcting for the positive flux that enters the area during this time. The red line in the same figure shows the light curve of the bright point (calculated in a box of $1.1^{\\prime\\prime} \\times 1.1^{\\prime\\prime}$ surrounding it), whose onset happens $\\sim 2$ minutes after the two polarities start cancelling out. The intensity increase stretches up to $7\\,\\%$ above the average quiet Sun level.\n\n\n\\subsection{Photospheric temperature}\\label{temperature_stratification}\n\n\n \\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.5]{spinor_temp.ps}\n\\includegraphics[angle=0,scale=.5]{temp_profile.ps}\n\\caption{Temperature profile at the sites of the two EFRs. Left: temperature at ${\\rm log}\\, \\tau = 0$ and ${\\rm log}\\, \\tau = -2.5$ for one instance in the observed sequence. Right: ${\\rm log}\\, \\tau = -2.5$ profile along the cut in the lower left-side panel (solid) and comparison to the temperature profile at the same location but 12 minutes later (dashed). The spatial coordinates refer to those of Figure \\ref{fig:evolution}. \\label{fig:temperature}}\n\\end{figure}\n\nOne of the disadvantages of using a ME inversion code to interpret the spectral line radiation is that it does not provide certain thermodynamical variables, such as temperature or pressure. Instead, it encodes the information in the parametrization of the source function and other variables, but it does not disentangle thermodynamical information. We have used the ME approach to extract the magnetic properties in the emergence site, but in this section, we will make use of spectral line inversions in the Local Thermodynamical Equilibrium (LTE) approximation to perform an analysis of the temperature stratification.\n\nThe {\\sc Sunrise} team provided LTE inversions of the Level 2 IMaX data (spatially reconstructed) carried out with the 1D version of the SPINOR code \\citep[for specific details of the inversion, see][]{frutiger2000, solanki2016}. The inversion strategy applied a global stray light correction of $25\\,\\%$ to Stokes $I$, and allowed temperature perturbations at 3 different heights along the LOS, whilst treating the rest of the parameters (the three components of the vector magnetic field, the line-of-sight velocity and the microturbulence) as height independent.\n\nThe left side of Figure \\ref{fig:temperature} shows the temperature at ${\\rm log}\\, \\tau = 0$ (upper panel) and ${\\rm log}\\, \\tau = -2.5$ (lower panel) for one instance in the time series. At ${\\rm log}\\, \\tau = 0$, the temperature is highly correlated with the granulation pattern, showing hot granules surrounded by cooler integranular lanes. The elongated granulation and the bright point in the confluence area are also evident. At this layer, the confluence area experiences a temperature increase of $\\sim 500$\\,K with respect to its pre-bright point phase. In the lower panel, the temperature at ${\\rm log}\\, \\tau = -2.5$ presents the expected signature of reversed granulation \\citep{cheung2007a}. The two EFRs show up as cooler areas surrounded by slightly hotter ridges, especially around the footpoints. Strands of very cool material ($T < 4000$ K), roughly aligned with the direction of the magnetic field, can be seen in the magnetized areas of the two emerging patches (more so in the West one).\nThe solid line in the right panel of Fig. \\ref{fig:temperature} shows the temperature profile along the vertical cut of the lower left panel. The dashed line corresponds to the temperature profile at the same location and height, but 12 minutes later, when the strength of the transverse magnetic field has decayed significantly. The large temperature dips captured by the solid line correspond to the strands of cool material in the West EFR. These strands appear only in the early stages of the emergence, when relatively strong magnetic field is traversing the surface layers. At the end of the sequence, the temperature profile across the EFR returns to its normal variablility expected at those heights (dashed line in right panel). \n\n\\section{The higher layers: Chromosphere} \\label{chromosphere}\n\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.35]{imax_east.ps}\n\\includegraphics[angle=0,scale=.35]{sufi_cont.ps}\n\\includegraphics[angle=0,scale=.35]{sufi_filaments.ps}\n\\caption{Images of the East EFR in IMaX continuum (left), SuFI 300 nm channel (middle) and SuFI 397 nm channel (right). The IMaX image uses the same convention as the left column of Figure \\ref{fig:evolution}. The UV continuum image (middle) presents a lot of similarities with the IMaX data, featuring the same elongated granule pattern, but displaying much more prominent bright points forming a chain around the flux emergence area. The chromospheric image (right) shows the confluence area as a bright region at the right edge of the figure. In this panel, the emerging flux patch appears as a dark bubble surrounded by bright points and traversed by a faint arching filament system that originates close to the footpoints. The spatial coordinates refer to those of Figure \\ref{fig:evolution}, and the white rectangle is the FOV of Figure \\ref{fig:fibrils}. \\label{fig:sufi_filaments}}\n\\end{figure}\n\n SuFI observed the entire emergence sequence from the time that IMaX started observing, to about an hour later. Unfortunately, it only captured the East EFR and the confluence area, leaving the West EFR almost entirely outside of its field of view. SuFI recorded the event in three channels, one in the UV continuum and two bandpasses around the Ca {\\rm II} H line at 396.8 nm \\citep{riethmueller2013}. The latter two sampled the mid-chromosphere with significant mid-to-upper photospheric contributions \\citep{danilovic2014}. \nFigure \\ref{fig:sufi_filaments} shows images of the East EFR in different wavelength channels. At photospheric layers, the IMaX continuum (left) and the UV continuum (middle) show the expected signatures of flux emergence: elongated granules flanked by dark lanes aligned in the direction of the magnetic field. The main difference between these two images is that the UV continuum shows a string of brighter points surrounding the slightly darker EFR. At chromospheric layers (right panel) the emerging flux appears clearly delineated by bright points surrounding a darker cell interior. The dark bubble, in turn, is traversed by a faint filament system that spans from footpoint to footpoint, along the direction of the magnetic field inferred from IMaX data (which is shown by the red dashes in the left panel, using the same convention as in Figure \\ref{fig:evolution}). The confluence area is the brightest region located close to the right border of the image (this is also the edge of the SuFI FOV). \n\n\\subsection{SuFI Ca{\\rm II} 397 nm}\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.45]{sufi_fibrils.ps}\n\\includegraphics[angle=0,scale=.45]{sufi_fibrils_bar.ps}\n\\caption{Time sequence of SuFI 397 brightness image inside the white rectangle in the right panel of Figure \\ref{fig:sufi_filaments} (although not co-temporal with it). The left side of each panel corresponds to the East-most footpoint, whilst the right side ends in the confluence area. A finger-like brightening protrudes from the confluence area and extends towards the East footpoint, becoming longer with time and then fading away again. \\label{fig:fibrils}}\n\\end{figure}\n\nThe SuFI images at 397 nm sample the emerging magnetic flux at mid Chromospheric layers. At the beginning of the observing sequence, the East EFR appears as a dark bubble surrounded by bright edges. But as time goes by, an arch filament system develops above the region, with bright filaments that span between the two footpoints. The filaments start to appear as the bright point in the confluence area develops. In fact, some of the filaments seem to protrude from this region and elongate towards the left-most footpoint (footpoint \\#4 in Fig. \\ref{fig:evolution}), as depicted in Figure \\ref{fig:fibrils}. There are many instances of this behavior, i.e., brightenings that start at the confluence footpoint and apparently travel towards the left-most footpoint of the East EFR (\\#4). Because the FOV of SuFI ends at the confluence, we cannot know if the same happens over the West EFR, i.e. chromospheric fibrils developing from the confluence towards the right-most footpoint (\\#1). The absence of spectroscopic measurements renders impossible the attribution of this apparent motion to outflows from the confluence point, however, it is in any case indicative of energy release, whether in the form of motion or heat (or both).\n\n\n\n\\subsection{AIA UV continuum at 160 and 170 nm}\n\n\\begin{figure}[!t]\n\\includegraphics[angle=0,scale=.5]{AIA1700.ps}\n\\includegraphics[angle=0,scale=.47]{aia_light_curve2.eps}\n\\caption{AIA 160 and 170 nm observations. The left panel shows an image of the full IMaX FOV (compare to Fig. \\ref{fig:context}) in the 160 nm channel of AIA, two minutes after the beginning of the {\\sc Sunrise} observations. The right panel shows light curves of the two UV channels in a $4.8^{\\prime\\prime} \\times 4.8^{\\prime\\prime}$ box around the confluence point. The light curve for the IMaX bright point is shown in red for comparison. \\label{fig:aia}}\n\\end{figure}\n\n\n\nThe AIA \\citep{aia_paper} instrument on board SDO also registered the emergence events. Despite the lower spatial resolution, the UV continuum channels at 160 and 170 nm show the brightenings delimiting the small emerging flux regions. This allows us to unequivocally identify them in the UV images. The left panel of Fig. \\ref{fig:aia} shows an AIA 160 nm image, cut out to approximately reproduce the IMaX FOV, two minutes into the {\\sc Sunrise} observations. Both emerging regions appear as dark bubbles surrounded by bright dots. The confluence area between the opposite polarity footpoints, also shows excess brightness. Throughout the 17 minute observation, the UV emission in the confluence area increases rapidly in both channels, developing a bright point that completely dominates the emission in the entire FOV. The right panel in Fig. \\ref{fig:aia} shows the light curves in the 160 nm and 170 nm channels. The light curves were computed by integrating the brightness in a small $4.8^{\\prime\\prime} \\times 4.8^{\\prime\\prime}$ box around the confluence area and were both normalized to its quiet Sun value. The brightness starts to increase approximately 2 minutes after the beginning of the IMaX observations and reaches its maximum 8 minutes later, increasing by a factor of 1.8 in 170 nm and by 2.6 in 160 nm.\n\nAs described in Section \\ref{sec:bp}, a bright point in the IMaX continuum image also appears in the confluence area.\nThe IMaX continuum light curve for this event (red solid line) has its onset about 100 s after the onset of the AIA event, and reaches its maximum intensity around the time that the AIA brightness starts to decay. Due to the lower cadence of the IMaX data, the time lag of the visible with respect to the UV brightening is subject to a large uncertainty, but the visible bright point lags the UV one by at least half a minute.\n\n\n\n\n\\section{Discussion} \\label{discussion}\n \nThe widely accepted picture of the emergence of Active Regions invokes the rise of an $\\Omega$-loop from a toroidal magnetic field that sits near the base of the convection zone, triggered by the buoyancy instabilities. When the field reaches subphotospheric layers it interacts with the convective flows, suffering a distortion that depends on the original strength and twist of the flux tube \\citep{cheung2007b}. This leads to a systematic undulation along the length tube (with typical wavelengths of 1 -2 Mm), where upflows aid the rise of the crests whilst downflows supress the rise of the troughs \\citep{pariat2004, cheung2010}.\nDuring the first day of its 2013 flight, {\\sc Sunrise} witnessed the rise of magnetic flux at the emergence site of an AR. The evolution of very small-scale processes was recorded in detail by the two onboard instruments: IMaX and SuFI. This enabled the retrieval of the full vector magnetic field in the photosphere and the chromospheric response to the flux emergence. \n\n {\\sc Sunrise} captured the evolution of 2 small EFRs during the 17m observation (see Fig. \\ref{fig:evolution}). Each one comprises a pair of opposite polarity footpoints connected by a horizontal magnetic field which is roughly aligned with the axis of the EFR and the polarity of the large-scale AR. The positive footpoint of the East EFR (\\#3 in Fig. \\ref{fig:evolution}) is contiguous to the negative footpoint of the West EFR, in an MDF configuration. Continuity arguments suggest a field topology that emerges from under the surface at \\#1, sinks at \\#2, threads up again through \\#3 and returns back through the surface at \\#4, in an undulating fashion. The velocities are such that all foopoints experience downflows whilst the horizontal field patches present upflows. \n The East EFR is already developed by the time the observation starts, while the West EFR is captured from its initial stages. At the beginning of its emergence process, the transverse field at the surface is weak ($<500$\\,G) and connects the footpoints without distorting the granulation. As stronger field ($>700$\\,G) rises up to the surface, the granulation adopts an elongated pattern aligned with the direction of the horizontal field. This lasts for $\\sim 5$\\,m, roughly a granule lifetime, after which the strong upflows cease, the transverse magnetic field weakens and the convective motions take charge again, creating slight dips in the field at the intergranular lanes. This results in the breakage of the elongated convection cells, returning to a granulation pattern that resembles that of the quiet Sun. The entire process happens in just over 10 minutes, which is similar to the normal granular time scale. \n \n Whether there is connectivity of the magnetic field below the surface between footpoints \\#2 and \\#3 is, of course, debatable. There is no evidence in the data aside from proximity and magnetic field continuity arguments. However, this small region where the two footpoints meet (referred to throughout the paper as the {\\em confluence area}), is the stage for some interesting happenings that take place in the following order:\n \n\n \n \\begin{tikzpicture}[snake=zigzag, line before snake = 5mm, line after snake = 5mm]\n \n \\draw (0,0) -- (14,0);\n \n\n \n \\foreach \\x in {0,2,4,6.6,8,10,11, 14}\n \\draw (\\x cm,3pt) -- (\\x cm,-3pt);\n\n \n \t\\draw (0,0) node[below=3pt] { 0s } node[above=3pt] { };\n \t\\draw (2,0) node[below=3pt] {Flux } node[above=3pt] {starts};\n \\draw (2,0) node[below=14pt] {cancellation} node[above=14pt] {AIA BP};\n \\draw (2,0) node[below=25pt] {starts} node[above=25pt] {};\n \\draw (4,0) node[below=3pt] {} node[above=14pt] {IMaX BP};\n \\draw (4,0) node[below=3pt] {200 s} node[above=3pt] {starts};\n \\draw (6.6,0) node[below=3pt] {} node[above=3pt] {system};\n \\draw (6.6,0) node[below=14pt] { } node[above=14pt] {filament};\n \\draw (6.6,0) node[below=25pt] { } node[above=25pt] {Arch};\n \\draw(8,0)node[below=3pt] {400s}; \n \\draw (10,0) node[below\t=14pt] { } node[above=14pt] {AIA BP};\n \\draw (10,0) node[below=3pt] { } node[above=3pt] {max};\n \\draw (11,0) node[below=3pt] {IMaX BP} node[above=3pt] {};\n \\draw (11,0) node[below=14pt] {max} node[above=14pt] {};\n \\draw (14,0) node[below=25pt] {} node[above=25pt] { Flux };\n \\draw (14,0) node[below=24pt] {} node[above=14pt] { cancellation};\n \\draw (14,0) node[below=3pt] {700 s} node[above=3pt] { ends};\n \\end{tikzpicture}\n \n\nAt the beginning of the observation the East EFR is fully developed. Its positive footpoint (\\#3) sits alone in the confluence region and strong downflows prevail in the area. The West EFR is surfacing, but still far from the confluence.\nAbout $100$\\,s later, the West EFR brings more magnetic flux up to the surface accompanied by strong upflows. At this time, its negative polarity footpoint (\\#2) develops and moves into the confluence area, and magnetic flux starts to cancel out. Almost simultaneously, a bright point (BP) develops in the 160 and 170 nm channels of AIA.\nAround $200$\\,s a bright point also starts to develop in the visible continuum. Later, at $330$\\,s, an arch-filament system over the East EFR becomes visible in SuFI's 396.8 nm channels, and apparent outflows that start at the confluence area seem to travel to the East-most footpoint (\\#4).\nThe AIA bright point reaches its maximum brightness at around $500$\\,s, followed by its analogous event in the IMaX continuum a few seconds later. At $700$\\,s the flux cancellation at the photosphere tapers off, the transverse magnetic field at the surface has weakened notably in both EFRs and the granulation has almost returned to its quiet Sun pattern. The chromospheric images, however, still show a bundle of bright strands overlying the East EFR.\n\n\nThe magnetic flux cancellation and the ensuing photospheric and chromospheric brightenings strongly suggest that magnetic reconnection is taking place at the confluence point. The flux removal process is expected to show transverse field signatures at the neutral line between the cancelling polarities, however, the spatial resolution of the IMaX data might still not be enough to see this \\citep[see][for an analysis of expected polarization signatures at flux cancellation sites]{kubo2014}. The downflows in the photosphere and the apparent outflows from the confluence into the arch filament system above the East EFR suggest bidirectional flows and\/or energy transfer originating somewhere in the upper photosphere, rather than below the surface. This is supported by the asymmetric photospheric spectral profiles (see Fig. \\ref{fig:brightpoint}) and the fact that the bright point in the visible continuum appears later than the bright point in the UV continuum and SuFI's 396.8 nm band. Unfortunately, the lack of spectroscopic measurements in Ca {\\sc ii} H prevent us from attributing the apparent chromospheric outflows to actual plasma motions.\n\n\\cite{vissers2015} argue that the AIA 160 and 170 nm channels are great indicators to identify EBs, since they show up as brightenings of a point-like nature (rather than filamentary strands). However, \\cite{rutten2016} states that they should not be visible in the optical continuum. Rather, that aside from the Balmer lines, other spectral indicators of EBs are brightenings in strong Ca {\\sc ii} H \\& K, other UV lines and UV continuum. \nAll our observations point towards an EB happening at the confluence: 1. emergence of new flux in a developing AR; 2. there is an MDF (bipole of the opposite polarity to that of the AR; the MDF harbors a magnetic U-loop under the surface) at the confluence; 3. magnetic flux cancellation takes places in the MDF; 4. strong brightnings ensue in the UV continuum and Ca {\\sc ii} H; 5. there are downflows at the photosphere and apparent outflows in the chromosphere, suggesting bi-directional flows; 6. the photospheric temperature increases by $\\sim 500$\\,K at the location of the bright point. All of these indicate that the reconnection starts somewhere in the upper photosphere and travels down towards the surface, where the flux cancellation is visible in the magnetograms.\nThe absence of H$_{\\alpha}$ spectra in our dataset prevents us from establishing whether the confluence point is host and witness to an EB; however, the magnetic flux cancellation and the photospheric and chromospheric responses are, strong indicators of reconnection and heating.\n\nEBs are estimated to release between $10^{23}$ and $10^{26}$ ergs of radiative energy per event \\citep{reid2016} and have typical durations of 10 minutes \\citep{georgoulis2002}. During the emergence of ARs, MDFs are a common occurence. They take place every time the field lines that span the distance between the main footpoints of the AR dip below the surface. Estimates from several authors \\citep{pariat2004, centeno2012} suggest that this happens every 5 to 10$^{\\prime\\prime}$ ($\\sim 3 - 8 $\\,Mm), and several strings of MDFs can exist in parallel, along the main direction of the AR. As the distance between the main footpoints of the AR becomes larger, more MDFs will form whenever new flux emerges to the surface, and each one of them is a likely scenario for an EB to happen (\\cite{pariat2004} associated $\\sim70\\%$ of their EBs to dipped U-loops). In this way, magnetic field reconnection naturally takes place in the resistive emergence scenario of developing ARs, and could account for a lot of the heating seen in the higher layers.\n\n\n\n\\section{Conclusions}\n\nTo our knowledge, the {\\sc Sunrise II} data presented in this paper have provided the most detailed observation of the small-scale emergence of magnetic flux in developing ARs to date, unveiling the topology of the full vector magnetic field at the photosphere and the response of the low chromospheric layers. Signatures of reconnection (magnetic flux cancellation and ensuing photospheric and chromospheric brightenings, temperature enhancements and possible bi-directional flows) take place around the U-loops where the emerging magnetic field remains trapped below the surface. This is a common occurrence during the formation phases of ARs that could account for a lot of the heating in and around the arch filament systems seen at higher layers.\n\n\n\n\\begin{acknowledgements}\nThe National Center for Atmospheric Research is sponsored by the National Science Foundation.The German contribution to \\textsc{Sunrise} and its reflight was funded by the Max Planck Foundation, the Strategic Innovations Fund of the President of the Max Planck Society (MPG), DLR, and private donations by supporting members of the Max Planck Society, which is gratefully acknowledged. The Spanish contribution was funded by the Ministerio de Econom\\'ia y Competitividad under Projects ESP2013-47349-C6 and ESP2014-56169-C6, partially using European FEDER funds. The HAO contribution was partly funded through NASA grant number NNX13AE95G. This work was partly supported by the BK21 plus program through the National Research Foundation (NRF) funded by the Ministry of Education of Korea. The HMI and AIA data used in this work are courtesy of NASA\/SDO and the HMI and AIA science teams. The National Solar Observatory (NSO) is operated by the Association of Universities for Research in Astronomy (AURA) Inc. under a cooperative agreement with the National Science Foundation. \n\n\\end{acknowledgements}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this note, we address the following question of Andr\\'es Sambarino and provide a positive answer when $d=4q+2$ for some $q \\in \\mathbb{N}$.\n\\medskip\n\n\\noindent \\textbf{Sambarino's Question:} \\emph{Suppose that $\\Gamma$ is a torsion free word hyperbolic group which admits a Borel Anosov representation into $\\mathsf{SL}(d,\\mathbb{R})$. Is $\\Gamma$ necessarily free or a surface group?.}\n\\medskip\n\n\\par Anosov representations of word hyperbolic groups into real semisimple Lie groups were introduced by Labourie \\cite{labourie-invent} in his study of the Hitchin component. They are discrete subgroups of real reductive Lie groups which generalize convex cocompact subgroups of rank one Lie groups. A representation $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(d,\\mathbb{R})$ is called \\emph{$P_{k}$-Anosov}, where $1\\leqslant k\\leqslant \\frac{d}{2}$, if it is Anosov with respect to the pair of opposite parabolic subgroups of $\\mathsf{GL}(d,\\mathbb{R})$ defined as the stabilizers of a $k$-plane and a complementary $(d-k)$-plane (see subsection \\ref{definition}). The representation $\\rho$ is called \\emph{Borel Anosov} if $\\rho$ is $P_k$-Anosov for every $k$. Labourie in \\cite{labourie-invent} proved that every Hitchin representation into $\\mathsf{PSL}(d,\\mathbb{R})$ is irreducible and admits a lift into $\\mathsf{GL}(d,\\mathbb{R})$ which is Borel Anosov. The only known examples of Borel Anosov representations are constructed from representations of free or surface groups. By a surface group we mean the fundamental group of a closed surface of negative Euler characteristic. Hitchin representations are the only known examples of Borel Anosov representations of surface groups in even dimensions. In all odd dimensions, Barbot's construction \\cite{Barbot} can be used to produce reducible examples.\n\\begin{comment} All known examples of Borel Anosov representations are constructed from representations of free or surface groups.\\end{comment}\n\nA positive answer to Sambarino's question was given in \\cite{CT} for $d=3$ or $4$. By using results of Benoist in \\cite{benoist-divisible0,benoist-divisible3}, we prove that a torsion free word hyperbolic group admitting a $P_{2q+1}$-Anosov representation into $\\mathsf{GL}(4q+2,\\mathbb{R})$ has to be either free or a surface group. Moreover, by using Wilton's result \\cite{Wilton} on the existence of quasiconvex surface groups or rigid subgroups in one ended-word hyperbolic groups and a theorem of Kapovich-Leeb-Porti in \\cite{KLP2} (see also \\cite[Theorem 6]{KLP3}), we prove the following stronger statement: \n\n\\begin{theorem} \\label{maintheorem} Let $\\Gamma$ be a word hyperbolic group and $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(4q+2, \\mathbb{R})$ a representation. Suppose that there exists a continuous, $\\rho$-equivariant dynamics preserving map $\\xi:\\partial_{\\infty}\\Gamma \\rightarrow \\mathsf{Gr}_{2q+1}(\\mathbb{R}^{4q+2})$. Then $\\Gamma$ is virtually free or virtually a surface group. \\end{theorem}\n\n\\noindent The group $\\Gamma$ is virtually free (resp. a surface group) if it contains a finite-index subgroup which is free (resp. a surface group). The map $\\xi$ is called dynamics preserving whenever $\\gamma \\in \\Gamma$ is an infinite order element, $\\rho(\\gamma)$ is $P_k$-proximal and $\\xi(\\gamma^{+})$ is its attracting fixed point in $\\mathsf{Gr}_{2q+1}(\\mathbb{R}^{4q+2})$. An analogue of Theorem \\ref{maintheorem} does not hold in dimensions which are multiples of $4$, see Section \\ref{examples}.\n\\medskip\n\n\\begin{corollary}\\label{cor1} Let $\\mathsf{G}_{4q+2}$ be either $\\mathsf{GL}(4q+2,\\mathbb{R})$ or $\\mathsf{PGL}(4q+2,\\mathbb{R})$. If $\\Gamma$ is a word hyperbolic group and $\\rho:\\Gamma \\rightarrow \\mathsf{G}_{4q+2}$ is a $P_{2q+1}$-Anosov representation, then $\\Gamma$ is virtually free or virtually a surface group. \\end{corollary}\n\nLet $\\tau^{+}_{k}:\\mathsf{Gr}_{k}(\\mathbb{R}^d) \\rightarrow \\mathbb{P}(\\wedge^{k}\\mathbb{R}^d)$ be the Pl${\\textup{\\\"u}}$cker embedding (see subsection \\ref{proximality}). By using the connectedness properties of the boundary of a rigid hyperbolic group with the methods of the proof of Theorem \\ref{maintheorem} we have:\n\n\\medskip\n\n\\begin{corollary} \\label{rigid} Let $\\Gamma$ be a torsion free rigid word hyperbolic group and $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(4q+2,\\mathbb{R})$ a representation. Suppose there exists a continuous $\\rho$-equivariant map $\\xi:\\partial_{\\infty}\\Gamma \\rightarrow \\mathsf{Gr}_{2q+1}(\\mathbb{R}^{4q+2})$. Then the map $\\xi$ is nowhere dynamics preserving and $\\tau_{2q+1}^{+}\\circ \\xi$ is not spanning. \\end{corollary}\n\n\\noindent The map $\\xi$ is called nowhere dynamics preserving if for every infinite order element $\\gamma \\in \\Gamma$, the restriction of $\\xi$ on $\\{\\gamma^{-},\\gamma^{+}\\}$ is not dynamics preserving.\n\n\\vspace{0.3cm}\n\n\\noindent \\textbf{Acknowledgements.} I would like to thank my advisor Richard Canary for his support and many useful comments on earlier versions of this paper and Andr\\'es Sambarino for his question. This work was partially supported by grants DMS-1564362 and DMS-1906441 from the National Science Foundation.\n\n\\section{Background}\nIn this section, we provide some background on proximality, define Anosov representations and state Benoist's results that we are going to use for the proof of the main theorem.\n\n\\subsection{Proximality.} \\label{proximality} Let $d \\geqslant 2$ and $e_1,..,e_d$ be the canonical basis of $\\mathbb{R}^d$. For an element $g \\in \\mathsf{GL}(d,\\mathbb{R})$ we denote by $\\lambda_{1}(g)\\geqslant \\lambda_{2}(g) \\geqslant ... \\geqslant \\lambda_{d}(g)$ the moduli of its eigenvalues. For $1 \\leqslant k \\leqslant \\frac{d}{2}$, we denote by $P_{k}$ the stabilizer of the plane $\\langle e_1,..,e_k \\rangle$ and by $P_{k}^{-}$ the stabilizer of the complementary $(d-k)$-plane $\\langle e_{k+1},...,e_d \\rangle$. The Grassmannian of $k$-planes, $\\mathsf{Gr}_{k}(\\mathbb{R}^d)$ is identified with the quotient manifold $\\mathsf{GL}(d,\\mathbb{R})\/P_{k}$. Similarly $ \\mathsf{Gr}_{d-k}(\\mathbb{R}^d)$ is identified with $\\mathsf{GL}(d,\\mathbb{R})\/P_{k}^{-}$. A pair of planes $(V^{+},V^{-}) \\in \\mathsf{Gr}_{k}(\\mathbb{R}^d) \\times \\mathsf{Gr}_{d-k}(\\mathbb{R}^d)$ is \\emph{transverse} if there exists $h \\in \\mathsf{GL}(d,\\mathbb{R})$ such that $V^{+}=h\\langle e_1,...,e_{k}\\rangle$ and $V^{-}=h\\langle e_{k+1},..,e_{d}\\rangle$. An element $g \\in \\mathsf{GL}(d, \\mathbb{R})$ is called $P_{k}$-\\emph{proximal} if $\\lambda_{k}(g)>\\lambda_{k+1}(g)$. Equivalently, $g$ has two fixed points $x_{g}^{+}\\in \\mathsf{Gr}_{k}(\\mathbb{R}^d)$ and \\hbox{$V_{g}^{-} \\in \\mathsf{Gr}_{d-k}(\\mathbb{R}^d)$} such that the pair $(x_{g}^{+},V_{g}^{-})$ is transverse and for every $k$-plane $V_{0}$ transverse to $V_{g}^{-}$, we have $\\lim_{n}g^nV_{0}=x_{g}^{+}$. The element $g$ is called $P_{k}$-\\emph{biproximal} if $g$ and $g^{-1}$ are $P_{k}$-proximal. We denote by $x_{g}^{-}$ the attracting fixed point of $g^{-1}$ in $ \\mathsf{Gr}_{k}(\\mathbb{R}^d)$.\nFor $k=1$, a $P_1$-proximal element $g \\in \\mathsf{GL}(d, \\mathbb{R})$ in $\\mathbb{P}(\\mathbb{R}^d)$ has a unique eigenvalue, $\\ell_1(g)$, of maximum modulus with multiplicity exactly one. The repelling hyperplane of $g$ is denoted by $V_{g}^{-}$. The matrix $g$ is called $P_1$-\\emph{positively proximal} if $\\ell_1(g)>0$.\n\\par The Pl${\\textup{\\\"u}}$cker embeddings $\\tau_{k}^{+}:\\mathsf{Gr}_{k}(\\mathbb{R}^d) \\rightarrow \\mathbb{P}(\\wedge^k \\mathbb{R}^d)$ and $\\tau_{k}^{-}:\\mathsf{Gr}_{d-k}(\\mathbb{R}^d) \\rightarrow \\mathsf{Gr}_{d_k-1}(\\wedge^{k}\\mathbb{R}^d)$, $d_k=\\binom{d}{k}$, where $$\\tau_{k}^{+}(gP_{k})=[ge_1\\wedge...\\wedge ge_k] \\ \\ \\ \\tau^{-}_{k}(gP_{k}^{-})=[\\wedge^{k}g (e_1 \\wedge ... \\wedge e_{k})^{\\perp}]$$ are embeddings and an element $g$ is $P_k$-proximal if and only if $\\tau_{k}^{+}(g)$ is $P_1$-proximal (see also \\cite[Proposition 3.3]{GGKW} for more details).\n\\medskip\n\nFrom now, unless specified, proximal (resp. positively proximal) will refer to $P_1$-proximality (resp. positive $P_1$-proximality) in the projective space.\n\n\\subsection{Dynamics preserving maps.} Let $\\Gamma$ be a word hyperbolic group and denote by $\\partial_{\\infty}\\Gamma$ its Gromov boundary. Every infinite order element $\\gamma \\in \\Gamma$ has exactly two fixed points $\\gamma^{+}$ and $\\gamma^{-}$ on $\\partial_{\\infty}\\Gamma$ called the attracting and repelling fixed points of $\\gamma$ respectively. Let $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ be a representation and $1 \\leqslant k \\leqslant d-1$. Suppose there exists a continuous $\\rho$-equivariant map $\\xi:\\partial_{\\infty}\\Gamma \\rightarrow \\mathsf{Gr}_{k}(\\mathbb{R}^d)$. The map $\\xi$ is called \\emph{dynamics preserving} if for every element $\\gamma \\in \\Gamma$ of infinite order, $\\rho(\\gamma)$ is $P_k$-proximal and $\\xi(\\gamma^{+})=x_{\\rho(\\gamma)}^{+}$. The map $\\xi$ is called \\emph{nowhere dynamics preserving} if for every $\\gamma \\in \\Gamma$ the restriction of $\\xi$ on $\\partial_{\\infty}\\langle \\gamma \\rangle=\\{\\gamma^{-},\\gamma^{+}\\}$ is not dynamics preserving.\n\\medskip\n\n\\subsection{Anosov representations.} \\label{definition} The dynamical definition of Anosov representations (see \\cite{GW, labourie-invent}) involves the geodesic flow of a word hyperbolic group. Characterizations of Anosov representations into real reductive Lie groups, without involving flow spaces, have been established in several papers, see \\cite{BPS, GGKW, KLP1, KP}. Here we define Anosov representations by using a characterization of Kapovich-Leeb-Porti in \\cite{KLP1} and Bochi-Potrie-Sambarino \\cite{BPS}. For a finitely generated group $\\Gamma$ we always fix a left-invariant word metric and for $\\gamma \\in \\Gamma$, $|\\gamma|_{\\Gamma}$ is the distance of $\\gamma$ from the identity element of $\\Gamma$. For an element $g \\in \\mathsf{GL}(d,\\mathbb{R})$ let $\\sigma_{1}(g) \\geqslant \\sigma_{2}(g) \\geqslant... \\geqslant \\sigma_{d}(g)$ be the singular values of $g$. Recall that for each $i$, $\\sigma_{i}(g)=\\sqrt{\\lambda_{i}(gg^t)}$, where $g^t$ is the transpose of $g$. Notice that for an element $[h] \\in \\mathsf{PGL}(d,\\mathbb{R})$ the ratio $\\frac{\\sigma_{i}(h)}{\\sigma_{i+1}(h)}$ does not depend on the choice of the representative $h \\in \\mathsf{GL}(d,\\mathbb{R})$.\n\\par Let $\\mathsf{G_d}$ be either $\\mathsf{GL}(d,\\mathbb{R})$ or $\\mathsf{PGL}(d,\\mathbb{R})$, $\\rho:\\Gamma \\rightarrow \\mathsf{G}_d$ a representation and $1 \\leqslant k \\leqslant \\frac{d}{2}$. Then $\\rho$ is $P_{k}$-Anosov if and only if there exist $C,\\alpha >0$ such that $$ \\frac{\\sigma_{k}(\\rho(\\gamma))}{\\sigma_{k+1}(\\rho(\\gamma))} \\geqslant Ce^{\\alpha |\\gamma|_{\\Gamma}}$$ for every $\\gamma \\in \\Gamma$.\n\\par It is clear from the previous definition that for every quasiconvex subgroup $H$ of $\\Gamma$ the restriction $\\rho|_{H}$ is $P_{k}$-Anosov. The following theorem summarizes some of the properties of Anosov representations. \n\\medskip\n\n\\begin{theorem}\\label{mainproperties} \\cite{GW,labourie-invent} Let $\\mathsf{G}_d$ be either $\\mathsf{GL}(d,\\mathbb{R})$ or $\\mathsf{PGL}(d,\\mathbb{R})$ and $\\Gamma$ be a word hyperbolic group. Suppose $1 \\leqslant k \\leqslant \\frac{d}{2}$ and $\\rho:\\Gamma \\rightarrow \\mathsf{G}_{d}$ is a $P_{k}$-Anosov representation. Then:\n\\medskip\n\n\\noindent \\textup{(i)} $\\rho$ is a quasi-isometric embedding, i.e. there exist constants $A,C>0$ such that for every $\\gamma \\in \\Gamma$ $$\\frac{1}{C}|\\gamma|_{\\Gamma}-A \\leqslant \\log \\frac{\\sigma_1(\\rho(\\gamma))}{\\sigma_{d}(\\rho(\\gamma))} \\leqslant C|\\gamma|_{\\Gamma}+A$$ \n\\medskip\n\n\\noindent \\textup{(ii)} There exist continuous $\\rho$-equivariant maps $$\\xi_{\\rho}^{k}:\\partial_{\\infty}\\Gamma \\rightarrow \\mathsf{Gr}_{k}(\\mathbb{R}^d) \\ \\ \\ \\xi_{\\rho}^{d-k}:\\partial_{\\infty}\\Gamma \\rightarrow \\mathsf{Gr}_{d-k}(\\mathbb{R}^d)$$ which are dynamics preserving and for distinct points $x,y \\in \\partial_{\\infty}\\Gamma$ the pair $(\\xi_{\\rho}^{k}(x),\\xi_{\\rho}^{d-k}(y))$ is transverse.\n\\medskip\n\n\\noindent \\textup{(iii)} The set of $P_{k}$-Anosov representations of $\\Gamma$ in $\\mathsf{G}_d$ is open in $\\textup{Hom}(\\Gamma,\\mathsf{G}_d)$.\n\\end{theorem}\n\\medskip\n\nNotice that by the previous definition, the representation $\\rho$ is $P_{k}$-Anosov if and only if $\\wedge^k \\rho$ is \\hbox{$P_{1}$-Anosov}. The Anosov limit maps of $\\wedge^k \\rho$ are $\\tau_{d,k}^{+}\\circ \\xi_{\\rho}^{k}$ and $\\tau_{d,k}^{-}\\circ \\xi_{\\rho}^{d-k}$.\n\\medskip\n\nWe also need the following fact which implies the continuity of the first eigenvalue among $P_1$-Anosov representations.\n\n\\begin{fact} \\label{continuity} Let $\\{A_{t}\\}_{t \\in [0,1]}$ be a continuous family of proximal elements of $\\mathsf{GL}(d,\\mathbb{R})$. Then, the function $t \\mapsto \\ell_1(A_t)$ is continuous.\\end{fact}\n\n\\begin{proof} \\begin{comment} Let $t_{0}\\in [0,1]$ and suppose that $(t_n)_{n \\in \\mathbb{N}}$ is a sequence converging to $t_0$. Then $\\lim_{n}\\rho_{t_n}(\\gamma)=\\rho_{t_0}(\\gamma)$. Suppose that $(t_{k_n})_{n \\in \\mathbb{N}}$ is a subsequence so that $\\lim_{n}\\ell_1(\\rho_{t_{k_n}}(\\gamma))=m_1 \\in \\mathbb{R}$. Up to further extracting we may assume that the other eigenvalues of $\\rho_{t_{k_n}}(\\gamma)$ converge to the $m_2,...,m_{d} \\in \\mathbb{C}$. We have that $|m_1|\\geqslant |m_2| \\geqslant ...\\geqslant |m_d|$ and the set of eigenvalues of $\\rho_{t_0}(\\gamma)$ is $\\{m_1,m_2,..,m_d\\}$ since the characteristic polynomial of $\\rho_{t_0}(\\gamma)$ is the limit of the characteristic polynomials of $\\rho_{t_{k_n}}(\\gamma)$. Since $\\rho_{t_0}(\\gamma)$ is a proximal matrix, it has a unique eigenvalue of maximum modulus $\\ell_1(\\rho_{t_0}(\\gamma))$ which should be $m_1$. Since $(t_n)_{n \\in \\mathbb{N}}$ was arbitrary, we deduce that $\\lim_{s \\rightarrow t_0}\\ell_1(\\rho_{s}(\\gamma))=\\ell_1(\\rho_{t_0}(\\gamma))$.\\end{comment}\n\nThe conclusion follows immediately from the continuity of the characteristic polynomial of matrices. \\end{proof}\n\n\\subsection{The work of Benoist.} We summarize here some results that we use from \\cite{benoist-divisible0} and \\cite{benoist-divisible3}. An open cone $C \\subset \\mathbb{R}^d$ is called \\emph{properly convex} if it does not contain an affine line. A domain $\\Omega \\subset \\mathbb{P}(\\mathbb{R}^d)$ is called \\emph{properly convex} if it is contained in some affine chart of $\\mathbb{P}(\\mathbb{R}^d)$ in which $\\Omega$ is bounded and convex. An element $g \\in \\mathsf{GL}(d, \\mathbb{R})$ is called positively semi-proximal if $\\lambda_1(g)$ is an eigenvalue of $g$. A subgroup $\\Gamma$ of $\\mathsf{GL}(d, \\mathbb{R})$ is called \\emph{positively proximal} if it contains a proximal element and every proximal element of $\\Gamma$ is positively proximal. \n\n\\begin{lemma}\\cite[Lemma 3.2]{benoist-divisible3} \\label{cone} Let $\\Gamma$ be a subgroup of $\\mathsf{GL}(d, \\mathbb{R})$ which preserves a properly convex open cone $C$ in $\\mathbb{R}^d$. Then every $\\gamma \\in \\Gamma$ is positively semi-proximal. In particular, every proximal element $\\gamma \\in \\Gamma$ is positively proximal. \\end{lemma}\n\nBenoist characterized irreducible subgroups of $\\mathsf{GL}(d,\\mathbb{R})$ which preserve a properly convex cone in $\\mathbb{R}^d$ as follows:\n\n\\begin{theorem}\\cite[Proposition 1.1]{benoist-divisible0} \\label{positivity3} Let $\\Gamma$ be an irreducible subgroup of $\\mathsf{GL}(d, \\mathbb{R})$. Then $\\Gamma$ preserves a properly convex open cone $C$ in $\\mathbb{R}^d$ if and only if $\\Gamma$ is positively proximal.\\end{theorem}\n\nWe also have the following fact for subgroups of $\\mathsf{GL}(d,\\mathbb{R})$ which preserve properly convex domains in $\\mathbb{P}(\\mathbb{R}^d)$:\n\n\\begin{fact} \\label{index2} \\normalfont Let $\\Gamma$ be a subgroup of $\\mathsf{GL}(d, \\mathbb{R})$ which preserves a properly convex domain $\\Omega \\subset \\mathbb{P}(\\mathbb{R}^d)$. There exists a representation $\\widetilde{\\iota}:\\Gamma \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ and a group homomorphism $\\varepsilon:\\Gamma \\rightarrow \\mathbb{Z}\/2$ such that: $\\widetilde{\\iota}(\\gamma)=(-1)^{\\varepsilon(\\gamma)}\\gamma$ for every $\\gamma \\in \\Gamma$ and $\\widetilde{\\iota}(\\Gamma)$ preserves a properly convex open cone $C$ lifting $\\Omega$. Thus, if $\\Gamma$ is also finitely generated the group $\\Gamma_2:=\\bigcap\\{H:[\\Gamma:H]\\leqslant2\\}$ has finite-index in $\\Gamma$ and preserves the properly convex cone $C$.\\end{fact}\n\n\nWe will also use the following fact:\n\n\\begin{proposition} \\label{discrete} Let $\\Gamma$ be a word hyperbolic group and $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(d,\\mathbb{R})$ be a representation. If there exists a continuous $\\rho$-equivariant non-constant map $\\xi:\\partial_{\\infty}\\Gamma \\rightarrow \\mathbb{P}(\\mathbb{R}^d)$, then $\\rho$ is discrete and $\\textup{ker}(\\rho)$ is finite. \\end{proposition}\n\n\\begin{proof}\nAssume that there exists an infinite sequence $(\\gamma_n)_{n \\in \\mathbb{N}}$ of elements of $\\Gamma$ with $\\lim_{n}\\rho(\\gamma_n)=I_{d}$. The group $\\Gamma$ acts on $\\partial_{\\infty}\\Gamma$ as a convergence group, hence up to subsequence, there exists $\\eta, \\eta' \\in \\partial_{\\infty}\\Gamma$ with $\\lim_{n}\\gamma_n x=\\eta$ for $x \\neq \\eta'$ and $\\xi(x)=\\xi(\\eta)$, $x \\neq \\eta'$. Since $\\partial_{\\infty}\\Gamma$ is perfect, $\\xi$ has to be constant, a contradiction. In particular, $\\textup{ker}(\\rho)$ is a torsion subgroup of $\\Gamma$, hence finite.\\end{proof}\n\nLet $F_k$ be the free group on $k$ generators. We close this section with the following proposition which follows by the work of Breuillard-Green-Guralnick-Tao (see \\cite[Theorem 4.1]{BGGT}):\n\n\\begin{proposition}\\cite{BGGT} \\label{density} The set of Zariski dense representations from $F_2$ in $\\mathsf{SL}(d, \\mathbb{R})$ is dense in the representation variety $\\textup{Hom}(F_k, \\mathsf{SL}(d, \\mathbb{R}))$. \\end{proposition}\n\n\\begin{comment} \\begin{proof} Notice that $\\textup{Hom}(F_k, \\mathsf{SL}(d, \\mathbb{R}))=\\mathsf{SL}(d, \\mathbb{R})\\times...\\times \\mathsf{SL}(d, \\mathbb{R})$ is a smooth manifold of dimension $k(d^2-1)$. By \\cite[Lemma 3.1]{Kim-Pansu}, the space of proper algebraic subgroups of $\\mathsf{SL}(d, \\mathbb{R})$ can be covered by countably many manifolds of dimension at most $d^4$. By continuing similarly as in \\cite[Corollary 3.2]{Kim-Pansu}, the dimension of the set $Q$ of non Zariski dense representations $F_k \\rightarrow \\mathsf{SL}(d, \\mathbb{R})$ is at most $d^4+k(d^2-2)$. Then, since $k(d^2-1)-d^4-k(d^2-2)=k-d^4>0$, the interior of $Q$ is empty and its complement dense in $\\textup{Hom}(F_k,\\mathsf{SL}(d,\\mathbb{R}))$. \\end{proof} \\end{comment}\n\n\\section {Proof of the main result}\nIn this section we give the proof of Theorem \\ref{maintheorem}. First, we need the following lemma which is proved using a theorem of Kapovich-Leeb-Porti \\cite{KLP2} (see also \\cite{CLS}).\n\n\\begin{lemma} \\label{free} Let $\\Gamma$ be a torsion free non-elementary word hyperbolic group and $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ be a representation which admits a continuous $\\rho$-equivariant map $\\xi:\\partial_{\\infty}\\Gamma \\rightarrow \\mathbb{P}(\\mathbb{R}^d)$. Suppose there exists $\\gamma \\in \\Gamma$ such that $\\rho(\\gamma)$ is biproximal, $\\xi(\\gamma^{+})=x_{\\rho(\\gamma)}^{+}$ and $\\xi(\\gamma^{-})=x_{\\rho(\\gamma)}^{-}$. Then, there exist $a,b \\in \\Gamma$ such that $\\langle a, b\\rangle$ is a free quasiconvex subgroup of $\\Gamma$ of rank $2$ and the restricted representation $\\rho: \\langle a,b \\rangle \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ is $P_1$-Anosov with Anosov limit map $\\xi$. \\end{lemma}\n\n\\begin{proof} By Proposition \\ref{discrete}, the representation $\\rho$ is discrete and faithful. Let $t \\in \\Gamma$ be an infinite order element such that $\\{\\gamma^{+},\\gamma^{-}\\}\\cap \\{t^{+},t^{-}\\}$ is empty. Up to conjugating $\\rho$ we may assume that $x_{\\rho(\\gamma)}^{+}=[e_1], x_{\\rho(\\gamma^{-1})}^{+}=[e_d]$ and $V_{\\rho(\\gamma)}^{-}=\\langle e_2,...,e_d \\rangle$, $V_{\\rho(\\gamma^{-1})}^{-}=\\langle e_1,...,e_{d-1} \\rangle$. Then we notice that $$\\rho(t^{\\pm1})x_{\\rho(\\gamma)}^{+} \\notin \\mathbb{P}(V_{\\rho(\\gamma)}^{-})\\cup \\mathbb{P}(V_{\\rho(\\gamma^{-1})}^{-}) \\ \\ \\ \\textup{and} \\ \\ \\ \\rho(t^{\\pm 1})x_{\\rho(\\gamma)}^{-} \\notin \\mathbb{P}(V_{\\rho(\\gamma)}^{-}) \\cup \\mathbb{P}(V_{\\rho(\\gamma^{-1})}^{-}) $$ For example, suppose that $\\rho(t)x_{\\rho(\\gamma)}^{+} \\in \\mathbb{P}(V_{\\rho(\\gamma)}^{-})$, then $\\lim_{n}\\rho(\\gamma^n)\\rho(t)x_{\\rho(\\gamma)}^{+}=\\lim_{n}\\xi(\\gamma^{n}t\\gamma^{+})=\\xi(\\gamma^{+})=[e_1]$ has to be in $\\mathbb{P}(V_{\\rho(\\gamma)}^{-})$, a contradiction. Since, $\\lim_{n}\\gamma^n t^{-1}\\gamma^{+}=\\gamma^{+}$ we have $\\lim_{n}\\rho(\\gamma^n t^{-1})\\xi(\\gamma^{+})=x_{\\rho(\\gamma)}^{+}$ and $\\rho(t^{-1})x_{\\rho(\\gamma^{-1})}^{+} \\notin \\mathbb{P}(V_{\\rho(\\gamma)}^{-})$. Then, by \\cite[Theorem 7.40]{KLP2} (see also \\cite[Theorem A2]{CLS}), there exists $N>0$ such that the group $H=\\langle \\gamma^N,t\\gamma^nt^{-1}\\rangle$ is a free group of rank $2$ and the restriction $\\rho|_{H}$ is $P_1$-Anosov. The restriction $\\rho|_{H}$ is also a quasi-isometric embedding hence $H$ is a quasiconvex subgroup of $\\Gamma$ and its Anosov limit map is the restriction of $\\xi$ on $\\partial_{\\infty}H$ considered as a subset of $\\partial_{\\infty}\\Gamma$. \\end{proof}\n\nRecall that for a finitely generated group $\\Gamma$, $\\Gamma_2$ is defined to be the intersection of all finite-index subgroups of $\\Gamma$ of index at most $2$. \n\n\\begin{lemma} \\label{positive1} Let $\\Gamma$ be a torsion free one-ended word hyperbolic group and $\\rho:\\Gamma \\ast \\mathbb{Z} \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ be a representation which admits a $\\rho$-equivariant continuous map $\\xi:\\partial_{\\infty}(\\Gamma \\ast \\mathbb{Z}) \\rightarrow \\mathbb{P}(\\mathbb{R}^d)$. Suppose that $\\delta \\in \\Gamma_2$ is a non-trivial element such that $\\rho(\\delta)$ is biproximal and $\\xi(\\delta^{+})=x_{\\rho(\\delta)}^{+}$ and $\\xi(\\delta^{-})=x_{\\rho(\\delta)}^{-}$. Then $\\rho(\\delta)$ is positively proximal. \\end{lemma}\n\n\\begin{proof} Let $s$ be a generator of the free cyclic factor, $t=s\\delta s^{-1} \\in \\Gamma$ and notice that $\\rho(t)$ is proximal with $\\rho(s)x_{\\rho(\\delta)}^{+}=x_{\\rho(t)}^{+}=\\xi(t^{+})$ and $t^{\\pm} \\notin \\partial_{\\infty}\\Gamma$. If $x \\in \\partial_{\\infty}\\Gamma$, $\\lim_{n}\\rho(t^n)\\xi(x)=\\lim_{n}\\xi(t^n x)=\\xi(t^{+})$. Since $\\rho(t)$ preserves $V_{\\rho(t)}^{-}$ and $\\lim_{n}t^{n}x=t^{+}$, $\\xi(x)$ cannot lie in $\\mathbb{P}(V_{\\rho(t)}^{-})$. It follows that $\\xi(\\partial_{\\infty}\\Gamma)$ lies in the affine chart $\\mathbb{P}(\\mathbb{R}^d)-\\mathbb{P}(V_{\\rho(t)}^{-})$. Let $V=\\langle \\xi(\\partial_{\\infty}\\Gamma) \\rangle$ and we consider the representation $\\rho':\\Gamma \\rightarrow \\mathsf{GL}(V)$ where $\\rho'(\\gamma)=\\rho|_{V}(\\gamma)$, $\\gamma \\in \\Gamma$. The map $\\xi$ is not constant, hence $\\rho'$ is discrete and faithful. The map \\hbox{$\\xi: \\partial_{\\infty}\\Gamma \\rightarrow \\mathbb{P}(V)$} is $\\rho'$-equivariant, $\\rho'(\\delta)$ is proximal with attracting fixed point $\\xi(\\delta^{+})$ and $\\ell_1(\\rho(\\delta))=\\ell_1(\\rho'(\\delta))$. \n\\par Then we notice that $\\xi(\\partial_{\\infty}\\Gamma)$ also lies in the affine chart $A=\\mathbb{P}(V)-\\mathbb{P}(V\\cap V_{\\rho(t)}^{-})$ of $\\mathbb{P}(V)$. Since $\\Gamma$ is one-ended, $\\partial_{\\infty}\\Gamma$ and $\\xi(\\partial_{\\infty}\\Gamma)$ are connected. The convex hull of $\\xi(\\partial_{\\infty}\\Gamma)$ in $A$, say $\\mathcal{C}$, is bounded and convex in $A$ and has non-empty interior since $\\xi(\\partial_{\\infty}\\Gamma)$ spans $V$. Then $\\rho'(\\Gamma)$ preserves $\\xi(\\partial_{\\infty}\\Gamma)$ and by \\cite[Proposition 2.8]{CT} it also preserves $\\mathcal{C}$. It follows that $\\rho'(\\Gamma)$ preserves the non-empty properly convex set $\\Omega=\\textup{Int}(\\mathcal{C}) \\subset \\mathbb{P}(V)$. Fact \\ref{index2} shows that there exists a representation $\\widetilde{\\rho}':\\Gamma \\rightarrow \\mathsf{GL}(V)$ which preserves a properly convex cone $C \\subset V$ and $\\rho'(\\gamma)=\\widetilde{\\rho}'(\\gamma)$ for every $\\gamma \\in \\Gamma_2$. By Lemma \\ref{cone}, $\\rho(\\delta)$ is positively proximal in $\\mathbb{P}(V)$ and hence in $\\mathbb{P}(\\mathbb{R}^d)$.\\end{proof}\n\nA torsion free word hyperbolic group $\\Gamma$ is called \\emph{rigid} if it does not admit a non-trivial splitting over a cyclic subgroup. For example, the fundamental group of a closed negatively curved Riemannian manifold of dimension at least $3$ is rigid. By a theorem of Bowditch \\cite{Bowditch} the Gromov boundary $\\partial_{\\infty}\\Gamma$ of a rigid hyperbolic group $\\Gamma$ does not contain local cut points. \n\n\\begin{lemma} \\label{positive2} Let $\\Gamma$ be a torsion free rigid one-ended word hyperbolic group. Let $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ be a representation which admits a continuous $\\rho$-equivariant map $\\xi:\\partial_{\\infty}\\Gamma \\rightarrow \\mathbb{P}(\\mathbb{R}^d)$. Suppose that $\\delta \\in \\Gamma_2$ is a non-trivial element such that $\\rho(\\delta)$ is biproximal and $\\xi(\\delta^{+})=x_{\\rho(\\delta)}^{+}$ and $\\xi(\\delta^{-})=x_{\\rho(\\delta)}^{-}$. Then $\\rho(\\delta)$ is positively proximal. \\end{lemma}\n\n\\begin{proof} Since $\\partial_{\\infty}\\Gamma$ does not have any local cut points, the set $\\partial_{\\infty}\\Gamma-\\{\\delta^{+},\\delta^{-}\\}$ is connected. For \\hbox{$x \\neq \\delta^{+},\\delta^{-}$} we have that $\\lim_{n}\\delta^{\\pm n} x=\\delta^{\\pm}$ and, as in Lemma \\ref{positive1}, the conected set $\\xi \\big(\\partial_{\\infty}\\Gamma-\\{\\delta^{+},\\delta^{-}\\}\\big)$ is contained in \\hbox{$\\mathbb{P}(\\mathbb{R}^d)-\\mathbb{P}(V_{\\rho(\\delta)}^{-})\\cup\\mathbb{P}(V_{\\rho(\\delta^{-1})}^{-})$}. Note that the two $(d-1)$-planes $V_{\\rho(\\delta)}^{-}$ and $V_{\\rho(\\delta^{-1})}^{-}$ are distinct, hence by the connectedness of $\\partial_{\\infty}\\Gamma-\\{\\delta^{+},\\delta^{-}\\}$ we can find a hyperplane $V_0$ such that $\\xi(\\partial_{\\infty}\\Gamma)$ is contained in $\\mathbb{P}(\\mathbb{R}^d)-\\mathbb{P}(V_0)$. Then we consider the restriction $\\rho':\\Gamma \\rightarrow \\mathsf{GL}(V)$, $V=\\langle \\xi(\\partial_{\\infty}\\Gamma)\\rangle$, whose image preserves the compact connected subset $\\xi(\\partial_{\\infty}\\Gamma)$ of the affine chart $\\mathbb{P}(V)-\\mathbb{P}(V\\cap V_0)$ of $\\mathbb{P}(V)$. The element $\\rho'(\\gamma)$ is proximal in $\\mathbb{P}(V)$ and $\\ell_1(\\rho(\\gamma))=\\ell_1(\\rho'(\\gamma))$. We similarly conclude that $\\rho'(\\Gamma)$ preserves a properly convex domain $\\Omega$ of $\\mathbb{P}(V)$. Again, Fact \\ref{index2} guarantees that $\\rho'(\\Gamma_2)$ preserves a properly convex cone of $V$ and $\\ell_1(\\rho'(\\delta))>0$. \\end{proof}\n\nNow we combine the previous results to prove Theorem \\ref{maintheorem}.\n\n\\medskip \\noindent {\\bf Theorem \\ref{maintheorem}:} {\\em Let $\\Gamma$ be a word hyperbolic group and $\\rho:\\Gamma \\rightarrow \\mathsf{GL}(4q+2, \\mathbb{R})$ a representation. Suppose that there exists a continuous, $\\rho$-equivariant dynamics preserving map $\\xi:\\partial_{\\infty}\\Gamma \\rightarrow \\mathsf{Gr}_{2q+1}(\\mathbb{R}^{4q+2})$. Then $\\Gamma$ is virtually free or virtually a surface group.}\n\n\\begin{proof} We first assume that $\\Gamma$ is a torsion free hyperbolic group. By Proposition \\ref{discrete}, $\\rho$ is faithful and we may assume that $\\rho(\\Gamma)$ is a subgroup of $\\mathsf{SL}(4q+2, \\mathbb{R})$. If not, we replace $\\rho$ with the representation $\\hat{\\rho}:\\Gamma \\rightarrow \\mathsf{SL}^{\\pm}(n,\\mathbb{R})$, $\\hat{\\rho}(\\gamma)=|\\textup{det}(\\rho(\\gamma))|^{-1\/(4q+2)}\\rho(\\gamma)$ and $\\Gamma$ with a finite-index subgroup $\\Gamma_0$ such that $\\hat{\\rho}(\\Gamma_0)$ is a subgroup of $\\mathsf{SL}(4q+2,\\mathbb{R})$. Notice that $\\hat{\\rho}$ has to be faithful since $\\xi$ is $\\hat{\\rho}$-equivariant and dynamics preserving for $\\hat{\\rho}$. \\par \nLet $V_q=\\wedge^{2q+1}\\mathbb{R}^{4q+2}$, and notice by assumption that $\\xi_{q}=\\tau_{2q+1}^{+}\\circ \\xi$ is $\\wedge^{2k+1}\\rho$-equivariant and dynamics preserving. We consider the following two cases:\n\n\\par \\emph{Case 1.} Suppose that $\\Gamma$ has infinitely many ends. Then we show that $\\Gamma$ is free. If not, by Stallings' theorem \\cite{Stallings}, there exists a splitting $\\Gamma=\\Gamma_1\\ast...\\ast\\Gamma_k \\ast F_{s}$, where $s \\geqslant 0$ and for $1 \\leqslant i \\leqslant k$, $\\Gamma_i$ is an one-ended word hyperbolic group. In particular, there exists a quasiconvex subgroup of $\\Gamma$ of the form $\\Delta \\ast \\mathbb{Z}$, with $\\Delta$ one-ended. Lemma \\ref{free}, shows that there exists a quasiconvex free subgroup $H_0$ of $\\Delta_2$ such that $\\wedge^{2q+1}\\rho(H_0)$ is $P_1$-Anosov in $\\mathsf{SL}(V_q)$ and its limit map is the restriction $\\xi_{q}:\\partial_{\\infty}H_0 \\rightarrow \\mathbb{P}(V_q)$. \\par Since $\\wedge^{2q+1}\\rho(\\delta)$ is proximal for every $\\delta \\in H_0 \\subset \\Delta_2$, by Lemma \\ref{positive1}, $\\ell_1(\\wedge^{2q+1}(\\rho(\\delta)))>0$. The representation $\\rho:H_0 \\rightarrow \\mathsf{SL}(4q+2,\\mathbb{R})$ is $P_{2q+1}$-Anosov and $\\wedge^{2q+1}\\rho(\\gamma)$ is positively proximal for every non-trivial $\\gamma \\in H_0$. By Theorem \\ref{mainproperties} \\textup{(iii)}, we can find a path connected open neighbourhood $U$ of $\\rho_{0}:=\\rho|_{H_{0}}$ in $\\textup{Hom}(H_0,\\mathsf{SL}(4q+2,\\mathbb{R}))$ consisting of entirely of $P_{2q+1}$-Anosov representations. Proposition \\ref{density} guarantees that there exists $\\rho_{1} \\in U$ such that $\\rho_{1}(F_k)$ is Zariski dense in $\\mathsf{SL}(4q+2,\\mathbb{R})$. Let $\\{\\rho_{t}\\}_{0\\leqslant t\\leqslant 1}$ be a continuous path between $\\rho_{0}$ and $\\rho_{1}$ contained entirely in $U$. By Fact \\ref{continuity}, for every $\\gamma \\in H_0$, the map $t \\mapsto \\ell_1(\\wedge^{2q+1}\\rho_{t}(\\gamma))$ is continuous with real values and nowhere vanishing. Hence $\\ell_1(\\wedge^{2q+1}\\rho_{1}(\\gamma))>0$ for every $\\gamma \\in H_0$. Therefore, since $\\wedge^{2k+1}$ is an irreducible representation, the group $\\wedge^{2q+1}\\rho_1(H_0)$ is a strongly irreducible subgroup of $\\mathsf{SL}(V_q)$ which is positively proximal. By Theorem \\ref{positivity3}, the group $\\wedge^{2q+1}\\rho_{1}(H_0)$ preserves a properly convex cone and hence a properly convex domain of $\\mathbb{P}(V_q)$. On the other hand, the group $\\wedge^{2q+1}\\mathsf{SL}(4q+2,\\mathbb{R})$ (and hence $\\wedge^{2q+1}\\rho_{1}(H_0)$) preserves the symplectic non-degenerate form $\\omega_{q}:V_q \\times V_q \\rightarrow \\mathbb{R}$ given by the formula $\\omega_{q}(a,b)=a\\wedge b \\in \\langle e_1 \\wedge...\\wedge e_{4q+2} \\rangle$. However, by \\cite[Corollary 3.5]{benoist-divisible0}, a strongly irreducible subgroup of $\\mathsf{SL}(d, \\mathbb{R})$ which preserves a symplectic form cannot preserve a properly convex domain of $\\mathbb{P}(\\mathbb{R}^d)$. We have reached a contradiction, so $\\Gamma$ cannot contain any non-trivial one-ended factors in its free product decomposition. Therefore, $\\Gamma$ is free.\n\n\\par \\emph{Case 2.} Suppose that $\\Gamma$ is one-ended and not virtually a surface group. Wilton's result \\cite[Corollary B]{Wilton} ensures that $\\Gamma$ contains a quasiconvex subgroup $\\Delta$ which is either isomorphic to a surface group or rigid. If $\\Delta$ has infinite index in $\\Gamma$, then there exists a quasiconvex subgroup of $\\Gamma$ isomorphic to $\\Delta \\ast \\mathbb{Z}$. However, by the previous case we obtain a contradiction. Therefore, we may assume that $\\Delta$ is rigid and has finite index in $\\Gamma$. By Lemma \\ref{free}, there exists $H_1$ a quasiconvex free subgroup of $\\Delta_2$ such that the restriction $\\wedge^{2q+1}\\rho|_{H_1}$ is $P_1$-Anosov. By Lemma \\ref{positive2}, for every $h \\in H_1$, $\\wedge^{2q+1}\\rho(h)$ is positively proximal in $\\mathbb{P}(V_q)$. By continuing as previously, we obtain a $P_{2q+1}$-Anosov, Zariski dense deformation $\\rho_1$ of $\\rho|_{H_1}$ such that $\\wedge^{2q+1}\\rho_1(H_{k})$ is positively proximal. Again, by Theorem \\ref{positivity3}, $\\wedge^{2q+1}\\rho_1(H_{k})$ preserves a properly convex domain and the symplectic form $\\omega_{q}$, a contradiction.\n\\par If $\\rho$ is not faithful, Proposition \\ref{discrete} shows that $\\textup{ker}(\\rho)$ is finite. The group $\\Gamma'=\\Gamma\/ \\textup{ker}\\rho$ is word hyperbolic, $\\partial_{\\infty}\\Gamma'=\\partial_{\\infty}\\Gamma$, so $\\xi$ is a $\\rho'$-equivariant dynamics preserving map, where $\\rho':\\Gamma' \\rightarrow \\mathsf{GL}(4q+2,\\mathbb{R})$ is the faithful representation induced by $\\rho$. By Selberg's lemma, there exists a torsion free finite-index subgroup $\\Gamma_1$ of $\\Gamma'$. The previous arguments imply that $\\Gamma_1$ is either a surface group or a free group. Therefore, $\\Gamma$ is either a finite extension of a virtually free group or a virtually surface group. In the second case, its boundary is the circle and by \\cite{Gabai}, $\\Gamma$ is virtually a surface group. In the first case, by \\cite{Dunwoody}, $\\Gamma$ has infinitely many ends and splits as the fundamental group of a finite graph of groups with finite edge groups and vertex groups of at most one end. The vertex groups of this splitting are also finite extensions of a virtually free group hence finite. It follows that $\\Gamma$ is virtually free.\\end{proof}\n\nBy following the argument of case 1 in of the proof of Theorem \\ref{maintheorem} we obtain the following conclusion:\n\n\\begin{theorem}\\label{positive} Let $F_2$ be the free group on two generators and $\\rho:F_2 \\rightarrow \\mathsf{GL}(4q+2,\\mathbb{R})$ a representation. Suppose that $\\rho$ is $P_{2q+1}$-Anosov. Then $\\wedge^{2q+1}\\rho(F_2)$ is not a positively proximal subgroup of $\\mathsf{GL}(\\wedge^{2q+1}\\mathbb{R}^{4q+2})$.\\end{theorem}\n\\medskip\n\nFor the proof of Corollary \\ref{cor1} we need the following proposition for the existence of lifts of \\hbox{$P_{2k+1}$-Anosov} representations into $\\mathsf{PGL}(d,\\mathbb{R})$. The proof is similar to Lemma \\ref{positive1} and \\ref{positive2}. In the case $\\rho$ is irreducible and $k=0$, Zimmer has proved the existence of lifts in \\cite[Theorem 3.1]{Zimmer}.\n\\medskip\n\n\\begin{proposition} Let $\\Gamma$ be a torsion free word hyperbolic group and $\\rho:\\Gamma \\rightarrow \\mathsf{PGL}(d, \\mathbb{R})$ is a $P_{2k+1}$-Anosov representation, where $0 \\leqslant k \\leqslant \\frac{d-1}{4}$. \n\\medskip\n\n\\noindent \\textup{(i)} Suppose that $\\Delta$ is an infinite index, one-ended quasiconvex subgroup of $\\Gamma$ and $\\rho_{0}$ is the restriction of $\\rho$ on $\\Delta$. There exists a lift $\\widetilde{\\rho_0}:\\Delta \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ such that $\\wedge^{2k+1}\\widetilde{\\rho_0}(\\Delta)$ is positively proximal.\\\\\n\n\n\\noindent \\textup{(ii)} If $\\Gamma$ is a rigid word hyperbolic group then there exists a lift $\\widetilde{\\rho}:\\Gamma \\rightarrow \\mathsf{GL}(d, \\mathbb{R})$ of $\\rho$ such that $\\wedge^{2k+1}\\rho(\\Gamma)$ is positively proximal.\\end{proposition}\n\\medskip\n\n\\begin{proof} We begin with the following observation: suppose that $\\varphi:\\Gamma \\rightarrow \\mathsf{PGL}(V_1\\oplus V_2)$ is a representation such that $\\varphi(\\gamma)$ preserves $V_1$ for every $\\gamma \\in \\Gamma$. If $\\rho(\\gamma)=[g_{\\gamma}]$ then the map $\\varphi_{0}(\\gamma)=[g_{\\gamma}|_{V_1}]$ is a well defined representation $\\varphi_{0}:\\Gamma \\rightarrow \\mathsf{PGL}(V_1)$. If $\\varphi_0$ admits a lift $\\widetilde{\\varphi}_{0}$, then there exists a lift $\\widetilde{\\varphi}$ of $\\varphi$ such that $\\widetilde{\\varphi}(\\gamma)|_{V_1}=\\widetilde{\\varphi}_{0}(\\gamma)$ for every $\\gamma \\in \\Gamma$. The lift $\\widetilde{\\varphi}$ is defined as follows: for $\\gamma \\in \\Gamma$, $\\widetilde{\\varphi}(\\gamma)$ is the unique element $h_{\\gamma} \\in \\mathsf{GL}(V_1\\oplus V_2)$ such that the restriction of $h_{\\gamma}$ on $V_1$ is $\\widetilde{\\varphi}_{0}(\\gamma)$ and $\\varphi(\\gamma)=[h_{\\gamma}]$. \n\\par Notice that we may asssume that $k=0$, because the exterior power $\\wedge^{2k+1}:\\mathsf{GL}(d,\\mathbb{R})\\rightarrow \\mathsf{GL}(\\wedge^{2k+1}\\mathbb{R}^{d})$ is faithful. For part \\textup{(i)}, we may consider $\\delta \\in \\Gamma$ with $\\delta^{\\pm} \\notin \\partial_{\\infty}\\Delta$ and $\\xi(\\partial_{\\infty}\\Delta)$ is a connected compact subset of the affine chart $\\mathbb{P}(\\mathbb{R}^{d})-\\mathbb{P}(V_{\\rho(\\delta)}^{-})$. In particular, $\\xi(\\partial_{\\infty}\\Delta)$ lies in the affine chart $A=\\mathbb{P}(V)-\\mathbb{P}(V\\cap V_{\\rho(\\delta)}^{-})$ of $\\mathbb{P}(V)$, where $V=\\langle \\xi(\\partial_{\\infty}\\Delta) \\rangle$. Since $\\rho_{0}(\\Delta)$ preserves $V$ there exists a well defined representation $\\rho_{1}:\\Delta \\rightarrow \\mathsf{PGL}(V)$. The image $\\rho_{1}(\\Delta)$ preserves the connected compact set $\\xi(\\partial_{\\infty}\\Delta)$ and hence the interior of the convex hull of $\\xi(\\partial_{\\infty}\\Delta)$ in $A$. There exists a lift $\\widetilde{\\rho_{1}}$ of $\\rho_{1}$ into $\\mathsf{GL}(V)$ such that $\\widetilde{\\rho_1}(\\Delta)$ preserves a properly convex cone $C$ of $V$. The representation $\\widetilde{\\rho_1}$ is $P_1$-Anosov, faithful and by Lemma \\ref{cone}, $\\widetilde{\\rho_1}(\\gamma)$ is positively proximal for every $\\gamma \\in \\Delta$ non-trivial. By our initial observation we obtain a lift $\\widetilde{\\rho_{0}}:\\Delta \\rightarrow \\mathsf{GL}(d,\\mathbb{R})$ of $\\rho_{0}$ with $\\widetilde{\\rho_{0}}(\\gamma)|_{V}=\\widetilde{\\rho_1}(\\gamma)$. The representation $\\widetilde{\\rho_1}$ is $P_{1}$-Anosov with Anosov limit map $\\xi$. For every non-trivial $\\gamma \\in \\Delta$, the attracting fixed point of $\\widetilde{\\rho_0}(\\gamma)$ is in $V$ and $\\ell_1(\\widetilde{\\rho_0}(\\gamma))=\\ell_1(\\widetilde{\\rho_1}(\\gamma))>0$.\n\\par The proof of \\textup{(ii)} follows by observing, as in Lemma \\ref{positive2}, that the image of $\\partial_{\\infty}\\Gamma$ under the Anosov limit map $\\xi$ lies in an affine chart of $\\mathbb{P}(\\mathbb{R}^d)$. Then we continue as previously to obtain the lift $\\widetilde{\\rho}$. \\end{proof}\n\\medskip\n\n\\noindent \\emph{Proof of Corollary \\ref{cor1}}. We assume that $\\Gamma$ is torsion free. If $\\Gamma$ contains a quasiconvex infinite index one-ended subgroup $\\Gamma_0$, there exists a lift $\\widetilde{\\rho}_0$ of $\\rho|_{\\Gamma_0}$ such that the group $\\wedge^{2k+1}\\widetilde{\\rho}_{0}(\\Gamma_0)$ is positively proximal, contradicting Theorem \\ref{positive}. Also $\\Gamma$ cannot be rigid again by part (ii) of the previous proposition. Therefore, $\\Gamma$ is either free or has one end and by \\cite[Corollary B]{Wilton} there exists a quasiconvex surface subgroup which has to be of finite index in $\\Gamma$. In every case, since $\\textup{ker}(\\rho)$ is finite, the boundary $\\partial_{\\infty}\\Gamma$ is either a circle or totally disconnected so $\\Gamma$ is virtually free or virtually a surface group. $ \\ \\ \\square$\n\\medskip\n\n\\noindent \\emph{Proof of Corollary \\ref{rigid}}. Suppose that there exists a continuous $\\rho$-equivariant map $\\xi$ and $\\rho(\\gamma) \\in \\rho(\\Gamma)$ a $P_{2q+1}$-proximal element with $\\xi(\\gamma^{+})=x_{\\rho(\\gamma)}^{+}$ and $\\xi(\\gamma^{-})=x_{\\rho(\\gamma)}^{-}$. The map $\\xi^{+}:=\\tau_{2q+1}^{+}\\circ \\xi$ is $\\wedge^{2q+1}\\rho$-equivariant and by Lemma \\ref{free} there exist a free quasiconvex subgroup $H$ of $\\Gamma_2$ such that $\\wedge^{2q+1}\\rho|_{H}$ is $P_1$-Anosov. Lemma \\ref{positive2} shows that $\\wedge^{2q+1}\\rho(H)$ is positively proximal, a contradiction by Theorem \\ref{positive}.\n\\par Let $V_q=\\wedge^{2q+1}\\mathbb{R}^{4q+2}$ and $\\xi^{-}=\\tau^{-}_{2q+1}\\circ \\xi$. We show that the map $\\xi^{+}$ cannot be spanning. Suppose that $\\xi^{+}$ is spanning and $x_1,...,x_r \\in \\partial_{\\infty}\\Gamma$ with $V_q=\\oplus_{i=1}^{r}\\xi^{+}(x_i)$, $r=\\textup{dim}(V_q)$. Since $\\Gamma$ acts minimally on $\\partial_{\\infty}\\Gamma$, every open subset $U$ of $\\partial_{\\infty}\\Gamma$, $\\xi^{+}(U)$ spans $V_q$ and the union $\\cup_{i=1}^{r}\\xi^{-}(x_i)$ cannot contain $\\xi^{+}(\\partial_{\\infty}\\Gamma)$. There exists $y \\in \\partial_{\\infty}\\Gamma$ and $1 \\leqslant j \\leqslant r$ with $V_q=\\xi^{+}(x_j)\\oplus \\xi^{-}(y)=\\xi^{+}(y)\\oplus \\xi^{-}(x_j)$. By the density of pairs $\\{ (\\delta^{+},\\delta^{-}): \\delta \\in \\Gamma\\}$ in the set of $2$-tuples of $\\partial_{\\infty}\\Gamma$, we can find $\\gamma \\in \\Gamma$ such that $V_q=\\xi(\\gamma^{+})\\oplus \\xi^{-}(\\gamma^{-})=\\xi^{+}(\\gamma^{-})\\oplus \\xi^{-}(\\gamma^{+})$. \n\\par Then we claim that $g=\\wedge^{2q+1}\\rho(\\gamma)$ is a biproximal matrix. Up to conjugating $g$ we may assume that $\\xi^{+}(\\gamma^{+})=[e_1]$, $\\xi^{-}(\\gamma^{-})=[e_{1}^{\\perp}]$ and write $g=\\begin{bmatrix}\na(g) & 0\\\\ \n0 & A\n\\end{bmatrix}$ for some matrix $A \\in \\mathsf{GL}(e_1^{\\perp})$. Suppose that $\\lambda_1(A)\\geqslant |a(g)|$. Let $p\\geqslant 1$ be the largest possible dimension of a complex Jordan block corresponding to an eigenvalue of maximum modulus of $A$. Then there exists a subsequence $(k_n)_{n \\in \\mathbb{N}}$, $A_{\\infty}$ a non-zero matrix and $b \\in \\mathbb{R}$ with $$\\lim_{n \\rightarrow \\infty}\\frac{1}{k_{n}^{p-1}\\lambda_{1}(A)^{k_n}}g^{k_n}=\\begin{bmatrix}\nb & 0\\\\ \n0 & A_{\\infty}\n\\end{bmatrix}$$ Since $\\partial_{\\infty}\\Gamma$ is perfect and $\\xi^{+}(\\partial_{\\infty}\\Gamma)$ spans $V_q$, we may choose $x \\in \\partial_{\\infty}\\Gamma-\\{ \\gamma^{-} \\}$ such that the projection of $\\xi^{+}(x)$ in $e_{1}^{\\perp}$ is not in $\\textup{ker}(A_{\\infty})$. Thus, $\\lim_{n} g^{k_n}\\xi^{+}(x)=\\lim_{n}\\xi^{+}(\\gamma^{k_n}x)=\\xi^{+}(\\gamma^{+})$ cannot be the line $[e_1]$, a contradiction. It follows that $|a(g)|>\\lambda_1(A)$ and $\\wedge^{2q+1}\\rho(\\gamma)$ is proximal with attracting fixed point $\\xi^{+}(\\gamma^{+})$. Since $V_q=\\xi^{+}(\\gamma^{-})\\oplus \\xi^{-}(\\gamma^{+})$, the same argument shows that $\\wedge^{2q+1}\\rho(\\gamma^{-1})$ is proximal with attracting fixed point $\\xi^{+}(\\gamma^{-})$. The map $\\xi^{+}$ (and hence $\\xi$) preserves the dynamics of $\\{\\gamma^{-},\\gamma^{+}\\}$. This contradicts the fact that $\\xi$ is nowhere dynamics preserving. Therefore, $\\tau_{2q+1}^{+}(\\xi(\\partial_{\\infty}\\Gamma))$ lies in some proper vector subspace of $V_q$. $\\ \\ \\square$\\\\\n\n\\section{Examples}\\label{examples}\nIn this section we provide an example showing that the analogue of Theorem \\ref{maintheorem} does not hold in dimensions which are multiples of $4$. Also, we give an example of a surface group representation $\\rho$ into $\\mathsf{SL}(4q+2,\\mathbb{R})$ which is not $P_{2q+1}$-Anosov but admits a $\\rho$-equivariant continuous dynamics preserving map $\\xi$ into $\\mathsf{Gr}_{2q+1}(\\mathbb{R}^{4q+2})$. Let $S$ be a closed orientable hyperbolic surface and $\\tau_{2}:\\mathsf{SL}(2,\\mathbb{C}) \\rightarrow \\mathsf{SL}(4,\\mathbb{R})$ be the standard inclusion defined as $\\tau_{2}(g)=\\begin{bmatrix}\n\\textup{Re}(g) &-\\textup{Im}(g) \\\\ \n\\textup{Im}(g) & \\textup{Re}(g)\n\\end{bmatrix}$ for $g \\in \\mathsf{SL}(2,\\mathbb{C})$.\n\n\\medskip\n\n\\begin{Example} \\normalfont Let $F_2$ be the free group on two generators. The group $\\Gamma=\\pi_1(S)\\ast F_{2}$ admits an Anosov representation $\\rho$ into $\\mathsf{SL}(2,\\mathbb{C})$ and hence $\\tau_2\\circ \\rho$ is a $P_{2}$-Anosov representation into $\\mathsf{SL}(4,\\mathbb{R})$. For $k \\in \\mathbb{N}$, the representation $\\rho_{k}=\\oplus_{i=1}^{k}(\\tau_{2}\\circ \\rho)$ of $\\Gamma$ into $\\mathsf{SL}(4k,\\mathbb{R})$ is $P_{2k}$-Anosov. In fact, by Theorem \\ref{mainproperties} (iii) and Proposition \\ref{density} there exists a deformation $\\rho_{k}'$ of $\\rho_{k}$ which is Zariski dense and $P_{2k}$-Anosov. \\end{Example}\n\\medskip\n\n\\begin{Example} \\normalfont Let $M$ be the mapping torus of the closed hyperbolic surface $S$ with respect to a fixed pseudo-Anosov homeomorphism $\\phi: S \\rightarrow S$. The group $\\pi_1(M)$ contains a normal infinite index subgroup $\\Gamma$ isomorphic with $\\pi_1(S)$. By a theorem of Thurston \\cite{Thurston} (see also Otal \\cite{Otal}), the group $\\pi_1(M)$ admits a convex cocompact representation $\\iota$ into $\\mathsf{PSL}(2,\\mathbb{C})$. In fact, by \\cite{Culler}, $\\iota$ lifts to a quasi-isometric embedding $\\widetilde{\\iota}:\\pi_1(M)\\rightarrow \\mathsf{SL}(2,\\mathbb{C})$. By composing $\\tau_2$ with $\\widetilde{\\iota}$, we obtain a $P_2$-Anosov representation $\\rho_1: \\pi_1(M) \\rightarrow \\mathsf{SL}(4,\\mathbb{R})$. The Cannon-Thurston map (see \\cite{CannonThurston}), $\\theta: \\partial_{\\infty}\\pi_1(S)\\rightarrow \\partial_{\\infty}\\pi_1(M)$ composed with the Anosov limit map $\\xi^{2}_{\\rho_1}:\\partial_{\\infty}\\pi_1(M) \\rightarrow \\mathsf{Gr}_{2}(\\mathbb{R}^4)$ provides a $\\rho_1|_{\\Gamma}$-equivariant dynamics preserving map $\\xi_0:\\partial_{\\infty}\\Gamma \\rightarrow \\mathsf{Gr}_{2}(\\mathbb{R}^4)$. Note that the representation $\\rho_1|_{\\Gamma}$ is not a quasi-isometric embedding, in particular not $P_2$-Anosov, since $\\Gamma$ is not a quasiconvex subgroup of $\\pi_1(M)$. Let $\\rho_{F}:\\Gamma \\rightarrow \\mathsf{SL}(2,\\mathbb{R})$ be a Fuchsian representation with limit map $\\xi_{\\rho_F}^{1}$. The representation $\\rho=(\\oplus_{i=1}^{q} \\rho_{1}|_{\\Gamma}) \\oplus \\rho_{F}$ into $\\mathsf{SL}(4q+2,\\mathbb{R})$ is not $P_{2q+1}$-Anosov, however the $\\rho$-equivariant map $\\xi=(\\oplus_{i=1}^{r}\\xi_0)\\oplus \\xi_{\\rho_F}^{1}$ is dynamics preserving. \\end{Example}\n\\vspace{0.4cm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results}\\label{sec_intro}\n\n Let $\\Omega\\subset\\mathbb{R}^N$ ($N\\geq 3$) be a bounded domain with a\n $C^{1,1}$ boundary $\\partial \\Omega$ and such that $0\\in \\Omega$. \nFor any $T>0$, consider the following parabolic equation with time-independent coefficients and homogeneous conormal Neumann boundary condition \n\\begin{equation}\\label{yu-6-24-1}\n\\begin{cases}\n\tu_t-\\mbox{div}(A(x)\\nabla u)+b(x)u=0&\\mbox{in}\\;\\;\\Omega\\times(0,T),\\\\\n\tA\\nabla u\\cdot\\nu=0\\;\\;&\\mbox{on}\\;\\;\\partial\\Omega\\times(0,T),\\\\\n\tu(\\cdot,0)=u_0\\in L^2(\\Omega),\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is the exterior unit normal vector on $\\partial\\Omega$, the symmetric matrix-valued function $A: \\overline\\Omega\\rightarrow \\mathbb{R}^{N\\times N}$ is Lipschitz continuous and satisfies the uniform ellipticity condition, i.e.,\n\t there is a constant $\\Lambda_1>1$ such that\n\\begin{equation}\\label{yu-11-28-2}\n\\begin{cases}\n\t|a_{ij}(x)-a_{ij}(y)|\\leq \\Lambda_1|x-y|\\;\\;\\mbox{for all}\\;\\;x,y\\in\\Omega\\;\\;\\mbox{and each}\\;\\;i,j=1,\\cdots, N,\\\\\n \\Lambda_1^{-1}|\\xi|^2\\leq A(x)\\xi\\cdot\\xi \\leq \\Lambda_1|\\xi|^2\\;\\;\\mbox{for a.e.}\\;x\\in\\Omega\\;\\;\\mbox{and all}\\;\\;\\xi\\in\\mathbb{R}^N,\n\\end{cases}\n\\end{equation}\nthe unbounded potential $b(\\cdot)$ verifies one of the following two assumptions: \n\\begin{equation}\\label{yu-6-24-1-1-b}\n\\begin{cases}\n(i)~~\\|b(\\cdot)\\|\n_{L^{N+\\delta}(\\Omega)}\\leq\\Lambda_2\\;\\;\\text{for some} \\;\\delta>0; \\\\\n(ii)~~|b(x)|\\leq \\dfrac{\\Lambda_2}{|x|}\\;\\;\\;\\;\\mbox{for a.e.}\\; x\\in\\Omega\n\\end{cases}\n\\end{equation}\nwith $\\Lambda_2>0$.\n\nThe first goal of the present paper is to establish a H\\\"older-type interpolation inequality at one time point for all solutions $u$ to \\eqref{yu-6-24-1}. Roughly speaking, for any $t>0$, there exist constants $C>0$ and $\\theta\\in (0,1)$ such that\n$$\n\\|u(\\cdot,t)\\|_{L^2(\\Omega)}\\leq C\\|u(\\cdot,t)\\|_{L^2(B_R(x_0)\\cap\\Omega)}^{\\theta}\\|u_0\\|_{L^2(\\Omega)}^{1-\\theta}\\quad\\text{for all}\\;\\;u_0\\in L^2(\\Omega).\n$$\n Such a kind of interpolation inequality have been established \nfor solutions of parabolic equations either in convex bounded domains or in bounded $C^2$-smooth domains but with homogeneous Dirichlet boundary conditions; See for instance \\cite{Bardos-Phung, Phung-2017, Phung-Wang-2010, Phung-Wang-2013, Phung-Wang-Zhang, Zhang-2017}. In these papers, the approach for \nthe desired interpolation inequality is mainly based on the parabolic-type Almgren frequency function method, which is essentially adapted from \\cite{Escauriaza-Fernandez-Vessella-2006,Poon}. \n\nThe second goal of this paper is to deduce an observability inequality from measurable sets in time.\nThis can be immediately obtained from the above-mentioned interpolation inequality combined with\nthe telescoping series method developed in \\cite{Apraiz-Escauriaza-Wang-Zhang, Phung-Wang-2013}.\n\n\nMore precisely, the main results of this paper can be stated as follows.\n\n\\begin{theorem}\\label{jiudu4}\nLet $T>0$ and $\\omega\\subset \\Omega$ be a non-empty open subset. Then there are constants $C=C(\\Lambda_1,\\Lambda_2,N,\\delta,\\Omega,\\omega)>0$ and $\\sigma=\\sigma(\\Lambda_1,\\Lambda_2,N,\\delta,\\Omega,\\omega)\\in(0,1)$ such that \n\tfor any solution $u$ of (\\ref{yu-6-24-1}) with the initial value $u_0\\in L^2(\\Omega)$,\n\t\\begin{equation}\\label{yu-7-12-1}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\Omega)}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\\|u(\\cdot,t_0)\\|_{L^2(\\omega)}^\\sigma\\|u_0\\|_{L^2(\\Omega)}^{1-\\sigma}\\;\\;\\mbox{for all}\\;\\;t_0\\in (0,T).\n\\end{equation}\n\\end{theorem}\n\\begin{remark}\n\tIn \\cite{Phung-Wang-2013}, the authors have obtained the global interpolation inequality (\\ref{yu-7-12-1}) for the heat equation with zero Dirichlet boundary condition and $L^\\infty(0,T;L^p(\\Omega))$ potential under the assumption $p>N$. \n\tThis is coincident with the assumption (i) in (\\ref{yu-6-24-1-1-b}). However, in view of the electric potential $O(|x|^{-1})$, one could see that the assumption $p>N$ is not optimal (see also \\cite{Zhang-2017}). \n\\end{remark}\n\n\n\n\\begin{theorem}\\label{yu-main-1}\nAssume $\\omega\\subset\\Omega$ is a non-empty open subset. Let $T>0$ and $E\\subset [0,T]$ be a subset of positive measure. Then there is a constant $C=C(\\Lambda_1,\\Lambda_2,N,\\delta,\\Omega,\\omega,T,E)>0$ such that for any solution $u$ of \n(\\ref{yu-6-24-1}) with the initial value $u_0\\in L^2(\\Omega)$,\n\\begin{equation}\\label{yu-6-24-2}\n\t\\|u(\\cdot,T)\\|_{L^2(\\Omega)}\\leq C\\int_E\\|u(\\cdot,t)\\|_{L^2(\\omega)}dt.\n\\end{equation}\n\tIn particular, when $E=[0,T]$, the constant $C$ in the above inequality can be taken the form \n\t$$C(\\Lambda_1,\\Lambda_2,N,\\delta,\\Omega,\\omega)e^{\\frac{C(\\Lambda_1,\\Lambda_2,N,\\delta,\\Omega,\\omega)(T^2+1)}{T}}.$$\n\\end{theorem}\n\\bigskip\nIt follows from the classical Hilbert uniqueness method (HUM) that (see, e.g., \\cite{Apraiz-Escauriaza-Wang-Zhang})\n\\begin{corollary}\nLet $T>0$. Assume $\\omega\\subset\\Omega$ is a nonempty open subset and $E\\subset [0,T]$ is a subset of positive measure. Then, for any $u_0\\in L^2(\\Omega)$, there is a control $f\\in L^\\infty(0,T;L^2(\\Omega))$, with \n$$\\|f\\|_{L^\\infty(0,T;L^2(\\Omega))}\\leq C\\|u_0\\|_{L^2(\\Omega)} \\quad\\text{for the same constant}\\;C\\;\\mbox{appeared in (\\ref{yu-6-24-2})},$$\nsuch that the solution of \n\\begin{equation*}\n\\begin{cases}\n\tu_t-\\mbox{div}(A(x)\\nabla u)+b(x)u=\\chi_{E\\times \\omega}f &\\mbox{in}\\;\\;\\Omega\\times(0,T),\\\\\n\tA\\nabla u\\cdot\\nu=0&\\mbox{on}\\;\\;\\partial\\Omega\\times(0,T),\\\\\n\tu(\\cdot,0)=u_0\\in L^2(\\Omega)\n\\end{cases}\n\\end{equation*}\nsatisfies $u(x,T)=0$ for a.e. $x\\in\\Omega$. \n\\end{corollary}\n\n\\par\n\n The interpolation inequality (\\ref{yu-7-12-1}) at one time point in Theorem \\ref{jiudu4} is a quantitative form \n of strong unique continuation for the equation (\\ref{yu-6-24-1}). The study of unique continuation\n property for parabolic equation has a long history. \n For the works in this topic, one can see \n \\cite{Adolfsson-Escauriaza, Alessandrini-Vessella, Canuto-Rosset-Vessella, Chen-1996, Escauriaza-2000, \n Escauriaza-Fernandez-Vessella-2006, Escauriaza-Montaner-Zhang, Escauriaza-Vessella-2003, Ito-Yamabe-1958, \n Lin-1990, Lin-1991, Poon, Vessella-2009, Yamabe-1959} and references therein. Among these papers, it is worth mentioning particularly \\cite{Lin-1990} and \n \\cite{Canuto-Rosset-Vessella}. In the paper \\cite{Lin-1990}, F. H. Lin showed the strong unique continuation property for\n the equation (\\ref{yu-6-24-1}) when the potential $b(\\cdot)\\in L_{loc}^{(N+1)\/2}(\\Omega)$. Although it is a qualitative\n form of unique continuation, F. H. Lin constructed an important and smart strategy that deduces a strong unique continuation of \n parabolic equations with time-independent coefficients to the elliptic counterparts. \n Later, by following and quantifying this strategy, \n B. Canuto, E. Rosset and S. Vessella proved in \\cite{Canuto-Rosset-Vessella} the \n local quantitative unique continuation for time-independent parabolic equations but without potentials (i.e., $b(\\cdot)=0$).\n It seems to us that the results in \\cite{Canuto-Rosset-Vessella} are not enough to derive\n the interpolation inequality in Theorem \\ref{jiudu4}; See more discussions in Remark \\ref{d1} below. Further, \n the presence of potential term will lead to some difficulties if one follows the same argument used in \\cite{Canuto-Rosset-Vessella}. These difficulties force us to slightly improve the strategy used by B. Canuto, E. Rosset and S. Vessella (see Section \\ref{dujin1} below). \n \n \nWhen the boundary condition in \\eqref{yu-6-24-1} is homogeneous Dirichlet-type, through using the frequency function method, the global interpolation inequality in Theorem \\ref{jiudu4} has been studied in \\cite{Bardos-Phung, Phung-2017, Phung-Wang-2010, Phung-Wang-2013, Phung-Wang-Zhang, Zhang-2017}. However,\nto the best of our knowledge, this approach seems to be not applicable for the case of homogeneous Neumann boundary condition (at least we do not know). This forces us to find a new method to obtain the corresponding interpolation inequality. \n\n\n\n In order to overcome these difficulties mentioned above, in this paper we shall adopt and slightly modify the reduction method, as well as Carleman estimates of elliptic operators. Roughly speaking, the reduction method \\cite{Lin-1990} is to \n reduce \na parabolic equation into an elliptic equation by using the Fourier transformation and adding one more spatial variable. However, \n because of the appearance of potential term, we shall adopt a sinh-type weighted Fourier transformation, which is slightly different to \n the strategy used in \\cite{Lin-1990,Canuto-Rosset-Vessella}. \n Moreover, \n for the proof of stability estimate (see Lemma \\ref{yu-proposition-7-1-1} below), \n the authors of \\cite{Canuto-Rosset-Vessella} reduced the elliptic equation to a hyperbolic equation and used harmonic measure. This strategy, in our opinion, cannot be applied when the potential is nonzero. Instead, in this paper we shall use suitable Carleman estimates\n to deduce the corresponding stability estimate. \n Note that the reduction method is based on a representation formula for solutions of parabolic equations in terms \nof eigenfunctions of the corresponding elliptic operators, and therefore cannot be applied to \ngeneral parabolic equations with time-dependent coefficients. \n\nWe emphasis that in the case of heat equation with homogeneous Dirichlet boundary conditions, the authors in \\cite{Apraiz-Escauriaza-Wang-Zhang} first observed that the observability estimate at one time point is in fact equivalent to \n a type of spectral inequality in \\cite{Lebeau-Robbiano-1998} (see also \\cite{Phung-2017}). This type of spectral inequality, roughly speaking, is an observability inequality from a partial region on the finite sum of eigenfunctions of the principal elliptic operator. \n For related works, we refer the reader to \\cite{Chaves-Silva-Lebeau-2016, Rousseau-Robbiano-2012, Lebeau-Zuazua, LU-2013, Miller} and references therein. \nTherefore, if one could establish a type of spectral inequality as in \\cite{Lebeau-Robbiano-1998} (see also \\cite{Phung-2017}), the global interpolation inequality can also be deduced by the technique utilized in \n \\cite{Apraiz-Escauriaza-Wang-Zhang}. We refer \\cite{LU-2013} for the spectral inequality of elliptic equation with Neumann boundary condition and without any potential term. \n \nMeanwhile, we also refer \\cite{Escauriaza-Fernandez-Vessella-2006,Vessella-2009} for quantitative estimates of unique continuation\nof parabolic equations with time dependent coefficients, in which some parabolic-type Carleman estimates were established. \nWe believe that the Carleman method developed in \\cite{Vessella-2009} (or \\cite{Escauriaza-Fernandez-Vessella-2006}) may provide a possible approach for proving the corresponding interpolation inequality.\nHowever, this issue escapes the study of the present paper and is deserved to be investigated in the continued work. \n\n\n\n\nLast but not least, we would like to stress that \nthe observability estimate from measurable sets in the time variable established in Theorem \\ref{yu-main-1} has several \n applications in control theory. \n In particular, it\n implies bang-bang properties of minimal norm and minimal time optimal control problems (see for instance \\cite{Phung-Wang-2013,wang-zhang1}).\n\n\n\n\n\n \\smallskip\n \n \n\n \n \n \n The structure of this paper is organized as follows. \n In Section \\ref{mainproof}, we first present two quantitative estimates of unique continuation needed for proof of the main results, and then we prove Theorems \\ref{jiudu4} and \\ref{yu-main-1}, respectively. In Section \\ref{dujin1}, we are devoted to the proofs of the above-mentioned two quantitative estimates of unique continuation. In Appendix, the proofs of some results used in Section \\ref{dujin1} are given. \n\n\n\t\n\n\n\\paragraph{Notation.}Throughout the paper,\n$\\triangle_R(x_0)$ stands for a ball in $\\mathbb R^N$ with the center $x_0$ and of radius $R>0$,\n$B_R(x_0,0)$ stands for a ball in $\\mathbb R^{N+1}$ with the center $(x_0,0)$ and of radius $R>0$. \nDenote by $\\partial\\triangle_R(x_0)$ the boundary of $\\triangle_R(x_0)$, by $\\rho_0=\\sup\\{|x-y|:x,y\\in\\Omega\\}$ and $\\Omega_\\rho=\\{x\\in\\Omega\\,:\\,d(x,\\partial\\Omega)\\geq\\rho$\\} with $\\rho\\in(0,\\min\\{1,\\rho_0\\})$.\nWrite $\\bar z$ for the complex conjugate of a complex number $z\\in\\mathbb C$.\nThe letter $C$ denotes a generic positive\nconstant that depends on the a priori data but not on the solution and may vary from line to line.\nMoreover, we shall denote by $C(\\cdot)$ a positive constant if we need to emphasize the dependence on some parameters in the brackets. \n\n\n\n\n\n\t \n\t \n\n\n\\section{Proofs of main results}\\label{mainproof}\n\n\\subsection{Unique continuation estimates}\nIn order to present the proof of Theorem \\ref{jiudu4},\nwe first state two results concerning quantitative estimates of unique continuation: The first one is local, and the second one is global. Their proofs are postponed to give in Section \\ref{dujin1}.\n\n\\begin{proposition}\\label{yu-theorem-7-5-1}\nLet $T>0$. Suppose $\\rho\\in(0,\\min\\{1,\\rho_0\\})$ such that $\\Omega_\\rho\\neq \\emptyset$. Then there exist $R\\in(0,\\rho)$ and $\\kappa\\in(0,1\/2)$ such that for any $r\\in (0, \\kappa R)$, any $t_0\\in(0,T\/2)$ and any $x_0\\in\\Omega_\\rho$, we have\n\\begin{equation*}\\label{yu-7-5-12}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{2r}(x_0))}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{r}(x_0))}^{\\sigma}\\left(\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|_{H^1(\\triangle_R(x_0))}\\right)^{1-\\sigma}\n\\end{equation*}\nwith some constants $C=C(\\Lambda_1,\\Lambda_2, \\Lambda_3, N,\\delta,r,R)>0$ and $\\sigma=\\sigma(\\Lambda_1,\\Lambda_2,\\Lambda_3,N,\\delta,r)\\in(0,1)$, where $u\\in C([0,2T];H_{\\text{loc}}^1(\\Omega))$ satisfies \n\\begin{equation}\\label{heat capacity}\nl(x)\\partial_tu-\\mbox{div}(A(x)\\nabla u)+b(x)u=0\\;\\;\\;\\mbox{in}\\;\\;\\Omega\\times(0,2T).\n\\end{equation}\t\nHere $A$ and $b$ are the same as in \\eqref{yu-6-24-1}, and $l:\\Omega\\rightarrow\\mathbb R^+$ verifies \n\\begin{equation}\\label{yu-7-29-3}\n\\Lambda_3^{-1}\\leq l(x)\\leq \\Lambda_3,\\;\\;\n\t|l(x)-l(y)|\\leq \\Lambda_3|x-y|\n\\quad \\;\\;\\mbox{for a.e.}\\;\\;x,y\\in\\Omega\n\\end{equation}\nwith a constant $\\Lambda_3>1$. \n\\end{proposition} \n\n\\begin{proposition}\\label{yu-theorem-7-10-6}\nLet $T>0$ and $\\omega\\subset\\Omega$ be a non-empty open subset. \nThen there are constants $C=C(\\Lambda_1,\\Lambda_2,N,\\delta,\\Omega,\\omega)>0$ and $\\sigma=\\sigma(\\Lambda_1,\\Lambda_2,N,\\delta,\\Omega,\\omega)\\in(0,1)$ such that for any solution $u\\in C([0,T];H^1(\\Omega))$ of \\eqref{yu-6-24-1}, we have\n\\begin{equation*}\\label{yu-7-10-2}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\Omega)}\\leq C\n\te^{\\frac{C(T^2+1)}{t_0}}\\|u(\\cdot,t_0)\\|_{L^2(\\omega)}^\\sigma \n\t\\left(\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|_{H^1(\\Omega)}\\right)^{1-\\sigma}\\quad \\text{for all}\\;\\; t_0\\in(0,T\/2).\n\\end{equation*}\n\\end{proposition}\n\n\n\n\n\\begin{remark}\\label{d1}\n\tThe local interpolation inequality established in Proposition \\ref{yu-theorem-7-5-1}\n\tis slightly different from the two spheres and one cylinder inequality established in \\cite[Theorem 3.1.1']{Canuto-Rosset-Vessella}.\nActually, in \\cite{Canuto-Rosset-Vessella}, the bound for the parameter $r$ depends \n\ton the instant $t_0$. This, however, will lead some difficulties when one applies it to prove the global interpolation and \n\t observability inequalities. \n\t\\end{remark}\n\n\\begin{remark}\nEquations of type \\eqref{heat capacity} appear when one transforms the parabolic operator via a linear \nmapping from $\\mathbb R^N$ into $\\mathbb R^N$.\nIt is also worth mentioning that parabolic equations of form \\eqref{heat capacity} with positive coefficients in front of the time derivative are much more nature from the physical point of view. \nThey model the heat diffusion of the temperature in a non-isotropic and non-homogeneous material.\nIn fact, there are two relevant physical quantities in heat diffusion processes: the conductivity coefficients and the specific heat capacity. The latter appears in the equation in front of the time derivative.\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{jiudu4}}\n\nWe first recall the following well-known Hardy inequality (see e.g. \\cite{Davies}) and Sobolev interpolation theorem (see e.g. \\cite[Theorem 5.8]{Adams-Fournier}), which will be used frequently in our argument below.\n\\begin{lemma}\\label{hardy}\nLet $\\Omega$ be a bounded Lipschitz domain in $\\mathbb R^N$ ($N\\geq 3$). Then, it holds that\n\\vskip 5pt \n (i) (Hardy's inequality) \n$$\n\\int_{\\Omega}|x|^{-2}|f|^2dx\\leq \n\\frac{4}{(N-2)^2}\\int_{\\Omega}|\\nabla f|^2dx\\quad \\mbox{for any}\\;\\;\nf\\in H_0^1(\\Omega).\n$$\n\\vskip 5pt\n\t(ii) (Sobolev's interpolation theorem) For each $p\\in [2,\\frac{2N}{N-2}]$, there is a constant $\\Gamma_1(\\Omega, N, p)>0$ such that \n$$\n\t\\|f\\|_{L^{p}(\\Omega)}\\leq \\Gamma_1(\\Omega, N, p)\\|f\\|_{H^1(\\Omega)}^\\theta\\|f\\|_{L^2(\\Omega)}^{1-\\theta}\\;\\;\\mbox{for any}\\;\\;\nf\\in H^1(\\Omega).\n$$\n\twhere $\\theta=N(\\frac{1}{2}-\\frac{1}{p})$.\n\\end{lemma}\nAs a simple consequence of the above Sobolev interpolation theorem, we have \n\\begin{corollary}\\label{yu-corollary-1}\n\tLet $\\Omega$ be a bounded Lipschitz domain in $\\mathbb R^N$ ($N\\geq 3$). For each case of \n\t(\\ref{yu-6-24-1-1-b}) and for every $\\epsilon\\in\\left(0,\\frac{1}{2}\\right]$, it holds that \n\\begin{equation*}\\label{yu-9-26-1}\n\tb(\\cdot)\\in L^{\\frac{N}{2}+\\epsilon}(\\Omega).\n\\end{equation*}\nFurther, for each $\\eta>0$ there is a constant $\\Gamma_2(\\Omega, N, \\eta)>0$ such that, for any $h(\\cdot)\\in L^{\\frac{N}{2}+\\eta}(\\Omega)$ \n\t and $f(\\cdot)\\in H^1(\\Omega)$,\n\\begin{eqnarray}\\label{yu-9-26-2}\n\t\\int_{\\Omega}|h||f|^2dx\n\n\t\\leq \\Gamma_2(\\Omega, N,\\eta)\\|h\\|_{L^{\\frac{N}{2}+\\eta}(\\Omega)}\\|f\\|_{L^2(\\Omega)}^{\\frac{4\\eta}{N+2\\eta}}\\|f\\|_{H^1(\\Omega)}^{\\frac{2N}{N+2\\eta}}.\n\\end{eqnarray}\n\\end{corollary}\n\n\\medskip\n\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{jiudu4}}]\tThe proof is divided into two steps. \n\\vskip 5pt\n\t\\textbf{Step 1. Energy estimates.} \n\tIn this step, we shall prove the following two claims:\n\\begin{itemize}\n \\item If $u(\\cdot,0)\\in L^2(\\Omega)$, then for each $t\\in[0,6T]$ we have\n\\begin{equation}\\label{yu-7-12-2-b}\n\t\\|u(\\cdot,t)\\|_{L^2(\\Omega)}\\leq e^{Ct}\\|u(\\cdot,0)\\|_{L^2(\\Omega)}\n\\end{equation}\n\tand\n\\begin{equation}\\label{yu-7-12-2}\n\t\\|u(\\cdot,t)\\|_{H^1(\\Omega)}\\leq \\frac{Ce^{Ct}}{\\sqrt{t}}\\|u(\\cdot,0)\\|_{L^2(\\Omega)}.\n\t\\end{equation}\n \\item If $u(\\cdot,0)\\in H^1(\\Omega)$, then we have\n \\begin{equation}\\label{yu-7-12-3}\n \t\\|u(\\cdot,t)\\|_{H^1(\\Omega)}\\leq e^{Ct}\\|u(\\cdot,0)\\|_{H^1(\\Omega)}\\;\\;\\mbox{for each}\\;\\;\n\tt\\in[0,6T].\n \\end{equation}\n \\end{itemize}\nIndeed, multiplying the first equation \n\t in (\\ref{yu-6-24-1}) by $u$ and then integrating by parts over $\\Omega$, we get \n\\begin{equation}\\label{yu-7-15-1}\n\t\\frac{1}{2}\\frac{d}{dt}\\int_{\\Omega}|u|^2dx\n\t+\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx=-\\int_{\\Omega}b|u|^2dx.\n\\end{equation}\nNote that, from (\\ref{yu-9-26-2}) (by letting $\\eta=\\frac{1}{2}$ there) and (\\ref{yu-11-28-2}), we have \n\\begin{eqnarray*}\\label{yu-7-24-1}\n\t-\\int_\\Omega b|u|^2dx&\\leq& C\\|b\\|_{L^{\\frac{N+1}{2}}(\\Omega)}\\|u\\|^{\\frac{2}{N+1}}_{L^2(\\Omega)}\n\t\\|u\\|^{\\frac{2N}{N+1}}_{H^1(\\Omega)}\\leq C\\Lambda_2\\|u\\|_{L^2(\\Omega)}^{\\frac{2}{N+1}}\\|u\\|_{H^1(\\Omega)}^{\\frac{2N}{N+1}}\\nonumber\\\\\n\t&\\leq& C\\epsilon^{-N}\\Lambda_2^{N+1}\\|u\\|_{L^2(\\Omega)}^2+\\epsilon\\|u\\|_{H^1(\\Omega)}^2\n\t=(C\\epsilon^{-N}\\Lambda_2^{N+1}+\\epsilon)\\|u\\|^2_{L^2(\\Omega)}+\\epsilon\\|\\nabla u\\|^2_{L^2(\\Omega)}\\nonumber\\\\\n\t&\\leq&(C\\epsilon^{-N}\\Lambda_2^{N+1}+\\epsilon)\\|u\\|^2_{L^2(\\Omega)}+\\epsilon\\Lambda_1\\int_\\Omega\\nabla u\\cdot(A\\nabla u)dx.\n\\end{eqnarray*}\n\tTaking $\\epsilon=\\frac{1}{2\\Lambda_1}$ in the above inequality, we obtain that \n\n\\begin{equation*}\\label{yu-7-24-2}\n\t-\\int_{\\Omega}b|u|^2dx\\leq \n\t \\left[C\\Lambda_1^N\\Lambda_2^{N+1}+\\frac{1}{2\\Lambda_1}\\right]\n\t\\int_{\\Omega}|u|^2dx+\\frac{1}{2}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx.\n\\end{equation*}\n\tThis, along with (\\ref{yu-7-15-1}), yields\n\\begin{equation}\\label{yu-7-15-2}\n\t\\frac{1}{2}\\frac{d}{dt}\\int_{\\Omega}|u|^2dx\n\t+\\frac{1}{2}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx\\leq \\left[C\\Lambda_1^N\\Lambda_2^{N+1}\n\t+\\frac{1}{2\\Lambda_1}\\right]\\int_{\\Omega}|u|^2dx.\n\\end{equation}\n\tThen \n\\begin{equation}\\label{yu-7-15-3}\n\t\\frac{d}{dt}\\left(e^{-\\left[C\\Lambda_1^N\\Lambda_2^{N+1}\n\t+\\Lambda_1^{-1}\\right]t}\\int_{\\Omega}|u|^2dx)\\right)\\leq 0.\n\\end{equation}\n\tThis gives (\\ref{yu-7-12-2-b}). Moreover, by (\\ref{yu-7-15-2}) and (\\ref{yu-7-15-3}), we obtain \n\\begin{eqnarray}\\label{yu-7-15-4}\n\t&\\;&\\int_0^t\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dxds\\nonumber\\\\\n\t&\\leq&\n\t\\left[t\\left(C\\Lambda_1^N\\Lambda_2^{N+1}+\\Lambda_1^{-1}\\right)e^{(C\\Lambda_1^N\\Lambda_2^{N+1}+\\Lambda_1^{-1})t}+1\n\t\\right]\n\t\\|u(\\cdot, 0)\\|^2_{L^2(\\Omega)}. \n\\end{eqnarray}\n\n\tNext, we show (\\ref{yu-7-12-3}). Here we divide our proof into two cases based on the assumptions in (\\ref{yu-6-24-1-1-b}). \n\\vskip 5pt\n\t\\emph{Case I. $|b(x)|\\leq \\frac{\\Lambda_2}{|x|}$ a.e. $x\\in\\Omega$.}\n\tIn this case, we take $r_0\\in(0,d(0,\\partial\\Omega))$ and $\\eta\\in C^\\infty(\\mathbb{R}^N;[0,1])$ such that \n\\begin{equation*}\\label{yu-7-24-4}\n\\begin{cases}\n\t\\overline{\\triangle_{r_0}(0)}\\subset\\Omega,\\\\\n\t\\eta=1&\\mbox{in}\\;\\;\\triangle_{\\frac{r_0}{2}}(0),\\\\\n\t\\eta=0&\\mbox{in}\\;\\;\\mathbb{R}^N\\backslash\\triangle_{r_0}(0)\\\\\n\t|\\nabla\\eta |\\leq \\frac{C}{r_0}&\\mbox{in}\\;\\;\\mathbb{R}^N. \n\\end{cases}\n\\end{equation*}\n Multiplying the first equation of (\\ref{yu-6-24-1}) by $-\\mbox{div}(A\\nabla u)\\eta^2$\n\t and integrating by parts over \n\t$\\Omega$, by Lemma \\ref{hardy} we have\n\\begin{eqnarray}\\label{yu-7-24-5}\n\t&\\;&\\frac{1}{2}\\frac{d}{dt}\\int_{\\Omega}\\nabla u\\cdot (A\\nabla u)\\eta^2dx\n\t+\\int_{\\Omega}|\\mbox{div}(A\\nabla u)|^2\\eta^2dx\\nonumber\\\\\n\t&\\leq&2\\int_{\\Omega}|\\mbox{div}(A\\nabla u)||(A\\nabla u)\\cdot \\nabla\\eta|\\eta dx\n\t+2\\int_{\\Omega}|b||u||(A\\nabla u)\\cdot\\nabla\\eta|\\eta dx\n\t+\\int_{\\Omega}|b||u|\\eta^2|\\mbox{div}(A\\nabla u)|dx\\nonumber\\\\\n\t&\\leq&\\frac{1}{2}\\int_{\\Omega}|\\mbox{div}(A\\nabla u)|^2\\eta^2dx+5\\int_{\\Omega}|(A\\nabla u)\\cdot\\nabla \\eta|^2dx\n\t+2\\int_{\\Omega}|b|^2|u|^2\\eta^2dx\\nonumber\\\\\n\t&\\leq&\\frac{1}{2}\\int_{\\Omega}|\\mbox{div}(A\\nabla u)|^2\\eta^2dx\n\t+\\frac{5C\\Lambda_1}{r_0^2}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx\n\t+\\frac{16\\Lambda_1\\Lambda_2^2}{(N-2)^2}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)\\eta^2dx\\nonumber\\\\\n\t&\\;&+\\frac{16C\\Lambda_2^2}{(N-2)^2r_0^2}\\int_{\\Omega}|u|^2dx. \n\\end{eqnarray}\n\tFurther, multiplying the first equation of (\\ref{yu-6-24-1}) by $-\\mbox{div}(A\\nabla u)(1-\\eta^2)$\n\t and integrating by parts over \n\t$\\Omega$, we get\t\n\\begin{eqnarray*}\\label{yu-7-24-6}\n\t&\\;&\\frac{1}{2}\\frac{d}{dt}\\int_{\\Omega}\\nabla u\\cdot (A\\nabla u)(1-\\eta^2)dx\n\t+\\int_{\\Omega}|\\mbox{div}(A\\nabla u)|^2(1-\\eta^2)dx\\nonumber\\\\\n\t&\\leq&2\\int_{\\Omega}|\\mbox{div}(A\\nabla u)||(A\\nabla u)\\cdot \\nabla\\eta|\\eta dx\n\t+2\\int_{\\Omega}|b||u||(A\\nabla u)\\cdot\\nabla\\eta|\\eta dx\n\t+\\int_{\\Omega}|b||u|(1-\\eta^2)|\\mbox{div}(A\\nabla u)dx|\\nonumber\\\\\n\t&\\leq&\\frac{1}{4}\\int_{\\Omega}|\\mbox{div} (A \\nabla u)|^2\\eta^2dx+5\\int_{\\Omega}|\\nabla\\eta\\cdot(A\\nabla u)|^2dx\n\t+\\int_{\\Omega}|b|^2|u|^2\\eta^2dx\\nonumber\\\\\n\t&\\;&+\\frac{\\Lambda_2}{r_0}\\int_{\\Omega}|u|(1-\\eta^2)|\\mbox{div}(A\\nabla u)|dx\\nonumber\\\\\n\t&\\leq&\\frac{1}{4}\\int_{\\Omega}|\\mbox{div} (A \\nabla u)|^2\\eta^2dx\n\t+\\frac{5C\\Lambda_1}{r_0^2}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx\n\t+\\frac{8\\Lambda_1\\Lambda_2^2}{(N-2)^2}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)\\eta^2dx\\nonumber\\\\\n\t&\\;&+\\frac{8C\\Lambda_2^2}{(N-2)^2r_0^2}\\int_{\\Omega}|u|^2dx\n\t+\\frac{\\Lambda_2^2}{3r_0^2}\\int_{\\Omega}|u|^2(1-\\eta^2)dx\n\t+\\frac{3}{4}\\int_{\\Omega}|\\mbox{div}(A\\nabla u)|^2(1-\\eta^2)dx. \n\\end{eqnarray*}\n\tThis, together with (\\ref{yu-7-24-5}), gives that \n\\begin{eqnarray}\\label{yu-7-24-7}\n\t&\\;&\\frac{1}{2}\\frac{d}{dt}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx+\\frac{1}{4}\\int_{\\Omega}|\\mbox{div}(A\\nabla u)|^2dx\\nonumber\\\\\n\t&\\leq&\\left(\\frac{10C\\Lambda_1}{r_0^2}+\\frac{24\\Lambda_2^2\\Lambda_1}{(N-2)^2}\\right)\n\t\\int_{\\Omega}\\nabla u\\cdot (A\\nabla u)dx\n\t+\\left(\\frac{24C}{(N-2)^2}+\\frac{1}{3}\\right)\\frac{\\Lambda_2^2}{r_0^2}\n\t\\int_{\\Omega}|u|^2dx.\n\\end{eqnarray}\n\tBy (\\ref{yu-7-15-2}) and (\\ref{yu-7-24-7}), we get \n\\begin{eqnarray*}\\label{yu-7-24-8}\n\t\\frac{1}{2}\\frac{d}{dt}\\left(\\int_{\\Omega}|u|^2dx\n\t+\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx\\right)\n\t\\leq C(r_0) \\left(\\int_{\\Omega}|u|^2dx\n\t+\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx\\right),\n\\end{eqnarray*}\n\twhere\n$$\n\tC(r_0):=C\\Lambda_1^N\\Lambda_2^{N+1}+\\frac{1}{2\\Lambda_1}+\\left(\\frac{24C}{(N-2)^2}+\\frac{1}{3}\\right)\\frac{\\Lambda_2^2}{r_0^2}\n\t+\\frac{10C\\Lambda_1}{r_0^2}\n\t+\\frac{24\\Lambda_2^2\\Lambda_1}{(N-2)^2}.\n$$\n\tTherefore, we have\n\\begin{equation}\\label{yu-7-24-9}\n\t\\frac{d}{dt}\\left[e^{-2C(r_0)t}\\int_\\Omega\\left(|u|^2+\\nabla u\\cdot(A\\nabla u)\\right)dx\\right]\\leq 0.\n\\end{equation}\n\tThis gives \n\\begin{eqnarray}\\label{yu-7-15-8}\n\t\\|u(\\cdot,t)\\|_{H^1(\\Omega)}^2\\leq \\Lambda_2^2e^{2C(r_0)t}\\|u(\\cdot,0)\\|^2_{H^1(\\Omega)}.\n\\end{eqnarray}\n\t Hence (\\ref{yu-7-12-3}) holds in this case. \n\\vskip 5pt\n\t\\emph{Case II. $b(\\cdot)\\in L^{N+\\delta}(\\Omega)$ and $\\|b(\\cdot)\\|_{L^{N+\\delta}(\\Omega)}\\leq \\Lambda_2$.} \n\tMultiplying the first equation of (\\ref{yu-6-24-1}) by $-\\mbox{div}(A\\nabla u)$ and integrating by parts over $\\Omega$, by \n\tLemma \\ref{hardy}, we get \t \n\\begin{eqnarray*}\\label{yu-9-28-1}\n\t&\\;&\\frac{1}{2}\\frac{d}{dt}\\int_{\\Omega}\\nabla u\\cdot(A\\nabla u)dx\n\t+\\frac{1}{2}\\int_{\\Omega}|\\mbox{div}(A\\nabla u)|^2dx\\nonumber\\\\\n\t&\\leq&2\\|b\\|^2_{L^N(\\Omega)}\\|u\\|^2_{L^{\\frac{2N}{N-2}}(\\Omega)}\n\t\\leq 2\\Gamma_1(\\Omega, N, 2)\\|b\\|^2_{L^N(\\Omega)}\\|u\\|^2_{H^1(\\Omega)}\\nonumber\\\\\n\t&\\leq&C\\Lambda_1\\|b\\|^2_{L^N(\\Omega)}\\int_{\\Omega}(|u|^2+\\nabla u\\cdot (A\\nabla u))dx.\n\\end{eqnarray*}\n\t This, along with (\\ref{yu-7-15-2}), gives that \t \n\\begin{equation*}\\label{yu-9-28-2}\n\t\\frac{1}{2}\\frac{d}{dt}\\int_{\\Omega}(|u|^2+\\nabla u\\cdot (A\\nabla u))dx\n\t\\leq \\left(C\\Lambda_2^2\\Lambda_1+C\\Lambda^N_1\\Lambda_2^{N+1}+\\frac{1}{2\\Lambda_1}\\right)\n\t\\int_{\\Omega}(|u|^2+\\nabla u\\cdot (A\\nabla u))dx.\n\\end{equation*}\n\tThis implies that \n\\begin{equation}\\label{yu-9-28-3}\n\t\\frac{d}{dt}\\left[e^{-(C\\Lambda_2^2\\Lambda_1+C\\Lambda_1^N\\Lambda_2^{N+1}+\\Lambda_1^{-1})t}\n\t\\int_{\\Omega}(|u|^2+\\nabla u\\cdot (A\\nabla u))dx\\right]\\leq 0.\n\\end{equation}\n\t Similar to the proof of (\\ref{yu-7-15-8}), we obtain (\\ref{yu-7-12-3}) in this case. \n\\par\nMoreover, using (\\ref{yu-7-24-9}) and (\\ref{yu-9-28-3}) respectively in each case analyzed above, we obtain that there exists $C>0$ such that \n\\begin{eqnarray*}\\label{yu-7-15-9}\n\\int_0^t\\int_{\\Omega}(|u|^2+\\nabla u\\cdot (A\\nabla u))dxds\n\t\\geq \\int_0^te^{-C(t-s)}ds\\, \\|u(\\cdot,t)\\|_{H^1(\\Omega)}^2\t\\geq \\Lambda_1^{-1}te^{-Ct}\\|u(\\cdot,t)\\|_{H^1(\\Omega)}^2.\n\\end{eqnarray*}\n\tThis, together with (\\ref{yu-7-15-4}) and (\\ref{yu-7-12-2-b}),\n\t yields (\\ref{yu-7-12-2}). \n\n\n\n\n\n\\vskip 5pt\n\t\\textbf{Step 2. Completing the proof.} \n\tWe arbitrarily fixed $t_0\\in(0,T)$ and consider the following equation\n\\begin{equation*}\\label{yu-7-12-12}\n\\begin{cases}\n\tv_t-\\mbox{div}(A(x)\\nabla v)+bv=0&\\mbox{in}\\;\\;\\Omega\\times(0,4T),\\\\\n\tA\\nabla v\\cdot\\nu=0&\\mbox{on}\\;\\;\\partial\\Omega\\times(0,4T),\\\\\n\tv(\\cdot,0)=u(\\cdot,\\frac{t_0}{2})&\\mbox{in}\\;\\;\\Omega.\n\\end{cases}\n\\end{equation*}\n\tIt is obvious that $v(\\cdot,t)=u(\\cdot,t+\\frac{t_0}{2})$ when $t\\in[0,4T]$. Moreover, by (\\ref{yu-7-12-2}) we have \n\t$u(\\cdot,\\frac{t_0}{2})\\in H^1(\\Omega)$, \n\twhich means that $v\\in C([0,4T];H^1(\\Omega))$. \n\tFrom Proposition \\ref{yu-theorem-7-10-6}, it follows that there are $C>0$ and $\\sigma\\in(0,1)$ such that \n\\begin{equation*}\\label{yu-7-13-1}\n\t\\left\\|v\\left(\\cdot,\\frac{t_0}{2}\\right)\\right\\|_{L^2(\\Omega)}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\n\t\\left\\|v\\left(\\cdot,\\frac{t_0}{2}\\right)\\right\\|^\\sigma_{L^2(\\omega)}\\left(\\sup_{s\\in[0,T]}\\|v(\\cdot,s)\\|_{H^1(\\Omega)}\\right)^{1-\\sigma}. \n\\end{equation*}\n\tThis, along with (\\ref{yu-7-12-3}), gives that \n\\begin{equation*}\\label{yu-7-13-2}\n\t\\left\\|v\\left(\\cdot,\\frac{t_0}{2}\\right)\\right\\|_{L^2(\\Omega)}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\n\t\\left\\|v\\left(\\cdot,\\frac{t_0}{2}\\right)\\right\\|^\\sigma_{L^2(\\omega)}\\|v(\\cdot,0)\\|^{1-\\sigma}_{H^1(\\Omega)}. \\end{equation*}\nWhich is \n\\begin{equation*}\\label{yu-7-13-3}\n\t\\left\\|u\\left(\\cdot,t_0\\right)\\right\\|_{L^2(\\Omega)}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\n\t\\left\\|u\\left(\\cdot,t_0\\right)\\right\\|^\\sigma_{L^2(\\omega)}\\left\\|u\\left(\\cdot,\\frac{t_0}{2}\\right)\\right\\|^{1-\\sigma}_{H^1(\\Omega)}.\n\\end{equation*}\n\tThis, together with (\\ref{yu-7-12-2}), implies (\\ref{yu-7-12-1}) and completes the proof.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{yu-main-1}}\n\nTo make the paper self-contained, we here provide the proof of Theorem \\ref{yu-main-1} in detail, although \nit is almost the same as the proof of \\cite[Theorem 1.1]{Phung-Wang-2013} or \\cite[Theorem 1]{Apraiz-Escauriaza-Wang-Zhang}. \n\n\n\n\\begin{lemma}\\label{yu-lemma-7-13-1}\n (\\cite[Proposition 2.1]{Phung-Wang-2013})\tLet $E\\subset(0,T)$ be a measurable set of positive measure, \n\t$\\ell$ be a density point of $E$. Then for each $z>1$, there exists $\\ell_1\\in(\\ell,T)$\n\tsuch that $\\{\\ell_m\\}_{m\\in\\mathbb{N}^+}$ given by\n\\begin{equation}\\label{yu-7-13-b-1}\n\t\\ell_{m+1}=\\ell+\\frac{1}{z^m}(\\ell_1-\\ell)\n\\end{equation}\nverifies \n\\begin{equation}\\label{yu-7-13-b-2}\n\t\\ell_m-\\ell_{m+1}\\leq 3|E\\cap(\\ell_{m+1},\\ell_m)|.\n\\end{equation}\t \n\\end{lemma}\n\n\\medskip\n\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{yu-main-1}}]\n\n\nBy (\\ref{yu-7-12-1}), one can show that, for arbitrary fixed $\\epsilon>0$ and any $t_0\\in(0,T)$, \n\\begin{equation*}\\label{yu-7-13-5}\n\t\\left\\|u\\left(\\cdot,t_0\\right)\\right\\|_{L^2(\\Omega)}\n\t\\leq \\frac{Ce^{\\frac{C(T^2+1)}{t_0}}}{\\epsilon^{\\gamma}}\n\t\\left\\|u\\left(\\cdot,t_0\\right)\\right\\|_{L^2(\\omega)}\n\t+\\epsilon\\|u(\\cdot,0)\\|_{L^2(\\Omega)},\n\\end{equation*}\n\twhere $\\gamma>0$ is a constant. By a translation in time, one has for each $0\\leq t_10.\n\\end{equation*}\n\tLet $0<\\ell_{m+2}<\\ell_{m+1}\\leq t<\\ell_m0.\n\\end{equation}\n\tNoting that, by (\\ref{yu-7-12-2-b}), \n\\begin{equation*}\\label{yu-7-13-8}\n\te^{-CT}\\|u(\\cdot,\\ell_m)\\|_{L^2(\\Omega)}\\leq \\|u(\\cdot,t)\\|_{L^2(\\Omega)}. \n\\end{equation*}\n\tThis, along with (\\ref{yu-7-13-7}), yields that for any $\\epsilon>0$,\n\\begin{equation*}\\label{yu-7-13-9}\n\t\\|u(\\cdot,\\ell_m)\\|_{L^2(\\Omega)}\n\t\\leq \\frac{Ce^{\\frac{C(T^2+1)}{\\ell_{m+1}-\\ell_{m+2}}}}{\\epsilon^\\gamma}\n\t\\|u(\\cdot,t)\\|_{L^2(\\omega)}+\\epsilon\\|u(\\cdot,\\ell_{m+2})\\|_{L^2(\\Omega)}.\n\\end{equation*}\n\tIntegrating over $E\\cap(\\ell_{m+1},\\ell_{m})$, we get \n\\begin{eqnarray*}\\label{yu-7-13-10}\n\t\\|u(\\cdot,\\ell_m)\\|_{L^2(\\Omega)}\n\t\\leq\\frac{Ce^{\\frac{C(T^2+1)}{\\ell_{m+1}-\\ell_{m+2}}}}{|E\\cap (\\ell_{m+1},\\ell_{m})|\\epsilon^\\gamma}\n\t\\int_{\\ell_{m+1}}^{\\ell_m}\\chi_E\\|u(\\cdot,t)\\|_{L^2(\\omega)}dt\n\t+\\epsilon \\|u(\\cdot,\\ell_{m+2})\\|_{L^2(\\Omega)}.\n\\end{eqnarray*}\n\tThis, together with (\\ref{yu-7-13-b-1}) and (\\ref{yu-7-13-b-2}), gives \n\\begin{eqnarray*}\\label{yu-7-13-11}\n\t\\epsilon^\\gamma e^{-\\eta z^{m+2}}\\|u(\\cdot,\\ell_m)\\|_{L^2(\\Omega)}\n\t-\\epsilon^{1+\\gamma}e^{-\\eta z^{m+2}}\\|u(\\cdot,\\ell_{m+2})\\|_{L^2(\\Omega)}\n\t\\leq C\\int_{\\ell_{m+1}}^{\\ell_m}\\chi_E\\|u(\\cdot,t)\\|_{L^2(\\omega)}dt,\n\\end{eqnarray*}\n\twhere $\\eta:=\\frac{C(T^2+1)}{z(z-1)(\\ell_1-\\ell)}$. \n\tLetting $\\epsilon:=e^{-\\eta z^{m+2}}$ and $z:=\\sqrt{\\frac{2+\\gamma}{1+\\gamma}}$, we have \n\\begin{eqnarray}\\label{yu-7-13-12}\n\te^{-\\eta(2+\\gamma)z^m}\\|u(\\cdot,\\ell_m)\\|_{L^2(\\Omega)}\n\t-e^{-\\eta(2+\\gamma)z^{m+2}}\\|u(\\cdot,\\ell_{m+2})\\|_{L^2(\\Omega)}\n\t\\leq C\\int_{\\ell_{m+1}}^{\\ell_m}\\chi_E\\|u(\\cdot,t)\\|_{L^2(\\omega)}dt.\n\\end{eqnarray}\n\tBy taking first $m=2m'$ and then summing the estimate (\\ref{yu-7-13-12}) from \n\t$m'=1$ to infinity, we obatin\n\\begin{eqnarray*}\\label{yu-7-14-1}\n\t&\\;&\\sum_{m'=1}^\\infty\\left[e^{-\\eta(2+\\gamma)z^{2m'}}\\|u(\\cdot,\\ell_{2m'})\\|_{L^2(\\Omega)}\n\t-e^{-\\eta(2+\\gamma)z^{2m'+2}}\\|u(\\cdot,\\ell_{2m'+2})\\|_{L^2(\\Omega)}\\right]\\nonumber\\\\\n\t&\\leq& C\\sum_{m'=1}^\\infty\\int_{E\\cap(\\ell_{2m'+1},\\ell_{2m'})}\\|u(\\cdot,t)\\|_{L^2(\\omega)}dt\n\t\\leq C\\int_{E}\\|u(\\cdot,t)\\|_{L^2(\\omega)}dt.\n\\end{eqnarray*}\n\tNote that $e^{-\\eta(2+\\gamma)z^{2m'}}\\to 0$ as $m'\\to \\infty$. Therefore, \n\\begin{equation*}\\label{yu-7-14-2}\n\t\\|u(\\cdot,\\ell_2)\\|_{L^2(\\Omega)}\\leq Ce^{\\eta(2+\\gamma)z^2}\n\t\\int_{E}\\|u(\\cdot,t)\\|_{L^2(\\omega)}dt.\n\\end{equation*}\n\tThis, along with (\\ref{yu-7-12-2-b}), leads to the desired observability inequality. \n\\par\n\tFinally, when $E=[0,T]$, we can take $\\ell=0$ and $\\ell_1=T$ in the above argument to conclude the desired result. \n\\end{proof}\n\t\n\n\n\\section{Proofs of quantitative estimates of unique continuation}\\label{dujin1}\n\n\t\t\n\\subsection{Preliminary lemmas}\n\n\\subsubsection{Local energy estimates and exponential decay}\n\nSuppose $\\rho\\in(0,\\min\\{1,\\rho_0\\})$ such that $\\Omega_\\rho\\neq \\emptyset$, $T>0$, $t_0\\in(0,T)$ and $x_0\\in\\Omega_\\rho$. Let $u\\in C([0,2T];H^1(\\triangle_{ \\rho}(x_0)))$ be a solution of \n\\begin{equation}\\label{yu-11-29-3}\n\\begin{cases}\n l(x)u_t-\\mbox{div}(A(x)\\nabla u)+b(x)u=0&\\mbox{in}\\;\\;\\triangle_{\\rho}(x_0)\\times(0,2T),\\\\\n u(\\cdot,0)=0&\\mbox{in}\\;\\;\\triangle_{ \\rho}(x_0).\n\\end{cases}\n\\end{equation}\nAssume $\\eta\\in C^\\infty(\\mathbb{R}^+;[0,1])$ is a cutoff function satisfying \n\\begin{equation}\\label{yu-6-6-6}\n\\begin{cases}\n\t\\eta\\equiv 1 &\\mbox{in}\\;\\;(0,t_0),\\\\\n\t\\eta\\equiv0 &\\mbox{in}\\;\\; [T,+\\infty),\\\\\n\t|\\eta_t|\\leq \\frac{C}{T-t_0}&\\mbox{in}\\;\\;(t_0,T)\n\\end{cases}\n\\end{equation}\nwith a generic positive constant $C$ independent of $t_0$ and $T$. Set\n\\begin{equation}\\label{du8141}\n\tR_0=\n\\begin{cases}\n\t\\Theta_N^{-N}\\left(8\\sqrt{2}\\Lambda_1\\Lambda_2\\Gamma_2(\\triangle_1(0),N,\\frac{N}{2})\\right)^{-\\frac{N+\\delta}{\\delta}}\n\t&\\mbox{if (i) in (\\ref{yu-6-24-1-1-b}) holds},\\\\\n\t\\frac{N(N-2)}{4\\sqrt{2}\\Lambda_1\\Lambda_2}&\\mbox{if (ii) in (\\ref{yu-6-24-1-1-b}) holds},\n\\end{cases}\n\\end{equation} \nand take $R\\in(0,\\min\\{R_0,\\rho\\})$. Here $\\Theta_N=|\\triangle_1(0)|$. \nLet $v$ be the solution of\n \\begin{equation}\\label{yu-11-29-4}\n\\begin{cases}\n l(x)v_t-\\mbox{div}(A(x)\\nabla v)+b(x)v=0&\\mbox{in}\\;\\;\\triangle_{R}(x_0)\\times\\mathbb{R}^+,\\\\\n v=\\eta u&\\mbox{on}\\;\\;\\partial\\triangle_R(x_0)\\times\\mathbb{R}^+,\\\\\n v(\\cdot,0)=0&\\mbox{in}\\;\\;\\triangle_R(x_0),\n\\end{cases}\n\\end{equation}\nwhere $u$ satisfies \\eqref{yu-11-29-3} and $\\eta$ verifies \\eqref{yu-6-6-6}. Then, we have\nthe following exponential decay estimate of $H^1$-energy for \\eqref{yu-11-29-4}.\n \n\\begin{lemma}\\label{yu-lemma-6-10-1}\n There exists\na generic constant $C>0$ such that\n\\begin{equation*}\\label{yu-6-18-1}\n \t\\|v(\\cdot,t)\\|_{H^1(\\triangle_R(x_0))}^2\\leq CT^{-1}e^{CR^{1-N}T\\left(1+\\frac{1}{T-t_0}\\right)-CR^{-2}(t-T)^+}F^2(R)\\quad\\text{for all}\\;\\; t\\in\\mathbb{R}^+,\n \\end{equation*}\nwhere $(t-T)^+=\\max\\{0,t-T\\}$ and\n\t$F(R)=\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|_{H^1(\\triangle_R(x_0))}$.\n\\end{lemma}\n\n\n\n\\begin{proof\nWe proceed the proof into two steps as follows.\n \n\\par\n\\vskip 5pt\n \\textbf{Step 1. To prove \\eqref{yu-6-18-1} when $t\\in[0,T]$}. \nSetting $w=v-\\eta u$ in $\\triangle_R(x_0)\\times\\mathbb R^+$, we find that $w$ verifies that \n\\begin{equation}\\label{yu-6-6-7}\n\\begin{cases}\n\tl(x)w_t-\\mbox{div}(A(x)\\nabla w)+b(x)w=-l(x)\\eta_tu\n\t&\\mbox{in}\\;\\;\\triangle_R(x_0)\\times\\mathbb R^+,\\\\\n\tw=0&\\mbox{on}\\;\\;\\partial\\triangle_R(x_0)\\times\\mathbb R^+,\\\\\n\tw(\\cdot,0)=0&\\mbox{in}\\;\\;\\triangle_R(x_0).\n\\end{cases}\n\\end{equation}\n\tWe now prove that for each $t\\in[0,T]$, \n\\begin{equation}\\label{yu-6-7-6}\n\t\\|w(\\cdot,t)\\|^2_{L^2(\\triangle_R(x_0))}\\leq \\frac{CT}{T-t_0}e^{C\\left(R^{1-N}+\\frac{1}{T-t_0}\\right)T}\n\t\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}\n\\end{equation}\nwith a generic constant $C>0$. We divide our proof into two cases. \n\\vskip 5pt\n\t\\emph{Case I. $|b(x)|\\leq \\frac{\\Lambda_2}{|x|}$ a.e. $x\\in\\Omega$.}\n\tIndeed, \nmultiplying first (\\ref{yu-6-6-7}) by $w$ and integrating by parts over $\\triangle_R(x_0)\\times(0,t)$, \nalong with the Hardy inequality in Lemma \\ref{hardy}, we have \n\\begin{eqnarray*}\\label{yu-6-7-1}\n\t&\\;&\\frac{1}{2}\\int_{\\triangle_R(x_0)}l|w(\\cdot,t)|^2dx+\\int_0^t\\int_{\\triangle_R(x_0)}\\nabla w\\cdot(A\\nabla w)dxds\\nonumber\\\\\n\t&\\leq&\\int_0^t\\int_{\\triangle_R}|b||w|^2dxds\n\t+\\int_0^t\\int_{\\triangle_R(x_0)}l\\eta_tuwdxds\\nonumber\\\\\n\t&\\leq&\\Lambda_2\\int_0^t\\int_{\\triangle_{R}(x_0)}|x|^{-1}|w|^2dxds\n\t+\\frac{1}{2}\\int_0^t\\int_{\\triangle_R(x_0)}l|\\eta_t||u|^2dxds\n\t+\\frac{1}{2}\\int_0^t\\int_{\\triangle_R(x_0)}l|\\eta_t||w|^2dxds\\nonumber\\\\\n\t&\\leq&\\frac{1}{2}\\epsilon\n\t\\int_0^t\\int_{\\triangle_R(x_0)}|x|^{-2}|w|^2dxds\n\t+\\frac{CT}{T-t_0}\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|_{L^2(\\triangle_R(x_0))}^2\\nonumber\\\\\n\t&\\;&+\\frac{1}{2}\\left(\\epsilon^{-1}\\Lambda_2^2+\\frac{C}{T-t_0}\\right)\n\t\\int_0^t\\int_{\\triangle_R(x_0)}|w|^2dxds\\nonumber\\\\\n\t&\\leq&\\frac{2\\epsilon\\Lambda_1}{(N-2)^2}\n\t\\int_0^t\\int_{\\triangle_R(x_0)}\\nabla w\\cdot(A\\nabla w)dxds\n\t+\\frac{CT}{T-t_0}\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|_{L^2(\\triangle_R(x_0))}^2\\nonumber\\\\\n\t&\\;&+\\frac{1}{2}\\left(\\epsilon^{-1}\\Lambda_2^2+\\frac{C}{T-t_0}\\right)\n\t\\int_0^t\\int_{\\triangle_R(x_0)}|w|^2dxds.\n\\end{eqnarray*}\n\tTaking $\\epsilon=\\frac{(N-2)^2}{2\\Lambda_1}$\n \tin the above inequality, we obtain \n\\begin{eqnarray*}\\label{yu-6-7-3}\n\t\\|w(\\cdot,t)\\|_{L^2(\\triangle_R(x_0))}^2\\leq\n\tC\\left(1+\\frac{1}{T-t_0}\\right)\\int_0^t\\|w(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}ds\n\t+\\frac{CT}{T-t_0}\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}\n\\end{eqnarray*}\n \twith a generic constant $C>0$. By the Gronwall inequality, we get (\\ref{yu-6-7-6}) immediately. \n\\vskip 5pt\n\\emph{Case II. $b(\\cdot)\\in L^{N+\\delta}(\\Omega)$ and $\\|b(\\cdot)\\|_{L^{N+\\delta}(\\Omega)}\\leq\\Lambda_2$.} We first note that, by using a standard scaling technique\nto (\\ref{yu-9-26-2}), without lose of generality, one has\n\\begin{equation}\\label{yu-9-29-1}\n\t\\Gamma_2(\\triangle_r(x_0),N,\\eta)=\\Gamma_2(\\triangle_1(0),N,\\eta) r^{-\\frac{2N}{N+2\\eta}}\n\t\\;\\;\\mbox{for each}\\;\\;r\\in(0,1).\n\\end{equation}\n\tMultiplying first (\\ref{yu-6-6-7}) by $w$ and then integrating by parts over $\\triangle_R(x_0)\\times(0,t)$, along with (\\ref{yu-9-26-2}) (by letting $\\epsilon=\\frac{1}{2}$ there) and (\\ref{yu-9-29-1}), we have \n \\begin{eqnarray*}\\label{yu-6-7-1}\n\t&\\;&\\frac{1}{2}\\int_{\\triangle_R(x_0)}l(x)|w(x,t)|^2dx+\\int_0^t\\int_{\\triangle_R(x_0)}\\nabla w\\cdot(A\\nabla w)dxds\\nonumber\\\\\n\t&\\leq&CR^{-\\frac{2N}{N+1}}\\|b\\|_{L^{\\frac{N+1}{2}}(\\triangle_R(x_0))}\\int_0^t\n\t\\|w\\|^{\\frac{2}{N+1}}_{L^2(\\triangle_R(x_0))}\\|w\\|^{\\frac{2N}{N+1}}_{H^1(\\triangle_R(x_0))}ds\n\t+\\int_0^t\\int_{\\triangle_R(x_0)}l\\eta_tuwdxds\\nonumber\\\\\n\t&\\leq&C\\epsilon^{-N}R^{N+1}\\|b\\|_{L^{N}(\\triangle_R(x_0))}^{N+1}\\int_0^t\\|w\\|^2_{L^2(\\triangle_R(x_0))}ds\n\t+\\epsilon R^{-2}\\int_0^t\\|w\\|_{H^1(\\triangle_R(x_0))}^2ds\\nonumber\\\\\n\t&\\;&+\\frac{1}{2}\\int_0^t\\int_{\\triangle_R(x_0)}l|\\eta_t||u|^2dxds\n\t+\\frac{1}{2}\\int_0^t\\int_{\\triangle_R(x_0)}l|\\eta_t||w|^2dxds\\nonumber\\\\\n\t&\\leq&\\left(C\\epsilon^{-N}R^{N+1}\\Lambda_2^{N+1}+\\frac{C}{2(T-t_0)}\n\t+\\epsilon R^{-2}\\right)\\int_0^t\\|w\\|^2_{L^2(\\triangle_R(x_0))}ds\\nonumber\\\\\n\t&\\;&+\\epsilon R^{-2}\\Lambda_1\\int_0^t\\int_{\\triangle_R(x_0)}\\nabla w\\cdot (A\\nabla w)dxds\n\t+\\frac{CT}{T-t_0}\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}\n\t\\qquad \\text{for any}\\;\\;\\epsilon>0.\n\\end{eqnarray*}\n\t Taking\n\t$\\epsilon=\\frac{R^2}{2\\Lambda_1}$\n \tin the above inequality, we obtain \n\\begin{multline*}\\label{yu-6-7-3}\n\t\\|w(\\cdot,t)\\|_{L^2(\\triangle_R(x_0))}^2\\leq\n\tC\\left(R^{1-N}+\\frac{1}{T-t_0}\\right)\\int_0^t\\|w(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}ds\\\\\n\t+\\frac{CT}{T-t_0}\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}\n\\end{multline*}\nwith a generic constant $C>0$.\nBy the Gronwall inequality, we get (\\ref{yu-6-7-6}) immediately. \nHence, from (\\ref{yu-6-7-6}) and the definition of $w$, we know that for any $t\\in[0,T]$,\n\\begin{equation}\\label{yu-6-7-11-1}\n\t\\|v(\\cdot,t)\\|^2_{L^2(\\triangle_R(x_0))}\\leq C\\left(1+\\frac{T}{T-t_0}\\right)e^{C\\left(R^{1-N}+\\frac{1}{T-t_0}\\right)T}\n\t\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}\n\\end{equation}\nwith a generic constant $C>0$.\n\\par\n\tNext, we show that \n\\begin{eqnarray}\\label{yu-6-8-3}\n\t\\|\\nabla w(\\cdot,t)\\|_{L^2(\\triangle_R(x_0))}^2\n\t\\leq \\left(1+\\frac{1}{T-t_0}\\right)\\frac{CT}{T-t_0}e^{CR^{1-N}T\\left(1+\\frac{1}{T-t_0}\\right)}\n\t\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}.\n\\end{eqnarray}\nWhich, along with the definition of $w$, gives that for each $t\\in[0,T]$, \n\\begin{multline}\\label{yu-6-8-4}\n\t\\|\\nabla v(\\cdot,t)\\|_{L^2(\\triangle_R(x_0))}^2\\leq 2\\|\\nabla w(\\cdot,t)\\|^2_{L^2(\\triangle_R(x_0))}+2\\|\\nabla u(\\cdot,t)\\|_{L^2(\\triangle_R(x_0))}^2\\\\\n\t\\leq \\left(1+\\frac{1}{T-t_0}\\right)\\frac{CT}{T-t_0}e^{C\\left(R^{1-N}+\\frac{1}{T-t_0}\\right)T}\n\t\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{H^1(\\triangle_R(x_0))}\n\\end{multline}\nwith a generic constant $C>0$.\nHence, the desired estimate \\eqref{yu-6-18-1} follows from (\\ref{yu-6-7-11-1}) and (\\ref{yu-6-8-4})\nwhen $t\\in[0,T]$. We also divide the proof of (\\ref{yu-6-8-3}) into two cases under the assumptions in \n(\\ref{yu-6-24-1-1-b}). \n\\vskip 5pt\n\t\\emph{Case I. $|b(x)|\\leq \\frac{\\Lambda_2}{|x|}$ a.e. $x\\in\\Omega$.}\nMultiplying first (\\ref{yu-6-6-7}) by $w_t$ and then integrating by parts over $\\triangle_R(x_0)\\times(0,t)$, \n\twe find \n\\begin{eqnarray*}\\label{yu-6-7-12}\n\t&\\;&\\int_0^t\\int_{\\triangle_R(x_0)}l|w_t|^2dxds\n\t+\\frac{1}{2}\\int_0^t\\int_{\\triangle_R(x_0)}[\\nabla w\\cdot(A\\nabla w)]_tdxds\\nonumber\\\\\n\t&\\leq&\\frac{1}{2}\\epsilon\\int_0^t\\int_{\\triangle_R(x_0)}\n\t|x|^{-2}|w|^2dxds+\\frac{\\Lambda_2^2\\Lambda_3\n\t+\\frac{C}{T-t_0}}{2\\epsilon}\\int_0^t\\int_{\\triangle_R(x_0)}l(x)|w_t|^2dxds\\nonumber\\\\\n\t&\\;&+\\frac{C\\epsilon}{2(T-t_0)}\\int_0^t\\int_{\\triangle_R(x_0)}|u|^2dxds\\qquad \\text{for any}\\;\\;\\epsilon>0\n\\end{eqnarray*}\nwith a generic constant $C>0$.\n\tLetting\n\t$\\epsilon=\\frac{\\Lambda_2^2\\Lambda_3+\\frac{C}{T-t_0}}{2}$\n\tin the inequality above, combined with the Hardy inequality in Lemma \\ref{hardy}, leads to \n\\begin{eqnarray*}\\label{yu-6-8-1}\n\t&\\;&\\int_{\\triangle_R(x_0)}\\nabla w(x,t)\\cdot(A(x)\\nabla w(x,t))dx\\nonumber\\\\\n\t&\\leq&C\\left(1+\\frac{1}{T-t_0}\\right)\n\t\\int_0^t\\int_{\\triangle_R(x_0)}|\\nabla w|^2dxds+\\left(1+\\frac{1}{T-t_0}\\right)\\frac{CT}{T-t_0}\n\t\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))},\n\\end{eqnarray*}\t\nfor a generic constant $C>0$.\n\tThis, together with the uniform ellipticity condition \\eqref{yu-11-28-2}, means that \n\\begin{eqnarray*}\\label{yu-6-8-2}\n\t\\|\\nabla w(\\cdot,t)\\|_{L^2(\\triangle_R(x_0))}^2&\\leq&\n\tC\\left(1+\\frac{1}{T-t_0}\\right)\n\t\\int_0^t\\|\\nabla w(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}ds\\nonumber\\\\\n\t&\\;&+\\left(1+\\frac{1}{T-t_0}\\right)\\frac{CT}{T-t_0}\n\t\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|^2_{L^2(\\triangle_R(x_0))}.\n\\end{eqnarray*}\nBy the Gronwall inequality, we get (\\ref{yu-6-8-3}). \n\\vskip 5pt\n\\emph{Case II. $b(\\cdot)\\in L^{N+\\delta}(\\Omega)$ and $\\|b(\\cdot)\\|_{L^{N+\\delta}(\\Omega)}\\leq\\Lambda_2$.} \n\tMultiplying (\\ref{yu-6-6-7}) by $w_t$ and integrating by parts over \n\t$\\triangle_R(x_0)\\times (0,t)$, we have \n\\begin{eqnarray*}\\label{yu-9-30-1}\n\t&\\;&\\int_0^t\\int_{\\triangle_R(x_0)}l|w_t|^2dxds+\\frac{1}{2}\\int_0^t\\int_{\\triangle_R(x_0)}\n\t[\\nabla w\\cdot (A\\nabla w)]_tdxds\\nonumber\\\\\n\t&\\leq&\\frac{\\epsilon}{2}\\int_0^t\\int_{\\triangle_R(x_0)}|b|^2|w|^2dxds+\\frac{\\Lambda_3+\\frac{C}{T-t_0}}{2\\epsilon}\n\t\\int_0^t\\int_{\\triangle_R(x_0)}l|w_t|^2dxds\\nonumber\\\\\n\t&\\;&+\\frac{C\\epsilon}{2(T-t_0)}\\int_0^t\\int_{\\triangle_R(x_0)}|u|^2dxds\\nonumber\\\\\n\t&\\leq&\\frac{C\\epsilon R^{-2}\\Lambda_2^{\\frac{1}{2}}}{2}\\int_0^t\\int_{\\triangle_R(x_0))}|w|^2dxds\n\t+\\frac{C\\epsilon R^{-2}\\Lambda_1\\Lambda_2^{\\frac{1}{2}}}{2}\n\t\\int_0^t\\int_{\\triangle_R(x_0)}\\nabla w\\cdot (A\\nabla w)dxdt\\nonumber\\\\\n\t&\\;&+\\frac{\\Lambda_3+\\frac{C}{T-t_0}}{2\\epsilon}\\int_0^t\\int_{\\triangle_R(x_0)}l|w_t|^2dxds\n\t+\\frac{C\\epsilon}{2(T-t_0)}\\int_0^t\\int_{\\triangle_R(x_0)}|u|^2dxds. \n\\end{eqnarray*}\n\tHere, we used (\\ref{yu-9-26-2}) and (\\ref{yu-9-29-1}). Taking $\\epsilon=\\frac{\\Lambda_3+\\frac{C}{T-t_0}}{2}$ in the above inequality, \n\tby (\\ref{yu-6-7-6}) we get \n\\begin{eqnarray*}\\label{yu-9-30-2}\n\t&\\;&\\int_{\\triangle_R(x_0)}\\nabla w(x,t)\\cdot(A\\nabla w(x,t))dx\\nonumber\\\\\n\t&\\leq&CR^{-2}\\left(1+\\frac{1}{T-t_0}\\right)\\int_0^t\\int_{\\triangle_R(x_0)}(|w|^2+\\nabla w\\cdot (A\\nabla w))dxds\\nonumber\\\\\n\t&\\;&+\\frac{CT}{T-t_0}\\left(1+\\frac{1}{T-t_0}\\right)\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|_{L^2(\\triangle_R(x_0)}^2\\nonumber\\\\\n\t&\\leq&CR^{1-N}\\left(1+\\frac{1}{T-t_0}\\right)\\int_0^t\\int_{\\triangle_R(x_0)}\\nabla w\\cdot (A\\nabla w)dxds\\nonumber\\\\\n\t&\\;&+C\\frac{T}{T-t_0}\\left(1+\\frac{1}{T-t_0}\\right)e^{C\\left(R^{1-N}+\\frac{1}{T-t_0}\\right)T}\n\t\\sup_{s\\in[0,T]}\\|u(\\cdot,s)\\|_{L^2(\\Omega)}^2. \n\\end{eqnarray*}\n\tBy the Gronwall inequality, we get (\\ref{yu-6-8-3}). \n\n\\medskip\n\t\t\n\t\n\t\n\t\n\\vskip 5pt\n \\textbf{Step 2. To prove \\eqref{yu-6-18-1} when $t\\geq T$.}\n \\vskip 5pt\n \nDefine for any $f,g\\in C_0^\\infty(\\triangle_R(x_0))$,\n$$\n\\langle f,g\\rangle_{\\mathcal L^2(\\triangle_R(x_0))} :=\\int_{\\triangle_R(x_0)}l(x)f(x)g(x)dx \\quad \\text{and}\\quad\n\\|f\\|_{\\mathcal L^2(\\triangle_R(x_0)}:=\\langle f,f\\rangle_{\\mathcal L^2(\\triangle_R(x_0))}^{1\/2}.$$\nSet\n$\\mathcal{L}^2(\\triangle_{R}(x_0))=\\overline{C_0^\\infty(\\triangle_R(x_0))}^{\\|\\cdot\\|_{\\mathcal L^2(\\triangle_R(x_0))}}\n$.\nSince $l$ is positive, it is clear that $\\mathcal{L}^2(\\triangle_R(x_0))=L^2(\\triangle_R(x_0))$ with an equivalent norm. \n Denoting $\\mathcal{A}=-l^{-1}[\\mbox{div}(A\\nabla)-b]$,\nwe claim that \n\tthere is a generic constant $C>0$ (independent of $R$) such that \n \\begin{equation}\\label{yu-6-7-9}\n \t\\langle \\mathcal{A}f,f\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\\geq CR^{-2}\\|f\\|^2_{L^2(\\triangle_R(x_0))}\n\t\\;\\;\\mbox{for each}\\;\\;f\\in H_{0}^1(\\triangle_R(x_0))\\cap H^2(\\triangle_R(x_0)).\n \\end{equation}\n \tWe also divide its proof into two cases. \n\\vskip 5pt\n \\emph{Case I. $|b(x)|\\leq \\frac{\\Lambda_2}{|x|}$ a.e. $x\\in\\Omega$.}\n In this case, by Lemma \\ref{hardy}, we find that for each $\\epsilon>0$, \n$$\n\t\\langle \\mathcal{A} f,f\\rangle_{\\mathcal{L}(\\triangle_R(x_0))}\\geq \\Lambda_1^{-1}\\int_{\\triangle_{R}(x_0)}\n\t|\\nabla f|^2dx-\\frac{2\\epsilon}{(N-2)^2}\\int_{\\triangle_R(x_0)}|\\nabla f|^2dx-\\frac{\\Lambda_2^2}{2\\epsilon}\n\t\\int_{\\triangle_R(x_0)}|f|^2dx,\n$$\n for any $f\\in H_0^1(\\triangle_R(x_0))\\cap H^2(\\triangle_R(x_0))$. Letting \n $\\epsilon=\\frac{(N-2)^2}{4\\Lambda_1}$ in the above inequality, by \n the Poincar\\'e inequality \n\\begin{equation}\\label{yu-11-30-b-1}\n \\int_{\\triangle_R(x_0)}|f(x)|^2dx\\leq \\left(\\frac{2R}{N}\\right)^2\\int_{\\triangle_R(x_0)}|\\nabla f(x)|^2dx\\;\\;\\mbox{for each}\\;\\;f\\in H_0^1(\\triangle_R(x_0)),\n\\end{equation}\t \n we derive \n\\begin{equation*}\\label{yu-10-12-1}\n\t\\langle\\mathcal{A}f,f\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\\geq\n\t\\left[\\frac{\\Lambda_1^{-1}}{2}-\\frac{8\\Lambda_1\\Lambda_2^2R^2}{(N-2)^2N^2}\\right]\\int_{\\triangle_R(x_0)}\n\t|\\nabla f|^2dx. \n\\end{equation*}\n \tFrom the definition of $R_0$ given in (\\ref{du8141}), and (\\ref{yu-11-30-b-1}), we can conclude\n\tthe claim (\\ref{yu-6-7-9}). \n\\vskip 5pt\n\\emph{Case II. $b(\\cdot)\\in L^{N+\\delta}(\\Omega)$ and $\\|b(\\cdot)\\|_{L^{N+\\delta}(\\Omega)}\\leq \\Lambda_2$.} \t\n\tBy using (\\ref{yu-9-26-2}), (\\ref{yu-9-29-1}) and (\\ref{yu-11-30-b-1}), we have \n\\begin{eqnarray*}\\label{yu-10-12-2}\n\t\\int_{\\triangle_R(x_0)}|b||f|^2dx&\\leq& \\Gamma_2\\left(\\triangle_R(x_0),N,\\frac{N}{2}\\right)\n\t\\|b\\|_{L^{N}(\\triangle_R(x_0))}\\|f\\|_{L^2(\\triangle_R(x_0))}\n\t\\|f\\|_{H^1_0(\\triangle_R(x_0))}\\nonumber\\\\\n\t&\\leq&\\sqrt{2}\\Gamma_2\\left(\\triangle_1(0),N,\\frac{N}{2}\\right)\\|b\\|_{L^{N}(\\triangle_R(x_0))}\\|\\nabla f\\|_{L^2(\\triangle_R(x_0))}^2\\nonumber\\\\\n\t&\\leq&\\sqrt{2}\\Theta_N^{\\frac{N\\delta}{N+\\delta}}\\Gamma_2\\left(\\triangle_1(0),N,\\frac{N}{2}\\right)R^{\\frac{\\delta}{N+\\delta}}\\|b\\|_{L^{N+\\delta}(\\Omega)}\\|\\nabla f\\|_{L^2(\\triangle_R(x_0))}^2.\n\\end{eqnarray*}\n\tFrom the definition of $R_0$, we have \n$$\n\t\\int_{\\triangle_R(x_0)}|b||f|^2dx\\leq \\frac{\\Lambda_1^{-1}}{8}\\|\\nabla f\\|^2_{L^2(\\triangle_R(x_0))}.\n$$\n\tThis implies \n$$\n\t\\langle \\mathcal{A}f, f\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\\geq \\frac{7\\Lambda_1^{-1}}{8}\\int_{\\triangle_R(x_0)}|\\nabla f|^2dx.\n$$\n\tand then (\\ref{yu-6-7-9}) holds. \n\t\n\t\n\n As a consequence of \\eqref{yu-6-7-9}, we see that the inverse of $\\mathcal A$ is positive, self-adjoint and compact in $\\mathcal L^2(\\triangle_R(x_0))$.\nBy the spectral theorem for compact self-adjoint operators,\tthere are eigenvalues \n\t$\\{\\mu_i\\}_{i\\in\\mathbb{N}^+}\\subset \\mathbb{R}^+$ and eigenfunctions \n\t$\\{f_i\\}_{i\\in\\mathbb{N}^+}\\subset H_0^1(\\triangle_R(x_0))$, which make up an orthogonal basis of $\\mathcal{L}^2(\\triangle_R(x_0))$,\nsuch that \n \\begin{equation}\\label{yu-6-7-10}\n \\begin{cases}\n \t-\\mathcal{A}f_i=\\mu_if_i\\;\\;\\mbox{and}\\;\\;\\|f_i\\|_{\\mathcal{L}^2(\\triangle_R(x_0))}=1&\\mbox{for each}\\;\\;i\\in\\mathbb{N}^+,\\\\\nCR^{-2}< \\mu_1\\leq \\mu_2\\leq \\cdots\n \\leq \\mu_i\\to+\\infty&\\mbox{as}\\;\\;i\\to+\\infty.\n \\end{cases}\n \\end{equation}\n Then, by the formula of Fourier decomposition, \n\tthe solution $w$ of (\\ref{yu-6-6-7}) in $[T,+\\infty)$ is given by\n\\begin{equation*}\\label{yu-6-12-3}\n\tw(\\cdot,t)=\\sum_{i=1}^\\infty\\langle w(\\cdot,T),f_i\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}e^{-\\mu_i(t-T)}f_i\t\\quad\\text{in}\\;\\;\\triangle_R(x_0)\\;\\;\\mbox{for each}\\;\\;t\\in[T,+\\infty).\n\\end{equation*}\n \tHence, we deduce that for each $t\\in[T,+\\infty)$,\n\\begin{eqnarray}\\label{yu-6-12-4}\n\t\\|w(\\cdot,t)\\|^2_{\\mathcal{L}^2(\\triangle_R(x_0))}\n\t\\leq e^{-CR^{-2}(t-T)}\\|w(\\cdot,T)\\|_{\\mathcal{L}^2(\\triangle_R(x_0))}^2\n\\end{eqnarray}\n and\n\\begin{eqnarray*\n\tw_t(\\cdot,t)=-\\sum_{i=1}^\\infty\\mu_i\\langle w(\\cdot,T),f_i\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}e^{-\\mu_i(t-T)}f_i.\n\\end{eqnarray*}\n It follows that\n\\begin{eqnarray}\\label{yu-6-13-1}\n\t-\\langle w(\\cdot,t),w_t(\\cdot,t)\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\n\t=\\sum_{i\\in\\mathbb{N}^+}\\mu_i|\\langle w(\\cdot,T),f_i\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}|^2\n\te^{-2\\mu_i(t-T)},\n\\end{eqnarray}\nfor each $t\\in[T,+\\infty)$. In particular, taking $t=T$ in the above identify leads to \n\\begin{equation}\\label{yu-6-13-2}\n\t-\\langle w(\\cdot,T),w_t(\\cdot,T)\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\n\t=\\sum_{i\\in\\mathbb{N}^+}\\mu_i|\\langle w(\\cdot,T),f_i\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}|^2.\t\n\\end{equation} \n Meanwhile, it follows from (\\ref{yu-6-6-7}) and Lemma \\ref{hardy} that \n\\begin{eqnarray}\\label{yu-6-13-3}\n-\\langle w(\\cdot,T),w_t(\\cdot,T)\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\\leq C\\|w(\\cdot,T)\\|^2_{H_{0}^1(\\triangle_R(x_0))}\n\\end{eqnarray}\nwith a generic constant $C>0$.\n \tFrom (\\ref{yu-6-13-2}) and (\\ref{yu-6-13-3}), we have \n\\begin{equation*}\\label{yu-6-14-1}\n\t\\sum_{i=1}^\\infty\\mu_i|\\langle w(\\cdot,T),f_i\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}|^2\n\t\\leq C\\|w(\\cdot,T)\\|^2_{H_{0}^1(\\triangle_R(x_0))}.\n\\end{equation*} \n \tThis, together with (\\ref{yu-6-13-1}), gives \n\\begin{equation}\\label{yu-6-14-2}\n\t-\\langle w(\\cdot,t),w_t(\\cdot,t)\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\n\t\\leq Ce^{-CR^{-2}(t-T)}\\|w(\\cdot,T)\\|^2_{H_{0}^1(\\triangle_R(x_0))},\n\\end{equation} \n\tfor each $t\\in[T,+\\infty)$. On the other hand, by (\\ref{yu-6-6-7}) and (\\ref{yu-6-7-9}), we see that for each $t\\in[T,+\\infty)$,\n \\begin{equation}\\label{yu-6-14-3}\n \t-\\langle w(\\cdot,t),w_t(\\cdot,t)\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}=\\langle w(\\cdot,t),-\\mathcal{A}w(\\cdot,t)\\rangle_{\\mathcal{L}^2(\\triangle_R(x_0))}\n\t\\geq C\\|\\nabla w(\\cdot,t)\\|^2_{L^2(\\triangle_R(x_0))}.\n \\end{equation}\n \tBy (\\ref{yu-6-14-2}) and (\\ref{yu-6-14-3}), we find that for each $t\\in[T,\\infty)$, \n \\begin{equation*}\\label{yu-6-14-4}\n \t\\|\\nabla w(\\cdot,t)\\|^2_{L^2(\\triangle_R(x_0))}\n\t\\leq C\n\te^{-CR^{-2}(t-T)}\\|w(\\cdot,T)\\|^2_{H_{0}^1(\\triangle_R(x_0))} \n\\end{equation*}\n \tThis, together with (\\ref{yu-6-12-4}), means that \n\\begin{equation*}\\label{yu-6-18-2}\n\t\\|w(\\cdot,t)\\|_{H_{0}^1(\\triangle_R(x_0))}^2\\leq Ce^{-CR^{-2}(t-T)}\\|w(\\cdot,T)\\|^2_{H_{0}^1(\\triangle_R(x_0))}.\n\\end{equation*}\nBy the fact that $w(\\cdot,t)=v(\\cdot,t)$ for each $t\\geq T$, we conclude the desired result.\n\\end{proof}\n\\medskip\nWe next define \n\\begin{equation*}\\label{yu-6-18-5}\n\t\\tilde{v}(\\cdot,t)=\n\\begin{cases}\n\tv(\\cdot,t)&\\mbox{if}\\;\\;t\\geq 0,\\\\\n\t0&\\mbox{if}\\;\\;t<0,\n\\end{cases}\n\\end{equation*}\nwhere $v$ is the solution of \\eqref{yu-11-29-4}.\nBy Lemma \\ref{yu-lemma-6-10-1}, we can take the Fourier transform of\t$\\tilde{v}$ with respect to the time variable $t\\in\\mathbb R$\n\\begin{equation*}\\label{yu-6-18-6}\n\t\\hat{v}(x,\\mu)=\\int_{\\mathbb{R}}e^{-i\\mu t}\\tilde{v}(x,t)dt\\quad\\text{for}\\;\\;(x,\\mu)\\in\\triangle_R(x_0)\\times\\mathbb R.\n\\end{equation*}\nThen, we have\n\\begin{lemma}\\label{yu-lemma-6-18-1}\nThere exists a generic constant $C>0$ such that, for each $\\mu\\in\\mathbb{R}$, \nthe following two estimates hold:\n\\begin{equation}\\label{yu-6-23-5}\n\t\\|\\nabla \\hat{v}(\\cdot,\\mu)\\|_{L^2(\\triangle_r(x_0))}\\leq \\frac{C(1+\\sqrt{|\\mu|})}{R-2r}\\|\\hat{v}(\\cdot,\\mu)\\|_{L^2(\\triangle_{\\frac{R}{2}}(x_0))}\\quad \\text{for all} \\;\\;00$ and $\\epsilon_2>0$, \n\\begin{eqnarray*}\\label{yu-6-23-3}\n\t&\\;&\\Lambda_1^{-1}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\leq&2\\Lambda_1\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla\\hat{v}||\\hat{v}||\\nabla\\psi||\\psi|dx\n\t+\\Lambda_3|\\mu|\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx\n\t+\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|b||\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\leq&\\epsilon_1\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla \\hat{v}|^2\\psi^2dx\n\t+\\frac{\\Lambda_1^2}{\\epsilon_1}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2|\\nabla\\psi|^2dx\n\t+\\epsilon_2\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|x|^{-2}|\\hat{v}\\psi|^2dx\\nonumber\\\\\n\t&&\\;\\;\\;\\;+\\left(\\frac{\\Lambda_2^2}{4\\epsilon_2}+\\Lambda_3|\\mu|\\right)\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\leq&\\left(\\epsilon_1+\\frac{8\\epsilon_2}{(N-2)^2}\\right)\n\t\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla\\hat{v}|^2\\psi^2dx\n\t+\\left(\\frac{8\\epsilon_2}{(N-2)^2}+\\frac{\\Lambda_1^2}{\\epsilon_1}\\right)\n\t\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2|\\nabla \\psi|^2dx\\nonumber\\\\\n\t&\\;&+\\left(\\frac{\\Lambda_2^2}{4\\epsilon_2}+\\Lambda_3|\\mu|\\right)\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx.\n\\end{eqnarray*}\nTaking $\\epsilon_1=\\frac{1}{4\\Lambda_1}$ and $\\epsilon_2=\\frac{(N-2)^2}{16\\Lambda_1}$ in the above inequality, we derive\n (\\ref{yu-6-23-5}).\n\\vskip 5pt\n\\emph{Case II. $b(\\cdot)\\in L^{N+\\delta}(\\Omega)$ and $\\|b(\\cdot)\\|_{L^{N+\\delta}(\\Omega)}\\leq\\Lambda_2$.}\n\tBy (\\ref{yu-11-28-2}), (\\ref{yu-7-29-3}) and (\\ref{yu-9-26-2}) with $\\eta=\\frac{N}{2}$, we get \n\\begin{eqnarray*}\\label{yu-6-23-3}\n\t&\\;&\\Lambda_1^{-1}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\leq&2\\Lambda_1\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla\\hat{v}||\\hat{v}||\\nabla\\psi||\\psi|dx\n\t+\\Lambda_3|\\mu|\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx\n\t+\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|b||\\hat{v}|^2\\psi^2dx\\nonumber\\\\\t\n\t&=&2\\Lambda_1\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla\\hat{v}||\\hat{v}||\\nabla\\psi||\\psi|dx\n\t+\\Lambda_3|\\mu|\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\;&+\\Gamma_2\\left(\\triangle_1(0),N,\\frac{N}{2}\\right)\\|b\\|_{L^{N}(\\triangle_{\\frac{R}{2}}(x_0))}\n\t\\left(\\frac{R}{2}\\right)^{-1}\\|\\hat{v}\\psi\\|_{L^2(\\triangle_{\\frac{R}{2}}(x_0))}\n\t\\|\\hat{v}\\psi\\|_{H^1(\\triangle_{\\frac{R}{2}}(x_0))}.\\nonumber\\\\\n\\end{eqnarray*}\t\nThen for any $\\epsilon>0$, we have\n\\begin{eqnarray*}\t\n&\\;&\\Lambda_1^{-1}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\leq&\\epsilon\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla \\hat{v}|^2\\psi^2dx\n\t+\\frac{\\Lambda_1^2}{\\epsilon}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2|\\nabla \\psi|^2dx\n\t+\\Lambda_3|\\mu|\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\;&+2\\sqrt{2}\\Gamma_2\\left(\\triangle_1(0),N,\\frac{N}{2}\\right)\\|b\\|_{L^N(\\triangle_{\\frac{R}{2}}(x_0))}\n\t\\int_{\\triangle_{\\frac{R}{2}}(x_0)}\\left(|\\nabla \\hat{v}|^2\\psi^2\n\t+|\\hat{v}|^2|\\nabla\\psi|^2\\right)dx\\nonumber\\\\\n\t&\\leq&\\epsilon\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla \\hat{v}|^2\\psi^2dx\n\t+\\frac{\\Lambda_1^2}{\\epsilon}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2|\\nabla \\psi|^2dx\n\t+\\Lambda_3|\\mu|\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\;&+2\\sqrt{2}\\Theta_N^{\\frac{N\\delta}{N+\\delta}}\\Gamma_2\\left(\\triangle_1(0),N,\\frac{N}{2}\\right)\n\t\\Lambda_2R^{\\frac{\\delta}{N+\\delta}}\n\t\\int_{\\triangle_{\\frac{R}{2}}(x_0)}\\left(|\\nabla \\hat{v}|^2\\psi^2\n\t+|\\hat{v}|^2|\\nabla\\psi|^2\\right)dx\\nonumber\\\\\n\t&\\leq&\\epsilon\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\nabla \\hat{v}|^2\\psi^2dx\n\t+\\frac{\\Lambda_1^2}{\\epsilon}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2|\\nabla \\psi|^2dx\n\t+\\Lambda_3|\\mu|\\int_{\\triangle_{\\frac{R}{2}}(x_0)}|\\hat{v}|^2\\psi^2dx\\nonumber\\\\\n\t&\\;&+\\frac{\\Lambda_1^{-1}}{4}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}\\left(|\\nabla \\hat{v}|^2\\psi^2\n\t+|\\hat{v}|^2|\\nabla\\psi|^2\\right)dx.\n\\end{eqnarray*}\t\n\tHere, we used (\\ref{yu-9-29-1}) and the definition of $R_0$. Taking $\\epsilon=\\frac{\\Lambda_1^{-1}}{4}$ in the above inequality and using (\\ref{yu-6-23-1}) lead to (\\ref{yu-6-23-5}).\t\n\t\n\t\n\t\n\t\n\\medskip\n\nNote that, when $\\mu=0$, by Lemma \\ref{yu-lemma-6-10-1} we have\n\\begin{eqnarray}\\label{yu-6-22-15}\n\t\\|\\hat{v}(\\cdot,0)\\|_{L^2(\\triangle_{R}(x_0))}\\leq CT^{-\\frac{1}{2}}e^{CR^{1-N}\\left(1+\\frac{1}{T-t_0}\\right)T}\n\tF(R).\n\\end{eqnarray}\nThus it suffices to prove \\eqref{yu-6-22-16} in the case that $\\mu\\neq0$.\nTo this end, define for each $\\mu\\in\\mathbb{R}\\setminus\\{0\\}$,\n\\begin{equation*}\\label{yu-6-19-1}\n\tp(x,\\xi,\\mu)=e^{i\\sqrt{|\\mu|}\\xi}\\hat{v}(x,\\mu)\\quad\\text{for a.e.}\\quad (x,\\xi)\\in\\triangle_R(x_0)\\times\\mathbb R.\n\\end{equation*}\nThen, $p(\\cdot,\\cdot,\\mu)$ verifies \n\\begin{equation*}\\label{yu-6-19-2}\n\t\\mbox{div}(A\\nabla p(\\cdot,\\cdot,\\mu))+i\\mbox{sign}(\\mu)l\\partial_{\\xi\\xi}p(\\cdot,\\cdot,\\mu)\n\t-bp(\\cdot,\\cdot,\\mu)=0\\;\\;\\mbox{in}\\;\\;\\triangle_R(x_0)\\times\\mathbb{R}.\n\\end{equation*}\nHere \n\\begin{equation*}\\label{yu-6-19-3}\n \\mbox{sign}(\\mu):=\n\\begin{cases}\n\t1&\\mbox{if}\\;\\;\\mu>0,\\\\\n\t-1&\\mbox{if}\\;\\;\\mu<0.\n\\end{cases}\n\\end{equation*}\n\t\n\n\tLet $m\\in\\mathbb{N}^+$ and $a_j=1-\\frac{j}{2m}$ for $j=0,1,\\dots,m+1$. For each $j\\in\\{0,1,\\cdots, m\\}$, we define a cutoff function \n\\begin{equation*}\\label{yu-6-19-4}\n\th_j(s):=\n\\begin{cases}\n\t0&\\mbox{if}\\;\\;|s|>a_j,\\\\\n\t\\frac{1}{2}\\left[1+\\cos\\left(\\frac{\\pi(a_{j+1}-s)}{a_{j+1}-a_j}\\right)\\right]\n\t&\\mbox{if}\\;\\;a_{j+1}\\leq |s|\\leq a_j,\\\\\n\t1&\\mbox{if}\\;\\;|s|0.\n\\end{eqnarray*}\n\tTherefore, \n\\begin{eqnarray*}\\label{yu-6-20-4}\n\tI_1&\\leq&\\frac{2^5\\epsilon_1^2}{(N-2)^4}\n\t\\left(\\int_{D_j}|\\nabla p_j|^2\\eta_j^2dxd\\xi\\right)^2\n +\\frac{[8\\pi^2m^2\\epsilon_1^2+\\Lambda_2^2R^2(N-2)^2]^2}{2\\epsilon_1^2R^4(N-2)^4}\\left(\\int_{D_j}|p_j|^2dxd\\xi\\right)^2,\n\\end{eqnarray*}\n Let $\\epsilon_1=\\frac{(N-2)^2}{2^5\\Lambda_1}$, we derive \n (\\ref{yu-10-15-1}). \n\\vskip 5pt\n\\emph{Case II. $b(\\cdot)\\in L^{N+\\delta}(\\Omega)$ and $\\|b(\\cdot)\\|_{L^{N+\\delta}(\\Omega)}\\leq\\Lambda_2$.} \n By (\\ref{yu-9-26-2}), (\\ref{yu-9-29-1}) and the definition of $R_0$, we have \n \\begin{eqnarray*}\\label{yu-10-15-2}\n \t&\\;&\\int_{D_j}|b||p_j|^2\\eta_j^2dxd\\xi\\nonumber\\\\\n\t&\\leq& \\Gamma_2\\left(\\triangle_{a_jR}(x_0),N,\\frac{N}{2}\\right)\n\t\\|b\\|_{L^N(\\triangle_{a_jR}(x_0))}\\int_{-a_jR}^{a_jR}\\|p_j\\eta_j\\|_{L^2(\\triangle_{a_jR}(x_0))}\n\t\\|p_j\\eta_j\\|_{H^1(\\triangle_{a_jR}(x_0))}d\\xi\\nonumber\\\\\n\t&\\leq&\\sqrt{2}\\Theta_N^{\\frac{N\\delta}{N+\\delta}}\\Gamma_2\\left(\\triangle_1(0),N,\\frac{N}{2}\\right)\n\tR^{\\frac{\\delta}{N+\\delta}}\\|b\\|_{L^{N+\\delta}(\\triangle_{a_jR}(x_0)}\n\t\\int_{-a_jR}^{a_jR}\\|\\nabla(p_j\\eta_j)\\|_{L^2(\\triangle_{a_jR}(x_0))}^2d\\xi\\nonumber\\\\\n\t&\\leq&2\\sqrt{2}\\Theta_N^{\\frac{N\\delta}{N+\\delta}}\\Gamma_2\\left(\\triangle_1(0),N,\\frac{N}{2}\\right)\n\t\\Lambda_2R_0^{\\frac{\\delta}{N+\\delta}}\\int_{D_j}\\left(|\\nabla p_j|^2\\eta_j^2+\\frac{m^2\\pi^2}{R^2}|p_j|^2\\right)dxd\\xi\\nonumber\\\\\n\t&\\leq&\\frac{\\Lambda_1^{-1}}{4}\\int_{D_j}|\\nabla p_j|^2\\eta_j^2dx\n\t+\\frac{\\Lambda_1^{-1}m^2\\pi^2}{4R^2}\\int_{D_j}|p_j|^2dx,\n \\end{eqnarray*}\n which gives (\\ref{yu-10-15-1}).\n \n \n Moreover, \n\\begin{eqnarray*}\\label{yu-6-21-1}\n\t&\\;&\\int_{D_j}|\\nabla \\eta_j^2\\cdot(A\\nabla p_j)||p_j|dxd\\xi\n\t\\leq 2\\Lambda_1\\int_{D_j}|\\nabla \\eta_j||\\eta_j||\\nabla p_j||p_j|dxd\\xi\\nonumber\\\\\n\t&\\leq&\\epsilon_2\\Lambda_1\\int_{D_j}|\\nabla p_j|^2\\eta_j^2dxd\\xi\n\t+\\frac{\\Lambda_1}{\\epsilon_2}\\int_{D_j}|p_j|^2|\\nabla\\eta_j|^2dxd\\xi\\nonumber\\\\\n\t&\\leq&\\epsilon_2\\Lambda_1\\int_{D_j}|\\nabla p_j|^2\\eta_j^2dxd\\xi\n\t+\\frac{\\Lambda_1\\pi^2m^2}{R^2\\epsilon_2}\\int_{D_j}|p_j|^2dxd\\xi\n\\end{eqnarray*}\nand\n\\begin{equation}\\label{yu-6-21-2}\n\tI_2\\leq 2\\Lambda^2_1\\epsilon_2^2\\left(\\int_{D_j}|\\nabla p_j|^2\\eta_j^2dxd\\xi\\right)^2\n\t+\\frac{2\\Lambda_1^2\\pi^4m^4}{R^4\\epsilon^2_2}\\left(\\int_{D_j}|p_j|^2dxd\\xi\\right)^2, \\;\\;\\forall \\epsilon_2>0.\n\\end{equation}\nFurther, \n\\begin{eqnarray*}\\label{yu-6-21-3}\n\t\\int_{D_j}l|p_{j+1}|p_j|\\partial_{\\xi}\\eta^2_jdxd\\xi\n\t&\\leq&\\epsilon_3\\int_{D_j}l|p_{j+1}|^2\\eta_j^2dxd\\xi\n\t+\\frac{1}{\\epsilon_3}\\int_{D_j}l|p_j|^2|\\partial_\\xi\\eta_j|^2dxd\\xi\\nonumber\\\\\n\t&\\leq&\\epsilon_3\\int_{D_j}|p_{j+1}|^2\\eta_j^2dxd\\xi\n\t+\\frac{m^2\\pi^2\\Lambda_3}{R^2\\epsilon_3}\\int_{D_j}|p_j|^2dxd\\xi\n\\end{eqnarray*}\t\nand\n\\begin{equation}\\label{yu-6-21-4}\n\tI_3\\leq 2\\epsilon_3^2\\left(\\int_{D_j}l|p_{j+1}|^2\\eta_j^2dxd\\xi\\right)^2\n\t+\\frac{2\\pi^4\\Lambda_3^2m^4}{R^4\\epsilon_3^2}\\left(\\int_{D_j}|p_j|^2dxd\\xi\\right)^2, \\;\\;\\forall \\epsilon_3>0.\n\\end{equation}\nTaking $\\epsilon_2=\\frac{\\sqrt{2}}{4\\Lambda_1^2}$, $\\epsilon_3=\\frac{1}{4}$\n in (\\ref{yu-6-21-2})\n\t\tand (\\ref{yu-6-21-4}), respectively, by (\\ref{yu-10-15-1}),\nwe derive that \n\\begin{eqnarray}\\label{yu-6-21-8}\n\t\\sum_{i=1}^3I_i&\\leq&\\frac{\\Lambda_1^{-2}}{8}\\left(\\int_{D_j}|\\nabla p_j|^2\\eta_j^2dxd\\xi\n\t\\right)^2+\\frac{1}{8}\\left(\\int_{D_j}l|p_{j+1}|^2\\eta_j^2dxd\\xi\\right)^2\\nonumber\\\\\n\t&\\;&+\\frac{M_1\n\t\t+M_2m^4}{R^4}\n\t\\left(\\int_{D_j}|p_j|^2\\eta_j^2dxd\\xi\\right)^2\n\\end{eqnarray}\t\nwith two positive constants $M_1$ and $M_2$. \n\tOn the other hand, by the uniform ellipticity condition (\\ref{yu-11-28-2}), we find that\n\\begin{equation*}\\label{yu-6-22-2}\n\t\\left(\\int_{D_j}\\nabla \\bar{p}_j\\cdot(A\\nabla p_j)\\eta_j^2dxd\\xi\\right)^2\n\t\\geq \\Lambda_1^{-2}\\left(\\int_{D_j}|\\nabla p_j|^2\\eta_j^2dxd\\xi\\right)^2.\n\\end{equation*}\n\tThis, together with (\\ref{yu-7-29-3}), (\\ref{yu-6-19-9}) and (\\ref{yu-6-21-8}), gives that for each $j\\in\\{0,1,\\cdots,m-1\\}$, \n\\begin{eqnarray*}\\label{yu-6-22-3}\n\t\\int_{D_{j+1}}|p_{j+1}|^2\\eta^2_jdxd\\xi&\\leq& \\frac{2\\Lambda_3\\sqrt{2(M_1+M_2m^4)}}{R^2}\n\t\\int_{D_j}|p_j|^2\\eta^2_jdxd\\xi\\nonumber\\\\\n\t&\\leq&\\frac{\\Pi(1+m^2)}{R^2}\\int_{D_j}|p_j|^2\\eta_j^2dxd\\xi,\n\\end{eqnarray*}\n\twhere $\\Pi=2\\Lambda_3\\sqrt{2(M_1+M_2)}$. \n\t Here, we used the definition of $D_j$. Iterating (\\ref{yu-6-22-3}) for each \n\t$j\\in\\{0,1,\\cdots,m-1\\}$, by the fact that $p_0=p=\\hat{v}$ we obtain \t\n\\begin{equation}\\label{yu-6-22-4}\n\t\\int_{\\triangle_{\\frac{R}{2}}(x_0)\\times(-\\frac{R}{2},\\frac{R}{2})}|p_m|^2dxd\\xi\n\t\\leq2R\\left[\\frac{\\Pi(1+m^2)}{R^2}\\right]^m\\int_{\\triangle_R(x_0)}|\\hat{v}(x,\\mu)|^2dx.\n\\end{equation}\nBy Lemma \\ref{yu-lemma-6-10-1}, we get that for each $\\mu\\in\\mathbb{R}$,\n\\begin{eqnarray}\\label{yu-6-22-5}\n\t\\|\\hat{v}(\\cdot,\\mu)\\|_{L^2(\\triangle_{R}(x_0))}\n\t&\\leq&CT^{-\\frac{1}{2}}e^{CR^{1-N}\\left(1+\\frac{1}{T-t_0}\\right)T}\n\tF(R).\n\\end{eqnarray}\n\tTherefore, by (\\ref{yu-6-22-4}) and (\\ref{yu-6-22-5}), we get that for each $m\\in\\mathbb{N}^+$, \n\\begin{equation}\\label{yu-6-22-6}\n\t\\int_{\\triangle_{\\frac{R}{2}}(x_0)\\times(-\\frac{R}{2},\\frac{R}{2})}|p_m|^2dxd\\xi\n\t\\leq CT^{-1}\\left[\\frac{\\Pi(1+m^2)}{R^2}\\right]^mRe^{CR^{1-N}\\left(1+\\frac{1}{T-t_0}\\right)T}\n\tF^2(R).\n\\end{equation}\n\\par\n\tFor any $\\varphi\\in L^2(\\triangle_{\\frac{R}{2}}(x_0);\\mathbb{C})$, we define \n\\begin{equation*}\\label{yu-6-22-7}\n\tP_{\\mu}(\\xi):=\\int_{\\triangle_{\\frac{R}{2}}(x_0)}p(x,\\xi,\\mu)\\bar{\\varphi}(x)dx, \\;\\;\\;\\xi\\in\\left(-\\frac{R}{2},\\frac{R}{2}\\right).\n\\end{equation*}\n\tIt is well known that the following interpolation inequality holds (See a proof in Appendix)\n\\begin{equation}\\label{yu-6-22-8}\n\t\\|f\\|_{L^\\infty(I)}\\leq C\\left(|I|\\|f'\\|_{L^2(I)}^2+\\frac{1}{|I|}\\|f\\|_{L^2(I)}^2\\right)^{\\frac{1}{2}}\n\t\\;\\;\\mbox{for each}\\;\\;f\\in H^1(I),\n\\end{equation}\n\twhere $I$ is an bounded nonempty interval of $\\mathbb{R}$ and $|I|$ is the length. Therefore, by \n\t(\\ref{yu-6-22-6}) we have that for any $\\xi\\in(-\\frac{R}{2},\\frac{R}{2})$ and $m\\in\\mathbb{N}^+$,\n\\begin{eqnarray}\\label{yu-6-22-9}\n\t|P_{\\mu}^{(m)}(\\xi)|&\\leq& C\\left(R\\int_{-\\frac{R}{2}}^{\\frac{R}{2}}|P_{\\mu}^{(m+1)}(\\xi)|^2d\\xi\n\t+\\frac{1}{R}\\int_{-\\frac{R}{2}}^{\\frac{R}{2}}|P_{\\mu}^{(m)}(\\xi)|^2d\\xi\\right)^{\\frac{1}{2}}\\nonumber\\\\\n\t&\\leq&C\\left(R\\int_{\\triangle_{\\frac{R}{2}}(x_0)\\times(-\\frac{R}{2},\\frac{R}{2})}\n\t|p_{m+1}|^2dxd\\xi+\\frac{1}{R}\\int_{\\triangle_{\\frac{R}{2}}(x_0)\\times(-\\frac{R}{2},\\frac{R}{2})}|p_m|^2dxd\\xi\\right)\n\t^{\\frac{1}{2}}\\|\\varphi\\|_{L^2(\\triangle_{\\frac{R}{2}}(x_0))}\\nonumber\\\\\n\t&\\leq&CT^{-\\frac{1}{2}}e^{CR^{1-N}\\left(1+\\frac{1}{T-t_0}\\right)T}F(R)\\frac{[2\\Pi(m+1)]^{m+1}}{R^m}\n\t\\|\\varphi\\|_{L^2(\\triangle_{\\frac{R}{2}}(x_0))}.\n\\end{eqnarray}\n\tThis implies that $P_{\\mu}(\\cdot)$ can be \n\tanalytically extended to the complex plane (still denoted by the same notation)\n\\begin{equation*}\\label{yu-6-22-10}\n\tE_0:=\\left\\{\\xi\\in\\mathbb{C}:\\mbox{Re}\\,\\xi\\in\\left(-\\frac{R}{2},\\frac{R}{2}\\right)\\;\\;\\mbox{and}\\;\\;\\mbox{Im}\\,\\xi\\in(-L_0,L_0)\\right\\},\n\\end{equation*}\n\twhere $L_0:=\\frac{R}{2e\\Pi}$. \nThen, \n\\begin{equation*}\\label{yu-6-22-11}\n\t|P_{\\mu}(\\xi)|\\leq \\sum_{m=0}^\\infty\\frac{|P^{(m)}(0)|}{m!}|\\xi|^m,\n\\end{equation*}\nwhen $\\xi\\in i\\mathbb{R}\\cap E_0$.\nTaking $\\xi_0=-\\frac{iR}{4e\\Pi}$, by (\\ref{yu-6-22-9}), we get that \n\\begin{equation}\\label{yu-6-22-12}\n\t|P_{\\mu}(\\xi_0)|\\leq C\\Pi\\sum_{m=0}^\\infty\\frac{(m+1)^{m+1}}{m!(2e)^m}\n\tT^{-\\frac{1}{2}}\n\te^{CR^{1-N}\\left(1+\\frac{1}{T-t_0}\\right)T}F(R)\n\t\\|\\varphi\\|_{L^2(\\triangle_{\\frac{R}{2}}(x_0))}.\n\\end{equation}\nWhile, by the definition,\n\\begin{eqnarray*}\\label{yu-6-22-13}\n\tP_{\\mu}(\\xi_0)=e^{\\frac{\\sqrt{|\\mu|}R}{4e\\Pi}}\\int_{\\triangle_{\\frac{R}{2}}(x_0)}\\hat{v}(x,\\mu)\\bar{\\varphi}(x)dx.\n\\end{eqnarray*}\n\tThis, together with (\\ref{yu-6-22-12}), means that, \n\t\\begin{equation}\\label{yu-6-22-14}\n\t\\|\\hat{v}(\\cdot,\\mu)\\|_{L^2(\\triangle_{\\frac{R}{2}}(x_0))}\\leq CT^{-\\frac{1}{2}}\n\te^{CR^{1-N}\\left(1+\\frac{1}{T-t_0}\\right)T-\\frac{\\sqrt{|\\mu|}R}{4e\\Pi}}F(R).\n\\end{equation}\n\t\tBy (\\ref{yu-6-22-14}) and (\\ref{yu-6-22-15}), we derive (\\ref{yu-6-22-16}) and complete the proof.\t\n\\end{proof}\n\n\n\\subsubsection{Stability estimate and three-ball inequality for elliptic equations}\\label{yu-section-7-26-3}\nSuppose $T>0$, $L>0$ and $\\triangle_R(x_0)\\subset\\Omega$ with $x_0\\in\\Omega$.\nLet $g\\in H^1(\\triangle_R(x_0)\\times(-L,L))$ be a solution of the following elliptic equation\t\n\\begin{equation}\\label{yu-6-23-9}\n\\begin{cases}\n\t\\mbox{div}(A(x)\\nabla g)+l(x) g_{x_{N+1}x_{N+1}}-b(x)g=0&\\;\\;\\;\\text{in}\\;\\;\\; \\triangle_R(x_0)\\times(-L,L),\\\\\n\tg(x,0)=f_1(x)&\\;\\;\\;\\text{in}\\;\\;\\; \\triangle_R(x_0),\\\\\n\tg_{x_{N+1}}(x,0)=f_2(x)&\\;\\;\\;\\text{in}\\;\\;\\; \\triangle_R(x_0),\n\\end{cases}\n\\end{equation}\n\twhere $f_1\\in H^1(\\triangle_R(x_0))$, $f_2\\in L^2(\\triangle_R(x_0))$, \n\t$A$, $b$ and $l$ satisfy the same assumptions as before. \n\t\n\t\n\\begin{lemma}[Stability estimate]\\label{yu-proposition-7-1-1}\nThere is $\\gamma\\in(0,1)$ such that for any $r\\in(0,\\min\\{R,L\\}\/3)$, \n{\\begin{equation}\\label{yu-7-3-b-2}\n\t\\|g\\|^2_{H^1(B_r(x_0,0))}\\leq Cr^{-4}\\|g\\|_{H^1(B_{2r}(x_0,0))}^{2\\gamma}\\left(\\|f_1\\|^2_{H^1(\\triangle_{2r}(x_0))}+\\|f_2\\|^2_{L^2(\\triangle_{2r}(x_0))}\\right)^{1-\\gamma}.\n\\end{equation}}\n\\end{lemma}\t\n\n\\medskip\n\nThe proof of Lemma \\ref{yu-proposition-7-1-1} is based on a point-wise estimate (see Lemma \\ref{yu-lemma-7-1-1} below).\nHere and in the sequel, for simplicity we denote\n\\begin{equation*}\\label{yu-7-1-6}\n\t\\bar{A}(x,x_{N+1})=\\left[\\bar{a}^{ij}(x,x_{N+1})\\right]_{(N+1)\\times (N+1)}:=\\mbox{diag}(A(x),l(x)),\n\\end{equation*}\n$$\n\\nabla=(\\nabla_x,\\partial_{x_{N+1}}),\\quad \\mbox{div}=\\mbox{div}_x+\\partial_{x_{N+1}}\n$$\nwhen they do not arise any confusion in the context.\t\n\t\n\\begin{lemma}\\label{yu-lemma-7-1-1}\nLet $s>0$, $\\lambda>0$, $\\varphi\\in C^2(\\overline{B_R}(x_0,0))$ and set $\\alpha=e^{\\lambda\\varphi}$, $\\theta=e^{s\\alpha}$.\nIf $V\\in C^2(\\triangle_R(x_0)\\times(-L,L))$ and $W=\\theta V$, then the following inequality holds:\n\\begin{eqnarray*}\\label{yu-7-2-1}\n\t&\\;&\\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2+\\mathcal{D}\\nonumber\\\\\n\t&\\geq&\\mathcal{B}_1|W|^2+\\mathcal{B}_2\\nabla W\\cdot(\\bar{A}\\nabla W)\n\t+2s\\lambda^2W\\nabla[\\alpha\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)]\\cdot(\\bar{A}\\nabla W)+2s\\lambda^2\\alpha|\\nabla W\\cdot (\\bar{A}\\nabla\\varphi)|^2\\nonumber\\\\\n\t&\\;&+2s\\lambda\\alpha(\\bar{A}\\nabla W)\\cdot[D^2\\varphi(\\bar{A}\\nabla W)]+2s\\lambda\\alpha\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_j\\varphi\\right)\\cdot(\\bar{A}\\nabla W)\\nonumber\\\\\n\t&\\;&-s\\lambda\\alpha\\left(\\sum_{i,j=1}^{N+1}\\partial_i W\\nabla \\bar{a}^{ij}\\partial_j W\\right)\\cdot(\\bar{A}\\nabla\\varphi),\n\\end{eqnarray*}\n\twhere\n\\begin{equation*}\n\\begin{cases}\n\t\\mathcal{B}_1=s^3\\lambda^4\\alpha^3|\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)|^2+s^3\\lambda^3\\alpha^3\\mbox{div}\\{\\bar{A}\\nabla\\varphi[\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)]\\}\\\\\n\t\\;\\;\\;\\;\\;\\;\\;\\;\\;-2s^2\\lambda^2\\alpha^2|\\mbox{div}(\\bar{A}\\nabla\\varphi)|^2-2s^2\\lambda^4\\alpha^2|\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)|^2\\\\\n\t\\;\\;\\;\\;\\;=s^3\\lambda^4\\alpha^3|\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)|^2+{s^3\\alpha^3O(\\lambda^3)}\n\t+s^2\\alpha^2O(\\lambda^4),\\\\\n\t\\mathcal{B}_2=s\\lambda^2\\alpha\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)-s\\lambda\\alpha\\mbox{div}(\\bar{A}\\nabla\\varphi)\\\\\n\t\\;\\;\\;\\;\\;=s\\lambda^2\\alpha\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)+s\\alpha O(\\lambda),\\\\\n\t\\mathcal{D}=2s\\lambda^2\\mbox{div}[\\alpha W\\bar{A}\\nabla W\\nabla\\varphi\\cdot(\\bar{A}\\nabla \\varphi)]\n\t+2s\\lambda\\mbox{div}[\\alpha \\bar{A}\\nabla W\\nabla W\\cdot(\\bar{A}\\nabla \\varphi)]\\\\\n\t\\;\\;\\;\\;\\;\\;\\;\\;\\;-s\\lambda\\mbox{div}[\\alpha\\nabla W\\cdot(\\bar{A}\\nabla W)\\bar{A}\\nabla\\varphi]\n\t+s^3\\lambda^3\\mbox{div}[\\alpha^3|W|^2\\bar{A}\\nabla\\varphi\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)].\n\\end{cases}\n\\end{equation*}\n\\end{lemma}\n\n\t\n\t\\smallskip\n\t\n\\begin{proof}[\\textbf{Proof of Lemma \\ref{yu-proposition-7-1-1}}] \n\t\nWith the same notation as above, (\\ref{yu-6-23-9}) can be rewritten as\n\\begin{equation*}\\label{yu-7-1-7}\n\t\\mbox{div}(\\bar{A}\\nabla g)-bg=0\\;\\;\\mbox{in}\\;\\;\\triangle_R(x_0)\\times(-L,L),\n\\end{equation*}\nwhere $\\bar A$ satisfies \n\\begin{equation*}\\label{yu-7-1-12}\n\t\\Lambda_4^{-1}|\\xi|^2\\leq \\bar{A}(x,x_{N+1})\\xi\\cdot\\xi\\leq \\Lambda_4|\\xi|^2\\;\\;\n\t\\mbox{for each}\\;\\;(x,\\xi)\\in(\\triangle_{R}(x_0)\\times(-L,L))\\times\\mathbb{R}^{N+1},\n\\end{equation*}\nwith $\\Lambda_4=\\max\\{\\Lambda_1,\\Lambda_3\\}$. \n\n\nWe next divide the proof into two steps as follows.\n\n\\vskip 5pt\n \\textbf{Step 1.} For each $r<\\min\\{R,L\\}$, let us set \n $$r_1=r,\\quad r_2=\\frac{3r}{2},\\quad r_3=2r,\\quad r_4=3r$$\n and \n \\begin{equation*}\\label{yu-7-2-10}\n \t\\omega_1=B^+_{r_1}(x_0,0),\\;\\;\\omega_2=B^+_{r_2}(x_0,0),\\;\\;\\omega_3=B^+_{r_3}(x_0,0),\\;\\;\\omega_4=\\triangle_{r_4}(x_0)\\times(0,3r).\n \\end{equation*}\n\tLet {$\\varphi\\in C^2(\\mathbb{R};[0,4])$} be such that\n\\begin{equation}\\label{yu-7-2-11}\n\\begin{cases}\n\t3<\\varphi<4&\\mbox{in}\\;\\;\\omega_1,\\\\\n\t0<\\varphi<1&\\mbox{in}\\;\\;\\omega_4\\backslash\\omega_2,\\\\\n\t|\\nabla\\varphi|>0&\\mbox{in}\\;\\;\\overline{\\omega_4}.\n\\end{cases}\n\\end{equation}\nTake a cutoff function $\\eta\\in C^\\infty(\\mathbb{R}^{N+1};[0,1])$ to be such that\n\\begin{equation*}\\label{yu-7-2-12}\n\\begin{cases}\n\t\\eta=1&\\mbox{in}\\;\\;\\overline{\\omega_2},\\\\\n\t\\eta=0&\\mbox{in}\\;\\;\\overline{\\omega_4}\\backslash\\omega_3,\\\\\n\t|\\mbox{div}(\\bar{A}\\nabla\\eta)|+|\\nabla\\eta|^2\\leq \\frac{C}{r^2}&\\mbox{in}\\;\\;\\mathbb{R}^{N+1},\n\\end{cases}\n\\end{equation*}\n\twhere $C$ is a generic constant independent of $r$.\n\t\tSetting $V=\\eta g$, we have\n\\begin{equation}\\label{yu-7-2-13}\n\\begin{cases}\n\t\\mbox{div}(\\bar{A}\\nabla V)-bV=\\mbox{div}(\\bar{A}\\nabla\\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)&\\mbox{in}\\;\\;\\omega_4,\\\\\n\t|\\nabla V|=V=0&\\mbox{on}\\;\\;\\partial\\omega_4\\backslash(\\triangle_{r_3}(x_0)\\times\\{0\\}).\n\\end{cases}\n\\end{equation}\nIt follows from Lemma \\ref{yu-lemma-7-1-1} that\n\\begin{eqnarray}\\label{yu-7-2-14}\n\t&&\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2dxdx_{N+1}\n\t+\\int_{\\omega_4}\\mathcal{D}dxdx_{N+1}\\nonumber\\\\\n\t&\\geq&\\int_{\\omega_4}\\mathcal{B}_1|W|^2dxdx_{N+1}+\\int_{\\omega_3}\\mathcal{B}_2\\nabla W\\cdot(\\bar{A}\\nabla W)dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+2s\\lambda^2\\int_{\\omega_4} W\\nabla[\\alpha\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)]\\cdot (\\bar{A}\\nabla W)dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+2s\\lambda\\int_{\\omega_4}\\alpha(\\bar{A}\\nabla W)\\cdot[D^2\\varphi(\\bar{A}\\nabla W)]dxdx_{N+1}\n\t+2s\\lambda^2\\int_{\\omega_4}\\alpha|\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+2s\\lambda\\int_{\\omega_4}\\alpha\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{A}^{ij}\\partial_j\\varphi\\right)\\cdot(\\bar{A}\\nabla W)dxdx_{N+1}\\nonumber\\\\\n\t&\\;&-s\\lambda\\int_{\\omega_4}\\alpha\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{A}^{ij}\\partial_jW\\right)\\cdot(\\bar{A}\\nabla\\varphi)dxdx_{N+1}.\n\\end{eqnarray}\n By the Cauchy-Schwarz inequality, we find\n\\begin{equation}\\label{yu-7-2-15}\n\t2s\\lambda^2\\left| W\\nabla[\\alpha\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)]\\cdot (\\bar{A}\\nabla W)\\right|\\leq C\\lambda^2(s^2\\lambda^2\\alpha|W|^2+\\alpha|\\nabla W|^2),\n\\end{equation}\n\\begin{equation}\\label{yu-7-2-16}\n\t2s\\lambda\\alpha\\left|(\\bar{A}\\nabla W)\\cdot[D^2\\varphi(\\bar{A}\\nabla W)]\\right|\\leq Cs\\lambda\\alpha|\\nabla W|^2,\n\\end{equation}\n\\begin{equation}\n\t2s\\lambda\\alpha\\left|\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_j\\varphi\\right)\\cdot(\\bar{A}\\nabla W)\\right|\\leq Cs\\lambda\\alpha|\\nabla W|^2\n\\end{equation}\n\tand \n\\begin{equation}\\label{yu-7-2-17}\n\ts\\lambda\\alpha\\left|\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_jW\\right)\\cdot(\\bar{A}\\nabla\\varphi)\\right|\n\t\\leq Cs\\lambda\\alpha|\\nabla W|^2.\n\\end{equation}\n\tBy definitions of $\\mathcal{B}_1$ and $\\mathcal{B}_2$, we get \n\\begin{equation}\\label{yu-7-2-18}\n\t\\mathcal{B}_1|W|^2\\geq (s^3\\lambda^4\\alpha^3\\Lambda^{-2}_1|\\nabla\\varphi|^2+s^3\\alpha^3O(\\lambda^3)+s^2\\alpha^2O(\\lambda^4))|W|^2\n\\end{equation}\n\tand \n\\begin{equation}\\label{yu-7-2-19}\n\t\\mathcal{B}_2|\\nabla W|^2\\geq {(s\\lambda^2\\alpha\\Lambda^{-1}_1|\\nabla\\varphi|^2+s\\alpha O(\\lambda))|\\nabla W|^2}.\n\\end{equation}\n\tFrom (\\ref{yu-7-2-14})--(\\ref{yu-7-2-19}) and the positivity of $|\\nabla\\varphi|$, we have \n\\begin{eqnarray*}\\label{yu-7-2-20}\n\t&\\;&\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2dxdx_{N+1}\n\t+\\int_{\\omega_4}\\mathcal{D}dxdx_{N+1}\\nonumber\\\\\n\t&\\geq&C\\int_{\\omega_4}(s^3\\lambda^4\\alpha^3+s^3\\alpha^3O(\\lambda^3)+s^2\\alpha^2 O(\\lambda^4)\n\t-{Cs^2\\lambda^4\\alpha})|W|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+C\\int_{\\omega_4}[s\\lambda^2\\alpha+s\\alpha O(\\lambda)-C(\\lambda^2+s\\lambda)\\alpha]|\\nabla W|^2dxdx_{N+1}.\n\\end{eqnarray*}\n\tTherefore, there is a constant $\\lambda_0>1$ such that for any $\\lambda\\geq \\lambda_0$, one can find $s_0>1$ such that \n\tfor any $s\\geq s_0$, \n\\begin{eqnarray}\\label{yu-7-2-21}\n\t\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2dxdx_{N+1}\t\n\t+\\int_{\\omega_4}\\mathcal{D}dxdx_{N+1}\\nonumber\\\\\n\t\\geq Cs^3\\lambda^4\\int_{\\omega_4}\\alpha^3|W|^2dxdx_{N+1}\n\t+Cs\\lambda^2\\int_{\\omega_4}\\alpha|\\nabla W|^2dxdx_{N+1}.\n\\end{eqnarray}\nBy the definition of $\\mathcal{D}$, we obtain \n\\begin{eqnarray}\\label{yu-7-2-22}\n\\int_{\\omega_4}\\mathcal{D}dxdx_{N+1} &=&2s\\lambda^2\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha W (\\bar{A}\\nabla W)\\cdot\\vec{n}\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)d\\Gamma\\nonumber\\\\\n\t&\\;&+2s\\lambda\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha(\\bar{A}\\nabla W)\\cdot \\vec{n}\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)d\\Gamma\\nonumber\\\\\n\t&\\;&-s\\lambda\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha\\nabla W\\cdot(\\bar{A}\\nabla W)(\\bar{A}\\nabla\\varphi)\\cdot\\vec{n}d\\Gamma\\nonumber\\\\\n\t&\\;&+s^3\\lambda^3\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha^3|W|^2(\\bar{A}\\nabla\\varphi)\\cdot\\vec{n}\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)d\\Gamma\\nonumber\\\\\n\t&\\leq& Cs\\lambda\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha |\\nabla W|^2d\\Gamma\n\t+Cs^3\\lambda^3\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha^3|W|^2d\\Gamma.\n\\end{eqnarray}\nFrom (\\ref{yu-7-2-21}) and (\\ref{yu-7-2-22}), we have \n\\begin{eqnarray}\\label{yu-7-3-1}\n\t&\\;&Cs^3\\lambda^4\\int_{\\omega_4}\\alpha^3|W|^2dxdx_{N+1}+Cs\\lambda^2\\int_{\\omega_4}\\alpha|\\nabla W|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq&\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2dxdx_{N+1}+Cs\\lambda\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha\n\t|\\nabla W|^2d\\Gamma \\nonumber\\\\\n&&\\;\\;+Cs^3\\lambda^3\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha^3|W|^2d\\Gamma.\n\\end{eqnarray}\n \\textbf{Step 2.} Now, we return $W$ in (\\ref{yu-7-3-1}) to $V$. Note that \n\\begin{eqnarray}\\label{yu-7-3-2}\n\t\\frac{1}{C}\\theta^2(|\\nabla V|^2+s^2\\lambda^2\\alpha^2|V|^2)\\leq\n\t|\\nabla W|^2+s^2\\lambda^2\\alpha^2|W|^2\\leq\n C\\theta^2(|\\nabla V|^2+s^2\\lambda^2\\alpha^2|V|^2).\n\\end{eqnarray}\n\t Based on the case of the potential $b$, by Lemma \\ref{hardy}, (\\ref{yu-9-26-2}) with $\\epsilon=0$, the first equation in (\\ref{yu-7-2-13}) and the fact that $\\omega_4$ is a rectangle domain, we have\n\\begin{eqnarray}\\label{yu-7-3-b-1}\n\t&\\;&\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2dxdx_{N+1}\\nonumber\\\\\n\t&=&\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla\\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)+bV|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq&C\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla\\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)|^2dxdx_{N+1}\n\t+C\\int_{\\omega_4}\\theta^2|bV|^2dxdx_{N+1}\\nonumber\\\\\n\t&=&C\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla\\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)|^2dxdx_{N+1}\t+C\\int_{\\triangle_{r_4}(x_0)\\times(0,r_4)}|bW|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq&C\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla\\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)|^2dxdx_{N+1}\t+C\\int_{\\omega_4}(|\\nabla W|^2+|W|^2)dxdx_{N+1}.\n\\end{eqnarray}\n\tTherefore, by (\\ref{yu-7-3-1})--(\\ref{yu-7-3-b-1}) and taking $\\lambda_0>1$ large enough, we get \n{\\begin{eqnarray}\\label{yu-7-3-3}\n\t&\\;&Cs^3\\lambda^4\\int_{\\omega_4}\\alpha^3\\theta^2|V|^2dxdx_{N+1}+Cs\\lambda^2\\int_{\\omega_4}\\alpha\\theta^2|\\nabla V|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq&C\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla\\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+Cs\\lambda\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha\\theta^2|\\nabla V|^2d\\Gamma\n\t+Cs^3\\lambda^3\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha^3\\theta^2|V|^2d\\Gamma.\n\\end{eqnarray}}\n\tBy the definition of $\\varphi$ (see (\\ref{yu-7-2-11})), we know that \n\\begin{equation*}\n\\begin{cases}\n\t\\alpha\\geq e^{3\\lambda}\\;\\;\\mbox{and}\\;\\;\\theta\\geq e^{se^{3\\lambda}}&\\mbox{in}\\;\\;\\omega_1,\\\\\n\t\\alpha\\leq e^{\\lambda}\\;\\;\\mbox{and}\\;\\;\\theta\\leq e^{se^{\\lambda}}&\\mbox{in}\\;\\;\\overline{\\omega_4}\\backslash\\omega_2. \n\\end{cases}\n\\end{equation*}\n\tMoreover, by the definition of $\\eta$, we have \n\\begin{equation*}\\label{yu-7-3-4}\n\t\\nabla \\eta=0\\;\\;\\mbox{in}\\;\\;\\omega_2\\cup (\\overline{\\omega_4}\\backslash\\omega_3).\n\\end{equation*}\n\tBy the fact that $V=\\eta g$, one can get \n\\begin{eqnarray}\\label{yu-7-3-5}\n\t&\\;&Cs^3\\lambda^4\\int_{\\omega_4}\\alpha^3\\theta^2|V|^2dxdx_{N+1}+Cs\\lambda^2\\int_{\\omega_4}\\alpha \\theta^2|\\nabla V|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\geq&Cs^3\\lambda^4\\int_{\\omega_1}\\alpha^3\\theta^2|g|^2dxdx_{N+1}+Cs\\lambda^2\\int_{\\omega_1}\\alpha \\theta^2|\\nabla g|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\geq&Cs^3\\lambda^4e^{9\\lambda}e^{2se^{3\\lambda}}\\int_{\\omega_1}|g|^2dxdx_{N+1}\n\t+Cs\\lambda^2e^{3\\lambda}e^{2se^{3\\lambda}}\\int_{\\omega_1}|\\nabla g|^2dxdx_{N+1}.\n\\end{eqnarray}\n\tMoreover, \n\\begin{eqnarray}\\label{yu-7-3-6}\n\t&\\;&\\int_{\\omega_4}\\theta^2|\\mbox{div}(\\bar{A}\\nabla\\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)\\eta|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq&\\frac{C}{r^4}\\int_{\\omega_3\\backslash\\omega_2}\\theta^2(|g|^2+|\\nabla g|^2)dxdx_{N+1}\t\\leq\\frac{C}{r^4}e^{2se^\\lambda}\\int_{\\omega_3\\backslash\\omega_2}(|g|^2+|\\nabla g|^2)dxdx_{N+1}.\n\\end{eqnarray}\n\tFurther, \n{\\begin{eqnarray}\\label{yu-7-3-7}\n\t&\\;&Cs\\lambda\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha \\theta^2|\\nabla V|^2d\\Gamma+Cs^3\\lambda^3\\int_{\\triangle_{r_4}(x_0)\\times\\{0\\}}\\alpha^3\\theta^2|V|^2d\\Gamma\\nonumber\\\\\n\t&\\leq&\\frac{Cs^3\\lambda^3}{r^2}e^{3\\lambda}e^{2se^{4\\lambda}}\n\t\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}|g|^2d\\Gamma+Cs\\lambda e^\\lambda e^{2se^{4\\lambda}}\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}\n\t|\\nabla g|^2d\\Gamma.\n\\end{eqnarray}}\n\tCombining (\\ref{yu-7-3-3})--(\\ref{yu-7-3-7}), we have \n\\begin{eqnarray*}\\label{yu-7-3-8}\n\tCe^{3\\lambda}e^{2se^{3\\lambda}}\\int_{\\omega_1}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\t&\\leq&\\frac{1}{r^4}e^{2se^{\\lambda}}\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+\\frac{s^3\\lambda^3}{r^2}e^{3\\lambda}e^{2se^{4\\lambda}}\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+\n\t|\\nabla g|^2)d\\Gamma.\n\\end{eqnarray*}\nHence,\n{\\begin{eqnarray}\\label{yu-7-3-9}\n\tC\\int_{\\omega_1}(|g|^2+|\\nabla g|^2)dxdx_{N+1}&\\leq&\\frac{1}{r^4}e^{-3\\lambda}e^{2s(e^\\lambda-e^{3\\lambda})}\n\t\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+\\frac{s^3\\lambda^3}{r^2}e^{2s(e^{4\\lambda}-e^{3\\lambda})}\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+|\\nabla g|^2)d\\Gamma.\n\\end{eqnarray}}\n\tFix $\\lambda:=\\lambda_0 >1$ and define\n\\begin{equation*}\\label{yu-7-3-10}\n\t\\epsilon:=e^{-3\\lambda_0}e^{2s(e^{\\lambda_0}-e^{3\\lambda_0})},\\;\\;\n\t\\mu:=\\frac{2s(e^{4\\lambda_0}-e^{3\\lambda_0})+3(\\ln s+\\ln \\lambda_0)}{2s(e^{3\\lambda_0}-e^{\\lambda_0})\n\t+3\\lambda_0},\n\\end{equation*}\n\\begin{equation*}\\label{yu-7-3-11}\n\t\\epsilon_0:=e^{-3\\lambda_0}e^{2s_0(e^{\\lambda_0}-e^{3\\lambda_0})}.\n\\end{equation*}\n\tSo, (\\ref{yu-7-3-9}) can be rewritten by \n\\begin{multline}\\label{yu-7-3-12}\n\tC\\int_{\\omega_1}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\\\\n\t\\leq \\frac{\\epsilon}{r^4}\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\t+\\frac{\\epsilon^{-\\mu}}{r^2}\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+|\\nabla g|^2)d\\Gamma.\n\\end{multline}\n\n\tWe treat two cases separately. \n\t\n\t$\\bullet$ If \n\\begin{equation*}\\label{yu-7-3-13}\n\t\\left(\\frac{\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+|\\nabla g|^2)d\\Gamma}{\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\t}\\right)^{\\frac{1}{1+\\mu}}> \\epsilon_0. \n\\end{equation*}\n\tThen \n{\\begin{eqnarray}\\label{yu-7-3-14}\n\t&\\;&C\\int_{\\omega_1}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\nonumber\\\\\n\t&=&C\\left(\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\right)^{1-\\frac{1}{1+\\mu}}\n\t\\left(\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\right)^{\\frac{1}{1+\\mu}}\\nonumber\\\\\n\t&\\leq&C\\left(\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\right)^{1-\\frac{1}{1+\\mu}}\n\t\\left(\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+|\\nabla g|^2)d\\Gamma\\right)^{\\frac{1}{1+\\mu}}.\n\\end{eqnarray}}\n\n$\\bullet$\tIf \n{\\begin{equation*}\\label{yu-7-3-15}\n\t\\left(\\frac{\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+|\\nabla g|^2)d\\Gamma}{\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\t}\\right)^{\\frac{1}{1+\\mu}}\\leq\\epsilon_0. \n\\end{equation*}}\n\tIn this case, we choose a $s>s_0$ such that {$\\epsilon=\\left(\\frac{\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+|\\nabla g|^2)d\\Gamma}{\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\n\t}\\right)^{\\frac{1}{1+\\mu}}$} in (\\ref{yu-7-3-12}), we have \t\n{\\begin{eqnarray}\\label{yu-7-3-16}\t\n\t&\\;&C\\int_{\\omega_1}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\nonumber\\\\\n\t&\\leq&\\frac{1}{r^4}\\left(\\int_{\\omega_3}(|g|^2+|\\nabla g|^2)dxdx_{N+1}\\right)^{1-\\frac{1}{1+\\mu}}\n\t\\left(\\int_{\\triangle_{r_3}(x_0)\\times\\{0\\}}(|g|^2+|\\nabla g|^2)d\\Gamma\\right)^{\\frac{1}{1+\\mu}}.\n\\end{eqnarray}}\n\tCombining (\\ref{yu-7-3-14}) and (\\ref{yu-7-3-16}), we get that \n{\\begin{equation}\\label{yu-7-3-17}\n\t\\|g\\|^2_{H^1(B_{r_1}^+(x_0,0))}\\leq Cr^{-4}\\|g\\|_{H^1(B_{r_3}^+(x_0,0))}^{\\frac{2}{1+\\mu}}\n\t\\left(\\|g(\\cdot,0)\\|^2_{H^1(\\triangle_{r_3}(x_0))}+\\|\\partial_{{N+1}}g(\\cdot,0)\\|^2_{L^2(\\triangle_{r_3}(x_0))}\\right)^{\\frac{\\mu}{1+\\mu}}.\n\\end{equation}}\n\n\nNote that $g$ is an even function with respect to the variable ${x_{N+1}}$. So, by (\\ref{yu-7-3-17}), we have (\\ref{yu-7-3-b-2}) and the proof is completed. \n\\end{proof}\n\t\n\t\n\\par\n\n\\begin{lemma}[Three-ball inequality]\n\\label{yu-lemma-7-4-2}\n There is $\\beta\\in(0,1)$ such that for any\n\t$r\\in(0,\\frac{1}{12}\\min\\{R,L\\})$, the inequality \n\t\\begin{equation}\\label{yu-7-31-1-bb}\n\t\\|g\\|_{L^2(B_{6r}(x_0,0))}\\leq C(r)\\|g\\|^\\beta_{L^2(B_{r}(x_0,0))}\\|g\\|_{L^2(B_{8r}(x_0,0))}^{1-\\beta}\n\\end{equation}\nholds for all solutions of \n\\begin{equation*}\\label{yu-7-26-1}\n\t\\mbox{div}(A(x)\\nabla g)+l(x)g_{x_{N+1}x_{N+1}}-b(x)g=0\\;\\;\\mbox{in}\\;\\;\\triangle_R(x_0)\\times(-L,L).\n\\end{equation*}\n\\end{lemma}\n\n\n\n\\begin{proof}\n\t\nWe divide the proof into the following two steps.\n\\vskip 5pt\n \\textbf{Step 1.} \n For any $r<\\min\\{R,L\\}$, let us set \n $$r_1=r,\\quad r_2=6r,\\quad r_3=8r,\\quad r_4=12r.$$ \nTake \n\\begin{equation}\\label{yu-7-31-1}\n\t\\varphi(x,x_{N+1})=r_4^2-|x-x_0|^2-x_{N+1}^2, \\quad (x,x_{N+1})\\in B_{r_4}(x_0,0),\n\\end{equation}\n\t and set a cutoff function $\\eta\\in C^\\infty(\\mathbb{R}^{N+1};[0,1])$ to be such that \n\\begin{equation*}\\label{yu-7-31-2}\n\\begin{cases}\n\t\\eta=0&\\mbox{in}\\;\\;\\overline{B_{\\frac{r_1}{2}}(x_0,0)},\\\\\n\t\\eta=1&\\mbox{in}\\;\\;\\overline{B_{\\frac{r_2+r_3}{2}}(x_0,0)}\\backslash B_{\\frac{3r_1}{4}}(x_0,0),\\\\\n\t\\eta=0&\\mbox{in}\\;\\;\\overline{B_{r_4}(x_0,0)}\\backslash B_{\\frac{r_2+3r_3}{4}}(x_0,0)\\\\\n\t|\\mbox{div}\\bar{A}\\nabla \\eta|+|\\nabla\\eta|^2\\leq \\frac{C}{r^2}&\\mbox{in}\\;\\;\\mathbb{R}^{N+1} ,\n\\end{cases}\n\\end{equation*}\n\twhere $C>0$ is a positive constant independent of $r$. Let $V=\\eta g$. Then, \n\\begin{equation*}\\label{yu-7-31-2}\n\\begin{cases}\n\t\\mbox{div}(\\bar{A}\\nabla V)-bV=\\mbox{div}(\\bar{A}\\nabla\\eta) g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)&\\mbox{in}\\;\\;\n\tB_{r_4}(x_0,0),\\\\\n\t|\\nabla V|=V=0&\\mbox{on}\\;\\;\\partial B_{r_4}(x_0,0). \n\\end{cases}\n\\end{equation*}\nTaking $W:=\\theta V$ and repeating the proof of Step 1 in Lemma \\ref{yu-proposition-7-1-1}, one can claim that there is $\\lambda_0(r)>0$ such that for any $\\lambda\\geq \\lambda_0(r)$, one can find $s_0(r)>1$ such that \n\t$s\\geq s_0$,\n\\begin{eqnarray*}\\label{yu-7-31-3}\n\tC(r)s^3\\lambda^4\\int_{B_{r_4}(x_0,0)}\\alpha^3|W|^2dxdx_{N+1}+C(r)s\\lambda^2\\int_{B_{r_4}(x_0,0)}\\alpha|\\nabla W|^2dxdx_{N+1}\\nonumber\\\\\n\t\\leq \\int_{B_{r_4}(x_0,0)}\\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2dxdx_{N+1}.\n\\end{eqnarray*}\t\n\tSimilar to the proof of (\\ref{yu-7-3-3}), we can get \n\\begin{eqnarray}\\label{yu-7-31-4}\n\tC(r)s^3\\lambda^4\\int_{B_{r_4}(x_0,0)}\\alpha^3\\theta^2|V|^2dxdx_{N+1}\n\t+C(r)s\\lambda^2\\int_{B_{r_4}(x_0,0)}\\alpha\\theta^2|\\nabla V|^2dxdx_{N+1}\\nonumber\\\\\n\t\\leq \\int_{B_{r_4}(x_0,0)}\\theta^2|\\mbox{div}(\\bar{A}\\nabla \\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)|^2dxdx_{N+1}.\n\\end{eqnarray}\n\\vskip 5pt\n\t\\textbf{Step 2.} By the definition of $\\varphi$ (see (\\ref{yu-7-31-1})), we have \n\\begin{equation*}\\label{yu-7-31-5}\n\\begin{cases}\n\t\\alpha\\geq e^{108\\lambda r^2}\\geq 1,\\;\\;\\theta\\geq e^{se^{108\\lambda r^2}}\n\t&\\mbox{in}\\;\\;\\overline{B_{r_2}(x_0,0)}\\backslash B_{r_1}(x_0,0),\\\\\n\t\\theta\\leq e^{se^{144\\lambda r^2}}&\\mbox{in}\\;\\;\\overline{B_{\\frac{3r_1}{4}}(x_0,0)},\\\\\n\t\\theta\\leq e^{se^{95\\lambda r^2}}&\\mbox{in}\\;\\;\n\t\\overline{B_{\\frac{r_2+3r_3}{4}}(x_0,0)}\\backslash B_{\\frac{r_2+r_3}{2}}(x_0,0). \n\\end{cases}\n\\end{equation*}\n\tFurther, \n\\begin{equation*}\\label{yu-7-31-6}\n\t|\\mbox{div}(\\bar{A}\\nabla\\eta)|=|\\nabla \\eta|=0\\;\\;\\mbox{in}\\;\\;\\overline{B}_{\\frac{r_1}{2}(x_0,0)}\\bigcup \\left(\\overline{B_{\\frac{r_2+r_3}{2}}(x_0,0)}\\backslash B_{\\frac{3r_1}{4}}(x_0,0)\\right)\\bigcup \\left(\\overline{B_{r_3}(x_0,0)}\\backslash B_{\\frac{r_2+3r_3}{4}}(x_0,0)\\right).\n\\end{equation*}\n\tHence, from the fact $V=\\eta g$, we have \n\\begin{equation}\\label{yu-7-31-7}\n\tC(r)s^3\\lambda^4\\int_{B_{r_4}(x_0,0)}\\alpha^3\\theta^2|V|^2dxdx_{N+1}\n\t\\geq C(r)s^3\\lambda^4e^{2se^{108\\lambda r^2}}\\int_{B_{r_2}(x_0,0)\\backslash B_{r_1}(x_0,0)}|g|^2dxdx_{N+1},\n\\end{equation}\n\tand \n\\begin{eqnarray}\\label{yu-7-31-8}\n\t&\\;&\\int_{B_{r_4}(x_0,0)}\\theta^2|\\mbox{div}(\\bar{A}\\nabla \\eta)g+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq& e^{2se^{144\\lambda r^2}}\\int_{B_{\\frac{3r_1}{4}}(x_0,0)\\backslash B_{\\frac{r_1}{2}}(x_0,0)}\n\t\\left(\\frac{1}{r^4}|g|^2+\\frac{1}{r^2}|\\nabla g|^2\\right)dxdx_{N+1}\\nonumber\\\\\n\t&\\;&+e^{2se^{95\\lambda r^2}}\\int_{B_{\\frac{r_2+3r_3}{4}}(x_0,0)\\backslash B_{\\frac{r_2+r_3}{2}}(x_0,0)}\n\t\\left(\\frac{1}{r^4}|g|^2+\\frac{1}{r^2}|\\nabla g|^2\\right)dxdx_{N+1}.\n\\end{eqnarray}\n\tBy the interior estimate of elliptic equations\n\\begin{equation*}\\label{yu-7-31-9}\n\t\\int_{B_{\\frac{3r_1}{4}}(x_0,0)\\backslash B_{\\frac{r_1}{2}}(x_0,0)}|\\nabla g|^2dxdx_{N+1}\n\t\\leq\\frac{C}{r^2}\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\n\\end{equation*}\t\n\tand \n\\begin{equation*}\\label{yu-7-31-10}\n\t\\int_{B_{\\frac{r_2+3r_3}{4}}(x_0,0)\\backslash B_{\\frac{r_2+r_3}{2}}(x_0,0)}|\\nabla g|^2dxdx_{N+1}\n\t\\leq \\frac{C}{r^2}\\int_{B_{r_3}(x_0,0)\\backslash B_{r_2}(x_0,0)}|g|^2dxdx_{N+1}. \n\\end{equation*}\n\tThese, along with (\\ref{yu-7-31-8}), yield that \n\\begin{eqnarray}\\label{yu-7-31-11}\n\t&\\;&\\int_{B_{r_4}(x_0,0)}\\theta^2|\\mbox{div}(\\bar{A}\\nabla \\eta)g\n\t+2\\nabla\\eta\\cdot(\\bar{A}\\nabla g)|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq&C\\frac{1}{r^4}e^{2se^{144\\lambda r^2}}\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\n\t+C\\frac{1}{r^4}e^{2se^{95\\lambda r^2}}\\int_{B_{r_3}(x_0,0)\\backslash B_{r_2}(x_0,0)}|g|^2dxdx_{N+1}. \n\\end{eqnarray}\n\tFrom (\\ref{yu-7-31-4}), (\\ref{yu-7-31-7}) and (\\ref{yu-7-31-11}), we get \n\\begin{eqnarray}\\label{yu-7-31-12}\n\tC(r)\\int_{B_{r_2}(x_0,0)\\backslash B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\n\t\\leq e^{2s(e^{144\\lambda r^2}-e^{108\\lambda r^2})}\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\\nonumber\\\\\n\t+e^{2s(e^{95\\lambda r^2}-e^{108\\lambda r^2})}\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}. \n\\end{eqnarray}\n\tFix $\\lambda:=\\lambda_0>0$ and denote \n\\begin{equation*}\\label{yu-7-31-13}\n\t\\epsilon:=e^{2s(e^{95\\lambda_0r^2}-e^{108\\lambda_0 r^2})},\\;\\;\n\t\\epsilon_0:=e^{2s_0(e^{95\\lambda_0r^2}-e^{108\\lambda_0 r^2})}\n\\end{equation*}\n\tand \n\\begin{equation*}\\label{yu-7-31-14}\n\t\\mu:=\\min_{r> 0}\\frac{e^{144\\lambda_0r^2}-e^{108\\lambda_0 r^2}}{e^{108\\lambda_0 r^2}-e^{95\\lambda_0 r^2}}>0. \n\\end{equation*}\n\tNote that, this minimum can be taken by the fact \n$$\n\t\\lim_{r\\to 0}\\frac{e^{144\\lambda_0r^2}-e^{108\\lambda_0 r^2}}{e^{108\\lambda_0 r^2}-e^{95\\lambda_0 r^2}}=\\frac{36}{13}.\n$$\n\tThen, it follows from (\\ref{yu-7-31-12}) that \n\\begin{multline}\\label{yu-7-31-15}\n\tC(r)\\int_{B_{r_2}(x_0,0)\\backslash B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\\\\\n\t\\leq \\epsilon^{-\\mu} \\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\n\t+\\epsilon\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}.\n\\end{multline}\n\nWe treat in two cases separately.\n\n$\\bullet$ If \n\\begin{equation*}\\label{yu-7-31-16}\n\t\\left(\\frac{\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}}{\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}}\\right)^{\\frac{1}{1+\\mu}}>\\epsilon_0, \n\\end{equation*}\n\tthen \n\\begin{eqnarray}\\label{yu-7-31-17}\n\t&\\;&C(r)\\int_{B_{r_2}(x_0,0)\\backslash B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\\nonumber\\\\\n\t&\\leq& C\\left(\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}\\right)^{1-\\frac{1}{1+\\mu}}\n\t\\left(\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}\\right)^{\\frac{1}{1+\\mu}}\\nonumber\\\\\n\t&\\leq& C\\epsilon_0\\left(\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}\\right)^{\\frac{\\mu}{1+\\mu}}\n\t\\left(\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\\right)^{\\frac{1}{1+\\mu}}.\n\\end{eqnarray}\n\n$\\bullet$\tIf \n\\begin{equation*}\\label{yu-7-31-18}\n\t\\left(\\frac{\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}}{\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}}\\right)^{\\frac{1}{1+\\mu}}\\leq \\epsilon_0, \n\\end{equation*}\n\twe choose $s\\geq s_0$ such that $\\epsilon=\\left(\\frac{\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}}{\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}}\\right)^{\\frac{1}{1+\\mu}}$. \n\tThen, by (\\ref{yu-7-31-15}), we have \n\\begin{equation}\\label{yu-7-31-19}\n\tC(r)\\int_{B_{r_2}(x_0,0)\\backslash B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\n\t\\leq 2\\left(\\int_{B_{r_3}(x_0,0)}|g|^2dxdx_{N+1}\\right)^{\\frac{\\mu}{1+\\mu}}\n\t\\left(\\int_{B_{r_1}(x_0,0)}|g|^2dxdx_{N+1}\\right)^{\\frac{1}{1+\\mu}}.\n\\end{equation}\n\tSo, by (\\ref{yu-7-31-17}) and (\\ref{yu-7-31-19}), we get (\\ref{yu-7-31-1-bb}) \n\twith $\\beta=\\frac{1}{1+\\mu}$. \n\tThe proof is completed. \n\\end{proof}\n\n\n\n\n\\subsection{Proof of Proposition \\ref{yu-theorem-7-5-1}}\n\n\\begin{proof}[\\textbf{Proof of Proposition \\ref{yu-theorem-7-5-1}}]\n Arbitrarily take $R\\in(0,\\min\\{R_0,\\rho\\})$. Let $u_1$ and $u_2$ be accordingly the solution to \n\\begin{equation*}\\label{yu-7-4-4}\n\\begin{cases}\n\tl(x)\\partial_tu_{1}-\\mbox{div}(A(x)\\nabla u_1)+b(x)u_1=0&\\mbox{in}\\;\\;\\triangle_R(x_0)\\times(0,2T),\\\\\n\tu_1=u&\\mbox{on}\\;\\;\\partial \\triangle_R(x_0)\\times(0,2T),\\\\\n\tu_1(\\cdot,0)=0 &\\mbox{in}\\;\\;\\triangle_{R}(x_0)\n\\end{cases}\n\\end{equation*}\t\n\tand\n\\begin{equation*}\\label{yu-7-4-5}\n\\begin{cases}\n\tl(x)\\partial_tu_{2}-\\mbox{div}(A(x)\\nabla u_2)+b(x)u_2=0&\\mbox{in}\\;\\;\\triangle_R(x_0)\\times(0,2T),\\\\\n\tu_2=0&\\mbox{on}\\;\\;\\partial \\triangle_R(x_0)\\times(0,2T),\\\\\n\tu_2(\\cdot,0)=u(\\cdot,0)&\\mbox{in}\\;\\;\\triangle_{R}(x_0).\n\\end{cases}\n\\end{equation*}\nIt is clear that\n\t$u=u_1+u_2$ in $\\triangle_R(x_0)\\times[0,2T]$.\nBy a standard energy estimate for parabolic equations, we have\n\\begin{equation}\\label{yu-7-4-7}\n\t\\sup_{t\\in[0,T]}\\|u_2(\\cdot,t)\\|_{H^1(\\triangle_R(x_0))}\\leq Ce^{CT}\\|u(\\cdot,0)\\|_{H^1(\\triangle_R(x_0))}.\n\\end{equation}\nHence\n\\begin{equation}\\label{yu-7-4-8}\n\t\\sup_{t\\in[0,T]}\\|u_1(\\cdot,t)\\|_{H^1(\\triangle_R(x_0))}\\leq C(1+e^{CT})\\sup_{t\\in[0,T]}\\|u(\\cdot,t)\\|_{H^1(\\triangle_R(x_0))}.\n\\end{equation}\n\n\n\nFix arbitrarily $t_0\\in(0,\\frac{T}{2})$ and let $v_1$ be the solution of \n \\begin{equation*}\\label{yu-11-29-4-jia}\n\\begin{cases}\n l(x)\\partial_tv_1-\\mbox{div}(A(x)\\nabla v_1)+b(x)v_1=0&\\mbox{in}\\;\\;\\triangle_{R}(x_0)\\times\\mathbb{R}^+,\\\\\n v_1=\\eta u_1&\\mbox{on}\\;\\;\\partial\\triangle_R(x_0)\\times\\mathbb{R}^+,\\\\\n v_1(\\cdot,0)=0&\\mbox{in}\\;\\;\\triangle_R(x_0),\n\\end{cases}\n\\end{equation*}\nwhere $\\eta$ is given by \\eqref{yu-6-6-6}.\nClearly,\n\t $u=v_1+u_2$ in $\\triangle_R(x_0)\\times[0,t_0]$. In particular, \n\\begin{equation*}\\label{yu-7-4-6}\n\tu(\\cdot,t_0)=v_1(\\cdot,t_0)+u_2(\\cdot,t_0)\\;\\;\\mbox{in}\\;\\;\\triangle_R(x_0).\n\\end{equation*}\nDefine\n\\begin{equation*}\\label{yu-6-18-5jia}\n\t\\tilde{v}_1(\\cdot,t):=\n\\begin{cases}\n\tv_1(\\cdot,t)&\\mbox{if}\\;\\;t\\geq 0,\\\\\n\t0&\\mbox{if}\\;\\;t<0,\n\\end{cases}\n\\end{equation*}\nand\n\\begin{equation*}\\label{yu-6-18-6jia}\n\t\\hat{v}_1(x,\\mu)=\\int_{\\mathbb{R}}e^{-i\\mu t}\\tilde{v}_1(x,t)dt\\quad\\text{for}\\;\\;(x,\\mu)\\in\\triangle_R(x_0)\\times\\mathbb R.\n\\end{equation*}\nNote \tfrom Lemma \\ref{yu-lemma-6-10-1} that $\\hat{v}_1$ is well defined.\n\\par\nLet \n\\begin{equation*}\\label{yu-7-5-bb-1}\n\t\\kappa:=\\min\\left\\{\\frac{1}{2},\\frac{\\sqrt{2}}{4e\\Pi}\\right\\} \\quad\\text{with}\\;\\;\\Pi\\;\\;\\text{given in Lemma\n\t\\ref{yu-lemma-6-18-1}}.\n\\end{equation*}\nWe define\n$$V=V_1+V_2\\quad\\mbox{in}\\;\\;\\triangle_R(x_0)\\times(-\\kappa R,\\kappa R),\n$$\nwhere \n\\begin{equation}\\label{yu-6-23-6jia}\n\tV_1(x,y)=\\frac{1}{2\\pi}\\int_{\\mathbb{R}}e^{it_0\\mu}\\hat{v}_1(x,\\mu)\n\t\\frac{\\sinh(\\sqrt{-i\\mu}y)}{\\sqrt{-i\\mu}}d\\mu\\quad\\mbox{in}\\;\\;\\triangle_R(x_0)\\times(-\\kappa R,\\kappa R)\n\\end{equation}\nand \n\\begin{equation}\\label{yu-7-5-7}\n\tV_2(x,y)=\\sum_{i=1}^\\infty\\alpha_ie^{-\\mu_it_0}f_i(x)\\frac{\\sinh(\\sqrt{\\mu_i}y)}\n\t{\\sqrt{\\mu_i}}\\;\\;\\mbox{in}\\;\\;\\triangle_R(x_0)\\times(-\\kappa R,\\kappa R)\n\\end{equation}\nwith \n\\begin{equation*}\\label{yu-7-5-6}\n\t\\alpha_i=\\int_{\\triangle_R(x_0)}l(x)u_2(x,0)f_i(x)dx\\;\\;\\mbox{for each}\\;\\;i\\in\\mathbb{N}^+,\n\\end{equation*}\nand $\\{\\mu_i\\}_{i=1}^\\infty\\subset\\mathbb{R}^+$, $\\{f_i\\}_{i=1}^{\\infty}\\subset \\mathcal{L}^2(\\triangle_{R}(x_0))$ given by (\\ref{yu-6-7-10}). Here we note that from \nLemma \\ref{yu-lemma-6-18-1}, $V_1$ is also well defined. \nOne can readily check that \n\\begin{equation}\\label{yu-7-5-9}\n\\begin{cases}\n\t\\mbox{div}(A(x)\\nabla V)+l(x)V_{yy}-b(x)V=0&\\mbox{in}\\;\\;\n\t\\triangle_{\\frac{R}{2}}(x_0)\\times(-\\kappa R,\\kappa R),\\\\\n\tV(x,0)=0&\\mbox{in}\\;\\;\\triangle_{\\frac{R}{2}}(x_0),\\\\\n\tV_y(x,0)=u(x,t_0)&\\mbox{in}\\;\\;\\triangle_{\\frac{R}{2}}(x_0).\n\\end{cases}\n\\end{equation}\t\n\tBy Lemma \\ref{yu-lemma-7-4-2}, we have for any $r\\in (0,\\frac{1}{16} \\kappa R)$, \n\\begin{equation}\\label{yu-7-4-10}\n \\|V\\|_{L^2(B_{6r}(x_0,0))}\\leq C(r)\\|V\\|^\\beta_{L^2(B_{r}(x_0,0))}\\|V\\|_{L^2(B_{8r}(x_0,0))}^{1-\\beta}.\n\\end{equation}\nSince $V_y$ also satisfies the first equation of (\\ref{yu-7-5-9}),\tby the interior estimate of elliptic equations\nwe find\n\\begin{eqnarray}\\label{yu-7-4-11}\n\t\\int_{B_{6r}(x_0,0)}|V|^2dxdy&\\geq &Cr^2\\int_{B_{11r\/2}(x_0,0)}(|\\nabla V|^2+|V_y|^2)dxdy\\nonumber\n\t\\\\&\\geq&\\frac{Cr^2}{2}\n\t\\left(\\int_{B_{11r\/2}(x_0,0)}|V_y|^2dxdy+\\int_{B_{11r\/2}(x_0,0)}|V_y|^2dxdy\\right)\\nonumber\\\\\n\t&\\geq&Cr^2\\left(\\int_{B_{11r\/2}(x_0,0)}|V_y|^2dxdy+r^2\\int_{B_{5r}(x_0,0)}(|\\nabla V_y|^2+|V_{yy}|^2)dxdy\\right)\\nonumber\\\\\n\t&\\geq& Cr^3\\left(\\frac{1}{r}\\int_{B_{5r}(x_0,0)}|V_y|^2dxdy+r\\int_{B_{5r}(x_0,0)}|V_{yy}|^2dxdy\\right).\n\\end{eqnarray}\n As a simple corollary of \\cite[Lemma 9.9, page 315]{Brezis}, we have the following trace theorem \n\\begin{equation*}\\label{yu-7-4-12}\n\t\\int_{\\triangle_{4r}(x_0)}|Q(x,0)|^2dx\\leq C\\left(\\frac{1}{r}\\int_{B_{5r}^+(x_0,0)}|Q|^2dxdy+r\\int_{B^+_{5r}(x_0,0)}|Q_y|^2dxdy\\right)\n\\end{equation*}\n for any $Q\\in H^1(B_{5r}(x_0,0))$. Hence, by (\\ref{yu-7-5-9}) and (\\ref{yu-7-4-11}) we have \n\\begin{eqnarray}\\label{yu-7-4-13}\n\tCr^3\\int_{\\triangle_{4r}(x_0)}|u(x,t_0)|^2dx\\leq \\int_{B_{6r}(x_0,0)}|V|^2dxdy.\n\\end{eqnarray}\t\nBy Lemma \\ref{yu-proposition-7-1-1}, we obtain that there is $\\gamma\\in (0,1)$ such that for any $r\\in(0,\\frac{1}{3}\\kappa R)$, \n\\begin{equation}\\label{yu-7-4-2}\n\t\\|V\\|_{L^2(B_r(x_0,0))}\\leq Cr^{-2}\\|V\\|_{H^1(B_{2r}(x_0,0))}^{\\gamma}\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{2r}(x_0))}^{1-\\gamma}.\n\\end{equation}\nAgain, by the interior estimate, there is a constant $C>0$ such that \n\\begin{equation}\\label{yu-7-4-3}\n\t\\|V\\|_{H^1(B_{2r}(x_0,0))}\\leq Cr^{-1} \\|V\\|_{L^2(B_{3r}(x_0,0))}.\n\\end{equation}\nHence, it follows from (\\ref{yu-7-4-2}) and (\\ref{yu-7-4-3}) that\n\\begin{equation}\\label{yu-7-4-1}\n\t\\|V\\|_{L^2(B_r(x_0,0))}\\leq Cr^{-3}\\|V\\|_{L^2(B_{3r}(x_0,0))}^{\\gamma}\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{2r}(x_0))}^{1-\\gamma}.\n\\end{equation}\nIt follows from (\\ref{yu-7-4-13}), (\\ref{yu-7-4-10}) and \\eqref{yu-7-4-1} that \n\\begin{equation}\\label{yu-7-5-1}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{4r(x_0)})}\\leq C(r)r^{-3(\\frac{1}{2}+\\beta)}\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{2r}(x_0))}^{(1-\\gamma)\\beta}\n\t\\|V\\|^{1-(1-\\gamma)\\beta}_{L^2(B_{8r}(x_0,0))}.\n\\end{equation}\n\\par\n\\medskip\n\nTo finish the proof, it suffices to bound the term $\\|V\\|_{L^2(B_{8r}(x_0,0))}$. Recall that $V=V_1+V_2$, we will treat \n$V_1$ and $V_2$ separately.\n\nIn fact, we derive from (\\ref{yu-6-23-6jia}) that\n\tfor each $x\\in \\triangle_{8r}(x_0)\\subset \\triangle_{\\frac{R}{2}}(x_0)$ and $|y|<\\frac{\\kappa R}{8}$, \n\\begin{eqnarray*}\\label{yu-7-5-2}\n\t|V_1(x,y)|\n\t&=&\\left|\\frac{1}{2\\pi}\\int_{\\mathbb{R}}e^{it_0\\mu}\\hat{v}_1(x,\\mu)\\int_{-y}^ye^{\\sqrt{-i\\mu}s}dsd\\mu\\right|\n\t\\leq \\frac{1}{2\\pi}\\int_{\\mathbb{R}}|\\hat{v}_1(x,\\mu)|\\int_{-y}^y|e^{\\sqrt{-i\\mu}s}|dsd\\mu\n\t\\nonumber\\\\\n\t&\\leq&\\frac{\\kappa R}{8\\pi}\\int_{\\mathbb{R}}|\\hat{v}_1(x,\\mu)|e^{\\frac{1}{8\\sqrt{2}}\\kappa\\sqrt{|\\mu|}R}d\\mu\\nonumber\\\\\n\t&\\leq &\\frac{\\kappa R}{8\\pi}\\left(\\int_{\\mathbb{R}}|\\hat{v}_1(x,\\mu)|^2e^{\\frac{3}{4\\sqrt{2}}\\kappa\\sqrt{|\\mu|}R}d\\mu\\right)^{\\frac{1}{2}}\n\t\\left(\\int_{\\mathbb{R}}e^{-\\frac{1}{2\\sqrt{2}}\\kappa \\sqrt{|\\mu|}R}d\\mu\\right)^{\\frac{1}{2}}\\nonumber\\\\\n\t&=&\\frac{\\sqrt{2}}{4\\pi}\\left(\\int_{\\mathbb{R}}|\\hat{v}_1(x,\\mu)|^2e^{\\frac{3}{4\\sqrt{2}}\\kappa\\sqrt{|\\mu|}R}d\\mu\\right)^{\\frac{1}{2}}.\n\\end{eqnarray*}\n\tHence, by Lemma \\ref{yu-lemma-6-18-1} and (\\ref{yu-7-4-8}), we have for each $r<\\frac{R}{32}$,\n\\begin{eqnarray}\\label{yu-7-5-3}\n\t\\int_{\\triangle_{8r}(x_0)}|V_1(x,y)|^2dx\n\t&\\leq& CT^{-1}e^{CR^{1-N}(1+\\frac{1}{T-t_0})T}G^2(R)\n\t\\int_{\\mathbb{R}}e^{-\\frac{1}{4\\sqrt{2}}\\kappa \\sqrt{|\\mu|}R}d\\mu\\nonumber\\\\\n\t&\\leq&\\frac{CT^{-1}e^{CR^{1-N}(1+\\frac{1}{T-t_0})(1+T)}G^2(R)}{R^2}.\n\\end{eqnarray}\n\t\\par\n\nWhile, by (\\ref{yu-7-5-7}) and (\\ref{yu-7-4-7}) we obtain \n\\begin{eqnarray}\\label{yu-7-5-10}\n\t\\int_{B_{8r}(x_0,0)}|V_2|^2dxdy&\\leq&\\Lambda_3 \\int_{-8r}^{8r}\\int_{\\triangle_{R}(x_0)}l(x)|V_2|^2dxdy\n\t\\leq \\Lambda_3\\int_{-8r}^{8r}\\sum_{i=1}^\\infty\\alpha_i^2e^{-2\\mu_it_0}\\left|\\frac{\\sinh(\\sqrt{\\mu_i}y)}{\\sqrt{\\mu_i}}\\right|^2dy\\nonumber\\\\\n\t&\\leq&2^{8}r^2\\Lambda_3\\left(1+e^{\\frac{8\\rho^2}{t_0}}\\right)\\sum_{i=1}^\\infty\\alpha_i^2\\leq \n\tCr^2e^{\\frac{C(1+T+T^2)}{t_0}}\\int_{\\triangle_R(x_0)}|u(x,0)|^2dx\\nonumber\\\\\n\t&\\leq& Ce^{\\frac{C(1+T^2)}{t_0}}G^2(R).\n\\end{eqnarray}\n\tTherefore, by (\\ref{yu-7-5-3}) and (\\ref{yu-7-5-10}) we conclude that\n\\begin{equation*}\\label{yu-7-5-11}\n\t\\|V\\|_{L^2(B_{8r}(x_0,0))}\\leq CR^{-2}e^{\\frac{CR^{1-N}(T^2+1)}{t_0}}G(R).\n\\end{equation*}\n\tThis, together with (\\ref{yu-7-5-1}), means that \n\\begin{equation*}\\label{yu-7-5-12}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{4r})}\n\t\\leq C(r)R^{-2[1-(1-\\gamma)\\beta]}e^{\\frac{CR^{1-N}(T^2+1)}{t_0}[1-(1-\\gamma)\\beta]}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{2r})}^{(1-\\gamma)\\beta}G^{1-(1-\\gamma)\\beta}(R).\n\\end{equation*}\n\tTaking $\\sigma=(1-\\gamma)\\beta$ and using a scaling technique, the proof is immediately achieved.\n\t\\end{proof}\n\n\n\n\\subsection{Proof of Proposition \\ref{yu-theorem-7-10-6}}\n\n\n\n\\begin{proof}[\\textbf{Proof of Proposition \\ref{yu-theorem-7-10-6}}]\nWe proceed the proof with three steps as follows.\n\n\\textbf{Step 1. In the interior.} Let $K_1$ and $K_2$ be two compact subsets with non-empty interior of $\\Omega$.\nDenoting $G_{\\Omega}=\\sup_{t\\in[0,T]}\\|u(\\cdot,t)\\|_{H^1(\\Omega)}$, we shall show that \n\\begin{equation}\\label{du919}\n\\|u(\\cdot,t_0)\\|_{L^2(K_1)}\\leq e^{\\frac{C(T^2+1)}{t_0}}\\|u(\\cdot,t_0)\\|^{\\sigma}_{L^2(K_2)}G_{\\Omega}^{1-\\sigma}.\n\\end{equation}\nIn fact, there exists a sequence of balls $\\{\\triangle_{r}(x_i)\\}_{j=0}^{p}$ such that \n\\begin{equation*}\\label{covering}\nK_1\\subset\\bigcup_{i=1}^{p}\\triangle_{r}(x_i)\\subset\\Omega, \\;\\;\\;\\;\n\\triangle_{r}(x_0)\\subset K_2,\n\\end{equation*}\nand for each $1\\leq i\\leq p$, there exists a chain\nof balls $\\triangle_{r}(x_i^j)$, $1\\leq j\\leq n_i$, such that\n\\begin{equation*}\\label{chain}\n\\begin{split}\n&\\triangle_r(x_i^{1})=\\triangle_{r}(x_i),\\;\\;\\triangle_r(x_i^{n_i})=\\triangle_{r}(x_0),\\\\\n&\\triangle_{r}(x_i^j)\\subset \\triangle_{2r}(x_i^{j+1})\\subset\\Omega,\\;\\;1\\leq j\\leq\nn_i-1.\n\\end{split}\n\\end{equation*}\nBy Proposition~\\ref{yu-theorem-7-5-1}, we obtain that there are constants \n$N_i^j= N_i^j(r,p)\\geq 1$ and $\\theta_i^j=\\theta_i^j(r,p)\\in(0,1)$ such\nthat\n\\begin{equation*}\n\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_r(x_i^j))}\\leq \\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{2r}(x_i^{j+1}))}\\leq e^{\\frac{N_i^j(T^2+1)}{t_0}}\\|u(\\cdot, t_0)\\|_{L^2(\\triangle_{r}(x_i^{j+1}))}^{\\theta_i^j}\nG_\\Omega^{1-\\theta_i^j}.\n\\end{equation*}\nIterating the above procedure, we derive that there are constants \n$N_i=N_i(K_1,K_2,p)\\geq 1$ and $\\theta_i=\\theta_i(K_1,K_2,p)\\in(0,1)$\nsuch that\n\\begin{equation*}\n\\|u(\\cdot,t_0)\\|_{L^2(\\triangle_{r}(x_i))}\\leq e^{\\frac{N_i(T^2+1)}{t_0}}\\|u(\\cdot, t_0)\\|_{L^2(\\triangle_{r}(x_0))}^{\\theta_i}\nG_\\Omega^{1-\\theta_i}.\n\\end{equation*}\nHence, \\eqref{du919} follows.\n\n\n\\medskip\n\n\n\n\t\n\t\t\n\\textbf{Step 2. Flattening the boundary and taking the even reflection.}\nArbitrarily fix $x_0\\in\\partial\\Omega$. Without loss of generality, we may assume that $A(x_0)=I$.\nFollowing the arguments to flatten locally the boundary as in \\cite{Adolfsson-Escauriaza} (see also \n\\cite{Canuto-Rosset-Vessella}), we have that \nthere exists a $C^1$-diffeomorphism $\\Phi$ from $\\triangle_{r_2}(0)$ to $\\triangle_{r_1}(x_0)$ such that\n\\begin{equation*}\\label{yu-7-27-1}\n\t\\Phi(y', 0)\\in \\partial\\Omega\\cap \\triangle_{r_1}(x_0)\\;\\;\\mbox{for each}\\;\\;y'\\in\\triangle'_{r_2}(0),\n\\end{equation*}\n\\begin{equation*}\\label{yu-7-27-4}\n\t\\Phi(\\triangle_{r_2}^{+}(0))\\subset \\triangle_{r_1}(x_0)\\cap\\Omega,\n\\end{equation*} \n\\begin{equation}\\label{yu-10-10-1}\nC^{-1}\\leq\\text{det} J\\Phi(y)\\leq C\\quad\\text{for each}\\;\\;y\\in\\triangle_{r_2}(0),\n\\end{equation}\n\\begin{equation}\\label{yu-10-10-2}\n|\\text{det} J\\Phi(y)-\\text{det} J\\Phi(\\tilde y)|\\leq C|y-\\tilde y|\\quad\\text{for each}\\;\\;y,\\tilde y\\in\\triangle_{r_2}(0),\n\\end{equation}\n\\begin{equation*}\\label{yu-10-10-3}\nC^{-1}|y-\\tilde y|\\leq |\\Phi(y)-\\Phi(\\tilde y)|\\leq C|y-\\tilde y|\\quad\\text{for each}\\;\\;y,\\tilde y\\in\\triangle_{r_2}(0),\n\\end{equation*}\n\\begin{equation}\\label{du8101}\n \\tilde{a}_{jN}(y',0)=\\tilde a_{Nj}(y',0)=0\\;\\;\\;\\text{for each}\\;\\;y'\\in \\triangle_{r_2}'(0), \\;\\;j=1,\\dots, N-1,\n\\end{equation}\n\twhere \n\\begin{equation*}\\label{yu-7-8-1}\n\t\\tilde{A}(y)=[\\tilde a_{ij}]_{N\\times N}=\\mbox{det}J\\Phi(y)(J\\Phi^{-1})(\\Phi(y))^{tr}A(\\Phi(y))(J\\Phi^{-1})(\\Phi(y)),\\quad\ny\\in \\triangle_{r_2}^{+}(0).\n\\end{equation*} \n\\par\n By (\\ref{yu-10-10-1}) and (\\ref{yu-10-10-2}), one can check that $\\tilde A(\\cdot)$ satisfies the uniform ellipticity condition and the Lipschitz condition in $\\triangle_{r_2}^{+}(0)$. \n Denoting \n\\begin{equation*}\\label{yu-7-8-2}\nz(y,t)=u(\\Phi(y),t),\\;\\;\\tilde{b}(y)=\\mbox{det}J\\Phi(y)b(\\Phi(y))\\quad\\text{for each}\\;\\;y\\in \\triangle_{r_2}^{+}(0),\\;\\;t\\in(0,2T),\n\\end{equation*}\nby (\\ref{yu-10-10-1}) we have\n\\begin{equation*}\\label{yu-7-8-5}\n\t\\tilde{b}(\\cdot)\\;\\;\\mbox{satisfies (\\ref{yu-6-24-1-1-b}) in}\\;\\;\\triangle_{r_2}^+(0),\n\\end{equation*}\n\\begin{equation*}\\label{du8102}\n\\left\\{\n\\begin{split}\n\\mbox{det}J\\Phi(y)z_t(y,t)-\\text{div}(\\tilde A(y)\\nabla z(y,t))+\\tilde b(y)=0\\;\\;\\;\\;\\text{in}\\;\\; \\triangle^+_{r_2}(0)\t\\times(0,2T), \\\\\n\\frac{\\partial z}{\\partial y_N}=0\\;\\;\\;\\;\\text{on}\\;\\; (\\triangle'_{r_2}(0)\\times\\{0\\})\t\\times(0,2T). \\\\\n\\end{split}\\right.\n\\end{equation*}\nFor any $y=(y',y_N)\\in\\triangle_{r_2}(0)$, using the even reflection and denoting $\\check{A}(y)=[\\check{a}^{ij}(y)]_{N\\times N}$ by \n\\begin{equation*}\\label{yu-7-8-6}\n\\begin{cases}\n\t\\check{a}_{ij}(y',y_N)=\\tilde{a}_{ij}(y',|y_N|),&\\mbox{if}\\;\\;1\\leq i,j\\leq N-1,\\;\\;\\mbox{or}\\;\\;i=j=N,\\\\\n\t\\check{a}_{Nj}(y',y_N)=\\check{a}_{jN}(y',y_N)=\\mbox{sign}(y_N)\\tilde{a}_{jN}(y',|y_N|),\n\t&\\mbox{if}\\;\\;1\\leq j\\leq N-1,\n\\end{cases}\n\\end{equation*}\n\\begin{equation*}\\label{yu-7-8-8}\n\t\\check{b}(y',y_N)=\\tilde{b}(y',|y_N|),\\;\\;\t\\check{l}(y',y_N)=\\mbox{det}J\\Phi(y',|y_N|),\n\\end{equation*}\nand\n\\begin{equation*}\\label{yu-7-8-9}\n\tZ(y,t)=z(y',|y_N|,t)\\;\\;\\mbox{for each}\\;\\;(y,t)\\in\\triangle_{r_2}(0)\\times(0,2T),\n\\end{equation*}\n By (\\ref{du8101}), we see that $\\check{A}$ verifies the uniform ellipticity condition and the Lipschitz condition in $\\triangle_{r_2}(0)$, $\\check{b}(\\cdot)$ verifies (\\ref{yu-6-24-1-1-b}) in $\\triangle_{r_2}(0)$, \n\\begin{equation*}\\label{yu-7-8-8}\nC^{-1}\\leq l(y)\\leq C,\\;\\;\t\t|l(y)-l(\\tilde y)|\\leq C|y-\\tilde y|\\quad\\mbox{for each}\\;\\;y,\\tilde y\\in\\triangle_{r_2}(0),\n\\end{equation*}\n\tand that \n\\begin{equation}\\label{yu-7-8-10}\n\t\\check{l}(y)Z_t(y,t)-\\mbox{div}(\\check{A}(y)\\nabla Z(y,t))+\\check{b}(y)=0\\;\\;\\mbox{in}\\;\\;\\triangle_{r_2}(0)\\times(0,2T).\n\\end{equation}\n\t\nLet $\\hat y=(0',r_2\/2)$. \nFor each $00$ and $\\rho>0$ such that \n\\begin{equation*}\\label{yu-7-8-b-1}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\Omega\\cap\\triangle_{r_3}(x_0))}\n\t\\leq C(r)e^{\\frac{C(T^2+1)}{t_0}}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\Omega_{\\rho})}^{\\sigma_1} G_\\Omega^{1-\\sigma_1}.\n\\end{equation*}\t\n\n\n\\medskip\n\n\\textbf{Step 3. Completing the proof.}\nWhen $\\Gamma$ is a neighborhood of\n$\\partial\\Omega$ in $\\Omega$, there are a sequence $\\{x_j\\}_{j=1}^{p}\\subset\\partial\\Omega$ and a sequence $\\{\\triangle_{r_j}(x_j)\\}_{j=1}^{p}$ such that \n$$\\Gamma\\subset \\bigcup_{j=1}^{p}(\\Omega\\cap \\triangle_{r_j}(x_j)).$$\nBy the result in Step 2 and a finite covering argument, we first have \n\\begin{equation}\\label{dujiu1}\n\\|u(\\cdot,t_0)\\|_{L^2(\\Gamma)}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\Omega_{\\rho})}^{\\sigma_1} G_\\Omega^{1-\\sigma_1}\n\\end{equation}\nwith some $\\rho>0$.\nBy the result in Step 1, we then have\n\\begin{equation*}\n\\|u(\\cdot,t_0)\\|_{L^2(\\Omega_\\rho)}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\omega)}^{\\sigma_1} G_\\Omega^{1-\\sigma_1}.\n\\end{equation*}\nThis, together with \\eqref{dujiu1}, indicates that \n\\begin{equation*}\n\\|u(\\cdot,t_0)\\|_{L^2(\\Gamma)}\\leq Ce^{\\frac{C(T^2+1)}{t_0}}\n\t\\|u(\\cdot,t_0)\\|_{L^2(\\omega)}^{\\sigma_2} G_\\Omega^{1-\\sigma_2}.\n\\end{equation*}\nWhich, combined with the result in Step 1 again, implies the desired estimate and completes the proof.\n\\end{proof}\n\n\n\n\n\\section{Appendix}\n\t\n\\subsection{Proof of Lemma \\ref{yu-lemma-7-1-1}}\nBy the definition of $W$, we have \n\\begin{eqnarray*}\\label{yu-7-1-8}\n\t\\nabla V&=&\\nabla(\\theta^{-1}W)=W\\nabla \\theta^{-1}+\\theta^{-1}\\nabla W\\nonumber\\\\\n\t&=&-s\\lambda \\alpha\\theta^{-1}W\\nabla \\varphi+\\theta^{-1}\\nabla W.\n\\end{eqnarray*}\n\tTherefore, \n\\begin{eqnarray}\\label{yu-7-1-9}\n\t-\\theta\\mbox{div}(\\bar{A}\\nabla V)&=& s\\lambda^2\\alpha W\\nabla\\varphi\\cdot(\\bar{A}\\nabla \\varphi)\n\t-s^2\\lambda^2\\alpha^2W\\nabla\\varphi\\cdot(\\bar{A}\\nabla \\varphi)\\nonumber\\\\\n\t&\\;&+2s\\lambda\\alpha\\nabla W\\cdot(\\bar{A}\\nabla \\varphi)+s\\lambda\\alpha W\\mbox{div}(\\bar{A}\\nabla \\varphi)\n\t-\\mbox{div}(\\bar{A}\\nabla W).\n\\end{eqnarray}\n\tLet \n\\begin{equation*}\\label{yu-7-1-10}\n\\begin{cases}\n\tI_1:=-\\mbox{div}(\\bar{A}\\nabla W)-s^2\\lambda^2\\alpha^2W\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi),\\\\\n\tI_2:=2s\\lambda\\alpha\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)+2s\\lambda^2\\alpha W\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi),\\\\\n\tI_3:=-\\theta\\mbox{div}(\\bar{A}\\nabla V)-s\\lambda\\alpha W\\mbox{div}(\\bar{A}\\nabla \\varphi)+s\\lambda^2\\alpha W\\nabla \\varphi\\cdot(\\bar{A}\\nabla\\varphi).\n\\end{cases}\n\\end{equation*}\n\tBy (\\ref{yu-7-1-9}), it is clear that $I_1+I_2=I_3$. Then \n\\begin{equation}\\label{yu-7-1-11}\n\tI_1I_2\\leq \\frac{1}{2}|I_3|^2.\n\\end{equation}\n\tFor the term $|I_3|^2$, we have \n\\begin{eqnarray}\\label{yu-7-1-14}\n\t\\frac{1}{2}|I_3|^2&\\leq& \\theta^2|\\mbox{div}(\\bar{A}\\nabla V)|^2+2s^2\\lambda^2\\alpha^2|W|^2|\\mbox{div} (\\bar{A}\\nabla\\varphi)|^2\\nonumber\\\\\n\t&\\;&+2s^2\\lambda^4\\alpha^2|\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)|^2|W|^2.\n\\end{eqnarray}\n\tBy (\\ref{yu-7-1-10}), we have \n\\begin{eqnarray}\\label{yu-7-1-15}\n\tI_1I_2&=&-2s\\lambda\\alpha[\\mbox{div}(\\bar{A}\\nabla W)+s^2\\lambda^2\\alpha^2W\\nabla\\varphi\\cdot(\\bar{A}\\nabla \\varphi)][\\nabla W\\cdot(\\bar{A}\\nabla \\varphi)+\\lambda W\\nabla\\varphi\\cdot (\\bar{A}\\nabla \\varphi)]\\nonumber\\\\\n\t&=&-2s\\lambda^2\\alpha W[\\mbox{div}(\\bar{A}\\nabla W)+s^2\\lambda^2\\alpha^2W\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)]\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)\\nonumber\\\\\n\t&\\;&-2s\\lambda\\alpha\\mbox{div}(\\bar{A}\\nabla W)\\nabla W\\cdot(\\bar{A}\\nabla \\varphi)-2s^3\\lambda^3\\alpha^3W\\nabla W\\cdot (\\bar{A}\\nabla\\varphi)\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)\\nonumber\\\\\n\t&:=&\\sum_{i=1}^3J_i.\n\\end{eqnarray}\t\n\tNext, we compute the terms $J_i$ one by one. For the term $J_1$, we have \n\\begin{eqnarray}\\label{yu-7-1-16}\n\tJ_1&=&-2s^3\\lambda^4\\alpha^3|W|^2|\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)|^2\n\t+2s\\lambda^2\\alpha\\nabla W\\cdot(\\bar{A}\\nabla W)\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)\\nonumber\\\\\n\t&\\;&+2s\\lambda^2 W\\nabla[\\alpha\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)]\\cdot(\\bar{A}\\nabla W)\n\t-2s\\lambda^2\\mbox{div}[\\alpha W\\bar{A}\\nabla W\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)].\n\\end{eqnarray}\n\tMoreover, \n\\begin{eqnarray*}\\label{yu-7-1-17}\n\tJ_2&=&2s\\lambda\\alpha\\nabla[\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)]\\cdot(\\bar{A}\\nabla W)\n\t+2s\\lambda^2\\alpha|\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)|^2\\nonumber\\\\\n\t&\\;&-2s\\lambda\\mbox{div}[\\alpha \\bar{A}\\nabla W\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)].\n\\end{eqnarray*}\n\tNote that \n\\begin{eqnarray*}\\label{yu-7-1-19}\n\t&\\;&\\nabla[\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)]\\cdot(\\bar{A}\\nabla W)\\nonumber\\\\\n\t&=&(\\bar{A}\\nabla W)\\cdot[D^2W(\\bar{A}\\nabla\\varphi)]\n\t+(\\bar{A}\\nabla W)[D^2\\varphi (\\bar{A}\\nabla W)]\\nonumber\\\\\n\t&\\;&+\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_j\\varphi\\right)\\cdot(\\bar{A}\\nabla W)\\nonumber\\\\\n\t&=&\\frac{1}{2}\\nabla[\\nabla W\\cdot(\\bar{A}\\nabla W)]\\cdot(\\bar{A}\\nabla\\varphi)+(\\bar{A}\\nabla W)[D^2\\varphi (\\bar{A}\\nabla W)]\\nonumber\\\\\n\t&\\;&+\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_j\\varphi\\right)\\cdot(\\bar{A}\\nabla W)\n\t-\\frac{1}{2}\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_jW\\right)\\cdot(\\bar{A}\\nabla\\varphi).\n\\end{eqnarray*}\n\tHence\n\\begin{eqnarray}\\label{yu-7-1-20}\n\tJ_2&=&2s\\lambda^2\\alpha|\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)|^2\n\t-s\\lambda^2\\alpha\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)\\nabla W\\cdot(\\bar{A}\\nabla W)\\cdot\\nonumber\\\\\n\t&\\;&-s\\lambda\\alpha\\mbox{div}(\\bar{A}\\nabla\\varphi)\\nabla W\\cdot (\\bar{A}\\nabla W)+2s\\lambda\\alpha(\\bar{A}\\nabla W)\\cdot[D^2\\varphi(\\bar{A}\\nabla W)]\\nonumber\\\\\n\t&\\;&{+2s\\lambda\\alpha\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_j\\varphi\\right)\\cdot(\\bar{A}\\nabla W)-s\\lambda\\alpha\\left(\\sum_{i,j=1}^{N+1}\\partial_iW\\nabla \\bar{a}^{ij}\\partial_jW\\right)\\cdot(\\bar{A}\\nabla\\varphi)}\n\t\\nonumber\\\\\n\t&\\;&-2s\\lambda\\mbox{div}[\\alpha \\bar{A}\\nabla W\\nabla W\\cdot(\\bar{A}\\nabla\\varphi)]+s\\lambda\\mbox{div}[\\alpha\\nabla W\\cdot(\\bar{A}\\nabla W)\\bar{A}\\nabla\\varphi].\n\\end{eqnarray}\n Further, \n\\begin{eqnarray}\\label{yu-7-1-18}\n\tJ_3&=&3s^3\\lambda^4\\alpha^2|W|^2|\\nabla\\varphi\\cdot(\\bar{A}\\nabla\\varphi)|^2\n\t+s^3\\lambda^3\\alpha^3|W|^2\\mbox{div}\\{\\bar{A}\\nabla\\varphi[\\nabla\\varphi\\cdot (\\bar{A}\\nabla\\varphi)]\\}\\nonumber\\\\\n\t&\\;&-s^3\\lambda^3\\mbox{div}[\\alpha^3|W|^2\\bar{A}\\nabla\\varphi\\nabla\\varphi\\cdot (\\bar{A}\\nabla\\varphi)].\n\\end{eqnarray}\n\tFinally, by (\\ref{yu-7-1-11})--(\\ref{yu-7-1-18}) we obtain the desired identity and complete the proof.\n\n\n\n\\subsection{Proofs of some useful inequalities}\n\\subsubsection{Proof of (\\ref{yu-9-29-1})}\n\tFor each $h\\in L^{\\frac{N}{2}+\\eta}(\\triangle_r(x_0))$ and $f\\in H^1(\\triangle_r(x_0))$, we let $\\theta=\\frac{1}{r}(x-x_0)\\in\\triangle_1(0)$, \n\t$\\tilde{h}(\\theta)=h(r\\theta+x_0)=h(x)$ and $\\tilde{f}(\\theta)=f(x)$ similarly.\n\tOne can check that \n$$\n\t\\int_{\\triangle_1(0)}|\\tilde{h}||\\tilde{f}|^2d\\theta =r^{-N}\\int_{\\triangle_r(x_0)} |h||f|^2dx,\n$$\n$$\n\t\\|\\tilde{h}\\|_{L^{\\frac{N}{2}+\\eta}(\\triangle_1(0))}=r^{-\\frac{2N}{N+2\\eta}}\n\t\\|h\\|_{L^{\\frac{N}{2}+\\eta}(\\triangle_r(x_0))},\n$$\n\tand \n$$\n\t\\|\\tilde{f}\\|_{L^2(\\triangle_1(0))}=r^{-\\frac{N}{2}}\\|f\\|_{L^2(\\triangle_r(x_0))}.\n$$\n\tMoreover, when $r\\in(0,1)$, \n\\begin{eqnarray*}\n\t\\|f\\|_{H^1(\\triangle_r(x_0))}^2&=&r^N\\int_{\\triangle_1(0)}\\left(\\frac{1}{r^2}|\\nabla_\\theta\\tilde{f}|^2+|\\tilde{f}|^2\\right)d\\theta\\nonumber\\\\\n\t&\\geq& r^N\\int_{\\triangle_1(0)}(|\\nabla_\\theta \\tilde{f}|^2+|\\tilde{f}|^2)d\\theta\n\t=r^N\\|\\tilde{f}\\|^2_{H^1(\\triangle_1(0))},\n\\end{eqnarray*}\n\ti.e., \n$$\n\t\\|\\tilde{f}\\|_{H^1(\\triangle_1(0))}\\leq r^{-\\frac{N}{2}}\\|f\\|_{H^1(\\triangle_r(x_0))}. \n$$\n\tTherefore, by (\\ref{yu-9-26-2}), we have \n\\begin{eqnarray*}\n &\\;&r^{-N}\\int_{\\triangle_r(x_0)}|h||f|^2dx=\\int_{\\triangle_1(0)}|\\tilde{h}||\\tilde{f}|^2d\\theta\\nonumber\\\\\n &\\leq& \\Gamma_2(\\triangle_1(0),N,\\eta)\\|\\tilde{h}\\|_{L^{\\frac{N}{2}+\\eta}(\\triangle_1(0))}\n \\|\\tilde{f}\\|_{L^2(\\triangle_1(0))}^{\\frac{4\\eta}{N+2\\eta}}\\|\\tilde{f}\\|^{\\frac{2N}{N+2\\eta}}_{H^1(\\triangle_1(0))}\\nonumber\\\\\n &\\leq&\\Gamma_2(\\triangle_1(0), N,\\eta)r^{-\\frac{2N}{N+2\\eta}}r^{-N}\n \\|h\\|_{L^{\\frac{N}{2}+\\eta}(\\triangle_r(x_0))} \\|f\\|_{L^2(\\triangle_r(x_0))}^{\\frac{4\\eta}{N+2\\eta}}\\|f\\|^{\\frac{2N}{N+2\\eta}}_{H^1(\\triangle_r(x_0))}. \n\\end{eqnarray*}\n\tThis implies that \n$$\n\t\\int_{\\triangle_r(x_0)}|h||f|^2dx\\leq \\Gamma_2(\\triangle_1(0), N,\\eta)r^{-\\frac{2N}{N+2\\eta}}\n\t \\|h\\|_{L^{\\frac{N}{2}+\\eta}(\\triangle_r(x_0))} \\|f\\|_{L^2(\\triangle_r(x_0))}^{\\frac{4\\eta}{N+2\\eta}}\\|f\\|^{\\frac{2N}{N+2\\eta}}_{H^1(\\triangle_r(x_0))}.\n$$\n\tThen (\\ref{yu-9-29-1}) holds.\n\\subsubsection{Proof of (\\ref{yu-6-22-8})}\n\tIndeed, for any $f\\in H^1(I)$, taking any $x\\in I$, we have \n\\begin{eqnarray*}\n\t|f(x)|^2\\leq 2\\left|\\int_y^xf'(s)ds\\right|^2+2|f(y)|^2\n\t\\leq 2|I|\\int_I|f'(s)|^2ds+2|f(y)|^2\n\\end{eqnarray*}\n\tfor each $y\\in I$. Integrating it with respect to $y$ over $I$, we get \n$$\n\t|f(x)|^2\\leq 2|I|\\int_{I}|f'(s)|^2ds+\\frac{2}{|I|}\\int_{I}|f(s)|^2ds. \n$$\n\tThis implies (\\ref{yu-6-22-8}). \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction.}\n\nThe central question that we study in this paper is the following. Assume that $M$ is a smooth surface,\npossibly with a boundary, and that $\\gamma$ is a simple (non-self-intersecting) closed curve on $M$. Assume that $\\gamma$ can be contracted\nto a point via free loops (closed curves) of length at most $L$. We would like to contract $\\gamma$ over closed curves based at a point \n$q \\in \\gamma$ so that the maximal length of these curves is as small as possible.\nCan we estimate the required maximal length in terms of $L$ and the diameter $d$ of $M$? Such a result would have a large number of applications, some of which\nwill be discussed at the end of the introduction. This problem is already interesting when $M$ is a $2$-disc endowed with\na Riemannian metric and $\\gamma$ is its boundary. The simple example of a Riemannian metric\non $M$ that looks like a long thin finger shown in Figure 1 demonstrates that we cannot estimate the required length in terms of $L$ alone, and this\nexample suggests that at the very least we need to add a summand equal to $2d$. In this example, we can contract the boundary\nof the disc via short closed curves to a point $p$ far from $\\partial M$ (see Figure 1(a)). To replace this homotopy by one composed of\nloops based at a point $q\\in\\partial M$, we connect $p$ and $q$ by a minimizing geodesic $\\tau$. In the course of our new homotopy, we travel along\n$\\tau$ to one of the closed curves in the original homotopy, travel along this curve, and then return back along $\\tau$ (Figure 1(b)). At some moment we end up\nat a loop that consists of two copies of $\\tau$ traversed in opposite directions. This loop can then be contracted to $q$ along itself.\n\nOf course, there are other, more complicated Riemannian metrics\non the $2$-disc, such as the metric depicted in Figure \\ref*{monotoneintro2}. There is also the family of Riemannian metrics considered in [FK].\nFor these metrics, the connection between \nthe length of curves in the ``best'' free loop homotopy and\nthe length of curves in the ``best'' fixed point homotopy is not so evident.\n\\par\nOur first theorem asserts that adding the summand $2d$ and an arbitrarily small $\\epsilon$ to $L$ will always suffice. It is quite possible that one\ndoes not need $\\epsilon$, but this does not seem to follow from a compactness argument, as when $\\epsilon \\longrightarrow 0$, our homotopies can become\nwigglier and wigglier. (More formally, we have not established a control over the Lipschitz constants of our homotopies as $\\epsilon \\longrightarrow 0$.)\n\n\\begin{Thm} \\label{Theoremmain}\nLet $M$ be a Riemannian manifold with boundary diffeomorphic\nto the standard disc of dimension $2$. Denote its diameter by $d$. Suppose there exists a \nhomotopy connecting the boundary $\\partial M$ of $M$\nto some point $p \\in M$ such that\nthe length of every closed curve in this homotopy does not exceed a real number $L$. \n\nThen, for any $q \\in \\partial M$ and \nfor any $\\epsilon > 0$,\nthere exists a fixed point homotopy \nthat connects $\\partial M$ with $q$, \nand passes through loops that are based at $q$ and have length not exceeding $L+2d+\\epsilon$.\n\\end{Thm}\n\nOur second theorem deals with the general case of a simple contractible curve on a surface endowed with a Riemannian metric.\n\n\\begin{Thm} \\label{Theoremmain1}\nLet $M$ be a closed Riemannian surface of diameter $d$. Let\n$\\gamma:[0,1] \\longrightarrow M$ be a simple\n closed curve in $M$, and $q$ a point on $\\gamma$.\nIf there exists a homotopy between $\\gamma$ and a point that passes through closed curves of length not exceeding $L$,\nthen there exists a homotopy that contracts $\\gamma$ to $q$ through loops that are \nbased at $q$ and have length $\\leq 3L+2d+\\epsilon$.\n\\end{Thm}\n\nOur proof of Theorem \\ref*{Theoremmain} and our proof of Theorem \\ref*{Theoremmain1} each uses the following theorem of independent interest\nproven by Gregory R. Chambers and Yevgeny Liokumovich in [CL]:\n\n\n\\begin{Thm}{(G. R. Chambers, Y. Liokumovich [CL])} \\label{Theoremgy}\nLet $M$ be a Riemannian surface. \nLet $\\gamma(t): S^1 \\longrightarrow M$ \nbe a closed curve in $M$. Suppose there exists a homotopy\n$H(t,\\tau):S^1 \\times [0,1] \\longrightarrow M$ such that\n$H(t,0)=\\gamma(t)$; $H(t,1)=p \\in M$, and the length of $\\gamma_\\tau=H(*,\\tau)$ is\nat most $L$ for all $\\tau \\in [0,1]$. Then, for any $\\epsilon >0$, \nthere exists a homotopy\n$\\tilde{H}(t,\\tau):S^1 \\times [0,1] \\longrightarrow M$, such\nthat $\\tilde{H}(t,0)=\\gamma(t)$; $\\tilde{H}(t,1)=p$; \nthe length of $\\tilde{\\gamma}_\\tau=\\tilde{H}(*,\\tau)$ is at most\n$L+\\epsilon$, and $\\tilde{\\gamma}_\\tau(t)$ is a\nsimple closed curve for every $\\tau \\in [0,1]$.\n\\end{Thm} \n\n \nThis result immediately reduces both Theorem \\ref*{Theoremmain} and Theorem \\ref*{Theoremmain1} to the particular case of\nwhen the original homotopy contracting $\\partial M$ or $\\gamma$ is known to\npass through {\\it simple} closed curves of length $\\leq L$. (Note that we did not\nmake this assumption in the text of the theorem.) Let us state the particular\ncase of Theorem \\ref*{Theoremmain} that we are going to prove below, and which implies the general case:\n\n\n\\begin{Pro} \\label{Theorem B}\nLet $M$ be a Riemannian manifold with boundary diffeomorphic \nto the $2$-dimensional disc. Let \n$H(t, \\tau):S^1 \\times [0,1] \\longrightarrow M$ be a homotopy \nbetween $\\gamma(t)=\\partial M$ \nand $p \\in M$ which passes through \nsimple closed curves of length at most $L$. Then, for any $\\epsilon >0$ and\nany\n$q \\in \\partial M$, there exists a homotopy\n$H_q(t, \\tau): S^1 \\times [0,1] \\longrightarrow M$ \nover closed curves of length at most $L+2d+\\epsilon$, where $d$ is the diameter of $M$,\nsuch that $H_q(t,0)=\\gamma(t)$, $H_q(t,1)=q$, and $H_q(s_0,\\tau)=q$ for all \n$\\tau \\in [0,1]$ and a fixed base point $s_0\\in S^1$.\n\\end{Pro}\n\nTo reiterate, the first step in producing a ``short'' fixed point homotopy out\nof a free loop homotopy is to continuously deform all of the curves \nin the original homotopy so that the lengths of the resulting curves\ndo not increase by much, and so that the resulting curves no longer have\nself-intersections. This is also the first step in \nproving Theorem \\ref*{Theoremmain1}; the other main steps of its proof will be briefly discussed \ntowards the end of the introduction.\n\n\n\\realfig{monotoneintro1}{monotoneintro1.pdf}{Long finger}{0.7\\hsize}\n\n\nThe questions considered in this paper fall within the realm of questions\nof investigating geometric properties of ``optimal'' homotopies.\n\n\\realfig{monotoneintro2}{monotoneintro2.pdf}{``Cactus'' metric on the disc}{0.7\\hsize}\n\n\nAnother example of such a question was a long-standing question of S. Frankel\nand M. Katz posed at the end of their paper [FK]. They asked if one contract the boundary $\\partial M$ of a\nRiemannian $2$-disc $M$ so that the length of curves in the homotopy\nis majorized above in terms of the diameter $d$, area $A$, \nand the length of the boundary of the manifold?\nNote that this question is a modification of an earlier question asked by M. Gromov ([Gr], p. 100).\nThe positive answer to this question was given by \nY. Liokumovich, A. Nabutovsky and the second author of this paper in [LNR].\nIn particular, it was shown that $\\partial M$ can be contracted over\ncurves of length at most \n$|\\partial M|+200d\\max\\{1, ln\\frac{\\sqrt{A}}{d}\\}$, where $|\\partial M|$ is the length of $\\partial M$. This estimate is optimal up to a multiplicative factor in the second term. When \n$\\frac{\\sqrt{A}}{d} <<1$, [LNR] provides a better bound of $2|\\partial M|+2d+686 \\sqrt{A}$. This was \nimproved to the asymptotically tight upper bound $|\\partial M|+2d+O(\\sqrt{A})$ by P. Papasoglu\nin a recent paper [P].\nNote that it is impossible to bound the length of curves in \nthe best homotopy solely in terms of the area of $M$, as \nan example of a three-legged star fish with long tentacles depicted\nin Figure \\ref*{monotoneintro3} demonstrates. It is also impossible to bound the length\nsolely in terms of the diameter of $M$, as was proven by \nFrankel and Katz in [FK], answering the original version of the question of M. Gromov\nmentioned above.\n\nA closely related family of questions deals with establishing the existence of various upper bounds on\nthe maximal length of optimal sweep-outs and slicings of surfaces either by closed curves or,\nmore generally, by cycles. For example, it was shown by Y. Liokumovich that there does not\nexist a universal diameter bound for the maximal length of curves or cycles in an\noptimal sweep-out of a closed Riemannian surface (see [L1] and [L2]).\nOn the other hand, F. Balacheff and S. Sabourau have found an upper bound for the maximal\nlength of a cycle in an optimal sweep-out of a surface in terms of the genus and the area\nof the surface (see [BS]). Also, a ``short'' sweep-out of a Riemannian $2$-sphere is possible \nif one assumes that there is no ``short'' geodesics of index $0$. This follows from the results\nof C. B. Croke in [C]. In this case, the maximal length of a cycle can be bounded by the area or the diameter of the surface.\n\n\\realfig{monotoneintro3}{monotoneintro3.pdf}{Three legged star fish}{0.7\\hsize}\n\n\nAnother interesting problem concerning optimal homotopies is the following.\nAssume that the boundary of a Riemannian $2$-disc can be contracted to a point\nvia {\\it simple} closed curves of length $\\leq L$. Let us call such a homotopy\n{\\it monotone} if the $2$-discs bounded by the closed curves of this homotopy\n``decrease'' in the sense of the definition below. \nIs it always possible to find a {\\it monotone} homotopy contracting\nthe boundary of the $2$-disc via closed curves of length $\\leq L$?\nIn order to pose this question more formally we need the following definition.\n\n\n\n\\begin{Def} \\label{Monotone}\nLet $M$ be a Riemannian manifold with boundary diffeomorphic\nto that of the $2$-disc. Let \n$H(t, \\tau): S^1 \\times [0,1] \\longrightarrow M$ be \na smooth map such that $H(t,0)=\\gamma(t)=\\partial M$,\n$H(t,1)=p \\in M$, and $H(t, \\tau)=\\gamma_\\tau(t)$ is a simple closed curve\nparametrized by $t$ for each $\\tau \\in [0,1]$. We will say that $H$ is \na monotone homotopy if closed $2$-discs\n$D_\\tau\\subset M$ bounded by $\\gamma_\\tau$ \nsatisfy the inclusion\n$D_{\\tau_2} \\subset D_{\\tau_1}$\nfor every $\\tau_1$ and $\\tau_2$ with $\\tau_1 < \\tau_2$.\n If these discs satisfy a stronger condition that $D_{\\tau_2} \\subset int D_{\\tau_1}$,\nwhere $int D_{\\tau_1}$ is the interior of $D_{\\tau_1}$, then the\nhomotopy will be called \\emph{strictly monotone}.\n\\end{Def}\n\nFigure \\ref*{monotonehomotopy}(a) depicts a strictly monotone homotopy\nof $\\gamma(t)$ to the point $p$, while\nFigure \\ref*{monotonehomotopy}(b) depicts a homotopy that is not\nmonotone. This definition can be trivially extended to homotopies connecting \na simple closed contractible curve to a point on any closed\nRiemannian surface not diffeomorphic to the $2$-disc. There is one technicality, however; \nif $M$ is diffeomorphic to $S^2$, then there is an ambiguity due to non-uniqueness of $D_{\\tau}$.\nWe agree to resolve it by allowing any possible choice of the system of discs $D_\\tau$ that is\ncontinuously dependent on $\\tau$, and\nthat has the monotonicity (or strict monotonicity) property.\n\n\\realfig{monotonehomotopy}{monotonehomotopy.pdf}{Monotone and non-monotone homotopies}{0.7\\hsize}\n\n\nThe following theorem provides an answer to the above question about the existence\nof monotone homotopies passing through ``short curves\".\n\n\\begin{Thm} {{\\bf (Monotonicity Theorem)}}\\label{Theorem A} \nLet $M$ be a Riemannian manifold with \nboundary diffeomorphic to the $2$-dimensional disc $D$. \nLet $H(t, \\tau):S^1 \\times [0,1] \\longrightarrow M$ be a \nhomotopy between the boundary of $M$ and some point $p \\in M$\npassing through simple closed curves $\\gamma_\\tau (t)$ of length\nat most $L$. Then, for any $\\epsilon > 0$, there exists a strictly monotone homotopy\n$\\tilde{H}(t, \\tau):S^1 \\times [0,1] \\longrightarrow M$ between $\\gamma(t)$\nand $\\tilde{p} \\in M$ over curves of length at most $L+\\epsilon$. \n\\end{Thm}\n\nFrom a technical point of view this is the central result of the paper, as it can be used to give \nthe following short proof of Proposition \\ref*{Theorem B} (which implies Theorem 0.1, as we explained above).\n\n\n\n\n\\begin{Pf}{Proof of Proposition \\ref*{Theorem B}}\nBy Theorem \\ref*{Theorem A}, there exists a strictly monotone homotopy $H$ between\n$\\gamma(t)$ and $\\tilde{p} \\in M$ over simple curves of length \nat most $L + \\epsilon$. Fix a point $q$ on $\\partial M$, and let $\\alpha(s) : [0,1] \\rightarrow M$\nbe a minimal geodesic connecting $q$ to $\\tilde{p}$. The length of $\\alpha$ is at most $d$.\nFor each $\\tau \\in [0,1]$, there is exactly one $\\tau' \\in [0,1]$ such that the curve $H(*, \\tau')$\ngoes through $\\alpha(\\tau)$. Let this curve be denoted by $H_{\\tau}$. Note that, if $\\alpha$ intersects a curve in $H$ multiple times, then\nwe will be able to find multiple values for $\\tau$ that result in the same curve.\n\nOur new contraction $G: S^1 \\times [0,1] \\rightarrow M$ of $\\gamma$ through curves based at $q$ is now defined\nas follows. For each $\\tau \\in [0,1]$, define $G(*, \\tau)$ to be the curve\n\t$$ \\alpha|_{[0, \\tau]} * H_{\\tau} * \\overline{\\alpha|_{[0,\\tau]}},$$\nwhere $\\overline{\\alpha|_{[0,\\tau]}}$ is the segment of $\\alpha$ traversed from $\\tau$ to $0$.\nEach curve in this homotopy is bounded in length by $L + 2d + \\epsilon$. Furthermore,\nit ends at $\\alpha|_{[0,1]} * \\overline{\\alpha|_{[0,1]}}$, which can obviously be contracted\nto $q$ through curves based at $q$ of length no more than $2d$. This completes the proof.\n\\end{Pf}\n\n \nNote that the Monotonicity Theorem {\\it does not} necessarily hold\nwhen one has a simple curve on a surface, or even a simple closed curve in a disc that is not\nassumed to be the boundary of that disc. This fact makes proving Theorem \n\\ref*{Theoremmain1} more difficult than Theorem 0.1, and is the reason\nfor the appearance of the extra $2L$ in our upper bound. We are grateful to Yevgeny Liokumovich\nfor first attracting our attention to this fact in conjunction with the example shown in\nFigure \\ref*{notmonotone1}.\nThis figure depicts a metric on a Riemannian \n$2$-disc and a curve $\\alpha_0$ \nsuch that the optimal homotopy contracting the curve to a point is not\nmonotone. Notice that there are three bumps depicted in this figure:\ntwo of them are long and thin and the one in the middle is short and asymmetric.\nIt takes less length to go under the middle bump than over it.\nThe original curve $\\alpha_0$ winds around the two thin bumps,\nand goes over the short one.\nIn order to contract $\\alpha_o$ to a point, it has to be stretched\nover the thin bumps but, because they are long, the length of the \ncurve will necessarily increase in the process. Thus, to begin with, \nit makes sense to first homotope $\\alpha_0$ to $\\alpha_1$, which runs below\nthe middle bump. $\\alpha_1$ is \nshorter than $\\alpha_0$, so we can ``spend'' this ``excess'' length \non dragging the curve over the two thin bumps one at a time. This\ncorresponds to the curves $\\alpha_2$ and $\\alpha_3$ in Figure \\ref*{notmonotone1}.\nWe now have to push the curve over the middle bump. This homotopy\nresults in $\\alpha_4$, which can then be easily contracted to a point.\nThe resulting homotopy is not monotone. Note that $\\alpha_0$ is \nnot the boundary of the disc, and so this example does not contradict Theorem \\ref*{Theorem A}. \n\n\n\n\\realfig{notmonotone1}{notmonotone1.pdf}{$\\alpha_0$ cannot be contracted to $p$\nvia a ``short'' monotone homotopy}{0.7\\hsize}\n\n\nFor surfaces, we will need a weaker form of the\nMonotonicity Theorem. We will not prove\nthat $\\gamma$ can be contracted by a strictly monotone homotopy via ``short\" curves, but \nonly that there exists a strictly monotone homotopy contracting another curve $\\tilde\\gamma$\nvia curves of length $\\leq L+\\epsilon$ so that $\\gamma$ will be contained in the\ndisc formed by the image of this monotone homotopy. More formally:\n\n\n\\begin{Def} \\label{Defcov}\nLet $\\gamma$ be a simple closed contractible curve on a Riemannian\nsurface $M$. We will say that a homotopy\n$H(t, \\tau):S^1 \\times [0,1] \\longrightarrow M$ \\emph{covers} $\\gamma$ \nif the following conditions are satisfied:\n\n\\noindent (1) $H(t, \\tau)$ is a strictly monotone homotopy between some simple\n\nclosed curve $\\tilde{\\gamma}=H(*,0)$ and a point $p \\in M$;\n\n\\noindent (2) The disc $D_{\\tilde{\\gamma}}=H(S^1\\times [0,1])$ generated by \nthe image of this homotopy\ncontains $\\gamma$. \n(Note that $\\partial D_{\\tilde{\\gamma}}= \\tilde{\\gamma}$.)\n\n\\end{Def}\n\n\n\n\\begin{Thm} \\label{Theoremcover}\nLet $M$ be a two-dimensional Riemannian manifold or a\nRiemannian manifold with boundary. Let \n$H(t, \\tau):S^1\\times[0,1] \\longrightarrow M$ be a homotopy between\na simple closed contractible curve $\\gamma$ and a point $p \\in M$ over\nsimple closed curves of length at most $L$.\nThen, for any $\\epsilon >0$, there exists a homotopy \n$\\tilde{H}(t, \\tau)$ over curves of length at most $L+\\epsilon$ that\ncovers $\\gamma$ (in the sense of the above definition).\n\n\n\\end{Thm}\n\nNote that if $M$ is a $2$-disc, then this theorem is equivalent to the\nMonotonicity Theorem as the homotopy that covers $\\gamma = \\partial M$ can only begin \nwith $\\gamma$, and so Theorem \\ref*{Theoremcover} implies Theorem \\ref*{Theorem A}.\nThe above theorem will be proven in the next section. In order to deduce Theorem \\ref*{Theoremmain1} from Theorem \\ref*{Theoremcover}\nwe need the following theorem which, in our opinion, is of independent interest: \n\n\n\n\\begin{Thm} \\label{Theoremloop}\nLet $M$ be a closed Riemannian manifold of diameter $d$. Let $\\gamma$ \nbe a simple closed contractible curve in $M$. Assume that there exists a homotopy\n$H(t, \\tau):S^1\\times[0,1] \\longrightarrow M$ over curves of length at most $L$ that covers $\\gamma$. \nThen, for any point $q \\in \\gamma$ and for any $\\epsilon > 0$, there exists a contraction\n$\\tilde{H}(t, \\tau):S^1\\times[0,1] \\longrightarrow M$ of $\\gamma$ over loops\nof length at most $3L+2d + \\epsilon$ that are based at $q$. Furthermore, there is a specific point $q^\\star \\in \\gamma$ such that\nwe can find a contraction of $\\gamma$ through loops based at $q^\\star$ of length bounded by\n$2L + 2d + \\epsilon$.\n\\end{Thm} \n\nThis theorem will be proven in the last section. It is obvious that Theorem 0.2\nimmediately follows from Theorems 0.8 and 0.9.\n\n\\bigskip\n\n\\noindent{\\bf \\large Applications.} Theorem \\ref*{Theoremmain1} and\nTheorem \\ref*{Theorem A} \nhave many immediate and potential applications to the geometry \nof loop spaces of Riemannian $2$-spheres, to questions about\nthe lengths of geodesics, and to problems about \noptimal sweep-outs. \n\nIn particular, Theorem \\ref*{Theoremmain1} provides a canonical way\nof obtaining a ``short'' based loop homotopy out of a ``short'' free\nloop homotopy on a Riemannian surface. The second author has encountered\nthis problem many times, and each time it was solved using ad hoc methods.\nSpecifically, Theorems \\ref*{Theoremmain1} and \\ref*{Theorem A} can be applied \nin the following situations. The details of each of these applications will\nappear elsewhere.\n\n\\noindent (1) {\\bf Lengths of geodesics on Riemannian $2$-spheres.}\nLet $p,q$ be an arbitrary pair of points on a Riemannian $2$-sphere\n$M$ of diameter $d$. A. Nabutovsky together with the second author\nhave demonstrated that there exist at least $k$ geodesics joining them\nof length at most $22kd$ (see [NR2]). If $p = q$, then this bound becomes \n$20kd$ (see [NR1]). We have noticed that applying \nTheorem \\ref*{Theoremmain1} dramatically decreases the complexity \nof proofs in [NR1] and [NR2], and improves the bounds in [NR2] to $16kd$, and\nthe bounds in [NR1] to $14kd$. \nThese improvements are due to the fact that the main technical\ndifficulty encountered in [NR1] and especially in [NR2] is the possible formation of\nintersections between various closed curves in a homotopy between a closed curve and a point.\n\n\\noindent (2) {\\bf Geometry of the loop spaces of Riemannian $2$-spheres.}\n\n\\noindent (a) Applying Theorem \\ref*{Theoremmain1} will immediately \ngeneralize Theorem 1.1 in [NR3] to the {\\it free} loop space of\na Riemannian $2$-sphere $M$. To be more precise, one can show that\nany map $f:S^m \\longrightarrow \\Lambda M$, where $\\Lambda M$ is a free loop\nspace on $M$, is homotopic to a map $\\tilde{f}:S^m \\longrightarrow \\Lambda M$\nthat passes through curves of length that do not exceed $L=L(m,k,d)$. Here, \n$d$ is the diameter of $M$, $k$ is the number of distinct \nnon-trivial periodic geodesics\non $M$ of length at most $2d$, and $L$ is a function of $m, k, d$ that\ncan be written down explicitly. Moreover, one can explicitly majorize the \nlengths of loops in an ``optimal'' homotopy connecting $f$ and $\\tilde{f}$\nin terms of $n,k,d$ and $\\sup_{x \\in S^m} length(f(x))$.\n\n\\noindent (b) The first author and Y. Liokumovich have some concrete\nideas of how applications of Theorems \\ref*{Theoremgy} and \\ref*{Theorem A}\ncan lead to a solution to the following question posed by N. Hingston \nand H.-B. Rademacher: Let $\\alpha$ be a generator of $H_1(\\Lambda M)$, where\n$\\Lambda M$ is the free loop space of a manifold $M$ diffeomorphic to $S^2$.\nIs it possible for the minimax level of $2\\alpha$ (with respect to the length\nfunctional on $\\Lambda M$) to be lower than the minimax level of $\\alpha$?\n\n\\noindent (3) {\\bf A lower bound for the diastole on a Riemannian $2$-sphere\nof area $A$ and diameter $d$.} Y. Liokumovich points out that by using \nthe Monotonicity Theorem together with the examples of Frankel and Katz from [FK], \none can construct an example of Riemannian metrics on $S^2$ with $A >>d^2$ \nthat cannot\nbe sliced into simple closed curves of length $ \\leq \\frac{2}{\\ln 3}d \\ln \\frac{\\sqrt{A}}{d}$.\nThis result complements Theorem 1.3 in [LNR].\n\n\\section{Proof of Theorem \\ref*{Theoremcover}}\n\n\\par\n{\\bf 1.1. Scheme of the proof.} Our proof of Theorem \\ref*{Theoremcover} goes as follows.\nFirst, we construct a finite set of ``short\" monotone homotopies that\npass through simple closed curves which are\nhomotopic to $\\gamma$ and are of length bounded by $L + \\epsilon$. These monotone homotopies will either overlap\nor ``quasi-overlap\" in a certain way. This fact will enable us to ``glue\" (or, more precisely, ``process\") them one by one\nto obtain\na monotone homotopy that passes through simple curves of length $\\leq L+\\epsilon$, ends at a point, starts at a simple closed curve homotopic to $\\gamma$,\nand the disc generated by this homotopy contains $\\gamma$. Note that the\ngluing process is not straightforward. The image of the resulting homotopy\nis, in general, not the union of images of the two overlapping homotopies\nthat we are trying to combine.\n\\par\nThe homotopy that we are going to obtain will only be monotone and not\nstrictly monotone. The loss of monotonicity will however happen only along a finite set of curves. Therefore, one can then obtain a strictly monotone\nhomotopy by inserting collars along each of these curves, regarding\neach collar as a collection of copies of the original curve, and disengaging\ncurves in the monotone homotopy using the different copies. We will\nomit the rather obvious but tedious details of this step of the proof and\nconstruct only a monotone homotopy.\n\\par\n{\\bf 1.2. Filling contractible closed curves by discs.}\nA simple contractible closed curve $\\alpha$ on a surface always bounds a $2$-disc. This disc is unique, unless the surface is diffeomorphic to $S^2$.\nIn this case one has two choices for such a disc. Let $\\gamma$\nbe a simple closed curve and $H$ a homotopy that contracts $\\gamma$ \nto a point through simple closed curves. In this case, we use the following method to assign\ndiscs $D_{\\gamma_\\tau}$ bounded by $\\gamma_\\tau$\nto all curves $\\gamma_\\tau$ that appear during this homotopy.\nWhen $\\gamma_\\tau$ becomes very small for some\nvalue of $\\tau$, we choose the smaller of two discs. We then extend the choice\nof the disc by continuity. Note that making a choice between two discs bounded\nby a curve is equivalent to orienting this curve. Thus, we have just oriented\nall curves appearing in the homotopy $H$ between $\\gamma$ and a point.\nOur choice of orientation has nothing to do with how the closed curves are parametrized in the homotopy.\n\\par\nNote also that if a (smooth) simple closed curve $\\alpha$ has already been oriented, then we can\norient each simple closed curve $\\beta$ in a tubular neighbourhood of $\\alpha$\nby choosing $D_\\beta$ so that $D_\\beta\\setminus D_\\alpha$ is contained\nin the same tubular neighbourhood of $\\alpha$.\n\\par\nIf $\\alpha_1$ and $\\alpha_2$ are two contractible simple closed curves that have already been\noriented, and the closure of $D_{\\alpha_2}$ is contained in the closure\nof $D_{\\alpha_1}$, then we say that $\\alpha_2$ lies {\\it inside} $\\alpha_1$.\nFurther, we call the closure of $D_{\\alpha_1}\\setminus D_{\\alpha_2}$\nan {\\it oriented annulus} bounded by $\\alpha_1$ and $\\alpha_2$ and refer to\n$\\alpha_2$ as to its {\\it inner boundary} and $\\alpha_1$ as its\n{\\it outer boundary}. Note that $\\alpha_1$ and $\\alpha_2$ are allowed to touch \neach other, and if they do, the oriented annulus is not diffeomorphic to $S^1\\times [0,1]$. \n\\par\nGiven an oriented annulus, we can orient each simple closed curve $\\beta$ that both lies in this\nannulus and is homotopic to its inner boundary $\\alpha_1$ by\nchoosing $D_{\\beta}$ so that it contains $D_{\\alpha_1}$.\n\\par\nWhen $M$ is diffeomorphic to $S^2$ and we are given\na simple closed curve $\\gamma$ and a homotopy $H$ that contracts it to a point\nthrough simple closed curves, we will automatically\nadopt the conventions about orienting simple closed curves indicated\nabove to obtain a {\\it continuous} orientation on a class of simple\ncurves that will include all closed curves relevant to our proof.\n\\par\n\n\n{\\bf 1.3. Simple intersections.}\n\n\\begin{Def} \\label{simpleintersections}\nLet $\\beta_1(t): [0,1] \\longrightarrow M$ and \n$\\beta_2(t):[0,1] \\longrightarrow M$ be two closed curves in a \nRiemannian manifold $M$. \n\nWe will say that $\\beta_1(t)$ and $\\beta_2(t)$\nsatisfy the \\emph{simple intersection property} if, for every two points\nof intersection of $\\beta_1$ and $\\beta_2$, they are consecutive on $\\beta_1$\nif and only if they are consecutive on $\\beta_2$.\n\\end{Def}\n\n\n\n\n\n{\\bf 1.4. Monotone homotopy covers.}\n\nLet $\\gamma$ be a contractible simple closed curve on $M$, and suppose we are given a homotopy $H$\nfrom $\\gamma$ to a point that passes through simple closed curves. If $M$ is diffeomorphic to $S^2$, then this homotopy induces orientations on a family of closed curves as described in section 1.2. \nIf $M$ is not a $2$-sphere, all contractible curves automatically bound unique\ndiscs.\n\\par\nConsider an oriented annulus (in the sense of definition given in section 1.2).\nAssume that we can choose a monotone homotopy that starts at its outer boundary\nand ends at its inner boundary. Such an annulus together with the monotone homotopy will be called an {\\it oriented monotone annulus}. Let $A_1$ and $A_2$\nbe two oriented monotone annuli. If\nthe inner boundary of $A_1$ is contained inside of the outer boundary\nof $A_2$,\nwe say that the annuli $A_1$ and $A_2$ are {\\it related}. Note that\nthis definition is {\\it not} symmetric with respect to the order of $A_1$ and $A_2$. Also note that related annuli can, in principle, be disjoint as\n$A_1$ can be contained inside the disc bounded by the inner boundary of $A_2$.\n\\par\nAssume that for some $n$, a collection of $n$ oriented monotone\nannuli has the property that for each $i=1,\\ldots n-1$, $A_i$ and $A_{i+1}$\nare related. Assume further that the inner boundary of $A_n$ is a point.\nIf $\\gamma$ is the outer boundary of $A_1$ and $D_\\gamma$ is equal to the disc bounded by the outer boundary of $A_1$, \nwe say that $A_1,\\ldots, A_n$ form a {\\it monotone\nhomotopy cover} of $\\gamma$ corresponding to $H$.\nIf, instead,\nthe disc $D_\\gamma$ bounded by $\\gamma$ is contained in the $2$-disc\nbounded by the outer boundary of $A_1$,\nwe say that\n$A_1,\\ldots , A_n$ form a {\\it weak monotone homotopy cover} of $\\gamma$ corresponding to $H$.\n\nFor the convenience of the reader, we give the following self-contained \ndefinitions of monotone homotopy cover and weak monotone homotopy cover that do not use \nthe terminology introduced in section 1.2.\n\n\\begin{Def} \\label{monotonecover}\nLet $\\gamma_\\tau(t), \\tau \\in [0,1]$ be a continuous \n$1$-parametric family of \nsimple closed curves starting from\na closed curve $\\gamma=\\gamma_0(t)$ and\nending with a point $p$. Assume that all curves\n$\\gamma_\\tau$ were continuously oriented in the sense of section 1.2, that is,\nthere is a continuous family of discs $D_{\\gamma_\\tau}$ bounded\nby $\\gamma_\\tau$. (This condition is relevant only if $M$ is diffeomorphic to $S^2$.) We will say that $n$ homotopies $F_i:S^1\\times [0,1]\\longrightarrow M, i=1,\\ldots , n,$ \nform a monotone homotopy cover\nof $\\gamma$ corresponding to the family $\\gamma_\\tau(t)$ if the following conditions are satisfied:\n\n\\begin{enumerate}\n\n\\item\t$F_0(t,0)=\\gamma_0(t)$ and $F_n(t,1)=\\tilde{p}$\nfor some point $\\tilde p\\in M$.\n\n\\item\tFor each $i$ there is a continuous \n$1$-parametric family $D_i(x)$ of closed $2$-discs in $M$ satisfying the condition\nthat the boundary of $D_i(x)$ is the closed curve $F(S^1,x)$.\n$D_1(0)$ coincides with $D_{\\gamma_0}$. For each $i$, $D_i(x_1)$\nis contained in $D_i(x_2)$ if $x_1>x_2$. (The last condition means that, \nfor each $i$, $F_i$ is a monotone homotopy of closed curves.)\n\n\\item\tThese $n$ families of discs satisfy the following condition:\nFor all $i \\in \\{1,...,n-1\\}$\n$D_i(1) \\subset D_{i+1}(0)$.\nFigure \\ref*{monotonecover}(a) and Figure \\ref*{monotonecover}(b)\ndepict this condition.\n\\end{enumerate}\n\nIf, instead of assuming that $\\gamma_0=\\partial D_1(0)$ and $D_{\\gamma_0}=\nD_1(0)$, we assume only that $D_{\\gamma_0}$ is contained in\n$D_1(0)$, we say that the $n$ homotopies $F_i$ form\na weak monotone homotopy cover of $\\gamma$ corresponding to the family $\\gamma_\\tau$.\n\\end{Def}\n\n\\realfig{monotonecover}{monotonecover.pdf}{Monotone homotopy cover}{0.7\\hsize}\n\n\\begin{Def} \\label{lengthcover}\nLet $\\{F(t,x)\\}_{i=0}^n$ be a (weak) monotone homotopy cover. We will \ncall $M=\\max_{i \\in \\{0,...,n\\}} \\sup_{x \\in [0,1]} length (F_i(S^1,x))$\nits length.\n\n\\end{Def}\n\n\n\n\n\n\\noindent\n{\\bf 1.5. The existence of monotone homotopy covers.}\n\nHere is our first major technical result:\n\n\n\\begin{Lem} \\label{Lemma1} \n\n\\begin{enumerate}\n\\item\tLet $M$ be a Riemannian \nmanifold with boundary diffeomorphic to the\n$2$-disc.\nLet $\\gamma_\\tau(t)$, $\\tau \\in [0,1]$ be a $1$-parameter\nfamily of simple smooth curves in $M$ parametrized by\ntheir arclength with $\\gamma_0=\\gamma = \\partial M$. If \nthe length of curves in this family is bounded by $L$, then for any \n$\\epsilon > 0$,\nthere exists a monotone homotopy cover of $\\gamma_\\tau$ of length \nat most $L+\\epsilon$. \n\n\\item\tLet $M$ be a Riemannian surface. Let $\\gamma$ be\na closed simple contractible curve on $M$. Let \n$\\gamma_{\\tau}(t), \\tau \\in [0,1]$ \nbe a 1-parameter\nfamily of simple closed curves of length at most $L$, such \n$\\gamma_0(t)=\\gamma$, and $\\gamma_1(t) = p \\in M$. Then, for any $\\epsilon >0$, \nthere exists a weak homotopy cover of $\\gamma$ corresponding to $\\gamma_{\\tau}(t)$ of length \nat most $L+\\epsilon$.\n\\end{enumerate}\n\\end{Lem}\n\n\\begin{Pf}{Proof}\nThe proof of the lemma consists of the following steps. First we discretize\nthe given homotopy. We choose an increasing sequence of $n$ points $\\tau_i \\in [0,1]$, $i = 1, \\dots, n$, so that\nthe curves $\\gamma_{\\tau_i}$ satisfy the following conditions:\n\\begin{enumerate}\n\\item\tThe first curve in this collection is $\\gamma_0$ (that is, $\\tau_1=0$).\n\\item\tThe last curve in this collection is very short and is contained in a very small neighbourhood\nof $p$. In particular, it is smaller than the injectivity radius of $M$.\nWe choose the radius of this neighbourhood to be so small that $\\gamma_n$\ncould be contracted to a point inside the small disc that it bounds\nvia curves of length $t_{j_0}$.\nThus, $\\tilde{\\gamma}_x$ will locally be formed by continuously replacing $\\gamma_x|_{[t^x_{*}, t^x_{**}]}$ by an arc\nof $\\alpha$ between the same endpoints. Hence, in this case, \n$\\tilde{\\gamma}_x$ changes continuously when $x$ is near $x_0$. \nNote that while $s_{j_0-1}, s_{j_0},$ and $s_{j_0+1}$ can come in any order\nas is indicated in Figure \\ref*{monotonecase2a}, this fact does not affect\nthe above analysis.\n\n\\realfig{monotonecase2a}{monotonecase2a.pdf}{Outside arcs}{0.7\\hsize}\n\nThus, the only problematic case is when the arcs $a$ and $b$ are inside\n$D_\\alpha$ as in Figures \\ref*{monotonecase2b} and \\ref*{monotonecase2c}. \nLet $m,l,$ and $k$ be three points of intersection between \n$\\gamma_{x_0}$ and $\\alpha$, where $l$ is the tangential point, and \n$m$ and $k$ are its neighbours. By ``neighbours'' we will mean points of intersection\nthat are the closest to the tangential point along\n$\\gamma_{x_0}$. That is, if $l=\\gamma_{x_0}(t_{j_0})$, \nthen $m=\\gamma_{x_0}(t_{j_0-1})$ and $k=\\gamma_{x_0}(t_{j_0+1})$.\nIt is, however, quite possible that along $\\alpha$ there are intersection\npoints that are closer to $l$ than $m$ and $k$ as \nin Figure \\ref*{monotoneintersection}.\n\n\\realfig{monotoneintersection}{monotoneintersection.pdf}{Complicated intersections between $\\alpha$ and $\\gamma_{x_0}$}{0.7\\hsize}\n\nEach pair of points subdivides the curve \n$\\alpha$ into two segments connecting them.\nLet us denote the two arcs connecting $m$ with $l$ as $a_{ml}$ and\n$\\tilde{a}_{ml}$, the two arcs connecting $m$ and $k$ as $a_{mk}$ and\n$\\tilde{a}_{mk}$, and the two arcs connecting $l$ and $k$ as $a_{lk}$ and\n$\\tilde{a}_{lk}$. Also, let us select $a_{ml}, a_{mk}$, and $a_{lk}$ so\nthat their interiors do not intersect. Without loss of\ngenerality, let $a$ be the arc of $\\gamma_{x_0}$ between\n$m$ and $l$ in $D_\\alpha$, while $b$ is the arc between $l$ and $k$ in $D_\\alpha$.\n\nSince $a$ and $b$ are inside $D_\\alpha$, our algorithm requires\nthat we change them to the corresponding arcs of $\\alpha$.\nThe possible discontinuity is a result of the following situation. Let us define\ntwo options for replacing arcs of $\\gamma_{x_0}$ with arcs of $\\alpha$.\nNote that, as before, the arc of $\\alpha$ that we are replacing an arc of $\\gamma_{x_0}$ with is the one that is path homotopic to the arc of $\\gamma_{x_0}$ inside the closed\nannulus $cl(D_h \\setminus D_g)$.\n\n\\noindent {\\bf Option 1.} Separately replace $a$ and $b$ with the appropriate arcs of $\\alpha$.\n\n\\noindent {\\bf Option 2.} Let $c=a*b$. Replace $c$ with the appropriate arc of $\\alpha$.\n\nAs we will see, the two different options will some time result\nin the same curve, and some times not. Our algorithm, at point $x_0$, always uses\nOption 1. Let us, however, consider the curves that are formed by both options.\n\nLet us first consider Option 1. Without loss of generality, \nsuppose that arc $a$ is changed to $a_{ml}$. (The other option\nis $\\tilde{a}_{ml}$.) Now there are \ntwo possibilities for the arc $b$. It will either be changed to \n$a_{lk}$ or to $\\tilde{a}_{lk}$.\n\nNow let us consider Option 2. Note that if $b$ was\nchanged to $a_{lk}$, then \n$c$ must be changed to $\\tilde{a}_{mk}=a_{ml}*a_{lk}$.\nThus, in this case, it does not matter whether we used Option 1 or \nOption 2 (see Figure \\ref*{monotonecase2b}).\n\nHowever, if $b$ was changed to $\\tilde{a}_{lk}$, then \n$c$ must be changed to $a_{mk}$. \nWhile $a_{mk} \\neq a_{ml}*\\tilde{a}_{lk}$, we have that \n$a_{ml}*\\tilde{a}_{lk}=a_{ml}*\\bar{a}_{ml}*a_{mk}$.\n(Recall that $\\bar{a}_{ml}$ denotes $a_{ml}$ traversed in the opposite direction,\nfrom $l$ to $m$.)\nThus, $a_{mk}$ is path homotopic to $a_{ml}*\\tilde{a}_{lk}$ by \nsimply contracting $a_{ml}*\\bar{a}_{ml}$ to $m$ along itself.\nObserve that the length of the curve during this homotopy changes monotonically (see Figure \\ref*{monotonecase2c}).\n\nOne can see that while the former situation does not create\na discontinuity, the latter situation does. \nLet $\\delta>0$ be once again small enough so that\n$\\gamma_x$ and $\\alpha$ do not have any additional tangential \npoints on the interval $(x_0-\\delta, x_0+\\delta)$.\n\nIt is possible that, when $x' \\in (x_0-\\delta, x_0)$, \n$\\tilde{\\gamma}_{x'}$ will approach the curve obtained using\nOption 2 as $x'$ approaches $x_0$, and for $x'' \\in (x_0, x_0+\\delta)$,\n$\\tilde{\\gamma}_{x^{''}}$ will approach the curve obtained using Option 1\nas $x^{''}$ approaches $x_0$ (or the other way around). This is shown in Figure \\ref*{monotonecase2d}.\n\n\n\n\n\n\n\\realfig{monotonecase2b}{realfigmonotonecase2b.pdf}{Inside arcs}{0.7\\hsize}\n\n\n\\realfig{monotonecase2c}{monotonecase2c.pdf}{Two ways to change\narcs}{0.7\\hsize}\n\n\n\n\\realfig{monotonecase2d}{monotonecase2d.pdf}{Homotopy between the curves formed by Option 1 and Option 2}{0.7\\hsize}\n\n\nIf we include the homotopy between the curves formed by Option 1 and Option 2 as described\nabove,\nthe resulting family of curves\n$\\tilde{\\gamma}_x$ will become continuous and we will be done.\n\nNote that some of the curves obtained in the\nprocedure described above could have self-intersections, however, this happens\nonly when they include arcs of $\\alpha$ traversed twice in opposite\ndirections. It is easy to see that one can make all closed curves\nin $\\tilde G$ simple using an arbitrarily small perturbation.\n\n\\noindent {\\bf Step 2.}\nWe will modify the homotopy $H(t,x)$ to obtain $\\tilde{H}(t,x)$, a monotone homotopy with the following properties:\n$\\tilde{H}(*,0) = \\alpha$, and $\\tilde{H}(*,1)$ is contained inside the disc bounded by $H(*,1)$.\nMoreover, the maximal length of curves in the new homotopy will \nincrease by not more than a summand that can be made arbitrarily small. \n\nBy analogy with Step 1, the new curves in the homotopy will be\nconstructed by ``pushing in'' those segments of $\\beta_x$ that\nlie outside the disc $D_\\alpha$. Let $D_{\\beta, x}$ be the closed disc\nthat has $\\beta_x$ as its boundary, as in the hypotheses of the lemma. It will be a procedure that\nis dual to the one in Step 1.\n\nWe will denote the curves in the new homotopy by $\\tilde{\\beta}_x(t)=\\tilde{H}(t,x)$. In particular,\n$D_\\alpha \\subset D_{\\beta,0}$, so $\\tilde{\\beta}_0=\\alpha$.\n\nNow, let us describe the curve $\\tilde{\\beta}_1$. If $D_{\\beta_1} \\subset D_\\alpha$,\nthen we will let $\\tilde{\\beta}_1(t)=\\beta_1(t)$.\nIf $D_\\alpha \\subset D_{\\beta_1}$, then we will let $\\tilde{\\beta}_1=\\alpha(t)$.\nIf $D_\\alpha \\cap D_\\beta=\\emptyset$, then we will let $\\tilde{\\beta}_1(t)$\nbe some point $\\tilde{p}$, where $\\tilde{p}$ is obtained as follows.\nLet $\\tilde{x}_0=\\sup \\{ x \\in [0,1]$ such that \n$ D_{\\beta,x} \\cap D_\\alpha \\neq \\emptyset \\}$. Let \n$\\tilde{p}=D_{\\beta,\\tilde{x}_0} \\cap D_\\alpha$.\n\nFinally, suppose that $D_\\alpha \\cap D_{\\beta_1} \\neq \\emptyset$, but that one is not\na subset of the other. In this case, $\\tilde{\\beta_1}$ is constructed\nas follows.\nLet us consider arcs of $\\beta_1(t)$ that are outside\n$D_\\alpha$. That is, let $0=t_0 0$,\nwe wish to construct a contraction of \n$\\gamma$ through curves based at $q$ with the property that all curves are bounded in\nlength by\n\t$$ 3L + 2d + \\epsilon$$\nwhere $d$ is the diameter of the manifold.\nWe will also show that there is a specific point $q^\\star \\in \\gamma$ such that there is a contraction\nof $\\gamma$ through curves based at $q^\\star$ of length bounded by\n\t$$ 2L + 2d + \\epsilon.$$\n\n Throughout this proof, we produce curves of length less than\nor equal to $Q + \\epsilon$ for some $Q > 0$, where $\\epsilon > 0$ is chosen to be arbitrarily small.\nWhen we combine two curves of length bounded in this way,\nwe simply write that the result has length bounded by $2Q + \\epsilon$. Although \nnot strictly true, since we chose the original $\\epsilon$ to be as small as desired, we can just\ngo back and choose it to be $\\frac{\\epsilon}{2}$, in which case the new inequality $2Q + \\epsilon$\nholds. To improve readability, we do not mention this argument when it is used.\n\nWe will also be using the terms \\emph{interior} and \\emph{exterior} of $H_\\tau$ and of $\\gamma$, which we redefine here\nfor clarification:\n\n\\begin{Def}\n\\label{defn:interior_exterior}\nSince $H_\\tau$ is a monotone contraction, there is a disc $\\mathbb{D} \\subset M$ defined by the set of all points \nthat are in the image of some curve in $H$. For each point $\\tau$, $H_\\tau$ is simple and is contained in $\\mathbb{D}$, and as such divides $M$\ninto two open regions. Exactly one of these regions is entirely contained in $\\mathbb{D}$. This region is the interior\nof $H_\\tau$, and the other region is the exterior. Similarly, since $H$ covers $\\gamma$, $\\gamma$ is contained in $\\mathbb{D}$.\nSince it is simple, $\\gamma$ divides $M$ into $2$ regions, exactly one of which is entirely contained in $\\mathbb{D}$.\nThis region is the interior of $\\gamma$, and the other region is the exterior of $\\gamma$.\n\\end{Def}\n\nWe will prove two lemmas which, when combined, will allow us to prove this theorem. For each, we assume that $\\epsilon > 0$ is fixed.\n\n\\begin{Lem}\n\\label{lem:stage_one}\nThere exists a point $x \\in \\gamma$ and a point $\\tau^\\star$ such that there exists a homotopy $\\widetilde{H}$ from $\\gamma$ to either a curve formed by slightly perturbing $H_{\\tau^\\star}$ or to the point $x$\nthrough curves of length at most $2L + \\epsilon$. Additionally, $x$ lies on every curve in the homotopy $\\widetilde{H}$.\n\\end{Lem}\n\nSince the point $x \\in \\gamma$ has the aforementioned properties, $\\widetilde{H}$ is a based loop homotopy.\nOur second lemma takes $\\widetilde{H}$ and transforms it into a contraction of $\\gamma$ through curves based\nat $x$ of length at most $2L + 2d + \\epsilon$.\n\n\\begin{Lem}\n\\label{lem:stage_two}\nIf Lemma \\ref*{lem:stage_one} does not contract $\\gamma$ to $x$, then there exists a\ncontraction of the curve formed by slightly perturbing $H_{\\tau^\\star}$ through loops based at $x$ of length bounded by $2L + 2d + \\epsilon$.\nWe denote this contraction by $K$.\n\\end{Lem}\n\nWe will first demonstrate how these two lemmas can be used to prove Theorem 0.9, and will then prove each of them in turn.\n\n\\begin{Pf}{Proof of Theorem 0.9}\nLet $H$ be our original homotopy, $\\widetilde{H}$ be the homotopy generated by Lemma \\ref*{lem:stage_one},\nand let $K$ be the homotopy generated by Lemma \\ref*{lem:stage_two}.\nBy Lemma \\ref*{lem:stage_one}, either $\\widetilde{H}$ contracts $\\gamma$ to the point $x$, or it homotopes $\\gamma$ to a slight perturbation of $H_{\\tau^\\star}$.\nIf it contracts $\\gamma$ to the point $x$, then we are done. If it doesn't, then we have to use Lemma \\ref*{lem:stage_two}.\nWe do this by concatenating $\\widetilde{H}$ and $K$ to get a contraction of $\\gamma$ through curves based at $x$ of length at most\n$2L + 2d + \\epsilon$, as desired. Hence, the point $x$ is the special base point $q^\\star \\in \\gamma$\nmentioned above. Furthermore, this will complete the proof of the theorem: if we choose any point $q \\in \\gamma$, then\nwe can build the appropriate contraction based at $q$ as follows. Let $\\alpha$ be an arc of $\\gamma$ from $q$ to $x$ of length at most $\\frac{L}{2}$,\nand let $-\\alpha$ be the same arc, but with opposite orientation. Lastly,\nlet $\\beta$ be the curve formed by concatenating $\\alpha$ with $-\\alpha$. We can then take our contraction of $\\gamma$ based at\n$q^\\star$, and for each curve $\\gamma_\\tau$ in this contraction, we replace $\\gamma_\\tau$ with the curve that is formed by\ntraversing $\\alpha$, then $\\gamma_\\tau$, then $- \\alpha$. In this way, we produce\na homotopy from $\\gamma$ to $\\beta$ which is based at $q$, and which consists of curves of length at most $3L + 2d + \\epsilon$.\nSince $\\beta$ can be contracted through loops based at $q$ of length at most $L$, this completes the proof.\n\\end{Pf}\n\nWe are left now with proving each of the two lemmas outlined above.\n\n\\subsection{Proof of Lemma \\ref*{lem:stage_one}}\n\nTo prove Lemma \\ref*{lem:stage_one}, we will adopt an approach\nthat will be very similar to that used by Chambers and Liokumovich in \\cite{CL}. To begin with, we would like to\nperturb the homotopy $H$ so that only finitely many non-transverse intersections between $H$\nand $\\gamma$ occur, and so that they do not occur concurrently.\n\n\\begin{Lem}[Perturbation Lemma]\n\\label{lem:perturb}\nFor any $\\epsilon > 0$, we can perturb $H$, obtaining a new homotopy $\\overline{H}$ and points\n\t$$ 0 = \\tau_0 < \\dots < \\tau_n = 1 $$\nsuch that, for all $\\tau \\in [\\tau_i, \\tau_{i+1}]$, all intersections between $H_\\tau$ and $\\gamma$ are transverse, except for exactly\none intersection at one point $\\tau$. The two possible interactions are shown in Figure \\ref*{fig:reidemeister_move_type_B}.\n$\\overline{H}$ also has the following additional properties:\n\\begin{enumerate}\n\t\\item\t$\\overline{H}$ is a contraction that covers $\\gamma$.\n\t\\item\t$\\overline{H}$ is monotone.\n\t\\item\t$\\overline{H}$ consists of curves of length at most $L + \\epsilon$.\n\\end{enumerate}\n\\end{Lem}\n\nTo prove this lemma, we use the same technique as in Proposition 2.1 from \\cite{CL}; we apply the parametric\nversion of Thom's Multijet Transversality Theorem to the submanifold of the $2$-fold $1$-jet bundle corresponding to curves with singularities\nto show that a perturbation is possible which satisfies the above criteria.\nThis approach does not rule out other singular behaviour which involves self-intersections\nin $\\gamma$ or in $H_\\tau$, however, since both of these are simple, they have no self-intersections, and so the interactions\nbetween the two curves are limited to the isolated tangential intersections shown in Figure \\ref*{fig:reidemeister_move_type_B}.\nWe use the term \\emph{Reidemeister move} to describe this behaviour, this term\nbeing derived from the obvious relationship between this singularity and the knot moves used in Reidemeister's Theorem.\nWe also note that, since $\\overline{H}$ is a contraction that covers $\\gamma$, $n \\geq 2$. In other\nwords, there must be at least $2$ Reidemeister moves, once where $\\overline{H}$ transitions from a curve which lies completely in the exterior\nof $\\gamma$ to one which only partly lies in the exterior, and a Reidemeister move in which $H$ goes from being a curve which only partly\nlies in the interior of $\\gamma$, to a curve which lies either entirely in the interior of $\\gamma$, or entirely in the exterior of $\\gamma$.\n\n\n\\realfig{fig:reidemeister_move_type_B}{type_2.pdf}{Interactions between $\\gamma$ and $H$}{1.00\\textwidth}\n\nTo simplify this exposition, we will assume that $H$ has already been perturbed, and so it already has all of the properties\ndescribed in Lemma \\ref*{lem:perturb}.\n\nWe now want to prove Lemma \\ref*{lem:stage_one} for $H$ and $\\gamma$. We will define the point $x$ and the point $\\tau^\\star$, and then prove\nthat these points satisfy all of the required criteria.\n\n\\begin{Def}\n\\label{defn:x_and_t_star}\nLet $x$ be the last point at which $H$ and $\\gamma$ intersect, and let $\\tau^\\star$ be\nthe point at which this intersection occurs. Note that $\\tau^\\star \\in (\\tau_{n-1}, \\tau_n)$, and\n$H_{\\tau^\\star}$ and $\\gamma$ intersect tangentially at $x$.\n\\end{Def}\n\nThe idea to prove that Lemma \\ref*{lem:stage_one} holds for these values of $x$ and $\\tau^\\star$ is similar to that used by\nChambers and Liokumovich in \\cite{CL}. We construct\na certain graph $\\Gamma$ where the vertices represent curves, and the edges represent homotopies between curves. We then show that this graph\ncontains a certain path which represents a homotopy that easily implies the existence of the desired homotopy.\n\n\\noindent \\textbf{Vertices}\n\nWe begin to construct this graph $\\Gamma$ by defining its vertices. As above, each vertex will correspond to a certain curve.\nFor each $i \\in \\{1, \\dots, n - 1 \\}$, consider\n\t$$ U_i = \\gamma \\cup H_{\\tau_i}. $$\nWe will begin by identifying certain closed curves whose images lie in $U_i$. We will then eliminate some of these \ncurves based on several criteria. For each curve that remains, we will add a vertex. We begin by defining our large set of closed curves.\nWe will call these curves \\emph{subcurves} at $\\tau_i$.\n\n\\begin{Def}[Subcurves at $\\tau_i$]\n\\label{defn:curves}\nChoose any pair $(p_1, p_2)$ of distinct intersection points between $H_{\\tau_i}$ and $\\gamma$. We can write $\\gamma$\nas the disjoint union of $p_1$, $p_2$, and two open segments $\\rho_1$ and $\\rho_2$. Each of these segments can be used to join $p_1$ to\n$p_2$. Similarly, $H_\\tau$ can be written as the disjoint union of $p_1$, $p_2$, and two open segments $\\sigma_1$ and $\\sigma_2$. Each of these\nsegments can also be used to join $p_1$ to $p_2$.\n\nFor any piecewise smooth closed curve $\\alpha$ whose image lies in $U_i$, if we can find such a pair $(p_1,p_2)$ of intersection points such that\n$\\alpha$ can be written as the disjoint union of $p_1$, $p_2$, $\\sigma_i$ and $\\rho_j$ for $i, j \\in \\{ 1, 2 \\}$, then we say that $\\alpha$\nis a \\emph{subcurve} at $\\tau_i$.\n\nWe say that such a subcurve has endpoints $p_1$ and $p_2$, and we will denote the segment\nof the curve that comes from $\\gamma$ as $\\rho$, and the segment that comes from $H_{\\tau_i}$ as $\\sigma$. Both are open, contiguous segments\nof their respective curves.\n\\end{Def}\n\nBefore we define which subcurves we will use to generate vertices, we will need a few definitions first. To start, we want to define\ntwo open, disjoint, contiguous segments of $\\gamma$, which we will call $\\eta_{\\textrm{start}}$ and $\\eta_{\\textrm{end}}$. Note that\nthe monotonicity of $H$ guarantees that they are disjoint.\n\n\\begin{Def}[$\\eta_{\\textrm{start}}$ and $\\eta_{\\textrm{end}}$]\n\\label{defn:etas}\nWe define the segment $\\eta_{\\textrm{start}}$ as the segment of $\\gamma$ that is \\emph{not}\ncontained in the closure of the interior of $H_{\\tau_1}$. Since there are exactly two intersection points between\n$H_{\\tau_1}$ and $\\gamma$, this segment is well defined.\n\nWe define $\\eta_{\\textrm{end}}$ as the open segment of $\\gamma$\nthat is contained in the interior of $H_{\\tau_{n-1}}$. Since $H_{\\tau_{n-1}}$ and $\\gamma$ intersect\nin exactly two points, this segment is well defined. These are shown in Figure \\ref*{fig:etas}.\n\\end{Def}\n\n\\realfig{fig:etas}{etas.pdf}{$\\eta_{\\textrm{start}}$ and $\\eta_{\\textrm{end}}$}{1.00\\textwidth}\n\nWe have a simple property of $\\eta_{\\textrm{start}}$ and $\\eta_{\\textrm{end}}$ which results from the monotonicity of $H$:\n\\begin{Lem}\n\\label{lem:etas_no_intersect}\nFor every point $\\tau \\in [\\tau_1, \\tau_{n-1}]$, and for any intersection point $p$ between $H_\\tau$ and $\\gamma$,\n$p$ lies neither in $\\eta_{\\textrm{start}}$, nor does it lie in $\\eta_{\\textrm{end}}$.\n\\end{Lem}\n\nWe can now begin to define the set of subcurves that we will use to produce our vertices; we will define \nwhether or not a subcurve \\emph{respects} $\\gamma$.\n\n\\begin{Def}[Respects $\\gamma$]\n\\label{defn:respects_gamma}\nWe say that a subcurve $\\alpha$ at $\\tau_i$ \\emph{respects $\\gamma$} if the \nsegment $\\rho$ of $\\alpha$ (the segment that came from $\\gamma$) has the following two properties:\n\\begin{enumerate}\n\t\\item\t$\\eta_{\\textrm{start}} \\cap \\rho = \\emptyset$\n\t\\item\t$\\eta_{\\textrm{end}} \\subset \\rho$\n\\end{enumerate}\n\\end{Def}\n\nFor every subcurve $\\alpha$ at $\\tau_i$ that respects $\\gamma$, we give each endpoint of this curve a sign, either a $+$, or a $-$.\nLet $p$ be an endpoint of $\\gamma$. Orienting $\\gamma$, we can list the order in which we encounter intersection points.\nLet $q$ and $r$ be the intersection points which we encounter immediately before and after $p$, which may be the same point.\nSince $\\gamma$ is oriented, we can also produce two contiguous segments of $\\gamma$: the segment traversed from $q$ to $p$,\nand the segment traversed from $p$ to $r$ (with respect to the orientation of $\\gamma$).\nNeither segment contains any intersection points. Let them be $\\beta_1$ and $\\beta_2$.\n\nWe also see that exactly one of $\\beta_1$ and $\\beta_2$ must be contained in the interior of $H_{\\tau_i}$ since $p$ is a transverse intersection point\nof $H_{\\tau_i}$ and $\\gamma$. Let this component be $\\beta_j$.\nFurthermore, recalling that $\\rho$ is the segment of $\\alpha$ that comes from $\\gamma$, exactly one of $\\beta_1$ and $\\beta_2$ must be contained in $\\rho$.\nLet this component be $\\beta_k$.\n\nIf $k = j$, then we assign a $+$ sign to $p$. If not, then we assign a $-$ sign to $p$. Note that the sign of a point does not depend on how we orient $\\gamma$.\n\nFigure \\ref*{fig:intersection_sign} depicts curves $H_\\tau$ and $\\gamma$. It also depicts a subcurve $\\alpha$ which respects $\\gamma$\nand its endpoints. Here, $\\gamma$ is the same curve that appears in Figure \\ref*{fig:etas}, and we assume that $\\eta_{\\textrm{start}}$ and\n$\\eta_{\\textrm{end}}$ are as in this figure. The segment $\\rho$ of $\\alpha$ is shown, and the signs of both endpoints are displayed as well. Lastly,\n$\\sigma$ is shown with the tangent vector at each of its endpoints.\nWe see that the directions of these tangents\nwith respect to the interior of $\\gamma$ do not agree with the signs of both intersection points, as per Definition \\ref*{defn:valid}. Hence, $\\alpha$ is not a valid\nsubcurve.\n\n\n\\realfig{fig:intersection_sign}{sign_of_intersection.pdf}{From top to bottom, left to right: $H_\\tau$ and $\\gamma$, a subcurve $\\alpha$ that respects $\\gamma$, $\\rho$,\n\tthe signs of the endpoints of $\\alpha$, and $\\sigma$ with the tangent vector at each of its endpoints}{1.00\\textwidth}\n\n\nWe can now define the set of subcurves which we want to use to produce vertices. We call such a subcurve \\emph{valid}.\n\n\\begin{Def}[Valid subcurve]\n\\label{defn:valid}\nWe say that a subcurve $\\alpha$ at $\\tau_i$ is \\emph{valid} if it respects $\\gamma$, and if the following additional properties\nare true of $\\sigma$.\n\nTo begin, let $p_1$ and $p_2$ be the endpoints of $\\alpha$.\nWe can parametrize $\\sigma$ so that it goes from $p_1$ to $p_2$. Since $H_{\\tau_i}$ and $\\gamma$ intersect transversely at $p_1$ and at $p_2$,\nwe can categorize the tangent vector of $\\sigma$ at $p_1$ and at $p_2$ as being into the interior of $\\gamma$, or into the exterior\nof $\\gamma$.\n\nWe then require that, at $p_1$, the tangent of $\\sigma$ points into the interior of $\\gamma$ if the sign at $p_1$ is $+$,\nand that it points into the exterior of $\\gamma$ if it is $-$. We also require that, at $p_2$, the tangent of $\\sigma$ points into the exterior\nof $\\gamma$ if the sign at $p_2$ is $+$, and that the tangent points into the interior at $p_2$ if the sign is $-$.\nNote that this definition is independent of the order in which we choose the endpoints of $\\alpha$; a subcurve is valid with respect to one\norder of endpoints if and only if it is valid with respect to the other order.\n\\end{Def}\nFor each valid subcurve at $\\tau_i$ with $i \\in \\{ 1, \\dots, n - 1 \\}$, we add a vertex $v$ to the graph $\\Gamma$. We say that this vertex is generated\nfrom $\\tau_i$. We also have a length bound\nfor each valid subcurve, as a result of it being composed of a segment of $\\gamma$ and a segment of $H_{\\tau_i}$:\n\n\\begin{Lem}[Length bound for valid subcurves]\n\\label{lem:length_subcurves}\nFor each valid subcurve $\\alpha$, the length of $\\alpha$ is at most\n\t$$ 2L + \\epsilon.$$\n\\end{Lem}\n\n\n\\noindent \\textbf{Edges}\n\nWe now add edges to this graph. The idea will be that, for each $i \\in \\{1, \\dots, n - 2 \\}$, we will add a set of edges, denoted by\n$E_i$. We will specify an algorithm which takes any vertex $v$ generated from $\\tau_i$ or from $\\tau_{i+1}$, and produces a different vertex $w$, also\ngenerated from $\\tau_i$ or from $\\tau_{i+1}$. This algorithm is symmetric in that, if given vertex $w$, it will produce vertex $v$.\nWe then join each pair of vertices produced by this algorithm by an edge. $E_i$ will be the collection of these edges.\n\nTo define this algorithm, fix a vertex $v$ in $\\Gamma$ generated from $\\tau_i$.\nWe will define the algorithm in two parts, depending on whether the resulting vertex\n$w$ is generated from $\\tau_{i+1}$ (a ``vertical'' edge), or if it is generated from $\\tau_i$ (a ``horizontal'' edge).\n\nThroughout the definition of this algorithm, we say that two intersections $p$ and $q$ between $H$ and $\\gamma$ at $\\tau_i$ are ``involved'' or ``deleted'' in the \nReidemeister move between $\\tau_i$ and $\\tau_{i+1}$. By this, we mean the following. Let the point at which $H$ and $\\gamma$ become tangent to each other be $\\tau'$,\nwith $\\tau_i < \\tau' < \\tau_{i+1}$. Since all intersections between $H$ and $\\gamma$ are transverse on $(\\tau_i, \\tau')$, we can trace the path of $p$ and\n$q$ forward to $\\tau'$. When we do this, we see that $p$ gets traced to the tangential intersection at $\\tau'$ (which is deleted), and\n$q$ gets traced to the same intersection point. We use the same terminology to describe intersection points between $H$ and $\\gamma$ at $\\tau_{i+1}$ that can be traced backwards to\nthe tangential intersection at $\\tau'$.\n\n\\noindent \\textbf{Vertical Edges}\nRecall that, between $\\tau_i$ and $\\tau_{i+1}$, there is exactly one Reidemeister move. This move involves two intersection points\nbetween $H$ and $\\gamma$; it either creates two intersection points, or it deletes two of them.\nLet $\\alpha$ be the valid subcurve at $\\tau_i$ that produced the vertex $v$ and let $p_1$ and $p_2$ be the two distinct endpoints of $\\alpha$.\nIf neither of these points is involved in the Reidemeister move, then the algorithm to find the vertex $w$\ngenerated from $\\tau_{i+1}$ is simple. Since neither $p_1$ nor $p_2$ are deleted from $\\tau_i$ to $\\tau_{i+1}$, they both\nfollow continuous paths from $\\tau_i$ to $\\tau_{i+1}$. Let $\\widetilde{p_1}$ and $\\widetilde{p_2}$\nbe the points which we reach at $\\tau_{i+1}$. We also see that we can follow $\\sigma$ and $\\rho$ from $\\tau_i$ to $\\tau_{i+1}$\nin a similar fashion, arriving at $\\widetilde{\\sigma}$ and $\\widetilde{\\rho}$. Let $\\widetilde{\\alpha}$ be the subcurve formed by following\n$\\widetilde{\\sigma}$ from $\\widetilde{p_1}$ to $\\widetilde{p_2}$, followed by $\\widetilde{\\rho}$ from $\\widetilde{p_2}$ back to $\\widetilde{p_1}$.\nIf $\\widetilde{\\alpha}$ is a valid subcurve, then it corresponds a vertex $w$. We will show that it is indeed valid; $w$ is the \nvertex that is produced by the algorithm.\n\nAs a result of Lemma \\ref*{lem:etas_no_intersect}, $\\widetilde{\\rho}$ has all of the required inclusion\/exclusion properties with respect to\n$\\eta_{\\textrm{start}}$ and $\\eta_{\\textrm{end}}$, and so $\\widetilde{\\alpha}$ respects $\\gamma$. To show that it is valid, we notice \nthat the sign of $p_1$ is the same as that of $\\widetilde{p_1}$, and the sign of $p_2$ is the same as that of $\\widetilde{p_2}$. Let us\norient $\\sigma$ from $p_1$ to $p_2$, and $\\widetilde{\\sigma}$ from $\\widetilde{p_1}$ to $\\widetilde{p_2}$. We then have that\nthe direction of the tangent vector of $\\sigma$ at $p_1$ with respect to the interior of $\\gamma$ is the same as the direction of the tangent\nvector of $\\widetilde{\\sigma}$ at $\\widetilde{p_1}$ with respect to the interior of $\\gamma$. Similarly, the direction of the tangent vector\nat $p_2$ is the same as that at $\\widetilde{p_2}$. Hence, $\\widetilde{\\alpha}$ is a valid subcurve, and so we are done.\n\nIf $v$ is instead generated from $\\tau_{i+1}$, and neither of the endpoints of $\\alpha$ are involved in the Reidemeister move\nbetween $\\tau_i$ and $\\tau_{i+1}$, then we follow the exact same procedure as above, but in reverse.\n\n\\noindent \\textbf{Horizontal Edges}\nAgain, let the vertex $v$ be generated from $\\tau_i$, let $\\alpha$ be the valid subcurve which corresponds to $v$, \nand let $p_1$ and $p_2$ be the endpoints of $\\alpha$. If neither $p_1$ nor $p_2$ are involved in the\nReidemeister move between $\\tau_i$ and $\\tau_{i+1}$, then we use the algorithm described above.\nIn this component of the algorithm, we determine the resulting vertex $w$ if $p_1$ or $p_2$ are involved\nin the move. Furthermore, let $\\tau'$ be the point between $\\tau_i$ and $\\tau_{i+1}$ at which\n$H_{\\tau'}$ is tangent to $\\gamma$. This is the point at which the Reidemeister move ``occurs''.\n\nWe first rule out the possibility that both $p_1$ and $p_2$ are involved in the Reidemeister move:\n\\begin{Lem}\n\\label{lem:no_double_interaction}\n$p_1$ and $p_2$ cannot both be deleted in the Reidemeister move between $\\tau_i$ and $\\tau_{i+1}$.\n\\end{Lem}\n\\begin{Pf}{Proof}\nAssume that they are both involved in the \nReidemeister move. As in the definition of subcurves, we can break $\\gamma$ into two contiguous segments, each with endpoints $p_1$ and $p_2$.\nWe do this by starting at $p_1$, and then by traversing $\\gamma$ to $p_2$ in each of the two possible directions. Let these two components be\n$\\beta_1$ and $\\beta_2$.\nIf both $p_1$ and $p_2$ are involved in the Reidemeister move, then at least one of these two segments would have to contain no intersection points.\nThis is because no intersection points are deleted between $\\tau_i$ and $\\tau'$, and there is no way for intersection points to move through each other.\nAs such, until $\\tau'$, the order of intersection points as we traverse $\\gamma$ remains the same. Hence, if there were intersection points\nin both $\\beta_1$ and $\\beta_2$, then there would be no way for $p_1$ and $p_2$ to be deleted together, as there would have to be an interaction between\nat least one other pair of intersection points first. Let this intersection-free segment be $\\kappa$.\n\nWe can also choose this segment $\\kappa$ so that, for every $s \\in \\kappa$, $s$ is an intersection point between $H_{\\tau_s}$ and $\\gamma$\nfor some $\\tau_s \\in [\\tau_i, \\tau']$.\n\nFurthermore, since $\\alpha$ is a valid subcurve, it respects $\\gamma$, and so we see that exactly one $\\beta_j$ must \ncontain $\\eta_{\\textrm{start}}$, and the other must contain $\\eta_{\\textrm{end}}$. Hence, $\\kappa$ must contain one of these curves.\nBy Lemma \\ref*{lem:etas_no_intersect}, there are thus points in $\\kappa$ that are not realized as intersection points between $\\tau_i$ and $\\tau_{i+1}$.\nThis is a contradiction, and so $p_1$ and $p_2$ cannot both be deleted in the Reidemeister move between $\\tau_i$ and $\\tau_{i+1}$.\n\\end{Pf}\n\nLet us now move to the case where just one of $p_1$ or $p_2$ is deleted at $\\tau'$. Without loss of generality, let us assume that it is $p_1$, and let $q$ be\nthe other intersection point at $\\tau_i$ which is deleted with $p_1$ in the Reidemeister move. We adopt a similar approach to when we added vertical edges.\nWe can trace the path of $p_1$ forward until $\\tau'$, and we also trace the path of $q$ forward until $\\tau'$. We notice that, since both are\ndeleted at $\\tau'$, they merge at this point. We can thus trace a path from $p_1$ to $q$ by first going forward to $\\tau'$, tracing the path\nof $p_1$ forward, and then we can go backward, tracing the path of $q$ backwards.\n\nThis path from $p_1$ to $q$ induces a homotopy from $\\alpha$ to a subcurve $\\widetilde{\\alpha}$ at $\\tau_i$ with endpoints $q$ and $p_2$. $\\widetilde{\\alpha}$\nis formed from the segment $\\widetilde{\\rho}$ of $\\gamma$ and the segment $\\widetilde{\\sigma}$ of $H_{\\tau_i}$. The first is found by following $\\rho$ forward\nto $\\tau'$, then by going backwards to $\\tau_i$, using $q$ as an endpoint instead of $p_1$ as we go backwards; $\\widetilde{\\sigma}$ is found by doing the same, but with\n$\\sigma$. This is shown in Figure \\ref*{fig:back_and_forth}.\n\n\\realfig{fig:back_and_forth}{back_and_forth.pdf}{$\\alpha$ and $\\widetilde{\\alpha}$ if an endpoint is deleted}{1.00\\textwidth}\n\nThe question, as before, is if $\\widetilde{\\alpha}$ is a valid subcurve. We see that, since $\\alpha$ is a valid subcurve, it respects $\\gamma$, and so\n$\\rho$ has the proper inclusion\/exclusion properties with respect to $\\eta_{\\textrm{start}}$ and $\\eta_{\\textrm{end}}$. Lemma \\ref*{lem:etas_no_intersect} then\nimplies that $\\widetilde{\\rho}$ has similar properties, and so $\\widetilde{\\alpha}$ respects $\\gamma$.\n\nTo show that $\\widetilde{\\alpha}$ is valid, we must show that $\\widetilde{\\sigma}$ agrees with the signs of $q$ and $p_2$. We first observe that the sign of\n$q$ with respect to $\\widetilde{\\alpha}$ is opposite to the sign of $p_1$ with respect to $\\alpha$. On the other hand, the sign of $p_2$ remains unchanged.\nIf we orient $\\sigma$ from $p_1$ to $p_2$, and $\\widetilde{\\sigma}$ from $q$ to $p_2$, then we\nsee that the tangent vector of $\\widetilde{\\alpha}$ at $p_2$ has the same direction with respect to the interior of $\\gamma$ as the tangent vector of \n$\\sigma$ at $p_2$, and so this endpoint meets the necessary criteria. In terms of $q$, we see that the direction of the tangent vector of $\\widetilde{\\sigma}$ at $q$ with respect\nto the interior of $\\gamma$ is opposite to that of the tangent vector of $\\sigma$ at $p_1$. Hence, this endpoint meets the necessary criteria as well, and so $\\widetilde{\\alpha}$\nis valid. The rigorous proof of this is a case-by-case analysis on the segment of $H_{\\tau_i} \\cup \\gamma$ around $q$ and $p_1$. The cases are formed by\nconsidering all possible interiors of $H_{\\tau_i}$, and all possible arcs $\\rho$. This is shown in Figure \\ref*{fig:intersection_fix}.\nSince $\\widetilde{\\alpha}$ is valid, it corresponds to a vertex $w$, which is the desired vertex.\n\n\n\\realfig{fig:intersection_fix}{intersection_fix.pdf}{In order of rows, top to bottom: intersections $q$ and $p_1$ with the interior of $\\gamma$ shaded, $\\rho$ with the interior of $H_{\\tau_i}$ shaded,\n\t $\\widetilde{\\rho}$ with the interior of $H_{\\tau_i}$ shaded, $\\sigma$, and $\\widetilde{\\sigma}$}{1.00\\textwidth}\n\n\nIf $v$ is generated from $\\tau_{i+1}$, then we follow the above procedure, but in reverse. That is, if the Reidemeister move\nbetween $\\tau_i$ and $\\tau_{i+1}$ creates two intersection points of which one is an endpoint of $\\alpha$, then we follow the above steps\nto produce a vertex $w$. Note that for reasons analogous to those presented in the proof of Lemma \\ref*{lem:no_double_interaction}, both endpoints of $\\alpha$ cannot be created\nby the Reidemeister move between $\\tau_i$ and $\\tau_{i+1}$.\n\n\nBefore we complete the proof of Lemma \\ref*{lem:stage_one}, we prove some important properties of $\\Gamma$:\n\n\\begin{Lem}[Properties of $\\Gamma$]\n\\label{lem:gamma_properties}\nThe graph $\\Gamma$ has the following properties:\n\\begin{enumerate}\n\t\\item\tFor each set of edges $E_i$, $i \\in \\{1, \\dots, n - 2 \\}$, and for each vertex $v$ generated from $\\tau_i$ or $\\tau_{i+1}$,\n\t\t$v$ is the endpoint of exactly one edge in $E_i$.\n\t\\item\tAll vertices generated from $\\tau_1$ and all vertices generated from $\\tau_{n-1}$ have degree 1;\n\t\tall other vertices have degree $2$.\n\t\\item\tThere is exactly one vertex generated from $\\tau_1$, and one vertex generated from $\\tau_{n-1}$,\n\t\tand they correspond to the curves shown in Figure \\ref*{fig:endpoint_curves}.\n\t\\item\tIf two vertices are joined by an edge, then there is a homotopy of closed curves between\n\t\tthe curves corresponding to the vertices through closed curves of length at most $2L + \\epsilon$.\n\t\tFurthermore, all of these curves contain $\\eta_{\\textrm{end}}$.\n\\end{enumerate}\n\\end{Lem}\n\\begin{Pf}{Proof}\nThe first statement is a result of the fact that the algorithm used to add edges takes any vertex $v$ generated from $\\tau_i$ or from \n$\\tau_{i+1}$ and produces a vertex $w$, $v \\neq w$. Since we use this algorithm to add edges, there is an edge between $v$ and $w$. Additionally, it is easy to check\nthat this algorithm is symmetric in that the vertex $w$ will produce the vertex $v$. Hence, each vertex is the endpoint of exactly one\nedge in $E_i$.\n\nThe second statement results from the fact that, for each vertex $v$ that is generated from $\\tau_1$ or from $\\tau_{n-1}$, $v$ is the endpoint\nof exactly one edge from $E_1$ or $E_{n-1}$, respectively, and there is no other set $E_j$ which contains an edge that has $v$ as an endpoint.\nThus, the degree is $1$. For any vertex $v$ generated from $\\tau_i$ with $i \\in \\{ 1, \\dots, n-1 \\}$, $v$ is the endpoint of an edge from\n$E_i$, and is also the endpoint of an edge from $E_{i+1}$. Hence, it has degree $2$.\n\nThe third statement follows from looking at the set of all valid subcurves at $\\tau_1$ and $\\tau_{n-1}$. At each of these points, there are exactly two\nintersections between $H$ and $\\gamma$, and so it is a simple exercise to look at each of the four subcurves and to show that the only ones\nthat are valid are the ones depicted in Figure \\ref*{fig:endpoint_curves}.\n\nThe last statement comes from examining how we add edges. In all of the cases, we are tracing two intersection points back or forth, and\nkeeping track of one segment of $H_\\tau$ that connects these $2$ points and one segment of $\\gamma$ that also connects these two points, which generates a continuous homotopy. Since\nboth $\\gamma$ and $H_\\tau$ are bounded in length by $L + \\epsilon$, taking a segment of one and joining it with a segment from the other has length\nat most\n\t$$ 2L + \\epsilon.$$\nThe fact that they all contain $\\eta_{\\textrm{end}}$ is a result of two observations. First, all subcurves at any $\\tau_i$ respect $\\gamma$, and so contain $\\eta_{\\textrm{end}}$.\nSecond, as a result of Lemma \\ref*{lem:etas_no_intersect}, no intersections between $H$ and $\\gamma$ lie in $\\eta_{\\textrm{end}}$ for any $\\tau \\in [\\tau_1, \\tau_{n-1}]$.\n\\end{Pf}\n\n\n\\realfig{fig:endpoint_curves}{curve_endpoints.pdf}{The curve that corresponds to the only vertex generated from $\\tau_1$, and the curve that corresponds to the only vertex generated from $\\tau_{n-1}$}{1.00\\textwidth}\n\n\nWe can now prove Lemma \\ref*{lem:stage_one}.\n\n\\begin{Pf}{Proof of Lemma \\ref*{lem:stage_one}}\nFrom Lemma \\ref*{lem:gamma_properties}, we have that there is only $1$ vertex $v$ generated from $\\tau_1$, and one vertex $w$ generated from $\\tau_{n-1}$. Additionally, they have degree $1$, and all other\nvertices in $\\Gamma$ have degree $2$. As a result, we have that there is a path in $\\Gamma$ from $v$ to $w$.\nLet $\\alpha_1$ and $\\alpha_2$ be the subcurves that correspond to $v$ and $w$, respectively.\nDue to the property of $\\Gamma$ that edges represent homotopies over closed curves of length at most $2L + \\epsilon$,\nthere is thus a homotopy from $\\alpha_1$ to $\\alpha_2$ over such curves. Furthermore, every curve in this homotopy contains\n$\\eta_{\\textrm{end}}$.\n\nWe now observe that $\\gamma$ is homotopic to $\\alpha_1$ over curves of length at most $2L + \\epsilon$, and so we can homotope\n$\\gamma$ to $\\alpha_2$ over closed curves with the same length bound. All of these curves also contain $\\eta_{\\textrm{end}}$.\n\nThe rest of the proof depends on whether $H$ contracts $\\gamma$ to a point inside $\\gamma$ or outside $\\gamma$. If it contracts\n$\\gamma$ to a point outside $\\gamma$, then we see that $\\alpha_2$ can be contracted to the point $x \\in \\eta_{\\textrm{end}}$ on $\\gamma$\nthrough curves that contain $x$.\nSince $\\eta_{\\textrm{end}}$ is contained in all curves in this homotopy up to $\\alpha_2$, we conclude that $x$\nis contained in every curve in this entire contraction.\n\nIf $H$ contracts $\\gamma$ to a point inside $\\gamma$, then recalling that $\\tau^\\star$ is the last point at which $H$ intersects $\\gamma$, and \n$x$ is the point of tangential intersection at $\\tau^\\star$, we can homotope $\\alpha_2$ to $H_{\\tau^\\star}$ through curves containing $x$ and \nwhich are bounded in length by $2L + \\epsilon$. Since $\\gamma$ can also be homotoped to $\\alpha_2$ through such curves,\nthis gives us a desirable homotopy from $\\gamma$ to $H_{\\tau^\\star}$. This completes the proof.\n\\end{Pf}\n\nFinally, we illustrate this process using an explicit homotopy. This is shown in Figure \\ref*{fig:example}.\n\n\\realfig{fig:example}{example_homotopy.pdf}{A homotopy that covers $\\gamma$ and the resulting contraction of $\\gamma$}{1.00\\textwidth}\n\n\n\n\\subsection{Proof of Lemma \\ref*{lem:stage_two}}\n\nWe now prove Lemma \\ref*{lem:stage_two}. Given a curve $H_{\\tau^\\star}$ and a point $x \\in \\gamma \\cap H_{\\tau^\\star}$, we want to show that we can contract $H_{\\tau^\\star}$ through curves based at $x$,\nand of length at most $2L + 2d + \\epsilon$. The idea here is to employ\na method similar to that used in this article to produce a contraction of the boundary of a Riemannian disc from a monotone contraction of that boundary.\nTo do this, let $c$ be the point that $H$ contracts $H_{\\tau^\\star}$ to. Join $x$ to $c$ via a minimal geodesic, and let $y$ be the last point of intersection between this geodesic\nand $H_{\\tau^\\star}$. This is depicted in Figure \\ref*{fig:curve_pieces}.\n\n\\realfig{fig:curve_pieces}{curve_pieces.pdf}{$H$ and $\\gamma$ as per the hypotheses of Lemma \\ref*{lem:stage_two}}{1.00\\textwidth}\n\nOur homotopy now works as follows. One should refer to Figure \\ref*{fig:example_homotopy} for a visual reference.\nLet $\\beta$ be the segment of the minimal geodesic that connects $y$ to $c$ entirely in the interior of $H_{\\tau^\\star}$. Let the length of $\\beta$ be $B$; we of course\nhave that $B \\leq d$, where $d$ is the diameter of the manifold. Let $\\alpha$ be a segment of $H_{\\tau^\\star}$ that connects $x$ to $y$ which is of length at most $\\frac{L + \\epsilon}{2}$.\nWe now produce our contraction of $H_{\\tau^\\star}$ in three parts.\nThe first part is a homotopy from $H_{\\tau^\\star}$ to the curve formed by traversing $\\alpha$ from $x$ to $y$, following by traversing the entirety of $H_{\\tau^\\star}$ from $y$ to $y$, and then by traversing $- \\alpha$\nfrom $y$ back to $x$. This homotopy consists of curves bounded in length by $2L + \\epsilon$. Let us call this curve $\\eta$.\n\n\\realfig{fig:example_homotopy}{tangent_example_sequence.pdf}{Building the contraction}{1.00\\textwidth}\n\nThe second step is a homotopy from $\\eta$ to the curve formed by traversing $\\alpha$ from $x$ to $y$, then $\\beta$ from $y$ to $c$, then $- \\beta$ from $c$ back to $y$, then $- \\alpha$ from $y$ back to $x$.\nLet us call this curve $\\nu$. This homotopy, $P$, is defined on the interval $[0, B]$, where (as above) $B$ is the length of $\\beta$. For each\n$s \\in [0,B]$, let $\\rho_s$ be the segment of $\\beta$ from $y$ which has a length of $s$. Since $H$ is monotone, there is exactly one curve $\\delta$ corresponding to a curve in the homotopy\n$H$ which has the property that it goes through $y$ if $s = 0$, and that it goes through the endpoint of $\\rho_s$ which is not $y$ if $s > 0$. Now, we define $P(s)$ to be the curve formed by\ntraversing $\\alpha$, then $\\rho_s$, then $\\delta$, then $- \\rho_s$, then $- \\alpha$. Since $H$ is monotone, this produces a continuous homotopy of piecewise smooth simple curves\nof length at most $2L + 2d + \\epsilon$.\n\nThe third step is that we homotope $\\nu$ to the point $x$ by contracting it in the obvious way; since it is a curve traversed forward from $x$ to $y$ to $c$ and then backward from $c$ to $y$ to $x$,\nit is obvious how to do this without exceeding a length bound of $L + 2d + \\epsilon$.\n\nBy concatenating the above homotopies, we get a homotopy of closed curves with the desired properties, completing the proof.\n\n\n\n\n\n\\bigskip\n\n\n\\noindent {\\bf Acknowledgements:} This work was supported in part by an NSERC Discovery Grant (Rotman), by an NSERC Postgraduate Scholarship (Chambers), and\nby an Ontario Graduate Scholarship (Chambers). This\npaper was partially written during the authors' visit of the \nMax-Planck Institute for Mathematics in Bonn. The authors would like to\nthank the Max-Planck Institute for its kind hospitality.\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\n\n\n\n \n\n\n\n\n\n\n\n\\small \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nThe effect of localization in spatially extended linear systems\nsubject to a frozen random spacial inhomogeneity of parameters is\nknown as Anderson localization (AL). AL has been first discovered\nand discussed for quantum systems~\\cite{Anderson-1958}. Later on,\ninvestigations were extended to diverse branches of classical and\nsemiclassical physics: wave optics ({\\it\ne.g.},~\\cite{Rossum-Nieuwenhuizen-1999}), acoustics ({\\it\ne.g.},~\\cite{Maynard-2001}), {\\it etc}. The phenomenon has been\ncomprehensively studied and well understood mathematically for the\nSchr\\\"odinger equation and related mathematical models ({\\it\ne.g.},~\\cite{Froehlich-Spencer-1984,Lifshitz-Gredeskul-Pastur-1988,Gredeskul-Kivshar-1992}).\nAlso, the role of nonlinearity in these models has been addressed\nin the literature (for instance, destruction of AL by\nnonlinearity~\\cite{Pikovsky-Shepelyansky-2008,Gredeskul-Kivshar-1992}).\n\nBeing well studied for conservative media (or systems) the\nlocalization phenomenon did not receive a comparable attention for\nactive\/dissipative ones as, {\\it e.g.}, in problems of thermal\nconvection or reaction-diffusion. The main reason is that the\nphysical interpretations of formal solutions to the Schr\\\"odinger\nequation and governing equations for active\/dissipative media are\nessentially different and, therefore, the theory of AL may be\nextended to the latter only under certain strong restrictions\n(this statement is discussed in details in the end of the next\nsection). Nevertheless, effects similar to AL can be observed in\nfluid dynamical systems (\\cite{Goldobin-Shklyaeva-PRE-2008};\nin~\\cite{Hammele-Schuler-Zimmermann-2006} the effect of parametric\ndisorder on the excitation threshold in one-dimensional\nGinzburg--Landau equation has been studied, but without attention\nto localization effects). In this paper, we study an example: the\nproblem where localized thermoconvective currents excited under\nparametric disorder crucially influence the process of transport\nof a passive scalar ({\\it e.g.}, a pollutant).\n\nThe paper is organized as follows. In \\sref{sec1} we formulate the\nspecific physical problem we deal with, introduce the relevant\nmathematical model, and discuss physical background for the\nproblem.\n\\Sref{sec2} presents the results of a numerical simulation. In\n\\sref{sec3} we develop an analytical theory for a certain limit\ncase.\n\\Sref{concl} ends the paper with conclusions.\n\n\n\\section{Problem formulation and basic equations}\\label{sec1}\nThe modified Kuramoto--Sivashinsky equation\n\\begin{equation}\n\\dot{\\theta}(x,t)=-\\left(\\theta_{xxx}(x,t)\n +q(x)\\,\\theta_x(x,t)-(\\theta_x(x,t))^3\\right)_x\n\\label{eq1-01}\n\\end{equation}\n\\noindent\nis relevant for a broad variety of active media where pattern\nselection occurs. It governs two-dimensional (2D) large-scale\nnatural thermal convection in a horizontal fluid layer heated from\nbelow~\\cite{Knobloch-1990,Shtilman-Sivashinsky-1991} and holds\nvalid for a turbulent fluid~\\cite{Aristov-Frick-1989}, a binary\nmixture at small Lewis number~\\cite{Schoepf-Zimmermann-1989-1993},\na porous layer saturated with a\nfluid~\\cite{Goldobin-Shklyaeva-BR-2008,Goldobin-Shklyaeva-PRE-2008},\n{\\it etc}.\\footnote{In these fluid dynamical systems, except the\nturbulent one~\\cite{Aristov-Frick-1989}, the plates bounding the\nlayer should be nearly thermally insulating for a large-scale\nconvection to arise.} Specifically, in the problems mentioned,\ntemperature perturbations $\\theta$ are almost uniform along the\nvertical coordinate $z$ and obey~\\eref{eq1-01}.\n\nTo argue for a general validity of~\\eref{eq1-01}, let us note the\nfollowing. Basic laws in physics are the conservation ones. This\nfact quite often results in final equations having the form\n$\\partial_t[\\mbox{quantity}]+\\nabla\\!\\cdot\\![\\mbox{flux of\nquantity}]=0$. With such conservation laws either for systems with\nthe sign inversion symmetry of the fields, which is wide spread in\nphysics, or for description of a spatiotemporal modulation of an\noscillatory mode, the original Kuramoto--Sivashinsky equation\n({\\it e.g.}, see~\\cite{Michelson-1986}) should be rewritten in the\nform~\\eref{eq1-01}. On these grounds, we claim\nequation~\\eref{eq1-01} to describe pattern formation in a broad\nvariety of physical systems.\n\nIn the following we restrict our consideration to the case of\nconvection in a porous medium; nevertheless, the most of results\nmay be easily extended to the other physical systems mentioned.\n\\Eref{eq1-01}\nis already dimensionless and below we introduce all parameters and\nvariables in appropriate dimensionless forms.\n\nRecall, the large-scale (or long-wavelength) approximation is\nidentical to the approximation of a thin layer and assumes that\nthe characteristic horizontal scales are large against the layer\nheight $h$. For large-scale convection, $(21\/2)h^2q(x)$\n(cf.\\,\\eref{eq1-01}, \\cite{Goldobin-Shklyaeva-BR-2008}) represents\nrelative deviations of the heating intensity and of the\nmacroscopic properties of the porous matrix (porosity,\npermeability, heat diffusivity, {\\it etc.}) from the critical\nvalues for the spatially homogeneous case. Thus, for positive\nspatially uniform $q$, convection sets up, while for negative $q$,\nall the temperature inhomogeneities decay. For convection in a\nporous medium~\\cite{Goldobin-Shklyaeva-BR-2008}, the macroscopic\nfluid velocity field\n\\begin{equation}\n\\vec{v}=\\frac{\\partial\\Psi}{\\partial z}\\vec{e}_x\n -\\frac{\\partial\\Psi}{\\partial x}\\vec{e}_z\\,,\n \\qquad\n\\Psi=\\frac{3\\sqrt{35}}{h^3}\\,z(h-z)\\,\\theta_x(x,t)\\equiv f(z)\\,\\psi(x,t)\\,,\n\\label{eq1-02}\n\\end{equation}\n\\noindent\nwhere $\\psi(x,t)\\equiv\\theta_x(x,t)$ is the stream function\namplitude, the reference frame is such that $z=0$ and $z=h$ are\nthe lower and upper boundaries of the layer, respectively\n(\\fref{fig1}b). Though the temperature perturbations\nobey~\\eref{eq1-01} for diverse convective systems, function\n$f(z)$, which determines the relation between the flow pattern and\nthe temperature perturbation, is specific for each case.\n\n\n\\begin{figure}[!t]\n\\center{\n\\begin{tabular}{cc}\n \\sf (a)&\\hspace{-10mm}\\includegraphics[width=0.90\\textwidth]%\n {gold-shkl-08jstat-fig1a.eps}\\\\[5pt]\n \\sf (b)&\\hspace{-10mm}\\includegraphics[width=0.90\\textwidth]%\n {gold-shkl-08jstat-fig1b.eps}\n\\end{tabular}\n}\n \\caption{\n(a):\\,Establishing steady solutions to~\\eref{eq1-01} for $q_0$\nindicated in the plot are sets of exponentially localized patterns\n[shown for one and the same realization of random inhomogeneity\n$\\xi(x)$ and $\\eps=1$; $q(x)$ is represented by\n$q_\\pi(x)=\\pi^{-1}\\int_{x-\\pi\/2}^{x+\\pi\/2}q(x')\\rmd x'$].\n(b):\\,The stream lines corresponding to the solutions in graph~(a)\nare plotted for the case of convection in a porous layer\n[cf.\\,\\eref{eq1-02}].}\n \\label{fig1}\n\\end{figure}\n\n\nThough \\eref{eq1-01} is valid for a large-scale inhomogeneity\n$q(x)$, which means $h|q_x|\/|q|\\ll1$, one may set such a hierarchy\nof small parameters, namely $h\\ll(h|q_x|\/|q|)^2\\ll1$, that a\nfrozen random inhomogeneity may be represented by white Gaussian\nnoise $\\xi(x)$:\n\\[\nq(x)=q_0+\\xi(x),\\quad \\la\\xi(x)\\ra=0,\\quad\n\\la\\xi(x)\\xi(x')\\ra=2\\eps^2\\delta(x-x'),\n\\]\n\\noindent\nwhere $\\eps^2$ is the disorder intensity and $q_0$ is the mean\nsupercriticality ({\\it i.e.}\\ departure from the instability\nthreshold of the disorderless system). Numerical simulation\nreveals only steady solutions to establish in~\\eref{eq1-01} with\nsuch $q(x)$~\\cite{Goldobin-Shklyaeva-PRE-2008}.\n\nLet us now discuss some general points related to the physical\nproblem under consideration. Obviously, the linearized form of\nequation~\\eref{eq1-01} in the stationary case, {\\it i.e.},\n\\[\n-\\theta_{xxx}(x)-\\xi(x)\\,\\theta_x(x)=q_0\\,\\theta_x(x)\\,,\n\\]\nis a stationary Schr\\\"odinger equation for $\\psi=\\theta_x$ with\n$q_0$ instead of the state energy and $-\\xi(x)$ instead of the\npotential. Therefore, like for the Schr\\\"odinger equation ({\\it\ne.g.},\nsee~\\cite{Froehlich-Spencer-1984,Lifshitz-Gredeskul-Pastur-1988,Gredeskul-Kivshar-1992}),\nall the solutions $\\psi(x)$ to the stationary linearized\nequation~\\eref{eq1-01} are spatially localized for arbitrary\n$q_0$; asymptotically,\n\\[\n\\psi(x)\\propto\\exp(-\\gamma|x|),\n\\]\nwhere $\\gamma$ is the localization exponent. Such a localization\ncan be easily seen for a solution to the nonlinear\nproblem~\\eref{eq1-01} in\n\\fref{fig1}a for $q_0=-2.5$.\n\nOne should keep in mind, that, in the quantum Schr\\\"odinger\nequation, localized modes are bound states of, {\\it e.g.}, an\nelectron in a disordered media. Even the mutual nonlinear\ninteraction of these modes, which appears due to the\nelectron--electron interaction and leads to destruction of AL,\nshould be interpreted in the context of the specific physical\nmeaning of the quantum wave function. Therefore, the theory\ndeveloped for AL in quantum systems may not be directly extended\nto active\/dissipative media. Indeed, in~\\eref{eq1-01}, all excited\nlocalized modes of the linearized problem do mutually interact via\nnonlinearity in a way where they irreversibly lose their identity\n(unlike solitons in soliton bearing systems, which completely\nrecover after mutual collision). Thus, when the spatial density of\nexcited localized modes is large and these modes form an almost\neverywhere intense flow, localization properties of formal\nsolutions to the linearized problem do absolutely not manifest\nthemselves.\n\nNevertheless, when excited modes are spatially sparse, solitary\nexponentially localized patterns can be discriminated as reported\nin~\\cite{Goldobin-Shklyaeva-PRE-2008}. \\Fref{fig1} shows sample\npatterns for such a case. One can see that for negative $q_0$ the\nspatial density of excited modes rapidly decreases as $q_0$\ndecreases and the pattern localization becomes more pronounced.\nFor a small spatial density of excited modes, one can distinguish\nall these modes and introduce the observable quantifier $\\nu$ of\nthe established steady pattern, which measures the spatial density\nof the domains of excitation of convective flow; fortunately, an\nempiric formula fits perfectly the numerically calculated\ndependence~\\cite{Goldobin-Shklyaeva-PRE-2008},\n\\begin{equation}\n\\nu\\approx\n\\frac{1}{4\\sqrt{1.95\\,\\pi}\\,\\eps^{2\/3}|\\wq_0|}\n\\exp\\left(-\\frac{1.95\\,\\wq_0^2}{4}\\right),\n\\label{eq1-03}\n\\end{equation}\n\\noindent\nwhere $\\wq_0\\equiv\\eps^{-4\/3}q_0$.\n\nHere we would like to emphasize the fact of existence of\nconvective currents below the instability threshold of the\ndisorderless system. These currents may considerably and\nnontrivially affect transport of a pollutant (or other passive\nscalar), especially when its molecular diffusivity is small (for\ninstance, for microorganisms or suspensions the diffusion due to\nBrownian motion is drastically weak against the possible\nconvective transport). Transport of a nearly indiffusive passive\nscalar is the subject of our research, as a ``substance'' which is\nessentially influenced by these localized currents and, thus,\nprovides an opportunity to observe manifestation of\ndisorder-induced phenomena discussed\nin~\\cite{Goldobin-Shklyaeva-PRE-2008}.\n\nFrom the viewpoint of mathematical physics, there is one more\nnontrivial question which is worthy to be addressed. In AL an\nimportant topological effect takes place; while in 1D case all the\nsolutions are localized, in higher dimensions spatially unbounded\nsolutions appear ({\\it e.g.},\n\\cite{Froehlich-Spencer-1984}). The modification of~\\eref{eq1-01}\nfor the case of inhomogeneity in the both horizontal directions,\n$q=q(x,y)$, ({\\it e.g.}, see~\\cite{Goldobin-Shklyaeva-BR-2008})\ncannot be turned into the Schr\\\"odinger equation even after\nlinearization in the stationary case. Thus, there are no reasons\nfor any topological effects directly analogous to the one\nmentioned for AL. Nevertheless, one may speak of a percolation\nkind transition, where the domain of an intense convective flow\nbecomes globally connected for high enough $q_0$. Noteworthy, this\ntransition cannot be observed in 1D system~\\eref{eq1-01} as there\nis always a finite probability of a large domain of negative\n$q(x)$ where the flow is damped and the domain of an intense flow\nbecomes disconnected. Essentially, the flow damped never decays\nexactly to zero and, hence, one needs a formal quantitative\ncriterion for the absence of intense currents at a certain point.\nOn the other hand, this transition leads to a crucial enhancement\nof transport of a nearly indiffusive scalar along the layer, and\nthe intensity of this transport can be used to detect the\ntransition immediately in the context that arises applied interest\nto it. In this way, one also avoids introducing a formal\nquantitative criterion. Remarkably, in the context of transport of\na passive scalar, that is the subject of the study we present, the\ntransition from a set of spatially localized currents to an almost\neverywhere intense ``global'' flow can be observed in 1D\nsystem~\\eref{eq1-01} as well.\n\nLet us describe the transport of a passive ({\\it i.e.}, not\ninfluencing the flow in contrast, for instance,\nto~\\cite{Goldobin-Lyubimov-2007}) pollutant by a steady convective\nflow~\\eref{eq1-02}. The flux $\\vec{j}$ of the pollutant\nconcentration $C$ is\n\\begin{equation}\n\\vec{j}=\\vec{v}\\,C-D\\nabla C\\,,\n\\label{eq1-04}\n\\end{equation}\n\\noindent\nwhere the first term describes the convective transport and the\nsecond one represents the molecular diffusion, $D$ is the\nmolecular diffusivity. The establishing steady distributions of\nthe pollutant obey\n\\begin{equation}\n\\nabla\\cdot\\vec{j}=0\\,.\n\\label{eq1-05}\n\\end{equation}\n\\noindent\n\\Eref{eq1-05}\nyields a uniform along $z$ distribution of $C$ (see appendix),\n\\begin{equation}\n\\frac{\\rmd C(x)}{\\rmd x}=-\\frac{J}{\\displaystyle D+\\frac{21\\,\\psi^2(x)}{2h^2D}}\\,,\n\\label{eq1-06}\n\\end{equation}\n\\noindent\nwhere $J$ is the constant pollutant flux along the layer. Note,\nfor the other convective systems we mentioned above, the result\ndiffers only in the factor ahead of $\\psi^2\/D$.\n\n\n\\section{Effective diffusivity}\\label{sec2}\nIn this section we introduce and consider the effective\ndiffusivity (for general ideas on introducing the effective\ndiffusivity one can consult, {\\it\ne.g.},~\\cite{Frisch-1995,Majda-Kramer-1999}). Let us consider the\ndomain $x\\in[0,L]$ with the imposed concentration difference\n$\\delta C$ at the ends. Then the establishing pollutant flux $J$\nis defined by the integral [cf.\\,\\eref{eq1-06}],\n\\[\n\\delta C=-J\\int\\limits_0^L\\frac{\\rmd x}{\\displaystyle D+\\frac{21\\,\\psi^2(x)}{2h^2D}}\\,.\n\\]\n\\noindent\nFor a lengthy domain the specific realization of $\\xi(x)$ becomes\ninsignificant;\n\\[\n\\delta C=-J\\,L\\la\\left(D+\\frac{21\\,\\psi^2(x)}{2h^2D}\\right)^{-1}\\ra\n\\equiv-\\sigma^{-1}J\\,L\\,,\n\\]\n\\noindent\nHence,\n\\[\nJ=-\\sigma\\frac{\\delta C}{L},\n\\]\n\\noindent\n{\\it i.e.}\\ $\\sigma$ can be considered as an effective\ndiffusivity.\n\n\n\\begin{figure}[!t]\n\\center{\n\\includegraphics[width=0.89\\textwidth]%\n {gold-shkl-08jstat-fig2.eps}\n}\n \\caption{\nDependencies of effective diffusivity $\\sigma$ on mean\nsupercriticality $q_0$ for molecular diffusivity $D$ indicated in\nthe plot. The bold black line in the inner plot represents the\nanalytical dependence (see \\sref{sec3}).}\n \\label{fig2}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\center{\n\\includegraphics[width=0.67\\textwidth]%\n {gold-shkl-08jstat-fig3.eps}\n}\n \\caption{\nDependencies of the effective diffusivity on the molecular one for\nnonnegative $\\wq_0$}\n \\label{fig3}\n\\end{figure}\n\n\nThe effective diffusivity\n\\begin{equation}\n\\sigma=\\la\\left(D+\\frac{21\\,\\psi^2(x)}{2h^2D}\\right)^{-1}\\ra^{-1}\n\\label{eq2-01}\n\\end{equation}\n\\noindent\nturns into $D$ for vanishing convective flow. For small $D$ the\nregions of the layer, where the flow is damped, $\\psi\\ll1$, make\nlarge contribution to the mean value appearing in~\\eref{eq2-01}\nand diminish $\\sigma$, thus, leading to the locking of the\nspreading of the pollutant.\n\nNote, disorder strength $\\eps^2$ can be excluded from equations by\nthe appropriate rescaling of parameters and fields. Thus, the\nresults on the effective diffusivity can be comprehensively\npresented in the terms of $\\wD$, $\\ws$, and $\\wq_0$:\n\\[\n\\wD=\\sqrt{\\frac{2}{21}}\\,\\eps^{4\/3}hD,\\qquad\n\\ws=\\sqrt{\\frac{2}{21}}\\,\\eps^{4\/3}h\\sigma,\\qquad\n\\wq_0=\\frac{q_0}{\\eps^{4\/3}}\\,.\n\\]\n\\noindent\n\\Fref{fig2} provides calculated dependencies of effective\ndiffusivity $\\ws$ on $\\wq_0$ for moderate\nand small values of molecular diffusivity $\\wD$. Noteworthy,\n\\\\\n(i)\\;for small $\\wD$ a quite sharp transition of effective\ndiffusivity $\\ws$ between moderate values and ones comparable with\n$\\wD$ occurs near $q_0=0$, suggesting the transition from an\nalmost everywhere intense ``global'' flow to a set of localized\ncurrents to take place;\n\\\\\n(ii)\\;below the instability threshold of the disorderless system,\nwhere only sparse localized currents are excited, the effective\ndiffusion can be dramatically enhanced by these currents; {\\it\ne.g.}, for $\\wD=10^{-4}$, $\\wq_0=-1$, the effective diffusivity is\nincreased by one order of magnitude compared to the molecular\ndiffusivity.\n\n\\Fref{fig3}\nshows dependencies of the effective diffusivity on the molecular\none for nonnegative $q_0$. Remarkably, for $\\wq_0\\gtrsim0.3$, the\ndependencies possess a minimum which is in agreement with known\ngeneral results on interference between turbulent and molecular\ndiffusion~\\cite{Saffman-1960-Mazzino-Vergassola-1997}. The reason\nis the fact, that for the convective transport the molecular\ndiffusion plays a destructive role. Hence, for high $q_0$, where\nconvective flows are intense, the enhancement of the convective\ntransport prevails over the weakening of the diffusional one as\nthe molecular diffusivity tends to zero; on the contrary, for low\n$q_0$, where convective flows are weak, the decrease of the\nmolecular diffusivity leads to the weakening of the transport.\n\n\n\\section{Analytical theory}\\label{sec3}\nLet us now analytically evaluate the effective diffusivity for a\nsmall molecular one, $\\wD\\ll1$, and sparse domains of excitation\nof convective flow, $\\nu\\ll1$. We have to calculate the average\n\\[\n\\beta\\equiv\\la\\big(\\wD+\\psi^2(x)\/\\wD\\big)^{-1}\\ra\n\\]\n\\noindent\n[cf.\\ \\eref{eq2-01}] which, due to ergodicity, can be evaluated\nnot only as an average over $x$ for a given realization of\n$\\xi(x)$, but also as an average over realizations of $\\xi(x)$ at\na certain point $x_0$. Let us set the origin of the $x$-axis at\n$x_0$. Hence,\n$\\beta=\\langle\\big(\\wD+\\psi^2(0)\/\\wD\\big)^{-1}\\rangle_\\xi$.\n\n\\begin{figure}[!t]\n\\center{\n\\includegraphics[width=0.68\\textwidth]%\n {gold-shkl-08jstat-fig4.eps}\n}\n \\caption{\nSketch of two localized flow patterns being nearest to the origin}\n \\label{fig4}\n\\end{figure}\n\nWhen the two nearest to the origin excitation domains are distant\nand localized near $x_1>0$ and $x_2<0$ (see~\\fref{fig4}),\n\\begin{equation}\n\\psi(0)\\approx\\psi_1\\e^{-\\gamma x_1}+\\psi_2\\e^{-\\gamma|x_2|},\n\\label{eq3-01}\n\\end{equation}\n\\noindent\nwhere $\\psi_{1,2}$ characterize the amplitude of flows excited\naround $x_{1,2}$. For small $\\wD$ and density $\\nu$, the\ncontribution of the excitation domains to $\\beta$ is negligible\nagainst the one of the regions of a weak flow. Therefore, we do\nnot have to be very accurate with the former and may utilize\nexpression~\\eref{eq3-01} even for small $x_{1,2}$,\n\\begin{eqnarray}\n\\beta=\\la\\frac{1}{\\wD+\\psi^2(0)\/\\wD}\\ra_\\xi\\nonumber\\\\\n\\qquad\n =\\la\\int\\limits_0^{\\infty}\\rmd x_1\\int\\limits_0^{\\infty}\\rmd x_2\n \\frac{p(x_1)\\,p(x_2)}{\\wD+\\wD^{-1}(\\psi_1\\e^{-\\gamma x_1}+\\psi_2\\e^{-\\gamma x_2})^2}\n \\ra_{\\psi_1,\\psi_2}\n\\label{eq3-02}\n\\end{eqnarray}\n\\noindent\nwhere $p(x_1)$ [\\,$p(x_2)$] is the density of the probability to\nobserve the nearest right [left] excitation domain at $+x_1$\n[$-x_2$]. For probability distribution $P(x_1\\!>\\!x)$, one finds\n $P(x_1\\!>\\!x+\\rmd x)=P(x_1\\!>\\!x)\\,(1-\\nu\\rmd x)$,\n{\\it i.e.}, $\\frac{\\rmd}{\\rmd x}P(x_1\\!>\\!x)=-\\nu P(x_1\\!>\\!x)$.\nHence, $P(x_1\\!>\\!x)=\\e^{-\\nu x}$, and probability density\n$p(x)=|\\frac{\\rmd}{\\rmd x}P(x_1\\!>\\!x)|=\\nu\\e^{-\\nu x}$. With\nregard to averaging over $\\psi_{1,2}$, it is important that the\nmultiplication of $\\psi_{1,2}$ by factor $F$ is effectively\nequivalent to the shift of the excitation domain by\n$\\gamma^{-1}\\ln{F}$ which is insignificant for $F\\sim1$ in the\nlimit case we consider. Hence, one can assume $\\psi_{1,2}=\\pm1$\n(the topological difference between $\\psi_1\\psi_2<0$ and\n$\\psi_1\\psi_2>0$ is not to be neglected) and rewrite~\\eref{eq3-02}\nas\n\\begin{eqnarray}\n \\beta=\\frac{1}{2}\\int\\limits_0^{\\infty}\\rmd x_1\\int\\limits_0^{\\infty}\\rmd x_2\\,\n \\nu^2\\e^{-\\nu(x_1+x_2)}\n \\left[\\frac{1}{\\wD+\\wD^{-1}(\\e^{-\\gamma x_1}+\\e^{-\\gamma x_2})^2}\n \\right.\\nonumber\\\\\n \\hspace{70mm}\\left.\n {}+\\frac{1}{\\wD+\\wD^{-1}(\\e^{-\\gamma x_1}-\\e^{-\\gamma x_2})^2}\\right].\n \\nonumber\n\\end{eqnarray}\n\\noindent\nThese integrals can be evaluated for $\\nu\/\\gamma\\ll1$, and one\nfinds\n\\begin{equation}\n\\ws=\\frac{1}{\\beta}\\approx\\wD\\left(\\frac{2}{\\wD}\\right)^\\frac{2\\nu}{\\gamma}\\!\\!.\n\\label{eq3-03}\n\\end{equation}\n\\noindent\nFor $\\wq_0\\!<\\!-1$, one can use the asymptotic expressions for\n$\\nu$ [equation~\\eref{eq1-03}] and\n\\[\n\\gamma=\\eps^{-\\frac{2}{3}}\\left(|\\wq_0|^\\frac{1}{2}-\\frac{1}{4}|\\wq_0|^{-1}-\\frac{5}{32}|\\wq_0|^{-\\frac{5}{2}}+\\dots\\right).\n\\]\n\\noindent\nThe latter expression is known from the classical theory of AL\n(cf.\\,\\cite{Lifshitz-Gredeskul-Pastur-1988,Gredeskul-Kivshar-1992}).\n\nIn~\\fref{fig2}, one can see analytic expression~\\eref{eq3-03} to\nmatch the numerically evaluated $\\ws$ for $\\wD=10^{-4}$,\n$\\wq_0\\!<\\!-1$ quite well. With~\\eref{eq3-03}, one can evaluate\nthe convectional enhancement of the effective diffusivity below\nthe excitation threshold of the disorderless system, and it is\ngiven by factor $\\ws\/\\wD=(2\/\\wD)^{2\\nu\/\\gamma}$ which can be large\nfor small $\\wD$.\n\n\n\\section{Conclusion}\\label{concl}\nSummarizing, we have studied the transport of a pollutant in a\nhorizontal fluid layer by spatially localized 2D thermoconvective\ncurrents appearing under frozen parametric disorder. Though the\nspecific physical system we have considered is a horizontal porous\nlayer saturated with a fluid and confined between two nearly\nthermally insulating plates, our results can be trivially extended\nto a broad variety of fluid dynamical systems (like ones studied\nin~\\cite{Knobloch-1990,Shtilman-Sivashinsky-1991,Aristov-Frick-1989,Schoepf-Zimmermann-1989-1993}).\nWe have calculated numerically the dependence of the effective\ndiffusivity on the molecular one and the mean supercriticality\n(see figures~\\ref{fig2},\\,\\ref{fig3}). In particular, for a nearly\nindiffusive pollutant ($\\wD\\ll1$), first, we have observed the\ntransition from a set of localized flow patterns to an almost\neverywhere intense ``global'' flow, which results in a soar of the\neffective diffusivity from values comparable with the molecular\ndiffusivity up to moderate ones. Second, we have found convective\ncurrents to considerably enhance the effective diffusivity even\nbelow this transition. For the latter effect the analytical\ntheory, which perfectly describes the limit of $\\wD\\ll1$,\n$\\nu\\ll1$, has been developed [equation~\\eref{eq3-03}].\n\n\\ack{\nThe authors thank M.\\,Zaks and S.\\,Shklyaev for useful discussions\nand O.\\,Khlybov and A.\\,Alabuzhev for technical support with\ncomputational facilities.\\footnote{Calculation of statistical\nproperties of states of an extensive distributed stochastic\nsystem, like the one performed in this work, is extremely\nCPU-time-consuming.} DG acknowledges the Foundation ``Perm\nHydrodynamics,'' the BRHE--program (CRDF Grant no.\\,Y5--P--09--01\nand MESRF Grant no.\\,2.2.2.3.16038), and the VW--Stiftung for\nfinancial support.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjlif b/data_all_eng_slimpj/shuffled/split2/finalzzjlif new file mode 100644 index 0000000000000000000000000000000000000000..6c82653b4020bddbbe835953c64a8df859021c40 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzjlif @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\t\tIn its original form, the quantum Zeno effect is defined for closed finite quantum systems. \\citeauthor{Misra.1977} predicted that \"an unstable particle which is continuously observed to see whether it decays will never be found to decay!\" \\cite[Abst.]{Misra.1977}. In a more general setup, frequent measurements enable a change in the time evolution and convergence to the so called \\textit{Zeno dynamics}. Experimentally, the Zeno effect is verified for instance in \\cite{Itano.1990, Fischer.2001}. In addition to its theoretical value, the quantum Zeno effect is used in error correction schemes to suppress decoherence in open quantum systems \\cites{Hosten.2006}{Franson.2004}{Beige.2000}{Barankai.2018}{Luchnikov.2017}. The idea is to frequently measure the quantum state and thereby force the evolution to remain within the code space. With an appropriate measurement, one can even decouple the system from its environment \\cites{Facchi.2004}{Burgarth.2020} and show that appropriately encoded states can be protected from decoherence with arbitrary accuracy \\cites{Dominy.2013}{Erez.2004}. Moreover, the quantum Zeno effect has been used in commercial atomic magnetometers \\cite{Kominis.2009}.\n\t\t\n\t\tIntroduced by \\citeauthor{Beskow.1967} in 1967 and later named by \\citeauthor{Misra.1977} after the greek philosopher Zeno of Elea, the quantum Zeno effect in its simplest form can be stated as follows: given a projective measurement $P$ and a unitary time-evolution generated by a Hamiltonian $H$ acting on a finite dimensional Hilbert space $\\cH$ \\cite{Misra.1977}: For $n\\to\\infty$\n\t\t\\begin{equation}\\label{eq:zeno-misra-sudarshan}\n \t\t(Pe^{\\frac{it}{n}H})^n\\longrightarrow e^{it\\,PHP}\n\t\t\\end{equation}\n\t Since the seminal works \\cites{Beskow.1967}{Misra.1977}, the result was extended in many different directions (overviews can be found in \\cites{Facchi.2008}{Schmidt.2004}{Itano.1990}). Recently, the convergence in \\Cref{eq:zeno-misra-sudarshan} was proven in the strong topology for unbounded Hamiltonian under the weak assumption that $PHP$ is the generator of a $C_0$-semigroup \\cite{Exner.2021}. Earlier approaches used the so called \\textit{asymptotic Zeno condition} \\cites{Schmidt.2004}{Exner.1989}{Misra.1977}, which assumes $(\\1-P)e^{itH}P$ and $Pe^{itH}(\\1-P)$ to be Lipschitz continuous at $t=0$. This condition is natural in the sense that it is related to the boundedness of the first moment of the Hamiltonian in the initial state and is efficiently verifiable in practice. With the works \\cites{Burgarth.2020}{Mobus.2019}{Barankai.2018}, the quantum Zeno effect was generalized to open and infinite dimensional quantum systems equipped with general quantum operations and uniformly continuous time evolutions. Note that in open quantum systems, we are dealing with operators acting on the Banach space $\\mathcal{T}_1(\\cH)$ of trace-class operators. More recently, \\citeauthor{Becker.2021} generalized\n\t\tthe Zeno effect further and interpreted the Zeno sequence as a product formula consisting of a contraction $M$ (quantum operation) and a $C_0$-contraction semigroup (quantum time evolution) on an abstract Banach space. Under a condition of \\textit{uniform power convergence} of the power sequence $\\{M^k\\}_{k\\in\\mathbb{N}}$ towards a projection $P$ and boundedness of $M\\cL $ and $\\cL M$, they proved a quantitative bound on the convergence rate \\cite{Becker.2021}:\n\t\t\\begin{align}\\label{eq:Becker}\n\t\t\t\\|(Me^{\\frac{t}{n}\\cL})^nx-e^{tP\\cL P}Px\\|=\\cO(n^{-\\frac{1}{3}}(\\|x\\|+\\|\\cL x\\|))\\,,\n\t\t\\end{align} \n\t\tfor $n\\rightarrow\\infty$ and all $x\\in\\cD(\\cL)$. However, the optimality of \\eqref{eq:Becker} was left open.\\footnote{Note that we found an inconsistency in the proof of \\cite[Lem.~2.1]{Zagrebnov.2017} (see \\cite{Zagrebnov.2022}), which slightly reduces the convergence rate found in \\Cref{eq:Becker} (more details are given in \\Cref{sec:alternative-chernoff-lemma-trotter-product-formula}).}\n\t\t\n\t\t\\subsubsection*{Main contributions:}\n\t\t\tIn this paper, we achieve the optimal convergence rate $\\mathcal{O}(n^{-1})$ of the Zeno sequence consistent with the finite-dimensional case \\cite{Burgarth.2020} by providing an explicit bound which recently attracted interest in finite closed quantum systems \\cite[Thm.~1]{Burgarth.2021}. Moreover, we generalize the results of \\cite{Becker.2021} in two complementary directions:\n\t\t\t\n\t\t\tIn \\Cref{thm:spectral-gap} below, we assume a special case of the \\textit{uniform power convergence} assumption on $M$, that is $\\|M^n-P\\|\\leq\\delta^n$ for some $\\delta\\in(0,1)$, and weaken the assumption on the semigroup to the \\textit{uniform asymptotic Zeno condition} inherited from the unitary setting of \\cite{Schmidt.2004}: for $t\\rightarrow0$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{(\\1-P) e^{t\\cL}P}_\\infty=\\cO(t)\\quad\\text{and}\\quad \\norm{Pe^{t\\cL}(\\1-P)}_\\infty=\\cO(t).\n\t\t\t\\end{equation*}\n\t\t\tTherefore, we prove the convergence of a non-trivial Zeno sequence in open quantum systems to a Zeno dynamics described by a possibly unbounded generator.\\\\\n\t\t\tSecond, \\Cref{thm:spectral-gap-uniform} is stated under slightly weaker assumptions as Theorem 3 in \\cite{Becker.2021} and improves the result to the optimal convergence rate and to the uniform topology.\n\t\t\n\t\t\tIn order to achieve these results, we prove a modified Chernoff $\\sqrt{n}$-Lemma in \\Cref{lem:improved-chernoff}, find a quantitative convergence rate for $\\operatorname{exp}\\big({nP(e^{\\frac{1}{n}t\\cL}-\\1)P}\\big)P-\\operatorname{exp}(tP\\cL P)P$ as $n \\rightarrow\\infty$, where $P\\cL P$ is possibly unbounded, and prove the upper semicontinuity of parts of the spectrum of $Me^{t\\cL}$ under tight assumptions.\n\t\t\n\t\t\\subsubsection*{Organization of the paper:}\n\t\t\tIn \\Cref{sec:prelim}, we provide a short recap on bounded and unbounded operator theory. We expose our main results in \\Cref{sec:assumptions-results}. \\Cref{sec:alternative-chernoff-lemma-trotter-product-formula} deals with the modified Chernoff $\\sqrt{n}$-Lemma and some of its implications as regards to Trotter-Kato's product formula. Then, we prove our main theorems under the weakest possible assumptions on the $C_0$-semigroup in \\Cref{sec:unbounded-generator-zeno-subspace}, and under the weakest possible assumptions on $M$ in \\Cref{sec:finitely-many-eigenvalues}. Our results are illustrated by three large classes of examples in finite and infinite dimensional quantum systems in \\Cref{sec:applications}. Finally, we discuss some remaining open questions in \\Cref{sec:discussion}.\n\t\t\n\t\t\n\t\\section{Preliminaries}\\label{sec:prelim}\n\t\tLet $(\\cX,\\|\\cdot\\|)$ be a Banach space and $(\\cB(\\cX),\\|\\cdot\\|_{\\infty})$ be the associated space of bounded linear operators over $\\C$ equipped with the operator norm, i.e.~$\\|C\\|_{\\infty}\\coloneqq\\sup_{x\\in\\cX\\backslash\\{0\\}}\\frac{\\|Cx\\|}{\\|x\\|}$, and the identity $\\1\\in\\cB(\\cX)$. By a slight abuse of notation, we extend all densely defined and bounded operators by the \\textit{bounded linear extension theorem} to bounded operators on $\\cX$ \\cite[Thm.~2.7-11]{Kreyszig.1989}. \n\t\tA sequence $(C_k)_{k\\in\\N}\\subset\\cB(\\cX)$ converges uniformly to $C\\in\\cB(\\cX)$ if $\\lim_{k\\rightarrow\\infty}\\|C_k-C\\|_\\infty=0$ and strongly if $\\lim_{k\\rightarrow\\infty}\\|C_kx-Cx\\|=0$ for all $x\\in\\cX$. The integral over bounded vector-valued maps, e.g.~$[a,b]\\rightarrow\\cX$ or $[a,b]\\rightarrow \\cB(\\cX)$ with $a0$ depending on $t$ and $b$, but independent of $n$ .\n\t\t\\end{manualthm}\n\t\t\\begin{rmk*}\n\t\t\tNote that the assumption on $M$ is a special case of the so-called \\textit{uniform power convergence} assumption (q.v.~\\Caref{unifpower-prev}) and the assumption (\\ref{eq:thm1-asympzeno-prev}) on the $C_0$-semigroup is a generalization of the uniform \\textit{asymptotic Zeno condition} which implies the convergence in the case of a unitary evolution frequently measured by a projective measurement \\cite[Sec.~3.1]{Schmidt.2004}. Note that in that specific case, \\cite{Exner.2021} recently managed to remove the asymptotic Zeno condition. Moreover, the assumption that $(P\\cL P,\\cD(\\cL P))$ is a generator can be relaxed to the assumption that $P\\cL P$ is closeable and its closure defines a generator (q.v.~remark after \\Caref{lem:proofthm1-term3}). The famous \\textit{Generation Theorem} by Hille and Yosida provides a sufficient condition under which $\\overline{P\\cL P}$ is a generator \\cite[Thm.~3.5-3.8]{Engel.2000}.\n\t\t\\end{rmk*}\n\t The following example confirms the optimality of the achieved convergence rate.\n\t\t\\begin{ex}\n\t\t\tLet $\\left\\{\\ket{1},\\ket{2},\\ket{3}\\right\\}$ be an orthonormal basis of $\\R^3$ and $\\delta\\in(0,1)$. We define, \n\t\t\t\\begin{equation*}\n\t\t\t\t\\cL\\coloneqq \\ket{1}\\bra{2}\\quad\\text{and}\\quad M\\coloneqq\\ket{1}\\bra{1}+\\delta\\ket{3}\\bra{3}.\n\t\t\t\\end{equation*}\n\t\t\tThen, $P=\\ket{1}\\bra{1}$, $(\\1-P) M=\\delta\\ket{3}\\bra{3}$, $M\\cL=\\cL M$, and $\\cL^2=0=\\cL P$, $\\|M^n-P\\|_\\infty\\leq\\delta^n$. Using these properties, $Me^{\\frac{t}{n}\\cL}=M +\\frac{t}{n}\\cL$ and for $t\\in[0,\\infty)$\n\t\t\t\\begin{align*}\n\t\t\t \\left(Me^{\\frac{t}{n}\\cL}\\right)^n&=\\left(M+\\frac{t}{n}\\cL\\right)^n\\\\\n\t\t\t &=\\delta^n\\ket{3}\\bra{3}+\\ket{1}\\bra{1}+\\frac{t}{n}\\ket{1}\\bra{2}\\\\\n\t\t\t &=\\left((\\1-P)M\\right)^ne^{t\\cL}+Pe^{tP\\cL P}+\\frac{t}{n}P\\cL.\n\t\t\t\\end{align*}\n\t\t\tTherefore, \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^\\frac{t}{n}\\cL\\right)^n-Pe^{tP\\cL P}}_\\infty=\\max\\{\\tfrac{t}{n},\\delta^n\\},\n\t\t\t\\end{equation*}\n\t\t\twhich shows the optimality of our convergence rate in \\Cref{thm:spectral-gap}.\n\t\t\\end{ex}\n Beyond the proven asymptotics, we find explicit error bounds in \\Cref{lem:proofthm1-term1}, \\ref{lem:proofthm1-term2}, and \\ref{lem:proofthm1-term3}, which simplify if $\\cL$ is bounded to the following explicit convergence bound depending on the generator $\\cL$, the projection $P$, the spectral gap $\\delta$, and the time $t$:\n\t\t\\begin{prop}\\label{cor:explicit-bound-thm1}\n\t\t\tLet $\\cL\\in\\cB(\\cX)$ be the generator of a contractive uniformly continuous semigroup and $M\\in\\cB(\\cX)$ a contraction satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\delta^n\n\t\t\t\\end{equation*}\n\t\t\tfor a projection $P\\in\\cB(\\cX)$, $\\delta\\in(0,1)$, and all $n\\in\\N$. Then, for all $t\\geq0$ and $n\\in\\N$,\n\t\t\t\\begin{align*}\n\t\t \\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq c_{p}\\frac{t\\|\\cL\\|_\\infty}{n} &+\\left(\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\right)\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t\t &+\\delta^n+\\frac{2\\delta}{1-\\delta}\\frac{e^{3t\\|\\cL\\|_\\infty c_{p}}}{n}\n\t \t\\end{align*}\n\t\t\twhere $c_{p}\\coloneqq\\|\\1-P\\|_\\infty$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$.\n\t\t\\end{prop}\n\t\tNote that the above proposition can be easily extended to the case of an unbounded generator with the assumption that $\\cL M$ and $M\\cL$ are densely defined and bounded. Another advantage of our setup is the freedom it provides for choosing the Banach space $\\cX$, which allows us to treat open quantum systems ($\\cX=\\cT(\\cH)$ the trace class operators over a Hilbert space) and closed quantum systems ($\\cX=\\cH$ a Hilbert space) on the same footing. In the case of finite dimensional closed quantum systems, \\Cref{cor:explicit-bound-thm1} reduces to the following bound, which was independently proven in \\cite[Thm.~1]{Burgarth.2021} (up to a change of the numerical constant in the quadratic term from $\\tfrac{5}{2}$ to $2$):\n\t\n\t\t\\begin{cor}\\label{cor:explicit-bound-thm1-closed-sys}\n\t\t\tLet $\\cH$ be a Hilbert space, $H\\in\\cB(\\cH)$ be a hermitian operator, and $P\\in\\cB(\\cH)$ a hermitian projection. Then,\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Pe^{-i\\frac{t}{n}H}\\right)^n-e^{-itPH P}P}_\\infty\\leq\\frac{1}{n}\\left(t\\norm{H}_\\infty+\\frac{5}{2}t^2\\norm{H}_\\infty^2\\right)\n\t\t\t\\end{equation*}\n\t\t\\end{cor}\n\t\tTo achieve the bound above, one inserts $\\delta=0$ and $\\cL=iH$ in \\Cref{cor:explicit-bound-thm1}. Note that $PHP$ is hermitian and $\\|e^{siPHP}\\|_\\infty=1$ for all $s\\geq0$. \n\t\t\n\t\tNext, we consider convergence rates under a slight weakening of the condition on the map $M$:\n\n\t\t\\begin{cor}\\label{cor:spectral-gap}\n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t\t\\end{equation*}\n\t\t\tfor some projection $P$, $\\delta\\in(0,1)$, $\\tilde{c}\\ge 0$ and all $n\\in\\N$. Moreover, assume there is $b\\geq0$ so that \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb.\n\t\t\t\\end{equation*}\n\t\t\tIf $(P\\cL P,\\cD(\\cL P))$ is the generator of a $C_0$-semigroup, then there exists a constant $c>0$ and $n_0\\in\\N$ so that $\\tilde{c}\\delta^{n_0}\\eqqcolon\\tilde{\\delta}<1$ and for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(M^{n_0}e^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+\\tilde{\\delta}^n\\norm{x}.\n\t\t\t\\end{equation*}\n\t\t\\end{cor}\n\t\tThe above \\namecref{cor:spectral-gap} follows by the choice $n_0$ such that $\\tilde{c}\\delta^{n_0}<1$ and applying \\Cref{thm:spectral-gap-prev} to $\\tilde{M}\\coloneqq M^{n_0}$. A more physically motivated result treating the same generalization as the \\namecref{cor:spectral-gap} above is provided in the next result:\n\t\t\n\t\t\\begin{manualprop}{II}[stated as~\\Caref{prop:spectral-gap-uniform-norm-power-convergence} in main text]\\label{prop:spectral-gap-uniform-norm-power-convergence-prev}\n Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction such that\n \\begin{equation}\\label{unifpowerc-prev}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t\t\\end{equation}\n\t\t\tfor some projection $P$, $\\delta\\in(0,1)$ and $\\tilde{c}\\ge 0$. Moreover, we assume that there is $b\\geq0$ so that \n\t\t\t\\begin{equation}\\label{key-prev}\n\t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb,\\quad\\norm{M^\\perp e^{t\\cL}-M^\\perp}_\\infty\\leq tb,\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb\n\t\t\t\\end{equation}\n\t\t\twhere $M^\\perp=(\\1-P)M$. If $(P\\cL P,\\cD(\\cL P))$ generates a $C_0$-semigroup, then there is an $\\epsilon>0$ such that for all $t\\geq0$, $n\\in\\N$ satisfying $t\\in[0,n\\epsilon]$, $\\tilde{\\delta}\\in(\\delta,1)$, and $x\\in\\cD((\\cL P)^2)$ \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c_1(t,b,\\tilde{\\delta}-\\delta)}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+c_2(\\tilde{c},\\tilde{\\delta}-\\delta)\\tilde{\\delta}^n\\norm{x}\\,,\n\t\t\t\\end{equation*}\n\t\t for some constants $c_1,c_2\\ge 0$ depending on $t$, $b$, the difference $\\tilde{\\delta}-\\delta$, and $\\tilde{c}$.\n\t\t\\end{manualprop}\n\t\t\n\t\tAs in \\Cref{cor:explicit-bound-thm1}, we also get a more explicit bound in the case of bounded generators in the following proposition:\n\t\t\\begin{prop}\\label{cor:explicit-bound-prop}\n\t\t\tLet $\\cL\\in\\cB(\\cX)$ be the generator of a contractive uniformly continuous semigroup and $M\\in\\cB(\\cX)$ a contraction satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq \\tilde{c}\\delta^n\n\t\t\t\\end{equation*}\n\t\t\tfor a projection $P\\in\\cB(\\cX)$, $\\delta\\in(0,1)$, $\\tilde{c}>1$, and all $n\\in\\N$. Then there is $\\epsilon>0$ such that for all $t\\geq0$, $n\\in\\N$ satisfying $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{align*}\n\t \\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq\\frac{tc_{p}\\|\\cL\\|_\\infty}{n}&+\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t &+\\frac{2\\tilde{c}}{\\tilde{\\delta}-\\delta}\\tilde{\\delta}^n+\\frac{2\\tilde{\\delta}}{1-\\tilde{\\delta}}\\frac{e^{\\frac{6tc_{p}\\tilde{c}\\|\\cL\\|_\\infty}{\\tilde{\\delta}-\\delta}}}{n}\n\t \\end{align*}\n\t where $c_{p}\\coloneqq\\|\\1-P\\|_\\infty$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$.\n\t\t\\end{prop}\n\t\tFinally, we extend the assumption on $M$ (q.v.~\\Caref{unifpower1-prev}, (\\ref{unifpowerc-prev})) to the \\textit{uniform power convergence} introduced in \\cite{Becker.2021}. Let $\\{P_j,\\lambda_j\\}_{j=1}^J$ be a set of projections satisfying $P_jP_k=1_{j=k} P_j$ and associated eigenvalues on the unit circle $\\partial\\D_1$. Then, $M$ is called uniformly power convergent with rate $\\delta\\in(0,1)$ if $M^n-\\sum_{j=1}^{J}\\lambda^n_jP_j=\\cO(\\delta^n)$ uniformly for $n\\rightarrow\\infty$. To prove our result in this case, we also need to assume that $M\\cL$ and $\\cL P_\\Sigma$ with $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$ are densely defined and bounded (cf.~\\cite{Becker.2021}) and $P_j$ is a contraction for all $j\\in\\{1,...,J,\\Sigma\\}$:\n\t\t\n \\begin{manualthm}{III}[stated as~\\Caref{thm:spectral-gap-uniform} in main text]\\label{thm:spectral-gap-uniform-prev}\n\t\t Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the following uniform power convergence: there is $\\tilde{c}>0$ so that\n\t\t\t\\begin{equation}\\label{unifpower-prev}\n\t\t\t \\norm{M^n-\\sum_{j=1}^{J}\\lambda^n_jP_j}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t \\end{equation}\n\t\t for a set of projections $\\{P_j\\}_{j=1}^J$ satisfying $P_jP_k=1_{j=k}P_j$, eigenvalues $\\{\\lambda_j\\}_{j=1}^J\\subset\\partial\\D_1$, and a rate $\\delta\\in(0,1)$. For $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$, we assume that $M\\cL$ and $\\cL P_\\Sigma$ are densely defined and bounded by $b\\geq0$ and $\\|P_j\\|_\\infty=1$ for all $j\\in\\{1,...,J,\\Sigma\\}$. Then, there is an $\\epsilon>0$ such that for all $n\\in\\N$, $t\\geq0$ satisfying $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\leq\\frac{c_1 }{n}+c_2\\tilde{\\delta}^n\\,,\n\t\t\t\\end{equation*}\n\t\t\tfor some constants $c_1,c_2\\ge 0$ depending on all involved parameters except from $n$.\n \\end{manualthm}\n \n\t\tIn comparison to Theorem 3 in \\cite{Becker.2021}, \\Cref{thm:spectral-gap-uniform} achieves the optimal convergence rate and is formulated in the uniform topology under slightly weaker assumptions on the generator.\n\t\t\\begin{rmk*}\n\t\t\tA natural way to weaken the above assumption is to assume that the power converges is in the strong topology (cf.~\\cite[Thm.~2]{Becker.2021}).\n\t\t\\end{rmk*}\n\t\t\n\t\t\n\t\\section{Chernoff \\texorpdfstring{$\\sqrt{n}$-}{}Lemma and Trotter-Kato's Product Formula}\\label{sec:alternative-chernoff-lemma-trotter-product-formula}\n\t\tIn previous works \\cite{Mobus.2019,Becker.2021}, Chernoff's $\\sqrt{n}$-Lemma \\cite[Lem.~2]{Chernoff.1968}, which we restate here, is used as a proof technique to approximate the Zeno product by a semigroup (q.v.~\\Caref{eq:proofthm1-term2}).\n\t\t\\begin{lem}[Chernoff \\texorpdfstring{$\\sqrt{n}$-}{square root }Lemma]\\label{lem:chernoff}\n\t\t\tLet $C\\in\\cB(\\cX)$ be a contraction. Then, $(e^{t(C-\\1)})_{t\\geq0}$ is a uniformly continuous contraction semigroup and for all $x\\in\\cX$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{C^nx-e^{n(C-\\1)}x}\\leq\\sqrt{n}\\norm{(C-\\1)x}.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\t\t\\begin{rmk*}\n\t\t\tIn Lemma 2.1 in \\cite{Zagrebnov.2017}, the dependence on $n$ is improved to $n^\\frac{1}{3}$. This is crucial in the proof of the convergence rate in \\cite[Lem.~5.4-5.5]{Becker.2021}. Unfortunately, we found an inconsistency in the proof of \\cite[Lem.~2.1]{Zagrebnov.2017}, i.e.~Inequality 2.3 is not justified. An update and more Chernoff bounds can be found in \\cite{Zagrebnov.2022}. Following the proof by \\citeauthor{Becker.2021}, one can achieve a convergence rate of order $\\tfrac{1}{\\sqrt{n}}$ in the bounded generator case \\cites[Thm.~1]{Becker.2021} and of order $\\tfrac{1}{\\sqrt[4]{n}}$ in the unbounded generator case \\cites[Thm.~3]{Becker.2021}.\n\t\t\\end{rmk*}\n\t\tIn the case of the quantum Zeno effect (see \\Caref{lem:proofthm1-term2} and \\ref{lem:proofthm2-term2}) for bounded generators, the contraction $C$ is a vector-valued map $t\\mapsto C(t)$ on $\\cX$ satisfying $\\|C(\\tfrac{1}{n})-\\1\\|_\\infty=\\cO(n^{-1})$. By Chernoff's $\\sqrt{n}$-Lemma \n\t\t\\begin{equation*}\n\t\t\t\\norm{C^n(\\tfrac{1}{n})-e^{n(C(\\tfrac{1}{n})-\\1)}}_\\infty\\leq\\frac{1}{\\sqrt{n}}.\n\t\t\\end{equation*}\n\t\tHere, we chose the bounded generator case for sake of simplicity. Nevertheless, the argument can be extended to unbounded generator as well (see \\Caref{lem:proofthm1-term2}, \\ref{lem:proofthm2-term2}, and \\ref{lem:approx-improved-chernoff}). Next, we prove a modified bound, which allows us to achieve the optimal rate in the quantum Zeno effect.\n\t\t\\begin{lem}[Modified Chernoff Lemma]\\label{lem:improved-chernoff}\n\t\t\tLet $C\\in\\cB(\\cX)$ be a contraction and $n\\in\\N$. Then, $(e^{t(C-\\1)})_{t\\geq0}$ is a contraction semigroup and for all $x\\in\\mathcal{X}$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(C^n-e^{n(C-\\1)}\\right)x}\\leq\\frac{n}{2}\\norm{(C-\\1)^2x}.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\t\t\\begin{rmk*}\n\t\t\tAt first glance, this seems to be worse than the original Chernoff $\\sqrt{n}$-Lemmas in \\cites{Chernoff.1968}. However, if $C$ is a vector-valued map satisfying $\\|C(\\tfrac{1}{n})-\\1\\|_\\infty=\\cO(n^{-1})$, then the modified Chernoff lemma gives\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{C(\\tfrac{1}{n})^n-e^{n(C(\\text{\\tiny{$\\tfrac{1}{n}$}})-\\1)}}_\\infty\\leq\\frac{n}{2}\\norm{C(\\tfrac{1}{n})-\\1}_{\\infty}^2=\\cO(n^{-1})\n\t\t\t\\end{equation*}\n\t\t\twhich is the key idea to prove the optimal convergence rate of the quantum Zeno effect for bounded generators and contractions $M$ satisfying the uniform power convergence (q.v.~\\Caref{lem:proofthm1-term2}).\n\t\t\\end{rmk*}\n\t\t\\begin{proof}[Proof of \\Cref{lem:improved-chernoff}]\n\t\t\tSimilar to Chernoff's proof \\cite[Lem.~2]{Chernoff.1968}, $(e^{t(C-\\1)})_{t\\geq0}$ is a contraction semigroup. We define $C_t\\coloneqq (1-t)\\1+tC=\\1+t(C-\\1)$ for $t\\in[0,1]$, which itself is a contraction as a convex combination of contractions, and we use the fundamental theorem of calculus so that\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{\\left(C^n-e^{n(C-\\1)}\\right)x}&\\leq\\int_{0}^{1}\\norm{\\frac{\\partial}{\\partial t}(C_t^ne^{(1-t)n(C-\\1)})x}dt\\\\\n\t\t\t\t&\\leq n\\int_{0}^{1}\\norm{C_t^{n-1}e^{(1-t)n(C-\\1)}}_\\infty\\norm{(\\1-C)(\\1-C_t)x}dt\\\\\n\t\t\t\t&\\leq\\frac{n}{2}\\norm{(C-\\1)^2x},\n\t\t\t\\end{align*}\n\t\t\twhich proves the \\namecref{lem:improved-chernoff}.\n\t\t\\end{proof}\n\t\tIn \\cite[p.~241]{Chernoff.1968}, Chernoff proves the convergence of Trotter's product formula by approximating the product using the Chernoff $\\sqrt{n}$-Lemma.\n\t\tFor bounded generators, Chernoff's proof gives a convergence rate of order $n^{-\\frac{1}{2}}$. Following his proof and using our modified Chernoff Lemma, we achieve the well-known optimal convergence rate of order $n^{-1}$ \\cites[Thm.~2.11]{Hall.2015}[p.~1-2]{Neidhardt.2018}:\t\t\n\t\t\\begin{prop}[\\texorpdfstring{\\cite[Thm.~1]{Chernoff.1968}}{[6, Thm.~1]}]\\label{prop:trotter-kato-product-formula}\n\t\t\tLet $F:\\R_{\\geq0}\\rightarrow\\cB(\\cX)$ be a continuously differentiable function (in the uniform topology) satisfying $\\sup_{t\\in\\R_{\\geq0}}\\|F(t)\\|_\\infty\\leq1$. Assume that $F(0)=\\1$ and denote the derivative at $t=0$ by $\\cL\\in\\cB(\\cX)$. Then, for all $t\\geq0$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{F\\left(\\tfrac{t}{n}\\right)^n-e^{t\\cL}}_\\infty\\leq \\norm{n\\left(F\\left(\\tfrac{t}{n}\\right)-\\1\\right)-t\\cL}_\\infty + \\frac{n}{2}\\norm{(F\\left(\\tfrac{t}{n}\\right)-\\1)}_\\infty^2.\n\t\t\t\\end{equation*}\n\t\t\\end{prop}\n\t\t\\begin{proof}\n\t\t\tThe case $t=0$ is clear. For $t>0$, applying \\Cref{lem:improved-chernoff}, we get\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{F\\left(\\tfrac{t}{n}\\right)^n-e^{t\\cL}}_\\infty&\\leq\\norm{F\\left(\\tfrac{t}{n}\\right)^n-e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}}_\\infty+\\norm{e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}-e^{t\\cL}}_\\infty\\\\\n\t\t\t\t&\\leq \\frac{n}{2}\\norm{F\\left(\\tfrac{t}{n}\\right)-\\1}_\\infty^2+\\norm{e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}-e^{t\\cL}}_\\infty.\n\t\t\t\\end{align*}\n\t\t\tFor the second term above, we apply \\Cref{lem:integral-equation-semigroups}:\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}-e^{t\\cL}}_\\infty&=\\int_{0}^{1}\\norm{e^{sn\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}\\left(n\\left(F\\left(\\tfrac{t}{n}\\right)-\\1\\right)-t\\cL\\right)e^{(1-s)t\\cL}}_\\infty ds\\\\\n\t\t\t\t&\\leq\\norm{n\\left(F\\left(\\tfrac{t}{n}\\right)-\\1\\right)-t\\cL }_\\infty,\n\t\t\t\\end{align*}\n\t\t\twhere we use $\\|F(\\tfrac{t}{n})\\|_\\infty\\leq1$ and that $e^{sn(F(\\text{\\tiny{$\\tfrac{t}{n}$}})-\\1)}$ is a contraction semigroup.\n\t\t\\end{proof}\n\t\tApplying the proposition to the case of Trotter's product formula, we achieve the well-known optimal convergence rate for bounded generators on Banach spaces \\cites[Thm.~2.11]{Hall.2015}[p.~1-2]{Neidhardt.2018}:\n\t\t\\begin{cor}[\\texorpdfstring{\\cites[Thm.~2.11]{Hall.2015}}{[20, Thm.~2.11]}]\\label{cor:trotter-kate-product-formula-convergence}\n\t\t\tLet $\\cL_1$ and $\\cL_2$ be bounded generators of two uniformly continuous contraction semigroups. Then, for $n\\rightarrow\\infty$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(e^{\\frac{1}{n}\\cL_1}e^{\\frac{1}{n}\\cL_2}\\right)^n-e^{\\cL_1+\\cL_2}}_\\infty=\\cO\\left(\\frac{1}{n}\\right).\n\t\t\t\\end{equation*}\n\t\t\\end{cor}\n\t\t\\begin{proof}\n\t\t\tWe define $F(\\tfrac{1}{n})\\coloneqq e^{\\frac{1}{n}\\cL_1}e^{\\frac{1}{n}\\cL_2}$ for which \n\t\t\t\\begin{align*}\n\t\t\t\tF(\\tfrac{1}{n})-\\1&=e^{\\frac{1}{n}\\cL_1}\\left(e^{\\frac{1}{n}\\cL_2}-\\1\\right)+e^{\\frac{1}{n}\\cL_1}-\\1\\\\\n\t\t\t\t&=\\left(e^{\\frac{1}{n}\\cL_1}-\\1\\right)\\left(e^{\\frac{1}{n}\\cL_2}-\\1\\right)+e^{\\frac{1}{n}\\cL_2}-\\1+e^{\\frac{1}{n}\\cL_1}-\\1\n\t\t\t\\end{align*}\n\t\t\tholds. Moreover,\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{e^{\\frac{1}{n}\\cL_1}-\\1}_\\infty=\\frac{1}{n}\\norm{\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL_1}\\cL_1d\\tau_1}_\\infty\\leq\\frac{1}{n}\\norm{\\cL_1}_\\infty\n\t\t\t\\end{equation*}\n\t\t\tand\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{n\\left(e^{\\frac{1}{n}\\cL_1}-\\1\\right)-\\cL_1}_\\infty=\\frac{1}{n}\\norm{\\int_{0}^{1}\\int_{0}^{1}\\tau_1e^{\\frac{\\tau_1\\tau_2}{n}\\cL_1}\\cL_1^2d\\tau_2d\\tau_1}_\\infty\\leq\\frac{1}{2n}\\norm{\\cL_1}^2_\\infty.\n\t\t\t\\end{equation*}\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{n\\left(F(\\tfrac{1}{n})-\\1\\right)-\\cL_1-\\cL_2}_\\infty\\leq\\frac{1}{n}\\left(\\norm{\\cL_1}_\\infty\\norm{\\cL_2}_\\infty+2\\norm{\\cL_1}^2_\\infty+2\\norm{\\cL_2}^2_\\infty\\right)\n\t\t\t\\end{align*}\n\t\t\tand the statement follows from \\Cref{prop:trotter-kato-product-formula}.\n\t\t\\end{proof}\n\t\n\t\n\t\\section{Strongly Continuous Zeno Dynamics}\\label{sec:unbounded-generator-zeno-subspace}\n\t\tWe proceed with the statement and proof of our first main result, namely \\Cref{thm:spectral-gap}, which we restate here for sake of clarity of conciseness:\n\t\t\n\t\t\\begin{thm}\\label{thm:spectral-gap}\n\t\t Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$, $M\\in\\cB(\\cX)$ a contraction, and $P$ a projection satisfying \n\t\t\t\\begin{equation}\\label{unifpower1}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\delta^n\n\t\t\t\\end{equation}\n\t for $\\delta\\in(0,1)$ and all $n\\in\\N$. Moreover, assume there is $b\\geq0$ so that for all $t\\ge 0$\n\t\t\t\\begin{equation}\\label{eq:thm1-asympzeno}\n\t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb.\n\t\t\t\\end{equation}\n\t\t\tIf $(P\\cL P,\\cD(\\cL P))$ is the generator of a $C_0$-semigroup, then for any $t\\ge 0$ and all $x\\in\\cD((\\cL P)^2)$ \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c(t,b)}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+\\delta^n\\norm{x}\n\t\t\t\\end{equation*}\n\t\tfor a constant $c(t,b)>0$ depending on $t$ and $b$, but independent of $n$ .\n\t\t\\end{thm}\n\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{thm:spectral-gap} }{Theorem 3.1}}\\label{subsec:proofthm1}~\\\\\n\t\t\tWe assume for sake of simplicity that $t=1$, and split our proof in three parts:\n\t\t\t\\begin{align}\n\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-e^{P\\cL P}Px}\\leq\\;&\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx}\\label{eq:proofthm1-term1}\\\\\t+\\;&\\norm{\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx-e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px}\\label{eq:proofthm1-term2}\\\\\n\t\t\t\t+\\;&\\norm{e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px}\\label{eq:proofthm1-term3}\n\t\t\t\\end{align}\n\t\t\tfor all $x\\in\\cD((\\cL P)^2)$.\n\t\t\t\n\t\t\t\\subsubsection{Upper bound on \\Cref{eq:proofthm1-term1}:}\n\t\t\t\tThe following \\namecref{lem:proofthm1-term1} uses similar proof strategies as Lemma 3 in \\cites{Burgarth.2020} and extends the result to infinite dimensions in the strong topology.\n\t\t\t\t\\begin{lem}\\label{lem:proofthm1-term1} \n\t\t\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the assumptions in \\Cref{thm:spectral-gap}. Then, for all $x\\in\\cX$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx}\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\\end{lem}\n\t\t\t\tThe proof of the above \\namecref{lem:proofthm1-term1} relies on a counting method: more precisely, we need to count the number of transitions in a binary sequence. This is related to the \\textit{urn problem}, where $k$ indistinguishable balls are placed in $l$ distinguishable urns \\cite[Chap.~1.9]{Stanley.1986}. Then, there are \n\t\t\t\t\\begin{equation}\\label{eq:urn-problem}\n\t\t\t\t\t\\binom{k-1}{l-1}\n\t\t\t\t\\end{equation}\n\t\t\t\tpossibilities to distribute the balls so that each urn contains at least one ball. \n\t\t\t\t\\begin{defi}\\label{defi:counting-patterns}\n\t\t\t\t\tLet $S=\\{A,B\\}$, $j,n,k\\in\\N$, and $n\\geq1$. We define\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\tS_{n,k}&\\coloneqq\\{s\\in S^n\\;|\\;\\text{$A$ appears $k$ times in $s$}\\}\\\\\n\t\t\t\t\t\tN(j,n,k)&\\coloneqq\\#\\{s\\in S_{n,k}\\;|\\;\\text{$s$ includes $j$ transitions $AB$ or $BA$}\\}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\\end{defi}\n\t\t\t\tIn words, $N(j,n,k)$ counts the number of sequences consisting of $k$ $A$'s and $n-k$ $B$'s with the restriction that $A$ changes to $B$ or vice versa $j$ times.\n\t\t\t\t\\begin{ex}\n\t\t\t\t\tLet $S=\\{A,B\\}$, $n=4$, and $k=2$. Then,\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\tN(0,n,k)&=\\#\\emptyset & &=0\\\\\n\t\t\t\t\t\tN(1,n,k)&=\\#\\{AABB,BBAA\\} & &=2\\\\\n\t\t\t\t\t\tN(2,n,k)&=\\#\\{ABBA,BAAB\\} & &=2\\\\\n\t\t\t\t\t\tN(3,n,k)&=\\#\\{ABAB,BABA\\} & &=2.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\\end{ex}\n\t\t\t\t\\begin{lem}\\label{lem:counting-patterns-binary-sequences}\n\t\t\t\t\tLet $S=\\{A,B\\}$ and $n,k,j\\in\\N$ with $k\\leq n$. Then,\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\tN(j,n,k)=\\begin{cases}\n\t\t\t\t\t\t\t2\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}& \\text{if }j=2l-1\\\\\n\t\t\t\t\t\t\t\\frac{n-2l}{l}\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}& \\text{if }j=2l\\\\\n\t\t\t\t\t\t\t1_{k\\in\\{0,n\\}}& \\text{if }j=0\n\t\t\t\t\t\t\\end{cases}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfor $j\\in\\{0,...,2\\min\\{k,n-k\\}-1_{2k=n}\\}$. Otherwise $N(j,n,k)=0$.\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tIf $j\\geq2\\min\\{n-k,k\\}-1_{2k=n}$, then $N(j,n,k)=0$ by \\Cref{defi:counting-patterns}. Next we assume that $j=0$, the only possible sequences are $A^n$ ($k=n$) and $B^n$ ($k=0$) so that $N(0,n,k)=1_{k\\in\\{0,n\\}}$. In the following, we assume that $1\\leq j\\leq2\\min\\{n-k,k\\}-1_{2k=n}$, then there is a $s\\in S_{n,k}$ so that $s$ includes exactly $j$ transitions $AB$ or $BA$ so that $N(j,n,k)>0$. In the odd case $j=2l-1$ for $l\\in\\{1,...,\\min\\{k,n-k\\}\\}$, the element $s$ is constructed by $l$ blocks of $A$'s and $l$ blocks of $B$'s:\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\ts&=\\underbrace{A...A}_{1}\\overbrace{B...B}^{1}\\underbrace{A...A}_{2}B\\quad.\\quad.\\quad.\\quad A\\overbrace{B...B}^{l-1}\\underbrace{A...A}_{l}\\overbrace{B...B}^{l},\\\\\n\t\t\t\t\t\ts&=\\underbrace{B...B}_{1}\\overbrace{A...A}^{1}\\underbrace{B...B}_{2}A\\quad.\\quad.\\quad.\\quad B\\overbrace{A...A}^{l-1}\\underbrace{B...B}_{l}\\overbrace{A...A}^{l}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tIdentifying these blocks with distinguishable urns and the elements $A$ and $B$ with indistinguishable balls (q.v.~\\Caref{eq:urn-problem}), the task is to count the possibilities of placing $k$ $A$'s in $l$ urns and vice versa $n-k$ $B$'s in $l$ urns with the additional assumption that each urn must contain at least one $A$ or one $B$. By changing the roles of $A$ and $B$, we get twice the number of possible combinations. Therefore, one of the \\textit{Twelvefold Ways} \\cite[Chap.~1.9]{Stanley.1986} shows\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\tN(j,n,k)=2\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tIn the even case $j=2l$ for $l\\in\\{1,...,\\min\\{k,n-k\\}-1_{2k=n}\\}$, we argue similarly to the odd case. The only difference is that s is constructed by $l+1$ blocks of $A$'s and $l$ blocks of $B$'s or vice versa:\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\ts&=\\underbrace{A...A}_{1}\\overbrace{B...B}^{1}\\underbrace{A...A}_{2}B\\quad.\\quad.\\quad.\\quad A\\overbrace{B...B}^{l-1}\\underbrace{A...A}_{l}\\overbrace{B...B}^{l}\\underbrace{A...A}_{l+1},\\\\\n\t\t\t\t\t\ts&=\\underbrace{B...B}_{1}\\overbrace{A...A}^{1}\\underbrace{B...B}_{2}B\\quad.\\quad.\\quad.\\quad A\\overbrace{A...A}^{l-1}\\underbrace{B...B}_{l}\\overbrace{A...A}^{l}\\underbrace{B...B}_{l+1}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tThen the \\textit{Twelvefold Ways} \\cite[Chap.~1.9]{Stanley.1986} proves the statement by\\\\\n\t\t\t\t\t\\begin{minipage}[b]{0.9\\textwidth}\n\t\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\tN(j,n,k)&=\\binom{n-k-1}{l-1}\\binom{k-1}{l}+\\binom{n-k-1}{l}\\binom{k-1}{l-1}\\\\\n\t\t\t\t\t\t\t&=\\frac{n-2l}{l}\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}.\n\t\t\t\t\t\t\\end{align*}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\\end{proof}\n\t\t\t\tWith the help of this counting method, we are ready to prove \\Cref{lem:proofthm1-term1}. In what follows, we identify the couple $(A,B)$ with the product $AB$ by slight abuse of notations.\n\t\t\t\t\\begin{proof}[Proof of \\Cref{lem:proofthm1-term1}]\n\t\t\t\t\tAssume w.l.o.g.~$P\\neq 0$, then $MP=PM=P$ because for all $n\\in\\N$\n\t\t\t\t\t\\begin{equation}\\label{eq:eigenprojection}\n\t\t\t\t\t\t\\|P-PM\\|_\\infty\\leq\\|(M^{n}-P)M\\|_\\infty+\\|P-M^{n+1}\\|_\\infty\\leq\\delta^n+\\delta^{n+1}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tand $\\norm{P}_\\infty\\leq1$ holds by a similar argument because for all $n\\in\\N$\n\t\t\t\t\t\\begin{equation}\\label{eq:projection-contraction}\n\t\t\t\t\t\t\\|P\\|_\\infty\\leq\\|M^n\\|_\\infty+\\|P-M^n\\|_\\infty\\leq 1+\\delta^n.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tThe main idea is to split $M=P+M^\\perp$ with $M^\\perp\\coloneqq P^\\perp M$ and order the terms after expanding the following polynomial appropriately. Let $A\\coloneqq M^\\perp e^{\\frac{1}{n}\\cL}$ and $B\\coloneqq Pe^{\\frac{1}{n}\\cL}$ so that\n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-polynomial-expansion}\n\t\t\t\t\t\t\\left((P+M^\\perp) e^{\\frac{1}{n}\\cL}\\right)^n= B^n+\\sum_{k=1}^{n-1}\\sum_{s\\in S_{n,k}}s+A^n\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\twhere elements in $S_{n,k}$ are identified with sequences of concatenated operators and denoted by $s$. Then, we partition summands by the number of transitions from $A$ to $B$ or vice versa and use\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{A}_\\infty\\le \\delta\\qquad \\text{ and }\\qquad \t\\norm{AB}_\\infty=\\norm{M^\\perp P^\\perp e^{\\frac{1}{n}\\cL}PM e^{\\frac{1}{n}\\cL}}_\\infty\\leq\\delta\\frac{b}{n}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tThe number of summands with $j$ transitions is equal to $N(j,n,k)$ given by \\Cref{lem:counting-patterns-binary-sequences} for $j\\in\\{1,...,m\\}$ and $m\\coloneqq2\\min\\{k,n-k\\}-1_{2k=n}$. Then, the inequality above shows \n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^n-\\left(Pe^{\\frac{1}{n}\\cL}\\right)^n}_\\infty&\\leq\\delta^n+\\sum_{k=1}^{n-1}\\sum_{j=1}^{m}\\delta^{k}N(j,n,k)\\left(\\frac{b}{n}\\right)^j\\\\\n\t\t\t\t\t\t\t&=\\delta^n\\begin{aligned}[t]\n\t\t\t\t\t\t\t\t&+\\sum_{k=1}^{n-1}\\sum_{l=1}^{\\ceil*{\\frac{m}{2}}}\\delta^{k}2\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}\\left(\\frac{b}{n}\\right)^{2l-1}\\\\\n\t\t\t\t\t\t\t\t&+\\sum_{k=1}^{n-1}\\sum_{l=1}^{\\floor*{\\frac{m}{2}}}\\delta^{k}\\frac{b(n-2l)}{nl}\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}\\left(\\frac{b}{n}\\right)^{2l-1}\n\t\t\t\t\t\t\t\\end{aligned}\\\\\n\t\t\t\t\t\t\t&\\overset{(1)}{\\leq}\\delta^n+\\frac{b(2+b)}{n}\\sum_{l=1}^{\\floor*{\\frac{n}{2}}}\\sum_{k=1}^{n-1}\\delta^{k}\\frac{n^{2l-2}}{(l-1)!^2}\\left(\\frac{b}{n}\\right)^{2l-2}\\\\\n\t\t\t\t\t\t\t&=\\delta^n+\\frac{b(2+b)}{n}\\frac{\\delta-\\delta^{n}}{1-\\delta}\\sum_{l=0}^{\\floor*{\\frac{n}{2}}-1}\\frac{b^{2l}}{l!^2}\\\\\n\t\t\t\t\t\t\t&{\\leq}\\delta^n+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}.\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tIn (1) above, we used the upper bound $\\binom{n}{k}\\leq\\frac{n^k}{k!}$ to show\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}\\leq\\frac{n^{2l-2}}{(l-1)!^2}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tAdditionally, we increase the upper index to $\\floor*{\\frac{n}{2}}$, and upper bound\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t2+\\frac{b(n-2l)}{nl}\\leq2+b.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tApplying the assumptions again to\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\left(Pe^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx=\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^{n-1}Pe^{\\frac{1}{n}\\cL}P^\\perp x\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfinishes the \\namecref{lem:proofthm1-term1}:\\\\\n\t\t\t\t\t\\begin{minipage}[b]{0.9\\textwidth}\n\t\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx}\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}.\n\t\t\t\t\t\t\\end{equation*}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tBy the counting method introduced above, we can approximate $(Me^{\\frac{1}{n}\\cL})^n$ by $(Pe^{\\frac{1}{n}\\cL}P)^n$, which is independent of $M^\\perp$. In previous works \\cites{Mobus.2019}{Becker.2021}, the operators considered in similar proof steps as \\Cref{eq:proofthm1-term2} and (\\ref{eq:proofthm1-term3}) depended on $M^\\perp$.\n\t\t\t\t\\end{rmk*}\n\t\t\t\n\t\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm1-term2}:}\n\t\t\t\tIn the next step, we apply our modified Chernoff \\Cref{lem:improved-chernoff}:\n\t\t\t\t\\begin{lem}\\label{lem:proofthm1-term2}\n\t\t\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $P\\in\\cB(\\cX)$ be a projection. Assume that both operators satisfy the same assumption as in \\Cref{thm:spectral-gap}. Then, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx-e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px}\\leq\\frac{1}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right).\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tThe proof relies on the modified Chernoff Lemma (q.v.~\\Caref{lem:improved-chernoff}) applied to the contraction $C(\\tfrac{1}{n})=Pe^{\\frac{1}{n}\\cL}P$ on $P\\cX$. Then, for all $x\\in\\cX$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx-e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px}\\leq \\frac{n}{2}\\norm{\\left(P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\right)^2x}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tMoreover, the asymptotic Zeno condition (\\ref{eq:thm1-asympzeno}) and the continuity of the norm imply\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{P^\\perp\\cL Px}=\\lim\\limits_{h\\rightarrow 0}\\frac{1}{h}\\norm{P^\\perp e^{h\\cL}Px}\\leq b\\norm{x}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfor all $x\\in\\cD(\\cL P)$. Hence $P^\\perp\\mathcal{L}P$ is a bounded operator with $\\|P^\\perp \\cL P\\|_\\infty\\leq b$. Next, given $x\\in\\cD(\\cL P)$, the $C_0$-semigroup properties (q.v.~\\Caref{lem:properties-semigroups}) imply\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\tn\\norm{\\left(P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\right)^2x}&=\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}(\\1-P+P)\\cL Pxd\\tau_1}\\\\\n\t\t\t\t\t\t&\\leq\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1}\\\\\n\t\t\t\t\t\t&\\quad+2\\int_{0}^{1}\\norm{Pe^{\\frac{\\tau_1}{n}\\cL}P^\\perp}_\\infty d\\tau_1\\norm{P^\\perp\\cL P}_\\infty\\norm{x}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tNote that $\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Px$ belongs to $\\cD(\\cL)$ by \\Cref{lem:properties-semigroups}, but not necessarily to $\\cD(\\cL P)$. However, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\tn\\norm{\\left(P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\right)^2x}&\\leq\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1}+\\frac{b^2}{n}\\norm{x}\\\\\n\t\t\t\t\t\t\t&\\leq\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1}\\\\\n\t\t\t\t\t\t\t&\\quad+\\frac{b^2}{n}\\norm{x}+\\norm{Pe^{\\frac{1}{n}\\cL}P^\\perp}_\\infty\\norm{\\cL Px}\\\\\n\t\t\t\t\t\t\t&\\leq\\frac{1}{n}\\norm{P\\int_{0}^{1}e^{\\frac{\\tau_2}{n}\\cL}\\cL\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1d\\tau_2}\\\\\n\t\t\t\t\t\t\t&\\quad+\\frac{b^2}{n}\\norm{x}+\\frac{b}{n}\\norm{\\cL Px}\\\\\n\t\t\t\t\t\t\t&\\leq\\frac{1}{n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right),\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhich proves \\Cref{lem:proofthm1-term2}.\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tAs regards to the convergence rate of the quantum Zeno effect, \\Cref{lem:proofthm1-term2} constitutes our main improvement compared to the work \\cites{Becker.2021}. The modified Chernoff lemma allows to improve the convergence rate to $n^{-1}$.\n\t\t\t\t\\end{rmk*}\n\t\t\t\n\t\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm1-term3}:}\n\t\t\t\tFinally, we prove an upper bound on \\Cref{eq:proofthm1-term3}, which can be interpreted as a modified \\textit{Dunford-Segal approximation} \\cite{Gomilko.2014}.\n\t\t\t\t\\begin{lem}\\label{lem:proofthm1-term3}\n\t\t\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $P\\in\\cB(\\cX)$ be a projection. Assume that both operators satisfy the assumptions of \\Cref{thm:spectral-gap}. Then, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px}\\leq\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\norm{x}+\\norm{(\\cL P)^2 x}\\right)\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twith $e^{\\tilde{b}}\\coloneqq\\sup_{s\\in[0,1]}\\|e^{sP\\cL P}P\\|_\\infty<\\infty$.\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tThe proof relies on the integral equation for semigroups from \\Cref{lem:integral-equation-semigroups}. We start by proving the continuity of \n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-integral-equation-continuity}\n\t\t\t\t\t\t[0,1]\\ni s\\mapsto -e^{sP\\cL P}P\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)e^{(1-s)nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tSince for all $s\\in[0,1]$ and $x\\in\\cD(\\cL P)$\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\te^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}P\\cL Px&=\\lim\\limits_{h\\rightarrow0}Pe^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}\\frac{P(e^{h\\cL}-\\1)P}{h}x\\\\\n\t\t\t\t\t\t&=\\lim\\limits_{h\\rightarrow0}\\frac{P(e^{h\\cL}-\\1)P}{h}e^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}Px=P\\cL Pe^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}Px,\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tthe vector-valued function defined in \\Cref{eq:proofthm1-integral-equation-continuity} is equal to\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t-e^{sP\\cL P}Pe^{(1-s)nP(e^{\\frac{1}{n}\\cL}-\\1)P}\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)x.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tand, thereby, well-defined and continuous in $s$. Therefore, \\Cref{lem:integral-equation-semigroups} gives for all $x\\in\\cD(\\cL P)$\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t&e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px\\\\\n\t\t\t\t\t\t&\\qquad\\quad=-\\int_{0}^{1}e^{sP\\cL P}Pe^{(1-s)nP(e^{\\frac{1}{n}\\cL}-\\1)P}\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)xds.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tMoreover, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation}\\label{eq:approx-generator}\n\t \t\t\t\t\\begin{aligned}\n\t \t\t\t\t nP(e^{\\frac{1}{n}\\cL}-\\1)Px-P\\cL Px&=P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}\\cL Px d\\tau_1-P\\cL Px\\\\\n\t \t\t\t\t &=\\frac{1}{n}P\\int_{0}^{1}\\int_{0}^{1}\\tau_1e^{\\frac{\\tau_1\\tau_2}{n}\\cL}(\\cL P)^2xd\\tau_2d\\tau_1+P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P^\\perp\\cL Px d\\tau_1.\n\t\t\t\t\t \\end{aligned}\n\t\t\t\t\t\\end{equation}\n\n\t\t\t\t\tFinally, we use $\\sup_{s\\in[0,1]}\\|e^{sP\\cL P}P\\|_\\infty<\\infty$, which holds by the \\textit{principle of uniform boundedness} (q.v.~proof of \\Caref{prop:trotter-kato-product-formula}), the property that $(e^{s nP(e^{\\frac{1}{n}\\cL}-\\1)P})_{s\\geq0}$ is a contraction, and the upper bounds $\\|P^\\perp\\cL P\\|_\\infty\\leq b$ and $\\|Pe^{s\\cL}P^\\perp\\|_\\infty\\leq sb$ so that\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\\norm{e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px}&\\leq e^{\\tilde{b}}\\int_{0}^{1}\\norm{\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)x}ds\\\\\n\t\t\t\t\t\t&\\leq\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\|x\\|+\\|(\\cL P)^2 x\\|\\right)\n\t\t\t\t\t\\end{align*} \n\t\t\t\t\tfor all $x\\in\\cD((\\cL P)^2)$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,1]}\\|e^{sP\\cL P}P\\|_\\infty$.\n\t\t\t\t\\end{proof}\n\t\t\t\tThe above approximation of $e^{P\\cL P}$ by $e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}$ is similar to the Dunford-Segal approximation, which would be given by $\\operatorname{exp}\\big({n(\\operatorname{exp}({\\frac{1}{n}P\\cL P})-\\1)}\\big)$: for the generator $(\\cK,\\cD(\\cK))$ of a bounded $C_0$-semigroup, \\citeauthor{Gomilko.2014} proved \\cite[Cor.~1.4]{Gomilko.2014}\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\norm{e^{nt(e^{\\frac{1}{n}\\cK}-\\1)}x-e^{t\\cK}x}\\leq8\\tilde{b}\\frac{t}{n}\\norm{\\cK^2x}\n\t\t\t\t\\end{equation*}\n\t\t\t\tfor all $x\\in\\cD(\\cK^2)$ and $\\tilde{b}\\coloneqq\\sup_{t\\geq0}\\|e^{t\\cK}\\|_\\infty$. In our case, it is not clear whether $(e^{sP\\cL P})_{s\\geq0}$ is uniformly bounded.\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tThe specificity of the last step stems from the fact that $(\\cL,\\cD(\\cL))$ is unbounded. In the previous works \\cites{Mobus.2019}{Becker.2021} a similar step exits but in both papers $\\cL$ was assumed to be bounded. Moreover, \\Cref{eq:approx-generator} is the only step in the proof of \\Cref{thm:spectral-gap}, which deals with the operator $P\\cL P$. If $P\\cL P$ is closable, $P\\cL Px=\\overline{P\\cL P}x$ for all $x\\in\\cD(\\cL P)$ so that it is enough to ask for the closure of $P\\cL P$ to define a generator. The same reasoning works for \\Cref{prop:spectral-gap-uniform-norm-power-convergence} and \\Cref{cor:spectral-gap}.\n\t\t\t\t\\end{rmk*}\n\t\t\t\n\t\t\t\\subsubsection*{End of the proof of \\Cref{thm:spectral-gap}:}\n\t\t\t\tWe combine \\Cref{lem:proofthm1-term1}, \\ref{lem:proofthm1-term2}, and \\ref{lem:proofthm1-term3} to prove \\Cref{thm:spectral-gap}.\n\t\t\t\t\\begin{proof}[Proof of \\Cref{thm:spectral-gap}]\n\t\t\t\t\tLet $x\\in\\cD((\\cL P)^2)$. Then,\n\t\t\t\t\t\\begin{flalign*}\n\t\t\t\t\t\t&&\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-e^{P\\cL P}Px}&\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}&&\\text{(\\Caref{lem:proofthm1-term1})}\\\\\n\t\t\t\t\t\t&& &+\\frac{1}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)&&\\text{(\\Caref{lem:proofthm1-term2})}\\\\\n\t\t\t\t\t\t&& &+\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\norm{x}+\\norm{(\\cL P)^2 x}\\right)&&\\text{(\\Caref{lem:proofthm1-term3})}.\n\t\t\t\t\t\\end{flalign*}\n\t\t\t\t\tRedefining $\\cL$ by $t\\cL$ and $b$ by $tb$, we achieve\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c}{n}\\left(\\norm{x}+\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)+\\delta^n\\norm{x}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twith an appropriate constant $c>0$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$.\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tThe upper bound in \\Cref{thm:spectral-gap} can be formulated for all $x\\in\\cD(\\cL P)$. For this, one must stop at an earlier stage of the proof and express the error terms by appropriate integrals. One possible bound would be the following \n\t\t\t\t\t\\begin{flalign*}\n\t\t\t\t\t\t&& \\|\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx&-e^{P\\cL P}Px\\| &&\\\\\n\t\t\t\t\t\t&& &\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}&&\\text{(\\Caref{lem:proofthm1-term1})}\\\\\n\t\t\t\t\t\t&& &+\\frac{1}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{P\\int_{0}^{1}\\int_{0}^{1}\\cL e^{\\frac{\\tau_1+\\tau_2}{n}\\cL}P\\cL Pxd\\tau_1d\\tau_2}\\right)&&\\text{(\\Caref{lem:proofthm1-term2})}\\\\\n\t\t\t\t\t\t&& &+\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\norm{x}+2\\norm{P\\int_{0}^{1}\\int_{0}^{1}\\tau_1\\cL e^{\\frac{\\tau_1\\tau_2}{n}\\cL}P\\cL Pxd\\tau_2d\\tau_1}\\right)&&\\text{(\\Caref{lem:proofthm1-term3})}.\n\t\t\t\t\t\\end{flalign*}\n\t\t\t\t\\end{rmk*}\n\t\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{cor:explicit-bound-thm1} and \\Cref{cor:explicit-bound-thm1-closed-sys}}{Proposition 3.1 and Corollary 3.2}}\\label{subsec:proofexplicitboundthm1}\n\t\t\t\t\\begin{proof}[Proof of \\Cref{cor:explicit-bound-thm1}]\n\t\t\t\t\tSince $\\|P\\|_\\infty\\leq1$ (\\ref{eq:projection-contraction}) and $t\\mapsto e^{t\\cL}$ is a uniformly continuous contraction semigroup, the generator is defined on $\\cX$ and bounded, i.e.~$\\|\\cL\\|_\\infty<\\infty$, so that\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{Pe^{t\\cL}(\\1-P)}_\\infty=\\norm{P(e^{t\\cL}-\\1)(\\1-P)}_\\infty\\leq t\\norm{\\cL\\int_0^1e^{ts\\cL}(\\1-P)ds}_\\infty\\leq t\\norm{\\cL}_\\infty c_{p}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $c_{p}\\coloneqq\\|\\1-P\\|_\\infty\\leq2$. Then, we simplify the bounds found in \\Cref{lem:proofthm1-term1}, \\ref{lem:proofthm1-term2}, and \\ref{lem:proofthm1-term3} to\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^n-e^{P\\cL P}P}_\\infty\\leq\\delta^n+\\frac{1}{n}\\left(b+\\frac{2\\delta}{1-\\delta}e^{3b}+\\frac{b^2}{2}+\\frac{b}{2}\\norm{\\cL}_\\infty+\\frac{1}{2}\\norm{\\cL}^2_\\infty+\\frac{e^{\\tilde{b}}b^2}{2}+\\frac{e^{\\tilde{b}}}{2}\\norm{\\cL}^2_\\infty\\right),\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$. By redefining $\\cL$ by $t\\cL$, $b$ by $tb$, and using $b\\leq\\|\\cL\\|_\\infty c_{p}$,\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq c_{p}\\frac{t\\|\\cL\\|_\\infty}{n} &+\\left(\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\right)\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t\t\t\t\t\t&+\\delta^n+\\frac{2\\delta}{1-\\delta}\\frac{e^{3t\\|\\cL\\|_\\infty c_{p}}}{n}\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\twhich proves the statement.\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{proof}[Proof of \\Cref{cor:explicit-bound-thm1-closed-sys}]\n\t\t\t\t\tIn closed quantum systems $\\cX=\\cH$ equipped with the operator norm induced by the scalar product, which shows $\\|U\\|_\\infty=1$ for all unitaries $U\\in\\cB(\\cH)$. Especially, $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty=1$ because $\\|P\\|_\\infty=1$ is equivalent to $P=P^\\dagger$ \\cite[Thm.~2.1.9]{Simon.2015} so that $PHP$ is hermitian. Moreover, $P=P^\\dagger$ implies $(\\1-P)^\\dagger=(\\1-P)$ which shows $c_{p}=\\|\\1-P\\|_\\infty\\leq1$. Finally, the choice $M=P$ implies $\\delta=0$ which proves the \\nameCref{cor:explicit-bound-thm1-closed-sys} by inserting the constants into \\Cref{cor:explicit-bound-thm1}.\n\t\t\t\t\\end{proof}\n\t\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{prop:spectral-gap-uniform-norm-power-convergence} }{Proposition 5.7}}\\label{subsec:thm1-prop}~\\\\\n\t\t\t\tIn this subsection, we weaken the assumptions (\\ref{unifpower1}) on the contraction $M$ at the cost of stronger assumptions on the $C_0$-semigroup. For that, we combine techniques from holomorphic functional calculus with the semicontinuity of the spectrum of $M$ perturbed by the semigroup under certain conditions. We refer to \\Cref{sec:appendix-holomorphic-fc-semicontinuity} for details on the tools needed to prove the main result of this section.\n\t\t\t\t\\begin{prop}\\label{prop:spectral-gap-uniform-norm-power-convergence}\n\t\t\t \tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction such that\n\t\t\t \t\\begin{equation}\\label{unifpowerc}\n\t\t\t \t\t\\norm{M^n-P}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t \t\t\\end{equation}\n\t \t\t\tfor some projection $P$, $\\delta\\in(0,1)$ and $\\tilde{c}\\ge 0$. Moreover, we assume that there is $b\\geq0$ so that \n\t \t\t\t\\begin{equation}\\label{key}\n\t \t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb,\\quad\\norm{M^\\perp e^{t\\cL}-M^\\perp}_\\infty\\leq tb,\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb\n \t\t\t\t\\end{equation}\n \t\t\t\twhere $M^\\perp=(\\1-P)M$. If $(P\\cL P,\\cD(\\cL P))$ generates a $C_0$-semigroup, then there is $\\epsilon>0$ such that for all $t\\geq0$, $n\\in\\N$ satisfying $t\\in[0,n\\epsilon]$, $\\tilde{\\delta}\\in(\\delta,1)$, and $x\\in\\cD((\\cL P)^2)$ \n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c_1(t,b,\\tilde{\\delta}-\\delta)}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+c_2(\\tilde{c},\\tilde{\\delta}-\\delta)\\tilde{\\delta}^n\\norm{x}\\,,\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfor some constants $c_1,c_2\\ge 0$ depending on $t$, $b$, the difference $\\tilde{\\delta}-\\delta$, and $\\tilde{c}$.\n\t\t\t\t\\end{prop}\n\t\t\t\tThe only difference to the proof of \\Cref{thm:spectral-gap} is summarized in the question: How can we upper bound $\\|(M^\\perp e^{\\frac{t}{n}\\cL})^k\\|_\\infty$ for all $k\\in\\{1,...,n\\}$ with the weaker assumption (\\ref{unifpowerc}) on $M$? For that, we replace the argument in the proof of \\Cref{lem:proofthm1-term1}, which only works for the case $\\tilde{c}=1$.\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tSince the bounds found in \\Cref{lem:proofthm1-term2} and \\ref{lem:proofthm1-term3} are independent of the value of $\\tilde{c}$, it is enough to improve \\Cref{lem:proofthm1-term1}:\n\t\t\t\t\t\\begin{flalign*}\n\t\t\t\t\t\t&&\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}&\\leq\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{t}{n}\\cL}P\\right)^nx}&&\\\\\n\t\t\t\t\t\t&& &+\\frac{t^2}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)&&\\text{(\\Caref{lem:proofthm1-term2})}\\\\\n\t\t\t\t\t\t&& &+\\frac{e^{\\tilde{b}}t^2}{2n}\\left(b^2\\norm{x}+\\norm{(\\cL P)^2 x}\\right)&&\\text{(\\Caref{lem:proofthm1-term3})}\n\t\t\t\t\t\\end{flalign*}\n\t\t\t\t\twith $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty<\\infty$. Since the assumption (\\ref{unifpowerc}) on $M$ is a special case of the uniform power convergence (q.v.~\\Caref{unifpower}), \\Cref{prop:equivalence-spectral-gap} shows the equivalence of the uniform\\\\[1ex]\n\t\t\t\t\t\\begin{minipage}[c]{0.69\\textwidth}\n\t\t\t\t\t\tpower convergence of $M$ to the spectral gap assumption, that is\n\t\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\t\\sigma(M)\\subset\\D_\\delta\\cup\\{1\\},\n\t\t\t\t\t\t\\end{equation*}\n\t\t\t\t\t\twhere the quasinilpotent operator corresponding to the eigenvalue $1$ vanishes. Therefore, the eigenprojection $P$ w.r.t.~$1$ satisfies $MP=PM=P$ and the curve \t$\\gamma:[0,2\\pi]\\rightarrow\\C,\\varphi\\mapsto\\tilde{\\delta}e^{i\\varphi}$ encloses the spectrum of $M^\\perp\\coloneqq MP^\\perp$ (q.v.~\\Caref{fig3}). Together with the second bound in \\eqref{key}, \\Cref{lem:semicontinuity-spectrum} shows that there exists $\\epsilon>0$ so that the spectrum of $M^\\perp e^{s\\cL}$ can be separated by $\\gamma$ for all $s\\in[0,\\epsilon]$.\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\t\\begin{minipage}[c]{0.3\\textwidth}\n\t\t\t\t\t\t\\begin{center}\n\t\t\t\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\t\t\t\\draw[dashed,tumivory] (0,0) ellipse (1.4cm and 1.4cm);\n\t\t\t\t\t\t\t\t\\draw[dashed,mygreen] (0,0) ellipse (1.15cm and 1.15cm);\n\t\t\t\t\t\t\t\t\\filldraw (1.4,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\t\t\\filldraw (0,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\t\t\\draw (1.7,0) node {$1$};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\\draw[<->] (0.0,0.05)--(0,0.95);\n\t\t\t\t\t\t\t\t\\draw[<->,mygreen] (0.0433,-0.025)--(0.9526cm,-0.55cm);\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\\fill[pattern=my north east lines] (0,0) ellipse (1cm and 1cm);\n\t\t\t\t\t\t\t\t\\filldraw[white] (-0.35,0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\t\t\\draw (-0.35,0.5) node {$\\delta$};\n\t\t\t\t\t\t\t\t\\filldraw[white] (0.35,-0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\t\t\\draw[mygreen] (0.35,-0.5) node {$\\tilde{\\delta}$};\n\t\t\t\t\t\t\t\t\\filldraw[white] (-1.06,-1.06) ellipse (0.15cm and 0.3cm);\n\t\t\t\t\t\t\t\t\\draw (-1.06,-1.06) node [mygreen]{$\\gamma$};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\\draw (1.5,1.2) node {$\\sigma(M)$};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\\end{tikzpicture}\\\\[1ex]\n\t\t\t\t\t\t\t\\captionof{figure}{}\\label{fig3}\n\t\t\t\t\t\t\\end{center}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\tTherefore, the holomorphic functional calculus (q.v.~\\Caref{prop:holomorphic-functional-calculus}) shows for all $t\\in[0,n\\epsilon], k\\in\\N$\n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-upper-bound-inner-part}\n\t\t\t\t\t\t\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k}=\\frac{1}{2\\pi i}\\oint_{\\gamma}z^{k}R(z,M^\\perp e^{\\frac{t}{n}})dz.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tLet $t\\geq0$, $n\\in\\N$ so that $t\\in[0,n\\epsilon]$. By the \\textit{principle of stability of bounded invertibility} \\cite[Thm.~IV.2.21]{Kato.1995}, $R(z,M^\\perp e^{s\\cL})$ is well-defined and bounded for all $z\\in\\gamma$ and $s\\in[0,\\epsilon]$. More explicitly, using the \\textit{second Neumann series} for the resolvent \\cite[p.~67]{Kato.1995}, we have \n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-perturbed-resolvent-uniform-boundedness}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\t\\norm{R(z,M^\\perp e^{s\\cL})}_\\infty&=\\norm{R(z,M^{\\perp})\\sum_{p=0}^{\\infty}\\left((M^\\perp e^{s\\cL}-M^\\perp)R(z,M^\\perp)\\right)^p}_\\infty\\\\\n\t\t\t\t\t\t\t&\\leq\\norm{R(z,M^\\perp)}_\\infty\\sum_{p=0}^{\\infty}\\left(sb\\norm{R(z,M^\\perp)}_\\infty\\right)^p\\\\\n\t\t\t\t\t\t\t&\\leq\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty \\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}\\eqqcolon c_2,\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\twhere we have applied the assumption \\eqref{key} and the following upper bound on $s$ (q.v.~\\Caref{eq:semicontinuity-time-bound}):\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\ts\\leq\\epsilon<\\frac{1}{2b}(1+\\tilde{\\delta}^2)^{-1}\\left(1+\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty^2\\right)^{-\\frac{1}{2}}\\leq\\frac{1}{2b}(1+\\tilde{\\delta}^2)^{-1}\\left(\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty\\right)^{-1},\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tto compute the geometric series. Combining \\Cref{eq:proofthm1-upper-bound-inner-part} and (\\ref{eq:proofthm1-perturbed-resolvent-uniform-boundedness}) shows for all $k\\in\\{1,...,n\\}$\n\t\t\t\t\t\\begin{equation}\\label{eq:proofprop-perturbed-measurement}\n\t\t\t\t\t\t\\norm{\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k-1}}_\\infty\\leq\\frac{1}{2\\pi}\\oint_{\\gamma}\\abs{z}^{k-1}\\norm{R(z,M^\\perp e^{\\frac{t}{n}\\cL})}_\\infty dz\\leq c_2\\tilde{\\delta}^{k}.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tNext, we define $A\\coloneqq M^\\perp e^{\\frac{{t}}{n}\\cL}$ and $B\\coloneqq Pe^{\\frac{{t}}{n}\\cL}$ and expand\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\left(Me^{\\frac{t}{n}\\cL}\\right)^n=\\left(PMe^{\\frac{t}{n}\\cL}+M^\\perp e^{\\frac{t}{n}\\cL}\\right)^n=(B+A)^n.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tThe above $n^{\\text{th}}$-power can be expanded in terms of sequences of the form \n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\tA...AB...BA...AB...\\quad\\text{or}\\quad B...BA...AB...BA...\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tSimilarly to \\Cref{lem:proofthm1-term1}, we can upper bound every sequence w.r.t.~the number of transitions $AB$ or $BA$ using the assumptions \\eqref{key} on the $C_0$-semigroup as well as the inequality (\\ref{eq:proofprop-perturbed-measurement}). The only difference to the proof of \\Cref{lem:proofthm1-term1} is the constant $c_2$ in the inequality so that\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(Pe^{\\frac{t}{n}\\cL}\\right)^n}_\\infty&\\leq c_2\\tilde{\\delta}^n+\\sum_{k=1}^{n-1}\\sum_{j=1}^{m}\\tilde{\\delta}^{k}N(j,n,k)\\left(\\frac{tbc_2}{n}\\right)^j\\\\\n\t\t\t\t\t\t\t&\\leq c_2\\tilde{\\delta}^n+\\frac{b}{n}+\\frac{1}{n}\\frac{bc_2(2+bc_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2bc_2},\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $m\\coloneqq2\\min\\{k,n-k\\}-1_{2k=n}$. Then, for all $x\\in\\cD((\\cL P)^2)$ and an appropriate $c_1\\geq0$\\\\\n\t\t\t\t\t\\begin{minipage}[b]{0.9\\textwidth}\n\t\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c_1}{n}\\left(\\norm{x}+\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)+c_2{\\tilde{\\delta}}^n\\norm{x}.\n\t\t\t\t\t\t\\end{equation*}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\\end{proof}\n\t\t\t\t\\vspace{-2ex}\n\t\t\t\t\\begin{proof}[Proof of \\Cref{cor:explicit-bound-prop}]\n\t\t\t\t\tSimilarly to \\Cref{cor:explicit-bound-thm1},\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq c_2\\tilde{\\delta}^n+\\frac{tc_{p}\\|\\cL\\|_\\infty}{n} +\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\frac{t^2\\|\\cL\\|_\\infty^2}{n}+\\frac{2\\tilde{\\delta}}{1-\\tilde{\\delta}}\\frac{e^{3tc_{p}c_2\\|\\cL\\|_\\infty}}{n}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$ and $c_2\\coloneqq\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty \\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}$ (see \\Caref{eq:proofthm1-perturbed-resolvent-uniform-boundedness}).\n\t\t\t\t\tThe constant $c_2$ can be bounded with the help of the \\textit{first von Neumann series} \\cite[p.~37]{Kato.1995} and the geometric series:\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}\\sup\\limits_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty=2\\sup\\limits_{z\\in\\gamma}\\|\\sum_{k=0}^{\\infty}z^{-(k+1)}(M^\\perp)^k\\|_\\infty\\leq2\\frac{\\tilde{c}}{\\tilde{\\delta}}\\sum_{k=0}^{\\infty}\\tilde{\\delta}^{-k}\\delta^k=\\frac{2\\tilde{c}}{\\tilde{\\delta}-\\delta}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tso that \n\t\t\t\t\t\\begin{align*}\n\t \t \\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq\\frac{tc_{p}\\|\\cL\\|_\\infty}{n}&+\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t \t &+\\frac{2\\tilde{c}}{\\tilde{\\delta}-\\delta}\\tilde{\\delta}^n+\\frac{2\\tilde{\\delta}}{1-\\tilde{\\delta}}\\frac{e^{\\frac{6tc_{p}\\tilde{c}\\|\\cL\\|_\\infty}{\\tilde{\\delta}-\\delta}}}{n}\n\t \t\\end{align*}\n \t\twhich finishes the proof of the \\namecref{cor:explicit-bound-prop}.\n\t\t\t\t\\end{proof}\n\t\t\n\t\\section{Uniform Power Convergence with Finitely Many Eigenvalues}\\label{sec:finitely-many-eigenvalues}\n\t\tIn this section, we weaken the assumption on $M$ to the uniform power convergence assumption (q.v.~\\Caref{unifpower}), that is we allow for finitely many eigenvalues $\\{\\lambda_j\\}_{j=1}^J$ and associated projections $\\{P_j\\}_{j=1}^J$ satisfying $P_jP_k=1_{j=k}P_j$. Similarly to Theorem 3 in \\cite{Becker.2021}, we strengthen the assumptions on the $C_0$-semigroup to $M\\cL$ and $\\cL P_\\Sigma$ being densely defined and bounded by $b\\geq0$, where $P_\\Sigma\\coloneqq \\sum_{j=1}^{J}P_j$. Under those assumptions, we can prove the Zeno convergence in the uniform topology:\n\n\t\t\\begin{thm}\\label{thm:spectral-gap-uniform}\n\t\t Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the following uniform power convergence: there is $\\tilde{c}>0$ so that\n\t\t\t\\begin{equation}\\label{unifpower}\n\t\t\t \\norm{M^n-\\sum_{j=1}^{J}\\lambda^n_jP_j}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t \\end{equation}\n\t\t for a set of projections $\\{P_j\\}_{j=1}^J$ satisfying $P_jP_k=1_{j=k}P_j$, eigenvalues $\\{\\lambda_j\\}_{j=1}^J\\subset\\partial\\D_1$, and a rate $\\delta\\in(0,1)$. For $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$, we assume that $M\\cL$ and $\\cL P_\\Sigma$ are densely defined and bounded by $b\\geq0$ and $\\|P_j\\|_\\infty=1$ for all $j\\in\\{1,...,J,\\Sigma\\}$. Then, there is an $\\epsilon>0$ such that for all $n\\in\\N$, $t\\geq0$ satisfying $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\leq\\frac{c_1 }{n}+c_2\\tilde{\\delta}^n\\,,\n\t\t\t\\end{equation*}\n\t\t\tfor some constants $c_1,c_2\\ge 0$ depending on all involved parameters except from $n$.\n\t\t\\end{thm}\n\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{thm:spectral-gap-uniform} }{Theorem 3.3 }}~\\\\\n\t\tSimilarly to the papers \\cite{Mobus.2019} and \\cite{Becker.2021}, we use the holomorphic functional calculus to\n\t\tseparate the spectrum of the contraction $Me^{t\\cL}$ appearing in the Zeno sequence. In contrast to \\cite{Becker.2021} where the $C_0$-semigroup is approximated by a sequence of uniformly continuous semigroups, we instead crucially rely upon the uniform continuity of the perturbed contraction to recover the optimal convergence rate. We upper bound the following terms: \n\t\t\\begin{align}\n\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty&\\leq\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n}_\\infty\\label{eq:proofthm2-term1}\\\\\n\t\t\t&+\\norm{\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\label{eq:proofthm2-term2}\\\\\n\t\t\t&+\\norm{\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\label{eq:proofthm2-term3}\n\t\t\\end{align}\n\t\twhere the definitions of the \\textit{perturbed spectral projection} $P_j(\\tfrac{t}{n})$ and the \\textit{Chernoff contraction} $C_{j}(\\tfrac{t}{n})$ are postponed to \\Cref{lem:proofthm2-term2}.\n\t\t\n\t\t\\subsubsection*{Approximation of the Perturbed Spectral Projection:}\n\t\t\t In the following result, we consider an operator $A$ uniformly perturbed by a vector-valued map $t\\mapsto B(t)$ in the following way:\n\t\t\t \\begin{equation*}\n\t\t\t t\\mapsto A+tB(t).\n\t\t\t \\end{equation*}\n\t\t\t Under certain assumptions on the perturbation controlled by $t$, we construct the associated perturbed spectral projection for which we obtain an approximation bound (cf.~\\cite[Lem.~5.3]{Becker.2021}). The key tools are the holomorphic functional calculus and the semicontinuity of the spectrum under \\textit{uniform perturbations}. The statements are summarized in \\Cref{prop:holomorphic-functional-calculus} and \\Cref{lem:semicontinuity-spectrum}.\n\t\t\t\\begin{lem}\\label{lem:quantitative-appro-riesz-projection}\n\t\t\t\tLet $A\\in\\cB(\\cX)$, $t\\mapsto B(t)$ be a vector-valued map on $\\cB(\\cX)$ which is uniformly continuous at $t=0$ with $\\sup_{t\\geq0}\\|B(t)\\|_\\infty\\leq b$, and $\\Gamma:[0,2\\pi]\\rightarrow\\rho(A)$ be a curve separating $\\sigma(A)$. Then, there exists an $\\epsilon>0$ so that for all $t\\in[0,\\epsilon]$\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\tP(t)\\coloneqq \\frac{1}{2\\pi i} \\oint_{\\Gamma}R(z,A+tB(t))dz\n\t\t\t\t\\end{equation*}\n\t\t\t\tdefines a projection with $\\norm{P(t)}_\\infty\\leq\\tfrac{d|\\Gamma|}{2\\pi}$ and derivative at $t=0$ given by\n\t\t\t\t\\begin{equation}\\label{eq:thm2-term1-derivative}\n\t\t\t\t\tP'\\coloneqq\\lim\\limits_{t\\rightarrow0}\\frac{P(t)-{P(0)}}{t}=\\frac{1}{2\\pi i}\\oint_{\\Gamma}R(z,A)B(0)R(z,A)dz.\n\t\t\t\t\\end{equation}\n\t\t\t\twith $\\|P'\\|_\\infty\\leq\\frac{R^2b|\\Gamma|}{2\\pi}$. The zeroth order approximation of $t\\mapsto P(t)$ can be controlled by\n\t\t\t\t\\begin{equation}\\label{eq:zeroth-order-approximation}\n\t\t\t\t \\|P(t)-P\\|_\\infty\\leq \\frac{tRbd|\\Gamma|}{2\\pi}\n\t\t\t\t\\end{equation}\n\t\t\t\tand the first order approximation by \n\t\t\t\t\\begin{equation}\\label{eq:first-order-approximation}\n\t\t\t\t\t\\norm{{P(t)-P-tP'}}_\\infty\\leq\\frac{tR^2|\\Gamma|}{2\\pi}\\left(tb^2d+\\norm{B(t)-B(0)}_\\infty\\right).\n\t\t\t\t\\end{equation}\n\t\t\t\tAbove $|\\Gamma|$ denotes the length of the curve $\\Gamma$, $P$ abbreviates the unperturbed spectral projection $P(0)$, $R\\coloneqq \\sup_{z\\in\\Gamma}\\|R(z,A)\\|_\\infty<\\infty$, and $d=R\\inf_{z\\in\\Gamma}\\frac{2+2|z|^2}{1+2|z|^2}$.\n\t\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tSince $t\\mapsto B(t)$ is uniformly continuous at $t=0$, the vector-valued map $t\\mapsto A+tB(t)$ is uniformly continuous as well. Then, \\Cref{lem:semicontinuity-spectrum} states that there exists an $\\epsilon>0$ such that $\\sigma(A+tB(t))$ is separated by $\\Gamma$ for all $t\\in[0,\\epsilon]$ and \\Cref{prop:holomorphic-functional-calculus} shows that \n\t\t\t\\begin{equation*}\n\t\t\t\tP(t)=\\frac{1}{2\\pi i}\\oint_{\\Gamma}R(z,A+tB(t))dz\n\t\t\t\\end{equation*}\n\t\t\tdefines a projection on $\\cX$ for all $t\\in[0,\\epsilon]$. Let $R\\coloneqq \\sup_{z\\in\\Gamma}\\|R(z,A)\\|_\\infty<\\infty$, then using the same steps as in \\Cref{eq:proofthm1-perturbed-resolvent-uniform-boundedness}, we have that for all $\\eta\\in\\Gamma$\n\t\t\t\\begin{equation}\\label{eq:proofthm2-perturbed-resolvent-uniform-boundedness}\n\t\t\t\t\\norm{R(\\eta,A+tB(t))}_\\infty\\leq R\\inf_{z\\in\\Gamma}\\frac{2+2|z|^2}{1+2|z|^2}\\eqqcolon d\\,.\n\t\t\t\\end{equation}\n\t Therefore, the perturbed resolvent is uniformly bounded. To prove the explicit representation of the derivative and the quantitative approximation, we follow the ideas of \\cite[Lem.~5.2-5.3]{Becker.2021}: \n\t\t\t\\begin{align*}\n\t\t\t\t\\frac{P(t)-P(0)}{t}&=\\frac{1}{t2\\pi i}\\left(\\oint_{\\Gamma}R(z,A+tB(t))dz-\\oint_{\\Gamma}R(z,A)dz\\right)\\\\\n\t\t\t\t&=\\frac{1}{2\\pi i}\\oint_{\\Gamma}R(z,A+tB(t))B(t)R(z,A)dz,\n\t\t\t\\end{align*}\n\t\t\twhich uses the \\textit{second resolvent identity}, i.e.~$R(z,A+tB(t))tB(t)R(z,A)=R(z,A)-R(z,A+tB(t))$ for all $z\\in\\Gamma$ and $t\\in[0,\\epsilon]$ and proves \\Cref{eq:thm2-term1-derivative} by Lebesgue's dominated convergence theorem \\cite[Thm.~3.7.9]{Hille.2000}. Moreover, the above equation proves \\Cref{eq:zeroth-order-approximation}. Finally,\n\t\t\t\\begin{equation}\\label{eq:prooflemapprox-constant}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{P(t)-P-tP'}_\\infty\n\t\t\t\t\t&\\leq \\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}(R(z,A+tB(t))-R(z,A))B(t)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}R(z,A)\\left(B(t)-B(0))\\right)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\leq \\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}R(z,A+tB(t))tB(t)R(z,A)B(t)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}R(z,A)\\left(B(t)-B(0))\\right)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\leq \\frac{tR^2|\\Gamma|}{2\\pi}\\left(tb^2d+\\norm{B(t)-B(0)}_\\infty\\right)\\\\\n\t\t\t\t\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\twhere $|\\Gamma|$ denotes the length of the curve $\\Gamma$.\n\t\t\\end{proof}\n\t\tNow, we are ready to prove \\Cref{thm:spectral-gap-uniform}.\n\t\t\n\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm2-term1}:}\n\t\t\\begin{lem}\\label{lem:proofthm2-term1} \n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction with the same assumption as in \\Cref{thm:spectral-gap-uniform} and $c_p\\coloneqq\\|\\1-P_\\Sigma\\|_\\infty$. Then, there is an $\\epsilon_1>0$ and $c_2\\geq0$ so that for all $t\\geq0$ and $n\\in\\N$ satisfying $t\\in[0,n\\epsilon_1]$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n}\\leq c_2\\tilde{\\delta}^n+\\frac{tb}{n}+\\frac{1}{n}\\frac{tbc_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2}\\,.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\n\t\t\\begin{proof}\n\t\t\tAs in the proof of \\Cref{prop:spectral-gap-uniform-norm-power-convergence}, \\Cref{prop:equivalence-spectral-gap} shows that the uniform power conver-\\linebreak\n\t\t\t\\begin{minipage}[c]{0.69\\textwidth}\n\t\t\t\tgence assumption (\\ref{unifpower}), that is $\\|M^n-\\sum_{j=1}^J\\lambda_j^{{ n}}P_j\\|_\\infty\\leq\\tilde{c}\\,\\delta^n$ for all $n\\in\\N$, is equivalent to the spectral gap assumption (q.v. \\Caref{sec:appendix-spectral-gap}),\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\sigma(M)\\subset\\D_\\delta\\cup\\{\\lambda_1,...,\\lambda_J\\},\n\t\t\t\t\\end{equation*}\n\t\t\t\twith corresponding quasinilpotent operators being zero. Therefore, the curve $\\gamma:[0,2\\pi]\\rightarrow\\C,\\varphi\\mapsto\\tilde{\\delta}e^{i\\varphi}$, with $\\tilde{\\delta}\\in (\\delta,1)$, encloses the spectrum of $M^\\perp\\coloneqq MP_\\Sigma^\\perp=P_\\Sigma^\\perp M$, where $P_\\Sigma=\\sum_{j=1}^JP_j$ and $P_\\Sigma^\\perp=\\1-P_\\Sigma$ (q.v.~\\Caref{fig1}). By \\Cref{lem:properties-semigroups}\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\norm{M^\\perp e^{s\\cL}-M^\\perp}_\\infty=s\\norm{M^\\perp\\cL\\int_{0}^{1}e^{\\tau s\\cL}d\\tau}_\\infty\\leq sc_{p}b\n\t\t\t\t\\end{equation*}\n\t\t\t with $c_{p}\\coloneqq\\|P_{\\Sigma}^\\perp\\|_\\infty$. Therefore, $M^\\perp e^{s\\cL}$ converges uniformly to $M^\\perp$ for $s\\downarrow0$. Hence, \\Cref{lem:semicontinuity-spectrum} shows that there exists an \\linebreak\\vspace{-2ex}\n\t\t\t\\end{minipage}\n\t\t\t\\begin{minipage}[c]{0.3\\textwidth}\n\t\t\t\t\\begin{center}\n\t\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\t\\draw[dashed,tumivory] (0,0) ellipse (1.4cm and 1.4cm);\n\t\t\t\t\t\t\\draw[dashed,mygreen] (0,0) ellipse (1.15cm and 1.15cm);\n\t\t\t\t\t\t\\filldraw (0,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (15:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (15:1.7) node {$\\lambda_1$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (70:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (70:1.7) node {$\\lambda_2$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (110:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (110:1.7) node {$\\lambda_3$};\n\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (130:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (130:1.7) node {$\\lambda_4$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (170:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (170:1.7) node {$\\lambda_5$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (195:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (195:1.7) node {$\\lambda_6$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (260:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (260:1.7) node {$\\lambda_7$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (300:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (300:1.7) node {$\\lambda_8$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (350:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (350:1.7) node {$\\lambda_J$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw [dotted,domain=310:340] plot ({1.7*cos(\\x)}, {1.7*sin(\\x)});\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw[<->] (0.0,0.05)--(0,0.95);\n\t\t\t\t\t\t\\draw[<->,mygreen] (0.0433,-0.025)--(0.9526cm,-0.55cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\fill[pattern=my north east lines] (0,0) ellipse (1cm and 1cm);\n\t\t\t\t\t\t\\filldraw[white] (-0.35,0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw (-0.35,0.5) node {$\\delta$};\n\t\t\t\t\t\t\\filldraw[white] (0.35,-0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw[mygreen] (0.35,-0.5) node {$\\tilde{\\delta}$};\n\t\t\t\t\t\t\\filldraw[white] (-1.06,-1.06) ellipse (0.15cm and 0.3cm);\n\t\t\t\t\t\t\\draw (-1.06,-1.06) node [mygreen]{$\\gamma$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw (1.5,1.2) node {$\\sigma(M)$};\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\\captionof{figure}{}\\label{fig1}\n\t\t\t\t\\end{center}\n\t\t\t\\end{minipage}\\\\\n\t\t\t$\\epsilon_1>0$ such that the spectrum of $M^\\perp e^{s\\cL}$ can be separated by $\\gamma$ for all $s\\in[0,\\epsilon_1]$. Therefore, we can apply the holomorphic functional calculus (\\Caref{prop:holomorphic-functional-calculus}) to conclude that for all $t\\in[0,n\\epsilon_1]$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k}=\\frac{1}{2\\pi i}\\oint_{\\gamma}z^{k}R(z,M^\\perp e^{\\frac{t}{n}})dz,\n\t\t\t\\end{equation*}\n\t\t\twhere $k\\in\\{1,..,n\\}$. By \\Cref{eq:proofthm1-perturbed-resolvent-uniform-boundedness} and with $c_2\\coloneqq\\sup_{z\\in\\gamma}\\|R(z,M^\\perp)\\|_\\infty \\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}$,\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k-1}}_\\infty\\leq\\frac{1}{2\\pi}\\oint_{\\gamma}\\abs{z}^{k-1}\\norm{R(z,M^\\perp e^{\\frac{t}{n}})}_\\infty dz\\leq c_2\\tilde{\\delta}^k.\n\t\t\t\\end{equation*}\n\t\t\tMoreover, by the assumptions $\\|P_\\Sigma\\|_\\infty=1$, $\\|M\\cL\\|_\\infty\\leq b$, $\\|\\cL P_\\Sigma\\|_\\infty\\leq b$, and \\Cref{lem:properties-semigroups}\n\t\t\t\\begin{equation*}\n\t\t\t \\norm{MP_\\Sigma e^{t\\cL}P_\\Sigma^\\perp M}_\\infty\\leq tb\\quad\\text{and}\\quad \\norm{MP_\\Sigma^\\perp e^{t\\cL}P_\\Sigma M}_\\infty\\leq tc_{p}b.\n\t\t\t\\end{equation*}\n\t\t\tBy the same expansion of $(Me^{\\frac{t}{n}\\cL})^n=(P_\\Sigma Me^{\\frac{t}{n}\\cL}+M^\\perp e^{\\frac{t}{n}\\cL})^n$ as in the proof of \\Cref{prop:spectral-gap-uniform-norm-power-convergence},\n\t\t\t\\begin{equation*}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}\\right)^n}_\\infty\\leq c_2\\tilde{\\delta}^n+\\frac{1}{n}\\frac{tbc_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2}.\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation*}\n\t\t\tFinally, \\Cref{lem:properties-semigroups} shows \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{P_\\Sigma Me^{\\frac{t}{n}\\cL}(\\1-P_\\Sigma)}_\\infty=\\frac{t}{n}\\norm{P_\\Sigma M\\cL\\int_{0}^{1} e^{\\tau\\frac{t}{n}\\cL}(\\1-P_\\Sigma)d\\tau}_\\infty\\leq\\frac{tbc_{p}}{n}\n\t\t\t\\end{equation*}\n\t\t\tso that\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n}\\leq c_2\\tilde{\\delta}^n+\\frac{tb}{n}+\\frac{1}{n}\\frac{tbc_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2},\n\t\t\t\\end{equation*}\n\t\t\twhich finishes the proof.\n\t\t\\end{proof}\n\t\t\n\t\t\n\t\\subsubsection*{Upper bound on \\texorpdfstring{\\Cref{eq:proofthm2-term2}}{Equation (2.1)}}\\label{subsec:proofthm2-term2}~\\\\\n\t\tAs in \\Cref{lem:proofthm1-term2}, we apply the modified Chernoff Lemma (\\Caref{lem:improved-chernoff}) to upper bound the second term (\\ref{eq:proofthm2-term2}). \n\t\tHowever, our proof strategy includes two crucial improvements compared to Theorem 3 in \\cite{Becker.2021}. Firstly, we show that the spectrum of the perturbed contraction is upper semicontinuous under certain assumptions on $M$ and the $C_0$-semigroup. Therefore, we can use the holomorphic functional calculus and apply the modified Chernoff Lemma with respect to each eigenvalue separately, which allows us to achieve the optimal convergence as in \\Cref{lem:proofthm1-term2}. \n\t\t\\begin{lem}\\label{lem:proofthm2-term2}\n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the same assumption as in \\Cref{thm:spectral-gap-uniform}. Then, there is an $\\epsilon_2>0$, and a $\\tilde{d}_1\\geq0$ so that for all $t\\geq0$ and $n\\in\\N$ satisfying $t\\in[0,n\\epsilon_2]$ \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\leq\\frac{J}{n}e^{t\\tilde{d}_1}\n\t\t\t\\end{equation*}\n\t\twhere $C_{j}(\\tfrac{1}{n})\\coloneqq\\bar{\\lambda}_jP_{j}(\\tfrac{1}{n})P_\\Sigma Me^{\\frac{1}{n}\\cL}P_\\Sigma P_{j}(\\tfrac{1}{n})$ and $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$.\n\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tBy \\Cref{prop:equivalence-spectral-gap}, the uniform power convergence (\\ref{unifpower}) shows that the $P_j$'s are the eigenprojections of $M$ so that $P_\\Sigma M=MP_\\Sigma=\\sum_{j=1}^{J}\\lambda_jP_j$ and the spectrum $\\sigma(P_\\Sigma M)$ consists of\\linebreak\n\t\t\t\\begin{minipage}[c]{0.69\\textwidth}\n\t\t\t $J$ isolated eigenvalues on the unit circle separated by the curves $\\Gamma_j:[0,2\\pi]\\rightarrow\\C,\\phi\\mapsto\\lambda_j+re^{i\\phi}$ (q.v.~\\Caref{sec:prelim} and \\Caref{fig2}) with radius\n\t\t\t\t\\begin{equation}\\label{eq:defr}\n\t\t\t\t\tr\\coloneqq\\min_{i\\neq j}\\left\\{\\frac{\\abs{\\lambda_i-\\lambda_j}}{3}\\right\\}.\n\t\t\t\t\\end{equation}\n\t\t\t\tNote that we use the curve interchangeably with its image and denote the formal sum of all curves around the eigenvalues $\\{\\lambda_j\\}_{j=1}^J$ by $\\Gamma$. In the following, we define the vector-valued function:\n\t\t\t\t\\begin{equation}\\label{eq:defB(t)}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\ts\\mapsto P_\\Sigma M+sB(s)&\\coloneqq P_\\Sigma M+sP_\\Sigma M\\cL\\int_0^1e^{s\\tau \\cL}P_\\Sigma d\\tau\\\\\n\t\t\t\t\t\t&\\;=P_\\Sigma Me^{s\\cL}P_\\Sigma\\,.\n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation}\n\t\t\t\\end{minipage}\n\t\t\t\\begin{minipage}[c]{0.3\\textwidth}\n\t\t\t\t\\begin{center}\n\t\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\t\\draw[dashed,tumivory] (0,0) ellipse (1.4cm and 1.4cm);\n\t\t\t\t\t\t\\draw[dashed,mygreen] (0,0) ellipse (1.15cm and 1.15cm);\n\t\t\t\t\t\t\\filldraw (0,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (15:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (15:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (15:1.9) node {$\\Gamma_1$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (70:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (70:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (70:1.9) node {$\\Gamma_2$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (110:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (110:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (110:1.9) node {$\\Gamma_3$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (130:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (130:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (130:1.9) node {$\\Gamma_4$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (170:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (170:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (170:1.9) node {$\\Gamma_5$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (195:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (195:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (195:1.9) node {$\\Gamma_6$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (260:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (260:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (260:1.9) node {$\\Gamma_7$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (300:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (300:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (300:1.9) node {$\\Gamma_8$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (350:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (350:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (350:1.9) node {$\\Gamma_J$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw [dotted,myorange,domain=310:340] plot ({1.9*cos(\\x)}, {1.9*sin(\\x)});\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw[<->] (0.0,0.05)--(0,0.95);\n\t\t\t\t\t\t\\draw[<->,mygreen] (0.0433,-0.025)--(0.9526cm,-0.55cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\fill[pattern=my north east lines] (0,0) ellipse (1cm and 1cm);\n\t\t\t\t\t\t\\filldraw[white] (-0.35,0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw (-0.35,0.5) node {$\\delta$};\n\t\t\t\t\t\t\\filldraw[white] (0.35,-0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw[mygreen] (0.35,-0.5) node {$\\tilde{\\delta}$};\n\t\t\t\t\t\t\\filldraw[white] (-1.06,-1.06) ellipse (0.15cm and 0.3cm);\n\t\t\t\t\t\t\\draw (-1.06,-1.06) node [mygreen]{$\\gamma$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw (1.5,1.2) node {$\\sigma(M)$};\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\\captionof{figure}{}\\label{fig2}\n\t\t\t\t\\end{center}\n\t\t\t\\end{minipage}\\\\[1ex]\n\t\t\tSince $M\\cL$ is bounded, the defined vector-valued map converges in the uniform topology to $P_\\Sigma M$. Moreover, \\Cref{lem:properties-semigroups} shows that $s\\mapsto B(s)$ is uniformly bounded and continuous in $s=0$ because\n\t\t\t\\begin{equation}\\label{eq:proofthm2-continuity-b}\n\t\t\t\t\\norm{B(s)-B(0)}_\\infty=s\\norm{P_\\Sigma M\\cL\\int_{0}^{1}\\int_0^{1}\\tau_1e^{\\tau_1\\tau_2s\\cL}\\cL P_\\Sigma d\\tau_2d\\tau_1}_\\infty\\leq s\\frac{b^2}{2}\n\t\t\t\\end{equation}\n\t\t\twhere we have used the assumption $\\|P_\\Sigma\\|_\\infty\\leq 1$. Then, we can apply \\Cref{lem:quantitative-appro-riesz-projection} which shows that there exists an $\\epsilon_2>0$ such that for all $s\\in[0,\\epsilon_2]$\n\t\t\t\\begin{equation*}\n\t\t\t\tP_j(s)\\coloneqq\\frac{1}{2\\pi i}\\oint_{\\Gamma_j}R(z,P_\\Sigma M+sB(s))dz\n\t\t\t\\end{equation*}\n\t\t\tdefines the perturbed spectral projection w.r.t.~$\\lambda_j$. Next, let $t\\geq0$, $n\\in\\N$ such that $t\\in[0,n\\epsilon_2]$. By \\Cref{lem:quantitative-appro-riesz-projection} and \\Cref{eq:proofthm2-continuity-b}, the perturbed spectral projection can be approximated by\n\t\t\t\\begin{align}\n\t\t\t \\norm{P_j(\\tfrac{t}{n})-P_j}_\\infty&\\leq\\frac{t}{n}R_jb\\left(d_j-\\frac{1}{2}\\right)r\\leq\\frac{t}{n}R_jbd_jr\\eqqcolon\\frac{t}{n}v_j \\label{eq:profthm2term2-approx-constant0}\\\\ \n\t\t\t \\norm{P_j(\\tfrac{t}{n})-P_j-\\tfrac{t}{n}P_j'}_\\infty&\\leq\\frac{t^2}{n^2}R_j^2b^2rd_j\\label{eq:profthm2term2-approx-constant}\n\t\t\t\\end{align}\n\t\t\twhere $R_j\\coloneqq\\sup_{z\\in\\Gamma_j}\\|R(z,P_\\Sigma M)\\|_\\infty$, $d_j\\coloneqq R_j\\inf_{z\\in\\Gamma_j}\\frac{2+2|z|^2}{1+2|z|^2}+\\frac{1}{2}$, and we use that $|\\Gamma_j|=2\\pi r$. Note that the defined $d_j$ is not exactly the $d$ in \\Cref{lem:quantitative-appro-riesz-projection}. Moreover, note that $\\|P_j(\\tfrac{t}{n})\\|_\\infty\\leq d_jr$ and $\\|P'_j\\|_\\infty\\leq R_j^2br$. By the spectral decomposition,\n\t\t\t\\begin{equation}\\label{eq:proof-thm2-spectral-decomp}\n\t\t\t\t\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma \\right)^n=\\sum_{j=1}^{J}\\left(P_j\\left(\\tfrac{t}{n}\\right)P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma P_j\\left(\\tfrac{t}{n}\\right)\\right)^n.\n\t\t\t\\end{equation}\n\t\t\tNext, we aim at applying the modified Chernoff lemma \\ref{lem:improved-chernoff} to $C_{j}(\\tfrac{t}{n})\\coloneqq\\bar{\\lambda}_jP_{j}(\\tfrac{t}{n})P_\\Sigma Me^{\\frac{1}{n}\\cL}P_\\Sigma P_{j}(\\tfrac{1}{n})$ for all $j\\in\\{1,..,J\\}$, which has to be adapted since it is no longer clear that $\\|C_j(\\tfrac{1}{n})\\|_\\infty=1$. We start by bounding the difference in norm between $C_t(\\tfrac{t}{n})$ and $P_j(\\tfrac{t}{n})$. By the fundamental theorem of calculus and the facts $\\|P_j(\\tfrac{t}{n})\\|_\\infty\\leq d_jr$, $\\|P_{\\Sigma}M\\cL\\|_\\infty\\leq b$, $\\|e^{s\\cL}\\|_\\infty\\leq 1$, $|\\lambda_j|=1$\n\t\t\t\\begin{equation}\\label{eq:proofthm2term2-approx1}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{C_{j}(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty&\\leq\\frac{t}{n}\\norm{\\bar{\\lambda}_jP_j(\\tfrac{t}{n})P_\\Sigma M\\cL\\int_{0}^{1}e^{\\frac{st}{n}\\cL}P_\\Sigma P_j(\\tfrac{t}{n})ds}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\norm{P_j(\\tfrac{t}{n})P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t\t&\\leq\\frac{t}{n}bd_j^2r^2+\\norm{\\bar{\\lambda}_jP_j(\\tfrac{t}{n})P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty.\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\tIn the next step, we focus on the second term and prove a higher order approximation then needed because in \\Cref{lem:proofthm2-term3} we will reuse this calculation. In the following calculation, we use the bounds from above, in particular \\Cref{eq:profthm2term2-approx-constant0} and (\\ref{eq:profthm2term2-approx-constant}). Moreover, we use the product rule for derivatives, which shows $P_j'=P_jP_j'+P_j'P_j$ by $\\tfrac{\\partial}{\\partial s}P_j(s)=\\tfrac{\\partial}{\\partial s}P_j(s)^2$ (cf.~\\cite[Lem.~3]{Mobus.2019}).\n\t\t\t\\begin{equation}\\label{eq:proofthm2term2-approx2}\n\t\t\t \\begin{aligned}\n\t\t\t\t\t&\\norm{\\bar{\\lambda}_jP_j(\\tfrac{t}{n})P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t\t&\\qquad=\\norm{\\left(P_j(\\tfrac{t}{n})-P_j-\\tfrac{t}{n}P_j'\\right)P_\\Sigma MP_j(\\tfrac{t}{n})}_\\infty+\\norm{P_jP_j(\\tfrac{t}{n})+\\frac{t}{n}P_j'P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t\t&\\qquad\\leq\\begin{aligned}[t]\n\t\t\t\t\t \\frac{t^2}{n^2}R_j^2b^2r^2d_j^2&+\\norm{P_j\\left(P_j(\\tfrac{t}{n})-P_j-\\frac{t}{n}P_j'\\right)}_\\infty+\\frac{t}{n}\\norm{P_j'P_\\Sigma M\\left(P_j(\\tfrac{t}{n})-P_j\\right)}_\\infty\\\\\n\t\t\t\t\t &+\\norm{P_j+\\frac{t}{n}\\left(P_jP_j'+P_j'P_j\\right)-P_j(\\tfrac{t}{n})}_\\infty\n\t\t\t\t\t\\end{aligned}\\\\\n\t\t\t\t\t&\\qquad\\leq\\frac{t^2}{n^2}R_j^2b^2r^2d_j^2+2\\frac{t^2}{n^2}R_j^2b^2rd_j+\\frac{t^2}{n^2}R_j^3b^2d_jr^2\\\\\n\t\t\t\t\t&\\qquad\\leq\\frac{t^2}{n^2}R_j^2b^2d_jr\\left(d_jr+R_jr+2\\right)\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\tCombining \\Cref{eq:proofthm2term2-approx1}, (\\ref{eq:proofthm2term2-approx2}), and $\\tfrac{t}{n}\\leq\\epsilon_2$ shows\n\t\t\t\\begin{equation}\\label{eq:proofthm2term2-approx}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{C_{j}(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty&\\leq\\frac{t}{n}bd_j^2r^2+\\frac{t^2}{n^2}R_j^2b^2d_jr\\left(d_jr+R_jr+2\\right)\\\\\n\t\t\t\t\t&\\leq\\frac{t}{n}bd_jr\\left(d_jr+\\epsilon_2R_j^2b\\left(d_jr+R_jr+2\\right)\\right)\\eqqcolon\\frac{t}{n}w_j\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\tIn \\Cref{eq:profthm2term2-approx-constant0} and (\\ref{eq:proof-thm2-chernoff}), we have proven that $\\|P_j(\\tfrac{t}{n})-P_j\\|\\leq\\tfrac{t}{n}v_j$, $\\|C_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})\\|\\leq\\tfrac{t}{n}w_j$ and note that $\\|P_j\\|_\\infty=1$, $P_j(\\tfrac{t}{n})C_j(\\tfrac{t}{n})=C_j(\\tfrac{t}{n})P_j(\\tfrac{t}{n})=C_j(\\tfrac{t}{n})$ holds by definition. Then we can apply the approximate version of the modified Chernoff \\Cref{lem:approx-improved-chernoff} to $C_j(\\tfrac{t}{n})$. This shows\n\t\t\t\\begin{equation}\\label{eq:proof-thm2-chernoff}\n\t\t\t \\norm{C_j(\\tfrac{t}{n})^n-e^{n(C_j(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_j(\\text{\\tiny{$\\tfrac{t}{n}$}}))}P_j(\\tfrac{t}{n})}_\\infty\\leq\\frac{t^2w_j^2}{2n}e^{t(v_j+w_j)}\\leq\\frac{1}{n}e^{t(v_j+2w_j)}.\n\t\t\t\\end{equation}\n\t\t\tCombining \\Cref{eq:proof-thm2-spectral-decomp} and (\\ref{eq:proof-thm2-chernoff}) and writing out the constants $v_j$ and $w_j$ gives\n\t\t\t\\begin{align*}\n\t\t\t\t&\\norm{\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t&\\hspace{10ex}\\leq J\\max\\limits_{j\\in\\{1,..,J\\}}\\norm{\\lambda_j^n\\left(C_j(\\tfrac{t}{n})\\right)^n-\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t&\\hspace{10ex}\\leq \\frac{J}{n}e^{t(v_j+2w_j)}\\\\\n\t\t\t\\end{align*}\n\t\t\tFinally, we define $\\tilde{d}_1=\\max\\limits_{j\\in\\{1,...,J\\}}v_j+2w_j$, which finishes the proof.\n\t\t\\end{proof}\n\t\t\n\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm2-term3}:}\n\t\t\\begin{lem}\\label{lem:proofthm2-term3}\n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction with the same assumption as in \\Cref{thm:spectral-gap-uniform}. Then, there exists a constant $\\tilde{d}_2\\geq0$ such that \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\leq\\frac{J}{n}e^{t\\tilde{d}_2}\\max\\limits_{s\\in[0,1]}\\norm{e^{stP_j\\cL P_j}}_\\infty.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tFor ease of notation, we absorb the time parameter $t$ into the generator $\\cL$ and $b$. In order to prove the convergence of the generator, \\Cref{eq:proofthm2term2-approx2} proves:\n\t\t\t\\begin{equation*}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{n\\left(C_{j}(\\tfrac{1}{n})-P_j(\\tfrac{1}{n})\\right)-P_j\\cL P_j}_\\infty&\\leq\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL\\int_{0}^{1}e^{\\frac{s}{n}\\cL}P_\\Sigma P_j(\\tfrac{1}{n})ds-P_j\\cL P_j}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\frac{1}{n}R_j^2b^2d_jr\\left(d_jr+R_jr+2\\right),\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation*}\n\t\twhere $R_j\\coloneqq\\sup_{z\\in\\Gamma_j}\\|R(z,P_\\Sigma M)\\|_\\infty$, $d_j\\coloneqq R_j\\inf_{z\\in\\Gamma_j}\\frac{2+2|z|^2}{1+2|z|^2}+\\frac{1}{2}$, and $r$ is the radius of the curves $\\Gamma_j$ defined in \\Cref{eq:defr}. Then, we apply \\Cref{lem:properties-semigroups} and \\Cref{lem:quantitative-appro-riesz-projection} on the first term:\n\t\t\t\\begin{align*}\n\t\t\t\t&\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P_\\Sigma P_j(\\tfrac{1}{n})d\\tau_1- P_j\\cL P_j}_\\infty\\\\\n\t\t\t &\\qquad \\leq\\frac{1}{n}\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL\\int_{0}^{1}\\int_{0}^{1}\\tau_1e^{\\frac{\\tau_1\\tau_2}{n}\\cL}\\cL P_\\Sigma P_j(\\tfrac{1}{n})d\\tau_2d\\tau_1}_\\infty\\\\\t\t&\\qquad\\quad+\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL P_\\Sigma P_j(\\tfrac{1}{n})-P_j\\cL P_j}_\\infty\\\\\n\t\t\t &\\qquad\\leq\\frac{1}{2n}b^2d_j^2r^2+\\norm{\\left(P_j(\\tfrac{1}{n})-P_j\\right)P_\\Sigma M\\cL P_\\Sigma P_j(\\tfrac{1}{n})}_\\infty+\\norm{P_j\\cL \\left(P_j(\\tfrac{1}{n})-P_j\\right)}\\\\\n\t\t\t &\\qquad\\leq\\frac{1}{n}b^2d_jr\\left(\\frac{1}{2}d_jr+R_jd_jr+R_j\\right)\\,,\n\t\t\t\\end{align*}\n\t\t\twhere we used \\Cref{eq:profthm2term2-approx-constant0} in the last step and the assumption that $M\\cL$ and $\\cL P_{\\Sigma}$ are bounded by $b$ and all the inequalities discussed before \\Cref{eq:proofthm2term2-approx}. In combination with \\Cref{lem:integral-equation-semigroups} \n\t\t\t\\begin{align*}\n\t\t\t\t&\\norm{e^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})\\right)}-e^{P_j\\cL P_j}}_\\infty\\\\\n\t\t\t\t&\\qquad\\leq\\max\\limits_{s\\in[0,1]}\\norm{e^{sP_j\\cL P_j}}_\\infty\\norm{e^{(1-s)n\\left(C_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})\\right)}}_\\infty\\norm{n\\left(C_{j}(\\tfrac{1}{n})-P_j(\\tfrac{1}{n})\\right)-P_j\\cL P_j}_\\infty\\\\\n\t\t\t\t&\\qquad\\leq\\frac{1}{n}\\max\\limits_{s\\in[0,1]}\\norm{e^{sP_j\\cL P_j}}_\\infty e^{w_j}b^2d_jr\\left(R_j^2\\left(d_jr+R_jr+2\\right)+\\frac{1}{2}d_jr+R_jd_jr+R_j\\right)\n\t\t\t\\end{align*}\n\t\t\tfor all $j\\in\\{1,..J\\}$ and where $w_j$ is defined in \\Cref{eq:proofthm2term2-approx}. With one more application of \\Cref{eq:profthm2term2-approx-constant0}, this shows\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})\\right)}P_{j}(\\tfrac{1}{n})-\\sum_{j=1}^{J}\\lambda_j^ne^{P_j\\cL P_j}P_j}_\\infty\\leq\\frac{J}{n}e^{t\\tilde{d}_2}\\max\\limits_{s\\in[0,1]}\\norm{e^{stP_j\\cL P_j}}_\\infty\n\t\t\t\\end{equation*}\n\t\t\twhere we choose $\\tilde{d}_2\\geq0$ appropriately and redefine $\\cL$ by $t\\cL$ and $b$ by $bt$.\n\t\t\\end{proof}\n\t\n\t\t\\subsubsection*{End of the Proof of \\Cref{thm:spectral-gap-uniform}:}\n\t\tFinally, we combine the upper bounds found in the lemmas in order to finish the proof of \\Cref{thm:spectral-gap-uniform}.\n\t\t\\begin{proof}[Proof of \\Cref{thm:spectral-gap-uniform}]\n\t\t\t\\Cref{lem:proofthm2-term1}, \\ref{lem:proofthm2-term2}, and \\ref{lem:proofthm2-term3} show for all $t\\in[0,n\\epsilon]$ with $\\epsilon\\coloneqq \\min\\{\\epsilon_1,\\epsilon_2\\}$\n\t\t\t\\begin{flalign*}\n\t\t\t\t&&\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty&\\leq c_2\\tilde{\\delta}^n+\\frac{tb}{n}+\\frac{tb}{n}\\frac{c_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2}&&\\text{(\\Caref{lem:proofthm2-term1})}\\\\\n\t\t\t\t&& &\\quad+\\frac{J}{n}e^{t\\tilde{d}_1}&&\\text{(\\Caref{lem:proofthm2-term2})}\\\\\n\t\t\t\t&& &\\quad+\\frac{J}{n}e^{t\\tilde{d}_2}\\max\\limits_{s\\in[0,1]}\\norm{e^{stP_j\\cL P_j}}_\\infty&&\\text{(\\Caref{lem:proofthm2-term3})}.\n\t\t\t\\end{flalign*}\n\t\t\tFor an appropriate constant $c_1\\geq0$, we finish the proof of \\Cref{thm:spectral-gap-uniform}.\n\t\t\\end{proof}\n\t\n\t\n\t\\section{Examples}\\label{sec:applications}\n\t\tIn this section, we present two classes of examples, which illustrate the range of applicability of our results. In the examples, we denote by $\\rho,\\sigma$ quantum states. \n\t\t\\begin{ex}[Finite dimensional quantum systems]\\label{ex:finite-dim}\n\t\t\tWe choose $\\cX=\\cB(\\cH)$ to be the algebra of linear operators over a finite dimensional Hilbert space $\\cH$ endowed with the trace norm $\\|x\\|_1=\\tr|x|$, $M:\\cB(\\cH)\\to \\cB(\\cH)$ a quantum channel, i.e.~a completely positive, trace preserving linear map, and $\\cL$ the generator of a semigroup of quantum channels over $\\cB(\\cH)$, also known as a quantum dynamical semigroup. In finite dimension, it is know that every quantum channel is a contraction \\cite[Cor.~3.40]{Watrous.2018}, the spectrum includes the eigenvalue $1$ \\cite[Thm.~3]{Wolf.2010}, and every linear operator in finite dimension has a discrete spectrum. Moreover, the nilpotent part of a quantum channel is zero \\cite[Lem.~A.1]{Hasenohrl.2020}. Therefore, there exist $\\delta\\in(0,1)$, $\\tilde{c}>0$, and a set of eigenvalues and projections $\\{\\lambda_j,P_j\\}_{j=1}^J$ such that for all $n\\in\\N$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^nx-\\sum_{j=1}^J\\lambda_jP_jx}_1\\leq\\tilde{c}\\delta^n\\norm{x}_1\n\t\t\t\\end{equation*}\n\t\t Note that the assumptions on the semigroup are satisfied due to the finiteness of the system and the contraction property of the $P_j$ must be assumed additionally.\n\t\t\\end{ex}\n\t\tIn the following example class, we calculate $\\delta$ directly.\n\t\t\\begin{ex}[{Power convergence via strong data processing inequalities}]\n\t\t\t{As in \\Cref{ex:finite-dim}, we define $\\cX=\\cB(\\cH)$ endowed with the trace norm $\\|x\\|_1$, $M:\\cB(\\cH)\\to \\cB(\\cH)$ a quantum channel, and $\\cL$ the generator of a quantum dynamical semigroup.}\n\t\t\tHere, we further assume the existence of a projection $P:\\cB(\\cH)\\to \\mathcal{N}$ onto a subalgebra $\\mathcal{N}\\subset \\cB(\\cH)$ with $MP=PM$ and such that the following \\textit{strong data processing inequality} holds for some $\\hat{\\delta}\\in(0,1)$: for all states $\\rho\\in\\cX$,\n\t\t\t\\begin{align}\\label{SDPI}\n\t\t\t\tD(M(\\rho)\\|M\\circ P(\\rho))\\le \\hat{\\delta}\\,D(\\rho\\|P(\\rho))\\,,\n\t\t\t\\end{align}\n\t\t\twhere we recall that the relative entropy between two quantum states, i.e.~positive, trace-one operators on $\\cH$, is defined as $D(\\rho\\|\\sigma):=\\tr[\\rho\\log\\rho-\\rho\\log\\sigma]$, whenever $\\operatorname{supp}(\\rho)\\subseteq\\operatorname{supp}(\\sigma)$. Equation \\eqref{SDPI} was recently shown to hold under a certain detailed balance condition for $M$ in \\cite{gao2021complete}: there exists a full-rank state $\\sigma$ such that for any two $x,y\\in\\cB(\\cH)$, $$\\tr[\\sigma\\,x^*M^*(y)]=\\tr[\\sigma\\,M^*(x^*)y].$$ Here $x^*$, resp. $M^*$, denotes the adjoint of $x$ w.r.t.~the inner product on $\\cH$, resp. the adjoint of $M$ w.r.t. the Hilbert-Schmidt inner product on $\\cB(\\cH)$. In finite dimensions, the quantity $\\sup_\\rho\\,D(\\rho\\|P(\\rho))<\\infty$ is called the Pimsner-Popa index of $P$ \\cite{pimsner1986entropy}. Using Pinsker's inequality, we see that the assumption of \\Cref{thm:spectral-gap-uniform} is satisfied: for all $x=x^*\\in\\cB(\\cH)$ with $\\|x\\|_1\\le 1$ and decomposition $x=x_+-x_-$ into positive and negative parts and corresponding states $\\rho_{\\pm}=x_{\\pm}\/\\tr[x_{\\pm}]$,\n\t\t\t\\begin{align*}\n\t\t\t\t\\|(M^n-M^n\\circ P)(x)\\|_1& \\le \\tr[x_+]\\,\\|(M^n-M^n\\circ P)(\\rho_+)\\|_1+ \\tr[x_-]\\,\\|(M^n-M^n\\circ P)(\\rho_-)\\|_1\\\\\n\t\t\t\t&= \\|x\\|_1\\,\\sqrt{2}\\,\\max_{\\rho\\in\\{\\rho_+,\\rho_-\\}}D(M^n(\\rho)\\|M^n\\circ P(\\rho))^{\\frac{1}{2}}\\\\\n\t\t\t\t&\\le \\sqrt{2}\\,\\sup_\\rho\\,D(\\rho\\|P(\\rho))^{\\frac{1}{2}}\\,\\hat{\\delta}^{\\frac{n}{2}}\\eqqcolon \\tilde{c}\\,\\delta^n\\,.\n\t\t\t\\end{align*}\n\t\t\t{Then, we can apply \\Cref{cor:explicit-bound-prop} which proves that there is an $\\epsilon>0$ such that for all $n\\in\\N$, $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty=\\cO\\left(\\frac{e^{t\\|\\cL\\|_\\infty}}{n}+\\frac{\\tilde{\\delta}}{n}e^{\\frac{8\\tilde{c}t\\norm{\\cL}_\\infty}{\\tilde{\\delta}-\\delta}}\\right)\n\t\t\t\\end{equation*}\n\t\t\tfor $n\\rightarrow\\infty$.}\n\t\t\\end{ex}\n\t\t\\begin{ex}[Infinite dimensional quantum systems and unbounded generators]\n\t\t\tHere, we pick $\\cH=L^2(\\mathbb{R})$, denote by $I$ the identity operator on $\\cH$, let $\\sigma$ be a quantum state on $\\cH$ and $M$ a generalized depolarizing channel of depolarizing parameter $p\\in (\\frac{1}{2},1)$ and fixed point $\\sigma$:\n\t\t\t\\begin{align}\n\t\t\t\tM(\\rho):=(1-p)\\rho +p\\tr(\\rho)\\,\\sigma\\,.\n\t\t\t\\end{align}\n\t\t\tIt is clear by convexity that $M$ satisfies the uniform strong power convergence with projection $P(\\rho)=\\tr(\\rho)\\,\\sigma$ and parameter $\\delta=2(1-p)<1$,\n\t\t\t\\begin{equation*}\n\t\t\t \\norm{M(\\rho)-P(\\rho)}_1=(1-p)\\norm{\\rho -\\tr(\\rho)\\,\\sigma}_1\\leq2(1-p)\\norm{\\rho}_1.\n\t\t\t\\end{equation*}\n\t\t\tLet $\\cL$ be a generator of a $C_0$-contraction semigroup such that $\\sigma\\in\\cD(\\cL)$. For example, let $\\cH$ be the Fock-space spanned by the Fock basis $\\{\\ket{0},\\ket{1},\\ket{2},...\\}$, $a$ and $a^\\dagger$ be the annihilation and creation operator defined by $a\\ket{0}=0$, $a\\ket{j}=\\sqrt{j}\\ket{j-1}$ for all $j\\in\\N_{\\geq 1}$, and $a^\\dagger\\ket{j}=\\sqrt{j+1}\\ket{j+1}$ for all $j\\in\\N_{\\geq0}$. Then, define $e^{t\\cL}(\\rho)\\coloneqq e^{-itH}\\rho e^{itH}$ where $H=a^\\dagger a+\\tfrac{1}{2}$ ($H=-\\Delta+x^2$) is the Hamiltonian of the harmonic oscillator as in \\cite{Becker.2021} and \n\t\t\t\\begin{equation*}\n\t\t\t \\sigma=\\frac{1}{3}(\\ket{0}\\bra{0}+\\ket{1}\\bra{1}+\\ket{2}\\bra{2})+\\frac{1}{10}(\\ket{0}\\bra{1}+\\ket{1}\\bra{0}).\n\t\t\t\\end{equation*}\n\t\t\tThen, we have that for all $t\\ge 0$: \n\t\t\t\\begin{align}\\label{upperbound1}\n\t\t\t\t\\|(\\1-P)e^{t\\cL}P\\|_{1\\to 1}=\\sup_{\\|x\\|_1\\le 1}|\\tr(x)|\\,\\|e^{t\\cL}(\\sigma)-\\sigma\\|_1\\le t\\,\\|\\cL(\\sigma)\\|_1\\,.\n\t\t\t\\end{align}\n\t\t\tMoreover, by duality and the unitality of the maps $e^{t\\cL^*}$ we have that\n\t\t\t\\begin{align}\\label{upperbound2}\n\t\t\t\t\\|Pe^{t\\cL}(\\1-P)\\|_{1\\to 1}=\\|(\\1-P)^*e^{t\\cL^*}P^*\\|_{\\infty\\to\\infty}=\\sup_{\\|y\\|\\le 1}|\\tr(\\sigma y)|\\|(\\1-P)(I)\\|=0\\,,\n\t\t\t\\end{align}\n\t\t\tTherefore, the assumptions of \\Cref{thm:spectral-gap} are satisfied and we find the convergence rate $\\mathcal{O}(n^{-1})$. Interestingly, this answers a conjecture of \\cite[Ex.~3, 5]{Becker.2021} for the Hamiltonian evolution generated by the one-dimensional harmonic oscillator. There, the authors had numerically guessed the optimal rate which we prove here. However their analytic bounds could only provide a decay of order $\\mathcal{O}(n^{-\\frac{1}{4}})$ (q.v.~remark after \\Caref{lem:chernoff}) and for a restriction of $H$ to a finite dimensional stable subspace, which effectively assumed the boundedness of the generator.\n\t\t\\end{ex}\n The depolarizing noise considered in the previous example is artificial. In an infinite dimensional bosonic system, a more natural model of noise is the photon loss channel, which we consider in the next example.\n\n\n\t\t\\begin{ex}[Bosonic beam-splitter]\n We define the bosonic one-mode system by the algebra generated by the creation and annihilation operators $a^*$ and $a$ which satisfy the canonical commutation relation (CCR):\n\t\t\t\\begin{equation*}\n\t\t\t\t[a,a^*]=\\1\\,.\n\t\t\t\\end{equation*}\n\t\t\tThe associated Fock basis $\\{\\ket{0},\\ket{1},\\ket{2},...\\}$ is orthonormal and defined by \n\t\t\t\\begin{equation*}\n\t\t\ta^*\\ket{j}=\\sqrt{j+1}\\ket{j+1}\\quad\\text{and}\\quad a\\ket{j}=\\sqrt{j}\\ket{j-1}\n\t\t\t\\end{equation*}\n\t\t\twhere the vacuum state $\\ket{0}$ satisfies $a\\ket{0}=0$. The Fock basis spans a Hilbert space called Fock space on which the operators from the CCR algebra are defined. A bosonic quantum state is a semidefinite operator in the CCR algebra with trace 1. \n\t\t\tA bosonic $2$-mode system is defined by the CCR-algebra generated by $\\{a,b,a^*,b^*\\}$, which satisfy, additionally to the canonical commutation relation, $[a,b]=0$. Next, we consider the \\textit{quantum beam-splitter} for $\\lambda\\in [0,1)$ \n\t\t\t\\begin{equation*}\n\t\t\t\tM_\\lambda(y)\\coloneqq\\tr_2[U_\\lambda y\\otimes\\sigma U_\\lambda^*] \\,,\n\t\t\t\\end{equation*}\n\t\t\twhere $\\tr_2$ denotes the partial trace over the second register, $U_\\lambda\\coloneqq e^{(a^* b-b^* a)\\operatorname{arcos}(\\sqrt{\\lambda})}$, an environment state $\\sigma$, and $y$ an element in the CCR algebra generated by $\\{a^*, a\\}$. Moreover, $P(y)\\coloneqq\\tr[y]\\sigma$ defines a projection which satisfies $PM_\\lambda=M_\\lambda P=P$ with the adjoint $P^*(x)=\\tr[\\sigma x]\\1$.\n \n In order to establish e.g.~the uniform power convergence of \\Cref{thm:spectral-gap} in the topology of the trace distance, we would need to consider a convergence in the form of $\\|M_\\lambda^n(\\rho)-\\sigma\\|_1\\to 0$ in the limit of large $n$ and uniformly in the initial state $\\rho$. Such property is notoriously hard to prove even in the classical setting \\cite{PW16}. Instead, we will consider a different metric on the set of quantum states which turns out to be more easy to work with.\n\t\t\t\n\t\t\tWe write $\\cB_{{N}}$ for the linear space of all ${N}$-bounded operators, where $ N= a^\\dagger a$ corresponds to the photon number operator. That is the vector space of linear operators $X$ on $L^2(\\mathbb{R})$ such that for any $|\\psi\\rangle\\in \\operatorname{dom}(N)$, $|\\psi\\rangle\\in\\operatorname{dom}(X)$ and there are some positive constants $a,b$ such that\n\t\t\t\\begin{align*}\n\t\t\t \\|X|\\psi\\rangle\\|\\le a\\|N|\\psi\\rangle\\|+b\\|\\psi\\|\\,.\n\t\t\t\\end{align*}\n\t\t\t We define the \\textit{Bosonic Lipschitz constant} of a $X\\in \\cB_{{N}}$ as \\cite{Cambyse.2021}\n \\begin{align}\n \\|\\nabla X\\|^2 := \\sup_{|\\psi\\rangle,|\\varphi\\rangle}\\,|\\langle \\psi|[a,X]|\\varphi\\rangle |^2+|\\langle \\psi|[a^*,X]|\\varphi\\rangle|^2\\,,\\nonumber\n \\end{align}\n where the suppremum is over all pure states $|\\psi\\rangle,|\\varphi\\rangle\\in\\operatorname{dom}({N})$ of norm $1$. By duality, we then define the \\textit{Bosonic Wasserstein norm} of a linear functional $f$ over $\\mathcal{B}_N$ with $f(\\1)=0$ as\n \\begin{align*}\n \\|f\\|_{W_1}:=\\sup_{\\|\\nabla X\\|\\le 1}\\,\\big|f(X)\\big|\\,.\n \\end{align*}\n where the supremum is over all ${N}$-bounded, self-adjoint operators $X$. We then choose our Banach space $\\cX$ as the closure of the set of such linear functionals such that $\\|f\\|_{W_1}<\\infty$. In particular, whenever $f\\equiv f_{\\rho-\\sigma}$ is defined in terms of the difference between two quantum states $\\rho,\\sigma$ as $f_{\\rho-\\sigma}(X)=\\tr((\\rho-\\sigma) X)$, we denote the Wasserstein distance associated to the norm $\\|.\\|_{W_1}$ as (see also \\cite{Cambyse.2021}):\n \\begin{align*}\n W_1(\\rho,\\sigma):=\\|f_{\\rho-\\sigma}\\|_{W_1}\\,.\n \\end{align*}\t\t\t\n\t\t\tThese definitions extend the classical Lipschitz constant $\\|\\nabla f\\|:=\\sup_{x\\in\\mathbb{R}^2}|\\nabla f(x)|$ of a real, continuously differentiable function $f$ of $2$ variables as well as the dual Wasserstein distance over probability measures on $\\mathbb{R}^2$. \n\t\t\t\n\t\t\tIn order to relate the Wasserstein distance to the statistically more meaningful trace distance, we seek for an upper bound on the trace distance in terms of $W_{1}$. By duality of both metrics, this amounts to finding an upper bound on the Lipschitz constant $\\|\\nabla X\\|$ of any bounded operator $X$ in terms of its operator norm $\\|X\\|_\\infty$. However, a bound of that sort does not exist (as classically, one can easily think of bounded observables which are not \\textit{smooth}). In the classical setting, the problem can be handled by first \\textit{smoothing} the function $f$, e.g. by convolving it with a Gaussian density $g$. In that case, one proves that there exists a finite constant $C>0$ such that $\\|\\nabla (f\\ast g)\\|\\le C\\|f\\|_\\infty$. In analogy with the classical setting, we can prove that for any two states $\\rho_1,\\rho_2$ and $\\lambda\\in [0,1)$ (see also \\cite[Proposition 6.4]{Cambyse.2021}),\n\t\t\t\\begin{align}\\label{tracetoWass}\n\t\t\t \\|M_\\lambda(\\rho_1-\\rho_2)\\|_1\\le \\,C\\, W_{1}(\\rho_1,\\rho_2)\\,,\n\t\t\t\\end{align}\n\t\t\twhere $C^2:=(\\|[a,\\sigma]\\|_1^2+\\|[a^*,\\sigma]\\|_1^2)\\lambda {(1-\\lambda)^{-1}}$.\n\t\t\t\n With a slight abuse of notations, we also write $M_\\lambda(f)$ for $f\\circ \\mathcal{B}^*_\\lambda$. It remains to prove the uniform power convergence. Proposition 6.2 from \\cite{Cambyse.2021} gives \n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{M_\\lambda(f)}_{W_1}&=\\sup_{\\|\\nabla X\\|\\leq1}\\abs{f\\circ M_\\lambda^*(X)}\\\\\n\t\t\t\t&=\\sup_{\\|\\nabla X\\|\\leq1}\\abs{f\\left(\\frac{M_\\lambda^*(X)}{\\|\\nabla M_\\lambda^*(X)\\|}\\right)}\\|\\nabla M_\\lambda^*(X)\\|\\\\\n\t\t\t\t&=\\sup_{\\|\\nabla X\\|\\leq1}\\abs{f(X)}\\sqrt{\\lambda}\\\\\n\t\t\t\t&=\\sqrt{\\lambda}\\,\\|f\\|_{W_1}\n\t\t\t\t\\end{align*}\n\t\t\t\tThe uniform power convergence follows by $P(f)(X)\\equiv f\\circ P^*(X)=\\tr(\\sigma X)f(\\1)=0$. Moreover, the asymptotic Zeno condition (\\ref{eq:thm1-asympzeno}) is satisfied if $\\sigma\\in\\cD(\\cL)$ so that \\Cref{thm:spectral-gap} is applicable.\n\t\t\\end{ex}\n\t\tAs illustrated here, our asymptotic Zeno condition is easily verifiable and provides a rich class of examples. More examples for which our optimal convergence rate holds can be found in \\cite{Becker.2021}.\n\t\t\n\t\t\n\t\\section{Discussion and Open Questions}\\label{sec:discussion}\n\t\tIn this paper, we proved the optimal convergence rate of the quantum Zeno effect in two results: \\Cref{thm:spectral-gap} focuses on weakening the assumptions of the $C_0$-semigroup to the so-called asymptotic Zeno condition. Hence, \\Cref{thm:spectral-gap} allows strongly continuous Zeno dynamics which is novel for open systems. In \\Cref{thm:spectral-gap-uniform} instead, we weaken the assumption on $M$ to the uniform power convergence as in \\cite[Thm.~3]{Becker.2021}. Additionally, we presented an example which shows the optimality of the achieved convergence rate. This brings up the question whether our assumptions are optimal and how the assumption on the contraction correlates with the assumption on the $C_0$-semigroup. For example, is it possible to weaken the uniform power convergence in \\Cref{thm:spectral-gap} or \\Cref{prop:spectral-gap-uniform-norm-power-convergence} to finitely many eigenvalues on the unit circle without assuming stronger assumption on the semigroup? Following our proof strategy (q.v.~\\Caref{lem:proofthm1-term1}), this question is related to the conjecture of a generalized version of \\textit{Trotter's product formula for finitely many projections} under certain assumptions on the generator,\n\t\t\\begin{equation*}\n\t\t\t\\norm{\\left(\\sum_{j=1}^{J}\\lambda_jP_je^{\\frac{1}{n}\\cL}\\right)^nx-\\sum_{j=1}^{J}\\left(\\lambda_jP_je^{\\frac{1}{n}\\cL}\\right)^nx}\\overset{?}{=}\\cO\\left(\\frac{1}{n}(\\norm{x}+\\norm{\\cL x}+\\|\\cL^2 x\\|)\\right),\n\t\t\\end{equation*}\t\t\n\t\tAnother line of generalization would be to weaken the assumption on $M$ to the strong topology as in Theorem 2 in \\cite{Becker.2021}. There, the authors assume that $M^n$ converges to $P$ in the strong topology and that the semigroup is uniformly continuous. It would be interesting to know whether an extension to $C_0$-semigroups is possible. \n\t\tFinally, another important line of generalization would be to extend our results to time-dependent semigroups as in \\cite{Mobus.2019}.\n\n\n\t\\emph{Acknowledgments:} We would like to thank Michael Wolf and Markus Hasen\\\"ohrl for their support on this project. Moreover, we would like to thank Valentin A. Zagrebnov for his helping out with questions on the Chernoff Lemma and the anonymous reviewers for their constructive and detailed feedback. T.M. and C.R. acknowledge the support of the Munich Center for Quantum Sciences and Technology, and C.R. that of the Humboldt Foundation.\n\t\n\n\t\\setlength{\\bibitemsep}{0.5ex}\n\t\\printbibliography[heading=bibnumbered]\n\t\\vspace{2ex}\n\t\\addresseshere\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{introduction}\n\nPhotometric redshifts (Connolly {\\it et al.~}~1995, Hogg {\\it et al.~}~1998, Benitez\n2000) are of paramount importance for current and planned multi-band\nimaging surveys. With photometric redshifts, surveys can\ninexpensively gather information about structure along the line of\nsight, without resorting to expensive spectroscopic followup.\nTherefore, it is important to understand systematic errors and\nlimitations in this method. For example, Ma {\\it et al.~}~2006 and Huterer\n{\\it et al.~}~2006 have examined the required photometric redshift accuracy\nfor surveys which plan to use weak lensing (cosmic shear) to constrain\ndark energy. For this application and also for baryon acoustic\noscillations (Zhan \\& Knox 2006), reducing photometric redshift errors\nis less important than knowing the error distribution accurately.\nThus, careful attention must be paid to systematic differences between\nthe photometric survey and the spectroscopic sample used to evaluate\nphotometric redshift performance. For most surveys, photometric S\/N\nis one of the systematic differences.\n\nThe most well-known test case for photometric redshifts is the blind\ntest in the Hubble Deep Field North (HDFN) conducted by Hogg {\\it et al.~}\\\n(1998). The best methods then yielded $\\sigma_{{\\Delta z}}\\sim0.1$, where\n${\\Delta z}\\equiv {{z_{\\rm spec}} - {z_{\\rm phot}} \\over 1 + {z_{\\rm spec}}}$, using Hubble Space Telescope\n(HST) photometry in UBVI bands and ground JHK (Dickinson ~1998). More\nrecently, with improved photometry and spectral redshift\nclassification, an accuracy of $\\sigma{\\Delta z}\\sim0.06$ is achieved over\nthe redshift range 0---6 (Fernandez-Soto {\\it et al.~}~1999, 2001; Benitez\n2000). Ground-based surveys suffer from less precise photometry but\nusually do not have to deal with such a large redshift range. Ilbert\n{\\it et al.~}~2007 cite an accuracy of $\\sigma_{{\\Delta z}} = 0.029$ after clipping\noutliers with ${\\Delta z}>0.15$ (3.8\\% of the sample). Ilbert {\\it et al.~}~2007\nalso find a decrease in precision at fainter magnitudes, but made no\neffort to separate the effects of S\/N from the other effects operating\non faint galaxies, such as a weaker magnitude prior and greater SED\nevolution. In this paper, we examine the impact of these effects\nseparately, focusing on photometric S\/N. \nThe quantitative results presented here are specific to the BVRz\nfilter set used in the Deep Lens Survey ({DLS}, Wittman {\\it et al.~}~2002).\nMore filters, covering a wider range in wavelengths, will do better\n(Abdalla {\\it et al.~}~2007). However, the trends with S\/N are broadly\napplicable.\n\n\n\\section{Method}\\label{method}\n\nWe use the {\\it BPZ}~ Bayesian photometric redshift code developed by\nBenitez (2000). We also tested the HyperZ code (Bolzonella {\\it et al.~}\\\n2000) with additional priors roughly equivalent to the default BPZ\npriors, and found similar performance. For clarity we present only\nresults from BPZ here. We did not test training-set methods, in which\na spectroscopic and photometric training set is used to perform a fit\nor to train a neural network, for two reasons. First, training set\nmethods are unlikely to be employed for surveys planning to push the\nphotometric sample deeper than the spectroscopic sample. Second, the\ntwo methods seem to be roughly equivalent in performance on the data\nsets in which they have been compared (e.g. Hogg {\\it et al.~}~1998), so the\ntrends presented here should be applicable to both methods.\n\nWe use the six spectral energy distribution ({\\it SED}) templates from\nBenitez (2000): E, Sbc, Scd, Irr, SB3, and SB2, modified as described\nbelow. For the simulations, the same templates are used to simulate\nthe photometry and to infer the photometric redshifts; there is no\nallowance for cosmic variance of the templates or ``template noise''.\nFor the data, it is important that the templates reflect real SEDs.\nTherefore, we use the photometry of objects with spectroscopic\nredshifts to optimize the templates (Csabai {\\it et al.~}~2000; Benitez\n{\\it et al.~}~2004; Ilbert {\\it et al.~}~2007). Section~\\ref{tempopt} describes the\nprocedure and shows the corrected templates. Clearly, even the\noptimized templates do not represent all types of SEDs in the\nuniverse. For both simulations and data, we start by demonstrating the\nperformance with as nearly perfect a data set as possible. After\nillustrating the best-case scenarios, we proceed to degrade the\nsimulations and data to successively lower S\/N, repeating the analysis\nfor each step.\n\nFor each galaxy, we identify the peak of its redshift probability\ndensity function (PDF) as its {\\it photometric redshift} or $z_{\\rm\nphot}$. This greatly simplifies the analysis and presentation of the\nresults, at the cost of some precision. Specifically, ``catastrophic\noutliers'' will appear, whose $z_{\\rm phot}$ differs greatly from\ntheir true redshift. In many cases, this may be an artifact of not\nconsidering the full PDF, a point argued forcefully in the case of the\nHDF by Fernandez-Soto\n{\\it et al.~} (2001, 2002). The full PDF may contain additional peaks, or\notherwise be broad enough to be consistent with the true redshift. In\nthis paper, we wish to focus on the trends with photometric S\/N rather\nthan the characterization of outliers. As will be seen in the tables\nand figures, the trends with S\/N are not substantively changed if\n``outliers'' are removed. Therefore we judge this simplification to\nbe acceptable. ``Outlier'' in this paper thus refers to difference\nbetween $z_{\\rm phot}$ and true redshift, without implying anything\nabout the full PDF.\n\nWe do consider characteristics of the PDF when using BPZ's ODDS\nparameter. BPZ assumes a natural error (template noise) of\n$0.067(1+z)$, and defines ODDS as the fraction of the area enclosed by\nthe PDF between $zphot\\pm n\\times0.067(1+z)$, where $n$ is a\nuser-settable parameter which we set to 1. ODDS values close to unity\nindicate that most of the area under the redshift probability density\nfunction (PDF) is within $Z_B\\pm0.067(1+z)$. In this paper, we\npresent results both for the entirety of a given sample, and after a\ncut of $ODDS>0.9$, which eliminates many of the ``outliers.'' We\nalso investigate the tradeoff between ODDS cut, number of usable\ngalaxies, and photometric redshift accuracy.\n\nThe error distributions are typically non-Gaussian, often highly so.\nThe rms or standard deviation is extremely sensitive to even a few\nnon-Gaussian events, so in the photometric redshift literature,\nresults are usually quoted as an rms after excluding a certain (small)\nfraction of galaxies as ``catastrophic outliers.'' The fraction\nvaries from paper to paper, making comparison difficult. The field of\nrobust statistics suggests several less sensitive metrics of\nvariation, such as the median or mean absolute deviation. However,\noutliers {\\it should} be included in the performance analysis with\nsome weight, because they will be included when using the entire\nphotometric sample for science. We therefore clip conservatively,\n$|{\\Delta z}|<0.5$, to avoid overly optimistic results. This threshold\nis at least five, and usually many more, times the clipped rms. We\nalso present, in many cases, differential and cumulative distributions\nas well. To make the connection with forecasts for, say, weak lensing\ntomography, we suggest these distributions be fit with double\nGaussians. Gaussians are analytically tractable, and a double\nGaussian can fit both the core and wings (but not truly catastrophic\noutliers).\n\n\n\\section{Simulations}\\label{simulations}\n\nWe simulate a mix of ellipticals, spirals, irregulars and starburst\ngalaxies (specifically, E, Sbc, Scd, Irr, SB3, and SB2 templates)\nfollowing the priors for galaxy type fraction as a function of\nmagnitude, $P(T|m_0)$, and for the redshift distribution for galaxies\nof a given spectral type and magnitude, $P(z|T,m_0)$, that are used in\n{\\it BPZ} 's Bayesian photometric redshift code. \nWe found that in Table 1 of Benitez (2000), two numbers were\ninadvertently switched, but the numbers were correct in the publicly\ndownloadable code. Benitez (private communication) has confirmed that\nthe table should read $k_t=0.450$ for E\/SO and $k_t=0.147$ for\nSbc\/Scd. Figure~\\ref{fig-priors} shows in solid red lines the priors\nused in this paper (same as {\\it BPZ} 's code); in dashed red lines are the\npriors quoted in BPZ's paper; and green lines represent Ilbert\n{\\it et al.~}~(2007) priors.\n\nIn order to have a realistic galaxy luminosity function, $N(mag)$, we\nstart our simulations from R-band magnitudes of 87260 objects detected\nin one of our $\\sim40^{\\prime}\\times40^{\\prime}$ Deep Lens Survey\nsub-fields (Wittman {\\it et al.~} 2002). The typical BVRz magnitude\ndistributions for the DLS are shown in Figure~\\ref{fig-nmagdls}. We\ntake this magnitude as the {\\it true} ($\\ne observed$) R-band\nmagnitude of a new object to be simulated. From the $P(T|m_0)$ prior\nwe select a {\\it SED}, and from $P(z|T,m_0)$ we choose a ${z_{\\rm input}}$ redshift for\nthe galaxy. The resulting ``true'' redshift distribution in the\nsimulations is shown in Figure~\\ref{fig-nzinput}. This distribution\nhas a larger tail to high redshift than usually found in the\nliterature (e.g. LeFevre et al 2005) and can be approximately described\nas $z^{2} exp\\big(-1(\\frac{z}{0.05})^{0.54}\\big)$. Magnitudes (with or\nwithout noise) in any other photometric bands can then be computed. We\nuse {\\it BPZ}~itself to compute synthetic colors, so there is no issue of\nminor differences in the k-corrections, priors, etc. We assume that\nthere are only six {\\it SED} `s of galaxies in the universe and make no\nattempt to introduce template noise in these simulations. We then\nperform three sets of simulations in the BVRz filter set of the\nDLS. In the first simulation (SIM1) we assume perfect, infinite\n{S\/N}~photometry. In the second set of simulations (SIM2) we\nsuccessively degrade the {S\/N}~ of the photometry but maintain constant\nthe {S\/N}~of all galaxies in all 4 bands (same magnitude error for all\ngalaxies in all 4 bands). In the third simulation (SIM3), we reproduce\nthe {S\/N}~distribution and completeness of the DLS.\n\n\n\\subsection{SIM1}\\label{sim1}\n\nThe first simulation (SIM1) has perfect photometry and represents the\nbest possible case. The ${z_{\\rm phot}}-{z_{\\rm spec}}$ scatter-plot for this simple\nsimulation is shown in Figure~\\ref{fig-sim1}, and the distribution of\n${\\Delta z} \\equiv {{z_{\\rm spec}} - {z_{\\rm phot}} \\over 1 + {z_{\\rm spec}}}$ is shown in\nFigure~\\ref{fig-sim1dz}. Note that Figure~\\ref{fig-sim1} contains\n87260 objects, distributed in redshift according to\nFigure~\\ref{fig-nzinput}, and that the ${z_{\\rm phot}}={z_{\\rm spec}}$ line is saturated\nwith objects. It is clear from Figure~\\ref{fig-sim1dz} that the\nmajority of objects have $|{\\Delta z}|\\sim0.0$. Table~\\ref{tab:sim12}\nindicates: (1) signal-to-noise of photometry (same in all bands); (2)\nfraction of galaxies with $|{\\Delta z}|<0.5$; (3) mean ${\\Delta z}$ for galaxies\nwith $|{\\Delta z}|<0.5$; (4) rms in ${\\Delta z}$ for galaxies with $|{\\Delta z}|<0.5$; (5)\nfraction of objects with $ODDS>0.9$; (6) fraction of objects with\n$ODDS>0.9$ and $|{\\Delta z}|<0.5$; (7) mean ${\\Delta z}$; and (8) rms in ${\\Delta z}$ for\nthese galaxies. \n\nThere are still catastrophic outliers, despite being the best possible\ncase in terms of noise, perfectly known templates, etc. This is\nbecause each galaxy is assigned a single ${z_{\\rm phot}}$ based on the peak of\nits PDF. Consider a degeneracy such that the same colors come from\n{\\it SED}~ A at $z_1$ or {\\it SED}~ B at $z_2$. In the absence of priors, this\nwould result in a PDF with two equal peaks. Now add priors encoding\nour astrophysical knowledge, such as that an apparently bright galaxies\nare likely to be at low redshift, or that ellipticals are rare at high\nredshift. This usually helps select the correct peak, but sometimes\nit will select the wrong peak because unlikely events do happen: some\nhigh-redshift galaxies are bright, or are ellipticals. As noted\nabove, this ignores the full PDF, which may be broad or multi-modal in\na way that is consistent with the true redshift. As our purpose is\nonly to establish SIM1 as a baseline for investigating the impact of\nphotometric S\/N, we do not pursue this here.\n\n\n\\subsection{SIM2}\\label{sim2}\n\nIn the second set of simulations (SIM2) we degrade the initially\nperfect photometry in SIM1 successively to {S\/N}~ of 250 ($R\\sim20.5^m$\nin the DLS, and the magnitude limit of the spectroscopic sample\npresented in Section \\ref{data}), 100, 60, 30, 10 and 5, and repeat\nthe analysis at each step. In these unrealistic simulations all\ngalaxies have the same photometric {S\/N}~ in all bands. The\nscatter-plots are shown in Figure~\\ref{fig-sim2}, and ${\\Delta z}$\ndistributions are shown in Figure~\\ref{fig-sim2dz}. We also present\nthe cumulative fraction of objects with ${\\Delta z}$ smaller than a given\nvalue, as a function of ${\\Delta z}$ (Figure~\\ref{fig-sim2frac}). This plot\nhas several advantages. First, multiple simulations can be\nover-plotted without obscuration. Second, the asymmetry in the\ndistribution of ${\\Delta z}$ is easily read off by looking at the fraction\nwith ${\\Delta z}<0$ (dashed vertical line). Third, the fraction of outliers\ncan also be directly read off the plot at any ${\\Delta z}$. The left panel of\nFigure~\\ref{fig-sim2frac} shows the cumulative fraction for all\nobjects, while the right panel shows $ODDS>0.9$ galaxies. The number\nof galaxies in the right panel is smaller than the number in the left\n(see Table~\\ref{tab:sim12}) but the accuracy of photo-zs is clearly\nbetter.\n\nBecause all realizations of SIM2 have the redshift distribution shown\nin Figure~\\ref{fig-nzinput}, even if all galaxies have colors measured\nat very high {S\/N}~, some objects will have degenerate colors and the\nsample will contain some fraction of catastrophic\noutliers. Spectroscopic samples typically have a much lower mean\nredshift than these simulations, so catastrophic outliers are likely to\nbe underrepresented in direct ${z_{\\rm phot}}-{z_{\\rm spec}}$ comparisons, if the full\nphotometric sample is very deep.\n\nTable~\\ref{tab:sim12} presents the statistics for the SIM2 objects\nshown in Figures~\\ref{fig-sim2}, ~\\ref{fig-sim2dz} and\n~\\ref{fig-sim2frac}. Clearly, the precision of photometric redshifts\nis a strong function of photometric {S\/N}. BPZ's ODDS parameter is very\neffective at removing outliers, and almost 100\\% of the objects with\n$ODDS>0.9$ have $|{\\Delta z}|<0.1$ regardless of {S\/N}~(right panel in\nFigure~\\ref{fig-sim2frac}). However, the fraction of objects with\n$ODDS>0.9$ decreases dramatically with decreasing {S\/N}.\n\nPerformance is, counter-intuitively, slightly worse for the infinite\nS\/N galaxies in SIM1 than for the high S\/N galaxies in SIM2. This is\nbecause the priors have too much power when there is no noise in color\nspace, and is not of concern in more realistic situations.\n\n\n\\subsection{SIM3}\\label{sim3}\n\nThe third simulation has the same {S\/N}~ distribution and completeness as\nthe DLS data. Again, the priors used assure that the galaxy type\nmixture and redshift distribution should be close to the real\nuniverse. The idea is to measure how well we can recover true ${z_{\\rm input}}$\nredshifts for a realistic photometric data set. This simulation is\nstill optimistic because no template noise is added---we derive colors\nfrom the same six templates used in the determination of photometric\nredshifts. The effect of template noise will be presented in the real\ndata analysis in Section \\ref{data}.\n\nAs a sanity check we compare the BVz magnitude distributions of our\nSIM3 simulation with the observed $N(mag)$ and find good\nagreement. The R magnitude distribution is by definition the same\nwithin the added photometric noise. We also compare the distribution\nof BPZ galaxy types in DLS fields with the one derived from the SIM3\nsimulation and find very good agreement. Figure~\\ref{fig-priors2}\nshows the galaxy type fraction as a function of magnitude for two\n$40^{\\prime}\\times40^{\\prime}$ DLS fields. The field with the higher\nfraction of ellipticals contains the richness class 2 galaxy cluster\nAbell 781 at $z=0.298$ (``$+$''), and the other is a more typical\n``blank'' field (``$\\times$''). The simulation input distribution is\nindicated by solid circles, which by definition agree with the red\nline, and the output BPZ types are indicated by open circles. SIM3 and\ndata show the same magnitude dependence.\n\nA third sanity check is a comparison between the redshift distribution\nderived in SIM3 and $N(z)$ for the entire DLS survey.\nFigure~\\ref{fig-dlsnz} shows both distributions and also the input\nredshift distribution used in the simulations (same as shown in\nFigure~\\ref{fig-nzinput}). The agreement is pretty good. The mean\ndensity of galaxies with photometric redshifts of any quality is\n$47\/arcmin^2$ and $11\\%$ of those objects have $ODDS>0.9$.\n\nThe photometric redshift performance on SIM3 is shown on\nFigures~\\ref{fig-sim3}, ~\\ref{fig-sim3dz} and~\\ref{fig-sim3frac}, just\nas in Figures~\\ref{fig-sim2}, ~\\ref{fig-sim2dz} and~\\ref{fig-sim2frac}\nfor SIM2. The summary statistics for SIM3 are presented in\nTable~\\ref{tab:sim3}. As in SIM2, the precision of photometric\nredshifts is a strong function of {S\/N}, and ODDS does a good job of\ncleaning up, at the cost of losing many low {S\/N}~galaxies.\n\nThere are two notable differences with SIM2. First, in SIM3, there is\na realistically strong correlation between high S\/N and bright\nmagnitudes. A bright magnitude implies a strong prior (most bright\ngalaxies are at low redshift), whereas a faint galaxy has a weak prior\n(it could be at any redshift). The high S\/N galaxies in SIM2 were\n(artificially) at all magnitudes, and therefore had generally looser\npriors. Therefore, the highest S\/N galaxies in SIM3 do better than\nthose in SIM2. We can see the effect of the tight priors directly by\ncomparing the $S\/N=250$ line of Table ~\\ref{tab:sim12}\n($\\sigma_{{\\Delta z}}=0.042$ after clipping 4\\% which had $|{\\Delta z}|>0.5$) with\nthat of Table~\\ref{tab:sim3} ($\\sigma_{{\\Delta z}}=0.031$ with no need to\nclip any outliers). This difference vanishes when low S\/N galaxies\nfrom SIM3 are included.\n\nIn fact, the $S\/N=5$ galaxies in SIM2 outperform the $S\/N>5$ galaxies\nin SIM3, despite the latter cut being only a lower bound. This is due\nto the second salient difference between SIM2 and SIM3: A given S\/N in\nSIM2 describes {\\it each} galaxy in {\\it each} band. In SIM3, the S\/N\nvaries with filter in a realistic way, and the cut applies to R band.\nMost galaxies will have lower S\/N in other bands. For $S\/N=30$ in R,\nthe median $S\/N$ in B, V, and z over the whole sample is 10, 18, and\n10 respectively.\n\n\nWhat S\/N is required for good photometric redshift performance?\nFirst, consider performance without any ODDS cut. At each step in\nTable~\\ref{tab:sim3} from $S\/N>100$ to $S\/N>10$, there is a 30--50\\%\nincrease in $\\sigma_{{\\Delta z}}$, so there is no natural breakpoint.\n$\\sigma_{{\\Delta z}}$ appears to stop this dramatic growth when stepping down\nfrom $S\/N>10$ to $S\/N>5$, but this is likely an artifact of clipping\nat $|{\\Delta z}|>0.5$, which is roughly three times the clipped rms at that\npoint. Even at $S\/N>10$, $\\sigma_{{\\Delta z}}$ may be artificially low due to\nclipping, as more than 10\\% of galaxies were clipped. Most survey\nusers would find the precision offered by the $S\/N>30$ cut acceptable,\nbut the $S\/N>10$ cut unacceptable. If we set $\\sigma_{{\\Delta z}}=0.1$ as\nthe limit of acceptability, we find an S\/N cut at 17 is required.\n\nNow consider using the ODDS cut at 0.9. $\\sigma_{{\\Delta z}}$ is always 0.04\nor less, regardless of S\/N. We suspect that for a given\n$\\sigma_{{\\Delta z}}$, the ODDS cut will provide more galaxies than the S\/N\ncut, because ODDS responds to the properties of the color space as\nwell as to S\/N. For example, high-precision S\/N is not required if\nthe galaxy is in a distinctive region of color space. In addition,\nODDS can take proper account of different S\/N in different bands,\nwhich a simple S\/N cut in R does not. We investigate this by finding\nthe ODDS cut which yields the same $\\sigma_{{\\Delta z}}$ as the $S\/N>30$ cut\n(0.076). We find that $ODDS>0.57$ is required, which yields 30\\% of\nall detected galaxies, vs. the 13\\% yielded by the S\/N cut.\n\nWe repeat this procedure for $\\sigma_{{\\Delta z}}=0.1$. The required ODDS\ncut is $>0.40$, yielding 45\\% of all detected galaxies, while the\nrequired S\/N cut at 17 yields only 26\\% of detected galaxies.\n\nThese fractions can all be read off\nFigure~\\ref{fig-sim3snodds} which summarizes the results from\nSIM3. The three left panels in Figure~\\ref{fig-sim3snodds} show: (1)\nthe cumulative fraction of objects with {S\/N}~greater than a given\nvalue; (2) mean ${\\Delta z}$; and (3) $\\sigma_{{\\Delta z}}$ for these objects. The\nthree right panels are the same but for a cut in $ODDS$.\n\nIn short, we recommend an ODDS cut. We recognize that an ODDS cut is\nnot easy to incorporate into survey forecasts of the number of usable\ngalaxies. Detailed simulations for a given filter set and depth as a\nfunction of wavelength must be performed. However, we hope that the\nabove numbers can serve as a rough guide for translation between\nphotometric redshift precision, S\/N threshold, and number of usable\ngalaxies.\n\n\n\\section{Data}\\label{data}\n\nWe take photometric data from the {DLS}~BVRz full-depth images in\nfields with spectroscopic redshifts from the {\\it Smithsonian\nHectospec Lensing Survey} (SHeLS, Geller {\\it et al.~}\\ 2005), and from the\n{\\it Caltech Faint Galaxy Redshift Survey} (CFGRS, Cohen {\\it et al.~}~1999)\nsurveys. Here, by definition, template noise is present. In Sections\n\\ref{shels} and \\ref{cfgrs} we present the spectroscopic data and the\nphotometric redshift accuracy for these two samples, but before that\nwe present our methodology for color measurement (Section\n\\ref{colors}), and template optimization (Section \\ref{tempopt}).\n\n\n\\subsection{Measuring Colors}\\label{colors}\n\nWe performed simulations to determine the best photometry method in\nthe face of different point-spread function (PSF) sizes in the\ndifferent filters. We added galaxies with De Vaucouleurs (elliptical)\nand exponential disk (spirals) light profiles to the {DLS}~BVRz data\nusing standard IRAF-Artdata routines, ran SExtractor (Bertin \\&\nArnouts 1996) and measured colors with many different types of\nmagnitudes. Figure~\\ref{fig-magRmagerr} shows the results for\ngalaxies added to the R images. The B, V and z results are\nqualitatively the same, but because of differences in {S\/N}~and PSF\nthere is a shift in the magnitude axis, and slightly different\nscatter. The left panels show the results using $MAG_{iso}$ and right\npanels show $MAG_{auto}$. The top panels show the difference between\nmeasured $MAG$ and input $MAG_{input}$. De Vaucoleurs galaxies are\nmeasured to be $\\sim0.15^m$ fainter than their true magnitudes both by\n$MAG_{iso}$ and $MAG_{auto}$. The bottom panels show the distribution\nof $(MAG-MAG_{input})\/MAGerr$ as a function of magnitude. As noted by\nBenitez {\\it et al.~}~(2004), $MAG_{auto}$ gives better results for\nmagnitudes, but for photometric redshifts we are interested in good\ncolors as deep as possible.\n\nFigures~\\ref{fig-colorerr} and \\ref{fig-cc} \nshow the distribution of {\\it color} errors,\nwhich, for photometric redshifts, are more important than magnitude\nerrors. Again, $MAG_{iso}$ is on the left and $MAG_{auto}$ on the\nright. The systematic magnitude errors tend to cancel when\nconsidering colors, and $MAG_{iso}$ is now slightly better. It is\nimportant to note that the errors in magnitude errors are not driven\nby faint galaxies, and that in fact the discrepancies between real and\nestimated colors errors are significantly worse for bright\nobjects. \n\nIn summary, $MAG_{iso}$ gives slightly more precise colors at a given\nmagnitude. This translates to more galaxies being detected above a\ngiven S\/N threshold, providing another benefit. However, for either\n$MAG_{auto}$ or $MAG_{iso}$, the error estimates provided by\nSExtractor are optimistic, especially at the bright end. The solid\nlines in Figure~\\ref{fig-frac} show the cumulative fraction of objects\nas a function of magnitude and color error, normalized by the nominal\nerror from SExtractor. Much less than $68(95)\\%$ of the galaxies have\nactual errors within the nominal 1(2)$\\sigma$ magnitude error. Actual\ncolor errors are closer to nominal, but still optimistic. (Caveat:\nunlike most real galaxies, the simulated galaxies had zero color.)\nFrom this analysis we determine an ad hoc correction to the magnitude\nerrors estimated by SExtractor: we first multiply $MAGerr_{iso}$ by\n$1.5$, and then add in quadrature an error of $0.02^m$. The dashed\nlines in both panels of Figure~\\ref{fig-frac} show the results of this\ncorrection. This single correction puts the 68th and 95th percentiles\nof all the color distributions in the correct place, with the\nexception of the 68th percentile of $R-z$ color. This adjustment to\nthe magnitude errors should in principle depend on galaxy color, but\nwe found that variations about this correction made little difference\nin the results.\n\nWe performed all the real-data tests in this paper with both\n$MAG_{iso}$ and $MAG_{auto}$. The differences in the results were\nvery minor, except that more galaxies were detected at a given S\/N\nwith $MAG_{iso}$, and about 20\\% more survived the ODDS cut with\n$MAG_{iso}$. We therefore adopt $MAG_{iso}$ for the remainder of this\npaper.\n\nAnother factor to consider is the quality of the survey's photometric\ncalibration, which was determined by observations of standard stars in\nLandolt's (1992) fields during photometric nights. The R and V DLS\nbands are very similar to Landolt's filter transmissions and yield\naccurate calibration. The DLS B-band however differs significantly\nfrom Landolt's and requires a color term correction which decreases\nthe accuracy of calibration in this band. Also, the DLS z-band\nphotometry derived from Sloan Digital Sky Survey standards (Smith\n{\\it et al.~}~2002) is also not as good as R and V. For this reason we add an\nextra $0.01^m$ to the magnitude error measurements in B and z\nbands. \n\n\n\\subsection{Template Optimization}\\label{tempopt}\n\nWe use spectroscopic redshifts and the DLS photometry to empirically\ncorrect the {\\it BPZ}~set of templates and to test our filter+instrument\nresponse knowledge with the methodology described in Ilbert {\\it et al.~}\n2007. We find optimized templates for El, Sbc, Scd, Im, and SB3 {\\it SED}\ns. The SB2 template was left unchanged because there were not enough\ngalaxies of this type to fit a correction. The biggest modifications\nwere found for the El {\\it SED}, which shows a less strong 4000\\AA~ break\nin the optimized template; and for the Sbc {\\it SED}, which has a stronger\n4000\\AA~ break than in the original BPZ template (See\nFigure~\\ref{fig-seds}). Because most of our galaxies are at low\nredshift, we cannot constrain the longest and shortest SED wavelengths\nand therefore we force them to agree with the initial templates.\n\n\n\\subsection{Comparison with Spectroscopic Data: SHeLS Survey}\\label{shels}\n\nThe SHeLS survey has a limiting magnitude of $R=20.3$, so that the DLS\nphotometry, which is complete to about five magnitudes fainter, is\nvery high {S\/N}. Being a bright magnitude-limited survey, SHeLS\ncontains overwhelmingly low-redshift ($z<0.6$) galaxies. However, our\nsubsample of 1,000 was chosen to provide a nearly uniform redshift\ndistribution so that characterization accuracy would be roughly\nredshift-independent. At a given redshift, selection was random.\n\nWe further cut the sample, requiring ${S\/N}>100$ in the R band, and\nexcluding objects in exclusion zones around bright stars, or with\nsaturated pixels in any band, or with SExtractor $flags\\ge4$\n(compromised photometry). The final sample contains 860 galaxies.\nThe top left panels of Figures~\\ref{fig-shelsdatasn} and\n~\\ref{fig-shelsdatasndz} show the ${z_{\\rm phot}}-{z_{\\rm spec}}$ scatter-plot, and ${\\Delta z}$\ndistribution for the maximum {S\/N}~ photometry. The distribution of\ngalaxy types assigned by BPZ to this spectroscopic sample is in\nagreement to the type distribution of all galaxies at $R=20\\pm0.5^{m}$\nin the entire DLS survey, suggesting that the spectroscopic sample is\nrepresentative of galaxies at this magnitude.\n\nThe SHeLS sample is expected to show evidence of template noise and\nhave somewhat higher $\\sigma_{{\\Delta z}}$ than the bright end of SIM3, and\nthis is in fact observed. Objects with ${S\/N}>100$ in SIM3 have\n$\\sigma_{{\\Delta z}}=0.037$, and $89.4\\%$ of the galaxies have $ODDS>0.9$\nwith $\\sigma_{{\\Delta z}}=0.026$. For the SHeLS survey, $\\sigma_{{\\Delta z}}=0.050$,\nand $85.6\\%$ have $ODDS>0.9$ with $\\sigma_{{\\Delta z}}=0.044$. The difference\nsuggests a template noise of $\\sigma_{{\\Delta z}}\\sim0.035(1+z)$ which is\nsmaller than the $0.065(1+z)$ estimated by Fernandez-Soto {\\it et al.~}~(1999)\nfor galaxies in the Hubble Deep Field, but expected given the much\nlower redshift of galaxies in the SHeLS survey.\n\nWe now degrade the photometry successively to ${S\/N}=100,60,30,10,5$ in\nall bands. If a galaxy has, for example ${S\/N}=50$ in the B band, its\nmagnitude and magnitude error are left unchanged in this band for the\nsimulations with ${S\/N}=100$ and ${S\/N}=60$, but noise is added to the\nother ones. The ${z_{\\rm phot}}-{z_{\\rm spec}}$ scatter-plots are shown in\nFigure~\\ref{fig-shelsdatasn}. ${\\Delta z}$ distributions are shown in\nFigure~\\ref{fig-shelsdatasndz}, and cumulative fraction as a function\nof ${\\Delta z}$ is shown in Figure~\\ref{fig-shelsfrac}. Statistics in\ndifferent {S\/N}~regimes are presented in Table~\\ref{tab:shelsdata}.\nThe trends with S\/N which were observed in the simulations are\nreproduced here. \n\nBecause the magnitude prior remains tight despite the {S\/N}~degradation,\nwe observe lower $\\sigma_{{\\Delta z}}$ at the low {S\/N}~end of the SHeLS\nsimulations than is observed for SIM2 at the same {S\/N}. At ${S\/N}=10$,\n$\\sigma_{{\\Delta z}}=0.080$, and $8.3\\%$ of galaxies in the SHeLS survey have\n$ODDS>0.9$, while $\\sigma_{{\\Delta z}}=0.121$, and $6.4\\%$ of the have\n$ODDS>0.9$ for the SIM2 galaxies.\n\nThe effectiveness of the ODDS cut is again evident. The fraction of\ngalaxies passing this cut at low S\/N is less than in SIM3 because the\ndata here are uniformly at low S\/N, whereas for SIM3 the given S\/N is\na lower limit. The fraction with $ODDS>0.9$ at low S\/N is more\ndirectly comparable with, and more consistent with, the fractions in\nSIM2, which were also at constant S\/N.\n\n\n\\subsection{Comparison with Spectroscopic Data: CFGRS Survey}\\label{cfgrs}\n\nThe CFGRS (Cohen {\\it et al.~}~1999) survey is about $2^m$ deeper than SHeLS\nand therefore the DLS photometry is not as high {S\/N}. We select\ngalaxies with quality=1 (multiple spectral features, Cohen {\\it et al.~}~1999)\nspectroscopic redshifts and divide the data in 2 equally sized\nsubsamples of 111 galaxies each: one with galaxies of photometric\n${S\/N}(R)>106$, and another with ${S\/N}(R)<106$. Note that the\nsignal-to-noise in the low\n{S\/N}~ sample is still fairly high, with 28 being the lowest value, and\na median of 69, but the difference in the quality of photometric\nredshifts is clear. Figure~\\ref{fig-cfgrs} shows the ${z_{\\rm phot}}-{z_{\\rm spec}}$\nscatter-plot for the two sub-samples. For the high {S\/N}~sample,\n${\\Delta z}=0.027\\pm0.084$, and ${\\Delta z}=0.021\\pm0.060$ if we exclude 1\ncatastrophic outlier with $|{\\Delta z}|>0.5$. For the lower\n{S\/N}~sample, ${\\Delta z}=0.033\\pm0.166$, and ${\\Delta z}=0.041\\pm0.095$ if we exclude\n2 objects with $|{\\Delta z}|>0.5$. However this includes the effect of\ndifferent redshift ranges. To isolate the\n{S\/N}~effect, we compute results using only galaxies between\n$0.40.5$\nare found in this redshift range.\n\n\n\n\\section{Selection in Galaxy Type and Redshift Range}\n\nFigure~\\ref{fig-priors2} suggests that faint Irr\/SB2\/SB3 galaxies are\noften misclassified as Sbc\/Scd. In this section we explore dependence\non type in more detail. Figures~\\ref{fig-sim1typ},\n~\\ref{fig-sim3typ}, and ~\\ref{fig-datatyp} show the ${z_{\\rm phot}}-{z_{\\rm spec}}$\nscatter-plot as a function of inferred {\\it BPZ}~galaxy type ($T_B$) for\nSIM1, SIM3, and the SHeLS galaxies respectively. Ellipticals form the\ntightest relation, while the redshift of irregular galaxies show a\nscatter more than twice as large. Figure~\\ref{fig-sim3typ} shows that\nsome of the scatter in ellipticals must be due to misclassifications,\nbecause there are no E-type galaxies at $z\\sim3-4$ in the simulations.\n\nWe look at type misclassification in SIM3 directly in\nFigures~\\ref{fig-types} and ~\\ref{fig-typesall}. The left column of\npanels shows the $T_B$ distribution for each of the true input types,\nwith the true type distribution overlaid like a diagonal matrix in red\nto guide the eye. The right column of panels shows the true type\ndistribution for each of the inferred types, with the inferred type\ndistribution overlaid in red to guide the eye. The overall\ndistribution of inferred (true) types is shown by the unshaded\nhistogram which is repeated in each panel in the left (right) column.\nFigures~\\ref{fig-types} shows galaxies with ${S\/N}\\ge30$ or $R\\le23$.\nFor example, the fourth panel down in the left column shows that\ngalaxies classified as $T_B=4$ (Irr), have in fact almost the same\nprobability of being of types 4, 5 or 6 (irregular or starburst).\nLikewise, starburst galaxies tend to be misclassified at irregulars\neven at high {S\/N}.\n\nThe types in decreasing order of reliability are E, Sbc, Scd, Irr,\nSB3, and SB2. Type reliability translates to redshift reliability,\nbecause type misclassification usually implies a large, if not\ncatastrophic, redshift error. These figures also demonstrate that\nalthough the ODDS cut appears to lose many high high-redshift galaxies\nand shrink the usable redshift range, in fact most of the\n``high-redshift'' galaxies lost were type misclassifications, and\ntherefore unreliable redshifts. Although the loss of these\n``high-redshift'' galaxies is painful if one wants as large a redshift\nrange as possible, it is necessary if one wants the sample to be\nreliable.\n\nIn Figure~\\ref{fig-typesall} we extend the analysis to lower\n{S\/N}~galaxies, and include all ``detected'' galaxies. The\nrate of misclassification is much higher. The insertion of these\nobjects in the sample creates new types of misclassification. For\nexample, a fraction of type 1 (E) galaxies is assigned $T_B=2$ and\nvice-versa. Also, a significant fraction of types 4, 5, and 6\n(irregular and starburst) are classified as types 2 or 3 (spirals).\n\n\n\\section{Summary and Discussion}\n\nWe have examined the dependence of photometric redshift performance on\nphotometric S\/N, using both simulations and data. For concreteness,\nwe have used the DLS filter set, but the general trends should apply\nto any filter set. As a reminder, SIM2 simulated galaxies at a range\nof magnitudes drawn from the DLS photometry, but at a series of\nconstant S\/N levels, while SIM3 strongly couples magnitude and S\/N as\nthey are in the DLS photometry. Thus, {\\it bright} is distinct from\n{\\it high S\/N} in SIM2 and in the noise-augmented SHeLS data because\n{\\it bright} implies a more effective magnitude prior. An additional\ndistinction between SIM3 and the other cases is that in SIM3 a given\nS\/N cut is performed in R, and for most galaxies that implies a lower\nS\/N in the other bands. For SIM2 and noise-augmented SHeLS data, a\ngiven S\/N describes each galaxy in each filter.\n\nWe therefore expect the smallest $\\sigma_{{\\Delta z}}$ for very high S\/N in\nSIM3, where the high S\/N galaxies automatically have a tight magnitude\nprior. This is what is observed, $\\sigma_z=0.031$ (0.037) for\n$S\/N>250$ (100) in SIM3. Degeneracies in color space determine this\nperformance limit, which is therefore highly filter-set dependent.\nHowever, it sets a baseline for what follows. At $S\/N=100$ in the\nSHeLS data, $\\sigma_{{\\Delta z}}$ is about 35\\% larger than this baseline,\nsuggesting a cosmic variance or template noise component of\n$\\sigma_{{\\Delta z}}=0.035(1+z)$. For SIM2, $\\sigma_{{\\Delta z}}$ is also about\n32\\% larger than this baseline, presumably due to the looser magnitude\npriors on average. The deeper the survey, the less effective the\nmagnitude prior, but performance is still quite good at this high S\/N.\n\nFrom this baseline, lowering the S\/N smoothly increases $\\sigma_{{\\Delta z}}$\nin SIM3, by 30--50\\% at each S\/N step in Table~\\ref{tab:sim3} until\n$\\sigma_{{\\Delta z}}$ is no longer trustworthy due to the clipping at\n$|{\\Delta z}|>0.5$. SIM2 degrades a bit more slowly due to its higher\nbaseline $\\sigma_{{\\Delta z}}$. The noise-augmented SHeLS data degrades even\nmore slowly, because magnitude priors always remain tight. Although\n$\\sigma_{{\\Delta z}}$ looks reasonably good even at $S\/N=5$ for the degraded\nSHeLS data, we expect SIM3 to be more representative of true\nperformance for this reason.\n\n\nSIM3 indicates that without an ODDS cut, $S\/N=17$ in R is likely to be\nthe lowest acceptable S\/N for reasonable photometric redshift\nperformance ($\\sigma_{{\\Delta z}}=0.1$) in a survey with the DLS\nspecifications (filter set and depth). A shallower survey may be able\nto go to lower S\/N because the magnitude prior remains helpful to\nlower S\/N in such a survey. In fact, the bright spectroscopic sample\nhas $\\sigma_{{\\Delta z}}<0.1$ even at $S\/N=5$, although we caution that this\nmeans $S\/N=5$ in {\\it each} filter. If we impose an ODDS cut rather\nthan an S\/N cut, $ODDS>0.40$ cut yields twice as many galaxies for the\nsame $\\sigma_{{\\Delta z}}$ as the $S\/N>17$ cut in R. Alternatively, survey\nusers could use ODDS to decrease $\\sigma_{{\\Delta z}}$ while sacrificing\ngalaxy counts; an $ODDS>0.9$ cut yields $\\sigma_{{\\Delta z}}=0.04$ averaged\nover all S\/N.\n\nWe caution that there are some unmodeled effects which, if included,\nwould result in a larger $\\sigma_{{\\Delta z}}$. First, template noise is not\nincluded in the simulations. $\\sigma_{{\\Delta z}}$ is larger in the SHeLS\ndata than in SIM3 for $S\/N>60$, which we attribute to template noise.\nTemplate noise becomes less important at lower photometric S\/N, but\nthe template noise in the SHeLS data may be artificially low. The\ntemplates were originally derived from bright galaxies like those in\nSHeLS, and further optimized on the SHeLS sample itself. A\nphotometric sample which pushes to higher redshift may thus incur more\ntemplate noise, and in fact Fernandez-Soto {\\it et al.~}~(1999) estimates\n$\\sigma_{{\\Delta z}}=0.065(1+z)$ for galaxies in the Hubble Deep Field.\n\nSecond, because galaxy counts are rising beyond the limiting magnitude\nfor detection, an additional source of photometry noise must be taken\ninto account. A source detected at S\/N of a few is much more likely\nto be an ``up-scattered'' fainter galaxy than a ``down-scattered''\nbrighter galaxy. As pointed out by Hogg \\& Turner (1998, hereafter\nHT98), this is distinct from Malmquist bias, which is the\nover-representation of high-{\\it luminosity} galaxies in a flux-limited\nsample. Although the resulting bias can be computed and corrected for\nif the galaxy count slope is known, the additional photometric\nuncertainty is unavoidable. In fact, HT98 conclude that ``sources\nidentified at signal-to-noise ratios of four or less are practically\nuseless.'' This source of noise was not reproduced in our\nsimulations, so extrapolation to $S\/N<5$ would be extremely dangerous.\nOur results for $S\/N=5$ are still valid if five is interpreted as the\neffective S\/N in the presence of this additional source of noise. For\nthe no-evolution, Euclidean slope of $q=1.5$, the HT98 formulae\nindicate that this requires a detection at $S\/N=5.64$. For $S\/N=10$\nand higher, the corrections are very small.\n\nIn addition to these dependences on S\/N, several other lessons can be\ndrawn:\n\\begin{itemize}\n\n\\item When forecasting photometric redshift performance for a survey,\nit is important to include realistic photometry errors. \n\n\\item Estimating photometric redshift performance with spectroscopic\nsamples can lead to optimistic results if the spectroscopic sample is\nnot representative of the photometric sample. If the spectroscopic\nsample is brighter, matching the S\/N is easily accomplished by adding\nphotometry noise, but accounting for the larger redshift range of the\nphotometric sample requires detailed modeling which must account for\ncosmic variance.\n\n\\item The BPZ $ODDS$ parameter is very effective at identifying\nphotometric redshifts which are likely to be poor. An $ODDS$ cut is\nmore efficient than an S\/N cut, because $ODDS$ takes account of the\nlooser photometry requirements in distinctive regions of color space.\nStill, our simulations and artificially noisy data show that of the\ngalaxies with $ODDS<0.9$, the ones with poor photometric redshifts may\nbe in the minority. The tradeoff between $ODDS$ cut and usable\nnumbers of galaxies must be assessed in light of the specific science\ngoal. For example, if the science analysis weights each galaxy by its\nphotometric S\/N, a strict $ODDS$ cut may cut most of the galaxies but\nnot most of the total weight. For weak lensing, shape noise limits\nthe maximum weight of a galaxy, so a strict $ODDS$ cut may cut most of\nthe weight. Finally, biases must be considered, as ellipticals are\noverrepresented in the set of galaxies with high $ODDS$. This may not\naffect weak lensing but will be important for studies of galaxy\nevolution and baryon acoustic oscillations.\n\n\\end{itemize}\n\nWe also explored cutting in type (as identified by BPZ) and redshift\nrange. As expected, ellipticals do better than any other type, but we\nfound that the $ODDS$ cut was still useful for ellipticals. As long\nas the $ODDS$ cut was being used, other types could safely be used as\nwell. Therefore, we recommend cutting on ODDS rather than type. \n\n\n\\acknowledgments\n\nWe thank NOAO for supporting survey programs and the CFGRS project for\nmaking data publicly available. DLS observations were obtained at\nCerro Tololo Inter-American Observatory (CTIO) and Kitt Peak National\nObservatory (KPNO). CTIO and KPNO are part of the National Optical\nAstronomy Observatory (NOAO), which is operated by the Association of\nUniversities for Research in Astronomy, Inc., under cooperative\nagreement with the National Science Foundation. We also would like to\nthank Margaret Geller and Michael Kurtz for providing us with 1,000\nSHeLS redshifts, which were observed with Hectospec at the MMT\nTelescope.\n\nWe thank Ian Dell'Antonio and Tony Tyson for comments that led to\nimprovements to the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDuring the past few years, the theory of frames have been growing\nrapidly and new topics about them are discovered almost every year.\nFor example, generalized frames (or g-frames), subspaces of frames\n(or fusion frames), continuous frames (or c-frames), $k$-frames,\ncontrolled frames and the combination of each two of them, lead\n to c-fusion frames, g-c-frames, c-g-frames, c$k$-frames,\nc$k$-fusion frames and etc. The purpose of this paper is to\nintroduce and review some of the generalized fusion frames (or g-fusion\nframes) and their operators. Then, we will get some useful\npropositions about these frames and finally, we will study g-fusion\nframe sequences.\n\nThroughout this paper, $H$ and $K$ are separable Hilbert spaces and $\\mathcal{B}(H,K)$ is the collection of all the bounded linear operators of $H$ into $K$. If $K=H$, then $\\mathcal{B}(H,H)$ will be denoted by $\\mathcal{B}(H)$. Also, $\\pi_{V}$ is the orthogonal projection from $H$ onto a closed subspace $V\\subset H$ and $\\lbrace H_j\\rbrace_{j\\in\\Bbb J}$ is a sequence of Hilbert spaces where $\\Bbb J$ is a subset of $\\Bbb Z$. It is easy to check that if $u\\in\\mathcal{B}(H)$ is an invertible operator, then (\\cite{ga})\n$$\\pi_{uV}u\\pi_{V}=u\\pi_{V}.$$\n\\begin{definition}\\textbf{(frame)}.\nLet $\\{f_j\\}_{j\\in\\Bbb J}$ be a sequence of members of $H$. We say that $\\{f_j\\}_{j\\in\\Bbb J}$ is a frame for $H$ if there exists $00$ for any $j\\in\\Bbb J$). We say that $(W_j, v_j)$ is a fusion frame for $H$ if there exists $00$ and $\\Lambda_j\\in\\mathcal{B}(H,H_j)$ for each $j\\in\\Bbb J$. We say $\\Lambda:=(W_j, \\Lambda_j, v_j)$ is a \\textit{generalized fusion frame} (or \\textit{g-fusion frame} ) for $H$ if there exists $00$. Now, we can write for each $f\\in H$\n$$\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2=\\langle S_{\\Lambda}f, f\\rangle\\geq C\\Vert f\\Vert^2$$\nand\n$$\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2=\\langle S_{\\Lambda}f, f\\rangle\\leq \\Vert S_{\\Lambda}\\Vert \\Vert f\\Vert^2.$$\nIt follows that $\\Lambda$ is a g-fusion frame for $H$.\n\\end{proof}\n\\begin{theorem}\nLet $\\Lambda:=(W_j, \\Lambda_j, v_j)$ and $\\Theta:=(W_j, \\Theta_j, v_j)$ be two g-fusion Bessel sequence for $H$ with bounds $B_1$ and $B_2$, respectively. Suppose that $T_{\\Lambda}$ and $T_{\\Theta}$ be their analysis operators such that $T_{\\Theta}T^*_{\\Lambda}=I_H$. Then, both $\\Lambda$ and $\\Theta$ are g-fusion frames.\n\\end{theorem}\n\\begin{proof}\nFor each $f\\in H$ we have\n\\begin{align*}\n\\Vert f\\Vert^4&=\\langle f, f\\rangle^2\\\\\n&=\\langle T^*_{\\Lambda}f, T^*_{\\Theta}f\\rangle^2\\\\\n&\\leq\\Vert T^*_{\\Lambda}f\\Vert^2 \\Vert T^*_{\\Theta}f\\Vert^2\\\\\n&=\\big(\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2\\big)\\big(\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Theta_j \\pi_{W_j}f\\Vert^2\\big)\\\\\n&\\leq\\big(\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2\\big) B_2 \\Vert f\\Vert^2,\n\\end{align*}\nthus, $B_2^{-1}\\Vert f\\Vert^2\\leq\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2$. This means that $\\Lambda$ is a g-fusion frame for $H$. Similarly, $\\Theta$ is a g-fusion frame with the lower bound $B_1^{-1}$.\n\\end{proof}\n\\section{Dual g-Fusion Frames}\nFor definition of the dual g-fusion frames, we need the following theorem.\n\\begin{theorem}\\label{dual}\nLet $\\Lambda=(W_j, \\Lambda_j, v_j)$ be a g-fusion frame for $H$. Then $(S^{-1}_{\\Lambda}W_j, \\Lambda_j \\pi_{W_j}S_{\\Lambda}^{-1}, v_j)$ is a g-fusion frame for $H$.\n\\end{theorem}\n\\begin{proof}\n Let $A,B$ be the g-fusion frame bounds of $\\Lambda$ and $f\\in H$, then\n\\begin{align*}\n\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_{j}\\pi_{W_j}S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2&=\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_{j}\\pi_{W_j}S_{\\Lambda}^{-1}f\\Vert^2\\\\\n&\\leq B\\Vert S_{\\Lambda}^{-1}\\Vert^2 \\Vert f\\Vert^2.\n\\end{align*}\nNow, to get the lower bound, by using (\\ref{3}) we can write\n\\begin{align*}\n\\Vert f\\Vert^4&=\\big\\vert\\langle\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_J\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}f, f\\rangle\\big\\vert^2\\\\\n&=\\big\\vert\\sum_{j\\in\\Bbb J}v_j^2\\langle\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}f, \\Lambda_j\\pi_{W_j}f\\rangle\\big\\vert^2\\\\\n&\\leq\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}f\\Vert^2 \\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j}f\\Vert^2\\\\\n&\\leq \\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2\\big(B\\Vert f\\Vert^2\\big),\n\\end{align*}\ntherefore\n\\begin{align*}\nB^{-1}\\Vert f\\Vert^2\\leq \\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2.\n\\end{align*}\n\\end{proof}\nNow, by Theorem \\ref{dual}, $\\tilde{\\Lambda}=(S^{-1}_{\\Lambda}W_j, \\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}, v_j)$ is a g-fusion frame for $H$. Then, $\\tilde{\\Lambda}$ is called the \\textit{(canonical) dual g-fusion frame} of $\\Lambda$. Let $S_{\\tilde{\\Lambda}}=T_{\\tilde{\\Lambda}}T^*_{\\tilde{\\Lambda}}$ is the g-fusion frame operator of $\\tilde{\\Lambda}$. Then, for each $f\\in H$ we get\n$$T^*_{\\tilde{\\Lambda}}f=\\lbrace v_j\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}\\pi_{S^{-1}_{\\Lambda W_j}}f\\rbrace=\\lbrace v_j\\Lambda_j\\pi_{W_j} S^{-1}_{\\Lambda}f\\rbrace=T^*_{\\Lambda}(S^{-1}_{\\Lambda}f),$$\nso $T_{\\Lambda}T^*_{\\tilde{\\Lambda}}=I_H$. Also, we have for each $f\\in H$,\n\\begin{align*}\n\\langle S_{\\tilde{\\Lambda}}f, f\\rangle&=\\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j \\pi_{W_j}S_{\\Lambda}^{-1}f\\Vert^2\\\\\n&=\\langle S_{\\Lambda}(S_{\\Lambda}^{-1}f), S_{\\Lambda}^{-1}f\\rangle\\\\\n&=\\langle S_{\\Lambda}^{-1}f, f\\rangle\n\\end{align*}\nthus, $S_{\\tilde{\\Lambda}}=S_{\\Lambda}^{-1}$ and by (\\ref{3}), we get for each $f\\in H$\n\\begin{align}\\label{frame}\nf=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f=\n\\sum_{j\\in\\Bbb J}v_j^2\\pi_{\\tilde{W_j}}\\tilde{\\Lambda_j}^*\\Lambda_j\\pi_{W_j}f,\n\\end{align}\nwhere $\\tilde{W_j}:=S^{-1}_{\\Lambda}W_j \\ , \\ \\tilde{\\Lambda_j}:=\\Lambda_j \\pi_{W_j}S_{\\Lambda}^{-1}.$\n\nThe following Theorem shows that the canonical dual g-fusion frame\ngives rise to expansion coefficients with the minimal norm.\n\\begin{theorem}\\label{min}\nLet $\\Lambda$ be a g-fusion frame with canonical dual $\\tilde{\\Lambda}$. \nFor each $g_j\\in H_j$, put $f=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j g_j$. Then\n$$\\sum_{j\\in\\Bbb J}\\Vert g_j\\Vert^2=\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2+\\sum_{j\\in\\Bbb J}\\Vert g_j-v_j^2\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2.$$\n\\end{theorem}\n\\begin{proof}\nWe can write again\n\\begin{align*}\n\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2&=\\langle f, S^{-1}_{\\Lambda}f\\rangle\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\langle\\pi_{W_j}\\Lambda^*_j g_j, S_{\\Lambda}^{-1}f \\rangle\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\Lambda_j\\pi_{W_j}S_{\\Lambda}^{-1}f \\rangle\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}} f \\rangle.\n\\end{align*}\nTherefore, $\\mbox{Im}\\Big(\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}} f \\rangle\\Big)=0$. So\n\\begin{align*}\n\\sum_{j\\in\\Bbb J}\\Vert g_j-v_j^2\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2=\\sum_{j\\in\\Bbb J}\\Vert g_j\\Vert^2 -2\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}} f \\rangle+\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2\n\\end{align*}\nand the proof completes.\n\\end{proof}\n\\section{Gf-Complete and g-Fusion Frame Sequences}\n\\begin{definition}\nWe say that $(W_j, \\Lambda_j)$ is \\textit{gf-complete} , if\n$\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace=H.$\n\\end{definition}\nNow, it is easy to check that $(W_j, \\Lambda_j)$ is gf-complete if and only if\n$$\\lbrace f: \\ \\Lambda_j \\pi_{W_j}f=0 , \\ j\\in\\Bbb J\\rbrace=\\lbrace 0\\rbrace.$$\n\\begin{proposition}\\label{p3}\nIf $\\Lambda=(W_j, \\Lambda_j, v_j)$ is a g-fusion frame for $H$, then $(W_j, \\Lambda_j)$ is a gf-complete.\n\\end{proposition}\n\\begin{proof}\nLet $f\\in(\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace)^{\\perp}\\subseteq H$. For each $j\\in\\Bbb J$ and $g_j\\in H_j$ we have\n$$\\langle \\Lambda_j\\pi_{W_j}f, g_j\\rangle=\\langle f, \\pi_{W_j}\\Lambda^*_j g_j\\rangle=0,$$\nso, $\\Lambda_j\\pi_{W_j}f=0$ for all $j\\in\\Bbb J$. Since $\\Lambda$ is a g-fusion frame for $H$, then $\\Vert f\\Vert=0$. Thus $f=0$ and we get $(\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace)^{\\perp}=\\lbrace0\\rbrace$.\n\\end{proof}\nIn the following, we want to check that if a member is removed from a g-fusion frame, will the new set remain a g-fusion frame or not?\n\\begin{theorem}\\label{del}\nLet $\\Lambda=(W_j, \\Lambda_j, v_j)$ be a g-fusion frame for $H$ with bounds $A, B$ and $\\tilde{\\Lambda}=(S^{-1}_{\\Lambda}W_j, \\Lambda_j \\pi_{W_j}S^{-1}_{\\Lambda}, v_j)$ be a canonical dual g-fusion frame. Suppose that $j_0\\in\\Bbb J$.\n\\begin{enumerate}\n\\item If there is a $g_0\\in H_{j_0}\\setminus\\lbrace 0\\rbrace$ such that $\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=g_0$ and $v_{j_0}=1$, then $(W_j, \\Lambda_j)_{j\\neq j_0}$ is not gf-complete in $H$.\n\\item If there is a $f_0\\in H_{j_0}\\setminus\\lbrace0\\rbrace$ such that $\\pi_{W_{j_0}}\\Lambda^*_{j_0}\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0=f_0$ and $v_{j_0}=1$, then $(W_j, \\Lambda_j)_{j\\neq j_0}$ is not gf-complete in $H$.\n\\item If $I-\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_{j_0}}}\\tilde{\\Lambda}^*_{j_0}$ is bounded invertible on $H_{j_0}$, then $(W_j, \\Lambda_j, v_j)_{j\\neq j_0}$ is a g-fusion frame for $H$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\\textit{(1).} Since $\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\in H$, then by (\\ref{frame}),\n$$\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0.$$\nSo,\n$$\\sum_{j\\neq j_0}v_j^2\\pi_{W_j}\\Lambda^*_j\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=0.$$\nLet $u_{j_0, j}:=\\delta_{j_0, j}g_0$, thus\n$$\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j u_{j_0, j}.$$\nThen, by Theorem \\ref{min}, we have\n$$\\sum_{j\\in\\Bbb J}\\Vert u_{j_0, j}\\Vert^2=\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\Vert^2+\\sum_{j\\in\\Bbb J}\\Vert v_j^2\\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0-u_{j_0, j}\\Vert^2.$$\nConsequently,\n$$\\Vert g_0\\Vert^2=\\Vert g_0\\Vert^2+2\\sum_{j\\neq j_0}v_j^2\\Vert \\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\Vert^2$$\nand we get $\\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=0$.\nTherefore,\n$$\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=0.$$\nBut, $g_0=\\tilde\\Lambda_{j_0}^*\\pi_{\\tilde{W}_{j_0}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\Lambda_{j_0}\\pi_{W_{j_0}}S^{-1}_{\\Lambda}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\neq0$, which implies that $S^{-1}_{\\Lambda}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\neq0$ and this means that $(W_j, \\Lambda_j)_{j\\neq j_0}$ is not gf-complete in $H$.\n\n\\textit{(2).} Since $\\pi_{W_{j_0}}\\Lambda^*_{j_0}\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0=f_0\\neq0$, we obtain $\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0\\neq0$ and\n$$\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0=\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0.$$\nNow, the conclusion follows from \\textit{(1)}.\n\n\\textit{(3)}. Using (\\ref{frame}), we have for any $f\\in H$\n$$\\Lambda_{j_0}\\pi_{W_{j_0}}f=\\sum_{j\\in\\Bbb J}v_j^2\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_j}}\\tilde{\\Lambda}^*_j\\Lambda_j\\pi_{W_j}f.$$\nSo,\n\\begin{equation}\\label{com}(I-\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_{j_0}}}\\tilde{\\Lambda}^*_{j_0})\\Lambda_{j_0}\\pi_{W_{j_0}}f=\\sum_{j\\neq j_0}v_j^2\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_j}}\\tilde{\\Lambda}^*_j\\Lambda_j\\pi_{W_j}f.\n\\end{equation}\nOn the other hand, we can write\n\\begin{small}\n\\begin{align*}\n\\big\\Vert\\sum_{j\\neq j_0}v_j^2\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_j}}\\tilde{\\Lambda}^*_j\\Lambda_j\\pi_{W_j}f\\big\\Vert^2&=\\sup_{\\Vert g\\Vert=1}\\big\\vert\\sum_{j\\neq j_0}v_j^2\\big\\langle \\Lambda_j\\pi_{W_j}f, \\tilde{\\Lambda}_j\\pi_{\\tilde{W}_j}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g \\big\\rangle\\big\\vert^2\\\\\n&\\leq\\big(\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2\\big)\\sup_{\\Vert g\\Vert=1}\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda}_j\\pi_{\\tilde{W}_j}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g\\Vert^2\\\\\n&\\leq\\tilde{B}\\Vert\\Lambda_{j_0}\\Vert^2(\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2)\n\\end{align*}\n\\end{small}\nwhere, $\\tilde{B}$ is the upper bound of $\\tilde\\Lambda$. Now, by (\\ref{com}), we have\n\\begin{equation*}\n\\Vert\\Lambda_{j_0}\\pi_{W_{j_0}}f\\Vert^2\\leq\\Vert(I-\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_{j_0}}}\\tilde{\\Lambda}^*_{j_0})^{-1}\\Vert^2 \\tilde{B}\\Vert\\Lambda_{j_0}\\Vert^2(\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2).\n\\end{equation*}\nTherefore, there is a number $C>0$ such that\n$$\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2\\leq C\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2$$\nand we conclude for each $f\\in H$\n$$\\frac{A}{C}\\Vert f\\Vert^2\\leq\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2\\leq B\\Vert f\\Vert^2.$$\n\\end{proof}\n\\begin{theorem}\n$\\Lambda$ is a g-fusion frame for $H$ with bounds $A,B$ if and only if the following two conditions are satisfied:\n\\begin{enumerate}\n\\item[(I)] The pair $(W_j, \\Lambda_j)$ is gf-complete.\n\\item[(II)] The operator\n$$T_{\\Lambda}: \\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\mapsto \\sum_{j\\in\\Bbb J}v_j\\pi_{W_j}\\Lambda_j^* f_j$$\nis a well-defined from $\\mathscr{H}_2$ into $H$ and for each $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathcal{N}^{\\perp}_{T_{\\Lambda}}$,\n\\begin{equation}\\label{e7}\nA\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2\\leq \\Vert T_{\\Lambda}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\Vert^2\\leq B\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2.\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nFirst, suppose that $\\Lambda$ is a g-fusion frame. By Proposition \\ref{p3}, (I) is satisfied. By Theorem \\ref{t2}, $T_{\\Lambda}$ is a well-defined from $\\mathscr{H}_2$ into $H$ and $\\Vert T_{\\Lambda}\\Vert^2\\leq B$. Now, the right-hand inequality in (\\ref{e7}) is proved.\n\nBy Theorem \\ref{2.3}, $T_{\\Lambda}$ is surjective. So, $\\mathcal{R}_{T^*_{\\Lambda}}$ is closed. Thus\n$$\\mathcal{N}^{\\perp}_{T_{\\Lambda}}=\\overline{\\mathcal{R}_{T^*_{\\Lambda}}}=\\mathcal{R}_{T^*_{\\Lambda}}.$$\nNow, if $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathcal{N}^{\\perp}_{T_{\\Lambda}}$, then\n$$\\lbrace f_j\\rbrace_{j\\in\\Bbb J}=T^*_{\\Lambda}g=\\lbrace v_j\\Lambda_j \\pi_{W_j}g\\rbrace_{j\\in\\Bbb J}$$\nfor some $g\\in H$. Therefore\n\\begin{align*}\n(\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2)^2&=(\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_j \\pi_{W_j}g\\Vert^2)^2\n\\vert\\langle S_{\\Lambda}(g), g\\rangle\\vert^2\\\\\n&\\leq\\Vert S_{\\Lambda}(g)\\Vert^2 \\Vert g\\Vert^2\\\\\n&\\leq\\Vert S_{\\Lambda}(g)\\Vert^2 \\big(\\frac{1}{A}\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_j \\pi_{W_j}g\\Vert^2\\big).\n\\end{align*}\nThis implies that\n$$A\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2\\leq\\Vert S_{\\Lambda}(g)\\Vert^2=\\Vert T_{\\Lambda}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\Vert^2$$\nand (II) is proved.\n\nConversely, Let $(W_j, \\Lambda_j)$ be gf-complete and inequality\n(\\ref{e7}) is satisfied. Let $\\lbrace t_j\\rbrace_{j\\in\\Bbb\nJ}=\\lbrace f_j\\rbrace_{j\\in\\Bbb J}+\\lbrace g_j\\rbrace_{j\\in\\Bbb J}$,\nwhere $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathcal{N}_{T_{\\Lambda}}$\nand $\\lbrace g_j\\rbrace_{j\\in\\Bbb\nJ}\\in\\mathcal{N}_{T_{\\Lambda}}^{\\perp}$. We get\n\\begin{align*}\n\\Vert T_{\\Lambda}\\lbrace t_j\\rbrace_{j\\in\\Bbb J}\\Vert^2&=\\Vert T_{\\Lambda}\\lbrace g_j\\rbrace_{j\\in\\Bbb J}\\Vert^2\\\\\n&\\leq B\\sum_{j\\in\\Bbb J}\\Vert g_j\\Vert^2\\\\\n&\\leq B\\Vert \\lbrace f_j\\rbrace+\\lbrace g_j\\rbrace\\Vert^2\\\\\n&=B\\Vert\\lbrace t_j\\rbrace_{j\\in\\Bbb J}\\Vert^2.\n\\end{align*}\nThus, $\\Lambda$ is a g-fusion Bessel sequence.\n\nAssume that $\\lbrace y_n\\rbrace$ is a sequence of members of $\\mathcal{R}_{T_{\\Lambda}}$ such that $y_n\\rightarrow y$ for some $y\\in H$. So, there is a $\\lbrace x_n\\rbrace\\in\\mathcal{N}_{T_{\\Lambda}}$ such that $T_{\\Lambda}\\{x_n\\}=y_n$. By (\\ref{e7}), we obtain\n\\begin{align*}\nA\\Vert\\lbrace x_n-x_m\\rbrace\\Vert^2&\\leq\\Vert T_{\\Lambda}\\lbrace x_n-x_m\\rbrace\\Vert^2\\\\\n&=\\Vert T_{\\Lambda}\\lbrace x_n\\rbrace -T_{\\Lambda}\\lbrace x_m\\rbrace\\Vert^2\\\\\n&=\\Vert y_n-y_m\\Vert^2.\n\\end{align*}\nTherefore, $\\lbrace x_n\\rbrace$ is a Cauchy sequence in $\\mathscr{H}_2$. Therefore $\\lbrace x_n\\rbrace$ converges to some $x\\in \\mathscr{H}_2$, which by continuity of $T_{\\Lambda}$, we have $y=T_{\\Lambda}(x)\\in\\mathcal{R}_{T_{\\Lambda_j}}$. Hence $\\mathcal{R}_{T_{\\Lambda}}$ is closed. Since $\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda_{j}^{\\ast}(H_j)\\rbrace\\subseteq\\mathcal{R}_{T_{\\Lambda}}$, by (I) we get $\\mathcal{R}_{T_{\\Lambda}}=H$.\n\n Let $T_{\\Lambda}^\\dagger$ denotes the pseudo-inverse of $T_{\\Lambda}$. By Lemma \\ref{Ru}(4), $T_{\\Lambda}T_{\\Lambda}^{\\dagger}$ is the orthogonal projection onto $\\mathcal{R}_{T_{\\Lambda}}=H$. Thus for any $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathscr{H}_2 $,\n\\begin{eqnarray*}\nA\\Vert T_{\\Lambda}^{\\dagger}T_{\\Lambda}\\lbrace f_j\\rbrace\\Vert^2\\leq\\Vert T_{\\Lambda}T_{\\Lambda}^{\\dagger}T_{\\Lambda}\\lbrace f_j\\rbrace \\Vert^2=\\Vert T_{\\Lambda}\\lbrace f_j\\rbrace\\Vert^2.\n\\end{eqnarray*}\nBy Lemma \\ref{Ru} (4), $\\mathcal{N}_{{T}_{\\Lambda}^{\\dagger}}=\\mathcal{R}^{\\bot}_{T_{\\Lambda}}$, therefore\n\\begin{eqnarray*}\n\\Vert T_{\\Lambda}^\\dagger\\Vert^2\\leq\\frac{1}{A}.\n\\end{eqnarray*}\nAlso by Lemma \\ref{Ru}(2), we have\n$$ \\Vert(T_{\\Lambda}^\\ast)^{\\dagger}\\Vert^2\\leq\\frac{1}{A}.$$\nBut $(T_{\\Lambda}^\\ast)^{\\dagger}T_{\\Lambda}^\\ast$ is the\northogonal projection onto\n\\begin{eqnarray*}\n\\mathcal{R}_{(T_{\\Lambda}^\\ast)^\\dagger}=\\mathcal{R}_{(T_{\\Lambda}^\\dagger)^\\ast}=\\mathcal{N}_{T_{\\Lambda}^\\dagger}^{\\bot}=\\mathcal{R}_{T_{\\Lambda}}=H.\n\\end{eqnarray*}\nSo, for all $f\\in H$\n\\begin{align*}\n\\Vert f\\Vert^2&=\\Vert(T_{\\Lambda}^\\ast)^{\\dagger}T_{\\Lambda}^\\ast f\\Vert^2\\\\\n&\\leq \\frac{1}{A}\\Vert T_{\\Lambda}^\\ast f\\Vert^2\\\\\n&=\\frac{1}{A}\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2.\n\\end{align*}\nThis implies that $\\Lambda$ satisfies the lower g-fusion frame condition.\n\\end{proof}\nNow, we can define a g-fusion frame sequence in the Hilbert space.\n\\begin{definition}\nWe say that $\\Lambda$ is a \\textit{g-fusion frame sequence} if it is a g-fusion frame for $\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace$.\n\\end{definition}\n\\begin{theorem}\\label{2.6}\n$\\Lambda$ is a g-fusion frame sequence if and only if the operator\n \\begin{align*}\nT_{\\Lambda}&:\\mathscr{H}_2\\longrightarrow H\\\\\nT_{\\Lambda}(\\lbrace f_j\\rbrace_{j\\in\\Bbb J})&=\\sum_{j\\in\\Bbb J}v_j \\pi_{W_j}\\Lambda_{j}^{*}f_j\n\\end{align*}\nis a well-defined and has closed range.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{2.3}, it is enough to prove that if $T_{\\lambda}$ has closed range, then $\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace=\\mathcal{R}_{T_{\\Lambda}}$.\nLet $f\\in\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace$, then\n$$f=\\lim_{n\\rightarrow\\infty}g_n , \\ \\ \\ g_n\\in\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace\\subseteq \\mathcal{R}_{T_{\\Lambda}}=\\overline{\\mathcal{R}}_{T_{\\Lambda}}.$$\nTherefore, $\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace\\subseteq\\overline{\\mathcal{R}}_{T_{\\Lambda}}=\\mathcal{R}_{T_{\\Lambda}}$. On the other hand, if $f\\in\\mathcal{R}_{T_{\\Lambda}}$, then\n$$f\\in\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace\\subseteq\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace$$\n and the proof is completed.\n\\end{proof}\n\\begin{theorem}\n $\\Lambda$ is a g-fusion frame sequence if and only if\n\\begin{equation}\\label{4}\nf \\longmapsto \\lbrace v_j \\Lambda_j \\pi_{W_j}f\\rbrace_{j\\in\\Bbb J}\n\\end{equation}\ndefines a map from $H$ onto a closed subspace of $\\mathscr{H}_2$.\n\\end{theorem}\n\\begin{proof}\nLet $\\Lambda$ be a g-fusion frame sequence. Then, by Theorem \\ref{2.6}, $T_{\\lambda}$ is well-defined and $\\mathcal{R}_{T_{\\Lambda}}$ is closed. So, $T^*_{\\Lambda}$ is well-defined and has closed range. Conversely, by hypothesis, for all $f\\in H$\n$$\\sum_{j\\in\\Bbb J}\\Vert v_j \\Lambda_j \\pi_{W_j}f\\Vert^2<\\infty.$$\nLet\n$$B:=\\sup\\big\\lbrace \\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2 : \\ \\ f\\in H, \\ \\Vert f\\Vert=1\\big\\rbrace$$\nand suppose that $g_j\\in H_j$ and $\\Bbb I\\subseteq\\Bbb J$ be finite. We can write\n\\begin{align*}\n\\Vert\\sum_{j\\in\\Bbb I}v_j \\pi_{W_j}\\Lambda^*_j g_j\\Vert^2&=\\Big(\\sup_{\\Vert f\\Vert=1}\\big\\vert\\langle\\sum_{j\\in\\Bbb I}v_j \\pi_{W_j}\\Lambda^*_j g_j, f\\rangle\\big\\vert\\Big)^2\\\\\n&\\leq\\Big(\\sup_{\\Vert f\\Vert=1}\\sum_{j\\in\\Bbb I}v_j\\big\\vert\\langle g_j, \\Lambda_j \\pi_{W_j}f\\rangle\\big\\vert\\Big)^2\\\\\n&\\leq\\big(\\sum_{j\\in\\Bbb I}\\Vert g_j\\Vert^2\\big)\\big(\\sup_{\\Vert f\\Vert=1}\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2\\big)\\\\\n&\\leq B\\big(\\sum_{j\\in\\Bbb I}\\Vert g_j\\Vert^2\\big)\n\\end{align*}\nThus, by Corollary \\ref{cor}, $\\Lambda$ is a g-fusion Bessel sequence for $H$. Therefore, $T_{\\Lambda}$ is well-defined and bounded. Furthermore, if the range of the map (\\ref{4}) is closed, the same is true for $T_{\\Lambda}$. So, by Theorem \\ref{2.6}, $\\Lambda$ is a g-fusion frame sequence.\n\\end{proof}\n\\begin{theorem}\nLet $\\Lambda=(W_j, \\Lambda_j, v_j)$ be a g-fusion frame sequence Then, it is a g-fusion frame for $H$ if and only if the map\n\\begin{equation}\\label{5}\nf \\longmapsto \\lbrace v_j \\Lambda_j \\pi_{W_j}f\\rbrace_{j\\in\\Bbb J}\n\\end{equation}\nfrom $H$ onto a closed subspace of $\\mathscr{H}_2$ be injective.\n\\end{theorem}\n\\begin{proof}\nSuppose that the map (\\ref{5}) is injective and $v_j \\Lambda_j \\pi_{W_j}f=0$ for all $j\\in\\Bbb J$. Then, the value of the map at $f$ is zero. So, $f=0$. This means that $(W_j, \\Lambda_j)$ is gf-complete. Since, $\\Lambda$ is a g-fusion frame sequence, so, it is a g-fusion frame for $H$.\n\nThe converse is evident.\n\\end{proof}\n\\begin{theorem}\nLet $\\Lambda$ be a g-fusion frame for $H$ and $u\\in\\mathcal{B}(H)$. Then $\\Gamma:=(uW_j, \\Lambda_j u^*, v_j)$ is a g-fusion frame sequence if and only if $u$ has closed range.\n\\end{theorem}\n\\begin{proof}\nAssume that $\\Gamma$ is a g-fusion frame sequence. So, by Theorem \\ref{2.6}, $T_{\\Lambda u^*}$ is a well-defined operator from $\\mathscr{H}_2$ into $H$ with closed range. If $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathscr{H}_2$, then\n\\begin{align*}\nuT_{\\Lambda}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}&=\\sum_{j\\in\\Bbb J}v_ju\\pi_{W_j}\\Lambda_j^* f_j\\\\\n&=\\sum_{j\\in\\Bbb J}v_j\\pi_{uW_j}u\\Lambda_j^* f_j\\\\\n&=\\sum_{j\\in\\Bbb J}v_j\\pi_{uW_j}(\\Lambda_j u^*)^* f_j\\\\\n&=T_{\\Lambda u^*}\\lbrace f_j\\rbrace_{j\\in\\Bbb J},\n\\end{align*}\ntherefore $uT_{\\Lambda}=T_{\\Lambda u^*}$. Thus $uT_{\\Lambda}$ has closed range too. Let $y\\in\\mathcal{R}_u$, then there is $x\\in H$ such that $u(x)=y$. By Theorem \\ref{2.3}, $T_{\\Lambda}$ is surjective, so there exist $\\{f_j\\}_{j\\in\\Bbb J}\\in\\mathscr{H}_2$ such that \n$$y=u(T_{\\Lambda}\\{f_j\\}_{j\\in\\Bbb J}).$$\nThus, $\\mathcal{R}_{u}=\\mathcal{R}_{uT_{\\Lambda}}$ and $u$ has closed range.\n\nFor the opposite implication, let\n\\begin{align*}\nT_{\\Lambda u^*}:&\\mathscr{H}_2\\longrightarrow H\\\\\nT_{\\Lambda u^*}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}&=\\sum_{j\\in\\Bbb J}v_j\\pi_{uW_j}(\\Lambda_j u^*)^* f_j.\n\\end{align*}\nHence, $T_{\\Lambda u^*}=uT_{\\Lambda}$. Since, $T_{\\Lambda}$ is surjective, so $T_{\\Lambda u^*}$ has closed range and by Theorem \\ref{t2}, is well-defined. Therefore, by Theorem \\ref{2.6}, the proof is completed.\n\\end{proof}\n\\section{Conclusions}\nIn this paper, we could transfer some common properties in general frames to g-fusion frames with the definition of the g-fusion frames and their operators. Afterward, we reviewed a basic theorem about deleting a member in Theorem \\ref{del} with the definition of the dual g-fusion frames and the gf-completeness. In this theorem, the defined operator in part \\textit{3} could be replaced by some other operators which are the same as the parts \\textit{1} and \\textit{2}; this is an open problem at the moment. Eventually, the g-fusion frame sequences and their relationship with the closed range operators were defined and presented.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n In 1992, the pioneering work of Allen and coworkers~\\cite{Allen1992} showed that twisted light beams also posses orbital angular momentum (OAM) in addition to spin angular momentum. Since then, research on twisted (or vortex) light beams has been a burgeoning research area in the scientific community \\cite{Yao,opticaloam1,opticaloam2}. The presence of extra angular momentum in the twisted light beam is associated with an azimuthally varying phase of the beam \\cite{padgett2004light}. In addition to OAM, twisted light beams exhibit helical phase fronts and phase singularity along the beam axis \\cite{twistedlight}. In contrast to the plane waves, the twisted light beams have a non-uniform intensity distribution across the beam cross section. In particular, we observe concentric rings in the beam cross section of the twisted light beams \\cite{molina2007twisted}. \n\\par Twisted light beams offer a plethora of applications because of their well-defined projection of the total angular momentum (TAM) upon the propagation axis. For example, twisted light beams have been utilized as alphabets to encode information beyond one bit per single photon \\cite{twistalphabets}. This makes twisted light beams such as Laguerre-Gaussian (LG) beams very suitable for a number of applications such as high-dimensional quantum information \\cite{highquantuminfo}, quantum memories \\cite{quantummemory}, quantum cryptography \\cite{quantumcrypto}. In addition, LG beams are also used in high harmonic generation \\cite{Willi} and optical tweezers \\cite{opticaltweezers}. \n\\par Atomic processes with twisted light beams, such as photo-ionization of atoms \\cite{photoionization1,photoionization2} and the scattering of twisted light beams by ions \\cite{rayleighscatt} or electrons \\cite{comptonscattering} have also gained much attention. It was shown that OAM of the twisted light beam can strongly modify the angular distribution of the emitted photo-electrons during the ionization process \\cite{angulardistribution}. In addition to photo-ionization and scattering processes, the photo-excitation of atoms by twisted light beams has attracted much interest in recent years. The transfer of OAM from the LG beams to the bound electrons in an atomic system during the excitation process was observed for the first time by Schmiegelow \\textit{et al.} \\cite{schmiegelow2016}. This experiment investigated the electric-quadrupole transition in the Ca$^{+}$ ion positioned on the beam axis and demonstrated the suppression of AC-Stark shift in the dark centre of the LG beams. Later, a theoretical study on the photo-excitation of atoms by twisted light beams showed that the magnetic sub-level population and fluorescence following the excitation process varied significantly with OAM of the LG beams \\cite{peshkov}.\n\\par While most studies on the photo-excitation of atoms have focused so far on circularly polarized LG beams, twisted light beams support many different polarization states, such as cylindrical polarization. However, photo-excitation of atoms by cylindrically polarized LG beams remains less explored. In this work, we therefore analyze the multipole distribution of the cylindrically polarized LG beams. We especially explore the dependence of the multipole distribution of cylindrically polarized LG beams on the beam waist and the radial distance from the beam axis. We show that the varying multipole distribution in the beam cross section influences the magnetic sub-level population of the target atom. As an example, we consider the electric-quadrupole transition $4s\\;^{2}S_{1\/2} \\rightarrow 3d\\;^{2}D_{5\/2}$ in the Ca$^{+}$ ion. This particular transition has already been observed in the past experimental as well as studied in theoretical works and thus serves as an ideal case for a comparison of our theoretical work. \n\\par This paper is structured as follows. In Sec. \\ref{subsec: Circularly polarized LG beams} we first recall circularly polarized LG beams. We then use the knowledge of circularly polarized LG beams to understand the cylindrical polarization and to construct the vector potential of cylindrically polarized LG beams in Sec. \\ref{subsec:Cylindrically polarized LG beams}. Moreover, we expand the vector potential of LG beam into its multipole components to obtain complex weight factors for a given polarization in Sec. \\ref{subsec:Multipole expansion of a vector potential of the LG beams}. We evaluate strength of the electric-quadrupole field in the beam cross section of a radially polarized LG beam with respect to circularly polarized LG beam in Sec. \\ref{subsec:Distribution of electric-quadrupole component in the beam cross section of cylindrically polarized LG beam}. Furthermore, in Sec. \\ref{subsec:Effect of the beam waist and radial position on the multipole distribution of LG beams} we analyze distribution of the projection of electric-quadrupole field in the beam cross section with respect to the beam waist $w_{o}$ and radial distance $b$ from the beam axis of cylindrically polarized LG beams. In the Sec. \\ref{subsec:Interaction of LG beams with the target atom}, we analyze the photo-excitation of the target Ca$^{+}$ ion by LG beams using the outcome of the multipole distribution and the selection rules. Finally, a summary of the paper is given in the Sec. \\ref{sec:summary}\n\n\\section{Theoretical background}\n\\subsection{Circularly polarized LG beams} \n\\label{subsec: Circularly polarized LG beams}\nLG beams are paraxial twisted light beams whose amplitude distribution $u(r)$ is known to satisfy the paraxial wave equation \\cite{siegman1986lasers}\n \\begin{equation}\n \\nabla^{2}\\;u\\;+2\\;i\\;k\\;\\frac{\\partial}{\\partial z}u \\; = \\; 0.\n \\end{equation}\nThis \\textit{paraxial wave} approximation is valid, if the amplitude distribution $u(\\bm{r})$ changes slowly with the distance $z$ and this $z$ dependence is less compared to variations of $u(\\bm{r})$ in the transverse direction \n\\begin{equation}\n |\\frac{\\partial^{2}u}{\\partial z^{2}}| \\ll |2k \\; \\frac{\\partial u}{\\partial z}| \\; , \\; |\\frac{\\partial^{2}u}{\\partial z^{2}}| \\ll |\\partial_{t}^{2}u|.\n\\end{equation}\nIn cylindrical coordinates the amplitude distribution of the LG beam is given by\n \\begin{widetext}\n \\begin{equation}\\label{eq:amplitude}\n u(\\bm{r}) = \\frac{1}{w(z)} \\left( \\frac{\\sqrt{2}r}{w(z)} \\right)^{m_{l}} \\textrm{exp}\\left(- \\frac{r^{2}}{w^{2}(z)} \\right) L^{m_{l}}_{p}\\left(\\frac{2r^{2}}{w^{2}(z)} \\right)\n \\textrm{exp}\\left[ im_{l}\\phi + \\frac{ikr^{2}z}{2(z^{2}+z^{2}_{R})} - i(2p+m_{l}+1) \\textrm{arctan}\\left(\\frac{z}{z_{R}}\\right) \\right],\n \\end{equation}\nwhere $m_{l}$ is the projection of the OAM upon the propagation axis, $p$ is the radial index of the LG beam, $z_{R}$ is the (so-called) Rayleigh range, $w(z)$ is the beam width of the LG beam. The beam width $w(z)$ of the LG beam varies along the propagation distance and is minimum for $z = 0$. This minimum beam width of a LG beam is known also as beam waist $w_{o}$ $\\equiv w(z = 0)$. Equation (\\ref{eq:amplitude}) can be used to obtain the intensity distribution $|u(r)|^{2}$ of the LG beam which exhibit a concentric ring-like structure in the beam cross section.\n\\par The wave amplitude of the circularly polarized LG beam in momentum space is expressed as a Fourier transformation of the amplitude distribution (\\ref{eq:amplitude}). Then the vector potential of a circularly polarized LG beam in Coulomb gauge is given by (see \\cite{peshkov} for a detailed derivation)\n \\begin{equation} \\label{eq:vectorpotential}\n \\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}\\bm{(r)} = \\int d^{2}\\bm{k}_{\\bot} v_{pm_{l}}(k_{\\bot})\\; e^{i(m_{l}+\\lambda)\\phi_{k}}\\; \\mathbf{e}_{k,\\lambda} \\;e^{i \\mathbf{k}\\cdot \\mathbf{r}} ,\n\\end{equation}\nwhere $v_{pm_{l}}(k_{\\bot})\\; e^{i(m_{l}+\\lambda)\\phi_{k}}$ is the momentum space wave function and $\\mathbf{e}_{k,\\lambda} \\;e^{i \\mathbf{k}\\cdot \\mathbf{r}}$ is the vector potential of a circularly polarized plane waves. In the above equation momentum space wave function $v_{pm_{l}}(k_{\\bot})$ is given by\n \\begin{equation}\n v_{pm_{l}}(k_{\\bot}) = \\frac{(-i)^{m_{l}}}{ w_{o} 4\\pi} \\; e^{-k_{\\bot}^{2}w^{2}_{o}\/4} \\; \\left(\\frac{k_{\\bot} w_{o}}{2}\\right)^{m_{l}} \n \\sum^{p}_{\\beta=0} (-1)^{\\beta} \\; 2^{\\beta + m_{l}\/2} \\; \\left( p+m_{l} \\atop p-\\beta \\right) \\;L^{m_{l}}_{\\beta}\\left( \\frac{k_{\\bot}^{2}w^{2}_{o}}{4} \\right).\n\\end{equation}\n\\end{widetext}\nAs seen from equation (\\ref{eq:vectorpotential}), the LG beams with circular polarization can be expressed as a coherent superposition of circularly polarized plane waves in the momentum space. The momentum vector $\\bm{k}$ of these plane waves lie on the surface of a cone in the momentum space with an opening angle of $\\theta_{k}$ = arctan$(k_{\\bot}\/k_{z})$. \n\n\\subsection{Cylindrically polarized LG beams}\n\\label{subsec:Cylindrically polarized LG beams}\nCylindrically polarized LG beams can be constructed as a linear combination of two circularly polarized LG modes \\cite{cylindrica}. The superposition of two LG modes results in non-separable spatial and polarization modes \\cite{andrews2012} which affects the state of polarization across the beam cross section. In contrast to LG beams with circular polarization, the state of polarization across the beam cross section in cylindrically polarized LG beams is spatially in-homogeneous \\cite{nonuniformsop}. Beams which are linear combinations of two LG modes are known as vector beams \\cite{vectorsolution} and constitute a vector solutions to the paraxial wave equation. \n\\par Radial and azimuthal polarizations are two special cases of a cylindrical polarization. In particular, radially and azimuthally polarized LG beams are linear combinations of two LG modes with the projection of OAM $m_{l} = \\pm1$ and helicity $\\lambda = \\pm1$ \\cite{linearcombinationofcircular}. The radially polarized LG beams have a state of linear polarization aligned along the radial direction \\cite{radiallypol,sabrina}. That is, the electric field of the radially polarized LG beam always points in the radial direction and is perpendicular to the beam axis. The vector potential of a radially polarized LG beam is constructed as a linear combination of vector potential of right circularly polarized $\\mathbf{A}^{\\mathrm{cir}}_{m_{l}=-1,\\lambda=1,p}(\\mathbf{r})$ and left circularly polarized $\\mathbf{A}^{\\mathrm{cir}}_{m_{l}=1,\\lambda=-1,p}(\\mathbf{r})$ LG beams and is given by\n\\begin{equation}\n \\mathbf{A}^{\\mathrm{rad}}_{p}(\\mathbf{r}) = \\frac{-i}{\\sqrt{2}} \\left[ \\bm{A}^{\\mathrm{cir}}_{m_{l}=1,\\lambda =-1,p}(\\mathbf{r}) + \\bm{A}^{\\mathrm{cir}}_{m_{l}=-1,\\lambda =1,p}(\\mathbf{r}) \\right].\n\\end{equation}\n\n\\par For an azimuthally polarized LG beam, the state of linear polarization is always aligned tangential \\cite{azimuthal,sabrina} to the ring of the beam. The electric field direction of the azimuthally polarized LG beams is always perpendicular to the radial direction and its vector potential is given by\n \\begin{equation}\n \\mathbf{A}^{\\mathrm{azim}}_{p}(\\mathbf{r}) = \\frac{1}{\\sqrt{2}} \n \\left[ \\bm{A}^{\\mathrm{cir}}_{m_{l}=1,\\lambda=-1,p}(\\mathbf{r}) - \\bm{A}^{\\mathrm{cir}}_{m_{l}=-1,\\lambda=1,p}(\\mathbf{r}) \\right].\n\\end{equation}\nFurther, we use the vector potential of LG beam of a given polarization to obtain the complex weight factors using the multipole expansion in the next section.\n\n\\subsection{Multipole expansion of a vector potential of the LG beam }\n\\label{subsec:Multipole expansion of a vector potential of the LG beams}\nA multipole expansion of the radiation field enables us generally to expand the vector potential in angular momentum basis \\cite{brinkangularmomentum,rose1995elementary}. It helps us to analyze the contributions of the individual multipole components to the photo-excitation of the atoms. Since the intensity distribution of the LG beam is nonuniform in the beam cross section, we perform multipole expansion of the vector potential of a LG beam at a radial distance $b$ from the beam axis. That is, when the z axis is translated by a vector $\\bm{b} = b\\bm{e_{x}}$ from the beam axis.\nThe vector potential of a circularly polarized LG beam at a position $b$ from the beam axis is given by\n\\begin{equation}\n \\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}(\\bm{r}; b,w_{o}) = \\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}(\\bm{r}) \\; e^{i\\bm{p}\\cdot\\bm{b}}.\n\\end{equation}\nThe multipole expansion of the circularly polarized LG beam is given by\n\\begin{equation}\n \\small{\\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}(\\bm{r};b,w_{o} ) \\; = \\; \\sum_{L,M,\\Lambda} W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L,M,\\Lambda; b,w_{o}) \\; \\mathbf{a}^{\\Lambda}_{L,M}\\mathbf{(r)}},\n\\end{equation}\nwhere $L$ and $M$ are the eigenvalues of the TAM and the projection of TAM operators, respectively. The $W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L,M,\\Lambda;b,w_{o})$ is the complex weight factor of the expansion which depends on the radial distance from the beam axis $b$, the beam waist $w_{o}$, projection of orbital angular momentum $m_{l}$, the radial index $p$ and the helicity $\\lambda$. The multipole expansion expresses the vector potential of the LG beam as a linear combination of electric ($\\Lambda = 1$) and magnetic ($\\Lambda = 0$) multipole components $\\mathbf{a}^{\\Lambda}_{L,M}\\mathbf{(r)}$. Mathematically, $\\mathbf{a}^{\\Lambda}_{L,M}\\mathbf{(r)}$ are expressed in terms of vector spherical harmonics of rank $L$ \\cite{reftovectorspherical,johnson2007atomic}. For circularly polarized LG beams, the complex weight factors is given by\n\\begin{widetext}\n\\begin{align}\n W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L,M,\\Lambda;b,w_{o}) &= \\sum_{L,M,\\Lambda} \\;\\sum^{p}_{\\beta=0} \\;(i \\lambda)^{\\Lambda}\\;(-1)^{\\beta} \\; 2^{\\beta +\\frac{m_{l}}{2}} \\; \\left( p+m_{l} \\atop p-\\beta \\right) \\; \\frac{(-i)^{m_{l}}w_{o}}{2\\pi}\\; \n (i)^{L+m_{l}+\\lambda-M} \\; (2 L + 1)^{1\/2} \\\\\\nonumber \n & \\times d^{L}_{M,\\lambda}(\\theta_{k}) \\;\\int_{0}^{\\infty} k_{\\bot} dk_{\\bot}\\; e^{-\\frac{k_{\\bot}^{2}w_{o}^{2}}{4}} \\left( \\frac{k_{\\bot} w_{o}}{2} \\right)^{m_{l}} \\; L^{m_{l}}_{\\beta}\\left( \\frac{(k_{\\bot} w_{o})^{2}}{4} \\right) \\; J_{m_{l}+\\lambda-M}(k_{\\bot} b).\n\\end{align}\n\\vspace{0.28in}\nUsing the complex weight factors of circularly polarized LG beams, the complex weight factor for the radially polarized LG beam reads as \n\\begin{align}\n W^{\\mathrm{rad}}_{p}(L,M,\\Lambda;b,w_{o}) = \\frac{-i}{\\sqrt{2}}\\left[ W^{\\mathrm{cir}}_{m_{l}= 1,\\lambda = -1,p}(L,M,\\Lambda;b,w_{o}) + W^{\\mathrm{cir}}_{m_{l} = -1,\\lambda = 1,p}(L,M,\\Lambda;b,w_{o}) \\right]\n\\end{align}\n\\vspace{0.28in}\nand for the azimuthally polarized LG beam as\n\\begin{align}\n W^{\\mathrm{azim}}_{p}(L,M,\\Lambda;b,w_{o}) = \\frac{1}{\\sqrt{2}}\\left[ W^{\\mathrm{cir}}_{m_{l}= 1,\\lambda = -1,p}(L,M,\\Lambda;b,w_{o}) - W^{\\mathrm{cir}}_{m_{l} = -1,\\lambda = 1,p}(L,M,\\Lambda;b,w_{o}) \\right].\n\\end{align}\n\\end{widetext}\nWith the help of complex weight factors, we can study strength of the individual multipole components of the radiation field. Since these complex weight factors depend on the radial distance $b$ and the beam waist $w_{o}$, we can control the multipole distribution of the LG beam by carefully choosing $b$ and $w_{o}$. In particular, we analyze the distribution of electric-quadrupole component and its individual projection component in the beam cross section of the LG beam in the next section. \n\n\n\\section{Results and Discussion}\n\\subsection{Distribution of electric-quadrupole field in the beam cross section of cylindrically polarized LG beam}\n\\label{subsec:Distribution of electric-quadrupole component in the beam cross section of cylindrically polarized LG beam}\n\nIn the last section, we performed multipole expansion of the vector potential of a LG beam to obtain the complex weight factors $W_{m_{l},\\lambda,p}(L,M,\\Lambda; b,w_{o})$ for both circularly and cylindrically polarized LG beams. The strength of a multipole component $L$, in particular electric-quadrupole component (L = 2), in the beam cross section is given by~$\\sum_{M=-L}^{+L}|W_{m_{l},\\lambda,p}(L=2,M,\\Lambda=1;b,w_{o})|^{2}$ for a corresponding polarization of the LG beam. We define relative strength of the electric-quadrupole~field\n\\begin{equation}\n S_{r}(E2) = \\frac{\\sum_{M=-L}^{+L}|W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L=2,M,\\Lambda=1;b,w_{o})|^{2}}{\\sum_{M=-L}^{+L}|W^{\\mathrm{rad}}_{p}(L=2,M,\\Lambda=1;b,w_{o})|^{2}}.\\label{eq:WrE2}\n\\end{equation}\nas a ratio of strength of the electric-quadrupole field between circularly polarized and radially polarized LG beams.\n\n\\par In Fig.~\\ref{fig:ratio}, we plot relative strength of the electric-quadrupole field $S_{r}(E2)$ against the radial distance $b$ from the beam axis of a LG beam for different radial index $p$ values in the top panel and the corresponding intensity profile of LG beams in bottom panel. For the radial index $p = 0$, we observe the value of $S_{r}(E2)$ to be greater than one near the beam axis ($b \\approx 0$) and decreases rapidly to less than one as radial distance $b$ increases. The behaviour of the ratio $S_{r}(E2)$ indicates that the strength of the electric-quadrupole field is suppressed near the beam axis of cylindrically polarized LG beam. However, for large $b$ values the ratio $S_{r}(E2)$ is less than one indicating a strong electric-quadrupole field in the beam cross section of cylindrically polarized LG beam. The radial index $p$ of the LG beam modifies strength of the electric-quadrupole field in the beam cross section as described by the Fig.~\\ref{fig:ratio} for $p = 1, 2$. Similar to $p = 0$ case, the ratio $S_{r}(E2)$ is greater than one near the beam axis, indicating the suppression of an electric-quadrupole field near the beam axis for cylindrically polarized LG beam. For large $b$ values, we observe the ratio $S_{r}(E2)$ to be lesser than one near the dark region in the beam cross section of the LG beam. The vertical lines in the Fig.~\\ref{fig:ratio} is used to denote the corresponding dark region in the beam cross section of LG beam with the help of intensity profile. In contrast to $p = 0$ case, cylindrically polarized LG beams possess a dominant electric-quadrupole field with respect to circularly polarized LG beam only near the dark region in the off-axis region.\n\n\\par The electric-quadrupole component can be associated with the electric field gradient of the light beam and is responsible for driving the electric-quadrupole transition in the target atoms. Therefore, the region in the Fig.~\\ref{fig:ratio} with value of $S_{r}(E2)$ less than one describes a strong electric field gradient region in the beam cross section of cylindrically polarized LG beams. This suggests that, if we were to place an atom in such a region the electric-quadrupole transition would be more efficiently driven by a cylindrical polarization over a circularly polarized LG beam.\n\\begin{figure}\n \\centering\n \\includegraphics[width=.48\\textwidth]{Figure1.pdf}\n \\caption{The relative strength of the electric-quadrupole field $S_{r}(E2)$ is plotted against the radial distance $b$ from the beam axis of the LG beam. The top plot shows variation of $S_{r}(E2)$ for three different radial indices $p = 0, 1,2$. The plot describes variation of strength of the electric-quadrupole field of radial polarization with respect to circular polarization in the beam cross section of the LG beam with beam waist of $w_{o} = 2.4$ $\\mu m$. The bottom plot describes the intensity profile of the LG beams with radial index $p = 0, 1, 2$} \n \\label{fig:ratio}\n\\end{figure}\n\\subsection{Effects of the beam waist and radial position on the projection of TAM of LG beams}\n\\label{subsec:Effect of the beam waist and radial position on the multipole distribution of LG beams}\nIn the last section, we analyzed the strength of the electric-quadrupole field in the beam cross section of the LG beam. Now, we shall discuss the strength of the projection of multipole component $M$ in the beam cross section of the LG beam with the help of modulus squared of the complex weight factor of circular or cylindrical polarization. As seen from the complex weight factors, the multipole distribution of the LG beams varies with the beam waist $w_{o}$ and the radial distance $b$ from the beam axis. The radial dependence is carried out by Bessel function $J_{m_{l}+\\lambda-M}(k_{\\bot} b)$ present in the complex weight factors. The properties of the bessel function dictates the variation of the projection of TAM $M$ across the beam cross section. For example, along the beam axis only $M = m_{l}+\\lambda$ component is non-zero because of the asymptotic property \\cite{watson1995treatise} of the Bessel function given by\n\\begin{equation}\n J_{m_{l}+\\lambda-M}(k_{\\bot} b = 0 ) = \\delta_{m_{l}+\\lambda-M,0}.\n\\end{equation}\nAs we increase the radial distance $b$ from the beam axis, the projection of TAM $M$ can have any value between $-L$ to $+L$.\n\n\\par To understand the variation of the projection of TAM across the beam cross section, we define relative weight of projection of the electric-quadrupole field as\n\\begin{equation}\n W_{r}(M) = \\frac{|W(L=2,M,\\Lambda =1;b,w_{o})|^{2}}{\\sum_{M=-L}^{+L}|W(L=2,M,\\Lambda=1;b,w_{o})|^{2}}\n\\end{equation}\nwhere $|W(L=2,M,\\Lambda=1;b,w_{o})|^{2}$ denotes the modulus squared of the complex weight factor of a circularly or cylindrically polarized LG beam.\n The $W_{r}(M)$ is plotted against the radial distance $b$ from the beam axis for both circularly and cylindrically polarized LG beams while keeping the beam waist $w_{o}$ fixed at 2.4 $\\mu m$. These plots describe the variation in the projection of the TAM across the beam cross section of the LG beams, as shown in the Fig.~\\ref{fig:cirrad} for circular and in Fig.~\\ref{fig:cylrad} for cylindrical polarization.\n\n\\par For a fixed radial distance $b$ from the beam axis, we analyze the variation of multipole distribution with respect to the beam waist $w_{o}$. For circularly polarized LG beams, $W_{r}(M)$ varies with respect to the beam waist $w_{o}$ for radial distance $b = 0.3$~$\\mu m$. However, for cylindrically polarized LG beams, $W_{r}(M)$ significantly varies with beam waist $w_{o}$ for large radial distance $b$ from the beam axis. Similarly, we plot $W_{r}(M)$ against the beam waist $w_{o}$ of the LG beams for both circular and cylindrical polarization. The Fig.~\\ref{fig:cirwaist} describes the variation in the projection of TAM $M$ in the beam cross section of circularly polarized LG beams with respect to the beam waist $w_{o}$, for $b = 0.3$ $\\mu m$. Similarly, the plots in Fig.~\\ref{fig:cylwaist} describes the variation of the projection of TAM $M$ in the beam cross section of cylindrically polarized LG beams with respect to the beam waist $w_{o}$, for $b = 2.2$ $\\mu m$.\n\n\\par By carefully selecting the beam waist $w_{o}$ and radial distance $b$ we can control the relative strength of the projection of TAM $M$ in the beam cross section of LG beam of a given polarization.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.48\\textwidth]{Figure2.pdf}\n \\caption{The relative weight of the projection of quadrupole component $W_{r}(M)$ is plotted against the radial distance $b$ from the beam axis for right circularly polarized LG beam. In the above plots radial index $p = 0$, $\\lambda = +1$ and the radial distance $b$ is fixed to $b = 2.2$ $\\mu m$.}\n \\label{fig:cirrad}\n\\end{figure}\n\n\n\\subsection{Interaction of LG beams with the target atom}\n\\label{subsec:Interaction of LG beams with the target atom}\nThe transition between two atomic bound states occurs if the charge distribution in atomic system matches with the multipole structure of the exciting light beam. Moreover, the transition between the magnetic sub-levels in an atomic system is characterized by the projection of the TAM of the exciting light beam. Thus, the photo-excitation of atoms by LG beams helps us to analyze the multipole distribution of the twisted light beam.\n\n \\par Mathematically, the transition between initial $|n_{i},j_{i},m_{i}\\rangle$ and final $| n_{f},j_{f},m_{f}\\rangle$ bound states in an effective one electron atom is given by the transition amplitude $M_{fi}$\n \\begin{equation}\\label{eq:transition}\n M_{fi} \\; = \\; \\langle n_{f},j_{f},m_{f}| \\bm{\\alpha}\\cdot\\mathbf{A}(\\mathbf{r}) |n_{i},j_{i},m_{i}\\rangle,\n\\end{equation}\nwhere $\\mathbf{A}(r)$ is the vector potential of either circularly polarized or cylindrically polarized LG beams, $\\bm{\\alpha}$ is the Dirac matrix and the atomic bound states are characterized by principle $n$, TAM $j$ and TAM projection $m$ quantum numbers. \n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{Figure3.pdf}\n \\caption{The relative weight of the projection of quadrupole component $W_{r}(M)$ is plotted against the radial distance $b$ from the beam axis for radially polarized (top) and azimuthally polarized LG beam (bottom). In the above plots radial index $p = 0$ and the beam waist is fixed to $w_{o} = 2.4$ $\\mu m$.}\n \\label{fig:cylrad}\n\\end{figure}\n\\par We substitute the multipole expansion of vector potential of LG beams into equation (\\ref{eq:transition}) to obtain the transition amplitude\n\\begin{align}\n M_{fi}(b,w_{o}) &= \\sum_{L,M,\\Lambda} W(L,M,\\Lambda;b,w_{o}) \\\\ \\nonumber \n & \\times \\langle n_{f},j_{f},m_{f}|\\bm{\\alpha} \\cdot\\mathbf{a}^{\\Lambda}_{L,M}(\\mathbf{r}) | n_{i},j_{i},m_{i} \\rangle,\n\\end{align}\nwhere $W(L,M,\\Lambda;b,w_{o})$ is the complex weight factor of either a circularly polarized or cylindrically polarized LG beam. The rest of the transition amplitude equation is solved using the Wigner-Eckart theorem \\cite{brinkangularmomentum,rose1995elementary}, which gives the transition amplitude \n \\begin{align}\\label{eq:tamplitude}\n M_{fi}(b,w_{o}) &= \\sum_{L,M,\\Lambda} W(L,M,\\Lambda;b,w_{o}) \\langle j_{i},m_{i},L,M| j_{f},m_{f}\\rangle \\\\ \\nonumber \n &\\times \\langle n_{f},j_{f}||\\bm{\\alpha}\\cdot \\mathbf{a}^{\\Lambda}_{L}(\\mathbf{r})||n_{i},j_{i}\\rangle,\n\\end{align}\nas the product of geometrical and atomic factors. The complex weight factor $W(L,M,\\Lambda;b,w_{o})$ and the Clebsch-Gordan (CG) coefficient represents the geometrical and the reduced matrix elements describe the atomic properties which influence the atomic excitation process. It is clear from the above equation (\\ref{eq:tamplitude}) that the transition amplitude depends on the complex weight factors $W(L,M,\\Lambda;b,w_{o})$, which together with the CG coefficients and reduced matrix elements determine the amplitudes of the individual transitions.\n\\par The transition amplitude of the atomic transition between two bound states must satisfy the set of following rules known as \\textit{selection rules} given by\n\\begin{equation}\n m_{i} + M = m_{f}\n\\end{equation}\n\\begin{equation}\n |j_{f}-j_{i}| \\leq L \\leq |j_{f}+j_{i}| \n\\end{equation}\n\\begin{equation}\n \\pi_{i} \\pi_{f} = (-1)^{L+p+1}\n\\end{equation}\nhere $\\pi_{i} \\pi_{f}$ are the parity of the initial and final atomic states. Selection rules are associated with the symmetry of the atomic system such as, rotational symmetry. Rotational symmetry of the atomic system dictates the conservation of TAM.\nMathematically, the selection rules are defined by symmetry relations of CG coefficients. According to the group theory, CG coefficients are a unitary transformation between an initial and final atomic states \\cite{sakurai1995modern}. These transformations or symmetry relations do not depend on the radial distance of the atoms from the beam axis. Hence, the selection rules for photo-excitation of atoms by twisted light beams are not position dependent as inferred in the previous works \\cite{selection1,duan2019,selection3}.\n\n\\subsection{Photo-excitation of Ca$^{+}$ ion by LG beams}\nWe investigate the interaction of a paraxial LG beam with beam waist $w_{o}$ and radial index $p = 0$ with the Ca$^{+}$ ion positioned at the focus along the $z$ axis, which is taken as the quantization axis. According to the electron configuration of the Ca$^{+}$ ion, only the single electron in the outer most shell has nonzero TAM. Since the lower shells have a closed configuration, their TAM is zero and does not couple with the exciting light beam. \nHence, we use one-electron notation in our work. However, to investigate the excitation of more complex atoms by twisted light beams we can use formalism of JAC \\cite{FRITZSCHE2019} to obtain the transition amplitude. \n\n\\par We consider the electric-quadrupole transition between an initial $|4s_{1\/2},m_{i}\\rangle$ of [Ar]4s state and final $|3d_{5\/2},m_{f}\\rangle$ of [Ar]3d state in Ca$^{+}$ ion. According to the selection rules, the allowed transitions between these two bound atomic states are electric-quadrupole (E2) and magnetic octopole. However, the magnetic octopole transitions are weak to be measured in an experiment and we restrict our discussion only to (E2) transitions.\n\\par The relative partial cross section $\\sigma_{r}$ is a usefull physical parameter to understand the excitation of the atoms \\cite{peshkov}, which is given by\n\\begin{equation}\n \\sigma_{r}(m_{f}) = \\frac{ |M_{fi}(b,w_{o})|^{2}} {\\sum_{m_{f}} |M_{fi}(b,w_{o})|^{2}}.\n\\end{equation}\nWe observe that the reduced matrix elements cancel in the above ratio and the above ratio now only corresponds to the product of CG coefficients and the complex weight factors. As discussed earlier, we plot only the $W_{r}(M)$ against $w_{o}$ and $b$. It is found that the extra term in the equation of relative partial cross section in the form of CG coefficients simply scales the plots of $W_{r}(M)$ against the beam waist $w_{o}$ and radial distance $b$ from the beam axis and does not change the nature of plots. Hence, studying the variation of $W_{r}(M)$ against the beam parameters is sufficient to explain the magnetic sub-level population of the target atom.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{Figure4.pdf}\n \\caption{The relative weight of the projection of quadrupole component $W_{r}(M)$ is plotted against the beam waist of right circularly polarized LG beam. In the above plots radial index $p = 0$, $\\lambda = +1$ and the radial distance is fixed to $b = 0.3$ $\\mu m$.}\n \\label{fig:cirwaist}\n\\end{figure}\n\\subsubsection{Circularly polarized LG beams}\nBefore investigating the atomic excitation by cylindrically polarized LG beams, we consider the circularly polarized LG beams interacting with the target Ca$^{+}$ ion. As mentioned in the last section, we consider the electric-quadrupole transition between 4s and 3d states of the target Ca$^{+}$ ion. The transitions between these two states are discussed with the help of plots and the selection rule $m_{i} + M = m_{f}$.\n\n\\par In Fig.~\\ref{fig:cirrad}, we investigate the radial dependence of the atomic transition. For the target Ca$^{+}$ ion positioned along the beam axis, right circularly polarized LG beam of beam waist 2.4 $\\mu m$ drives a transition between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=3\/2\\rangle$ magnetic sub-states. Now if the target Ca$^{+}$ ion is displaced from the beam axis by some distance $b$, we observe transition between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=3\/2\\rangle$ magnetic sub-states decreases rapidly while the transition occurs between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=1\/2\\rangle$ magnetic sub-states.\n\n\\par The beam waist $w_{o}$ of the interacting circularly polarized LG beam influences the transition between the magnetic sub-states of the target Ca$^{+}$ ion positioned very close to the beam axis, as shown in the Fig.~\\ref{fig:cirwaist}. For the target Ca$^{+}$ ion displaced from the beam axis by approximately $b = 0.3$ $\\mu m$, we observe transition between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=1\/2\\rangle$ magnetic sub-states. As the beam waist $w_{o}$ of the interacting LG beam is increased, transition occurs between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=3\/2\\rangle$. The above discussion of the transition between the magnetic sub-levels of the target Ca$^{+}$ ion by a circularly polarized LG beam is in agreement with the results obtained in \\cite{peshkov} \n\n\\subsubsection{Radially polarized LG beams}\nNow we focus our attention to our main aim of this paper, that is the excitation of target atoms by cylindrically polarized LG beams. In the beginning we consider the transitions between the $4s_{1\/2}$ state and $3d_{5\/2}$ state of the Ca$^{+}$ ion driven by radially polarized LG beams of beam waist $w_{o} = 2.4$ $\\mu m$. The target Ca$^{+}$ ion positioned along the beam axis has more probablity to undergo a transition between $|4s_{1\/2}, m_{f}=-1\/2\\rangle$ and the $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-state absorbing $M = -1$ component, as shown in the Fig.~\\ref{fig:cylrad}. As the radial distance $b$ between the target Ca$^{+}$ ion and the beam axis is increased, we observe transition between $|4s_{1\/2}, m_{f}=-1\/2\\rangle$ and the $|3d_{5\/2}, m_{f}= 1\/2\\rangle$ magnetic sub-state absorbing $M = +1$ component.\n\\par The influence of the varying beam waist $w_{o}$ of radially polarized LG beams on the transition between the magnetic substates is discussed in the Fig.~\\ref{fig:cylwaist}. For the target Ca$^{+}$ ion displaced from the beam axis by radial distance approximately $b = 2.2$ $\\mu m$, we observe transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= +1\/2\\rangle$ magnetic sub-state, absorbing $M = +1$ component. But as we increase the beam width, the strength of $M = +1$ component in the beam cross section of radially polarized LG beam decreases rapidly and the strength of $M = -1$ component increases. This results in the transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-states.\n\n\\subsubsection{Azimuthally polarized LG beams}\nIn Fig.~\\ref{fig:cylrad}, we discuss the photo-excitation of the target Ca$^{+}$ ion by azimuthally polarized LG beams. For the target Ca$^{+}$ ion placed along the beam axis, we observe transition between the initial $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and final $|3d_{5\/2}, m_{f}= -1\/2\\rangle$ absorbing $M = 0$ component. As the radial distance $b$ between the target Ca$^{+}$ ion is increased, strength of $M = -1$ component increases and a transition occurs between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-states. For large values of radial distance $b$, we observe a transition between the initial $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and final $|3d_{5\/2}, m_{f}= 1\/2\\rangle$ magnetic sub-states due to $M = +1$ component.\n\n\\par Similar to the radial polarization case, we discuss the influence of the beam waist $w_{o}$ on the transition between the magnetic sub-states in the target Ca$^{+}$ ion in the Fig.~ \\ref{fig:cylwaist}. For the target Ca$^{+}$ ion placed at a radial distance of approximately $b = 2.2$ $\\mu m$, transition occurs between initial $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and final $|3d_{5\/2}, m_{f}= 1\/2\\rangle$ magnetic sub-states absorbing $M = +1$ component. As the beam waist $w_{o}$ of the interacting azimuthally polarized LG beam is increased, strength of $M = -1$ component increases and it results in the transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-states of the target Ca$^{+}$ ion. Further, for larger values of beam waist $w_{o}$ we observe transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -1\/2\\rangle$ magnetic sub-states absorbing $M = 0$ component.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{Figure5.pdf}\n \\caption{The relative weight of the projection of quadrupole component is plotted against the beam waist $w_{o}$ of cylindrically polarized LG beam. The plots describe the variation of strength of the projection of TAM $M$ in the beam cross section of radially (top) and azimuthally polarized (bottom) LG beams. In the above plots, radial index $p = 0$ and the radial distance is fixed to $b = 2.2$~$\\mu m$. }\n \\label{fig:cylwaist}\n\\end{figure}\n\\section{Summary}\n\\label{sec:summary}\nWe have theoretically investigated the photo-excitation of atoms by LG beams especially for cylindrical polarization. To do so, we constructed the complex weight factor of cylindrically polarized LG beam as a linear combination of complex weight factors for circular polarization. We analyzed strength of the electric-quadrupole field across the beam cross section of cylindrically polarized LG beams. We observed strength of the electric-quadrupole field of cylindrical polarization to dominate over a circularly polarized LG beam in the dark region away from the beam axis. In addition, we observed that strength of the electric-quadrupole field in the beam cross section is sensitive to the radial index $p$ of the LG beam. The variation of the magnetic component of electric-quadrupole was analyzed as a function of beam waist $w_{o}$ and the radial distance $b$ from the beam axis. To better understand the variation of the magnetic component of electric-quadrupole field, we plotted the relative weight of the projection of electric-quadrupole $W_{r}(M)$ against the beam waist and the radial distance from the beam axis.\n\\par Furthermore, we used the multipole distribution of the cylindrically polarized LG beams to discuss the excitation of target atoms. As an example, we considered the electric-quadrupole transition ($4s\\; ^{2}S_{1\/2} \\rightarrow 3d\\; ^{2}D_{5\/2}$) in the target Ca$^{+}$ ion. We observed that the radial distance $b$ from the beam axis of the cylindrically polarized LG beam influences the magnetic transitions between the $4s_{1\/2}$ and $3d_{5\/2}$ state as in the circular polarization case. Our results explicitly shows that the beam waist of the LG beams significantly affects the transitions between the magnetic sub-states of the target atoms displaced from the beam axis. Furthermore, we explicitly showed that the magnetic sub-level population in the target atom following the excitation process by LG beam can be explained primarily using the multipole distribution of the twisted light beam.\n\n\n\\begin{acknowledgments}\nThis work has been funded by the Research School of Advanced Photon Science (RS-APS) of Helmholtz Institute Jena, Germany.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nEntanglement is usually presented as one of the weirdest features of\nquantum theory that depart strongly from our common\nsense~\\cite{Schrodinger:1935ys}. Since the seminal work of Einstein,\nPodolsky, and Rosen (EPR)~\\cite{Einstein:1935yt}, countless\ndiscussions on this subject have\npopped up~\\cite{Horodecki:2009vu}.\n\nA major step in the right direction is due to Bell~\\cite{Bell:1964gc}, who\nformulated the EPR dilemma in terms of an inequality which naturally\nled to a falsifiable prediction. Actually, it is common to use\nan alternative formulation, derived by Clauser, Horne, Shimony and\nHolt (CHSH)~\\cite{Clauser:1969it}, which is better suited for realistic\nexperiments.\n\nThe main stream of research~\\cite{Brunner:2014ys,Werner:2001fj} settled\nthe main concepts of this topic in the realm of quantum\nphysics. However, in recent years a general consensus has been reached\non the fact that entanglement is not necessarily a signature of the\nquantumness of a system. Actually, as aptly remarked in\nRefE.~\\cite{Toppel:2014jt}, one should distinguish between two types of\nentanglement: between spatially separated systems (inter-system\nentanglement) and between different degrees of freedom of a single\nsystem (intra-system entanglement). Inter-system entanglement occurs\nonly in truly quantum systems and may yield to nonlocal statistical\ncorrelations. Conversely, intra-system entanglement may also appear\nin classical systems and cannot generate nonlocal\ncorrelations~\\cite{Brunner:2005gv}; for this reason, it is often\ndubbed as ``classical entanglement''. Since its introduction by\nSpreeuw~\\cite{Spreeuw:1998ho}, this notion has been employed in a\nvariety of contexts~\\cite{Ghose:2014oe}. \n \nClassical entanglement has allowed to test Bell inequalities with\nclassical wave fields. The physical significance of this violation is\nnot linked to quantum nonlocality, but rather points to the\nimpossibility of constructing such a beam using other beams with\nuncoupled degrees of freedom. However, all the experiments conducted\nthus far to observe this violation have involved only discrete\nvariables, such as spin and beam path of single\nneutrons~\\cite{Hasegawa:2003hf}, polarization and transverse modes of\na laser beam~\\cite{Souza:2007fb,Simon:2010jk,Qian:2011jc,\n Gabriel:2011bn,Eberly:2014}, different transverse modes propagating\nin multimode waveguides~\\cite{Fu:2004fk}, polarization of two\nclassical fields with different frequencies~\\cite{Lee:2002uq}, orbital\nangular momentum~\\cite{Goyal:2013la,Chowdhury:2013}, and polarization\nand spatial parity~\\cite{Kagalwala:2013br}.\n\nIn this paper, we continue the analysis of this classical entanglement\nby focusing on the simple but engaging example of vortex beams. To\nthis end, in Sec.~\\ref{sec:Schmidt} we revisit a decomposition of\nLaguerre-Gauss (LG) beams in the Hermite-Gauss (HG) basis that can be\nrightly interpreted as a Schmidt decomposition. This immediately\nsuggests that many ideas ensuing from the quantum world may be\napplicable to these beams as well. In particular, in\nSec.~\\ref{sec:CHSH} we address the inseparability of the LG modes\nusing a CHSH violation that we quantify in terms of the associated\nWigner function. As this distribution can be understood as a measure\nof the displaced parity, in Sec.~\\ref{sec:exp} we discuss an\nexperimental realization which nicely agrees with the theoretical\npredictions. Finally, our conclusions are summarized in\nSec.~\\ref{sec:concl}.\n\n\n\\section{Optical vortices and Schmidt decomposition}\n\\label{sec:Schmidt}\n\nIt is well known that the beam propagation along the $z$ direction of\na monocromatic scalar field of frequency $\\omega$; i.e.,\n$E (\\mathbf{r}, t) = \\mathcal{E} (\\mathbf{r} ) \\exp [- i ( \\omega t -\nk z )] $, is governed by the paraxial wave equation\n\\begin{equation}\n \\frac{\\partial \\mathcal{E}}{\\partial z} = -\\frac{\\lambdabar}{2}\n \\left( \\frac{\\partial^2}{\\partial x^2} +\n \\frac{\\partial^2}{\\partial y^2}\\right) \\mathcal{E} \\, ,\n \\label{freespace}\n\\end{equation}\nwith $\\lambdabar = \\lambda\/2\\pi$ and $\\lambda $ is the\nwavelength. Equation~(\\ref{freespace}) is formally identical to the\nSchr\\\"{o}dinger equation for a free particle in two dimensions, with\nthe obvious identifications $t \\mapsto z$, $\\psi \\mapsto \\mathcal{E}$,\nand $\\hbar \\mapsto \\lambdabar$.\n\nAny optical beam can be thus expressed as a superposition of\nfundamental solutions of Eq.~(\\ref{freespace}). In Cartesian\ncoordinates, a natural orthonormal set is given by the Hermite-Gauss\n(HG) modes:\n\\begin{gather}\n \\mathrm{HG}_{mn} ( x, y ) = \\sqrt{\\frac{2}{\\pi n! m! 2^{n+m}}} \\left (\n \\frac{1}{w} \\right ) H_{m} \\left ( \\frac{\\sqrt{2} x}{w} \\right )\n H_{n } \\left ( \\frac{\\sqrt{2} y}{w} \\right ) \\nonumber \\\\\n \\times \\exp \\left ( - \\frac{x^{2} + y^{2}}{w^{2}} \\right ) ,\n \\label{waveHG}\n\\end{gather}\nwhere $w$ is the beam waist, and $H_{m}$ are the Hermite\npolynomials. Note that we are restricting ourselves to the plane\n$z=0$, since we are not interested here in the evolution.\n\nFor cylindrical symmetry, it is convenient to use the set of\nLaguerre-Gauss (LG) modes, which contain optical vortices with\ntopological singularities; they read\n\\begin{gather}\n \\mathrm{LG}_{mn} ( r,\\varphi ) = \\sqrt{\\frac{2}{\\pi m! n!}} \\min (m,n)!\n (-1)^{\\min (m,n)}\n \\left ( \\frac{1}{w} \\right ) \\nonumber \\\\\n \\times \\left ( \\frac{\\sqrt{2} r}{w} \\right )^{|m-n|} \\!\\! L_{\\min\n (m,n)}^{|m - n|} \\left ( \\frac{2 r^{2}}{w^{2}} \\right ) \\exp \\left\n (- \\frac{r^{2}}{w^{2}} \\right ) \\exp [ i (m-n) \\varphi ] \\, ,\n \\label{waveLG}\n\\end{gather}\nwhere $L_{p}^{|\\ell |} (x)$ are the generalized Laguerre\npolynomials. A word of caution seems to be in order: usually, these\nmodes are presented in terms of two different indices: the azimuthal\nmode index $\\ell = m - n$, which is a topological charge giving the\nnumber of $2 \\pi$-phase cycles around the mode circumference, and\n$p = \\min (m, n)$ is the radial mode index, which is related to the\nnumber of radial nodes~\\cite{Karimi:2014yu}. However, the form\n(\\ref{waveLG}) will be advantageous in what follows.\n\nThe crucial observation is that the LG modes can be represented as\nsuperpositions of HG modes, and viceversa. This can be compactly\nwritten down as~\\cite{Beijersbergen:1993eu}\n\\begin{equation}\n \\mathrm{LG}_{mn} (\\rho,\\varphi) = \\sum_{k=0}^{m+n} B_{mn}^{k} \\,\n\\mathrm{HG}_{m+n-k,k} (x,y)\n \\label{legherm}\n\\end{equation}\nwhere the coefficients are\n\\begin{equation}\nB_{mn}^{k} = \\sqrt{\\frac{k! (m +n-k)!}{m! n! 2^{n+m}}} \\frac{(-i)^{k}}{k!}\n\\frac{d^k}{dt^k} \\left . [ (1-t)^m(1+t)^n] \\right |_{t=0} .\n \\label{def11}\n\\end{equation}\nThis looks exactly the same as a Schmidt decomposition for a bipartite\nquantum system. It is nothing but a particular way of expressing a\nvector in the tensor product of two inner product\nspaces~\\cite{Peres:1993fk}. Alternatively, it can be seen as another\nform of the singular-value decomposition~\\cite{Stewart:1993uq}, which\nidentifies the maximal correlation directly. In quantum\ninformation, the Schmidt coefficients $B_{mn}^{k}$ convey complete\ninformation of the entanglement~\\cite{Agarwal:02}. Here, we intend to \n assess entanglement in LG beams via the violation of suitably\n formulated Bell inequalities. \n\n\\section{CHSH violation for Laguerre-Gauss modes}\n\\label{sec:CHSH}\n\nThe traditional form of the CHSH inequality applies to dichotomic\ndiscrete variables. For continuous variables, the sensible\nformulation is in terms of the Wigner function,\nwhich for a classical beam reads\n\\begin{equation}\n W(\\mathbf{x},\\mathbf{p}) = \\frac{1}{\\lambdabar^2 \\pi^2}\n \\int d^2\\mathbf{x}^{\\prime} \\, \n e^{2i \\mathbf{p} \\cdot \\mathbf{x}^{\\prime}\/\\lambdabar}\n \\langle E^{\\ast}(\\mathbf{x} - \\mathbf{x}^{\\prime}) E(\\mathbf{x} +\n \\mathbf{x}^{\\prime}) \\rangle \\, ,\n \\label{wigel}\n\\end{equation}\nthe angular brackets denoting statistical average. Although originally\nintroduced to represent quantum mechanical phenomena in phase\nspace~\\cite{Wigner:1932kx}, the Wigner distribution was established in\noptics~\\cite{Walther:1968qy} to relate partial coherence with\nradiometry. Since then, a great number of applications of this\nfunction have been reported~\\cite{Bastiaans:2009sp,\n Galleani:2002hb,Dragoman:1997rw,Mecklenbrauker:1997tx,Alonso:2011pi}.\nNote that $W$ has the dimensions of an intensity and it yields a\ndescription displaying both the position and the momentum (which in\nthe paraxial approximation has the significance of a scaled angular\ncoordinate) of the intensity of the wave field: in fact, one easily\nproves that\n\\begin{eqnarray}\n &\\displaystyle\n \\int W(\\mathbf{x},\\mathbf{p}) \\, d\\mathbf{p} = \n I ( \\mathbf{x} ) \\equiv \\langle E^{\\ast} (\\mathbf{x} )\n E(\\mathbf{x} \\rangle \\, , & \\nonumber \\\\\n& & \\\\\n& \\displaystyle\n \\frac{1}{\\lambdabar^2 \\pi^2} \\int W(\\mathbf{x},\\mathbf{p}) \\,\n d\\mathbf{x} = \n I ( \\mathbf{p} ) \\equiv \n \\langle E^{\\ast} (\\mathbf{p} ) E (\\mathbf{p}) \\rangle \\, , & \n\\nonumber\n\\end{eqnarray}\nwith \n\\begin{equation}\nE ( \\mathbf{p} )= \\frac{1}{\\lambdabar^2 \\pi^2} \n\\int E ( \\mathbf{x} ) \\, \\exp( i \\mathbf{p} \\cdot\n\\mathbf{x}\/\\lambdabar) \\, d\\mathbf{x} \\, .\n\\end{equation}\n Thus, the marginals of the Wigner function are the intensity\n distributions in $\\mathbf{x}$ or $\\mathbf{p}$ space, respectively. \n\n\nThe CHSH inequality can now be stated in terms of the Wigner function \n as~\\cite{Banaszek:1999mw}\n\\begin{equation}\n B = \\frac{\\pi^2}{4} | W (\\alpha, \\beta) +\n W(\\alpha,\\beta^{\\prime}) + W ( \\alpha^{\\prime},\\beta)\n - W(\\alpha^{\\prime},\\beta^{\\prime})| < 2,\n \\label{belin12}\n\\end{equation}\nwhere $\\alpha = (x, p_{x} )\/\\sqrt{2}$ and $\\beta = (y,\np_{y})\/\\sqrt{2}$. This also follows from\nthe work of Gisin~\\cite{Gisin1991201}, who formulated a Bell inequality for the\nset of observables with the property $\\hat{O}^{2} = \\openone$: as we\nshall see, the Wigner function appears as the average\nvalue of the parity, whose square is unity. Reference~\\cite{Chowdhury:2013}\npresents a detailed study of the violations of (\\ref{belin12}).\n\nFor the state $\\mathrm{LG}_{mn}$, the normalized\nWigner function can be written as~\\cite{Simon:2000xp}\n\\begin{gather}\n W_{mn}^{\\mathrm{LG}} ( X, P_{X}; Y, P_{Y} ) = \\frac{(-1)^{m+n}} {\\pi^{2}}\n \\exp(-4 Q_0) \\nonumber \\\\\n \\times L_{m} [4 (Q_0+Q_2)] L_{n} [4(Q_0-Q_2)] \\, ,\n \\label{WF_LG_n1_n2}\n\\end{gather}\nwhere \n\\begin{equation}\nQ_{0} = \\frac{1}{4} ( X^{2} + Y^{2} + P_{X}^{2} + P_{Y}^{2} ) \\, ,\n\\qquad \nQ_{2} = \\frac{1}{2} ( X P_{Y} - Y P_{X} ) \\, ,\n\\end{equation}\nand we have rescaled the variables as $ x \\mapsto (w\/\\sqrt{2}) X$\nand $p_x \\mapsto (\\sqrt{2} \\lambdabar\/w)~P_X$ (and analogously\nfor the $y$ axis). Let us first look at the simple case\nof the mode $\\mathrm{LG}_{10}$, which reduces to\n\\begin{gather}\n W^{\\mathrm{LG}}_{10} (X, P_X; Y, P_Y) = \\frac{1}{\\pi^2} \\exp(\n -P_X^2-P_Y^2-X^2-Y^2 ) \\nonumber \\\\\n\\times [ (P_X - Y)^2 + (P_Y+X)^2 -1 ] \\, .\n\\end{gather}\n The two measurement settings on one side are chosen to be\n$\\alpha = ( X = 0, P_{X}=0 ) $ and $ \\alpha^{\\prime} = (\nX^{\\prime} = X, P_{X}^{\\prime}=0 )$, and the corresponding\nsettings on the other side are $\\beta = ( Y=0, P_{Y} = 0 ) $\nand $\\beta^{\\prime} = (Y^{\\prime}=0, P_{Y}^{\\prime} =\nP_{Y})$~\\cite{Zhang:2007xb}, for which the Bell sum is\n\\begin{gather}\n B = e^{-P_Y^2} (P_Y^2 - 1 ) + e^{-X^2} ( X^2 - 1 ) \\nonumber \\\\\n - e^{-( P_Y^2 + X^2)} [ ( P_Y + X)^2 - 1] -1 \\, .\n \\label{BI_1_3}\n\\end{gather}\nUpon maximization with respect to $X$ and $P_{Y}$, we obtain the\nmaximum Bell violation, $|B_{\\mathrm{max}}| \\simeq 2.17$, which\nhappens for the choices\n$X \\simeq 0.45,~P_{Y} \\simeq 0.45$~\\cite{Chowdhury:2013}. For\ncomparison, note that the maximum Bell violation in quantum mechanics\nthrough the Wigner function for the two-mode squeezed vacuum state\nusing similar settings is given by\n$|B_{\\mathrm{max}}^{\\mathrm{QM}} | \\simeq 2.19$~\\cite{Banaszek:1999mw}.\n\nThe Bell violation may be further optimized by a more general choice\nof settings than those used here. For example, maximizing it with\nrespect to the parameters $\\alpha = (X,P_{X})$,\n$\\alpha^{\\prime}= (X^{\\prime}, P_{X}^{\\prime})$, $\\beta= (Y, P_{Y})$,\n$\\beta^{\\prime} = (Y^{\\prime}, P_{Y}^{\\prime})$, one obtains the\nabsolute maximum Bell violation, $|B_{\\max}| = 2.24$ and occurs for\nthe choices $X \\simeq -0.07,~P_{X} \\simeq 0.05,~X^{\\prime} \\simeq\n0.4,~P_{X}^{\\prime} \\simeq-0.26,~Y \\simeq-0.05,~P_{Y} \\simeq\n-0.07,~Y^{\\prime} \\simeq 0.26,~P_{Y}^{\\prime} \\simeq 0.4$.\nThe violation also increases with higher orbital angular\nmomentum. This increase with $n$ is analogous to the enhancement of\nnonlocality in quantum mechanics for many-particle\nGreenberger-Horne-Zeilinger states~\\cite{Mermin:1990nf}.\n\n\n\\section{Experimental results}\n\\label{sec:exp}\n\n\\begin{figure}[b]\n \\centerline{\\includegraphics[width=0.95\\columnwidth]{Figure1}}\n \\caption{(Color online) Scheme of the Bell measurement. The\n abbreviations are as follows: He-Ne: laser source, FC: fiber\n coupler, SMF: single mode fiber, CO: collimation optics, SLM:\n spatial light modulator, AS: aperture stop, BS: beam splitter,\n M1-M4: mirrors, CCD: camera}\n \\label{figSetup}\n\\end{figure}\n\nWe have carried a direct measurement of the Bell sums for optical\nbeams with different amount of nonlocal correlations. To understand\nthe measurement, we recall that the Wigner function in quantum optics\nis often regarded as the average of the displaced parity\noperator~\\cite{Royer:1977qf}. At the classical level, we can consider\nthe field amplitudes $\\mathcal{E} (X, Y)$ as vectors in the Hilbert\nspace of complex-valued functions that are square integrable over a\ntransverse plane. In this space we define linear Hermitian operators\n\\begin{equation}\n \\hat{X}: \\mathcal{E} (X, Y )\n \\mapsto X \\mathcal{E} (X, Y ) \\, ,\n \\qquad\n \\hat{P}_{x}: \\mathcal{E} (X, Y )\n \\mapsto - i \\frac{\\partial}{\\partial X}\n \\mathcal{E} (X, Y ) \\, ,\n\\end{equation}\nand analogous ones for the $Y$ variable. Formally, these operators\nsatisfy the canonical commutation relations\n$ [\\hat{X}, \\hat{P}_{X} ] = [\\hat{Y}, \\hat{P}_{Y} ] = i $. Therefore,\nthe unitary parity operator is \n\\begin{equation}\n\\hat{\\Pi}_{X} \\,\n\\hat{X} \\, \\hat{\\Pi}_{X} = - \\hat{X} \\, , \n\\qquad\n\\hat{\\Pi}_{X} \\, \n\\hat{P}_{X} \\,\\hat{\\Pi}_{X} = - \\hat{P}_{X} \\, ,\n\\end{equation}\nand changes $\\mathcal{E} (X, Y)$ into $\\mathcal{E} (-X, Y)$. The\ndisplacement operators are \n\\begin{equation}\n\\hat{D} (X, P_{X}) =\n\\exp [ i (P_{X} \\hat{X} - X \\hat{P}_{X} )] \\, . \n\\end{equation}\nIndeed, with these notations we have\n\\begin{align}\n W (X, P_{x}; Y, P_{y} ) & = \\nonumber \\\\\n& \\frac{4}{\\pi^2}\n \\langle\n \\hat{D} (X, P_{x}) \\hat{\\Pi}_{X} \\hat{D}^{\\dagger}(X, P_{x})\n\\, \\hat{D} (Y, P_{y}) \\hat{\\Pi}_{Y} \\hat{D}^{\\dagger}(Y, P_{y})\n \\rangle \\, .\n\\end{align}\n\n\\begin{figure}\n \\centerline{\\includegraphics[width=0.97\\columnwidth]{Figure2}}\n \\caption{(Color online) Snapshots of the CCD camera for the state\n $\\mathrm{LG}_{10}$ at the four settings $(X, P_{X}; Y, P_{Y})$ indicated. The scans are\n normalized to the peak intensity among the measurements and the\n area of interest for the intensity integration is marked by a red\n circle.}\n \\label{figCCD}\n\\end{figure}\n\n\nParity measurement can be, in turn, realized by a common-path\ninterferometer with a Dove prism inserted into the optical\npath~\\cite{Mukamel:2003qq}. In our setup, sketched in\nFig.~\\ref{figSetup}, the prism was substituted with an equivalent\nfour-mirror Sagnac arrangement~\\cite{Smith:2005vo}. The two copies of\nthe input signal obtained after the input beam splitter are\ntransformed by the mirrors so as to make one copy spatially inverted\nwith respect to the other, prior to combining the beams together. The\nresulting interference pattern is detected by a CCD camera:\nFigure~\\ref{figCCD} shows snapshots of the camera for the state\n$\\mathrm{LG}_{10}$ at the four settings indicated. The total intensity\nwitnessing parity of the measured beam is computed by spatial\nintegration and this is proportional to the desired Wigner\ndistribution sample after normalization to the overall intensity.\n\nThe target signal beams were prepared with digital holograms created\nby a spatial light modulator (SLM), which modulated a collimated\noutput of a single mode fiber coupled to a He-Ne laser. We also\nincluded a 4$f$-system, with an aperture stop, to filter the unwanted\ndiffraction orders produced by the SLM. To allow for a better\nflexibility, all the necessary shifts in the $X$, $Y$, $P_{X}$, and\n$P_{Y}$ variables were incorporated into the SLM, so that each\nBell measurement was associated with a separate hologram.\n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\columnwidth]{Figure3}}\n \\caption{(Color online) Experimental results for three different\n optical beams: a) $\\mathrm{HG}_{10}$, b)\n $0.4 \\, \\mathrm{HG}_{10}+ i 0.6 \\, \\mathrm{HG}_{01}$, and c) $\\mathrm{LG}_{10}$. At the\n top, we plot $\\frac{\\pi^{2}}{4} W(X, P_{X}; Y, P_{Y})$ at the\n values $(X,P_{X}; Y, P_{Y})$ indicated for each one. The next\n plot shows the measured Bell sums, all reported with $75\\%$ and\n $25\\%$ quartile (orange boxes) and the minimal and maximal\n measured values (error bars). The theoretical values\n $(-1.91, -2.15, -2.17)$ are the dots and the black bar is at\n $|B|=2$, which delimites the classically entangled states. The\n theoretical amplitude (top) and phase (bottom) distributions of\n the measured beams are plotted bellow the chart.}\n \\label{figResults}\n\\end{figure}\n\nThe measured beams were coherent superpositions of Hermite-Gaussian\nbeams in the form $a \\, \\mathrm{HG}_{10}+ i b \\, \\mathrm{HG}_{01}$ with\n$\\{a=1,\\, b=0\\}$, $\\{a=0.4,\\, b=0.6\\}$ and $\\{a=0.5,\\, b=0.5\\}$,\nrespectively. The first and the third are thus a pure Hermite-Gaussian\nbeam and a pure Laguerre-Gaussian vortex beam, respectively. For all\nthe beams we used the settings\n$X \\simeq 0.0,~P_{X} \\simeq 0.0,~X^{\\prime} \\simeq\n-0.45,~P_{X}^{\\prime} \\simeq 0.0,~Y \\simeq 0.0,~P_{Y} \\simeq\n0.0,~Y^{\\prime} \\simeq 0.0,~P_{Y}^{\\prime} \\simeq -0.45$\nfor the evaluation of the Bell sums. The theoretical values of the\nBell sums for these are $ (-1.91, -2.15, -2.17)$, respectively.\n\nEach measurement was repeated many times with slightly different\nreadings, due to laser intensity instabilities and CCD noise. These\neffects manifest as measurement errors, which can be estimated from\nthe sample statistics. As the parity measurement requires to normalize\nthe total measured intensity of the interference pattern with respect\nto the input beam intensity, a separate reading of the input beam\nintensity was performed. For each optical beam, the mean value of the\nBell sum is reported. The results are summarized in\nFig.~\\ref{figResults}. The Bell correlations grow with the coupling\nbetween the basis $\\mathrm{HG}_{10}$ and $\\mathrm{HG}_{01}$ modes, with statistically\nsignificant violation of CHSH inequality by the second and third\nbeams, as theoretically predicted.\n\n We also show the measured values of the Wigner function. For\n both, $\\mathrm{HG}_{10}$ and $\\mathrm{LG}_{10}$ modes, the values of\n $\\pi^2 W (0, 0; 0, 0)$ are quite close to $-1$. For classical\n beams, ours is one of the few measurements on the negativity of the\n Wigner function, though it has to be anticipated from the\n corresponding results in quantum optics~\\cite{Schleich:2000}. We note\n that very early, March and Wolf~\\cite{Wolf:1974} had constructed\n an example of a classical source which exhibited negative Wigner\n function.\n\n Finally, we have checked the violation of CHSH inequality for the\n beam $\\mathrm{LG}_{20}$. A beam with higher topological charge is more\n sensitive to setup imperfections, hence the Bell sum variation is\n significantly larger than in the case of $\\mathrm{LG}_{10}$. On the other\n hand, as shown in Fig.~\\ref{figResults20}, the increasing of the Bell\n sum for higher orbital angular momentum is clearly demonstrated:\n the theoretical value for $\\mathrm{LG}_{20}$ is $-2.24$, which agrees pretty\n well with the experimental results.\n \\footnote{A study of the Bell violations for LG beams is also presented by\n S. Prabhakar, S. G. Reddy, A. A. Chithrabhanu, P. G. K. Samantha, and\n R. P. Singh, arxiv:1406.6239, although the authors does not employ\n parity measurements, but Fourier transform.}. \nNote that the Wigner function at the origin for the $\\mathrm{LG}_{20}$ beam\nis positive, as expected. \n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\columnwidth]{Figure4}}\n \\caption{(Color online) Experimental results for the beam\n $\\mathrm{LG}_{20}$, as presented in Fig.~3. The values of $(X,0;0, P_{Y})$\n are also indicated. The Bell sum variation is significantly larger\n than in the case of $LG_{10}$. The plots on the right bottom panel\n are the amplitude and phase of $\\mathrm{LG}_{20}$.}\n \\label{figResults20}\n\\end{figure}\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec:concl}\n\nIn short, we have presented an experimental study of nonlocal\ncorrelations in classical beams with topological\nsingularities~\\cite{Chowdhury:2013}. These correlations between modes are\nmanifested through the violation of a CHSH inequality, which we have\ndetected via direct parity measurements. Such a violation is shown to increase with the value of\norbital angular momentum of the beam. As a byproduct of our\nmeasurements, we obtain negativity of the Wigner function at certain\npoints in phase space for the $\\mathrm{HG}_{10}$ and $\\mathrm{LG}_{10}$ beams. Note\nthat this has implications for similar studies with electron beams,\nfor which vortices have been reported~\\cite{Schattschneider:2010,Unguris:2011}.\n\nThough entanglement here does not bear any paradoxical meaning, such\nas ``spooky action on the distance'', it still represents a\npotential resource for classical signal processing.\nIt might be expected that future applications of quantum information\nprocessing can be tailored in terms of classical light: the research\npresented in this work explores one of those options.\n\nFurthermore, our results are relevant not only for a correct understanding\nof ``classical entanglement'', but also for bringing out different\nstatistical features of the optical beams, since it provides an\nalternative paradigm to the well developed optical coherence theory.\n\n\\begin{acknowledgments}\nWe acknowledge illuminating discussions with Gerd Leuchs, Elisabeth\nGiacobino, and Andrea Aiello. This work was supported by the Grant\nAgency of the Czech Republic (Grant 15-031945), the European Social\nFund and the State Budget of the Czech Republic POSTUP II (Grant\nCZ.1.07\/2.3.00\/30.0041), the IGA of the Palack\\'y University (Grant\nPrF-2015-002), the Spanish MINECO (Grant FIS2011-26786), and UCM-Banco\nSantander Program (Grant GR3\/14).\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzkmpr b/data_all_eng_slimpj/shuffled/split2/finalzzkmpr new file mode 100644 index 0000000000000000000000000000000000000000..3e298ffbae56b24316cc6fde3d526666aac07eb9 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzkmpr @@ -0,0 +1,5 @@ +{"text":"\n\n\\section{Introduction}\n\n\n\nWe present a method for automatically generating human-robot interaction (HRI) scenarios in shared autonomy. Consider as an example a manipulation task, where a user provides inputs to a robotic manipulator through a joystick, guiding the robot towards a desired goal, e.g., grasping a bottle on the table. The robot does not know the goal of the user in advance, but infers their desired goal in real-time by observing their inputs and assisting them by moving autonomously towards that goal. Performance of the algorithm is assessed by how fast the robot reaches the goal. However, different environments and human behaviors could cause the robot to fail, by picking the wrong object or colliding with obstacles. \n\n\n\n\n\n\\begin{figure}[!t]\n\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/Title-2.pdf}\n\\caption{An example archive of solutions returned by the quality diversity algorithm MAP-Elites. The solutions in red indicate scenarios where the robot fails to reach the desired user goal in a simulated shared autonomy manipulation task. The scenarios vary with the environment (y-axis: distance between the two candidate goals) and human inputs (x-axis: variation from optimal path).}\n\n\\label{fig:best}\n\\end{figure}\n\n Typically, such algorithms are evaluated with human subject experiments~\\cite{thomaz2016computational}. While these experiments are fundamental in exploring and evaluating human-robot interactions and they can lead to exciting and unpredictable behaviors, they are often limited in the number of environments and human actions that they can cover. Testing an algorithm in simulation with a \\textit{diverse} range of scenarios can improve understanding of the system, inform the experimental setup of real-world studies, and help avoid potentially costly failures ``in the wild.'' \n \n\n \n\n \n\n\n\n\n\n\nOne approach is to simulate agent behaviors by repeatedly sampling from models of human behavior and interaction protocols~\\cite{steinfeld2009oz}. While this approach will show the \\textit{expected} behavior of the system given the pre-specified models, it is unlikely to reveal failure cases that are not captured by the models or are in the tails of the sampling distribution. Exhaustive search of human actions and environments is also computationally prohibitive given the continuous, high-dimensional space of all possible environments and human action sequences. \n\nAnother approach is to formulate this as an optimization problem, where the goal is to find adversarial environments and human behaviors. But we are typically not interested in the maximally adversarial scenario, which is the single, global optimum of our optimization objective, since these scenarios are both easy to find and unlikely to occur in the real-world, e.g., the human moving the joystick of an assistive robotic arm consistently in the wrong direction. \n\nInstead, we are interested in answering questions of the form: how noisy can the human input be before the algorithm breaks? Or, in the aforementioned example task, how far apart do two candidate goals have to be for the robot to disambiguate the human intent? \n\n\n\n\n\n\nOur work makes the following contributions:\\footnote{We include an overview video: \\url{https:\/\/youtu.be\/9P3qomydMWk}}\n\n\\textbf{1.} We propose formulating the problem of generating human-robot interaction scenarios as a \\textit{quality diversity} (QD) problem, where the goal is not to find a single, optimal solution, but a collection of high-quality solutions, in our case failure scenarios of the tested algorithm, across a range of measurable criteria, such as noise in human inputs and distance between objects.\n\n\n\n\n\n\n\n\n\n\n\\textbf{2.} We adopt the QD algorithm MAP-Elites, originally presented in~\\cite{cully:nature15, mouret2015illuminating}, for the problem of scenario generation. Focusing on the shared autonomy domain, where a robotic manipulator attempts to infer the user's goal based on their inputs, we show that MAP-Elites outperforms two baselines: standard Monte Carlo simulation (random search), where we uniformly sample the scenario parameters, and CMA-ES~\\cite{hansen:cma16}, a state-of-the-art derivative-free optimization algorithm, in finding diverse scenarios that minimize the performance of the tested algorithm. We select to test the algorithm ``shared autonomy with hindsight optimization''~\\cite{javdani2015hindsight}, since it has been widely used and we have found it to perform robustly in a range of different environments and tasks. Additionally, in hindsight optimization, inference and planning are tightly coupled, which makes testing particularly challenging; simply testing each individual component is not sufficient to reveal how the algorithm will perform.\n\n \n\n\n\\textbf{3.} We show that Monte Carlo simulation does not perform well because of \\textit{behavior space distortion}: sampling directly from the space of environments and human actions covers only a small region in the space of measurable aspects (behavioral characteristics). For example, uniformly sampling object locations (scenario parameters) results in a non-uniform distribution of their distances (behavioral characteristic) with a very small variance near the mean. On the other hand, MAP-Elites focuses on exploring the space of the behavioral characteristics by retaining an archive of high-performing solutions in that space and perturbing existing solutions with small variations. Therefore, MAP-Elites performs a type of simultaneous search guided by the behavioral characteristics, where solutions in the archive are used to generate future candidate solutions~\\cite{mouret2015illuminating}.\n\n\\textbf{4.} We analyze the failure cases and we show that they result from specific aspects of the implementation of the tested algorithm, rather than being artifacts of the simulation environment. We use the same approach to contrast the performance of hindsight optimization with that of linear policy blending~\\cite{dragan2012formalizing} and generate a diverse range of scenarios that \nconfirm previous theoretical findings~\\cite{trautman2015assistive}. The generated scenarios transfer to the real-world; we reproduce some of the automatically discovered scenarios on a real robot with human inputs. While some of the scenarios are expected, e.g., the robot approaches the wrong goal if the human provides very noisy inputs, others are surprising, e.g, the robot never reaches the desired goal even for a nearly optimal user if the two objects are aligned in column formation in front of the robot (Fig.~\\ref{fig:best})! \n\n\n\n\n\n\nQD algorithms treat the algorithm being tested as a ``black box'', without any knowledge of its implementation, which makes them applicable to multiple domains. \nOverall, we are excited about the potential of QD to facilitate understanding of complex HRI systems, opening up a number of scientific challenges and opportunities to be explored in the future. \n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Problem Statement} \\label{sec:problem}\n\n\n\nGiven a shared autonomy system where a robot interacts with a human, our goal is to generate scenarios that minimize performance of the system, while ensuring that the generated scenarios cover a range of prespecified measures.\n\n\nWe let $R$ be a single robot interacting with a single human $H$. We assume a function $G_H$ that generates human inputs, a function $G_E$ that generates an environment, and an HRI algorithm $G_R$ that generates actions for the robot. The human input generator is parameterized by $\\theta \\in \\mathbb{R}^{n_{\\theta}}$, where $n_{\\theta}$ is the dimensionality of the parameter space, while the environment generator is parameterized by $\\phi \\in \\mathbb{R}^{n_{\\phi}}$. We define a \\textit{scenario} as the tuple $(\\theta, \\phi)$. \n\nIn shared autonomy, $G_E(\\phi)$ generates an initial environment (and robot) state $x_E$. The human observes $x_E$ and provides inputs to the system $u_H = G_H(x_E, \\theta)$ through some type of interface. The robot observes $x_E$ and the human input $u_H$ and takes an action \n $u_R = G_R(x_E, u_H)$. The state changes with dynamics: $\\dot{x}_E = h(x_E, u_R)$. H and R interact for a time horizon $T$, or until they reach a final state $x_f \\in X_E$. \n \n To evaluate a scenario, we assume a function \\\\ $f(x_E^{0..T}, u_R^{0..T},u_H^{0..T}) \\rightarrow \\mathbb{R}$ that maps the state and action history to a real number. We call this an \\textit{assessment} function, which measures the performance of the robotic system. We also assume $M$ user-defined functions, $b_i(x_E^{0..T}, u_R^{0..T},u_H^{0..T})\\rightarrow \\mathbb{R},~i\\in[M]$. These functions measure aspects of generated scenarios that should vary, e.g., noise in human inputs or distance between obstacles. We call these functions \\textit{behavior characteristics} (BCs), which induce a Cartesian space called a \\textit{behavior space}.\n\n\n\n\n\nGiven the parameterization of the environment and human input generators, we can map a value assignment of the parameters $(\\theta, \\phi)$ to a state and action history $(x_E^{0..T},u_R^{0..T},u_H^{0..T})$ and therefore to an assessment $f(\\theta, \\phi)$ and a set of BCs $b(\\theta,\\phi)$. We assume that the behavior space is partitioned into N cells, which form an \\textit{archive} of scenarios, and we let $(\\theta_i, \\phi_i)$ be the parameters of the scenario occupying cell $i \\in [N]$.\n\nThe objective of our scenario generator is to fill in as many cells of the archive as possible with scenarios of high assessment $f$: \\footnote{We note that the assessment function could be any performance metric of interest, such as time to completion or minimum robot's distance to obstacles. Additionally, while in this work we focus on minimizing performance, we could instead search for scenarios that maximize performance, or that achieve performance that matches a desired value. We leave this for future work.\n}\n\\begin{equation}\n \\mathcal{M}(\\theta_1, \\phi_1, ... , \\theta_N, \\phi_N) = \\max \\sum_{i=1}^N f(\\theta_i, \\phi_i)\n\\label{eq:objective}\n\\end{equation}\n\n\n\\section{Background} \\label{sec:background}\n\\noindent\\textbf{Automatic Scenario Generation }\nAutomatically generating scenarios is a long standing problem in human training~\\citep{hofer1998automated}, with the core challenge being the generation of \\textit{realistic} scenarios~\\citep{martin:pcg10}. Previous work~\\citep{zook2012automated} has shown that optimization methods can be applied to generate scenarios by maximizing a scenario quality metric. \n\nScenario generation has been applied extensively to evaluating autonomous vehicles~\\citep{arnold:safecomp13,mullins:av18, abey:av19, rocklage:av17, gambi:av19,sadigh2019verifying}. Contrary to model-checking and formal methods~\\cite{choi2013model,o2014automatic}, which require a model describing the system's performance such as a finite-state machine~\\cite{meinke2015learning} or process algebras~\\cite{o2014automatic},\n black-box approaches do not require access to a model. Most relevant are black-box falsification methods~\\cite{deshmukh2017testing,zhao2003generating,kapinski2016simulation,dreossi2019verifai} that attempt to find an input trace that minimizes the performance of the tested system. Rather than searching for a single global optimum~\\cite{deshmukh2017testing, deshmukh2015stochastic,sadigh2019verifying}, or attempting to maximize coverage of the space of scenario parameters~\\cite{zhao2003generating} or of performance boundary regions~\\cite{mullins:av18}, we propose a quality diversity approach where we optimize an archive formed by a set of behavioral characteristics, with a focus on the shared autonomy domain. This allows us to \\textit{simultaneously} search for human-robot interaction scenarios that minimize the performance of the system over a range of measurable criteria, e.g., over a range of variation in human inputs and distance between goal objects.\n \n\nFinally, scenario generation is closely related to the problem of generating video game levels in procedural content generation (PCG)~\\citep{hendrikx2013procedural,shaker:book16}. An approach gaining popularity is procedural content generation through quality diversity (PCG-QD)~\\citep{gravina2019procedural}, which leverages QD algorithms to drive the search for interesting and diverse content.\n\n\\noindent\\textbf{Quality Diversity and MAP-Elites. }\nQD algorithms differ from pure optimization methods, in that they do not attempt to find a single optimal solution, but a collection of good solutions that differ across specified dimensions of interest. For example, QD algorithms have generated video game levels of varying number of enemies or tile distributions~\\cite{khalifa2019intentional, fontaine2020illuminating}, and objects of varying shape complexity and grasp difficulty~\\cite{morrison2020egad}. \n\n\\mbox{MAP-Elites} \\citep{mouret2015illuminating,cully:nature15} is a popular QD algorithm that searches along a set of explicitly defined attributes called \\textit{behavior characteristics} (BCs), which induce a Cartesian space called a \\textit{behavior space}. The behavior space is tessellated into uniformly spaced grid cells. In each cell, the algorithm maintains the highest performing solution, which is called as \\textit{elite}. The collection of elites returned by the algorithm forms an \\textit{archive} of solutions. \n\n\n\n\n MAP-Elites populates the archive by first randomly sampling a population of solutions, and then selecting the elites -- which are the top performing solutions in each cell of the behavior space -- at random and perturbing them with small variations. The objective of the algorithm is two-fold: maximize the number of filled cells (coverage) and maximize the quality of the elite in each cell. Recent algorithms have focused on how the behavior space is tessellated~\\citep{smith:ppsn16,fontaine:gecco19}, as well as how each elite is perturbed~\\cite{vassiliades:gecco18}. Recent work~\\cite{fontaine2021differentiable} has also shown that, when the objective function and behavior characteristics are first-order differentiable, MAP-Elites via a Gradient Arborescence (MEGA) can result in significant improvements in search efficiency. \n\nBy retaining an archive of high-performing solutions and perturbing existing solutions with small variations, \\mbox{MAP-Elites} simultaneously optimizes every region of the archive, using existing solutions as ``stepping stones'' to find new solutions. Previous work has shown that \\mbox{MAP-Elites} variants~\\cite{mouret2020quality} and surrogate models~\\cite{gaier2017data} outperform independent single-objective constrained optimizations for each cell with \\mbox{CMA-ES}, with the same total budget of evaluations.\n\n \\noindent\\textbf{Coverage-Driven Testing in HRI.} Previous work~\\cite{araiza2015coverage,araiza2016systematic} explored test generation in human-robot interaction using Coverage-Driven Verification (CDV), emulating techniques used in functional verification of hardware designs. Human action sequences were randomly generated in advance and with a model-based generator which modeled the interaction with Probabilistic-Timed Automata. Instead, we focus on online scenario generation by searching over a set of scenario parameters; the generator itself is agnostic to the underlying HRI algorithm and human model. Previous work~\\cite{araiza2016intelligent} has also used Q-learning to generate plans for an agent in order to maximize coverage. Our focus is both on coverage and quality of generating scenarios, with respect to a prespecified set of behavioral characteristics that we want to cover. In contrast to previous studies that simulate human actions, \\textit{we jointly search for environments and human\/agent behaviors.}\n\n\n\\noindent\\textbf{Shared Autonomy.}\nShared autonomy (also: shared control, assistive teleoperation) combines human teleoperation of a robot with intelligent robotic assistance. The method has been applied in the control of robotic arms~\\cite{javdani2018shared,dragan2012formalizing,nikolaidis2017mutualadaptation,Muelling2017,herlant2016assistive,gopinath2016human, jain2019probabilistic,jeon2020shared, losey2019controlling,rakita2019shared,rakita2018shared}, the flight of UAVs~\\cite{reddy2018shared,gillula2011applications,lam2009artificial}, and robot-assisted surgery~\\cite{li2003recognition,ren2008dynamic}. Shared autonomy has been implemented through a variety of interfaces, such as whole body motions~\\cite{dragan2012formalizing}, natural language \\cite{doshi2007efficient}, laser pointers \\cite{veras2009scaled}, brain-computer interfaces \\cite{Muelling2017}, body-machine interfaces~\\cite{jain2015assistive}, and eye gaze~\\cite{bien2004,javdani2018shared}. A shared autonomy system first predicts the human's goal, often through machine learning methods trained from human demonstrations \\cite{hauser13,koppula16,wang13}, and then provides assistance, which often involves blending the user's input with the robot assistance to achieve the predicted goal \\cite{dragan2012formalizing,fagg04,kofman05}. Assistance can provide task-dependent guidance \\cite{aarno2005adaptive}, manipulation of objects~\\cite{jeon2020shared}, or mode switches \\cite{herlant2016assistive}.\n\n\n\n\n\\noindent\\textbf{Shared Autonomy via Hindsight Optimization.}\nIn shared autonomy via hindsight optimization~\\cite{javdani2015hindsight} assistance blends user input and robot control based on the confidence of the robot's goal prediction. The problem is formulated as a Partially Observable Markov Decision Process (POMDP), wherein the user's goal is a latent variable. The system models the user as an approximately optimal stochastic controller, which provides inputs so that the robot reaches the goal as fast as possible. The system treats the user's inputs as observations to update a distribution over the user's goal, and assists the user by minimizing the expected cost to go -- estimated using the distance to goal -- for that distribution. Since solving a POMDP exactly is intractable, the system uses the hindsight optimization (QMDP) approximation~\\cite{littman1995learning}. The system was shown to achieve significant improvements in efficiency of manipulation tasks in an object-grasping task~\\cite{javdani2015hindsight} and more recently in a feeding task~\\cite{javdani2018shared}. We empirically found this algorithm to perform robustly in a range of different environments, which motivates a systematic approach for testing. We refer to this algorithm simply as \\textit{hindsight optimization}.\n\n\n\n\\section{Scenario Generation with MAP-Elites}\n\nAlgorithm~\\ref{alg:map-elites} shows the MAP-Elites algorithm from~\\cite{mouret2015illuminating,cully:nature15}, adapted for scenario generation. The algorithm takes as input a function $G_H$ parameterized by $\\theta$ that generates human inputs, a function $G_E$ parameterized by $\\phi$ that generates environments, and an HRI algorithm $G_R$ that generates actions for the robot. The algorithm searches for scenarios $\\theta, \\phi$ of high assessment values $f$ that fill in the archive $\\mathcal{P}$. \n\nFor each scenario $(\\theta, \\phi)$, MAP-Elites instantiates the generator functions $G_H$ and $G_E$. For instance, $\\phi$ could be a vector of objects positions, and $\\theta$ could be a vector of waypoints representing a trajectory of human inputs, or parameters of a human policy. \n\nWhen a scenario is generated, MAP-Elites executes the scenario in a simulated environment and estimates the assessment function $f$ and the BCs $b$. MAP-Elites then updates the archive if (1) the cell corresponding to the BCs $\\mathcal{X}[b]$ is empty, or (2) the existing scenario (elite) in $\\mathcal{X}[b]$ has a smaller assessment function (lower quality) than the new scenario. This allows populating the archive to maximize coverage as well as improving the quality of existing scenarios.\n\nFor the first $N_{init}$ iterations, the algorithm generates scenarios $\\theta, \\phi$ by randomly sampling from the parameter space. These sampled parameters seed the archive with an initial set of scenarios. After the first $N_{init}$ iterations, MAP-Elites selects a scenario uniformly at random from the archive and perturbs it with a small variation. This allows for better exploration of the archive, compared to random search, as we show in section~\\ref{subsec:Analysis}.\n\nWe note that, while our experiments focus on the shared autonomy domain, the proposed scenario generation method is general and can be applied to multiple HRI domains. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{algorithm}[t!]\n \\caption{Scenario Generation with MAP-Elites}\n\\label{algorithm: known context}\n\\begin{algorithmic}\n\\STATE\\textbf{Input:} Human input generator $G_H$, environment generator $G_E$, HRI algorithm $G_R$, variations\n$\\sigma_{\\theta},\\sigma_{\\phi}$\n\\STATE\\textbf{Initialize:} Scenarios in archive $\\mathcal{X}\\leftarrow \\emptyset$, assessments $\\mathcal{F}\\leftarrow \\emptyset$\n\n\\FOR{$t=1,\\ldots,N $}\n\\IF{$t < N_{init}$}\n\\STATE Generate scenario $\\theta, \\phi = \\mathrm{random\\_generation()}$ \n\\ELSE\n\\STATE Select elite $\\theta', \\phi' = \\mathrm{random\\_selection}(\\mathcal{X})$\n\\STATE Sample $\\theta \\sim N(\\theta', \\sigma_\\theta)$ \n\\STATE Sample $\\phi \\sim N(\\phi', \\sigma_\\phi)$ \n\n\\ENDIF\n\n\\STATE Instantiate $G_H^{\\theta} = G_H(\\theta)$\n\\STATE Instantiate $G_E^{\\phi} = G_E(\\phi)$\n\\STATE Compute $f = \\textrm{assessment}(G_H^{\\theta},G_E^{\\phi}, G_R)$\n\\STATE Compute $b = \\textrm{behaviors}(G_H^{\\theta},G_E^{\\phi}, G_R)$\n \\IF{$\\mathcal{F}[b]= \\emptyset$ or $\\mathcal{F}[b] < f$ } \n \\STATE Update archive $\\mathcal{X}[b] \\leftarrow (\\theta, \\phi),\\mathcal{F}[b] \\leftarrow f$ \n \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\label{alg:map-elites}\n\\end{algorithm}\n\n\\section{Generating Scenarios in Shared Autonomy} \\label{sec:limiting}\n\nWe focus on a shared autonomy manipulation task, where a human user teleoperates a robotic manipulator through a joystick interface. The robot runs a hindsight optimization shared autonomy algorithm~\\cite{javdani2015hindsight}, which uses the user's input to infer the object the user wants the robot to grasp, and assists the user by moving autonomously towards that goal. \n\n\\subsection{Scenario Parameters} \\label{subsec:parameters}\nFollowing the specification of section~\\ref{sec:problem}, we define a human input generator $G_H$ parameterized by $\\theta$ and an environment generator $G_E$ parameterized by $\\phi$.\n\n\\noindent\\textbf{Environment Generator:} The environment generator $G_E$ takes as input the 2D positions $g_i$ of $n$ goal objects (bottles), so that $\\phi = (g^1, ..., g^n)$, and places them on top of a table. We specify the range of the coordinates $g_x \\in [0, 0.25]$ (in meters), $g_y \\in [0, 0.2]$ so that the goals are always reachable by the robot's end-effector. We position the robotic manipulator to face the objects (Fig.~\\ref{fig:elites}).\n\n\n\\noindent\\textbf{Human Input Generator:} Our pilot studies with the shared autonomy system have shown that user inputs are typically not of constant magnitude. Instead, inputs spike when users wish to ``correct'' the robot's path, and decrease in magnitude afterwards when the robot takes over. \n\nTherefore, we specify the human input generator $G_H$, so that it generates a set of equidistant waypoints in Cartesian space forming a straight line that starts from the initial position of the robot's end-effector and ends at the desired goal of the user. At each timestep, the generator takes as input the current state of the robot (and the environnment) $x_E$, and provides a translational velocity command $u_H$ for the robot's end-effector towards the next waypoint, proportional to the distance to that waypoint. \n\nWe allow for noise in the waypoints by adding a disturbance $d \\in [-0.05, 0.05]$ for each of the intermediate waypoints in the horizontal direction (x-axis in Fig.~\\ref{fig:elites}). We selected $m=5$ intermediate waypoints, and specified the human input parameter $\\theta$ as a vector of disturbances, so that $\\theta = (d_1, ... , d_5)$. We note that this is only one way, out of many, of simulating the human inputs. \n\n\n\\noindent\\textbf{HRI Algorithm and Simulation Environment:} We use the publicly available implementation of the hindsight optimization algorithm~\\cite{ada_code}, which runs on the OpenRAVE~\\cite{diankov2008openrave} simulation environment. Experiments were conducted with a Gen2 Lightweight manipulator. For each goal object we assume one target grasp location, on the side of the object that is facing the robot. \n\n\n\n\\subsection{Assessment Function.} The assessment function $f$ represents the quality of a scenario. We evaluate a scenario by simulating it until the robot's end-effector reaches the user's goal, or when the maximum time (10 $s$) has elapsed. We use as an assessment function time to completion, where longer times represent higher scenario quality, since we wish to discover scenarios that \\textit{minimize} performance. \n\n\\subsection{Behavioral Characteristics.} \nWe wish to generate scenarios that show the limits of the shared autonomy system: how noisy can the human be without the system failing to reach the desired goal? How does distance between candidate goals affect the system's performance? Intuitively, noisier human inputs and smaller distances between goals would make the inference of the user's goal harder and thus make the system more likely to fail. \n\nThese dimensions of interest are the behavioral characteristics (BC) $b$: attributes that we wish to obtain coverage for. We explore the following BCs:\n\n\n\\noindent\\textbf{Distance Between Goals:} How far apart the human goal is from other candidate goals in a scenario plays an important role in disambiguating the human intent when the robot runs the hindsight optimization algorithm. The reason is that the implementation of the algorithm models the human user as minimizing a cost function proportional to the distance between the robot and the desired goal. The framework then infers the user's goal by using the user inputs as observations; the more unambiguous the user input, the more accurate the inference of the system. Therefore, we expect that the further away the human goal object $g_H$ is from the nearest goal $g_N$, the better the system will perform. We define this BC as: \n\\begin{equation}\nBC1 = ||g_H - g_N||_2\n\\end{equation}\n\nGiven the range of the goal coordinates, the range of this BC is $[0, 0.32]$. In practice, there will be always a minimum distance between two goal objects because of collisions, but this does not affect our search, since we can ignore cases where the objects collide. We partitioned this behavior space to 25 uniformly spaced intervals\n\n\\noindent\\textbf{Human Variation:} \nWe expect noise in the human inputs to affect the robot's inference of the user's goal and thus the system's performance. \nWe capture variation from the optimal path using the root sum of the squares of the disturbances $d_i$ applied to the $m$ intermediate waypoints.\n\\begin{equation}\nBC2 = \\sqrt{\\sum_{i=1}^m d_i^2}\n\\end{equation}\n A value of 0 indicates a straight line to the goal. Since we have $d_i \\in [-0.05,0.05]$ (section~\\ref{subsec:parameters}), the range of this BC is $[0,0.11]$. We partitioned this behavior space to 100 uniformly spaced intervals\n\n\n\\noindent\\textbf{Human Rationality:} If we interpret the user's actions using a bounded rationality model~\\cite{baker2007goal,fisac2018probabilistically}, we can explain deviations from the optimal trajectory of human inputs as a result of how ``rational'' or ``irratonal'' the user is.\\footnote{We note that we use the human rationality model as one way, out of many, to \\textit{interpret} human inputs and not to as a way to \\textit{generate} inputs. Human inputs can be generated with any generator model. In this paper, we generate human inputs with the deterministic model described in section~\\ref{subsec:parameters}. We discuss extensions to stochastic human models in section~\\ref{sec:discussion}.}\n\n\n\n\n Formally, we let $x_R$ be the 3D position of the robot's end-effector and $u_H$ be the velocity controlled by the user in Cartesian space. We model the user as following Boltzmann policy $\\pi_H \\mapsto P(u_H|x_R,g_H, \\beta)$, where $\\beta$ is the rationality coefficient -- also interpreted as the expertise~\\cite{jeon2020shared}-- of the user and $Q_{g_H}$ is the value function from $x_R$ to the goal $g_H$.\n\\begin{equation}\nP(u_H|x_R,g_H, \\beta) \\propto \ne^{- \\beta Q_{g_H}(x_R, u_H)}\n\\label{eqn:human}\n\\end{equation}\n\nLet $Q_{g_H} = -||u_H||_2 - ||x_R + u_H - g_H||_2$~\\cite{fisac2018probabilistically}, so that the user minimizes the distance to the goal. Observe that if $\\beta \\rightarrow \\infty$, the human is rational, providing velocities exactly in the direction to the goal. If $\\beta \\rightarrow 0$, the human is random, choosing actions uniformly. \n\nWe can estimate the user's rationality, given their inputs, with Bayesian inference~\\cite{fisac2018probabilistically}: \n\n\\begin{equation}\nP(\\beta|x_R,g_H, u_H) \\propto \nP(u_H|x_R,g_H, \\beta) P(\\beta)\n\\label{eqn:human}\n\\end{equation}\n\n\n\nSince the human inputs change at each waypoint (section~\\ref{subsec:parameters}), we perform $m+1$ updates, at the starting position and at each intermediate waypoint, on a finite set of discrete values $\\beta$. Following previous work~\\cite{jeon2020shared}, we set the rationality range $\\beta \\in [0,1000]$. We then choose as behavioral characteristic the value with the maximum a posteriori probability at the end of the task:\n\n\\begin{equation}\nBC3 = \\argmax P(\\beta|x^{0..T}_R, g_H, u^{0..T}_H)\n\\end{equation}\n\nWe partitioned the space to 101 uniformly spaced intervals. \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\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\resizebox{.8\\linewidth}{!}{\n\\begin{tabular}{l|cc|cc|cc}\n\\hline\n & \\multicolumn{2}{l|}{BC1 \\& BC3, 2 goals} & \\multicolumn{2}{l|}{BC1 \\& BC2, 2 goals} & \\multicolumn{2}{l}{BC1 \\& BC2, 3 goals} \\ \\\\ \n \\toprule\nAlgorithm & Coverage & QD-Score & Coverage & QD-Score & Coverage & QD-score\\\\\n \\midrule\nRandom & 22.3\\% & 3464 & 48.4\\% &7782 & 41.9\\% & 7586\\\\\nCMA-ES & 24.8\\% & 4540& 38.9\\% & 7422 & 34.5\\% & 7265\\\\\nMAP-Elites & \\textbf{62.8\\%} & \\textbf{10128} & \\textbf{63.0\\%} & \\textbf{11216} & \\textbf{57.4\\%} & \\textbf{11204}\\\\\n \\bottomrule\n\\end{tabular}\n}\n\\caption{Results: Percentage of cells covered (coverage) and QD-Score after 10,000 evaluations, averaged over 5 trials.}\n\\label{tab:results}\n\\end{table*}\n\n\n\\begin{figure*}[t!]\n \\centering\n \\includegraphics[width=1.0\\textwidth]{figs\/maps_BC1_plot.pdf}\n \\includegraphics[width=1.0\\textwidth]{figs\/maps_BC2_plot.pdf}\n\\caption{QD-Scores over evaluations (generated scenarios) and example archives returned by the three algorithms for the first two behavior spaces of Table~\\ref{tab:results}. The colors of the cells in the archives represent time to task completion in seconds.}\n\\label{fig:maps}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\nWe compare different search algorithms in their ability to find diverse and high-quality scenarios in different behavior spaces.\n\n\n\\subsection{Independent Variables}\nThe experiment has two independent variables, the \\textit{behavior space} and the \\textit{search algorithm}.\n\n\\textit{Behavior Space}: (1) Distance between $n=2$ goal objects (BC1) and human rationality (BC3), (2) Distance between $n=2$ goal objects (BC1) and human variation (BC2), and (3) Distance between human goal and nearest goal for $n=3$ goals (BC1) and human variation (BC2).\\footnote{We note that the behavior spaces can be more than two-dimensional, e.g, we could specify a space with all three BCs. We include only 2D spaces since they are easier to visualize and inspect.}\n\n\n\n\\textit{Search Algorithm}: We evaluate three different search methods: MAP-Elites, CMA-ES and random search. The Covariance Matrix Adaptation Evolution Strategy (\\mbox{CMA-ES}) is one of the most competitive derivative-free optimizers for single-objective optimization of continuous spaces (see~\\citep{hansen:cma16,hansen2009benchmarking}) and it is commonly used for falsification of cyber-physical systems~\\cite{deshmukh2015stochastic,zhang2018time}. In random search we use Monte Carlo simulation where scenario parameters are sampled uniformly within their prespecified ranges.\n\nWe implemented a multi-processing system on an AMD Ryzen Threadripper 64-core (128 threads) processor, as a master search node and multiple worker nodes running separate OpenRAVE processes in parallel, which enables simultaneous evaluation of many scenarios. Random search and MAP-Elites run asynchronously on the master search node, while \\mbox{CMA-ES} synchronizes before each covariance matrix update. We generated 10,000 scenarios per trial, and ran 45 trials, 5 for each algorithm and behavior space. One trial parallelized into 100 threads lasted approximately 20 minutes. \n\n\\subsection{Algorithm Tuning}\nMAP-Elites first samples uniformly the space of scenario parameters $\\theta, \\phi$ within their prespecified ranges for an initial population of $100$ scenarios (Algorithm~\\ref{alg:map-elites}). The algorithm then \nrandomly perturbs the elites (scenarios from the archive) with Gaussian noise scaled by a $\\sigma$ parameter. The two scenario parameters, position of goal objects $\\phi$ and human waypoints $\\theta$, are on different scales, thus we specified a different $\\sigma$ for each: $\\sigma_\\phi = 0.01, \\sigma_\\theta = 0.005$. \n\nTo generate the scenarios for random search, we \nuniformly sample scenario parameters within their prespecified ranges, a method identical to generating the initial population of MAP-Elites. \n\nFor CMA-ES, we selected a population of $\\lambda = 12$ following the recommended setting from~\\cite{hansen:cma16}. To encourage exploration, we used the bi-population variant of CMA-ES with restart rules~\\cite{auger2005restart,hansen2009benchmarking}, where the population doubles after each restart, and we selected a large step size, $\\sigma = 0.05$. Since the two search parameters are in different scales, we initialized the diagonal elements of the covariance matrix $C$, so that $c_{ii} = 1.0, i \\in [2n]$ and $c_{ii} = 0.5, i \\in \\{2n+1, ..., 2n+m\\}$, with $2n$ and $m$ the dimensionality of the goal object and human input parameter spaces respectively. \n\nBoth CMA-ES and MAP-Elites may sample scenario parameters that do not fall inside their prespecified ranges. Following recent empirical results on bound constraint handling~\\cite{biedrzycki2020handling}, we adopted a resampling strategy, where new scenarios are resampled until they fall within the prespecified range. \n\\subsection{Measures} \nWe wish to measure both the diversity and the quality of scenarios returned by each algorithm. These are combined by the QD-Score metric~\\cite{pugh2015confronting}, which is defined as the sum of $f$ values of all elites in the archive (Eq.~\\ref{eq:objective} in section~\\ref{sec:problem}). Empty cells have 0 $f$ value. Therefore, QD-score is positively affected by both the coverage of the archive (the number of occupied cells in the archive divided by the total number of cells) and the assessment of the occupied cells. Similarly to previous work~\\cite{fontaine:gecco20}, we compute the QD-Score of CMA-ES and random search for comparison purposes by calculating the behavioral characteristics for each scenario and populating a pseudo-archive. We also include the coverage score as an additional metric of diversity.\n\n\n\\subsection{Hypothesis}\n\\textit{We hypothesize that MAP-Elites will result in larger QD-Score and coverage than both CMA-ES and random search.}\n\nPrevious work~\\cite{fontaine:gecco19,fontaine:gecco20} has shown that behavior spaces are typically distorted: uniformly sampling the search parameter space results in samples concentrated in small areas of the behavior space. Therefore, we expect random search to have small coverage of the behavior space. Additionally, since random search ignores the assessment function $f$, we expect the quality of the found scenarios in the archive to be low.\n\nCMA-ES moves with a single large population that has global optimization properties. Therefore, we expect it to concentrate in regions of high-quality scenarios, rather than explore the archive. On the other hand, MAP-Elites both expands the archive and maximizes the quality of the scenarios within each cell. \n\n\n\\begin{figure*}[t!]\n \\centering\n \\begin{tabular}{ccc}\n \\begin{subfigure}[t]{.25\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figs\/overlayed-side.pdf}\n \\end{subfigure} & \n \\begin{subfigure}[t]{.25\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figs\/overlayed-side-optimal.pdf}\n \\end{subfigure}&\n \\begin{subfigure}[t]{.25\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figs\/overlayed-line.pdf}\n \\end{subfigure} \\\\ \n \n \\end{tabular}\n\\caption{(Left) The robot fails to reach the user's goal $g_H$ because of the large deviation in human inputs from the optimal path. The waypoints of the human inputs are indicated with green color. (Center) We show for comparison how the robot would act if human deviation was 0 (optimal human). (Right) The robot fails to reach the user's goal $g_H$ (bottle furthest away from the robot), even though the human provides a near optimal input trajectory.}\n\\label{fig:elites}\n\\end{figure*}\n\n\n\n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{figs\/hists_BC12_plot.pdf}\n\n \n \n \n \n \n \n \n \n \\caption{Distribution of cells explored for random search and MAP-Elites. The cell colors represent frequency counts.}\n \\label{fig:hist}\n \n \\end{figure}\n\\subsection{Analysis} \\label{subsec:Analysis}\nTable~\\ref{tab:results} summarizes the performance of the three algorithms, for each of the three behavior spaces. We conducted a two-way ANOVA to examine the effect of the behavior space and the search algorithm on the QD-Score and coverage. There was a statistically significant interaction between the search algorithm and the behavior space for both QD-Score ($F(4,36) = 62.39, p < 0.001$) and coverage ($F(4,36) = 77.92, p < 0.001$). Simple main effects analysis with Bonferroni correction showed that \\mbox{MAP-Elites} outperformed \\mbox{CMA-ES} and random search in both QD-Score and coverage ($p<0.001$ in all comparisons). This result supports our hypothesis. \n\n\nFig.~\\ref{fig:maps} shows the improvement in the QD-Score over time and one example archive from each algorithm for the first two behavior spaces. MAP-Elites visibly finds more cells and of higher quality (red color), illustrating its ability to cover larger areas of the archive with high-quality scenarios. As expected, CMA-ES concentrates in regions of high-quality scenarios but has small coverage. \n\n\nRandom search covers a smaller area of the archive, compared to MAP-Elites, because of the \\textit{behavior space distortion}, shown in Fig.~\\ref{fig:hist}. Even though the search parameters are sampled uniformly, scenarios are concentrated on the left side of the archive specified by the human rationality and distance between goals BCs (Fig.~\\ref{fig:hist}-top). This occurs because if any of the sampled waypoints deviates from the optimal path, low values of rationality become more likely. In the human variation and distance between goals BCs, the distribution of scenarios generated by random search is concentrated in a small region near the center (Fig.~\\ref{fig:hist}-bottom). This is expected, since the two BCs are Euclidean norms of random vectors (see~\\cite{random2018}). On the other hand, MAP-Elites selects elite scenarios from the archive and perturbs them with Gaussian noise, instead of uniformly sampling the scenario parameters, resulting in larger coverage. \n\n\n\n\n\n\n\n\\subsection{Interpreting the Archives}\nIn the generated archives (Fig.~\\ref{fig:maps}), each cell contains an elite, which is the scenario that achieved the maximum assessment value (time to complete the task) for that cell. It is important to confirm that \\emph{the timeouts (red cells in the archive) occur because of the implementation of the tested algorithm (hindsight optimization), rather than being artifacts of the simulation environment}. \n\n\nTherefore, we replay the elites in different regions of the archives to explain the system's performance. We focus on the first two behavior spaces in Fig.~\\ref{fig:maps} using the archives generated with MAP-Elites, since MAP-Elites had the largest QD-Score and coverage.\n\n\n\n\n\n\n\nWe observe that if the distance between goals is large and the human is nearly optimal, the robot performs the task efficiently. This is shown by the blue color in the top-right of the first behavior space (distance and human rationality $\\beta$). We observe the same for large distance and small variation in the second behavior space.\n\n\n\nWe then explore different types of scenarios where the robot fails to reach the user's goal by the maximum time (10s), indicated by the red cells in the archives. When human variation is large (or equivalently rationality is low), the human may provide inputs that guide the robot towards the wrong goal. Since the robot updates a probability distribution over goals based on the user's input~\\cite{javdani2015hindsight} and the robot assumes that the user minimizes their distance to their desired goal, noisy inputs may result in assigning a higher probability to the wrong goal and the robot will move towards that goal instead. Fig.~\\ref{fig:elites}(left) shows the execution trace of one elite where this occurs. Fig.~\\ref{fig:best} shows the position of this elite in the archive. Fig.~\\ref{fig:elites}(center) shows how the robot would reach the desired goal if the human had behaved optimally, instead. \n\n\n\n\n\n\n\n\n\n\nWhat is surprising, however, is that the robot does not reach the user's goal even in parts of the behavior space where human variation is nearly 0 (or equivalently rationality is very high), that is when the human provides a nearly optimal input trajectory! Fig.~\\ref{fig:elites}(right) reveals a case, where the two goal objects are aligned one closely behind the other. The robot approaches the first object, on the way towards the second object, and stops there. \n\n\nWhat is interesting in both scenarios is that the robot gets ``stuck'' at the wrong goal, even when the simulated user continues providing inputs to their desired goal! Inspection of the publicly available implementation~\\cite{ada_code} of the algorithm shows that this results from the combination of two factors: the robot's cost function and the human inputs.\n\n\n\n\n\n\n\\noindent\\textbf{Cost Function.} The cost function that the robot minimizes is specified as a constant cost when the robot is far away from the goal and as a smaller linear cost when the robot is near the target~\\cite{javdani2018shared} (distance to target is smaller than a threshold). This makes the cost of the goal object near the robot significantly lower than the cost of the other goal objects, which results in the probability mass of the goal prediction to concentrate on that goal. While this can help the user align the end-effector with the object (see~\\cite{javdani2018shared}), it can also lead to incorrect inference, if the robot approaches the wrong goal on its way towards the correct goal or because of noisy human input. We confirmed that removing the linear term from the cost function results in the robot reaching the right goal in both examples\n\n\n\n\n\n\n\n\n\n\\noindent\\textbf{Human Inputs.} The hindsight optimization implementation minimizes a cost function specified as the sum of two quadratic terms, the expected cost-to-go to assist for a distribution over\ngoals, and a term penalizing disagreement with the user's input. The first-order approximation of the value function leads to an interpretation of the final robot action $u_R = u_R^A+u_R^u$ as the sum of two independent velocity commands, an ``autonomous'' action $u_R^A$ towards the distribution over goals and an action that follows the user input $u_R^u$, as if the user was directly teleoperating the robot (see~\\cite{javdani2018shared}).\n\nWe have simulated the human inputs, so that they provide a translational velocity command towards the next waypoint, proportional to the distance of the robot's end-effector to that waypoint (section~\\ref{subsec:parameters}). This results in a term $u_R^u$ of small magnitude when the end-effector is close to one of the waypoints. If at the same time the robot has high confidence on one of the goals, $u_R^A$ will point in the direction of that goal and it will cancel out any term $u_R^u$ that attempts to move the robot in the opposite direction.\n\nWe confirmed that, if the user instead applied the maximum possible input towards their desired goal, the robot would get ``unstuck,'' so a real user would always be able to eventually reach their desired goal. However, this requires effort from the user who would need to ``fight'' the robot. Overall, the archive reveals limitations that depend on \\textit{how the goal objects are aligned in the environment, the direction and magnitude of user inputs, and the cost function used by the implementation of the hindsight optimization algorithm.}\n\n\n\n\n \n \\begin{figure}[t!]\n \\begin{tabular}{cc}\n \\centering\n \\begin{subfigure}[t]{.49\\columnwidth}\n \\includegraphics[width=1.0\\columnwidth]{figs\/obstacle-collision-around.pdf}\n \n \\end{subfigure} &\n \\begin{subfigure}[t]{0.49\\columnwidth}\n \\includegraphics[width=1.0\\columnwidth]{figs\/obstacle-collision-noisy.pdf}\n \n \n \\end{subfigure}\n \\end{tabular}\n \\caption{Scenarios where the policy blending algorithm results in collision with an obstacle, approximated by a sphere. (Left) While the human and robot trajectories are each collision-free, blending the two results to collision when they point towards opposite sides of the obstacle. (Right) Blending with a very noisy human input results in collision. }\n \\label{fig:obstacle-examples}\n \\end{figure}\n \n\n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.6\\columnwidth]{figs\/example_two_goals.jpg}\n \\caption{We reproduced the generated scenarios in the real world with actual joystick inputs.}\n \\label{fig:real}\n \\end{figure}\n\\section{Comparing Algorithms}\\label{sec:comparing_short}\nGiven the effectiveness of quality diversity in automatically generating a diverse range of test scenarios, we can also use it to understand differences in performance between algorithms. In this work, we compare the performance of hindsight optimization~\\cite{javdani2015hindsight} with linear policy blending~\\cite{dragan2012formalizing}. We describe the experiment in Appendix~\\ref{sec:comparing}. We found that policy blending resulted in collisions, even for a nearly optimal human, in cases where human and robot inputs pointed towards opposite sides of an obstacle (Fig.~\\ref{fig:obstacle-examples}). This confirms previous theoretical findings~\\cite{trautman2015assistive} of unsafe behavior that linear blending has in the presence of obstacles. On the other hand, hindsight optimization avoids collisions, since it uses the human inputs as observations and the robot's motion is determined only by the robot's policy. \n\n\n\\section{Discussion} \\label{sec:discussion}\n\n\\noindent\\textbf{Experimental Findings.} We found that failure scenarios for hindsight optimization occur when the two goals are close to each other and the human inputs are noisy, or when one goal is in front of the other. In the latter case, failure occurs even if the human input is nearly optimal in minimizing the distance to the desired goal. In both cases, the robot becomes over-confident about the wrong goal and gets ``stuck'' there. \n\nAn important factor is the linear decrease of the cost in the vicinity of the goal objects. When specifying the cost function, it would be prudent to make the distance threshold for the linear decrease proportional to the distance between the goal objects, rather than setting it to an absolute value. \n\nOther potential measures to avoid the system's overconfidence towards the wrong goal are: (1) including the Shannon entropy with respect to all the goals in the cost function~\\cite{jeon2020shared} to penalize actions that result in very high confidence to one specific goal; (2) assigning a non-zero probability that the user changes their mind throughout the task and switches goals~\\cite{nikolaidis2017human, jain2018recursive}. It would be interesting to investigate the effect of more ``conservative'' assistance on subjective and objective metrics of the robot's performance.\n\nFinally, while linear policy blending naturally gives more control to the user and it is preferred by users in simple tasks~\\cite{javdani2018shared}, we empirically verified that the algorithm can generate unsafe trajectories, even if the individual human and robot inputs are safe. \n\nTo show that the presented scenarios can occur in deployed systems, we reproduce them in the real world with actual inputs through a joystick interface (Fig.~\\ref{fig:real}).\\footnote{We show different generated scenarios reproduced in the real world here: \\url{https:\/\/youtu.be\/2-JCO3dUHsA}} \n\n\\noindent\\textbf{Stochasticity in Scenarios.} In our experiments the generated scenarios are deterministic. One may wish to simulate scenarios where there is stochasticity in the robot's decision making or in the environment dynamics. A designer may also wish to test the system's performance under a stochastic human model, e.g., when human inputs are generated by a stochastic noisily rational human.\n\n\n\nThe most common approach in evolutionary optimization of noisy domains is \\textit{explicit averaging}, where we run multiple trials of the same scenario and then retain an aggregate measure of the assessment estimate, e.g, we compute the average to estimate the expected assessment $\\mathbb{E}[f(\\theta,\\phi)]$~\\cite{rakshit2017noisy,jin2005evolutionary}. We can follow the same process to estimate the behavior characteristics~\\cite{justesen2019map}. To improve the efficiency of the estimation, previous work has also employed implicit averaging methods, e.g., where the assessment of a scenario ($\\theta, \\phi$) is estimated by taking the assessments of previously evaluated scenarios in the neighborhood of $\\theta,\\phi$ into account. Previous work also includes adaptive sampling techniques, where the number of trials increases over time as the quality of the solutions in the archive improves~\\cite{justesen2019map}. A recent variant of MAP-Elites (Deep-Grid MAP-Elites) which updates a subpopulation of solutions for each cell in the behavior space has shown significant benefits in sample efficiency ~\\cite{flageat2020fast}. We leave these exciting directions for future work.\n\n\n\n\n\\noindent\\textbf{Limitations.} An important challenge is how to effectively characterize the behavior spaces. While we have assumed bounded behavior spaces, the rationality coefficient does not meet this assumption, which resulted in elites accumulating in the upper bound of the rationality in the archive (Fig.~\\ref{fig:maps}). Adapting the boundaries of the space dynamically based on the distribution of generated scenarios~\\cite{fontaine:gecco19} could improve coverage in this case.\n\nWhile distance between objects indeed played a role, our experiments showed the unexpected and surprising edge case where the two objects are in column formation and the human is nearly optimal. An interesting follow-up experiment would be to specify as a BC some metric of object alignment in column formation and investigate further the effect of this variable. In general, a practitioner can test the system with an initial design of BCs, observe the failure cases, create new BCs from newly observed insights and test the system further.\n \n We focused on how to effectively search the generative space of scenarios, but not on the generation methods themselves. Realism is an important future consideration, both in generating environments and human inputs. In human training, realism can be measured through a modified Turing test designed to require humans to distinguish generated scenarios from human authored ones~\\citep{martin:ucf}. Alternatively, we could run a user study where we place objects in the same locations as our failure scenarios and observe whether participants perform similar actions that cause failures.\n \n\n\n\\noindent\\textbf{Implications.}\nFinding failure scenarios of HRI algorithms will play a critical role in the adoption of these systems. We proposed quality diversity as an approach for automatically generating scenarios that assess the performance of HRI algorithms in the shared autonomy domain and we illustrated a path for future work. While real-world studies are essential in evaluating complex HRI systems, automatic scenario generation can facilitate understanding and tuning of the tested algorithms, as well as provide insights on the experimental design of real-world studies, whose findings can in turn inform the designer for testing the system further. We are excited about applications of quality diversity algorithms as test oracles in verification systems~\\cite{kress2020formalizing,porfirio2018authoring}, as well as in other domains where deployed robotic systems face a diverse range of interaction scenarios. \n\n\\section{Acknowledgements}\nWe thank Tapomayukh Bhattacharjee, David Hsu, Shen Li, Dylan Losey, Dorsa Sadigh, Rosario Scalice, Julian Togelius and our anonymous RSS reviewers for their feedback on early versions of this work.\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFor every separable exact $C^*$-algebra $\\Ac$, the lack of non-empty quasi-compact open subsets of the primitive ideal \nspace $\\Prim(\\Ac)$ ensures that $\\Ac$ is AF-embeddable, that is, $\\Ac$ is isomorphic to a closed subalgebra of an \nAF-algebra. \nThe converse also holds if $\\Ac$ is traceless (cf. \\cite[Cor.~B-C]{Ga20}), and for other $C^*$-algebras as well. \nFor instance, if $G$ is a generalized $ax+b$-group, hence a solvable Lie group $G = \\Vc \\rtimes \\RR$, where \n$\\Vc$ is a finite-dimensional real vector space, then $C^*(G)$ is not traceless. \nNevertheless, $C^*(G)$ is AF-embeddable if and only if $\\Prim(G)$ has no non-empty quasi-compact open subsets. \n(See Example~\\ref{ax+b} below.) \nHere and throughout this paper, we denote $\\Prim(G):=\\Prim(C^*(G))$ for any locally compact group~$G$, and a topological space is called quasi-compact if all its open covers have finite subcovers, without requiring any separation property. \n\n\nThe questions addressed in the present paper concern the relation between existence of non-empty open quasi-compact subsets of $\\Prim(G)$ and stably finiteness or even AF-embeddability of $C^*(G)$ for solvable Lie groups $G$.\n\nThe focus of our study is on exponential solvable Lie groups for two main reasons: \nFirstly, for all simply connected solvable Lie groups with polynomial growth, we have proved in \\cite{BB20} that there exist no non-empty quasi-compact open subsets in the primitive ideal spaces of their $C^*$-algebras. \nSecondly, there exist large classes of exponential solvable Lie groups $G$ for which $\\Prim(G)$ has finite (hence quasi-compact) open subsets or, more generally, the stabilized $C^*$-algebra $\\Kc \\otimes C^*(G)$ contains non-zero projections, that is, \n$C^*(G)$ is not stably projectionless. \nSee for instance \\cite{GKT92}, \\cite{KT96}, or Section~\\ref{Sect4}. \n\nOur main results for simply connected solvable Lie groups~$G$ could be summarized as follows: \n\\begin{enumerate}[(1)]\n\\item\\label{item_1} If $\\dim G\\in 4\\ZZ + 2$ or $\\dim G\\in 2 \\ZZ +1$ then $C^*(G)$ is stably finite if and only if it is stably projectionless \n(Theorem~\\ref{4n+2}). \n\\item\\label{item_2} If $\\dim G\\in 4\\ZZ$, we prove by examples that\nboth situations can appear: \nif $\\Prim(G) $ has finite open subsets, then $C^*(G)$ could be either AF-embeddable, \n or not even stably finite \n (Proposition~\\ref{Heis} and Theorem~\\ref{N6N15}).\n\\end{enumerate}\nWe also take the first steps towards describing the exponential solvable Lie groups $G$ whose nilradical is 1-codimensional and for which $C^*(G)$ is AF-embeddable while $\\Prim(G)$ contains finite open subsets (Corollary~\\ref{cf-cor8}). \n\nThe methods we use for obtaining some of these results owe much to the deep work \\cite{Sp88} on AF-embeddings of $C^*$-algebra extensions. \nIn addition, we have used the way the ``real'' structures of group $C^*$-algebras (in the sense of \\cite{Ka80} and \\cite{Ros16}) are encoded in the K-theory, \nand also the propagation of stably finiteness of the group $C^*$-algebras through suitable deformations of the constants of structure of the Lie algebras. \n\n\nIn more detail, this paper has the following contents. \nIn Section~\\ref{section1} we obtain some technical results that involve the link between stably finiteness and existence of projections. \nWe also investigate the interaction between ``real'' structures of $C^*$-algebras and Rieffel's construction \nof Connes' Thom isomorphism (Proposition~\\ref{tsigns}) with an application to solvable Lie groups (Corollary~\\ref{signs_solvable}). \nThen we establish that the property of continuous deformations preserve stably finiteness for certain continuous fields of $C^*$-algebras (Proposition~\\ref{prop-cf2}), and establish a condition for the failure of stably finiteness in terms of open points of the primitive ideal space (Proposition~\\ref{proj}). \n \nSection~\\ref{section4k} begins by our main stably-finiteness result on solvable Lie groups whose dimension is not divisible by~4 (Theorem~\\ref{4n+2}). \nWe then turn to groups whose nilradical is 1-codimensional. \nThese groups are basically determined by a derivation of a nilpotent Lie algebra $D\\in\\Der(\\ng)$ and the main task is to describe the stably finiteness or AF-embeddability properties of the $C^*$-algebra of the corresponding group $N\\rtimes_D\\RR$ in terms of the spectrum of~$D$. \nIn this connection, the technique of continuous fields allows us to establish a necessary condition\nfor stably finiteness in terms of the spectrum or the involved derivation (Theorem~\\ref{prop-cf4}). \nWe then obtain a technical result that explains the greater complexity of the behaviour of the groups whose dimension is divisible by~4 (Theorem~\\ref{cf-prop7}), \nand then we draw a key consequence that is effective by way of decreasing the dimensions in the study of the specific examples (Corollary~\\ref{cf-cor8}). \n\nFinally, in Section~\\ref{Sect4}, we use the techniques of Sections \\ref{section1} and \\ref{section4k} in order to study the $C^*$-algebras of some specific solvable Lie groups. \nWe focus on groups whose nilradical is 1-dimensional and 2-step nilpotent \nsince this is the simplest class of groups after the generalized $ax+b$-groups (Example~\\ref{ax+b}). \nOur most complete results are obtained in the cases when the nilradical is a Heisenberg group (Proposition~\\ref{Heis}) or is a central extension of the free 6-dimensional 2-step nilpotent Lie group (Theorem~\\ref{N6N15}), \nwhen we characterize the group $C^*$-algebra properties in terms of the spectral data of the derivation involved in the construction.\nIn particular, when the Heisenberg group has dimension $4k+3$, there are cases, \nwhen \nthere exists an open point in the unitary dual of $\\HH_n \\rtimes \\RR$, but its $C^*$-algebra \nis AF-embeddable.\nWe also discuss a class of Heisenberg-like groups associated to finite-dimensional real division algebras (Theorem~\\ref{N6N17}). \nThese last examples show, in particular,\nthat the necessary condition for stably finiteness is \nnot sufficient, that is, the lack of stably finiteness is not preserved by continuous deformations.\n\n\n\n\n \n \\section{$K$-theoretic tools, stably finiteness, and AF-embeddability}\\label{section1}\n\nIn this section we obtain some technical results that play a key role in the next sections. \n \n \n\\subsection{Notation related to the construction of $K$-groups}\n\\label{App_K}\n\nWe start by reminding notions and notation needed in our paper.\nThroughout the paper we use the notation from \\cite{RLL00}.\n\nFor any $C^*$-algebra $\\Ac$ its unitization is $\\widetilde{\\Ac}:=\\CC\\1\\dotplus A$. \nAlso, for any integers $m,n\\ge 1$ we denote by $M_{m,n}(\\Ac)\\subseteq M_{m,n}(\\widetilde{\\Ac})$ the $m\\times n$ matrix spaces with entries in $\\Ac$ and $\\widetilde{\\Ac}$, respectively, \nand for $m=n$ we write as usually $M_n(\\Ac)\\subseteq M_n(\\widetilde{\\Ac})$ for the corresponding matrix $C^*$-algebras. \nMoreover, $\\1_n\\in M_n(\\widetilde{\\Ac})$ \nthe identity matrix and $\\0_n\\in M_n(\\Ac)$ is the zero matrix. \n\n\nWe denote $\\Pg(\\Ac):=\\{p\\in A\\mid p=p^2=p^*\\}$ and $\\Pg_n(\\Ac):=\\Pg(M_n(\\Ac))$ for any $n\\ge 1$. \nThe disjoint union \n$$\\Pg_\\infty(\\Ac):=\\bigsqcup_{n\\ge 1}\\Pg_n(\\Ac)$$ \nhas the natural structure of a graded (noncommutative) semigroup \nwith its operation $\\oplus$ defined by \n$$\\Pg_n(\\Ac)\\times\\Pg_m(\\Ac)\\to\\Pg_{n+m}(\\Ac),\\quad \n(p,q)\\mapsto\\matto{p}{q}:=\\matt{p}{q}$$ \nfor all $m,n\\ge 1$. \nThe Cartesian projection $s\\colon \\widetilde{\\Ac}\\to\\CC\\1(\\subseteq\\widetilde{\\Ac})$ is extended to \n$s\\colon M_n(\\widetilde{A})\\to M_n(\\widetilde{\\Ac})$, \n$(a_{ij})_{i,j}\\mapsto (s(a_{ij}))_{i,j}$, for any $n\\ge 1$. \nWe recall that the equivalence relation $\\sim_0$ on $\\Pg_\\infty(\\widetilde{\\Ac})$ is defined in the following way: \nif $p\\in\\Pg_m(\\widetilde{\\Ac})$ and $q\\in\\Pg_n(\\widetilde{\\Ac})$, \nthen $p\\sim_0 q$ if and only if there exists $v\\in M_{m,n}(\\widetilde{\\Ac})$ with $v^*v=p$ and $vv^*=q$. \nThere is a natural additive map $\\Pg_\\infty(\\widetilde{\\Ac}) \\to K_0(\\widetilde{\\Ac})$, $p \\mapsto [p]_0$. \nThen, with the above notation, we have\n\\begin{align*}\nK_0(\\widetilde{\\Ac})\n&=\\{[p]_0-[q]_0\\mid p,q\\in \\Pg_\\infty(\\widetilde{\\Ac})\\}, \\\\\nK_0(\\Ac)\n&=\\{[p]_0-[s(p)]_0\\mid p\\in \\Pg_\\infty(\\widetilde{\\Ac})\\}.\n\\end{align*}\nMoreover, for any $C^*$-algebra $\\Bc$ and any $*$-morphism $\\varphi\\colon \\Ac\\to \\Bc$ there is a group morphism $K_0(\\varphi)\\colon K_0(\\Ac)\\to K_0(\\Bc)$, \n$[p]_0-[s(p)]_0\\mapsto[\\widetilde{\\varphi}(p)]_0-[s(\\widetilde{\\varphi}(p))]_0$. \n\n\n\nWe denote $\\Uc(\\widetilde{\\Ac}):=\\{u\\in \\widetilde{A}\\mid u^*u=uu^*=\\1\\}$ and $\\Uc_n(\\widetilde{\\Ac}):=\\Uc(M_n(\\widetilde{\\Ac}))$ for every $n\\ge 1$, which is the basic ingredient in the construction\nof the group $K_1(\\Ac)=K_1(\\widetilde{\\Ac})$.\n\n\n\n\\subsection{A $K$-theoretic condition for stably finiteness}\nThe next proposition is a partial generalization of \\cite[Lemma 1.5]{Sp88}, and\nit is one of the main tools in this paper. \n\n\\begin{proposition}\\label{P1}\nFor every $C^*$-algebra $\\Ac$ the following assertions are equivalent: \n\\begin{enumerate}[{\\rm(i)}]\n\t\\item\\label{P1_item1} \n\tThere exist $k\\ge 1$ and $p\\in\\Pg_k(\\Ac)\\setminus\\{\\0_k\\}$ with $[p]_0=0\\in K_0(\\Ac)$. \n\t\\item\\label{P1_item2} \n\tThere exist $r\\ge 1$ and $v\\in M_r(\\widetilde{\\Ac})$ with $vv^*=\\1_r\\ne v^*v$.\n\t\\item\\label{P1_item3} The $C^*$-algebra $\\Ac$ is not stably finite. \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\eqref{P1_item1}$\\Rightarrow$\\eqref{P1_item2}: \nWe have $p\\in\\Pg_k(\\Ac)\\subseteq M_k(\\Ac)$ hence $s(p)=\\0_k$. \nThen, by \\cite[4.2.2(ii)]{RLL00}, there exists $m\\ge 1$ with \n$\\matt{p}{\\1_m}\\sim\\matt{\\0_k}{\\1_m}$ in $\\Pg_{k+m}(\\widetilde{\\Ac})$. \nFurthermore, by \\cite[2.2.8(i)]{RLL00}, we obtain \n$\\mattt{p}{\\1_m}{\\0_{k+m}}\\sim_u\\mattt{\\0_k}{\\1_m}{\\0_{k+m}}$ in $\\Pg_{2(k+m)}(\\widetilde{\\Ac})$. \nThat is, there exists $w\\in\\Uc_{2(k+m)}(\\widetilde{\\Ac})$ with \n$$\\mattt{p}{\\1_m}{\\0_{k+m}}= w\\mattt{\\0_k}{\\1_m}{\\0_{k+m}}w^*\n\\in \\Pg_{2(k+m)}(\\widetilde{\\Ac}).$$\nNow, defining \n$$v:=w\\mattt{\\0_k}{\\1_m}{\\0_{k+m}}+\n\\mattt{\\1_k-p}{\\0_m}{\\1_{k+m}}\\in M_{2(k+m)}(\\widetilde{\\Ac})$$\nwe obtain \n\\begin{align*}\n\\allowdisplaybreaks\nvv^*&=w\\mattt{\\0_k}{\\1_m}{\\0_{k+m}}w^* \n+\\mattt{\\1_k-p}{\\0_m}{\\1_{k+m}} \n=\\mattt{\\1_k}{\\1_m}{\\1_{k+m}} \\\\\n&=\\1_{2(k+m)}\n\\end{align*}\nand \n\\allowdisplaybreaks\n\\begin{align*}\nv^*v\n=\n&\\mattt{\\0_k}{\\1_m}{\\0_{k+m}}w^* w\\mattt{\\0_k}{\\1_m}{\\0_{k+m}} \\\\\n&+ \\mattt{\\0_k}{\\1_m}{\\0_{k+m}}w^*\\mattt{\\1_k-p}{\\0_m}{\\1_{k+m}} \\\\\n&+ \\mattt{\\1_k-p}{\\0_m}{\\1_{k+m}}w\\mattt{\\0_k}{\\1_m}{\\0_{k+m}} \\\\\n&+ \\mattt{\\1_k-p}{\\0_m}{\\1_{k+m}} \\\\\n=\n&\n\\mattt{\\0_k}{\\1_m}{\\0_{k+m}}+ \\0_{2(k+m)}+\\0_{2(k+m)}\n+ \\mattt{\\1_k-p}{\\0_m}{\\1_{k+m}} \\\\\n=\n&\n\\mattt{\\1_k-p}{\\1_m}{\\1_{k+m}} \\\\\n\\ne \n&\\1_{2(k+m)}.\n\\end{align*}\n\n\\eqref{P1_item2}$\\Rightarrow$\\eqref{P1_item1}: \nFor $r\\ge 1$ and $v\\in M_r(\\widetilde{\\Ac})$ with $vv^*=\\1_r\\ne v^*v$ \nwe define $p:=\\1_r-v^*v\\in\\Pg_r(\\widetilde{\\Ac})\\setminus\\{\\0_r\\}$. \nSince $vv^*=\\1_r$, we obtain $\\1_r=s(vv^*)=s(v)s(v)^*$ in $M_r(\\CC\\1)$, \nhence $s(v)^*s(v)=\\1_r$, and this implies $s(p)=\\1_r-s(v^*v)=\\1_r-s(v)^*s(v)=\\0_r$ \nIt follows that $p \\in \\Pg_r(\\Ac)$. \n\nMoreover, we have $p+v^*v=vv^*$ and $p v^*v=\\0_r$ hence, by \\cite[3.1.7(iv)]{RLL00}, \n$[p]_0+[v^*v]_0=[vv^*]_0$ in $K_0(\\widetilde{A})$. \nHere we have $v^*v\\sim_0 vv^*$, hence $[v^*v]_0=[vv^*]_0$ in $K_0(\\widetilde{A})$, and we then obtain $[p]_0=0\\in K_0(\\Ac)$.\n\n\\eqref{P1_item2}$\\Rightarrow$\\eqref{P1_item3}: \nThis is just the definition of stably finite $C^*$-algebras. \n\\end{proof}\n\n\nThe following simple facts were already noted in \\cite[proof of Cor. D]{Ga20},\n but we prove them here for completeness. \n\n\\begin{corollary}\\label{rem-1.5} Let $\\Ac$ be a $C^*$-algebra. \n\\begin{enumerate}[{ \\rm (i) }]\n\\item \\label{rem-1.5_i} If $\\Ac$ is stably projectionless, then it is stably finite.\nMoreover, for a $C^*$-algebra $\\Ac$ with $K_0(\\Ac)=0$, $\\Ac$ is stably finite if and only if it is stably projectionless.\n\\item\\label{rem-1.5_ii} $\\Prim(\\Ac)$ has no nonempty quasi-compact open subset, \nthen $\\Ac$ is stably projectionless. \n\\item\\label{rem-1.5_iii} If $\\Prim(\\Ac)$ has no nonempty quasi-compact open subsets, \nthen $\\Ac$ is stably finite. \n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nAssertion \\eqref{rem-1.5_i} is a direct consequence of Proposition~\\ref{P1} (\\eqref{P1_item1}$\\iff$ \\eqref{P1_item3}.)\n\n\\eqref{rem-1.5_ii}\nIf $\\Ac$ is not stably projectionless, \nthere exist \n$k\\ge 1$ and $p\\in\\Pg_k(\\Ac)\\setminus\\{0\\}$, hence\nthere exists a nonempty quasi-compact open subset of $\\widehat{M_k(\\Ac)}$. \n(See for instance the proof of \\cite[Ex. 4.8((vii)$\\Rightarrow$(iv))]{BB20}.) \nSince $M_k(\\Ac)=M_k(\\CC)\\otimes \\Ac$, \nit follows that $\\widehat{M_k(\\Ac)}$ is homeomorphic to $\\widehat{\\Ac}$, \nby \\cite[Th. B.45(b)]{RaWi98}. \nTherefore $\\widehat{\\Ac}$ has a nonempty quasi-compact open subset. \nMoreover, the canonical mapping $\\widehat{\\Ac}\\to\\Prim(\\Ac)$, $[\\pi]\\mapsto\\Ker\\pi$, \nis continuous and open, hence it maps any nonempty quasi-compact \nopen subset of $\\widehat{\\Ac}$ onto a nonempty quasi-compact open subset of $\\Prim(\\Ac)$.\nIt follows that $\\Prim(\\Ac)$ has a nonempty quasi-compact open subset, \nwhich is a contradiction with the hypothesis. \n\nAssertion~\\eqref{rem-1.5_iii} follows immediately from \\eqref{rem-1.5_i} and \\eqref{rem-1.5_ii}. \n\\end{proof} \n\n\n\n\\begin{remark}\n\\normalfont\nIn the special case of separable exact $C^*$-algebras, \nCorollary~\\ref{rem-1.5} \\eqref{rem-1.5_ii} is a weak version of \\cite[Cor. B]{Ga20}, which gives AF-embeddability rather than just stable finiteness. \n\\end{remark}\n\n\n\\subsection{Action of ``real'' structures on $K$-groups} \n\nThe following terminology goes back to G.G. Kasparov \\cite{Ka80}.\n\\begin{definition}\\label{real} \n\\normalfont\nA \\emph{``real'' structure} of a $C^*$-algebra $\\Ac$ is an antilinear mapping $\\tau\\colon \\Ac\\to \\Ac$, satisfying $\\tau(ab)=\\tau(a)\\tau(b)$, $\\tau(a^*)=\\tau(a)^*$, \nand $\\tau(\\tau(a))=a$ for all $a,b\\in\\Ac$. \nA \\emph{``real'' $C^*$-algebra} is a $C^*$-algebra $\\Ac$ with a fixed ``real'' structure~$\\tau$.\nWe denote $\\overline{a}:=\\tau(a)$ for all $a\\in \\Ac$ when no confusion arise.\n\n\nA \\emph{``real'' ideal} of $\\Ac$ is a closed two-sided ideal $\\Jc\\subseteq \\Ac$ \nthat is invariant to the ``real'' structure of $\\Ac$. \nIn this case $\\Jc$ is a ``real'' $C^*$-algebra with respect to the ``real'' structure $\\tau\\vert_\\Jc\\colon \\Jc\\to \\Jc$. \n\nLet $\\Ac$, $\\Bc$ be $C^*$-algebras with ``real'' structures $\\tau_\\Ac$ and $\\tau_\\Bc$, respectvely. \nA \\emph{``real'' morphism} is a $*$-morphism $\\psi\\colon \\Ac\\to \\Bc$ satisyfing $\\psi(\\tau_\\Ac (a))=\\tau_\\Bc(\\psi(a))$ for all $a\\in \\Ac$. \n\\end{definition}\n\nFor every $n\\ge 1$ the matrix algebra $M_n(\\CC)$ has a canonical ``real'' structure;\nif \n$\\Ac$ is a ``real'' $C^*$-algebra then the $C^*$-algebra \n$M_n(\\Ac)=M_n(\\CC)\\otimes \\Ac$ has a canonical ``real'' structure $(a_{ij})\\mapsto (\\overline{a_{ij}})$. \nMoreover $\\widetilde{\\Ac}$ has a canonical ``real'' structure given by $\\overline{a+ z\\1}=\\overline{a}+\\overline{z}\\1$ for every $a\\in \\Ac$ and $z\\in\\CC$. \n\nIf $u,v\\in\\Uc_\\infty(\\widetilde{\\Ac})$ then $\\overline{u},\\overline{v}\\in \\Uc_\\infty(\\widetilde{\\Ac})$ and $\\overline{\\matto{u}{v}}=\\matto{\\overline{u}}{\\overline{u}}\\in\\Uc_\\infty(\\widetilde{\\Ac})$, and \n$$u\\sim_1 v\\iff \\overline{u}\\sim_1\\overline{v}.$$\n(See \\cite[8.1.1]{RLL00}.) \nTherefore we obtain a well-defined group homomorphism \n$$K_1(\\widetilde{\\Ac})\\to K_1(\\widetilde{\\Ac}),\\quad [u]_1\\mapsto \\overline{[u]_1}:=[\\overline{u}]_1$$\nwhich is actually an isomorphism and is equal to its own inverse. \n\nIf $p,q\\in\\Pg_\\infty(\\widetilde{\\Ac})$ then $\\overline{p},\\overline{q}\\in\\Pg_\\infty(\\widetilde{\\Ac})$ \nand \n$\\overline{\\matto{p}{q}}=\\matto{\\overline{p}}{\\overline{q}}\\in\\Uc_\\infty(\\widetilde{\\Ac})$, \n\\begin{align*}\np\\sim_0 q\\iff \\overline{p}\\sim_0\\overline{q}\n\\end{align*}\nand $s(\\overline{p})=\\overline{s(p)}$. \nWe then obtain a well-defined semigroup homomorphism \n$$\\Dc(\\widetilde{\\Ac})\\to \\Dc(\\widetilde{\\Ac}), \\quad [p]_\\Dc\\mapsto\\overline{[p]_\\Dc}:=[\\overline{p}]_\\Dc$$\nwhich is actually an isomorphism and is equal to its own inverse. \nThis further gives rise to a group isomorphism \n$$K_0(\\widetilde{\\Ac})=G(\\Dc(\\widetilde{\\Ac}))\\to G(\\Dc(\\widetilde{\\Ac}))=K_0(\\widetilde{\\Ac}), \\quad [p]_0\\mapsto\\overline{[p]_0}:=[\\overline{p}]_0$$\nwhich is equal to its own inverse, and satisfies \n$$[s(\\overline{p})]_0=\\overline{[s(p)]_0}\\text{ for all }p\\in\\Pg_\\infty(\\widetilde{\\Ac}).$$\nIf $\\psi\\colon \\Ac\\to \\Bc$ is a ``real'' morphism of ``real'' $C^*$-algebras, \nthen its corresponding group morphism \n$K_j(\\widetilde{\\psi})\\colon K_j(\\widetilde{\\Ac})\\to K_j(\\widetilde{\\Bc})$ \nsatisfies $K_j(\\widetilde{\\psi})(\\overline{x})=\\overline{K_j(\\widetilde{\\psi})(x)}$ for all $x\\in K_j(\\widetilde{\\Ac})$ and $j=0,1$. \n\nIn particular, for $j=0$, $\\Bc=\\{0\\}$, and $\\psi=0$, it follows that the subgroup $K_0(\\Ac)=\\Ker(K_0(\\widetilde{\\psi}))$ is invariant to the automorphism $x\\mapsto\\overline{x}$ of $K_0(\\widetilde{\\Ac})$. \n\n\\begin{lemma}\\label{deltas}\nLet $\\psi\\colon \\Ac \\to \\Bc$ be a ``real'' surjective morphism of ``real'' $C^*$-algebras. \nDenote $\\Jc:=\\Ker\\psi$, regarded as a ``real'' ideal of $\\Ac$ \nwith its corresponding inclusion map $\\varphi\\colon \\Jc\\hookrightarrow \\Ac$, \nand consider the six-term exact sequence \n\\begin{equation}\\label{hexagon}\n\\xymatrix{\nK_0(\\Jc) \\ar[r]^{K_0(\\varphi)} & K_0(\\Ac) \\ar[r]^{K_0(\\psi)} & K_0(\\Bc) \\ar[d]^{\\delta_0}\\\\ \nK_1(\\Bc) \\ar[u]^{\\delta_1} & K_1(\\Ac) \\ar[l]_{K_1(\\psi)} & K_1(\\Jc) \\ar[l]_{K_1(\\varphi)}\n}\n\\end{equation}\nThen we have \n\\begin{enumerate}[{\\rm(i)}]\n\\item\\label{deltas_item1} \n$\\delta_0(\\overline{x})=-\\overline{\\delta_0(x)}$ for all $x\\in K_0(\\Bc)$; \n\\item\\label{deltas_item2} \n$\\delta_1(\\overline{y})=\\overline{\\delta_1(y)}$ for all $y\\in K_1(\\Bc)$.\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\eqref{deltas_item1}\nFor arbitrary $x\\in K_0(\\Bc)$ there exist $n\\ge 1$ and $p\\in \\Pg_n(\\widetilde{\\Bc})$ with $x=[p]_0-[s(p)]_0$. \nThere also exist $a=a^*\\in M_n(\\widetilde{\\Ac})$ and $u\\in \\Uc_n(\\widetilde{\\Jc})$ \nwith \n$$\\widetilde{\\psi}(a)=p\\; \\text{ and }\\; \\widetilde{\\varphi}(u)=\\exp(2\\pi\\ie a)\\in\\Uc_n(\\widetilde{A}),$$ \nand $\\delta_0(x)=-[u]_1$ by \\cite[12.2.2(i)]{RLL00}. \n\nFurthermore $\\overline{x}=[\\overline{p}]_0-[\\overline{s(p)}]_0$ \nand $\\overline{a}=\\overline{a}^*\\in M_n(\\widetilde{A})$ satisfies $\\widetilde{\\psi}(\\overline{a})=\\overline{\\widetilde{\\psi}(a)}=\\overline{p}$. \nOn the other hand, \n$\\widetilde{\\varphi}(\\overline{u}^*)\n=\\overline{\\widetilde{\\varphi}(u)}^*\n=\\exp(2\\pi\\ie \\overline{a})$, \nhence, since $[u^*]_1=-[u]_1$ by \\cite[8.1.3]{RLL00}, we obtain\n$$\\delta_0(\\overline{x})=-[\\overline{u}^*]_1=[\\overline{u}]_1=\\overline{[u]_1}\n=-\\overline{\\delta_0(x)}.$$\n\\eqref{deltas_item2} \nFor arbitrary $y\\in K_1(\\Bc)=K_1(\\widetilde{\\Bc})$ \nthere exist $n\\ge 1$ and $u\\in \\Uc_n(\\widetilde{\\Bc})$ with $y=[u]_1$. \nThere also exist $v\\in \\Uc_{2n}(\\widetilde{\\Ac})$ and \n$p\\in\\Pg_{2n}(\\widetilde{\\Jc})$ \nwith \n$$\\widetilde{\\psi}(v)=\\matt{u}{u^*}\\; \\text{ and }\\;\n\\widetilde{\\varphi}(p)=v\\matt{\\1_n}{\\0_n}v^*, $$ \nand \nwe have $\\delta_1(y)=[p]_0-[s(p)]_0$ by \\cite[9.1.4]{RLL00}. \n\nFurthermore $\\overline{y}=[\\overline{u}]_1$ \nand $\\overline{u}\\in \\Uc_n(\\widetilde{B})$, \n$\\overline{v}\\in \\Uc_{2n}(\\widetilde{A})$, \n$\\overline{p}\\in\\Pg_{2n}(\\widetilde{J})$ satisfy \n$$\\widetilde{\\psi}(\\overline{v})\n=\\overline{\\widetilde{\\psi}(p)}\n=\\matt{\\overline{u}}{\\overline{u}^*}\\; \\text{ and }\\; \n\\widetilde{\\varphi}(\\overline{p})\n=\\overline{\\widetilde{\\varphi}(p)}\n=\\overline{v}\\matt{\\1_n}{\\0_n}\\overline{v}^*$$ \nhence \n$$\\delta_1(\\overline{y})\n=[\\overline{p}]_0-[s(\\overline{p})]_0\n=[\\overline{p}]_0-[\\overline{s(p)}]_0\n=\\overline{\\delta_1(y)}.$$\nThis completes the proof. \n\\end{proof}\n\n\\begin{remark}\n\\normalfont\nFor any locally compact group $G$ we regard its $C^*$-algebra $C^*(G)$ as a ``real'' $C^*$-algebra with its canonical ``real'' structure given by $\\overline{f}(x):=\\overline{f(x)}$ for every $f\\in\\Cc_c(G)\\hookrightarrow C^*(G)$. \nSee for instance \\cite[\\S 3.2]{Ros16}. \n\\end{remark}\n\n\\begin{lemma}\\label{dim1}\nWe have $[\\overline{u}]_1=[u]_1\\in K_1(C^*(\\RR))\\simeq\\ZZ$ \nfor all $u\\in\\Uc_\\infty(C^*(\\RR)^\\sim)$. \n\\end{lemma}\n\n\\begin{proof}\nWe use the well-known $*$-isomorphism given by the Fourier transform \n$$F\\colon C^*(\\RR)\\to\\Cc_0(\\ie\\RR),\\quad \n(F(f))(\\ie\\xi)=\\int_{\\RR}\\ee^{-\\ie\\xi x}f(x)\\de x \n\\text{ if }f\\in\\Cc_c(\\RR)$$\nwhere we regard $\\Cc_0(\\ie\\RR)$ as a commutative $C^*$-algebra with its pointwise operations and with the sup-norm, and \"real\" structure $\\tau_0\\colon \\Cc_0(\\ie\\RR)\\to\\Cc_0(\\ie\\RR)$, \n$(\\tau_0(g))(z)=\\overline{g(\\overline{z})}$ for all $g\\in\\Cc_0(\\ie\\RR)$ and $z\\in\\ie\\RR$. \nWe have \n$$F(\\overline{f})(z)=\\overline{F(f)(\\overline{z})}\\text{ for all }z\\in\\ie\\RR\\text{ and }f\\in\\Cc(\\RR)$$\nhence $F$ is a ``real'' isomorphism of ``real'' $C^*$-algebras. \nConsider the Cayley homeomorphism \n$$\\kappa\\colon \\ie\\RR\\to\\TT\\setminus\\{-1\\},\\quad \\kappa(\\ie\\xi)=\\frac{\\ie\\xi+1}{-\\ie\\xi+1}$$\nwith its inverse \n$\\kappa^{-1}(w)=\\frac{w-1}{w+1}$ for all $w\\in\\TT\\setminus\\{-1\\}$; then\n$$\\overline{\\kappa(z)}=\\kappa(\\overline{z})\\text{ for all }z\\in\\ie\\RR.$$ \nTherefore, the Cayley transform gives a ``real'' isomorphism from the unitization of the``real'' $C^*$-algebra $\\Cc_0(\\ie\\RR)$ \nonto $\\Cc(\\TT)$, when $\\Cc(\\TT)$ is endowed with the ``real'' structure \n$(\\tau(h))(w)=\\overline{h(\\overline{w})}$ for all $h\\in\\Cc(\\TT)$ and $w\\in\\TT$. \n\nFor the above reasons it suffices to prove that the action of the ``real'' structure~$\\tau$ on $K_1(\\Cc(\\TT))$ is the identity map. \nTo this end, we recall that, if we denote $u:=\\id_{\\TT}\\in\\Cc(\\TT,\\TT)=\\Uc_1(\\Cc(\\TT))$, then the mapping \n$$\\ZZ\\to K_1(\\Cc(\\TT)),\\quad m\\mapsto m[u]_1$$\nis a group isomorphism, hence for every $ y= [w]_1 \\in K_1(\\Cc(\\TT))$ there is a unique $m \\in \\ZZ$ such that \n$y = m [u]_1$. \nOn the other hand, it is clear that $\\tau(u)=u$, hence $[\\tau(w)]_1=m [u]_1\\in K_1(\\Cc(\\TT))$.\nThis directly shows that the action of $\\tau$ on $K_1(\\Cc(\\TT))$ is the identity map. \n\\end{proof}\n\n\n\n\\begin{definition}\\label{rdym}\n\\normalfont\nA \\emph{``real'' $C^*$-dynamical system} is a $C^*$-dynamical system $(\\Ac,T,\\alpha)$, where $\\Ac$ is a ``real'' $C^*$-algebra, $T$ is a locally compact group, and $\\alpha\\colon T\\to\\Aut \\Ac$, $t\\mapsto\\alpha_t$, satisfies $\\alpha_t(\\overline{a})=\\overline{\\alpha_t(a)}$ for all $t\\in T$ and $a\\in \\Ac$. \n \\end{definition}\n\n\\begin{lemma}\\label{rcrossed}\nFor every ``real'' $C^*$-dynamical system $(\\Ac,T,\\alpha)$ \nits corresponding crossed product $\\Ac\\rtimes_\\alpha T$ has a unique ``real'' structure satisfying $\\overline{f}(t):=\\overline{f(t)}$ for all $t\\in T$ and $f \\in\\Cc_c(T,\\Ac)$. \n\\end{lemma}\n\n\\begin{proof}\nUniqueness follows from the fact that $\\Cc_c(T,\\Ac)$ is dense in $\\Ac\\rtimes_\\alpha T$. \n\nTo prove the existence, we first note that \nthe antilinear mapping $f\\mapsto \\overline{f}$ defined as in the statement on $\\Cc_c(T,\\Ac)$ preserves the multiplication and the involution. \nIn fact, we recall that \n$$(f\\ast g)(t)=\\int_T f(r)\\alpha_r(g(r^{-1}t))\\de r\\text{ and }\nf^*(t):=\\Delta(t^{-1})\\alpha_t(f(t^{-1})^*)$$\nhence $\\overline{f\\ast g}=\\overline{f}\\ast \\overline{g}$ and $\\overline{f^*}=\\overline{f}^*$ for all $f,g\\in\\Cc_c(T,\\Ac)$. \n\nIt remains to show that $f\\mapsto \\overline{f}$ is isometric with respect to the $C^*$-norm on $\\Cc_c(T,\\Ac)$. \nTo this end let $(\\pi,U)$ be a covariant representation of $(\\Ac,T,\\alpha)$ on a complex Hilbert space~$\\Hc$, that is, $\\pi(\\alpha_t(a))=U_t \\pi(a)U_t^*\\in\\Bc(\\Hc)$ for all $t\\in T$ and $a\\in \\Ac$. \nFor any fixed antilinear involutive isometry $C\\colon \\Hc\\to\\Hc$ \nwe define $\\overline{\\pi}\\colon \\Ac\\to\\Bc(\\Hc)$, $\\overline{\\pi}(a):=C\\pi(\\overline{a})C$, and $\\overline{U}\\colon T\\to\\Bc(\\Hc)$, $\\overline{U}_t:=CU_tC$. \nThen it is straightforward to check that $(\\overline{\\pi},\\overline{U})$ is again a \ncovariant representation of $(\\Ac,T,\\alpha)$ on the complex Hilbert space~$\\Hc$, and moreover for every $f\\in\\Cc_c(T,\\Ac)$ we have \n\\begin{align*}\n(\\pi\\rtimes U)(\\overline{f})\n&=\\int_T\\pi(\\overline{f}(t))U_t\\de t\n=\\int_T\\pi(\\overline{f(t)})U_t\\de t\n=\\int_TC\\overline{\\pi}(f(t))CU_t\\de t \\\\\n&=C\\Bigl(\\int_T\\overline{\\pi}(f(t))\\overline{U}_t\\de t \\Bigr)C\n=C\\Bigl((\\overline{\\pi}\\rtimes \\overline{U})(f)\\Bigr)C.\n\\end{align*}\nThis shows that for every covariant representation $(\\pi,U)$ there exists a covariant representation $(\\overline{\\pi},\\overline{U})$ with \n$\\Vert (\\pi\\rtimes U)(\\overline{f})\\Vert=\\Vert (\\overline{\\pi}\\rtimes \\overline{U})(f)\\Vert$ and then, by the definition of the $C^*$-norm on $\\Cc_c(T,\\Ac)$, we obtain $\\Vert f\\Vert=\\Vert \\overline{f}\\Vert$ in $\\Ac\\rtimes_\\alpha T$.\nThis finishes the proof. \n\\end{proof}\n\n\\begin{remark}\\label{cross-morphism}\n\\normalfont\nLet $(\\Ac,T,\\alpha)$ and $(\\Bc,T,\\beta)$ be ``real'' $C^*$-dynamical systems and $\\psi\\colon \\Ac\\to\\Bc$ is an equivariant ``real'' morphism, then the corresponding $*$-morphism \n $\\psi\\rtimes\\iota \\colon \\Ac\\rtimes_\\alpha T\\to \\Bc\\rtimes_\\beta T$ \n satisfying $((\\psi\\rtimes \\iota)(f))(t)=\\psi(f(t))$ for all $t\\in T$ and $f\\in\\Cc_c(T,\\Ac)$ is a ``real'' morphism. \n\\end{remark}\n\n\nWe now study the interaction between ``real'' structures and some constructions from \\cite{Ri82}. \n\n\\begin{definition}\\label{WHdef}\n\\normalfont \nLet $(\\Ac,\\RR,\\alpha)$ be a ``real'' $C^*$-dynamical system. \nWe consider the $C^*$-algebra $\\cone \\Ac:=\\Cc_0(\\RR\\cup\\{+\\infty\\},\\Ac)$ \nwith its $*$-morphism $\\ev_{+\\infty}\\colon \\cone \\Ac\\to \\Ac$, $f\\mapsto f(+\\infty)$ and the ideal $\\susp \\Ac:=\\Ker(\\ev_{+\\infty})\\simeq \\Cc_0(\\RR,\\Ac)$. \nThen $\\cone \\Ac$ is a ``real'' $C^*$-algebra with its ``real'' structure defined by $\\overline{f}(t):=\\overline{f(t)}$ for all $t\\in\\RR\\cup\\{+\\infty\\}$ and $f\\in\\cone \\Ac$.\n Moreover $\\susp \\Ac$ is a ``real'' ideal, $\\ev_{+\\infty}$ is a ``real'' morphism, and we have the short exact sequence \n$$0\\to\\susp \\Ac\\hookrightarrow \\cone \\Ac\\mathop{\\longrightarrow}\\limits^{\\ev_{+\\infty}} \\Ac\\to 0.$$\nIf we define \n$$\\tau\\otimes \\alpha\\colon \\RR\\to\\Aut(\\cone \\Ac),\\quad ((\\tau\\otimes\\alpha)_r f)(t):=\\alpha_r(f(t-r))$$\nthen $(\\cone A,\\tau\\otimes\\alpha,\\RR)$ is a ``real'' $C^*$-dynamical system and $\\ev_{+\\infty}$ intertwines the actions of $\\RR$ on $\\cone \\Ac$ and $\\Ac$ via $\\tau\\otimes\\alpha$ and $\\alpha$, respectuvely. \nIn particular the ``real'' ideal $\\susp \\Ac$ is invariant to $\\tau\\otimes\\alpha$. \nWe further obtain the short exact sequence \n\\begin{equation}\\label{WHdef_eq1}\n0\\to\\susp \\Ac \\rtimes_{\\tau\\otimes\\alpha}\\RR\\hookrightarrow \n\\cone \\Ac\\rtimes_{\\tau\\otimes\\alpha}\\RR\\mathop{\\longrightarrow}\\limits^{\\psi} \\Ac\\rtimes_\\alpha\\RR\\to 0\n\\end{equation}\ncalled the \\emph{Wiener-Hopf extension} for $\\Ac\\rtimes_\\alpha\\RR$, \nwhere $\\psi:=\\ev_{+\\infty}\\rtimes\\RR$ is a ``real'' morphism. \n(See Remark~\\ref{cross-morphism}.)\n\\end{definition}\n\n\\begin{remark}\\label{SvN}\n\\normalfont\nIn the special case $\\Ac=\\CC$ we get the ``real'' $C^*$-dynamical system $(\\susp,\\tau,\\RR)$ with $\\susp:=\\susp\\CC=\\Cc_0(\\RR)$ and $\\tau\\colon\\RR\\to\\Aut(\\susp)$, \n$(\\tau_rf)(t):=f(t-r)$. \nIf the regular representation of the group $\\RR$ is again denoted by $\\tau\\colon \\RR\\to L^2(\\RR)$, $(\\tau_r\\xi)(t):=\\xi(t-r)$, \nand we define $M\\colon\\susp\\to\\Bc(L^2(\\RR))$, $M(f)\\xi=f\\xi$ for all $f\\in \\susp$ and $\\xi\\in L^2(\\RR)$, \nthen we obtain a covariant representation $(M,\\tau)$ of the $C^*$-dynamical system $(\\susp,\\tau,\\RR)$ whose integrated representation gives a $*$-isomorphism \n\\begin{equation}\\label{SvN_eq1}\nM\\rtimes\\tau\\colon \\susp\\rtimes_\\tau \\RR\\to\\Kc(L^2(\\RR)).\n\\end{equation} \nSee \\cite[Th. 4.24]{Wi07}. \nIf $h\\in\\Cc_c(\\RR)$ and $f\\in\\susp$, then the function $h(\\cdot)f$ (that is, $r\\mapsto h(r)f$) belongs to $\\Cc_c(\\RR,\\susp)\\subseteq\\susp\\rtimes_\\tau\\RR$ \nand we have \n$$(M\\rtimes\\tau)(h(\\cdot)f)=\\int_{\\RR}M(h(r)f)\\tau(r)\\de r=M(f)\\int_{\\RR}h(r)\\tau(r)\\de t$$\nhence \n$$((M\\rtimes\\tau)(h(\\cdot)f))\\xi=f\\cdot (h\\ast\\xi)\\text{ for }\\xi\\in L^2(\\RR)$$\nThe $*$-isomorphism~\\eqref{SvN_eq1} is a ``real'' isomorphism if we regard $\\Kc(L^2(\\RR))$ as a ``real'' $C^*$-algebra \nwith its ``real'' structure given by $\\overline{T}:=CTC$ for all $T\\in \\Kc(L^2(\\RR))$, where $C\\colon L^2(\\RR)\\to L^2(\\RR)$, $C(\\xi):=\\overline{\\xi}$. \nThus, if $T\\in \\Kc(L^2(\\RR))$ is an integral operator defined by an integral kernel $K_T\\colon \\RR\\times\\RR\\to\\CC$, \nthen $\\overline{T}$ is the integral operator defined by the integral kernel \n $K_{\\overline{T}}\\colon \\RR\\times\\RR\\to\\CC$, where $K_{\\overline{T}}(t,r):=\\overline{K_T(t,r)}$ for all $t,r\\in\\RR$. \n\\end{remark}\n\n\\begin{proposition}\\label{tsigns}\nFor every ``real'' $C^*$-dynamical system $(\\Ac,\\RR, \\alpha)$ there exist a group isomorphism \n$$\\Theta_0\\colon K_0(\\Ac\\rtimes_\\alpha\\RR)\\to K_1(\\Ac)$$ \nsatisfying $\\Theta_0(\\overline{x})=-\\overline{\\Theta_0(x)}$ for all $x\\in K_0(\\Ac\\rtimes_\\alpha \\RR)$ \nand \na group isomorphism \n$$\\Theta_1\\colon K_1(\\Ac\\rtimes_\\alpha\\RR)\\to K_0(\\Ac)$$ \nsatisfying $\\Theta_1(\\overline{x})=\\overline{\\Theta_1(x)}$ for all $x\\in K_1(\\Ac\\rtimes_\\alpha \\RR)$. \n\\end{proposition}\n\n\\begin{proof}\nAs proved in \\cite{Ri82}, we have $K_0(\\cone \\Ac\\rtimes_{\\tau\\otimes\\alpha}\\RR)=\\{0\\}$ and $K_1(\\cone \\Ac\\rtimes_{\\tau\\otimes\\alpha}\\RR)=\\{0\\}$. \nTherefore, in the six-term exact sequence \\eqref{hexagon} corresponding to the Wiener-Hopf extension~\\eqref{WHdef_eq1}, \nthe vertical arrows \n\\begin{align*}\n\\delta_0 & \\colon K_0(\\Ac\\rtimes_\\alpha\\RR)\\to \nK_1(\\susp \\Ac \\rtimes_{\\tau\\otimes\\alpha}\\RR), \\\\\n\\delta_1 & \\colon K_1(\\Ac\\rtimes_\\alpha\\RR)\\to \nK_0(\\susp \\Ac \\rtimes_{\\tau\\otimes\\alpha}\\RR),\n\\end{align*}\nare group isomorphisms. \n\nOn the other hand, \n$(\\susp \\Ac,\\tau\\otimes\\iota,\\RR)$\nis a ``real'' $C^*$-dynamical system and we have the $*$-isomorphism \n\\begin{equation}\n\\label{tsigns_proof_eq1}\n\\gamma\\colon \\susp \\Ac \\rtimes_{\\tau\\otimes\\alpha}\\RR\\to \n\\susp \\Ac \\rtimes_{\\tau\\otimes\\iota}\\RR\n\\end{equation}\nwhere \n$\\gamma(f)\\in\\Cc_c(\\RR,\\susp \\Ac)\\subseteq \\susp \\Ac \\rtimes_{\\tau \\otimes \\iota}\\RR$ is given by \n$\\bigl((\\gamma(f))(r)\\bigr)(t)=\\alpha_{-t}((f(r))(t))$ , $r,t\\in\\RR$, \nfor every $f\\in\\Cc_c(\\RR,\\susp \\Ac)\\subseteq \\susp A \\rtimes_{\\tau\\otimes\\alpha}\\RR$, \n(See \\cite[page 147]{Ri82}.)\nThen $\\gamma$ is a ``real'' isomorphism. \n\nMoreover, by \\cite[Lemma 2.75]{Wi07},\nwe have a $*$-isomorphism \n$$\\eta\\colon \\susp \\Ac \\rtimes_{\\tau \\otimes\\iota}\\RR \\to (\\susp \\Ac\\rtimes_{\\tau}\\RR)\\otimes \\Ac$$\nsatisfying $\\eta(h(\\cdot)(f\\otimes a))=(h(\\cdot)f)\\otimes a$ for all $h\\in\\Cc_c(\\RR)$, $f\\in\\Cc_c(\\RR)\\subseteq\\susp$, and $a\\in \\Ac$, \nwhere we regard $h(\\cdot)f$ as an element of $\\Cc_c(\\RR,\\susp)$ as in Remark~\\ref{SvN}. \nTaking into account the $*$-isomorphism $M\\rtimes\\tau$ from \\eqref{SvN_eq1}, \nwe further obtain the $*$-isomorphism\n\\begin{equation}\n\\label{tsigns_proof_eq2}\n\\kappa:=((M\\rtimes\\tau)\\otimes\\id_\\Ac)\\circ \\eta \\colon \\susp \\Ac \\rtimes_{\\tau\\otimes\\iota}\\RR \n\\to \\Kc(L^2(\\RR))\\otimes \\Ac\n\\end{equation}\nsatisfying \n$\\kappa(h(\\cdot)(f\\otimes a))=\\bigl((M\\rtimes\\tau)(h(\\cdot)f)\\bigr)\\otimes a$ \nfor all $h\\in\\Cc_c(\\RR)$, $f\\in\\Cc_c(\\RR)\\subseteq\\susp$, and $a\\in A$. \nIn particular, this shows that $\\Kc(L^2(\\RR))\\otimes \\Ac$ has the structure of a ``real'' $C^*$-algebra satisfying $\\overline{T\\otimes a}=\\overline{T}\\otimes\\overline{a}$ for all $T\\in\\Kc(L^2(\\RR))$ \n(cf. the end of Remark~\\ref{SvN}) and $a\\in \\Ac$.\nThen the above $*$-isomorphism $\\kappa$ is a ``real'' isomorphism. \n \nUsing \\eqref{tsigns_proof_eq1} and \\eqref{tsigns_proof_eq2}, \nwe now obtain the ``real'' isomorphism \n\\begin{equation*}\n\\kappa\\circ\\gamma\\colon \\susp \\Ac \\rtimes_{\\tau\\otimes\\alpha}\\RR\\to \n\\Kc(L^2(\\RR))\\otimes \\Ac.\n\\end{equation*}\nThis in turn gives the group isomorphisms\n$$K_j(\\kappa\\circ\\gamma)\\colon K_j(\\susp \\Ac \\rtimes_{\\tau\\otimes\\alpha}\\RR)\n\\to \nK_j(\\Kc(L^2(\\RR))\\otimes \\Ac)\n$$\nsatisfying $K_j(\\kappa\\circ\\gamma)(\\overline{x})=\\overline{K_j(\\kappa\\circ\\gamma)(x)}$ \nfor all $x\\in K_j(\\susp \\Ac \\rtimes_{\\tau\\otimes\\alpha}\\RR)$ and $j=0,1$. \n\nFinally, we select any $\\xi_0\\in L^2(\\RR)$ with $\\overline{\\xi_0}=\\xi_0$ and $\\Vert \\xi_0\\Vert=1$ and we consider its corresponding rank-one projection $p_0:=(\\cdot\\mid\\xi_0)\\xi_0\\in\\Kc(L^2(\\RR))$, \nso that $\\overline{p_0}=p_0$ in the ``real'' $C^*$-algebra $\\Kc(L^2(\\RR))$. \nThen the mapping \n$$\\mu_{p_0}\\colon \\Ac\\to \\Kc(L^2(\\RR))\\otimes \\Ac,\\quad a\\mapsto p_0\\otimes a$$\nis a ``real'' morphism \nhence the group morphism \n$$K_j(\\mu_{p_0})\\colon K_j(\\Ac)\\to K_j(\\Kc(L^2(\\RR))\\otimes \\Ac)$$ \nsatisfies $K_j(\\mu_{p_0})(\\overline{y})=\\overline{K_j(\\mu_{p_0})(y)}$ for all $y\\in K_j(\\Ac)$ and $j=0,1$. \nOn the other hand, $K_j(\\mu_{p_0})$ is actually a group isomorphism for $j=0,1$. \n(See \\cite[6.4.1 and 8.2.8]{RLL00}.)\nConsequently we obtain the group isomorphisms \n$$\\Theta_0:=K_1(\\mu_{p_0})^{-1}\\circ K_1(\\kappa\\circ\\gamma) \\circ\\delta_0\n\\colon K_0(\\Ac\\rtimes_\\alpha\\RR)\\to K_1(\\Ac)$$\nand \n$$\\Theta_1:=K_0(\\mu_{p_0})^{-1}\\circ K_0(\\kappa\\circ\\gamma) \\circ\\delta_1\n\\colon K_1(\\Ac\\rtimes_\\alpha\\RR)\\to K_0(\\Ac).$$\nLemma~\\ref{deltas} ensures that $\\Theta_1$ and $\\Theta_2$ have the required properties. \n\\end{proof}\n\n\\begin{corollary}\\label{signs_semid}\nLet $N$ be a locally compact group and $\\alpha\\colon \\RR\\to\\Aut(N)$ be a \ncontinuous action of $\\RR$ by automorphisms of $N$. \nThen there exist a group isomorphism \n$$\\Theta_0\\colon K_0(C^*(N\\rtimes_\\alpha\\RR))\\to K_1(C^*(N))$$ \nsatisfying $\\Theta_0(\\overline{x})=-\\overline{\\Theta_0(x)}$ for all $x\\in K_0(C^*(N\\rtimes_\\alpha\\RR))$ \nand \na group isomorphism \n$$\\Theta_1\\colon K_1(C^*(N\\rtimes_\\alpha\\RR))\\to K_0(C^*(N))$$ \nsatisfying $\\Theta_1(\\overline{x})=\\overline{\\Theta_1(x)}$ for all $x\\in K_1(C^*(N\\rtimes_\\alpha\\RR))$. \n\\end{corollary}\n\n\\begin{proof}\nThere exists a group morphism $\\beta\\colon \\RR\\to \\Aut(C^*(N))$ for which \n$(C^*(N),\\RR,\\beta)$ is a ``real'' $C^*$-dynamical system, and \nthe natural inclusion map \n$$\\Cc_c(\\RR,\\Cc_c(N))\\hookrightarrow\\Cc_c(N\\times \\RR)$$\nextends to a $*$-isomorphism $\\gamma\\colon C^*(N)\\rtimes_\\beta\\RR\\to C^*(N\\rtimes_\\alpha\\RR)$, \nby \\cite[Prop. 3.11]{Wi07}.\nThe above inclusion map intertwines the operation of taking the complex-conjugates of the functions on $\\RR$, $N$, and $N\\times\\RR$, \nhence $\\gamma$ is a ``real'' isomorphism. \nThen $K_j(\\gamma)\\colon K_j(C^*(N)\\rtimes_\\beta\\RR)\\to K_j(C^*(N\\rtimes_\\alpha\\RR))$ is a group isomorphism satisfying \n$K_j(\\gamma)(\\overline{x})=\\overline{K_j(\\gamma)(x)}$ for all $x\\in K_j(C^*(N)\\rtimes_\\beta\\RR)$ and $j=0,1$. \nNow the assertion follows by an application of Proposition~\\ref{tsigns}. \n\\end{proof}\n\n\n\n\\begin{corollary}\n\\label{signs_solvable}\nLet $G$ be a connected, simply connected, solvable Lie group and denote $n:=\\dim G$. \nThen the following assertion hold: \n\\begin{enumerate}[{\\rm(i)}]\n\t\\item If $n\\in 2\\ZZ$ then $K_1(C^*(G))= \\{0\\}$, $K_0(C^*(G))\\simeq\\ZZ$, and for every $x\\in K_0(C^*(G))$ we have \n\t$$\\overline{x}=\\begin{cases}\n\tx &\\text{ if }n\\in 4\\ZZ,\\\\\n\t-x &\\text{ if }n\\in 4\\ZZ+2.\n\t\\end{cases}$$\n\t\\item If $n\\in 2\\ZZ+1$ then $K_0(C^*(G))=\\{0\\}$, $K_1(C^*(G))\\simeq\\ZZ$, and for every $x\\in K_1(C^*(G))$ we have \n\t$$\\overline{x}=\\begin{cases}\n\tx &\\text{ if }n\\in 4\\ZZ+1,\\\\\n\t-x &\\text{ if } n\\in 4\\ZZ+3.\n\t\\end{cases}$$\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nThe group isomorphisms from the statement are well known. (See \\cite[Sect.~V, Cor.~7]{Co81}.)\nTo prove the assertions on $\\overline{x}$ \nwe recall that, since $G$ is a connected, simply connected, solvable Lie group, \nthere exists a Lie group isomorphism $G\\simeq G_1\\rtimes\\RR$ for a suitable connected, simply connected, solvable Lie group $G_1$.\nNow the conclusion follows by induction, using Corollary~\\ref{signs_semid} and Lemma~\\ref{dim1}. \n\\end{proof}\n\n\n\n\n\\subsection{Continuous fields of $C^*$-algebras }\n\nThe following lemma is implicitly used in the proof of \\cite[Thm.~3.1]{ENN93}. \n\n\\begin{lemma}\\label{lemma-cf1}\nLet $((\\Ac_t)_{t\\in S}, \\Theta)$ be a continuous field of $C^*$-algebras over the locally compact space $S$.\nAssume that for a $t_0\\in S$ there is a projection $p_0\\in \\Pg(\\Ac_{t_0})\\setminus \\{0\\}$. \nThen there is an open neighbourhood $V_0$ of $t_0$ in $S$ and a section \n$\\theta_0\\in \\Theta\\vert_{V_0}$ such that $\\theta_0(t_0) = p_0$ and $ \\theta_0(t) \\in \\Pg(\\Ac_{t})\\setminus \\{0\\}$\nfor every $t\\in V_0$. \n\\end{lemma}\n\n\\begin{proof}\nFor $\\delta \\in (0, 1\/2)$, define\n$$ U=\\{ z\\in \\CC\\mid |z|< \\delta\\} \\cup \\{z\\in \\CC\\mid |z-1|< \\delta\\}.$$\nThen $\\opn{Sp}'_{A_{t_0}}(p_0)=\\{0, 1\\}\\subset U$. \n(Here $\\opn{Sp}'_{\\Ac}(a)$ denotes the \nspectrum of $a$ in the non-necessarily unital $C^*$-algebra $\\Ac$; see \\cite[1.1.6]{Di64}.)\nThen, by \\cite[20.1.10]{Di64}, there exists $x_1\\in \\Theta$ such that $x_1(t_0)=p_0$.\nDefine $x_2 = \\frac{1}{2} (x_1 +x_1^*)$; \nthen $x_2 \\in \\Theta$, $x_2= x_2^*$ and $x_2(t_0)= p_0$. \nIt follows\nby \\cite[10.3.6]{Di64} that there is an open neighbourhood $V_0$ of $t_0$ in $S$ such that \n$\\opn{Sp}'_{A_{t}}(x_2(t))\\subset U$ for every $t\\in V_0$. \nHence, if $f\\in \\Cc (\\CC)$ is such that $f(t)=1$ for $t\\in \\{z\\in \\CC\\mid |z-1|< \\delta\\}$ and \n$f(t) =0$ for $t \\in \\{ z\\in \\CC\\mid |z|< \\delta\\}$, then $f(x_2(t)) \\in \\Pg(\\Ac_t)$.\nBy \\cite[10.3.3]{Di64}, $\\theta_0= f(x_2)\\in \\Theta$.\nSince the function $\\Vert \\theta_0(\\cdot)\\Vert$ is continuous on $V_0$ and $\\Vert \\theta_0(t)\\Vert\n\\in \\{0, 1\\}$, it follows that $\\Vert \\theta_0(t)\\Vert = \\Vert \\theta_0(t_0)\\Vert=1$ for every \n$t \\in V_0$. \nWe have thus obtained that $\\theta_0(t) \\in \\Pg(\\Ac_t) \\setminus \\{ 0\\}$ for every $ t \\in V_0$, $\\theta_0(t_0)= p_0$, hence $\\theta_0$ satisfies all the conditions in the statement.\n\\end{proof}\n\n\\begin{proposition}\\label{prop-cf2}\nLet $((\\Ac_t)_{t\\in [0, 1]}, \\Theta)$ be a continuous field of $C^*$-algebras, trivial away from $0$ (that is, trivial on $(0, 1]$). \nIf $\\Ac_t$ is stably finite for $t\\in (0, 1]$, then $\\Ac_0$ is stably finite.\n\\end{proposition}\n\n\n\\begin{proof}\nAssume that $\\Ac_0$ is not stably finite. \nThen by Proposition~\\ref{P1} \\eqref{P1_item1} it follows that there is $k\\ge 1$ and \n$p_0 \\in \\Pg(M_k(\\Ac_0))\\setminus \\{\\0_k\\}$ such that $[p_0] =0 \\in K_0(\\Ac)$. \nSince $(M_k(\\Ac_t))_{t\\in [0, 1]}$ is a continuous field of $C^*$-algebras, trivial away from $0$ \n(see \\cite[Thm.~2.4]{ENN93}), we may assume that $k=1$. \nBy Lemma~\\ref{lemma-cf1} and since $(\\Ac_t)_{t\\in [0, 1]}$ is trivial away from $0$, there is $\\theta_0\\in \\Theta$ such that $\\theta_0(0)=p_0$ and \n$\\theta_0(t)\\in \\Pg(\\Ac_t)\\setminus \\{0 \\}$ for every $t \\in [0, 1]$. \nIt follows by \\cite[Thm.~3.1 and its proof]{ENN93} that there is a group homomorphism \n$\\varphi \\colon K_0(\\Ac_0) \\to K_0(\\Ac_t)$ such that \n$\\varphi([p_0])= [\\theta_0(1)]$. \nSince we have assumed that $[p_0]=0$, we get that for $\\theta_0(1)\\in \\Pg(\\Ac_1)\\setminus \\{0\\}$\n we have $[\\theta_0(1)]=0$, thus by Proposition~\\ref{P1} \\eqref{P1_item1}, $\\Ac_1$ is not stably finite. \n This is a contradiction; thus $\\Ac_0$ must be stably finite.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{On open points in the primitive ideal spectrum}\n\n\n\\begin{proposition}\\label{P3}\n\tLet $\\Ac$ be a separable $C^*$-algebra\n\tIf $\\pi_0\\colon\\Ac\\to\\Bc(\\Hc_0)$ is a $*$-representation \n\twith its kernel $\\Pc_0:=\\Ker\\pi_0\\subseteq\\Ac$ \n\tand $\\Kc(\\Hc_0)\\subseteq\\pi_0(\\Ac)\\ne\\{0\\}$, \n\tthen the following conditions are equivalent: \n\t\\begin{enumerate}[{\\rm(i)}]\n\t\t\\item\\label{P3_item1} $\\{\\Pc_0\\}$ is an open subset of $\\Prim(\\Ac)$. \n\t\t\\item\\label{P3_item2} There exists a closed two-sided ideal $\\Jc_0\\subseteq\\Ac$ for which \n\t\t$\\pi_0\\vert_{\\Jc_0}\\colon \\Jc_0\\to\\Kc(\\Hc_0)$ is a $*$-isomorphism.\n\t\t\\end{enumerate}\n\t\tIf these conditions are satisfied, then \n\t\t \\begin{equation}\\label{P3_proof_eq7}\n\t\t\\Jc_0=\\bigcap\\limits_{\\Pc\\in\\Prim(\\Ac)\\setminus\\{\\Pc_0\\}}\\Pc\n\t\t\\end{equation}\n\t\tand moreover $\\Jc_0$ is a minimal closed two-sided ideal of $\\Ac$ \n\t\twith \t$\\Pc_0\\cap\\Jc_0=\\{0\\}$. \n\\end{proposition}\n\n\\begin{proof}\nThe hypothesis $\\Kc(\\Hc_0)\\subseteq\\pi_0(\\Ac)\\ne\\{0\\}$ implies that the $*$-representation $\\pi_0$ is irreducible and $\\Hc_0\\ne\\{0\\}$. \nThen, since $\\Ac$ is separable, the Hilbert space $\\Hc_0$ is separable, too.\nWe will show that both conditions in the statement are equivalent to the following: \n\\begin{enumerate}[{\\rm(i)}]\n \\setcounter{enumi}{2}\n\t\\item\\label{P3_item3} There exists a closed two-sided ideal $\\Jc_0\\subseteq\\Ac$ such that \n\t\\begin{equation}\\label{P3-eq1}\n\t\\Prim(\\Ac)=\\{\\Pc_0\\}\\sqcup\\{\\Pc\\in\\Prim(\\Ac)\\mid \\Jc_0\\subseteq\\Pc\\}.\n\t\\end{equation}\n\\end{enumerate}\n\t\n\\eqref{P3_item1}$\\iff$\\eqref{P3_item3}: \nThis and \\eqref{P3_proof_eq7} follow by the definition of the topology of $\\Prim(\\Ac)$. \n(See \\cite[3.1.1]{Di64}.) \n\n\\eqref{P3_item3}$\\implies$\\eqref{P3_item2}: \nThe hypothesis \\eqref{P3-eq1} implies $\\Jc_0\\not\\subseteq\\Pc_0=\\Ker\\pi_0$, \nthat is, $\\pi_0\\vert_{\\Jc_0}\\ne0$. \nMoreover, for every irreducible $*$-representation $\\pi\\colon\\Ac\\to\\Bc(\\Hc)$ we have \n$$\\pi\\vert_{\\Jc_0}\\ne0\\iff\\Jc_0\\not\\subseteq\\Ker\\pi\n\\mathop{\\iff}\\limits^{\\eqref{P3-eq1}}\\Ker\\pi=\\Pc_0\n\\iff[\\pi]=[\\pi_0]\\in\\widehat{\\Ac}$$\nwhere the last equivalence follows by \\cite[Cor. 4.1.10]{Di64} \nsince $\\Kc(\\Hc_0)\\subseteq\\pi_0(\\Ac)$. \nThen, by \\cite[Prop. 2.10.4]{Di64}, $\\widehat{\\Jc_0}$ consists of only one point, namely $\\widehat{\\Jc_0}=\\{[\\pi_0\\vert_{\\Jc_0}]\\}$. \nSince $\\Ac$ is separable it follows that $\\Jc_0$ is separable, too. \nThen $\\Jc_0$ is $*$-isomorphic to the $C^*$-algebra of all compact operators on a separable complex Hilbert space by \\cite[4.7.3]{Di64}. \nNow, since $\\pi_0\\vert_{\\Jc_0}\\ne0$ and $\\pi_0$ is an irreducible representation of~$\\Ac$, hence $\\pi_0\\vert_{\\Jc_0}\\colon \\Jc_0\\to\\Bc(\\Hc_0)$ is an irreducible representation, it follows that \n$\\pi_0\\vert_{\\Jc_0}\\colon \\Jc_0\\to\\Kc(\\Hc_0)$ is a $*$-isomorphism. \n(See \\cite[Cor. 4.1.5]{Di64}.) \nHence \\eqref{P3_item3} holds true with $\\Jc_1:=\\Jc_0$.\n\n\\eqref{P3_item2}$\\implies$\\eqref{P3_item3}: \nThe hypothesis \\eqref{P3_item2} implies $\\widehat{\\Jc_0}=\\{[\\pi_0\\vert_{\\Jc_0}]\\}$. \nThen, \nfor every irreducible $*$-representation $\\pi\\colon\\Ac\\to\\Bc(\\Hc)$, we have either $\\pi\\vert_{\\Jc_0}=0$ or $[\\pi\\vert_{\\Jc_0}]=[\\pi_0\\vert_{\\Jc_0}]\\in\\widehat{\\Jc_1}$. \nThat is, either $\\Jc_1\\subseteq\\Ker\\pi$ or $[\\pi]\n=[\\pi_0]\\in\\widehat{\\Ac}$ by \\cite[Prop. 2.10.4]{Di64}. \nThus\n$\\Prim(\\Ac)=\\{\\Ker\\pi_0\\}\\sqcup\\{\\Pc\\in\\Prim(\\Ac)\\mid \\Jc_1\\subseteq\\Pc\\}$, \nhence~\\eqref{P3_item2} holds true. \n\nFinally, if \\eqref{P3_item1}--\\eqref{P3_item3} hold true, then $\\Jc_0$ is $*$-isomorphic to $\\Kc(\\Hc_0)$, hence $\\Jc_0$ is a simple $C^*$-algebra, and then it is also a minimal ideal of $\\Ac$ with $\\Pc_0\\cap\\Jc_0=\\{0\\}$.\n\\end{proof}\n\n\\begin{remark}\\label{R5}\n\\normalfont\nIn Proposition~\\ref{P3} we have the short exact sequence \n$$0\\to\\Pc_0\\hookrightarrow\\pi_0^{-1}(\\Kc(\\Hc_0))\\mathop{\\longrightarrow}\\limits^{\\pi_0}\\Kc(\\Hc_0)\\to0$$\nand this extension is trivial in the sense that \n$\\pi_0\\vert_{\\pi_0^{-1}(\\Kc(\\Hc_0))}$ has a right inverse, namely $(\\pi_0\\vert_{\\Jc_0})^{-1}\\colon\\Kc(\\Hc_0)\\to\\Jc_0$ given by Proposition~\\ref{P3}\\eqref{P3_item3}. \nThis also shows the direct sum decomposition of ideals \n$$\\pi_0^{-1}(\\Kc(\\Hc_0))=\\Pc_0\\dotplus\\Jc_0.$$\n\\end{remark}\n\n\n\\begin{remark}\\label{P3_group}\n\\normalfont \nThe hypothesis $\\Kc(\\Hc_0)\\subseteq\\pi_0(\\Ac)$ in Proposition~\\ref{P3} is superfluous if $\\Ac=C^*(G)$ for an exponential Lie group~$G$. \nIn fact, let $\\pi_0\\colon G\\to\\Bc(\\Hc_0)$ be an irreducible unitary representation \nwith its corresponding irreducible $*$-representation $\\pi_0\\colon \\Ac\\to\\Bc(\\Hc_0)$ with $\\Pc_0:=\\Ker\\pi\\subseteq\\Ac$. \nSince $G$ is type~I, we have $\\Kc(\\Hc_0)\\subseteq\\pi_0(\\Ac)$, \nand on the other hand $\\{\\Pc_0\\}$ is an open subset of $\\Prim(\\Ac)$ if and only if the unitary representation~$\\pi_0$ is square integrable, \nby \\cite[Prop. 2.3 and 2.14]{Ros78} and \\cite[Cor. 2]{Gr80}. \nMoreover the irreducible unitary representation~$\\pi_0$ is square integrable if and only if its corresponding coadjoint orbit is open in~$\\gg^*$ by \n\\cite[Thm.~3.5]{Ros78}.\nThis provides an alternative argument for the fact that the Kirillov-Bernat correspondence gives a bijection between the open points of $\\Prim(G)$ and the open coadjoint orbits of~$G$, without using the more difficult and deep fact that the Kirillov-Bernat map is actually a homeomorphism. \n\\end{remark}\n\n\n\n\n\n\n\n\n\\begin{corollary}\\label{C4}\nLet $\\Ac$ be a $C^*$-algebra and, for $k=1,\\dots,n$, let $\\pi_k\\colon\\Ac\\to\\Bc(\\Hc_j)$ be a $*$-representation satisfying the hypotheses of Proposition~\\ref{P3}, with its corresponding ideal $\\Jc_k\\subseteq\\Ac$ for which \n$\\pi_k\\vert_{\\Jc_k}\\colon \\Jc_k\\to\\Kc(\\Hc_k)$ is a $*$-isomorphism, \nand $\\Pc_k:=\\Ker\\pi_k$. \nThen the following assertions hold: \n\\begin{enumerate}[{\\rm(i)}]\n\\item\\label{C4_item1} \nWe have $\\Jc_{k_1}=\\Jc_{k_2}$ if and only if $\\Pc_{k_1}=\\Pc_{k_2}$. \n\\item \\label{C4_item2} \nIf we assume $\\Pc_{k_1}\\ne\\Pc_{k_2}$ for $k_1\\ne k_2$, then\n\\begin{equation}\\label{C4_item2_eq1} \n\t\\text{$\\Jc:=\\Jc_1+\\dots+\\Jc_n$ is a direct sum of ideals of $\\Ac$}\n\\end{equation}\nand\n\\begin{equation}\\label{C4_item2_eq2}\n\t\\Prim(\\Ac)=\\{\\Pc_1\\}\\sqcup\\cdots\\sqcup\\{\\Pc_n\\}\\sqcup \\{\\Pc\\in\\Prim(\\Ac)\\mid\\Jc\\subseteq\\Pc\\}.\n\\end{equation}\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\n\\eqref{C4_item1} \nWe have $\\Prim(\\Ac)\\setminus\\{\\Pc_k\\}=\\{\\Pc\\in\\Prim(\\Ac)\\mid \\Jc_k\\subseteq\\Pc\\} $ by \\eqref{P3-eq1}, \nhence $\\Jc_{k_1}=\\Jc_{k_2}$ implies $\\Pc_{k_1}=\\Pc_{k_2}$. \nOn the other hand, by \\eqref{P3_proof_eq7}, \n$\\Pc_{k_1}=\\Pc_{k_2}$ implies $\\Jc_{k_1}=\\Jc_{k_2}$. \n\n\\eqref{C4_item2} \nIf $k_1\\ne k_2$, then $\\Pc_{k_1}\\ne\\Pc_{k_2}$ by hypothesis, hence $\\Jc_{k_1}\\ne\\Jc_{k_2}$ by \\eqref{C4_item1}. \nSince both $\\Jc_{k_1}$ and $\\Jc_{k_2}$ are distinct minimal ideals of $\\Ac$ \nand $\\Jc_{k_1}\\Jc_{k_2}\\subseteq\\Jc_{k_1}\\cap \\Jc_{k_2}$, \nwe obtain $\\Jc_{k_1}\\Jc_{k_2}=\\Jc_{k_1}\\cap \\Jc_{k_2}=\\{0\\}$, and then \n\\eqref{C4_item2_eq1} is straightforward.\n\n\nWe now prove \\eqref{C4_item2_eq2}. \nIn fact, by \\eqref{P3-eq1}, \nwe have \n$\\Prim(\\Ac)\\setminus\\{\\Pc_k\\}=\\{\\Pc\\in\\Prim(\\Ac)\\mid \\Jc_k\\subseteq\\Pc\\}$ for $k=1,\\dots,n$, hence \n\\allowdisplaybreaks\n\\begin{align*}\n\\Prim(\\Ac)\\setminus\\{\\Pc_1,\\dots,\\Pc_n\\}\n&=\\bigcap_{k=1}^n\\Prim(\\Ac)\\setminus\\{\\Pc_k\\} \\\\\n&=\\bigcap_{k=1}^n \\{\\Pc\\in\\Prim(\\Ac)\\mid \\Jc_k\\subseteq\\Pc\\} \\\\\n&=\\{\\Pc\\in\\Prim(\\Ac)\\mid \\Jc_1+\\cdots+\\Jc_n\\subseteq\\Pc\\}.\n\\end{align*}\nThis finishes the proof. \n\\end{proof}\n\nThe next result is needed in the proof of Corollary~\\ref{cf-cor8}.\n \n\\begin{proposition}\\label{proj}\n\tAssume the setting of Proposition~\\ref{P3} \n\tand, additionally, that \n\t\\begin{enumerate}[{\\rm(i)}]\n\t \\item\\label{proj_item1} $\\Ac$ is a ``real'' $C^*$-algebra; \n\t \\item\\label{proj_item2} $\\Pc_0\\cap\\overline{\\Pc_0}=\\{0\\}$; \n\t \\item\\label{proj_item3} if $p\\in\\Jc_0$ is a minimal projection, then \n\t $K_0(\\Ac)=\\{n[p]_0\\mid n\\in\\ZZ\\}$; \n\t \\item\\label{proj_item4} $\\Pg(A)\\setminus(\\Jc_0+\\overline{\\Jc_0})\\ne\\emptyset$. \n\t\\end{enumerate}\nThen $\\Ac$ is not stably finite. \n\\end{proposition}\n\n\\begin{proof}\nWe define\n$\\overline{\\pi_0}\\colon\\Ac\\to\\Bc(\\Hc_0)$, $\\overline{\\pi_0}(a):=C\\pi_0(\\overline{a})C$, \nfor a fixed antilinear involutive isometry $C\\colon \\Hc\\to\\Hc$. \nIt is easily seen that $\\overline{\\pi_0}$ is an irreducible $*$-representation \nwith $\\Ker\\overline{\\pi_0}=\\overline{\\Pc_0}$ and $\\{\\overline{\\Pc_0}\\}$ is an open subset of $\\Prim(\\Ac)$. \nMoreover, using~\\eqref{P3_proof_eq7}, one can show that \n$\\Jc_{\\overline{\\pi_0}}=\\overline{\\Jc_0}$, where $\\Jc_{\\overline{\\pi_0}}$ is the minimal ideal of $\\Ac$ that is given by the application of Proposition~\\ref{P3} for the representation $\\overline{\\pi_0}$. \nFurthermore, by Remark~\\ref{R5}, we then obtain \n\\begin{equation}\\label{proj_proof_eq1}\n\\overline{\\pi_0}^{-1}(\\Kc(\\Hc_0))\n=\\Jc_{\\overline{\\pi_0}}\\dotplus \\Ker\\overline{\\pi_0}\n=\\overline{\\Jc_0}\\dotplus \\overline{\\Ker\\pi_0} \n=\\overline{\\pi_0^{-1}(\\Kc(\\Hc_0))}.\n\\end{equation}\nSince $\\Pc_0\\cap\\overline{\\Pc_0}=\\{0\\}$ by hypothesis, we may use Corollary~\\ref{C4} for the representations $\\pi_0$ and $\\overline{\\pi_0}$, and we thus obtain $\\Jc_0\\cdot\\overline{\\Jc_0}=\\{0\\}$. \n\nNow let us denote $\\Jc:=\\Jc_0\\dotplus \\overline{\\Jc_0}$ \nand select any $q\\in \\Pg(A)\\setminus\\Jc$. \nWe show that \n\\begin{equation}\\label{proj_proof_eq2}\n\\dim(\\pi_0(q)\\Hc)=\\infty\\text{ and }\\dim(\\overline{\\pi_0}(q)\\Hc)=\\infty.\n\\end{equation}\nIn fact, since $\\pi_0(q),\\overline{\\pi_0}(q)\\in\\Bc(\\Hc_0)$ are projections, \nit suffices to show that $\\pi_0(q)\\not\\in \\Kc(\\Hc_0)$, and this will imply $\\overline{\\pi_0}(q)\\not\\in\\Kc(\\Hc_0)$ by~\\eqref{proj_proof_eq1}. \nWe argue by contradiction: \nAssuming $\\pi_0(q)\\in \\Kc(\\Hc_0)$, we obtain $\\overline{\\pi_0}(q)\\in\\Kc(\\Hc_0)$ by~\\eqref{proj_proof_eq1} and then, by Proposition~\\ref{P3} there exists uniquely determined projections $p_0\\in\\Pg(\\Jc_0)$ and $r_0\\in\\Pg(\\overline{\\Jc_0})$ with $\\pi_0(p_0)=\\pi_0(q)$ and $\\overline{\\pi_0}(r_0)=\\overline{\\pi_0}(q)$. \nSince $\\Pc_0\\ne\\overline{\\Pc_0}$ by the hypothesis~\\eqref{proj_item2}, \nwe have \n$\\Jc_0\\subseteq\\overline{\\Pc_0}$ and $\\overline{\\Jc_0}\\subseteq\\Pc_0$ by \\eqref{P3-eq1} in Proposition~\\ref{P3}, and we then obtain \n$(\\pi_0\\oplus\\overline{\\pi_0})(p_0+r_0)=\\pi_0(q)\\oplus \\overline{\\pi_0}(q)\n=(\\pi_0\\oplus\\overline{\\pi_0})(q)$. \nThe hypothesis $\\Pc_0\\cap\\overline{\\Pc_0}=\\{0\\}$ then implies $q=p_0+r_0\\in\\Jc_0\\dotplus \\overline{\\Jc_0}=\\Jc$, which is a contradiction with the way $q$ was selected. \nThus \\eqref{proj_proof_eq2} is proved. \n\nNow, if $p\\in\\Jc_0$ is a minimal projection, it follows by the hypothesis that there exists $n\\in\\ZZ$ with $[q]_0=n[p]_0\\in K_0(\\Ac)$. \nThere are three possible cases: \n\nCase 1: $n=0$. \nThen $[q]_0=0\\in K_0(\\Ac)$, and Proposition~\\ref{P1} shows that $\\Ac$ is not stably finite. \n\nCase 2: $n<0$. \nThen, denoting $k:=\\vert n\\vert$, we have \n$$0=[q]_0+k[p]_0=[q\\oplus\\underbrace{p\\oplus\\cdots\\oplus p}_{k\\text{ times}}]_0$$\nhence, since $q\\oplus p\\oplus\\cdots\\oplus p\\in M_{k+1}(\\Ac)\\setminus\\{0\\}$, \n Proposition~\\ref{P1} again shows that $\\Ac$ is not stably finite. \n \n Case 3: $n>0$. \n In this case, by \\eqref{proj_proof_eq2}, there exists $\\widetilde{p}_1\\in\\Pg(\\Kc(\\Hc_0))$ with $\\widetilde{p}_1\\le\\pi(q)$ and $\\dim(\\widetilde{p}_1(\\Hc))=n$. \n By Proposition~\\ref{P3}, there exists a unique $p_1\\in\\Pg(\\Jc_0)$ with $\\pi_0(p_1)=\\widetilde{p}_1$. \n We already noted above that $\\Jc_0\\subseteq\\overline{\\Pc_0}=\\Ker\\overline{\\pi_0}$, \n hence $\\overline{\\pi_0}(p_1)=0$, and then \n $$(\\pi_0\\oplus\\overline{\\pi_0})(p_1)=\\pi_0(p_1)\\oplus 0=\\widetilde{p}_1\\oplus 0\\le\\pi_0(q)\\oplus\\overline{\\pi_0}(q)=(\\pi_0\\oplus\\overline{\\pi_0})(q).$$\n As above, the hypothesis hypothesis $\\Pc_0\\cap\\overline{\\Pc_0}=\\{0\\}$ then implies \n $p_1\\le q$, hence $q-p_1\\in\\Pg(\\Ac)$ and $p_1(q-p_1)=0$. \n Now, by \\cite[3.1.7(iv)]{RLL00}, we obtain \n \\begin{equation}\\label{proj_proof_eq3}\n [q]_0=[p_1]_0+[q-p_1]_0\\in K_0(\\Ac)\\subseteq K_0(\\widetilde{\\Ac}).\n \\end{equation}\n On the other hand, since $\\dim(\\widetilde{p}_1(\\Hc))=n$ and $\\pi_0\\vert_{\\Jc_0}\\colon\\Jc_0\\to\\Kc(\\Hc_0)$ is a $*$-iso\\-mor\\-phism, we obtain $[p_1]_0=n[p]_0$ in $K_0(\\Jc_0)$. \n Denoting by $\\varphi\\colon \\Jc_0\\to\\Ac$ the inclusion map, \n it then follows that \n $K_0(\\varphi)([p_1]_0)=nK_0(\\varphi)([p]_0)$ in $K_0(\\Ac)$, \n that is, $[p_1]_0=n[p]_0$ in $K_0(\\Ac)$. \n Then, using \\eqref{proj_proof_eq3} and the way $n$ was chosen, we obtain \n $[q-p_1]_0=0\\in K_0(\\Ac)$. \n On the other hand, $q-p_1\\ne 0$ since $p_1\\in\\Jc_0\\subseteq\\Jc$, while $q\\in\\Pc(\\Ac)\\setminus\\Jc$. \n We may thus apply Proposition~\\ref{P1} to obtain that $\\Ac$ is not stably finite. \n\\end{proof}\n\n\n\n\n\n\n\n\\section{$C^*$-algebras of exponential Lie groups with open coadjoint orbits}\\label{section4k}\n\nThis section contains some of our results on the relation between the quasi-compact open subsets in the primitive ideal space of the $C^*$-algebra of a solvable Lie group and the finite approximation properties of that $C^*$-algebra (Corollaries \\ref{solv-4n+2}~and~\\ref{cf-cor8}). \nThese results mostly concern the exponential Lie groups that admit open coadjoint orbits, \nwhich we call exact symplectic Lie groups since they admit a left-invariant exact symplectic form. (They are elsewhere called as Frobenius Lie groups). \n\n\n\\subsection{Solvable Lie groups of dimension $\\not \\in 4 \\ZZ$}\n\\begin{theorem}\\label{4n+2}\nLet $G$ be \na\nsimply connected\n solvable Lie group with $\\dim G \\not \\in 4 \\ZZ$. \n Then $C^*(G)$ is stably finite if and only if it is \n is stably projectionless.\n \\end{theorem}\n\n\n\\begin{proof}\nThe fact that if $C^*(G)$ is stably projectionless then it is stably finite follows from Corollary~\\ref{rem-1.5} \n\\eqref{rem-1.5_i}. \n\nFor the reverse implication assume first that $\\dim G$ is odd. \nRecall that for a simply connected solvable Lie group $G$, the Connes' Thom isomorphism implies that $K_i(C^*(G))= K_i(\\RR^{\\dim G})$, $i =0, 1$. \nHence if $\\dim G$ is odd $K_0(C^*(G))=0$.\n(See \\cite[Sect. V, Cor. 7]{Co81}.)\nThen the statement\nis a direct consequence of Lemma~\\ref{rem-1.5}.\n\nIt remains to analyse the case when $\\dim G\\in 4 \\ZZ +2$. \nWe prove that if $C^*(G)$ is not stably projectionless then it is not stably finite. \nLet $0\\ne p\\in \\Pg_k(C^*(G))$. \nThen \n$ [p]_0+[\\overline{p}]_0= [p]_0+\\overline{[p]}_0 = 0$\nby Corollary~\\ref{signs_solvable}.\nIf $p=\\overline{p}$ it follows that $[p]_0=0$, hence $C^*(G)$ is not stably finite, \nby Proposition~\\ref{P1}.\nIf $p \\ne \\overline{p}$, define $ q:= p\\oplus \\overline{p}=\\opn{diag}(p, \\overline{p}) \\in \\Pg_{2k} (C^*(G))\\setminus\\{0\\}$.\nThen $[q]_0= 0$, hence, again by Proposition~\\ref{P1}, $C^*(G)$ is not stably finite. \n\\end{proof}\n\n\\begin{corollary}\\label{solv-4n+2}\nLet $G$ be an exponential solvable Lie group such that $\\dim G \\in 4\\ZZ+2$ and has open coadjoint orbits.\nThen $C^*(G)$ is not stably finite. \n\\end{corollary}\n\n\n\n\n\\begin{proof}\nFor an exponential Lie group $G$, an open coadjoint orbit corresponds to an open point $[\\pi]\\in \\widehat{G}\\simeq\\Prim(G)$. (See Remark~\\ref{P3_group}.) Moreover, since $G$ is separable and type I, $\\pi$ is square integrable and \n$\\pi(C^*(G))$ contains the compact operators, by \\cite[Cor. 1 and 2]{Gr80} and \\cite[Prop.~2.3]{Ros78}.\nThen by Proposition~\\ref{P3} there is a minimal ideal $\\Jc_0\\subseteq C^*(G)$ such that $\\Jc_0\\simeq \\Kc(\\Hc_0)$ for a Hilbert space $\\Hc_0$. \nHence there exists $p \\in \\Jc_0$, $0\\ne p= p^*= p^2$. \nThe corollary now follows from Theorem~\\ref{4n+2}.\n\\end{proof}\n\n\n\n\\subsection{Groups of the form $N \\rtimes \\RR$.}\n\nWe are going to see that the above result of Theorem~\\ref{4n+2} fails to be true for groups of dimension of the form \n$4 k$, $k\\in \\NN$. \nTo show this, to give a simple necessary condition for stably finiteness, and to study a little bit further the case \nof $\\dim G\\in 4 \\ZZ +2$, we restrict ourselves to the the groups of the form $G= N \\rtimes \\RR$, where $N$ is a connected simply connected nilpotent Lie group.\n \nBut first we consider the case of groups $G=N\\rtimes \\RR$ where $N$ is abelian, \nthat is, the case of \nthe generalized $ax+b$-groups, where there is a quite clean relation\n between quasi-compact open sets and approximation properties. \n\n\n\\subsubsection{The case of the generalized $ax+b$ groups}\nMost of the following example is already known (see \\cite[Ex.~4.8]{BB20}).\n\n\\begin{example}\\label{ax+b} \\normalfont (\\textit{Generalized $ax+b$-groups})\nLet $\\Vc$ be a finite-dimensional real vector space, $D\\in\\End(\\Vc)$, \nand $G_D:=\\Vc\\rtimes_{\\alpha_D}\\RR$ their corresponding generalized $ax+b$-group. \nThen we claim that the following assertions are equivalent: \n\\begin{enumerate}[{\\rm(i)}]\n\\item\\label{ax+b_item1} Either $\\mathrm{Re}\\, z>0$ for every $z\\in\\sigma(D)$ or $\\mathrm{Re}\\, z<0$ for every $z\\in\\sigma(D)$. \n\\item\\label{ax+b_item2} The $C^*$-algebra $C^*(G_D)$ is not quasidiagonal. \n\\item\\label{ax+b_item3} The $C^*$-algebra $C^*(G_D)$ is not AF-embeddable. \n\\item\\label{ax+b_item4} There exists a nonempty quasi-compact open subset of $\\widehat{G_D}$. \n\\item\\label{ax+b_item5} There exists a nonempty quasi-compact open subset of $\\Prim(G_D)$. \n\\item\\label{ax+b_item6} The set $\\widehat{G_D}\\setminus\\Hom(G_D,\\TT)$ is a nonempty quasi-compact open subset of $\\widehat{G_D}$.\n\\item\\label{ax+b_item8} There exist nonzero self-adjoint idempotent elements of $C^*(G_D)$. \n\\item\\label{ax+b_item1.5} The $C^*$-algebra $C^*(G_D)$ is not stably finite.\n\\end{enumerate}\n \n\n\\begin{proof}[Proof of claim]\nAssertions \\eqref{ax+b_item1} -- \\eqref{ax+b_item8} are equivalent by \\cite[Ex.~4.8]{BB20}. \nThe implication \\eqref{ax+b_item1.5} $\\implies$ \\eqref{ax+b_item2} is clear.\n\nIt remains to prove \\eqref{ax+b_item1} $\\implies$ \\eqref{ax+b_item1.5}.\nAssume that the condition in \\eqref{ax+b_item1} holds. \nIf \n$$\\alpha^*\\colon \\Cc_0(\\Vc^*) \\times \\RR \\to \\Cc_0(\\Vc^*), \\quad \\alpha^*(f, t) = f\\circ \\ee^{t D^*}, $$ \n then $C^*(G_D) \\simeq \\Cc_0(\\Vc^*) \\rtimes_{\\alpha^*} \\RR$.\n\nLet $\\overline{\\Vc^*}$ be one point compactification of $\\Vc^*$ and extend $\\alpha^*$ to \n$\\Vc^*$ by $\\alpha^*_t(\\infty)=\\infty$ for every $t\\in \\RR$.\nThen it follows from \\cite[Pro.~2.14]{BB18} and \\cite[Prop.~4.6]{Pi99} \nthat the $C^*$-algebra $\\Cc_0(\\overline{\\Vc^*})\\rtimes \\RR$ is not stably finite whenever \n\\eqref{ax+b_item1} is true.\n\nWe now use the same argument as in \\cite[Lemma~2.10]{BB18}:\nThe split exact sequence $0\\to \\Cc_0(\\Vc^*) \\to \\Cc(\\overline{\\Vc^*})\\to\\CC\\1\\to 0$ \nleads\n to the split exact sequence \n\t$$0\\to \\Cc_0(\\Vc^*) \\rtimes \\RR \\to \\Cc(\\overline{\\Vc^*})\\rtimes \\RR \\to C^*(\\RR)\\to 0.$$ \n\tThen if we assume that $\\Cc_0(\\Vc^*) \\rtimes \\RR$ is stably finite, since\n\tthe $C^*$-algebra $C^*(\\RR)$ is stably finite, \n\tit follows by \\cite[Lemma~1.5]{Sp88} that $\\Cc(\\overline{\\Vc^*})\\rtimes \\RR$ is stably finite.\n\tThis is a contradiction, hence $\\Cc_0(\\Vc^*) \\rtimes \\RR$ is not stably finite.\n\t\\end{proof}\n\\end{example}\n\n\n\n\n\n\n\\subsubsection{Continuous fields of nilpotent Lie groups} \n\n\nLet $(\\ng, [\\cdot, \\cdot])$ be a nilpotent Lie algebra and let $\\varphi\\colon (0, 1]\\to \\GL(\\ng)$, $t\\mapsto\\varphi_h$ be a continuous map. \nAssume the following conditions hold:\n\\begin{enumerate}\n\\item $\\varphi_1=\\id$; \n\\item There exists $[x, y]_0=\\lim_{h\\to 0 }\\varphi_h^{-1}([\\varphi_h (x), \\varphi_h(y)]_h)$, for every $x, y\\in \\ng$. \n\\end{enumerate}\n\n\nThen we define the bilinear map $\\ng \\times \\ng \\to \\RR$,\n\\begin{equation}\\label{defm}\n[x, y]_h =\\varphi_h^{-1}([\\varphi_h(x), \\varphi_h(y)]).\n\\end{equation}\n\n\n\\begin{remark}\\label{defm-rem}\n\\normalfont\n \\begin{enumerate}[\\rm (i)]\n \\item\\label{defm-rem_i} For all $h\\in [0, 1]$, $[\\cdot, \\cdot]_h$ is a Lie bracket on the vector space underlying $\\ng$, and we denote by $\\ast_h$ the corresponding Baker-Campbell-Hausdorff multiplication, and the corresponding connected and simply connected Lie group by\n$N_h= (\\ng, \\ast_h)$.\n \n \\item \\label{defm-rem_ii} \nFor every $h\\in (0, 1]$, $\\varphi_h \\colon (\\ng, [\\cdot, \\cdot]_h) \\to (\\ng, [\\cdot, \\cdot])$ is a Lie algebra isomorphism. \n\n\\item\\label{defm-rem_iii} For every $h\\in (0, 1]$ we have\n$$ \\ad_h x= \\varphi_h^{-1} \\circ \\ad (\\varphi_h(x)) \\circ \\varphi_h$$\nwhere \n$$ \\begin{aligned} \n\\ad\\, x\\colon \\ng \\to \\ng, & (\\ad\\, x)(y)= [x, y] =[x, y]_1, \\\\\n\\ad_h x\\colon \\ng \\to \\ng, & (\\ad_h x)(y)= [x, y]_h .\n\\end{aligned}\n$$\n\\end{enumerate}\n\\end{remark}\n\nWe consider the map \n\\begin{equation}\\label{defm-mult}\nm \\colon [0, 1] \\times \\ng \\times \\ng \\to \\ng, \\; \\; m(h, x, y) =(h, x\\ast_h y).\n\\end{equation}\nThen $m$ is continuous, by the assumptions above. \nConsider the groupoid with equal source and target maps\n$$ \n\\begin{gathered} \n\\Tc:=[0, 1] \\times \\ng \\stackrel{p}{\\rightarrow} S:=[0, 1], \\; p(h, x)=x, \\\\\n(h, x) \\cdot (h, y) := (h, m_h(x, y)) = (h, x \\ast_h y) \\; \\text{for all } \\;\n(h, x), (h, y) \\in \\Tc_h:=p^{-1}(h).\n\\end{gathered}\n $$\nHence $p$ is a group bundle (depending on the map $\\varphi$) with Haar system given by the Lebesgue measure. \nIf follows by \\cite[Lemma~3.3]{BB18} that $C^*(\\Tc)$ is a $\\Cc(S)$ -algebra that \nis $\\Cc(S)$-linearly $*$-isomorphic \nto the algebra of sections of an upper semi-continuous $C^*$-bundle over~$ S$ \nwhose fibre over any $s\\in S$ is $C^*(\\Tc_h)\\simeq C^*(N_h)$.\n\n\n\n\n\n\\begin{theorem}\\label{prop-cf4} \nFor $\\ng$ be a nilpotent Lie algebra and\n $D\\in \\Der(\\ng)$, define\nthe semi-direct product\n$G :=N \\rtimes_{\\alpha_D} \\RR$.\nIf\nthere exists $\\epsilon \\in \\{-1, 1\\}$ such that \n$\\epsilon \\Re\\, z> 0$ for all $z\\in \\sigma(D)$, \nthen \n$C^*(G)$ is not stably finite.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $\\Vc$ be the underlying real vector space of the Lie algebra $\\ng$, and \ndenote \n$G_0:= \\Vc\\rtimes_{\\alpha_D}\\RR$.\nIf we proved that\n \\begin{equation}\\label{prop-cf4-1}\n C^*(G) \\; \\text{stably finite } \\; \\Rightarrow \\; C^*(G_0) \\; \\text{stably finite,}\n\\end{equation}\nthen the statement follows from Example~\\ref{ax+b}.\n\nThus, it remains to prove \\eqref{prop-cf4-1}.\n\nFor every $h \\in [0, 1]$, let $\\varphi\\colon [0, 1]\\to \\GL(\\ng)$ be the map $\\varphi_h(x) = h x$, \nfor every $x\\in \\ng$. \nConsider the deformed nilpotent Lie algebra \n$ \\ng_h = (\\Vc, h [\\cdot, \\cdot]_\\ng)$ and the corresponding nilpotent Lie group\n$(N_ h , \\ast_{h})$, as above. Then $N_1=N$ and $N_0= \\Vc$. \nWe define $G_h:= N_h \\rtimes_{\\alpha_D} \\RR$, with the multiplication \n$ (x, t)\\cdot_h (y, s) = (x\\ast_h \\ee^{t D}y, t+s)$.\nThen $\\Gc:= \\sqcup_{ h \\in [0, 1]} G_h$ is a smooth bundle of Lie groups over $[0, 1]$. \nIt follows by \\cite[\\S 3]{BB18} that $C^*(\\Gc)= \\sqcup_{h \\in [0, 1]} C^*(G_\\hbar)$ is an upper-continuous bundle of $C^*$-algebras. \n\nOn the other hand, the action $\\alpha_D\\colon \\RR \\to \\Aut(N_{h})$ is independent of $h\\in [0, 1]$, and thus we may choose the Haar measure on $G_h$ to be independent of $h$ as well.\nHence, by \\cite[Def.~3.3, Thm.~3.5]{Ri89}, $(C^*(G_h))_{h \\in [0, 1]}$ is a continuous field of $C^*$-algebras, which is clearly trivial away from $0$. \nThe result then follows by Proposition~\\ref{prop-cf2}.\n\\end{proof}\n\n\n\n\n \n\n\n\n\\subsubsection{Exponential Lie groups with exact symplectic Lie algebras and codimension 1 nilradicals}\n\n\n\n\\begin{definition}\\label{exactsympl}\n\\normalfont \nA solvable Lie algebra $\\gg$ is said to be \\emph{exact symplectic} \nif there is $\\xi_0\\in \\gg^*$ with $\\gg(\\xi_0)=\\{0\\}$. \nEquivalently, if $G$ is any connected Lie group with Lie algebra $\\gg$, then the coadjoint of $\\xi_0$ is an open subset of $\\gg^\\ast$. \n\\end{definition}\n\n\n\n\n\\begin{lemma}\\label{extra}\nLet $\\gg$ be a \nsolvable Lie algebra \nsuch that its nilradical~$\\ng$ has codimension 1. \nLet $\\zg$ be the centre of $\\ng$. \nLet $G$ be a connected simply connected Lie group with its Lie algebra $\\gg$ and $N\\subseteq G$ be the connected subgroup corresponding to the subalgebra $\\ng\\subseteq\\gg$.\nThen the following assertions are equivalent: \n\\begin{enumerate}[\\rm(i)]\n\\item\\label{extra-i} $\\gg$ is exact symplectic.\n\\item\\label{extra-i-2} $\\gg$ is exact symplectic\nand has exactly two open coadjoint orbits.\n\\item\\label{extra2-i} There is an open point in $\\widehat{G}$. \n\\item\\label{extra2-i-2} There are exactly two open points in $\\widehat{G}$. \n\\item \\label{extra-ii}\n$[\\gg, \\zg]\\ne \\{0\\}$, $\\dim \\zg =1$ and the nilpotent Lie group $N=\\exp \\ng$ has generic flat coadjoint orbits\n{\\rm (}or equivalently, there is $\\ell \\in \\ng^*$ such that $\\ng(\\ell) =\\zg$.\\rm{)}\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\nThe equivalences \\eqref{extra2-i} $ \\iff$ \\eqref{extra2-i-2} $\\iff$ \\eqref{extra-ii} follows from \\cite[Thm.~4.5]{KT96}.\n\nThe implication \\eqref{extra-i-2} $\\Rightarrow$ \\eqref{extra-i} is trivial. \nIt remains to prove \\eqref{extra-i} $\\Rightarrow$ \\eqref{extra-ii} $\\Rightarrow$ \\eqref{extra-i-2}. \n For every $\\xi\\in\\gg^*$ we denote by $\\Oc_\\xi\\subseteq\\gg^*$ its corresponding coadjoint orbit. \n\n\\eqref{extra-ii} $\\Rightarrow$\\eqref{extra-i-2}: \nWe prove by contradiction the following assertion: \n\\begin{equation}\\label{extra_proof_eq0}\n\\text{if }\\xi\\in\\gg^*\\text{ and }\\ng(\\xi\\vert_\\ng)=\\zg\\text{ then }\n\\gg(\\xi)=\\{0\\}. \n\\end{equation}\nHence let us assume $\\gg(\\xi)\\ne\\{0\\}$ and let us denote $\\ell:=\\xi\\vert_\\ng\\in\\ng^*$. \nThe hypothesis~\\eqref{extra-ii} implies that that $\\dim\\ng$ is an odd integer and $\\dim\\gg$ is an even integer. \nSince $\\dim\\Oc_\\xi=\\dim\\gg\/\\gg(\\xi)$ and this is an even integer, it follows that $\\dim\\gg(\\xi)$ is also an even integer, hence $\\dim\\gg(\\xi)\\ge2$. \nThen $\\dim\\ng+\\dim\\gg(\\xi)>\\dim\\gg$, hence $\\ng\\cap\\gg(\\xi)\\ne\\{0\\}$. \nOn the other hand \n$$\\ng\\cap\\gg(\\xi)=\\{X\\in\\ng\\mid\\langle\\xi,[X,\\gg]\\rangle=\\{0\\}\\}\n\\subseteq\\ng(\\xi\\vert_\\ng)=\\ng(\\ell)=\\zg$$\nhence, since $\\dim\\zg=1$, we obtain $\\ng\\cap\\gg(\\xi)=\\zg$. \nIn particular $\\zg\\subseteq\\gg(\\xi)$. \n\nNow, selecting any $Y\\in\\gg\\setminus\\ng$, \nthe centre $\\zg\\subseteq\\ng$ is invariant to the derivation $(\\ad_\\gg Y)\\vert_\\ng\\in\\Der(\\ng)$. \nSpecifically, for any $X_0\\in\\zg\\setminus\\{0\\}$ \nand all $X\\in\\ng$ we have \n$$0=[Y,[X,X_0]]=[X,[Y,X_0]]+[[Y,X],X_0]]=[X,[Y,X_0]],$$\nwhere we used that $[X,X_0]=[[Y,X],X_0]]=0$ since $X,[Y,X]\\in\\ng$. \nIt then follows that $[Y,X_0]\\in\\zg$. \nSince $\\gg=\\ng\\dotplus\\RR Y$ and $[\\ng,\\zg]=\\{0\\}$, the hypothesis $[\\gg,\\zg]\\ne\\{0\\}$ \nimplies $[Y,X_0]\\ne0$, hence there exists $a\\in\\RR^\\times$ with $[Y,X_0]=aX_0$. \nThus $\\langle\\xi,[Y,X_0]\\rangle=\\langle\\xi,aX_0\\rangle\n=a\\langle\\ell,X_0\\rangle\\ne0$, \nusing the assumption $\\ng(\\ell)=\\zg=\\RR X_0$. \nThis shows that $X_0\\not\\in\\gg(\\xi)$, which is a contradiction \nwith the above conclusion $\\zg\\subseteq\\gg(\\xi)$. \nThis completes the proof of~\\eqref{extra_proof_eq0}, which shows that $\\gg$ has open coadjoint orbits. \n\nIn order to prove that there are exactly two open coadjoint orbits, \nwe consider the set of generic points \n$$\\gg^*_{\\rm gen}:=\\{\\xi\\in\\gg^*\\mid\\gg(\\xi)=\\{0\\}\\}.$$ \nThe set $\\gg^*_{\\rm reg}$ is the union of the open coadjoint orbits of $\\gg$, which are connected and mutually disjoint, \nhence are also relatively closed in $\\gg^*_{\\rm gen}$. \nThus the open coadjoint orbits are exactly the connected components of $\\gg^*_{\\rm gen}$. \nOn the other hand, by \\eqref{extra_proof_eq0}, we have \n$$\\gg^*_{\\rm gen}=\\{\\xi\\in\\gg^*\\mid \\xi\\vert_{\\zg}\\in\\zg^*\\setminus\\{0\\}\\}=\\gg^*\\setminus\\zg^\\perp$$\nsince for any $\\ell\\in\\ng^*$ the equality $\\ng(\\ell)=\\zg$ is equivalent to $\\ell\\vert_\\zg\\in\\zg^*\\setminus\\{0\\}$, \nby the hypothesis on~$\\ng$ and $\\zg$. \nThus $\\gg^*_{\\rm gen}$ is the complement of a hyperplane in $\\gg^*$, \nand then $\\gg^*_{\\rm gen}$ is the union of two open half-spaces in \n $\\gg^*$. \nThe above remarks then show that these open half-spaces are just \n the open coadjoint orbits of $\\gg^*$, hence there are exactly two such orbits. \n\n\n\\eqref{extra-i}$\\Rightarrow$\\eqref{extra-ii}: \nFix $\\xi_0 \\in \\gg^*$ with $\\gg(\\xi_0)=\\{0\\}$, which exists by hypothesis. \nAssume that\n$[\\gg, \\zg] =\\{ 0\\}$; then $\\zg \\subseteq \\gg(\\xi_0)$, and $\\dim \\zg \\ge 1$ since $\\ng$ is nilpotent. \nThus $\\dim \\gg(\\xi_0)\\ge 1$ and \nthis is a contradiction with \\eqref{extra-i}, therefore\n$[\\gg, \\zg]\\ne \\{0\\}$. \n\nFor $\\xi_0\\in \\gg^*$ as above, define the bilinear functional \n$$B_{\\xi_0} \\colon \\gg\\times \\gg\\to \\RR, \n\\quad B_{\\xi_0}(X, Y) = \\langle \\xi_0, [X, Y]\\rangle.$$ \nWe have that $\\zg \\subseteq \\ng^{\\perp_{B_{\\xi_0}}}$, therefore $\\dim \\ng^{\\perp_{B_{\\xi_0}}}\\ge 1$.\nIf there exists $X_0\\in \\ng^{\\perp_{B_{\\xi_0}}}\\setminus \\ng$ then, since $\\dim(\\gg\/\\ng)=1$, \n$\\gg =\\ng \\dot{+} \\RR X_0$. \nOn the other hand, $X_0 \\perp_{B_{\\xi_0}} \\ng$, hence $X_0\\perp_{B_{\\xi_0}}\\zg$, while \n$\\ng\\perp_{B_{\\xi_0}}\\zg$; it follows that $\\gg \\perp_{B_{\\xi_0}}\\zg$. \nSince $\\zg \\ne \\{0\\}$ this is a contradiction with the fact that $B_{\\xi_0}$ is symplectic. \nThus $\\ng^{\\perp_{B_{\\xi_0}}}\\subseteq \\ng$.\nFor arbitrary $X\\in \\gg \\setminus \\ng$, $\\dim (X^{\\perp_{B_{\\xi_0}}}) =\\dim \\gg -1$.\nHence \nif $\\dim(\\ng^{\\perp_{B_{\\xi_0}}}) \\ge 2$, then $X^{\\perp_{B_{\\xi_0}}}\\cap \\ng^{\\perp_{B_{\\xi_0}}}\\ne 0$, that is, $\\gg^{\\perp_{B_{\\xi_0}}}\\ne 0$, which is again a contradiction.\nIt follows that $\\dim(\\ng^{\\perp_{B_{\\xi_0}}})=1$, hence\n $\\ng (\\xi_0\\vert_{\\ng}) =\\zg$ and $\\dim \\zg =1$. \nThis completes the proof. \n \\end{proof}\n\n\\begin{corollary}\\label{extra-cor}\nLet $G$ be an exponential Lie group with its Lie algebra $\\gg$ and nilradical $\\ng$ such that $\\dim \\gg\/\\ng =1$.\nThen the following assertions are equivalent. \n\\begin{enumerate}[\\rm(i)]\n\\item \\label{cor_extra2-0} There is an open point in $\\Prim(G)$.\n\\item\\label{cor_extra2-i} $\\gg$ is exact symplectic.\n\\item \\label{cor_extra-ii}\n\\begin{itemize}\n\\item{} There is a continuous action $\\alpha \\colon \\RR \\to \\Aut N$ such that $G = N \\rtimes_\\alpha \\RR$ \nand $\\alpha$ acts non-trivially on the centre $Z$ of $N$.\n\\item{} The nilpotent Lie group $N$ has generic flat coadjoint orbits and its centre is 1-dimensional.\n\\end{itemize}\n\\end{enumerate}\n\\end{corollary}\n\n\n\\begin{proof}\nThe assertion is a consequence of Lemma~\\ref{extra} and Remark~\\ref{P3_group}.\n\\end{proof}\n\n\\subsubsection{More on exact symplectic groups $G= N\\rtimes \\RR$}\nLet $N$ be a nilpotent Lie group, connected and simply connected, and $Z$ be the centre of $N$. \nLet $\\ng$ and $\\zg$ be the Lie algebras of $N$ and $Z$, respectively.\nWe assume that $\\dim Z=1$. \nLet $\\Ic$ be the ideal \n$$ \\Ic:= \\bigcap_{\\sigma \\in \\widehat{N\/Z}} \\Ker_{C^*(N)}(\\sigma), $$\nand let $\\Psi \\colon C^*(N) \\to C^{*}(N\/Z)$ be the surjective morphism given by \n$$ (\\Psi(f))(xZ) = \\int_Z f(xz) \\de z.$$\nThen, by \\cite[Prop.~8.C.8]{BkHa20}, we have the short exact sequence \n\\begin{equation}\\label{cf-0}\n 0 \\longrightarrow \\Ic \\longrightarrow C^*(N) \\stackrel{\\Psi}{\\longrightarrow} C^*(N\/Z) \\longrightarrow 0.\n \\end{equation}\n\n\nNow assume that the group $N$ has generic flat coadjoint orbits, and denote $\\dim N = 2d +1$, $d \\in \\NN$.\nLet $\\alpha \\colon \\RR\\to \\Aut(N)$ be a continuous action that acts non-trivially on $Z$, that is, \n$\\alpha_t\\ne \\id_Z$ for some, hence all, $t \\in \\RR\\setminus \\{0\\}$. \nThen there is $\\tau_0 \\in \\RR\\setminus \\{0\\}$ such that $\\de \\alpha_t\\vert_{\\zg} = \\ee^{\\tau_0 t} \\id_\\zg$.\n\nFix $X_0\\in \\zg \\setminus \\{0\\}$ and let $\\pi_1\\colon N \\to \\Bc(L^2(\\RR^d))$ be the unitary irreducible representation such that \n$$\n\\pi_1(\\exp_N (s X_0))=\\ee^{\\ie s} \\id_{L^2(\\RR^d)}. \n$$\nThen \n\\begin{equation}\\label{cf-4}\n(\\pi_1 \\circ \\alpha_t)(\\exp_N(sX_0))= \\pi_1(\\exp_N(s\\ee^{\\tau_0t}X_0))= \n\\ee^{\\ie s\\ee^{\\tau_0t}} \\id_{L^2(\\RR^d)}.\n\\end{equation}\n\nLet $C\\colon L^2(\\RR^d) \\to L^2(\\RR^d)$ be the usual complex conjugation $v \\mapsto \\overline{v(\\cdot)}$. \nFor every $t\\in \\RR$ define \n\\begin{equation}\\label{cf-4.5} \n\\pi_t\\colon N \\to \\Bc(L^2(\\RR^d)),\n \\quad \\pi_t:= \n \\begin{cases} \\pi_1\\circ \\alpha_{\\log t} & \\text{if } t >0, \\\\\n C \\pi_{-t} C & \\text{if } t < 0.\n \\end{cases}\n \\end{equation}\n Then \\eqref{cf-5} and \\eqref{cf-4} give \n $$\n \\pi_t (\\exp_N (sX_0))= \\begin{cases} \\ee^{\\ie t^{\\tau_0} s} \\id_{L^2(\\RR^d)} & \\text{if } t >0, \\\\\n \\ee^{-\\ie \\vert t\\vert^{\\tau_0} s} \\id_{L^2(\\RR^d)} & \\text{if } t < 0.\n \\end{cases}\n$$\nHence the map \n$$ \\RR^{\\times} \\to \\what{N}\\setminus \\what{N\/Z}, \\quad t \\mapsto [\\pi_t]$$\nis a homeomorphism. \n\nWe have thus obtained that there is a $*$-isomorphism \n\\begin{equation}\\label{cf-5}\n \\Phi\\colon \\Ic \\to \\Cc_0(\\RR^{\\times}, \\Kc(L^2(\\RR^d))), \\;\\;\n a\\mapsto (s\\mapsto \\pi_s(a)=\\Phi(a)(s))\n \\end{equation}\n that satisfies \n \\begin{equation}\\label{cf-6}\n ( \\Phi(\\overline{a}))(t)= \\pi_t(\\overline{a}) = C\\overline{\\pi_t}(a) C = C\\pi_{-t}(a) C,\\; \\text{\n for all } t\\in \\RR^{\\times}.\n \\end{equation}\n(Compare with the proof of Lemma~\\ref{rcrossed}.)\n\nDenote \n$$ \\Ic_{\\pm}:=\\Ic \\cap \\bigcap\\limits_{t>0} \\Ker_{C^*(N)} \\pi_{\\mp t}, $$\nwhich are ideals of $\\Ic$.\nThen by \\eqref{cf-5}, \\eqref{cf-6} we get \n\\begin{gather}\n\\Ic = \\Ic_{+} \\dot{+} \\Ic_{-}, \\label{cf-7}\\\\\n\\overline{\\Ic_{+}}=\\Ic_{-}, \\label{cf-8}\\\\\n\\alpha_t(\\Ic_{\\pm})=\\Ic_{\\pm}, \\quad \\text{for all } \\, t \\in\\RR.\\nonumber\n\\end{gather}\n\n\\begin{lemma}\\label{cf-lemma6}\nWith the notation and in the conditions above, \n$$ \\Ic_{+}\\rtimes_{\\alpha} \\RR \\simeq \\Kc(L^2(\\RR^d))\\otimes \\Kc(L^2(\\RR^d)).$$\n\\end{lemma}\n\n\\begin{proof}\nThe restriction of $\\Phi$ to $\\Ic_{+}$ gives a $*$-isomorphism \n$$\\Phi\\vert_{\\Ic_+}\\colon \\Ic_{+} \\, \\widetilde{\\longrightarrow}\\, \\Cc_0(\\RR^\\times_+, \\Kc(L^2(\\RR^d))).$$ \nBy \\eqref{cf-6} and \\eqref{cf-4.5}, for every $t\\in \\RR$ and $s>0$ we have\n\\begin{equation}\\label{-cf-10}\n\\begin{aligned}\n\\Phi(\\alpha_t(a))(s) & = \\pi_s(\\alpha_s(a))\\\\\n & =\\pi_1(\\alpha_{\\log s})(\\alpha_t(a))\\\\\n & =\\pi_1(\\alpha_{\\log (s \\ee^t)}(a))\\\\\n & = \\pi_{s\\ee^t}(a) \n \\\\ \n &=\\Phi(a)(s\\ee^t).\n\\end{aligned}\n\\end{equation}\nOn the other hand, if we denote\n$$ \\rho\\colon \\RR_+^\\times \\times \\Cc_0(\\RR^\\times_+) \\to \\Cc_0(\\RR^\\times_+), \\quad \n(\\rho_t (f))(s)= f(st),$$\nwe have that \n\\begin{equation}\\label{cf-11}\n\\Cc_0(\\RR^\\times_+) \\rtimes_\\rho \\RR^\\times_+ \\simeq \\Kc(L^2(\\RR^\\times_+)).\n\\end{equation}\nThen the assertion in the statement follows from the commutative diagram\n$$ \\xymatrix{\n\\RR \\times \\Ic_+ \\ar[r]^{\\alpha} \\ar[d]_{\\exp \\times \\Phi} & \\Ic_+\\ar[d]^{\\Phi}\\\\\n\\RR_+^\\times \\times (\\Cc_0(\\RR^\\times_+)\\otimes \\Kc(L^2(\\RR^d)))\\ar[r]^{\\;\\; \\rho\\otimes \\id_\\Kc}\n & \\Cc_0(\\RR^\\times_+)\\otimes \\Kc(L^2(\\RR^d))\n}\n$$\nand \\eqref{cf-11}. \n\\end{proof}\n\nDenote \n\\begin{equation}\\label{cf-Jc}\n\\Jc: = \\Ic_+\\rtimes_\\alpha \\RR.\\end{equation}\nThen\n$\\Jc$ is an elementary $C^*$-algebra, by Lemma~\\ref{cf-lemma6}. \n\n\\begin{theorem}\\label{cf-prop7}\nLet $G$ is a solvable Lie group with exact symplectic Lie algebra $\\gg$ of dimension $2d+2$. \nAssume that the nilradical $N$ of $G$ is of codimension 1, and \nlet $Z$ be the centre of $N$.\n\\begin{enumerate}[\\rm (i)]\n\\item\\label{cf-prop7_i} There is an ideal $\\Jc$ of $C^*(G)$ \n such that \n$\\Jc \\simeq \\Kc(L^2(\\RR^d)) \\otimes \\Kc(L^2(\\RR^d))$ and\nthere is the following short exact sequence\n\\begin{equation}\\label{cf-prop7-eq1} \n0\\longrightarrow \\Jc \\dot{+} \\overline{\\Jc} \\stackrel{\\iota}{\\longrightarrow} C^*(G ) \\stackrel{\\psi}{\\longrightarrow} \nC^*(G\/Z) \\rightarrow 0\n\\end{equation}\n\\item\\label{cf-prop7_ii} $K_0(\\Jc\\dot{+} \\overline{\\Jc})^+ \\cap \\Ker K_0(\\iota) =\\{0\\}$ if and only if $\\dim(G) \\in 4 \\ZZ$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma~\\ref{extra}, we have that $\\dim Z=1$, $\\gg\/\\ng\\simeq \\RR$, the continuous action\n$\\alpha\\colon \\RR \\to \\Aut(N)$ is non-trivial on the centre $Z$ of $N$ and \n$G= N\\rtimes \\RR$.\n\n\nIt follows from \\eqref{cf-0} \nthat we have the short exact sequence\n$$ 0\\longrightarrow \\Ic \\rtimes \\RR \\stackrel{\\iota}\\longrightarrow C^*(N \\rtimes \\RR) \\longrightarrow \nC^*((N\/Z) \\rtimes \\RR) \\longrightarrow 0,\n$$\nwhere $\\iota$ is the inclusion map.\nThen assertion \\eqref{cf-prop7_i} is a consequence of \\eqref{cf-7}, \\eqref{cf-8} and of Lemma~\\ref{cf-lemma6}.\n\n\\eqref{cf-prop7_ii}\nFirst note that, $\\dim G= \\dim(N\\rtimes \\RR)=2d+2$, hence there are two possibilities, either \n$\\dim G\\in 4\\ZZ +2 $ or $\\dim G\\in 4\\ZZ$. \nThen in the six term exact sequence corresponding to \\eqref{cf-prop7-eq1}\n$$ \\xymatrix{\nK_0(\\Jc \\dot{+} \\overline{\\Jc}) \\ar[r]^{K_0(\\iota)}\n & K_0 (C^*(N\\rtimes \\RR))\\ar[r]^{K_0(\\psi)} \n& K_0 (C^*((N\/Z)\\rtimes \\RR))\\ar[d]^{\\exp}\\\\\nK_1 (C^*((N\/Z)\\rtimes \\RR)) \\ar[u]^{\\text{ind}} & \\ar[l]^{K_1(\\psi)}\nK_1 (C^*(N\\rtimes \\RR)) &\\ar[l]^{K_1(\\iota)}\nK_1(\\Jc\\dot{+} \\overline{\\Jc})\n}\n$$\nwe have \n$$ K_0 (C^*((N\/Z)\\rtimes \\RR))=K_1(\\Jc\\dot{+} \\Jc) = K_1 (C^*(N\\rtimes \\RR))=0$$\nand \n$$ K_0(\\Jc \\dot{+} \\overline{\\Jc})\\simeq \\ZZ \\times \\ZZ, \\; \\; K_0 (C^*(N\\rtimes \\RR))\\simeq K_1 (C^*((N\/Z)\\rtimes \\RR))\\simeq \\ZZ.$$\nMore specifically, there is an isomorphism\n$$ \n\\chi\\colon K_0(\\Jc) \\times K_0(\\overline{\\Jc}) \\, \\wtilde{\\longrightarrow} \\, \nK_0(\\Jc \\dot{+} \\overline{\\Jc}),\n$$ \nsuch that $p$ is a minimal projection in $\\Jc$, we have\n$$K_0(\\Jc) \\times K_0(\\overline{\\Jc}) =\\{ (m[p]_{0, \\Jc}, \nn [\\overline{p}]_{0, \\overline{\\Jc}})\\mid m, n \\in \\ZZ \\}\\simeq \\ZZ \\times \\ZZ, $$\nand $\\chi (m[p]_{0, \\Jc}, n[\\overline{p}]_{0, \\overline{\\Jc}})\n = [mp+n\\overline{p}]_{0, \\Jc\\dot{+}\\overline{\\Jc}}\n$ if $m,n\\in\\{0,1\\}$.\nThus\n\\begin{equation}\\label{cf-prop7-eq2}\n( K_0(\\iota) \\circ \\chi ) (m[p]_{0, \\Jc}, n[\\overline{p}]_{0, \\overline{\\Jc}})= \nm[p]_{0} + n[\\overline{p}]_0 \n\\text{ for all }m,n\\in\\ZZ.\n\\end{equation} \n\\textit{Case 1: $\\dim (N\\rtimes \\RR)\\in 4 \\ZZ +2$.} \n\nIn this case $[p]_{0} =- [\\overline{p}]_0$, hence \n $$ [p+\\overline{p}]_{0, \\Jc\\dot{+}\\overline{\\Jc}}\\in\n ( K_0(\\Jc \\dot{+} \\overline{\\Jc})^+ \\cap \\Ker K_0(\\iota) ) \\setminus \\{0\\}.$$\n\\textit{Case 2: $\\dim (N\\rtimes \\RR)\\in 4 \\ZZ$.} \n\nIn this case $[p]_{0} =[\\overline{p}]_0$, hence the morphism\n$ K_0(\\iota) \\circ \\chi\\colon \\ZZ \\times \\ZZ \\to K_0(C^*(N\\rtimes\\RR) ) \\simeq \\ZZ$, \nis given by \n$$ (K_0(\\iota) \\circ \\chi ) (m[p]_{0, \\Jc}, n[\\overline{p}]_{0, \\overline{\\Jc}})= \n(m+n) [p]_{0}.$$\nThus \n$$ K_0(\\Jc \\dot{+} \\overline{\\Jc})^+ \\cap \\Ker K_0(\\iota) = \\{0\\}.$$\nThis finishes the proof.\n\\end{proof}\n\n\n \n\n\n\\begin{corollary}\\label{cf-cor8}\nLet $G$ Let $G$ is a solvable Lie group with exact symplectic Lie algebra $\\gg$. \nAssume that the nilradical $N$ of $G$ is of codimension 1 and let \n$Z$ is the centre of the nilradical of $G$.\nThen the following assertions hold true:\n\\begin{enumerate}[\\rm (i)]\n\\item\\label{cf-cor7-i} $C^*(G)$ is stably finite \nif and only if $C^*(G\/Z)$ is. \n\\item\\label{cf-cor7-ii} $C^*(G)$ is AF-embeddable\nif and only if $C^*(G\/Z)$ is.\n\\end{enumerate}\n\\end{corollary}\n\n\n\\begin{proof} By Lemma~\\ref{extra} it is enough to prove the corollary for groups of the form $G= N \\rtimes \\RR$ \nand $G\/Z = (N\/Z) \\rtimes \\RR$ such that the nilradical $N$ of $G$ and the action of $\\RR$ by automorphisms of $N$ satisfies the hypotheses of Theorem~\\ref{cf-prop7}.\n\n\\eqref{cf-cor7-i} \"$\\Leftarrow$\" The assertion follows from Theorem~\\ref{cf-prop7}\nand \\cite[Lemma~1.5]{Sp88}.\n\n \"$\\Rightarrow$\" Assume that $C^*((N\/Z)\\rtimes \\RR)$ is not stably finite. \n Then, by Proposition~\\ref{P1}, there is $0\\ne q _0\\in \\Pc(M_k \\otimes C^*((N\/Z)\\rtimes \\RR)$. \n(Note that $\\dim ((N\/Z) \\rtimes \\RR)$ is odd, so that $K_0( C^*((N\/Z)\\rtimes \\RR ))=0$; hence\n $[q]_{0, C^*((N\/Z)\\rtimes \\RR)}=0$ anyway.) \n Then, by \\cite[Thm.~1]{Ch83}, there exists $0\\ne q \\in \\Pc(M_k \\otimes C^*( N\\rtimes \\RR))$ with \n $(1\\otimes \\psi)(q) =q_0$; hence in particular, $q\\not \\in M_k \\otimes (J \\dot{+} \\overline{J})$.\nThe result then follows from Proposition~\\ref{proj}.\n\n\\eqref{cf-cor7-ii} Use Theorem~\\ref{cf-prop7}\\eqref{cf-prop7_i}, \\eqref{cf-cor7-i} and \\cite[Thm.~1.15]{Sp88}.\n\\end{proof}\n\n\n\n\n\n\\section{Examples}\n\\label{Sect4}\n\nIn this final section we illustrate the results from Section~\\ref{section4k}, effectively using them for establishing existence or lack of various finite approximation properties for the $C^*$-algebras of several concrete solvable Lie groups. \nFor instance, we study exponential solvable Lie groups that have exactly two open coadjoint orbits, and whose nilradical is either a Heisenberg group (Proposition~\\ref{Heis}), or the free 2-step nilpotent Lie group with 3 generators (Theorem~\\ref{N6N15}), or a Heisenberg-like group associated to a finite-dimensional real division algebra (Theorem~\\ref{N6N17}). \n\n\n\\subsection{Some semidirect products} \nIn Lemma~\\ref{NC_lemma} and Proposition~\\ref{NC} below we use some notation related to quasi-orbits of group actions (see \\cite[\\S 2.1]{BB20}). \nSpecifically, for any group action $\\Gamma \\times X \\to X$ on the topological space $X$, we denote by \n$$ (X\/\\Gamma)^\\approx :=\\{\\overline{\\Gamma x}\\mid x\\in X\\}$$\nthe set of all orbit closures, regarded as a topological subspace of the space $\\text{Cl}(X)$ of all closed subsets of $X$, endowed with the upper topology. \n\n\n\n\\begin{lemma}\\label{NC_lemma}\n\tLet $\\Vc$ be a finite-dimensional real vector space and $T\\in\\End(\\Vc)$ with $\\sigma(T)\\cap\\ie\\RR\\subseteq\\{0\\}$, for which there exist $w_1,w_2\\in\\sigma(T)$ with $\\Re\\, w_1\\le 0\\le\\Re\\, w_2$. \n\tIf we define the abelian group $\\exp(\\RR T):=\\{\\ee^{sT}\\mid s\\in\\RR\\}\\subseteq\\End(\\Vc)$ with its natural action on~$\\Vc$, then there exists a continuous open mapping $\\Psi\\colon(\\Vc\/\\exp(\\RR T))^\\approx\\to\\RR$. \n\\end{lemma}\n\n\\begin{proof}\n\tCase 1: $0\\in\\sigma(T)$, that is, $\\Ker T\\ne\\{0\\}$. \n\t\n\tThen, using the Jordan decomposition, we can obtain a linear subspace $\\Vc_0\\subsetneqq\\Vc$ with $T(\\Vc)\\subseteq\\Vc_0$. \n\tIn particular $T(\\Vc_0)\\subseteq\\Vc_0$, and this implies that \t\n\tthe group $\\exp(\\RR T)$ naturally acts on $\\Vc\/\\Vc_0$, and the quotient map $q\\colon\\Vc\\to\\Vc\/\\Vc_0$ is $\\exp(\\RR T)$-equivariant. \n\tWe then obtain the commutative diagram \n\t$$\\xymatrix{\n\t\t\\Vc \\ar[r]^q \\ar[d] & \\Vc\/\\Vc_0 \\ar[d] \\\\\n\t\t(\\Vc\/\\exp(\\RR T))^\\approx \\ar[r]^{q^\\approx}& ((\\Vc\/\\Vc_0)\/\\exp(\\RR T))^\\approx\n\t}$$\n\twhose vertical arrows are quasi-orbit maps, hence they are continuous and open. \n\t(See for instance \\cite[Lemma 2.3]{BB20}.) \n\tSince $q$ is also continuosu and open, it then directly follows that $q^\\approx$ is continuous and open. \n\t\n\tRecalling that $T(\\Vc)\\subseteq\\Vc_0$, the action of $\\exp(\\RR T)$ on $\\Vc\/\\Vc_0$ is trivial, hence the right-most arrow in the the above diagram is actually a homeomorphism and the composition of its inverse with $q^\\approx$ is a continuous open map \n\t$$\\Psi_1\\colon(\\Vc\/\\exp(\\RR T))^\\approx\\to\\Vc\/\\Vc_0.$$\n\tThe real vector space $\\Vc\/\\Vc_0$ is different from $\\{0\\}$, hence there exists a non-zero linear functional $\\xi\\colon\\Vc\/\\Vc_0\\to\\RR$, \n\tand then the mapping $\\Psi:=\\xi\\circ\\Psi_1$ has the required properties. \n\t\n\tCase 2: $0\\not \\in\\sigma(T)$. \n\t\n\tWe then have the direct sum decomposition $\\Vc=\\Vc_+\\dotplus\\Vc_-$, \n\twhere $\\Vc_\\pm$ is the direct sum of the real generalized eigenspaces corresponding to all $w\\in\\sigma(T)$ with $\\pm\\Re\\,w>0$. \n\t(See \\cite[Sect. 2]{BB18b}.) \n\tDenoting $n_\\pm:=\\dim\\Vc_\\pm$, we obtain $n_-n_+\\ne0$ by hypothesis. \n\tLet $n:=n_++n_-=\\dim\\Vc$ and define \n\t$$T_0:=\\begin{pmatrix}\n\t\\1_{n_+} & 0 \\\\\n\t0 & -\\1_{n_-}\n\t\\end{pmatrix}\n\t\\in M_n(\\RR).$$\n\tIt follows by \\cite[Lemma 2.1]{BB18b} \n\tthat there exists a homeomorphism $\\Theta\\colon \\Vc\\to\\RR^n$ \n\tsatisfying $\\Theta\\circ\\ee^{sT}=\\ee^{sT_0}\\circ\\Theta$ for all $s\\in\\RR$, \n\thence we obtain a homeomorphism \n\t$$\\Theta^\\approx\\colon(\\Vc\/\\exp(\\RR T))^\\approx\n\t\\to(\\RR^n\/\\exp(\\RR T_0))^\\approx.$$\n\tNow define the injective linear map \n\t$$p\\colon \\RR^n=\\RR^{n_+}\\times\\RR^{n_-}\\to\\RR^2,\\quad \n\t((x_1,\\dots,x_{n_+}),(y_1,\\dots,y_{n_-}))\\mapsto(x_1,y_1)$$\n\t(which makes sense since $n_-n_+\\ne0$)\n\tand let \n\t$$S:=\\begin{pmatrix}\n\t1 & \\hfill 0 \\\\\n\t0 & -1\n\t\\end{pmatrix}\\in M_2(\\RR).$$\n\tWe have $p\\circ T_0=S\\circ p$ and $p$ is continuous and open, hence we obtain a continuous open mapping \n\t$$p^\\approx\\colon (\\RR^n\/\\exp(\\RR T_0))^\\approx\\to \n\t(\\RR^2\/\\exp(\\RR S))^\\approx.$$\n\tFurthermore, we note that the mapping \n\t$$\\varphi\\colon\\RR^2\\to\\RR,\\quad \\varphi(x,y):=xy$$\n\tis continuous and open, since its restriction to $\\RR^2\\setminus\\{(0,0)\\}$ is actually a submersion while $\\varphi((-a,a)^2)=(-a^2,a^2)$ for all $a\\in(0,\\infty)$. \n\tOn the other hand, we have $\\varphi\\circ\\ee^{tS}=\\varphi$ for all $t\\in\\RR$, \n\thence there exists a commutative diagram \n\t$$\\xymatrix{\n\t\t\\RR^2 \\ar[r]^\\varphi \\ar[d] & \\RR \\\\\n\t\t(\\RR^2\/\\exp(\\RR S))^\\approx \\ar@{.>}[ur]_{\\varphi^\\approx}\n\t}\n\t$$\n\twhose vertical arrow is a quasi-orbit map, hence is continuous and open, and this directly implies that $\\varphi^\\approx$ is continuous and open as well. \n\tFinally, the composition \n\t$$\\Psi:=\\varphi^\\approx\\circ p^\\approx\\circ\\Theta^\\approx\\colon \n\t(\\Vc\/\\exp(\\RR T))^\\approx\\to\\RR $$ \n\tis a continuous open mapping, as required. \n\\end{proof}\n\n\\begin{proposition}\\label{NC}\n\tLet $\\ng$ be a nilpotent Lie algebra with its center~$\\zg$, and $D\\in\\Der(\\ng)$ \n\tsatisfying the conditions \n\t\\begin{equation}\\label{NC_eq1}\n\t\\sigma(D)\\cap\\ie\\RR\\subseteq\\{0\\}\n\t\\end{equation}\n\tand \n\t\\begin{equation}\\label{NC_eq2}\n\t\\text{there exist }w_1,w_2\\in\\sigma(D\\vert_\\zg)\\text{ with }\n\t\\Re\\, w_1\\le 0\\le\\Re\\, w_2.\n\t\\end{equation}\n\tIf $G$ is a simply connected Lie group with its Lie algebra $\\gg:=\\ng\\rtimes\\RR D$, then $G$ is an exponential solvable Lie group and there exists a continuous open mapping $\\Phi\\colon\\Prim(G)\\to\\RR$. \n\\end{proposition}\n\n\\begin{proof}\n\tThe hypothesis \\eqref{NC_eq1} ensures that $G$ is an exponential solvable Lie group. \n\t\n\tStep 1: \n\tIf $0\\in\\sigma(D\\vert_\\zg)$, then $0\\ne\\Ker (D\\vert_\\zg)=\\zg\\cap\\Ker D$. \n\tOn the other hand, it is easily seen that $\\zg\\cap\\Ker D$ is contained in the center of $\\gg$, hence it follows that the center $Z_G$ of the exponential solvable Lie group $G$ satisfies~$\\dim Z_G\\ge 1$, and then the assertion follows at once using the (continuous open) restriction mapping ${\\rm Res}^G_{Z^G}\\colon \\Prim(G)\\to\\widehat{Z_G}$ given by \\cite[Lemma 2.11]{BB20} \n\talong with the fact that $\\widehat{Z_G}$ is homeomomorphic to $\\RR^k$ for $k:=\\dim Z_G\\ge 1$. \n\t\n\tHence we may assume $0\\not\\in\\sigma(D\\vert_\\zg)$ from now on, without any loss of generality. \n\t\n\tStep 2: \n\tLet $A:=\\{\\ee^{tD}\\mid t\\in\\RR\\}\\hookrightarrow\\Aut(\\ng)=\\Aut(N)$, \n\twhere $N:=(\\ng,\\cdot)$ is the simply connected Lie group associated with~$\\ng$. \n\tWe regard $A$ as an abelian Lie group that is isomorphic either to $(\\RR,+)$ since $D\\ne0$ by Step~1. \n\t(See also \\eqref{NC_eq1}.) \n\tAlso let $Z=(\\zg,\\cdot)=(\\zg,+)$, the center of $N$. \n\t\n\tWe have the semidirect product of Lie groups $G=N\\rtimes A$, hence $C^*(G)=C^*(N)\\rtimes A$, which carries a natural dual action $\\widehat{A}\\times C^*(G)\\to C^*(G)$. \n\tWe then obtain the composition of continuous open maps \n\t$$\\Phi_1\\colon \\Prim(G) \\to(\\Prim(G)\/\\widehat{A})^\\approx\n\t\\simeq(\\Prim (N)\/A)^\\approx \n\t\\to (\\widehat{Z}\/A)^\\approx,$$\n\twhere the left-most map is the quasi-orbit map corresponding to the natural action $\\widehat{A}\\times\\Prim(G)\\to\\Prim(G)$, \n\tthe middle homeomorphism is given by \\cite[Cor. 2.5]{GL86}, \n\twhile the right-most map is obtained as in the proof of \\cite[Prop. 4.7]{BB20} using the fact that the restriction mapping $R^N\n\t\\colon \\Prim(N) \\to\\widehat{Z}$ is not only continuous and open by \\cite[Lemma 2.11]{BB20}, but also $\\Aut(N)$-equivariant. \n\t\n\tStep 3: \n\tThe canonical homeomorphism $E\\colon \\zg^*\\to \\widehat{Z}$, $\\xi\\mapsto\\ee^{\\ie\\xi}$, intertwines the group actions \n\t$$\\RR\\times\\zg^*\\to\\zg^*,\\quad (t,\\xi)\\mapsto\\xi\\circ\\ee^{tD}$$\n\tand \n\t$$\\RR\\times\\widehat{Z}\\to\\widehat{Z},\\quad (t,\\chi)\\mapsto\\chi\\circ\\ee^{tD}\\vert_\\zg$$\n\thence we obtain the homeomorphism \n\t$$E^\\approx\\colon(\\zg^*\/\\RR)^\\approx\\to(\\widehat{Z}\/\\RR)^\\approx.$$\n\tOn the other hand, the hypothesis~\\eqref{NC_eq2} show that we may use Lemma~\\ref{NC_lemma} for $T:=(D\\vert_\\zg)^*\\in\\End(\\zg^*)$ to obtain a continuous open mapping $\\Psi\\colon (\\widehat{Z}\/\\RR)^\\approx\\to\\RR$, \n\tand then the mapping $\\Phi_2:=\\Psi\\circ (E^\\approx)^{-1}\\colon (\\widehat{Z}\/\\RR)^\\approx\\to\\RR$ is continuous and open. \n\tFInally, using the mapping $\\Phi_1$ from Step~2, we obtain the continuous open mapping $\\Phi:=\\Phi_2\\circ\\Phi_1\\colon\\Prim (C^*(G))\\to\\RR$, as required. \n\\end{proof}\n\n\\begin{corollary}\\label{NC_cor1}\n\tIn Proposition~\\ref{NC}, the topological space $\\Prim(G)$ contains no non\\-empty quasi-compact open subsets, and the $C^*$-algebra $C^*(G)$ is AF-embeddable. \n\\end{corollary}\n\n\\begin{proof}\n\tAny nonempty quasi-compact open subset of $\\Prim(G)$ would be mapped via $\\Phi$ onto a nonempty compact open subset of $\\RR$, but there are no such subsets of the connected noncompact space~$\\RR$. \n\tMoreover, $C^*(G)$ is nuclear since $G$ is an amenable group. \n\tOne can then use \\cite[Cor. B]{Ga20} to see that $C^*(G)$ is AF-embeddable. \n\\end{proof}\n\n\n\n\\subsection{The semidirect product $H_n\\rtimes \\RR$}\nWe now give the simplest example of exponential solvable Lie group whose primitive ideal space has finite open subsets and whose $C^*$-algebra is nevertheless AF-embeddable. (See \\eqref{Heis_item4} in Proposition~\\ref{Heis}.)\n\nLet $ H_n$ be the $(2n+1)$-dimensional Heisenberg group with its Lie algebra\n$\\hg:=\\hg_n=\\spa\\{Z,Y_1,\\dots,Y_n,X_1,\\dots,X_n\\}$, \nwhere\n$$\n[Y_j,X_j]=Z\n$$ \nfor $j=1,\\dots,n$.\nDenote by $\\zg: =\\RR Z$ the centre of $\\hg_n$. \nFor any $D\\in\\Der(\\hg_n)$ there exists $d_\\zg\\in\\RR$ with \n$D\\vert_\\zg=d_\\zg\\id_\\zg$, hence there exists an operator \n $$D\/\\zg\\colon\\hg_n\/\\zg\\to\\hg_n\/\\zg, \\quad V+\\zg\\mapsto D(V)+\\zg.$$\nWe also define $\\alpha_D\\colon\\RR\\to\\Aut(H_n)$, \n$t\\mapsto\\exp (tD)$. \n\n\\begin{proposition}\\label{Heis} \nLet $D\\in\\Der(\\hg_n)$ be such that $\\sigma(D)\\cap\\ie\\RR\\subseteq\\{0\\}$, \nand let $G_{n,D}:=H_n \\rtimes_{\\alpha_D}\\RR$ be the corresponding semidirect product.\nThen we have \n\\begin{enumerate}[{\\rm(i)}]\n\\item\\label{Heis_item1} If $d_\\zg=0$ then $C^*(G_{n,D})$ is AF-embeddable for every $n\\ge 1$ and there is no nonempty quasi-compact open subset of $\\Prim(G_{n,D}) $.\n\\item\\label{Heis_item1.5}\n If $d_\\zg\\ne 0$, then there are two open points in $\\Prim(G_{n,D})$, for every $n\\ge 1$. \n\\item\\label{Heis_item2} \nIf $d_\\zg\\ne 0$ and there exists $\\epsilon \\in \\{-1, 1\\}$ such that \n$\\epsilon \\Re\\, z> 0$ for all $z\\in \\sigma(D\/\\zg)$,\n then $C^*(G_{n,D})$ is not stably finite for every $n\\ge 1$. \n\\item\\label{Heis_item3} If $d_\\zg\\ne 0$ and $n\\in 2\\ZZ$, then $C^*(G_{n,D})$ is not stably finite. \n\\item\\label{Heis_item4} If $d_\\zg\\ne 0$, $n\\in 2 \\ZZ +1$, and there are $z_1, z_2\\in \\sigma(D\/\\zg)$ with \n$\\Re\\, z_1\\le 0\\le\\Re\\, z_2$, then $C^*(G_{n,D})$ is AF-embeddable. \n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nAssertion \\eqref{Heis_item1} follows for Corollary~\\ref{NC_cor1}, \\eqref{Heis_item1.5} from Lemma~\\ref{extra}, \n\\eqref{Heis_item2} is a consequence of Corollary~\\ref{cf-cor8} and Example~\\ref{ax+b}.\nAssertion \\eqref{Heis_item3} follows from \\eqref{Heis_item1.5} along with Corollary~\\ref{solv-4n+2}, and \\eqref{Heis_item4} can be obtained using Corollary~\\ref{cf-cor8} and Example~\\ref{ax+b}.\n\\end{proof}\n\n\n\n\n\n\\subsection{Two more classes of examples}\n\n\nWe start with a lemma that is essentially a by-product of \\cite{Sp88}.\nWe prove it here for completeness, as we do not have a reference for this very result, and it is needed for Example~\\ref{N6N15}, via\nLemma~\\ref{special}\nWe denote by $\\mathcal{N}$ the class of separable nuclear $C^*$-algebras to which the universal coefficient theorem applies. \n(See for instance \\cite{RoSc87}.)\nThe class~$\\mathcal{N}$ contains the $C^*$-algebras of all simply connected solvable Lie groups, since they are obtained by iterated crossed products by actions of the group~$\\RR$, starting from the 1-dimensional $C^*$-algebra. \n\n\\begin{lemma}\\label{embed}\nLet $0\\to\\Ic\\to\\Ac\\to\\Ac\/\\Ic\\to0$ be an exact sequence of $C^*$-algebras satisfying the following conditions: \n\\begin{enumerate}[{\\rm(i)}]\n\t\\item\\label{embed_item1} The $C^*$-algebra $\\Ac\/\\Ic$ belongs to the class $\\mathcal{N}$. \n\t\\item\\label{embed_item2} The $C^*$-algebras $\\Ic$ and $\\Ac\/\\Ic$ are AF-embeddable. \n\t\\item\\label{embed_item3} The index map $\\delta_1\\colon K_1(\\Ac\/\\Ic)\\to K_0(\\Ic)$ vanishes. \n\\end{enumerate}\nThen the $C^*$-algebra $\\Ac$ is AF-embeddable. \n\\end{lemma}\n\n\\begin{proof}\nStep 1 (reducing to essential ideals): By \\cite[Lemma 1.12]{Sp88} and its proof we obtain a $C^*$-algebra $\\Ac'$ that fits in a commutative diagram \n$$\\xymatrix{\n\\Ic \\ar[r] \\ar@{^{(}->}[d] & \\Ac\\ar[r] \\ar@{^{(}->}[d] & \\Ac\/\\Ic \\ar[d] \\\\\n\\Ic\\otimes\\Kc \\ar[r] & \\Ac'\\ar[r] & \\Ac\/\\Ic\n}\n$$\nwhere $\\Ic\\otimes\\Kc$ embeds as an essential ideal of $\\Ac'$, \nthe first two vertical arrows give rise to group isomorphisms $K_*(\\Ic)\\simeq K_*(\\Ic\\otimes\\Kc)$ ($\\simeq K_*(\\Ic)$) and $K_*(\\Ac)\\simeq K_*(\\Ac')$, while the right-most vertical arrow is an automorphism of $\\Ac\/\\Ic$. \nIt then follows by the hypothesis~\\eqref{embed_item3} along with the naturality of the index map (cf. \\cite[Prop. 9.1.5]{RLL00}) \nthat the index map $\\delta_1\\colon K_1(\\Ac\/\\Ic)\\to K_0(\\Ic\\otimes\\Kc)$ of the bottom horizontal line in the above diagram vanishes. \n\nStep 2 (reducing to AF ideals): \nSince $\\Ic$ is AF-embeddable, it follows that $\\Ic\\otimes\\Kc$ is AF-embeddable, hence there exists an embedding $\\Ic\\otimes\\Kc\\hookrightarrow\\widetilde{\\Jc}$, where $\\widetilde{\\Jc}$ is an AF-algebra. \nLet $\\Jc$ be the hereditary sub-$C^*$-algebra of $\\widetilde{\\Jc}$ generated by $\\Ic\\otimes\\Kc$. \nSince $\\widetilde{\\Jc}$ is an AF-algebra, it then follows by \\cite[Th. 3.1]{Ell76} that $\\Jc$ is an AF-algebra. \nOn the other hand $\\Jc=\\{bcb\\mid 0\\le b\\in\\Ic\\otimes\\Kc,\\ c\\in\\widetilde{\\Jc}\\}$ by \\cite[Cor. II.5.3.9]{Bl06}, \nwhich directly implies that every approximate unit of $\\Ic\\otimes\\Kc$ \nis an approximate unit for~$\\Jc$, too. \nThat is, the embedding $\\Ic\\otimes\\Kc\\hookrightarrow\\Jc$ is approximately unital in the sense of \\cite[Def. 1.10]{Sp88}. \nTherefore we may use \\cite[Rem. 1.11]{Sp88} \nto obtain a commutative diagram \n$$\\xymatrix{\n\t\\Ic\\otimes\\Kc \\ar[r] \\ar@{^{(}->}[d] & \\Ac'\\ar[r] \\ar@{^{(}->}[d] & \\Ac\/\\Ic \\ar[d] \\\\\n\t\\Jc \\ar[r] & \\Ac'+\\Jc\\ar[r] & \\Ac\/\\Ic\n}\n$$\nwhere the right-most vertical arrow is an automorphism of $\\Ac\/\\Ic$. \nSince the index map $\\delta_1\\colon K_1(\\Ac\/\\Ic)\\to K_0(\\Ic\\otimes\\Kc)$ \nof the upper line vanishes by Step~1, \nit then follows by the naturality of the index map \nthat the index map $\\delta_1\\colon K_1(\\Ac\/\\Ic)\\to K_0(\\Jc)$ of the bottom horizontal line in the above diagram vanishes as well. \nNow, since we have seen above that $\\Jc$ is an AF-algebra, it follows by \\cite[Lemma 1.13]{Sp88} applied to the short exact sequence \n$0\\to\\Jc \\to \\Ac'+\\Jc\\to \\Ac\/\\Ic\\to0$ \nthat $\\Ac'+\\Jc$ is AF-embeddable. \n(The $C^*$-algebra $\\Ac'+\\Jc$ belongs to the class~$\\mathcal{N}$ by the two-out-of-three property of that class mentioned in \\cite[V.1.5.4]{Bl06}, so all the hypotheses of \\cite[Lemma 1.13]{Sp88} are satisfied as stated.)\n\nStep 3: The above Steps 1--2 give the embeddings $\\Ac\\hookrightarrow\\Ac'\\hookrightarrow\\Ac'+\\Jc$ \nalong with the fact that $\\Ac'+\\Jc$ is AF-embeddable, \nhence $\\Ac$ is AF-embeddable as well. \n\\end{proof}\n\n \n\n\n\n\n\\begin{lemma}\\label{free_L1}\nLet $\\ng$ be a nilpotent Lie algebra with its centre $\\zg$, \nand assume that $D\\in\\Der(\\ng)$. \nIf there exists $\\xi\\in\\ng^*\\setminus\\zg^\\perp$ \nwith $\\xi\\circ(\\exp D)\\in \\Oc_\\xi$, then \n$\\sigma(D\\vert_\\zg)\\cap 2\\pi\\ie \\ZZ\\ne\\emptyset$. \n\\end{lemma}\n\n\\begin{proof}\nBy hypothesis, there exists $X\\in\\gg$ with $\\xi\\circ(\\exp D)=\\xi\\circ\\exp(\\ad_\\gg X)\\in\\gg^*$. \nThis implies \n$\\xi\\circ(\\exp D)\\vert_{\\zg}=\\xi\\circ\\exp(\\ad_\\gg X)\\vert_\\zg\\in\\zg^*$. \nFor every $Y\\in\\zg$ we have $\\exp(\\ad_\\gg X)Y=Y$ \nand on the other hand since $D$ is a derivation, $D(\\zg)\\subseteq\\zg$. \nTherefore $\\xi\\circ \\exp(D\\vert_\\zg)=\\xi\\vert_\\zg$. \nSince $\\xi\\in\\ng^*\\setminus\\zg^\\perp$, that is, $\\xi\\vert_\\zg\\ne0$, \nwe then obtain $1\\in\\spec(\\exp(D\\vert_\\zg))$. \nTherefore, by the spectral mapping theorem, there exists $w\\in\\sigma(D\\vert_\\zg)$ with $\\exp w=1$, that is, $w\\in2\\pi\\ie \\ZZ$. \n\\end{proof}\n\n\\begin{lemma}\\label{free_L2}\nAssume that $\\ng$ is a nilpotent Lie algebra with its centre $\\zg$, \nand let $X:=(\\ng^*\\setminus\\zg^\\perp)\/N$ be endowed with its quotient topology. \nThen for arbitrary $D\\in\\Der(\\ng)$ the map\n$$\\alpha\\colon X\\times \\RR\\to X,\\quad \n(\\Oc_\\xi,t)\\mapsto \\alpha_t(\\Oc_\\xi):=(\\exp(tD))^*(\\Oc_\\xi)=\\Oc_{\\xi\\circ\\exp(tD)}.$$\nis well defined and a continuous right action. \nMoreover, \n\\begin{enumerate}[\\rm (i)]\n\\item\\label{free_L2_i} if\n$\\sigma(D\\vert_\\zg)\\cap \\ie \\RR=\\emptyset$, \nthen the group action $\\alpha$ is free; \n\\item\\label{free_L2_ii} if $X$ is Hausdorff and \n there exists $\\epsilon \\in \\{-1, 1\\}$ such that \n$\\epsilon \\Re\\, z> 0$ for every $z\\in \\sigma(D\\vert_\\zg)$, \n the \n action \n$\\alpha$ is proper.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\tIn order to check the equality \n\t\\begin{equation}\\label{free_L2_proof_eq1}\n\t(\\exp(tD))^*(\\Oc_\\xi)=\\Oc_{\\xi\\circ\\exp(tD)}\n\t\\end{equation}\nwe use that for every $Y\\in \\ng$ and every $\\gamma\\in\\Aut(\\ng)$ one has $\\gamma\\circ \\exp(\\ad_\\ng Y)\\circ\\gamma^{-1}=\\exp(\\ad_\\ng \\gamma(Y))$. \nTherefore, for $\\gamma:=\\exp(tD)^{-1}$, \n$$\\xi\\circ\\exp(\\ad_\\ng Y)\\circ \\gamma^{-1}\n=\\xi\\circ\\gamma^{-1}\\circ \\exp(\\ad_\\ng \\gamma(Y))\\in\\Oc_{\\xi\\circ\\gamma^{-1}}.$$\nSince the mapping $\\gamma\\colon\\ng\\to\\ng$ is bijective, \nwe then directly obtain~\\eqref{free_L2_proof_eq1}. \n\nIt is clear that $\\alpha$ is a group action of the abelian group $(\\RR,+)$, and its continuity follows by the commutative diagram \n$$\\xymatrix{\n(\\ng^*\\setminus\\zg^\\perp)\\times\\RR \\ar[d]_{q\\times\\id_{\\RR}} \\ar[r]& \\ng^*\\setminus\\zg^\\perp \\ar[d]^{q} \\\\\nX \\times\\RR \\ar[r]^{\\alpha} & X\n}$$\nwhere $q\\colon \\ng^*\\setminus\\zg^\\perp\\to X$, $q(\\xi):=\\Oc_\\xi$ is the quotient map defined by the coadjoint action of $N$, \nwhile the upper horizontal arrow is defined by $(\\xi,t)\\mapsto\\xi\\circ\\exp(tD)$ and is clearly continuous.\n\n\\eqref{free_L2_i} Assume that the group action $\\alpha$ is not free, \nthat is, there exist $t\\in\\RR^\\times$ and $\\xi\\in \\ng^*\\setminus\\zg^\\perp$ \nwith $\\alpha_t(\\Oc_\\xi)=\\Oc_\\xi$. \nBy \\eqref{free_L2_proof_eq1}, we then have $\\Oc_{\\xi\\circ\\exp(tD)}=\\Oc_\\xi$, \nthat is, $\\xi\\circ\\exp(tD)\\in\\Oc_\\xi$. \nThen Lemma~\\ref{free_L1} shows that $\\sigma(tD\\vert_\\zg)\\cap 2\\pi\\ie \\ZZ\\ne\\emptyset$, in particular $\\sigma(D\\vert_\\zg)\\cap \\ie \\RR\\ne\\emptyset$. \n\n\\eqref{free_L2_ii} \n$X$ is a locally compact Hausdorff space.\n\nWithout losing the generality we assume that $\\Re\\, z > 0$ for every $z\\in \\sigma(D\\vert_\\zg)$. \nLet $\\xi,\\eta\\in \\ng^*\\setminus \\zg^{\\perp}$, $ (\\xi_j)_{j\\ge 1}$ be a sequence in \n $\\ng^*\\setminus \\zg^\\perp $, and $(t_j)_{j\\ge1}$ be a sequence in~$\\RR$ such that \n $\\Oc_{\\xi_j} \\to \\Oc_\\xi $ \n\n\nand $\\alpha_{t_j}(\\Oc_{\\xi_j})\\to \\Oc_\\eta $ in~$X$. \n\nAssume that $(t_j)_{j\\ge 1}$ has no limit points, hence it is not bounded. \n It follows that there is a subsequence $(t_{j_k})_{k\\ge 1}$ such that $t_{j_k} \\to +\\infty$ or $t_{j_k} \\to -\\infty$.\n Since $\\Oc_{\\xi_j} \\to \\Oc_\\xi $ in $X$, there is $\\xi'_j \\in \\Oc_{\\xi_j} $ such that \n $\\xi'_j \\to \\xi$, thus $\\xi'_j \\vert_{\\zg}= \\xi_j\\vert_{\\zg} \\to \\xi\\vert_{\\zg}$. \n Similarly, since $\\alpha_{t_j}(\\Oc_{\\xi_j})= \\Oc_{\\xi_j \\circ \\ee^{t_j D}}\\to \\Oc_\\eta $ in $X$, \n it follows that $\\xi_j \\circ \\ee^{t_j D}\\vert_{\\zg} \\to \\eta\\vert_{\\zg}$. \n Assume that $t_{j_k}\\to -\\infty$. \n Then $\\xi_{j_k} \\circ \\ee^{t_{j_k} D}\\vert_{\\zg} \\to 0$ and we get that $\\eta\\vert_{\\zg}=0$. \n This is not possible since $\\eta \\in \\ng^* \\setminus \\zg^\\perp$. \n If $t_{j_k}\\to +\\infty$,\nthen $\\xi_{j_k} \\circ \\ee^{t_{j_k} D}\\vert_{\\zg}\\to +\\infty$. \nThis is again impossible since $\\xi_{j_k} \\circ \\ee^{t_{j_k} D}\\vert_{\\zg}\\to \\eta\\vert_\\zg$. \nTherefore the sequence $(t_j)_{j\\ge1}$ must have a limit point, hence the action $\\alpha$ is proper. \n(See \\cite[Lemma~3.42]{Wi07}.)\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{special}\nLet $H$ be a nilpotent Lie group with its Lie algebra $\\hg$ and $\\zg_\\hg$ the centre of $\\hg$.\nAssume the following conditions hold:\n\\begin{enumerate}[{\\rm (i)}]\n\\item\\label{cond_i} $\\hg$ is two-step nilpotent and $[\\hg, \\hg]=\\zg_\\hg$.\n\\item\\label{cond_ii} The non-trivial coadjoint orbits of $M$ have the same dimension $d$, that is, there is \n$d\\in \\NN$ such that \n$$ \\hg^*\/H= (\\hg^*\/H)_d \\sqcup [\\hg, \\hg]^{\\perp}.$$\nHere $(\\hg^*\/H)_d$ denotes the space of coajoint orbits of $H$ of dimension $d$. \n\\end{enumerate}\nLet $D\\in \\Der(\\hg)$ be such that there exist $z_1, z_2 \\in \\sigma(D)$ such that $(\\Re z_1) (\\Re z_2) \\le 0$.\nThen $C^*(H \\rtimes_D \\RR)$ is AF-embeddable. \n\\end{lemma}\n\n\\begin{proof}\nAssume first that there are $z_1, z_2 \\in \\sigma(D\\vert _{\\zg_\\hg})$ such that $(\\Re z_1) (\\Re z_2) \\le 0$. \nThen $C^*(H\\rtimes_D \\RR)$ is AF-embeddable, by Proposition~\\ref{NC}, and this proves the lemma in this case. \n\n\nNext assume that $\\epsilon\\in \\{-1, 1\\}$ such that \n\\begin{equation}\\label{roots_1}\n\\sigma(D \\vert_{\\zg_\\hg}) \\subset \\epsilon (0, \\infty) +\\ie \\RR. \n\\end{equation}\nDenote by $D\/\\zg_\\hg$ the derivation of $\\hg \/\\zg_\\hg$ obtained from $D$. \nThen the hypothesis and \\eqref{roots_1} show that there exist\n\\begin{equation}\\label{roots_2}\nz_1, z_2 \\in \\sigma(D\/\\zg_\\hg) \\quad \\text{such that} \\; \\; \\Re z_1\\le 0\\le \\Re z_2.\n\\end{equation}\nThen there is a short exact sequence\n$$ 0 \\rightarrow \\Ic \\rightarrow C^*(H)\\rightarrow C^*(H\/Z_H)\\rightarrow 0, \n$$\nwhere $\\widehat{\\Ic} = (\\hg^*\/H)_d$ and $Z_M=\\exp \\zg_\\hg$ is the centre of $H$.\nSince $\\hg$ is two step nilpotent, hence it has only flat orbits, it follows from \\cite[Lemma~6.8]{BBL17} that $\\Ic$ has continuous trace. \nSince $ C^*(H\/Z_H)$ is invariant under automorphisms of $H$, $\\Ic$ is $\\Aut(H)$ invariant as well. \nWe thus obtain the short exact sequence\n\\begin{equation}\\label{ses}\n0 \\rightarrow \\Ic\\rtimes \\RR \\rightarrow C^*(H\\rtimes_{D} \\RR) \\rightarrow C^*(H\/Z_H\\rtimes_{D\/\\zg_\\hg} \\RR )\\rightarrow 0, \n\\end{equation}\nHere, we have that $\\sigma(D) \\cap \\ie \\RR =\\emptyset$ \nand \\ref{roots_1}, hence, by Lemma~\\ref{free_L2}, the continuous action \n$\\alpha_{D} \\colon \\RR \\times \\widehat{\\Ic} \\to \\widehat{\\Ic}$ is free and proper. \nThus $\\Ic \\rtimes_{D} \\RR$ has continuous trace, hence it is AF-embeddable. \nIt follows from condition \\eqref{roots_2} and Example~\\ref{ax+b} that \n$C^*(H\/Z_H\\rtimes_{D\/\\zg_\\hg} \\RR)$ is AF-embeddable as well. \nIf in addition $\\dim(\\hg\/ \\zg_\\hg)$ is odd, then $K_1(C^*(H\/Z_M\\rtimes_{D\/\\zg_\\hg} \\RR) ) = K_0 (C^*(H\/Z_H))=0$, hence the index \nmap corresponding to \\eqref{ses} vanishes. \nTherefore, by Lemma~\\ref{embed}, $C^*(H\\rtimes_{D} \\RR)$ is AF-embeddable. \n\\end{proof}\n\n\n\\begin{remark}\n\\normalfont\nIn the above Lemma, if $\\dim(\\hg\/ \\zg_\\hg)$ is even, the index map corresponding to \\eqref{ses} may not vanish, and \n$C^*(H\\rtimes_{D} \\RR)$ may not be AF-embeddable, or even stably finite, as we will see below.\n\\end{remark}\n\n\n\n\n\n\\begin{remark}\\label{deriv-ext}\n\\normalfont\nLet $\\hg$ be a finite-dimensional real Lie algebra with a symplectic structure $\\omega\\colon\\hg\\times\\hg\\to\\RR$, and define the corresponding central extension \n$\\ng:=\\hg\\dotplus_\\omega\\RR$ with its Lie bracket $[(X_1,t_1),(X_2,t_2)]:=([X_1,X_2],\\omega(X_1,X_2))$ for all $X_1,X_2\\in\\hg$ and $t_1,t_2\\in\\RR$. \nFor any $D_0\\in\\Der(\\hg)$ and $a_0\\in\\RR$ we define the linear map $D\\colon\\ng\\to\\ng$, $D(X,t):=(D_0X,a_0t)$. \nThen $D\\in\\Der(\\ng)$ if and only if \n$$\\omega(D_0X_1,X_2)+\\omega(X_1,D_0X_2)=a_0\\omega(X_1,X_2) \n\\text{ for all }X_1,X_2\\in\\hg.$$\n\\end{remark}\n\n\\begin{lemma}\\label{N6N15-lemma}\n\tLet $\\hg$ be the real Lie algebra with a basis $X_1,X_2,X_3,Y_1,Y_2,Y_3$ satisfying the commutation relations $$[X_1,X_2]=Y_3,\\ [X_2,X_3]=Y_1,\\ [X_3,X_1]=Y_2.$$\n\t\\begin{enumerate}[{\\rm(i)}]\n\t\t\\item\\label{N6N15_item0} The centre of $\\hg$ is $\\zg:=\\spa\\{Y_1,Y_2,Y_3\\}$ and for every $\\xi\\in\\hg^*\\setminus\\zg^\\perp$ we have $\\dim\\Oc_\\xi=2$. \n\t\t\\item\\label{N6N15_item1} \n\t\tFor any $a_1,a_2,a_3\\in\\RR$ there exists a unique skew-symmetric bilinear functional \n\t\t$$\\omega\\colon\\hg\\times\\hg\\to\\RR$$ \n\t\tsatisfying $\\omega(X_j,Y_k)=\\delta_{jk}a_j$, $\\omega(X_j,X_k)=\\omega(Y_j,Y_k)=0$ for all $j,k\\in\\{1,2,3\\}$. \n\t\tMoreover, $\\omega$ is a symplectic structure of the Lie algebra $\\hg$ if and only if \n\t\t\\begin{equation}\\label{N6N15_item1_eq1}\n\t\ta_1+a_2+a_3=0\\text{ and }a_1a_2a_3\\ne0.\n\t\t\\end{equation}\n\t\t\\item\\label{N6N15_item2} \n\t\tFor any matrix $B=(b_{jk})_{1\\le j,k\\le3}\\in M_3(\\RR)$ there exists a unique derivation $D_B\\in\\Der(\\hg)$ satisfying $D_BX_j=\\sum\\limits_{k=1}^3b_{jk}X_k$ for $j=1,2,3$. \n\t\tIf $a_1,a_2,a_3\\in\\RR$ satisfy~\\eqref{N6N15_item1_eq1} and $\\omega$ is their corresponding symplectic structure of~$\\hg$ as in~\\eqref{N6N15_item1} above, then there exists $D\\in\\Der(\\hg\\dotplus_\\omega\\RR)$ with $D\\vert_\\hg=D_B$ \n\t\tif and only if \n\t\t\\begin{equation}\\label{N6N15_item2_eq2}\n\t\tb_{ij}(a_i-a_j)=0\\text{ for all }j,k\\in\\{1,2,3\\}\n\t\t\\end{equation}\n\t\tand if this is the case then $D(0,1)=(0,\\Tr B)$. \n\t\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\t\t\\eqref{N6N15_item0} This is well known. \n\n\\eqref{N6N15_item1} \nIt is straightforward to check that $\\omega$ is a 2-cocycle if and only if $a_1+a_2+a_3=0$, and on the other hand $\\omega$ is non-degenerate if and only if $a_1a_2a_3\\ne0$. \nTherefore, $\\omega$ is a symplectic structure of the Lie algebra $\\hg$ if and only if \\eqref{N6N15_item1_eq1} is satisfied. \n\n\\eqref{N6N15_item2} \nIn order to obtain a derivation $D_B\\colon\\hg\\to\\hg$ we define $D_BY_j:=[D_B X_r,X_s]+[X_r,D_BX_s]$ if $Y_j=[X_r,X_s]$. \nA straightforward computation then leads to the formula \n\\begin{equation}\\label{N6N15_proof_eq1}\nD_BY_j=-\\sum_{k\\ne j}b_{kj}Y_k+\\Bigl(\\sum_{k\\ne j}b_{kk})Y_j\n\\text{ for }j=1,2,3,\n\\end{equation}\nwhich further implies \n$$\\omega(D_B X_i,Y_j)+\\omega(X_i,D_B Y_j)=\n\\begin{cases}\n(\\Tr B)a_j=(\\Tr B)\\omega(X_j,Y_j)&\\text{ if }i=j,\\\\\nb_{ij}(a_i-a_j)&\\text{ if }i\\ne j.\n\\end{cases}\n$$\nSince $\\omega(X_i,Y_j)=0$ if $i\\ne j$, the assertion then follows by Remark~\\ref{deriv-ext}. \n\\end{proof}\t\t\n\t\t\n\t\t\n\t\t\\begin{theorem}\\label{N6N15}\nLet $\\hg$ be the real Lie algebra with a basis $X_1,X_2,X_3,Y_1,Y_2,Y_3$ satisfying the commutation relations $$[X_1,X_2]=Y_3,\\ [X_2,X_3]=Y_1,\\ [X_3,X_1]=Y_2, $$\n\t\tand , for $b_1, b_2, b_3 \\in \\RR$, $b_1+b_2+b_3\\ne 0$, let $D\\in \\Der(\\hg)$ be the unique derivation with \n\t\t$D X_j= b_{j}X_j$ for $j=1,2,3$.\n\t\tIf we denote $\\ng:=\\hg\\dotplus_\\omega\\RR$ and $N\\rtimes_D\\RR$ is the simply connected Lie group whose Lie algebra is $\\ng\\rtimes\\RR D$, \n\t\tthen the following assertions are equivalent: \n\t\t\\begin{itemize}\n\t\t\t\\item $C^*(N\\rtimes_D\\RR)$ is not AF-embeddable. \n\t\t\t\\item $C^*(N\\rtimes_D\\RR)$ is not stably finite. \n\t\t\t\\item There exists $\\epsilon\\in\\{\\pm1\\}$ with \n\t\t\t$\\epsilon z> 0$ for every $z\\in \\sigma(D)$. \n\t\t\t\\item There exists $\\epsilon\\in\\{\\pm1\\}$ with \n\t\t\t$\\epsilon b_j > 0$ for $j=1,2,3$. \n\t\t\\end{itemize}\n\\end{theorem}\n\n\\begin{proof} \nThe last two assertions in the statement are clearly equivalent since \n$$\\sigma(D)=\\{b_j\\mid j=1,2,3\\}\\cup\\Bigl\\{\\sum_{k\\ne j}b_k\\mid j=1,2,3\\Bigr\\}\n\\cup\\{b_1+b_2+b_3\\}$$\nby \\eqref{N6N15_item2} and \\eqref{N6N15_proof_eq1}. \n\nIf there exists $\\epsilon\\in\\{\\pm1\\}$ with \n$\\epsilon z> 0$ for every $z\\in \\sigma(D)$, then $C^*(N\\rtimes_D\\RR)$ is not stably finite by \nTheorem~\\ref{prop-cf4}. \n\nConversely, if there exists no $\\epsilon\\in\\{\\pm1\\}$ with \n$\\epsilon b_j > 0$ for $j=1,2,3$, then \nit suffices to show that $C^*(H\\rtimes_D\\RR)$ is AF-embeddable,\nsince $\\dim(N\\rtimes_D\\RR)=8\\in 4\\ZZ$ and by using Corollary~\\ref{cf-cor8}\\eqref{cf-cor7-ii}, this implies that \n$C^*(N\\rtimes_D\\RR)$ is AF-embeddable.\nNow the fact that $C^*(H\\rtimes_D\\RR)$ is AF-embeddable follows\n\\eqref{N6N15_item0} and Lemma~\\ref{special}, and this finishes the proof. \n\\end{proof}\n\nFor Theorem~\\ref{N6N17} below, we recall that a \\emph{finite-dimensional real division algebra} is a finite-dimensional real vector space $\\KK$ endowed with a bilinear map $\\KK\\times\\KK\\to\\KK$, $(v,w)\\mapsto v w$, \nwhose corresponding linear mappings $v\\mapsto v w_0$ and $w\\mapsto v_0 w$ are injective (hence bijective) for all $v_0,w_0\\in\\KK\\setminus\\{0\\}$. \nIf this is the case, then $\\dim_\\RR\\KK\\in\\{1,2,4,8\\}$ by \\cite[Cor. 1]{BoMi58}, \nand these values of $\\dim_\\RR\\KK$ are realized for instance if $\\KK$ is the real field~$\\RR$, the complex field~$\\CC$, the quaternion field~$\\mathbb{H}$, and the octonion (non-associative) algebra~$\\mathbb{O}$, respectively. \n\nLet $\\KK$ be a finite-dimensional real division algebra and define \n$$\n\\begin{gathered}\n\\omega\\colon(\\KK^n\\times\\KK^n)\\times(\\KK^n\\times\\KK^n)\\to\\KK,\\\\\n\\omega((v_1,w_1),(v_2,w_2)):=\\sum_{k=1}^n(v_{1k}w_{2k}-v_{2k}w_{1k})\n\\end{gathered}\n$$\nfor $v_j=(v_{j1},\\dots,v_{jn}), w_j=(w_{j1},\\dots,w_{jn})\\in\\KK^n$, $j=1,2$. \nIt is clear that $\\omega((v_1,w_1),(v_2,w_2))=-\\omega((v_2,w_2),(v_1,w_1))$, \nhence we may define the \\emph{real} 2-step nilpotent Lie algebra \n\\begin{equation}\\label{hgK}\n\\hg_\\KK:=\\KK^n\\times\\KK^n\\times\\KK\n\\end{equation}\nwith its Lie bracket \n$$[(v_1,w_1,z_1),(v_2,w_2,z_2)]:=[(0,0,\\omega((v_1,w_1),(v_2,w_2)))].$$\nLet $H_\\KK$ be a connected, simply connected nilpotent Lie group whose Lie algebra is $\\hg_\\KK$.\n\n\\begin{lemma}\\label{N6N17-lemma}\nLet $\\KK$ be a finite-dimensional real division algebra, and define \n $\\zg:=\\{0\\}\\times\\{0\\}\\times\\KK\\subseteq\\hg_\\KK$. Then the following assertions hold: \n\\begin{enumerate}[{\\rm(i)}]\n\t\\item\\label{N6N17_item1} We have $[\\hg_\\KK,\\hg_\\KK]=\\zg$ and $\\zg$ is the centre of $\\hg$. \n\t\\item\\label{N6N17_item2} For every $\\xi\\in\\hg_\\KK^*\\setminus[\\hg_\\KK,\\hg_\\KK]^\\perp$ we have $\\hg_\\KK(\\xi)=\\zg$ and $\\dim\\Oc_\\xi=2n\\dim_\\RR\\KK$. \n\t\\item\\label{N6N17_item3} \n\tThe mapping\n\t$r_\\zg\\colon (\\hg_\\KK^*\\setminus\\zg^\\perp)\/H_\\KK\\to \\zg^*\\setminus\\{0\\}\n\t,\\quad \\Oc_\\xi\\mapsto\\xi\\vert_{\\zg}$\n\tis a well-defined homeomorphism. \n\t\\item\\label{N6N17_item4} If $a=(a_1,\\dots,a_n),b=(b_1,\\dots,b_n)\\in\\RR^n$, $c\\in\\RR$, and $D\\colon\\hg_\\KK\\to\\hg_\\KK$ is the $\\RR$-linear mapping defined by \n\t$$D(v,w,z):=((a_1v_1,\\dots,a_nv_n),(b_1w_1,\\dots,b_nw_n),cz), $$ \n\tthen $D\\in\\Der(\\hg_\\KK)$ if and only if $a_k+b_k=c$ for $k=1,\\dots,n$. \n\t\\end{enumerate}\n \\end{lemma}\n \n\n\\begin{proof}\n \\eqref{N6N17_item1} \nIt is clear that $[\\hg_\\KK,\\hg_\\KK]=\\zg$ and $[\\hg_\\KK,\\zg]=\\{0\\}$.\nIn order to prove that $\\zg$ is actually equal to the centre of $\\hg_\\KK$, \nlet us assume that there exists $x_0:=(v_0,w_0,z)\\in\\hg_\\KK\\setminus\\zg$ with $[x_0,\\hg_\\KK]=\\{0\\}$ and $v_0=(v_{01},\\dots,v_{0n}), w_0=(w_{01},\\dots,w_{0n})\\in\\KK^n$. \nSince $x_0\\not\\in\\zg$, there exists $j\\in\\{1,\\dots,n\\}$ with $v_{0j}\\in\\KK\\setminus\\{0\\}$ or $w_{0j}\\in\\KK\\setminus\\{0\\}$. \nIf for instance $v_{0j}\\ne0$, then we define $x:=(v,w,0)\\in\\hg$ \nwhere $v=0\\in\\KK^n$ and $w=(w_1,\\dots,w_n)\\in\\KK^n$ \nis given by $w_k=0$ if $k\\in\\{1,\\dots,n\\}\\setminus\\{j\\}$ and $w_j=v_{0j}\\in\\KK$, \nand we obtain $[x_0,x]=(0,0,v_{0j}v_{0j})\\in\\hg\\setminus\\{0\\}$, which is a contradiction with the assumption $[x_0,\\hg]=\\{0\\}$. \nThe case $w_j\\in\\KK\\setminus\\{0\\}$ can be discussed similarly. \n\n\\eqref{N6N17_item2} \nWe have $\\hg_\\KK(\\xi)=\\{x\\in\\hg\\mid[x,\\hg_\\KK]\\subseteq\\Ker\\xi\\}$, hence the inclusion $\\zg\\subseteq\\hg_\\KK(\\xi)$ follows by~\\eqref{N6N17_item1}. \nFor the converse inclusion, assume there exists $x_0:=(v_0,w_0,z)\\in\\hg_\\KK(\\xi)\\setminus\\zg$. \nSince $x_0\\not\\in\\zg$, there exists $j\\in\\{1,\\dots,n\\}$ with $v_{0j}\\in\\KK\\setminus\\{0\\}$ or $w_{0j}\\in\\KK\\setminus\\{0\\}$. \nIf for instance $v_{0j}\\ne0$, then for any $y=(v,w,0)\\in\\hg$ with $v=0\\in\\KK^n$ and $w=(w_1,\\dots,w_n)\\in\\KK^n$ \nwith $w_k=0$ if $k\\in\\{1,\\dots,n\\}\\setminus\\{j\\}$ we have \n$[x_0,x]=(0,0,v_{0j}w_j)$. \nHere $w_j\\in\\KK$ is arbitrary and $v_{0j}\\in\\KK\\setminus\\{0\\}$ hence, \nsince $\\KK$ is a division algebra, it follows that $\\zg\\subseteq[x_0,\\hg_\\KK]$. \nOn the other hand, we have by assumption $x_0\\in\\hg_\\KK(\\xi)$, hence \n$[\\hg_\\KK,\\hg_\\KK]=\\zg\\subseteq[x,\\hg_\\KK]\\subseteq\\Ker\\xi$, which is a contradiction with the hypothesis $\\xi\\in\\hg_\\KK^*\\setminus[\\hg_\\KK,\\hg_\\KK]^\\perp$. \n\nThe second assertion follows by the general equality $\\dim\\Oc_\\xi=\\dim(\\hg_\\KK\/\\hg_\\KK(\\xi))$. \n\n\\eqref{N6N17_item3} \nFor every $\\xi\\in\\hg_\\KK^*\\setminus[\\hg_\\KK,\\hg_\\KK]^\\perp$ we have $\\Oc_\\xi=\\xi+\\zg^\\perp$ by~\\eqref{N6N17_item3} hence the mapping $r_\\zg$ is well-defined and bijective. \nMoreover, if we define $r\\colon \\hg_\\KK^*\\setminus\\zg^\\perp\\to \\zg^*\\setminus\\{0\\}$, $\\xi\\mapsto\\xi\\vert_\\zg$, and $q\\colon \\hg_\\KK^*\\setminus\\zg^\\perp\\to(\\hg_\\KK^*\\setminus\\zg^\\perp)\/H_\\KK$, $\\xi\\mapsto \\Oc_\\xi$, \nthen $r_\\zg\\circ q=r$ and, since $r$ is a continuous open mapping and $q$ is a quotient mapping, it follows that $r_\\zg$ is continuous and open. \n\n\\eqref{N6N17_item4} \nThis assertion is straightforward. \n\\end{proof}\n\n\n\\begin{theorem}\\label{N6N17}\nLet $\\KK$ be a finite-dimensional real division algebra.\nAssume $a=(a_1,\\dots,a_n),b=(b_1,\\dots,b_n)\\in\\RR^n$, $c\\in\\RR$,\nare such that $a_k+b_k=c \\ne 0$ for $k=1,\\dots,n$.\nand let $D\\in \\Der(\\hg_\\KK)$ be the $\\RR$-linear mapping defined by \n\t$D(v,w,z):=((a_1v_1,\\dots,a_nv_n),(b_1w_1,\\dots,b_nw_n),cz)$.\nLet $H_\\KK\\rtimes_D\\RR$ the simply connected Lie group whose Lie algebra is $\\hg_\\KK\\rtimes\\RR D$, then \n\t$\\Pg(C^*(H_\\KK\\rtimes_D \\RR))\\ne\\{0\\}$. \n\tIf moreover $\\KK\\ne\\RR$, then $C^*(H_\\KK\\rtimes_D \\RR)$ is not stably finite. \n \\end{theorem}\n\n\\begin{proof} \nSince $H_\\KK$ is a nilpotent Lie group, we may use Kirillov's homeomorphism \n$\\widehat{H_\\KK}\\simeq\\hg_\\KK^*\/H_\\KK$ along with \\eqref{N6N17_item3} to obtain a short exact sequence \n$$0\\to\\Ic\\hookrightarrow C^*(H_\\KK)\\to C^*(H_\\KK\/Z)\\to 0$$\nwhere one has $*$-isomorphisms \n$\\Ic\\simeq \\Cc_0(\\zg^*\\setminus\\{0\\})\\otimes\\Kc$ \nand $C^*(H_\\KK\/Z)\\simeq \\Cc_0(\\zg^\\perp)$. \nThe derivation $D$ gives rise to a group action $\\alpha\\colon \\hg_\\KK^*\/H_\\KK\\times\\RR\\to\\hg_\\KK^*\/H_\\KK$, $(\\Oc_\\xi,t)\\mapsto \\Oc_{\\xi\\circ\\ee^{tD}}$. \nIf we fix any real scalar product on $\\zg$ and we denote by $S_{\\zg^*}$ its corresponding unit sphere, then we have the homeomorphism \n$$S_{\\zg^*}\\times\\RR\\to\\zg^*\\setminus\\{0\\},\\quad (\\eta,t)\\mapsto \\ee^{tc}\\eta$$\nsince $c\\in\\RR\\setminus\\{0\\}$, \nand then we obtain a $*$-isomorphism $\\Cc_0(\\zg^*\\setminus\\{0\\})\\simeq\\Cc(S_{\\zg^*})\\otimes \\Cc_0(\\RR)$. \nWe then obtain $*$-isomorphisms \n\\begin{align*}\n\\Ic\\rtimes_\\alpha\\RR\n& \\simeq (\\Cc_0(\\zg^*\\setminus\\{0\\})\\otimes\\Kc)\\rtimes_\\alpha\\RR\n\\simeq \\Cc(S_{\\zg^*})\\otimes (\\Cc_0(\\RR)\\rtimes_c\\RR)\\otimes\\Kc \\\\\n& \\simeq \\Cc(S_{\\zg^*})\\otimes \\Kc(L^2(\\RR))\\otimes\\Kc \\\\\n& \\simeq \\Cc(S_{\\zg^*})\\otimes\\Kc.\n\\end{align*}\nThe above crossed product $\\Cc_0(\\RR)\\rtimes_c\\RR$ is defined via the group action \n$$\\Cc_0(\\RR)\\times\\RR\\to \\Cc_0(\\RR), \\quad (\\varphi,t)\\mapsto \\varphi_t, \\text{ where }\\varphi_t(s):=\\varphi(s\\ee^{tc}),$$ \nhence one has a $*$-isomorphism \n$\\Cc_0(\\RR)\\rtimes_c\\RR\\simeq \\Kc(L^2(\\RR))$ since $c\\in\\RR\\setminus\\{0\\}$. \nThe above $*$-isomorphisms show that $\\Pg(\\Ic\\rtimes_\\alpha\\RR)\\ne\\{0\\}$. \nSince $\\Ic\\rtimes_\\alpha\\RR$ is an ideal of $C^*(H_\\KK)\\rtimes_\\alpha\\RR$, \nand $C^*(H_\\KK)\\rtimes_\\alpha\\RR\\simeq C^*(H_\\KK\\rtimes_D\\RR)$, we obtain \n$\\Pg(C^*(H_\\KK \\rtimes_D \\RR))\\ne\\{0\\}$. \n\nIf moreover $\\KK\\ne\\RR$, then $\\dim_\\RR\\KK\\in 2\\ZZ$, hence $\\dim(H_\\KK\\rtimes_D\\RR)\\in 2\\ZZ+1$, \n it follows that $C^*(H_\\KK\\rtimes_D \\RR)$ is not stably finite by Theorem~\\ref{4n+2}. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSimple Temporal Problems (STPs) provide a formal structure to describe possible time relations between events. These time relations feature in a wide variety of real-world problems \\cite{art:time21} and include precedences, maximum elapsed time, and a single release and due date per event. A major advantage of STPs is that they are solvable in polynomial time with standard shortest path methods~\\cite{art:stp}. Nevertheless, researchers and practitioners are still fairly limited with respect to what can be modeled using STPs. Even an otherwise simple feature such as multiple release and due dates per event cannot be expressed with STPs.\n\nAlternatively, Disjunctive Temporal Problems (DTPs) offer a much broader framework for describing time relations. However, this expressiveness is offset by the fact that they usually represent NP-Complete problems \\cite{art:dtps}. Simple Disjunctive Temporal Problems (SDTPs) are a primitive type of DTP which generalize STPs and retain efficient polynomial-time solution methods. SDTPs extend STPs by enabling multiple, non-overlapping release and due dates per event.\n\nThroughout the academic literature, SDTPs are referred to by many names: \\textsc{Star} class of problems, zero-one extended STPs and t\\textsubscript{2}DTPs. While it is difficult to know for sure why exactly so much terminology exists for the same problem, one could speculate that it might be that different researchers have each arrived at SDTPs from different angles. Although the particular reason for this terminological variance is not our main concern, it is clear that it results in a highly inefficient scenario. Researchers and practitioners alike end up being held back by the burdensome task of needing to discover what is known about SDTPs when the findings are catalogued under different names. Moreover, even when one does locate existing literature concerning SDTPs, those research papers are primarily theoretical in nature and provide neither a practical implementation of the methods nor empirical insights concerning how the behavior of those algorithms compares under different scenarios.\n\nWe have experienced precisely the situation outlined when trying to compare different approaches for scheduling tasks with multiple time windows in logistical problems such as vehicle routing with synchronizations \\cite{art:vrpms,art:delsynch}, pickup and delivery with transshipment \\cite{art:pdpt}, dial-a-ride with transfers \\cite{art:darpt} and truck driver scheduling with interdependent routes \\cite{art:ssvb-1}. SDTPs are an excellent model for scheduling in these problems. Nevertheless, one needs extremely efficient SDTP methods when solving these logistical problems via local search heuristics, for example. Given the fact that the literature is not only difficult to navigate but also lacks empirical results, we needed to (i) find existing methods, (ii) implement them and (iii) evaluate their advantages and limitations in different cases before employing SDTPs in our applications.\n\nThe goal of this paper is therefore fourfold. First, we propose a standard nomenclature to refer to SDTPs so that future researchers can refer to the same problem by the same name. Second, we explore existing methods and develop new algorithms to solve SDTPs that are capable of not only reducing the theoretical asymptotic worst-case time and space complexities but also ensuring good performance in practice. Third, we provide an empirical study alongside open source implementations of all of the techniques and also make our instances publicly available with the aim of avoiding duplicate work\\footnote{The complete code repository will be made available at a later date.}. Fourth and finally, we hope this paper will serve as a foundation for researchers and practitioners who would like to apply SDTPs in their work and build upon our research and implementations to easily achieve their goals.\n\n\\section{The simple disjunctive temporal problem} \\label{sec:sdtp}\n\nLet us begin by formally defining simple temporal problems, which will be useful when introducing simple disjunctive temporal problems. \n\n\\begin{definition}[Simple Temporal Problem \\cite{art:stp}]\\label{def:stp}\nA \\textit{Simple Temporal Network} (STN) is denoted $N=(T,C)$ where $T$ is the set of \\textit{variables} or \\textit{time-points} and $C$ is the set of binary constraints relating variables of $T$. A time-point $i \\in T$ has a closed domain $[l_i,u_i],\\ l_i,u_i \\in \\mathbb{R}$. Constraints in $C$ are \\textit{simple temporal constraints} given as a tuple $(i,j,w_{ij}) \\in C,\\ i,j \\in T,\\ w_{ij} \\in \\mathbb{R}$ which corresponds to Equation~\\ref{eq:stc}:\n\\begin{equation}\n s_i - s_j \\leq w_{ij} \\label{eq:stc}\n\\end{equation}\n\\noindent where $s_i,s_j \\in \\mathbb{R}$ denote the solution values assigned to time-points $i$ and $j$. \n\n The STP involves determining whether its associated STN is consistent. A network $N$ is consistent iff a \\textit{feasible schedule} or \\textit{solution} $s$ can be derived such that times $s_i$ assigned to each $i \\in T$ respect all constraints present in $C$ and the domains of each time-point.\n\\end{definition}\n\n \\citet{art:stp} showed that an STN can be represented with a distance graph where time-points $T$ are nodes and constraints $C$ are arcs connecting these nodes. First, let us associate a special time-point $\\alpha$ with domain $[0,0]$ to represent the beginning of the time horizon $s_\\alpha = 0$. This fixed point can be used to write unary constraints such as domain boundaries over time-points in $T$ as simple temporal constraints. For example, the bound $[l_i,u_i]$ for $i \\in T$ can be written as:\n \\begin{align*}\n s_\\alpha - s_i &\\leq -l_i\\\\\n s_i - s_\\alpha &\\leq u_i\n \\end{align*}\n \n We can then associate two distance graphs with STN $N=(T,C)$: the \\textit{direct graph} $G_D=(V,A_D)$ and the \\textit{reverse graph} $G_R=(V,A_R)$. For both of these graphs $V=T \\cup \\{\\alpha\\}$. Arc set $A_D= C \\cup \\{(\\alpha,i,-l_i), (i,\\alpha,u_i)\\ \\forall\\ i \\in T\\}$ for which an element $(i,j, w_{ij}) \\in A_D$ denotes an arc from node $i$ to $j$ with weight $w_{ij}$. Meanwhile, $A_R$ is the same as $A_D$ but with the direction of each arc reversed: $(i,j,w_{ij}) \\in A_D$ is $(j,i,w_{ij}) \\in A_R$.\n\nDetermining consistency of an STN then reduces to verifying the existence of negative-cost cycles in either $G_D$ or $G_R$. If there is no negative-cost cycle, the shortest path distance $\\tau_{\\alpha i}$ from node $\\alpha \\in V$ to every other node $i \\in V \\backslash \\{\\alpha\\}$ provides a feasible schedule. When computed over $G_D$, $s_i = -\\tau_{\\alpha i}$ provides the \\textit{earliest feasible schedule}. Meanwhile, when computed over $G_R$, $s_i=\\tau_{\\alpha i}$ provides the \\textit{latest feasible schedule}. The earliest feasible schedule can be defined as the solution $s$ for which given any other feasible solution $s'$ to the SDTP, it holds that $s_i \\leq s'_i,\\ \\forall\\ i \\in T$. Similarly, for the latest feasible schedule it holds that $s_i \\geq s'_i,\\ \\forall\\ i \\in T$.\n\nThere are many algorithms that can be used to detect negative-cost cycles quickly in a distance graph \\cite{art:sp-fp}. One of the most simple is \\textsc{BellmanFord} \\cite{book:cormen}. Indeed, this is an algorithm employed by most methods to solve SDTPs. For the remainder of this paper, we always consider \\textsc{BellmanFord} to refer to its implementation as a label-correcting algorithm using a first-in, first-out queue \\cite{art:sp-fp}.\n\nSTPs can only accommodate time-points for which the domain is a single interval. In order to tackle problems where time-points may be assigned values in one of several disjunctive intervals, a more expressive model is required. Let us now turn our attention to the main problem in this paper: the simple disjunctive temporal problem.\n\n\\begin{definition}[Simple Disjunctive Temporal Problem]\\label{def:sdtp}\nA \\textit{Simple Disjunctive Temporal Network} (SDTN) is denoted $N=(T,C)$, where $T$ is the set of time-points and $C=C_1 \\cup C_2$ are the constraints over these time-points. Based on the classification introduced by \\citet{art:kra}, constraints in $C_1$ are Type 1 while those in $C_2$ are Type 2.\n\\begin{enumerate}\n \\item[] (Type 1) Simple temporal constraints $(i,j,w_{ij}) \\in C_1,\\ i,j \\in T$ representing Equation \\ref{eq:stc}\n \\item[] (Type 2) Simple disjunctive constraints $(i,D_i) \\in C_2$, where $i \\in T$ and $D_i$ is a list of \\textit{intervals} or \\textit{domains} denoted $[l^c_i,u^c_i] \\in D_i,\\ l^c_i,u^c_i \\in \\mathbb{R}$ representing the disjunction\n \\begin{equation*}\n \\bigvee_{[l^c_i,u^c_i] \\in D_i} (l^c_i \\leq s_i \\leq u^c_i) \n \\end{equation*}\n Note that $C_2$ includes unary constraints ($|D_i|=1$). $T_D \\subseteq T$ denotes the set of all time-points for which $|D_i| > 1$. \n\\end{enumerate}\n\nIn order to solve the SDTP, we need to determine whether SDTN $N$ is consistent. Similar to STPs, $N$ is consistent iff a feasible solution $s$ can be derived which respects both constraints $C_1$ and $C_2$. \n\n\\end{definition}\n\n\n\nWe assume that domains in $D_i$ are sorted in ascending order \\cite{art:kra,art:cra}. Let $K = \\max_{(i,D_i) \\in C_2} |D_i|$ denote the largest number of domains for any given time-point and $\\omega = \\sum_{(i,D_i) \\in C_2} |D_i|$ denote the total number of domains in the instance. Let us further denote by $L(D_i)$ and $U(D_i)$ the lower and upper bound values in all domains of $i \\in T$, respectively. The \\textit{global boundary} of $i$ is given by $[L(D_i),U(D_i)]$ such that $s_i$ must belong to this boundary. However, some values within these bounds can still be infeasible. In other words: the domains of time-points are not continuous.\n\nThe existence of Type 2 constraints means we cannot solve the problem directly via shortest paths. Nevertheless, we can use the global boundaries of the time-points to redefine graphs $G_D$ and $G_R$ with $A_D=C_1 \\cup \\{(\\alpha,i,-L(D_i)), (i,\\alpha,U(D_i))\\ \\forall\\ (i,D_i) \\in C_2\\}$ and $A_R$ ($A_D$ with all arc directions reversed). These graphs can be used to compute \\textit{lower-} and \\textit{upper-bound solutions} for the SDTP while employing shortest path algorithms in the same way as for STPs. If a negative cycle exists in $G_D$ or $G_R$ when considering these global boundaries, then the associated SDTP instance is definitely infeasible.\n\nOnce a domain $d_i \\in D_i$ has been selected for each time-point $(i,D_i) \\in C_2$, the SDTP reduces to an STP. Indeed, some of the special-purpose algorithms available in the literature exploit this problem structure to solve SDTP instances. Section \\ref{sec:algs} will discuss this further.\n\n\n\\subsection{Related problems and classification}\n\nAs noted in this paper's introduction, the SDTP has been referred to by various names in the literature. \\citet{art:ult} term it the \\textsc{Star} class of problems because the multiple domains per time-point create connections to the beginning of time $\\alpha$, which resembles the shape of a star. \\citet{art:kumar-estp} refers to the problem as zero-one extended STPs, where subintervals of a time-point's domain are associated with a weight that is either 0 when the interval is infeasible, or~1 when the interval is feasible. An SDTP solution has therefore been constructed when the sum of the weights of selected intervals is $|C_2|$. Meanwhile, \\citet{art:kra} introduced Restricted Disjunctive Temporal Problems (RDTPs) which contain constraints of Type 1, 2 and 3\\footnote{Type 3 constraints consider two different time-points $i,j \\in T,\\ i\\neq j$ and relate them via a disjunction with exactly two terms in the form $(l'_i \\leq s_i \\leq u'_i) \\lor (l'_j \\leq s_j \\leq u'_j)$, where $l'_i,u'_i,l'_j,u'_j \\in \\mathbb{R}$ denote bounds for time-points $i$ or $j$. This type of constraint is not handled in this paper but interested readers are referred to \\cite{art:kra,art:cra} for more information about them.}. The SDTP therefore arises as a special case of RDTPs when there are no Type 3 constraints. \\citet{art:cra} also refer to SDTPs as t\\textsubscript{2}DTPs, framing them as DTPs that only contain constraints of Type 1 and 2. \n\nIn a move to simplify and unify nomenclatures, we have decided to introduce the name \\textit{Simple Disjunctive Temporal Problem} following the same reasoning behind the naming of STPs \\cite{art:stp}. Figure~\\ref{fig:tcps-class} below situates the SDTP within the larger scheme of DTPs.\n\n\\begin{figure}[h]\n \\begin{center}\n \\resizebox{0.35\\linewidth}{!}{\n \\begin{tikzpicture}\n \\node[set,text width=5cm,label={[below=125pt of dtp,text opacity=1]DTP}] \n at (0,-0.8) (dtp) {};\n \\node[set,text width=4cm,label={[below=95pt of dtp,text opacity=1]RDTP}] \n at (0,-0.65) (rdtp) {};\n \\node[set,fill=gray!20,text width=3cm,label={[below=68pt of rdtp,text opacity=1]SDTP}] \n at (0,-0.4) (sdtp) {};\n \\node[set,fill=gray!40,text width=2cm,label={[below=40pt of sdtp]STPTE}] \n (stpt) at (0,-0.2) {};\n \\node[set,fill=gray!60,text width=1cm] (stp) at (0,0) {STP};\n \\end{tikzpicture}\n }\n \\caption{Classification of DTPs in a set diagram.}\n \\label{fig:tcps-class}\n \\end{center}\n\\end{figure}\n\nSDTPs generalize STPs since the latter can be cast as an SDTP for which $|D_i|=1,\\ \\forall\\ i \\in T$. They also generalize the Simple Temporal Problem with Taboo regions featuring both instantaneous Events and processes of constant duration (STPTE) \\cite{art:stpts}. This class of problems differs from SDTPs because STPTEs define common intervals when no time-point can be scheduled rather than individual intervals per time-point. This clearly demonstrates how SDTPs generalize STPTEs. However, when the duration of processes can vary within an interval in STPs with taboo regions, then SDTPs cannot generalize them because Type 3 constraints are needed \\cite{art:stpts}. Finally, RDTPs generalize all of the aforementioned problems while DTPs further generalize RDTPs. The gray area in Figure \\ref{fig:tcps-class} represents the problems that the models and algorithms in this paper address. \n\nAnother problem related to SDTPs is the time-dependent STP \\cite{art:td-stp}. While it might not be an obvious connection at first, in the time-dependent version of STPs the weight $w_{ij}$ in Type 1 constraints is not a constant but rather a function $f(s_i,s_j)$ which depends on the values assigned to the time-points. When $f(s_i,s_j)$ is a piecewise linear and partial function over the global boundary $[L(D_j),U(D_j)]$, it is possible to cast the SDTP as a time-dependent STP. In this case, the function is defined between $\\alpha$ and every $j \\in T$, that is $f(s_\\alpha,s_j)$. The pieces of function $f(s_\\alpha,s_j)$ represent the domains of time-point $j$. This relation has not previously been established in the literature and one of the possible reasons could be that \\citet{art:td-stp} originally focused more on total functions given that each time-point had a single domain in their application. The effect of partial functions in the development of algorithms will be discussed further in Section \\ref{sec:algs}. We opted not to include the time-dependent STP in Figure \\ref{fig:tcps-class} so as to maintain a clear relation between problems that are often considered together in the temporal reasoning literature, namely those that deal with disjunctions. Nevertheless, the connection we have established has important implications for computing solutions to SDTPs.\n\n\\subsection{Constraint programming model}\n\nConstraint Programming (CP) tools are widely used in planning and scheduling domains. Hence, it is worth considering whether CP is a good candidate for solving SDTPs in practice. The corresponding CP model is: \n\\begin{align}\n s_{i} - s_{j} \\leq w_{ij}, &\\quad \\forall\\ (i,j,w_{ij}) \\in C_1 \\label{cp:1}\\\\\n \\bigvee_{[l^k_i,u^k_i] \\in D_i} (l^k_i \\leq s_i \\leq u^k_i), &\\quad \\forall\\ (i,D_i) \\in C_{2} \\label{cp:2}\n\\end{align}\n\\noindent which is actually the same set of equations as those in Definition \\ref{def:sdtp}. This is very convenient because we essentially have a one-to-one mapping between the classic definition of SDTPs and their CP formulation. For simplicity, we will refer to model (\\ref{cp:1})-(\\ref{cp:2}) as CP.\n\nA simplified CP formulation can be written as follows:\n\\begin{align}\n s_{i} - s_{j} \\leq w_{ij}, &\\quad \\forall\\ (i,j,w_{ij}) \\in C_1\\label{cps:1}\\\\\n L(D_i) \\leq s_i \\leq U(D_i), &\\quad \\forall\\ (i,D_i) \\in C_2\\label{cps:2}\\\\\n s_i \\notin \\Phi_i, &\\quad \\forall\\ i \\in T\\label{cps:3}\n\\end{align}\n\nConstraints (\\ref{cps:1}) are the same as Equation \\ref{eq:stc}, while Constraints (\\ref{cps:2}) model the global boundaries of time-points. Constraints (\\ref{cps:3}) are the \\textit{compatibility constraints} and serve as a replacement for disjunctive Constraints (\\ref{cp:2}). Set $\\Phi_i$ contains the enumeration of all infeasible assignments to $i \\in T$ that belong to the interval $[L(D_i),U(D_i)]$. In other words: $\\Phi_i = \\{u^1_i+1,u^1_i+2,\\dots, l^2_i-1,u^2_i+1,\\dots,l^k_i-1\\},\\ k=|D_i|$. Adding these constraints is only possible if we make the additional assumption that $s_i \\in \\mathbb{Z}$. However, given that typical CP tools only operate with integer variables, this assumption is not necessarily restrictive in practice. We will refer to the formulation defined by (\\ref{cps:1})-(\\ref{cps:3}) as \\textit{Simplified Constraint Programming} (SCP).\n\n\n\\subsection{Integer linear programming model}\n\n Integer Linear Programming (ILP) is also often used in the planning and scheduling domains, which motivated us to also formulate the SDTP in ILP form. First, let us define the binary decision variable $x^c_i$ which takes value 1 whenever solution value $s_i$ belongs to domain $[l^c_i,u^c_i] \\in D_i,\\ (i,D_i) \\in C_2$ and 0 otherwise. The corresponding ILP model for SDTPs is:\n \n\\begin{align}\n s_i - s_{j} \\leq w_{ij}, &\\quad \\forall\\ (i,j,w_{ij}) \\in C_1\\label{ilp:1}\\\\ \n l^c_i - M^L_i(1-x^c_i) \\leq s_i, &\\quad \\forall\\ (i,D_i) \\in C_{2},\\ [l^c_i,u^c_i] \\in D_i \\label{ilp:2}\\\\ \n s_i \\leq u^c_i + M^U_i(1-x^c_i), &\\quad \\forall\\ (i,D_i) \\in C_{2},\\ [l^c_i,u^c_i] \\in D_i\\label{ilp:3}\\\\ \n \\sum_{[l^c_i,u^c_i] \\in D_i} x^c_i = 1, &\\quad \\forall\\ (i,D_i) \\in C_{2}\\label{ilp:4}\\\\ \n x^c_i \\in \\{0,1\\}, &\\quad \\forall\\ (i,D_i) \\in C_{2},\\ [l^c_i,u^c_i] \\in D_i \\label{ilp:5}\n\\end{align}\n\nConstraints (\\ref{ilp:1}) refer to the simple temporal constraints (Equation \\ref{eq:stc}). Meanwhile, Constraints (\\ref{ilp:2})-(\\ref{ilp:3}) restrict the values assigned to solution $s$ so that they belong to the active bounds defined by variables $x^c_i$. Note that Constraints (\\ref{ilp:2})-(\\ref{ilp:3}) are big-$M$ constraints. They can be tightened by setting, for each $(i, D_i) \\in C_2$:\n\\begin{align*}\nM^L_i &= \\max_{[l^c_i,u^c_i]\\in D_i} l^c_i - L(D_i)\\\\\nM^U_i &= U(D_i) - \\min_{[l^c_i,u^c_i]\\in D_i} u^c_i \n\\end{align*}\n\\noindent Constraints (\\ref{ilp:4}) ensure that exactly one domain is selected per time-point $i \\in T$. Finally, Constraints~(\\ref{ilp:5}) restrict $x^c_i$ variables to take binary values. Recall that an SDTP is a feasibility problem, with this explaining why there is no objective function present in this ILP.\n\n\n\n\\bigskip\n\nAll three models (ILP, CP and SCP) can be used to solve SDTPs by employing a state-of-the-art solver such as IBM's CPLEX. However, these solvers are often financially expensive. Furthermore, specific methods can provide guarantees concerning expected run times, such as asymptotic polynomial worst-case time complexity. In the following section, we describe many algorithms that can be used to quickly solve SDTPs in practice.\n\n\n\\section{Algorithms} \\label{sec:algs}\n\nA variety of special-purpose algorithms have been proposed for SDTPs. All of the algorithms that are presented in this section will be implemented for our computational experiments. It is worth noting that all of the algorithms provide a guaranteed polynomial asymptotic worst-case time complexity. Algorithms are presented in chronological order of publication date. In some cases, we adapted algorithms to ensure they could be implemented efficiently in practice. For that reason, we try to provide as many implementation details as possible. In all cases where details are missing, we refer interested readers to our code for deeper inspection.\n\nWe assume that each algorithm receives as input an SDTP instance containing network $N=(T,C)$ and associated graphs $G_D$ and $G_R$. Some algorithms also receive additional structures, which we detail for the individual method whenever necessary. All algorithms return a solution vector $s$. When the SDTP instance is feasible, each entry $s_i$ contains a time assigned to $i \\in T$ which in combination with the other entries renders the solution feasible (network $N$ consistent). Whenever the SDTP instance is infeasible, $s=\\emptyset$ is returned.\n\n\\subsection{Upper-Lower Tightening}\n\n\\citet{art:ult} introduced the \\textit{Upper-Lower Tightening} (ULT) algorithm to tackle general disjunctions in DTPs. The original intention behind ULT was to tighten disjunctive constraints and simplify DTP instances. However, \\citet{art:ult} were the first to show that SDTPs could be solved in polynomial time by means of ULT.\n\nThe ULT algorithm operates with constraints between two variables denoted as an interval. The first step is to therefore define set $H = \\{ (i,j)\\ \\forall\\ (i,j,w_{ij}) \\in C_1 \\} \\cup \\{ (\\alpha,i)\\ \\forall\\ (i,D_i) \\in C_2\\}$. Let us further assume that $(i,j) \\in H \\implies (j,i) \\notin H$. \\textit{Boundary} sets $B_{ij}$ are defined for $(i,j) \\in H\\ :\\ i \\neq \\alpha$, $B_{ij}=\\{[-w'_{ji}, w'_{ij}] \\}$ where $w'_{ij} = w_{ij}\\text{ if } (i,j,w_{ij}) \\in C_1$, otherwise $w'_{ij} = +\\infty$, and $w'_{ji} = w_{ji}\\text{ if } (j,i,w_{ji}) \\in C_1$, otherwise $w'_{ji} = +\\infty$. Meanwhile, for $(\\alpha,i) \\in H$ we relate $i$ to the beginning of time $\\alpha$ via $B_{\\alpha i}=D_i\\ :\\ (i,D_i) \\in C_{2}$. We will use the notation $L(B_{ij})$ and $U(B_{ij})$ to denote the lower and upper bounds in $B_{ij}$, respectively.\n\nAlgorithm \\ref{alg:ult} outlines how ULT works. First, a distance matrix $\\delta$ is initialized in line 1. The main loop of the algorithm spans lines 2-8. In lines 3-4, some entries of the distance matrix $\\delta$ are updated according to the current bounds $B$ of each pair $(i,j) \\in H$. \\textsc{FloydWarshall} \\cite{book:cormen} is then used to update matrix $\\delta$ by computing All-Pairs Shortest Paths (APSPs) using the current values in $\\delta$ as the arc weights (line 5). A temporary boundary set $B'$ is created in line 6 with the newly computed values in $\\delta$. Note that in the implementation itself we do not create $B'$ since we can use matrix $\\delta$ directly in its place whenever needed (for example, in line 7). The intersection of $B'$ and $B$ is computed in line 7. Here, we follow the definition of the $\\cap$ operation introduced by \\citet{art:ult}: it returns a set of intervals whose values are permitted by both $B'$ and $B$. ULT iterates so long as there are changes to the bounds in $B$, denoted by operation $\\textsc{Change}$, and no bound is either empty or infeasible. All checks in line 8 can be performed in $O(1)$ time by maintaining the correct flags after lines 6-7. Similarly, lines 3-4 can be performed during operation $\\cap$ in line 7 without increasing the asymptotic worst-case time complexity. Lines 9-11 prepare solution $s$ to be returned. If the instance is feasible then line 10 assigns the earliest feasible schedule to $s$, otherwise $\\emptyset$ is returned.\n\n\\begin{algorithm}[H]\n \\caption{ULT}\n \\label{alg:ult}\n \\footnotesize\n \\begin{algorithmic}[1]\n \\State $\\delta_{ij} \\gets +\\infty,\\ \\forall\\ i,j \\in T \\cup \\{\\alpha\\}$\n \\Do\n \\State $\\delta_{ij} \\gets U(B_{ij}),\\ \\forall\\ (i,j) \\in H$ \\Comment{Update current $\\delta$ entries with new bounds}\n \\State $\\delta_{ji} \\gets -L(B_{ij}),\\ \\forall\\ (i,j) \\in H$\n \\State $\\textsc{FloydWarshall}(\\delta)$ \\Comment{Update distance matrix $\\delta$}\n \\State $B'_{ij} \\gets \\{[-\\delta_{ji},\\delta_{ij}]\\},\\ \\forall\\ (i,j) \\in H$\n \\State $B \\gets B \\cap B'$ \\Comment{Tightens boundaries}\n \\DoWhile{$\\textsc{Change}(B) \\textbf{ and } (B_{ij} \\neq \\emptyset \\textbf{ and } L(B'_{ij}) \\leq U(B'_{ij}),\\ \\forall (i,j) \\in H)$}\n \\State $s \\gets \\emptyset$\n \\IfThen{$B_{ij} \\neq \\emptyset \\textbf{ and } L(B'_{ij}) \\leq U(B'_{ij}),\\ \\forall (i,j) \\in H$}{$s_i \\gets L(B_{\\alpha i}),\\ \\forall\\ i \\in T$}\n \\State \\Return $s$\n \\end{algorithmic}\n\\end{algorithm}\n\nThe asymptotic worst-case time complexity of ULT is $O(|T|^3|C|K + |C|^2K^2)$ \\cite{art:ult}, while its space complexity is $O(|T|^2)$ due to distance matrix $\\delta$. Despite its apparently high computational complexity, ULT is a polynomial time algorithm. Additionally, \\citet{art:ult} noted that even when a problem instance contains multiple disjunctions per constraint between time-points $i,j \\in T$, and is therefore not an SDTP instance, ULT may successfully remove sufficient disjunctions to reduce the problem to an SDTP. In this case, ULT is guaranteed to solve the problem exactly. This is the only algorithm in our study capable of such a reduction.\n\n\n\\subsection{Kumar's Algorithm}\n\n\\citet{art:kumar-estp} proposed a polynomial time algorithm to solve zero-one extended STPs, which essentially correspond to an SDTP. Algorithm \\ref{alg:kra} provides a pseudocode outline of how \\textit{Kumar's Algorithm} (KA) works. In line 1, a distance matrix $\\delta$ is constructed by computing APSPs over graph $G_R$. This step can detect infeasibilities such as if there exists a negative cycle formed by $C_1$ constraints and global boundaries, in which case $\\delta = \\emptyset$ is returned. \n\nMatrix $\\delta$ can be computed by employing (i) \\textsc{FloydWarshall}, (ii) repeated calls to \\textsc{BellmanFord} or (iii) Johnson's Algorithm \\cite{book:cormen}. \\citet{art:kumar-estp} did not specify which method should be used when computing $\\delta$ and therefore we will consider both options (ii) and (iii). Option (i) is disregarded due to its overall poor performance during our preliminary experiments.\n\n\\begin{algorithm}\n \\caption{KA}\n \\label{alg:kra}\n \\footnotesize\n \\begin{algorithmic}[1]\n \\State $\\delta \\gets \\textsc{ComputeDistanceMatrix}(G_R)$\n \\IfThen{$\\delta = \\emptyset$}{\\textbf{return} $\\emptyset$}\n \\State $G_C \\gets \\textsc{CreateConflictGraph}(\\delta, C_2)$ \\Comment{Graph $G_C=(E,A_C)$}\n \\IfThen{$G_C = \\emptyset$}{\\textbf{return} $\\emptyset$}\n \\State $G_B \\gets \\textsc{CreateBipartiteGraph}(G_C)$ \\Comment{Graph $G_B=(E,E',A_B)$, where $E'$ is a copy of $E$}\n \\State $G_F \\gets \\textsc{SolveMaxFlow}(G_B)$ \\Comment{From source $\\theta_1$ to sink $\\theta_2$, with $G_F$ corresponding to the residual graph}\n \\State $S \\gets \\{(\\theta_1,e^c_i)\\ :\\ e^c_i \\notin R(G_F, \\theta_1)\\} \\cup \\{(e^{k\\prime}_j,\\theta_2)\\ :\\ e^{k\\prime}_j \\in R(G_F, \\theta_1)\\}$ \\Comment{Minimum cut in $G_F$}\n \\State $S' \\gets \\{e^c_i\\ :\\ (\\theta_1,e^c_i) \\in S\\ \\lor (e^{c\\prime}_i,\\theta_2) \\in S\\}$ \\Comment{Vertex cover for $G_C$}\n \\State $S'' \\gets E \\backslash S'$\n \\IfThen{$|S''| \\neq |T|$}{\\textbf{return} $\\emptyset$}\n \\State $\\textsc{UpdateGraph}(G_R,S'')$\n \\State $s \\gets \\textsc{BellmanFord}(G_R, \\alpha)$ \\Comment{Solve STP}\n \\State \\Return $s$\n \\end{algorithmic}\n\\end{algorithm}\n\nLine 3 proceeds to create a conflict graph $G_C=(E,A_C)$ with the domains from the SDTP. First, set $E$ of intervals is defined as $E = \\{ e^c_{i}\\ :\\ (i,D_i) \\in C_2,\\ [l^c_i,u^c_i] \\in D_i \\}$. Hence, every element $e^c_{i} \\in E$ represents exactly one domain of a time-point. A domain $[l^c_i,u^c_i]$ has no corresponding element in $E$ if it produces a size-1 conflict, that is, if the following is true:\n\\begin{equation*}\n \\delta_{i\\alpha} + u^c_i < 0\\ \\lor\\ \\delta_{\\alpha i} - l^c_i < 0\n\\end{equation*}\n\nOnce vertex set $E$ has been created, arc set $A_C$ can be defined. An arc $(e^c_{i},e^k_{j}) \\in A_C$ denotes a size-2 conflict between two time-point domains $[l^c_i,u^c_i]$ and $[l^k_j,u^k_j]$. Such a conflict occurs whenever:\n\\begin{equation*}\n u^c_i + \\delta_{ij} - l^k_j < 0\n\\end{equation*}\nNote that size-2 conflicts are also defined between domains of the same time-point $i \\in T$. There is always a conflict $(e^c_{i},e^{c+1}_{i}) \\in A_C$ because $\\delta_{ii}=0$ and $u^c_i < l^{c+1}_i$ (recall from Section \\ref{sec:sdtp} that domains are in ascending order).\n\nProcedure \\textsc{CreateConflictGraph} returns either graph $G_C$ or $\\emptyset$. The latter is returned whenever all domains of a time-point $i \\in T$ produce size-1 conflicts. In this case no domain associated with $i$ is included in $E$, thereby implying that the SDTP instance is infeasible. Once graph $G_C$ has been constructed, line 5 creates a bipartite graph $G_B=(E,E',A_B)$ by copying every element $e^c_{i} \\in E$ to $e^{c\\prime}_{i} \\in E'$. For each $(e^c_{i},e^k_{j}) \\in A_C$ we create an arc $(e^c_{i},e^{k\\prime}_{j}) \\in A_B$. All arcs in $A_B$ connect an element of $E$ to an element of $E'$.\n\nLine 6 solves a maximum bipartite matching over $G_B$ as a maximum flow problem (max-flow), producing the residual graph $G_F$ \\cite{book:cormen}. To solve the problem in the form of a max-flow, we introduce a source node $\\theta_1$ and a sink node $\\theta_2$ to $G_B$. Arcs $(\\theta_1,e^c_{i}),\\ \\forall\\ e^c_{i} \\in E$ and $(e^{k\\prime}_{j},\\theta_2),\\ \\forall\\ e^{k\\prime}_{j} \\in E'$ are included in the graph together with all arcs in $A_B$. Additionally, all arcs are given unitary capacity. Then, it suffices to solve a max-flow from $\\theta_1$ to $\\theta_2$ to produce $G_F$. \n\nThe minimum-cut $S$ is computed in $G_F$ thanks to the max-flow min-cut theorem (line 7). $R(G_F,\\theta_1)$ denotes the set of nodes that are reachable from source $\\theta_1$ in $G_F$ (meaning there is a path with positive residual capacity). Line 8 merges node copies in $S$ to create $S'$, which is a vertex cover for $G_C$ when seen as an undirected graph. Since $S'$ is a vertex cover, if we take all elements in $E$ which are not part of $S'$ to create set $S''$ (line 9), there will be no two elements in $S''$ which have a conflict. In other words: all domains in $S''$ can be part of a feasible SDTP solution.\n\nIf $|S''| = |T|$ then every time-point has exactly one domain assigned to it, that is, $S''_i = e^c_{i},\\ \\forall\\ i \\in T$. If $|S''| < |T|$ the instance is infeasible (line 10). Line 11 continues to update graph $G_R$ with the information in $S''$ concerning the selected domain for each time-point:\n\\begin{align*}\n (\\alpha,i,w_{\\alpha i}) \\in A_R \\implies w_{\\alpha i}=U(S''_i)\\\\\n (i,\\alpha,w_{i\\alpha}) \\in A_R \\implies w_{i\\alpha}=-L(S''_i)\n\\end{align*}\n\\noindent where $U(S''_i)=u^c_i$ and $L(S''_i)=l^c_i$. The final solution $s$ is computed with standard \\textsc{BellmanFord} since the SDTP has now been reduced to a feasible STP (line 12).\n\nNote that in our implementation we do not explicitly create graphs $G_C$ and $G_B$. Instead, we directly create max-flow graph $G_F$. This graph is also modified by \\textsc{SolveMaxFlow} to produce the residual graph. By taking this approach, we reduce both KA's execution time and the amount of memory it requires. Conceptually, however, it is easier to explain how KA works by documenting the creation of each graph in a step-by-step fashion.\n\n\\citet{art:kumar-estp} did not provide the asymptotic worst-case time complexity of KA and instead suggested that KA runs in polynomial time because each step can be performed in polynomial time. Therefore, for the purpose of completeness, we will now explicitly analyse the time complexity of KA. The three algorithmic components which dictate KA's complexity can be found on lines 1, 3 and 6. All other parts of the algorithm can be completed in time which is never slower than these three main components.\n\nLine 1 takes time $O(|T|^2|C|)$ if computed with repeated \\textsc{BellmanFord} and time $O(|T||C| + |T|^2\\log |T|)$ if computed with Johnson's algorithm provided Dijkstra's algorithm \\cite{book:cormen} is implemented with Fibonacci Heaps \\cite{art:fib-heaps}. However, Fibonacci Heaps are often inefficient in practice due to pointer operations leading to poor cache locality and performance \\cite{art:splib,art:heaps}. We therefore opted to use Sequence Heaps \\cite{art:sequence-heaps} in our implementation, which increases the asymptotic worst-case time complexity to $O(|T||C|\\log |T|)$ but improves the performance of the algorithm in practical settings. Line 3 has complexity $O(\\omega^2)$ because we need to check every pair of intervals in $E$ and $|E| = O(\\omega)$. Line 6 solves a max-flow problem. While there are many algorithms to solve max-flow \\cite{art:max-flow}, we opted to use Dinic's Algorithm with complexity $O(\\omega^\\frac{5}{2})$ when applied to graphs from maximum bipartite matching. We have observed that max-flow is not the bottleneck in KA. Indeed, lines 1 and 3 are the most time-consuming steps (see Section \\ref{sec:discussion} for a full discussion).\n\nIn the remainder of the paper, we will refer to the version of KA using repeated \\textsc{BellmanFord} as KAB, and the one using Johnson's algorithm as KAJ. We will write KA when referring to the algorithm in a generic sense which covers both KAB and KAJ. The complexity of KAB is $O(|T|^2|C| + \\omega^\\frac{5}{2})$, while KAJ's is $O(|T||C|\\log |T| + \\omega^\\frac{5}{2})$. Meanwhile, the space complexity of KA is $O(|T|^2 + \\omega^2)$ due to distance matrix $\\delta$ and graph $G_F$.\n\n\n\\subsection{Comin-Rizzi Algorithm}\n\n\\citet{art:cra} introduced asymptotically faster algorithms to solve both SDTPs and RDTPs, making their methods the current state of the art for both problems. For SDTPs, they introduced an algorithm which resembles Johnson's Algorithm for APSPs. Their method begins by performing a first phase using \\textsc{BellmanFord} to detect negative cycles, while subsequent iterations use Dijkstra's Algorithm to correct computations over a graph that contains no negative cycles. However, no experimental study has been conducted using this method until now.\n\nThe \\textit{Comin-Rizzi Algorithm} (CRA) for SDTPs is detailed in Algorithm \\ref{alg:cra}. CRA begins by computing an initial earliest feasible solution $s^0$ considering $C_1$ constraints only. In our implementation, we partially consider $C_2$ constraints by using the global boundaries defined in Section \\ref{sec:sdtp} within $G_D$. The computation of $s^0$ then either produces the earliest possible solution or proves that one cannot exist because (i) there is a negative cycle formed by $C_1$ constraints or (ii) it is not possible to assign a time $s_i$ to at least one time-point $i \\in T$ while complying with the global bounds $[L(D_i),U(D_i)]$.\n\nIf $s^0 \\neq \\emptyset$ then CRA proceeds to its main loop. First, each time-point $i \\in T$ where the current solution $s^0_i$ does not belong to one of the domains $D_i$ is added to list $F$ of assignments that require fixing. While there are elements in $F$, the following steps are repeated (lines 5-12). A time-point $i$ is removed from $F$ (line 6). The first time $i$ is removed from $F$, we compute entry $\\delta_i$ of the distance matrix $\\delta$ from $i$ to all other nodes in the underlying graph $G^{1\\prime}_R$ containing only $C_1$ constraints (lines 7-9). In this graph, the weight $w_{ij}$ of each arc $(i,j,w_{ij}) \\in A_R$ is modified to $w'_{ij}=w_{ij} + s^0_j - s^0_i$. \\citet{art:cra} showed that $G^{1\\prime}_R$ cannot contain negative cycles because it is always true that $w'_{ij} \\geq 0$. Therefore, distances $\\delta_i$ can be computed using \\textsc{Dijkstra} instead of \\textsc{BellmanFord}, which greatly improves the performance of CRA. Each entry $\\delta_i$ is only computed once because $G^{1\\prime}_R$ remains unchanged during CRA's execution.\n\n\\begin{algorithm}\n \\caption{CRA}\n \\label{alg:cra}\n \\footnotesize\n \\begin{algorithmic}[1]\n \\State $s^0 \\gets \\textsc{BellmanFord}(G_D, \\alpha)$ \\Comment{Solve STP using SDTP global boundaries}\n \\IfThen{$s^0 = \\emptyset$}{\\textbf{return} $\\emptyset$}\n \\State $s \\gets s^0$\n \\State $F \\gets \\{i\\ :\\ (i,D_i) \\in C_2 \\land s_i \\notin D_i\\}$ \\Comment{Set of all time-points $i \\in T$ with assignment $s_i$ infeasible}\n \\While{$F \\neq \\emptyset \\textbf{ and } s \\neq \\emptyset \\textbf{ and } s_i \\leq U(D_i)\\ \\forall (i,D_i) \\in C_2$}\n \\State $i \\gets \\textsc{Pop}(F)$\n \\If{$\\delta_i \\textbf{ not yet computed}$}\n \\State $\\delta_i \\gets \\textsc{Dijkstra}(G^{1\\prime}_D, i)$ \\Comment{Lazy computation of $\\delta$}\n \\EndIf\n \\State $\\textsc{UpdateAssignments}(s,s^0,i,\\delta_i)$\n \\State $F \\gets \\{i\\ :\\ (i,D_i) \\in C_2 \\land s_i \\notin D_i\\}$\n \\EndWhile\n \\State \\Return $s$\n \\end{algorithmic}\n\\end{algorithm}\n\nFor each $i$ taken from $F$ in line 6, we update the assignment to $s_i$ by means of procedure \\textsc{UpdateAssignments} (line 10). First, the procedure performs the following operation\n\\begin{equation*}\n s_i \\leftarrow \\lambda(s_i,D_i)\\ :\\ (i,D_i) \\in C_2\n\\end{equation*}\n\\noindent where $\\lambda(s_i,D_i)$ is a function that either returns value $l^c_i$ belonging to the first domain in ascending order $[l^c_i,u^c_i] \\in D_i$ for which $s_i < l^c_i$, or it returns $\\perp$ if no such domain exists. Whenever $\\lambda(s_i,D_i) = \\perp$, CRA stops computations because this proves that the instance is infeasible. In this case, \\textsc{UpdateAssignments} sets $s=\\emptyset$. Alternatively, if $\\lambda(s_i,D_i) \\neq \\perp$ then the new assignment $s_i$ can cause changes to other time-point assignments since $s_i$ has necessarily increased. To correctly propagate these changes, \\citet{art:cra} introduced the following update rules\n\\begin{align*}\n \\rho_{ij} &\\leftarrow \\delta_{ij} + (s_j - s^0_j) - (s_i - s^0_i),\\ &\\forall\\ j \\in P(G^1_D, i)\\\\\n s_j &\\leftarrow s_j + \\max(0, \\lambda(s_i,D_i) - s_i - \\rho_{ij}),\\ &\\forall\\ j \\in P(G^1_D, i)\n\\end{align*}\n\\noindent where $P(G^1_D,i)$ denotes the set of all nodes $j \\in V$ which are reachable from $i$ in $G^1_R$. In other words: there is a path from $i$ to $j$ in $G^1_D$. These update rules can be applied in $O(1)$ time per $j \\in P(G^1_D,i)$ or $O(|T|)$ time in total.\n\nAfter fixing the assignment to $i$ and potentially other time-points, CRA constructs a new list $F$ (line 11). Once $F = \\emptyset$, the assignment in $s$ is feasible and corresponds to the earliest feasible solution. This assignment is then returned in line 13. For infeasible instances, $s=\\emptyset$ is returned instead.\n\nThe asymptotic worst-case time complexity of CRA is $O(|T||C| + |T|^2\\log |T| + |T|\\omega)$ when using Fibonacci Heaps for \\textsc{Dijkstra}'s computations. The asymptotic complexity increases to $O(|T||C|\\log |T| + |T|\\omega)$ when using Sequence Heaps instead, however the empirical performance improves significantly \\cite{art:sequence-heaps}. Regardless of the heap implementation, CRA's space complexity is $O(|T|^2)$ due to distance matrix $\\delta$.\n\nIn their original description of CRA, \\citet{art:cra} precomputed distance matrix $\\delta$ before beginning the main loop in Algorithm \\ref{alg:cra}. For our implementation, we describe the computation as a \\textit{lazy computation} of entries in $\\delta$ given that we only compute them when strictly necessary (lines 7-9). Although both approaches exhibit the same asymptotic worst-case time complexity, in practice the lazy computation performs significantly better since many unnecessary computations are avoided. Additionally, we have incorporated the creation of list $F$ at line 11 into procedure \\textsc{UpdateAssignments}. Whenever the assignment $s_j$ to a time-point $j \\in T$ is modified, we check whether $j$ should be added to or removed from $F$. This avoids reconstructing list $F$ every iteration of the main loop (lines 5-12), thus speeding up computations.\n\n\\subsection{Reduced Upper-Lower Tightening}\n\nThe \\textit{Reduced Upper-Lower Tightening} (RULT) method is a speedup of ULT, specifically targeted towards SDTPs. One can easily derive RULT from ULT by exploiting the structure of SDTPs. Recall that in ULT, we must compute APSPs using \\textsc{FloydWarshall} with complexity $O(|T|^3)$ because \\citet{art:ult} assumed the input was a general DTP with possibly multiple disjunctions per constraint between two time-points $i,j \\in T$. \n\nHowever, SDTPs feature a structure that only contains simple temporal constraints between time-points in $T$. It is therefore sufficient to compute single-source shortest paths twice: first to determine the earliest feasible assignment for each time-point and a second time to determine the latest feasible assignment for each time-point. This creates a single interval per time-point denoting a possibly tighter global boundary concerning their assignments. Similar to ULT, we can use this global boundary to reduce $C_2$ disjunctions in every iteration, thereby reducing the number of disjunctions.\n\nAlgorithm \\ref{alg:rult} outlines RULT. First, boundary set $B$ is initialized with the domains of each time-point (lines 1-2). In contrast to ULT, we only have to maintain boundaries per $i \\in T$ rather than per constraint. The main loop (lines 3-11) runs for as long as there are changes to $B$ and the bounds remain feasible. In every iteration graph $G_D$ is changed with \\textsc{UpdateGraph}, which replaces the weight of arcs connected to $\\alpha$: \n\\begin{align*}\n (\\alpha,i,w_{\\alpha i}) \\in A_D \\implies w_{\\alpha i}=-L(B_i)\\\\\n (i,\\alpha,w_{i\\alpha}) \\in A_D \\implies w_{i\\alpha}=U(B_i)\n\\end{align*}\n\\noindent The same procedure takes place for $G_R$ but outgoing arcs from $\\alpha$ get the upper bound $U(B_i)$ while the incoming arcs get the lower bound $-L(B_i)$. During RULT's execution, values $L(B_i)$ are non-decreasing and $U(B_i)$ are non-increasing. Hence, updating the graphs tightens the global boundary $B_i$ of each time-point $i \\in T$.\n\n\n\n\\begin{algorithm}\n \\caption{RULT}\n \\label{alg:rult}\n \\footnotesize\n \\begin{algorithmic}[1]\n \\State $B_i \\gets \\{[-\\infty,+\\infty]\\},\\ \\forall\\ i \\in T$\n \\State $B_i \\gets D_i,\\ \\forall\\ (i,D_i) \\in C_2$\n \\Do\n \\State $\\textsc{UpdateGraph}(G_D,B)$ \\Comment{Update arc weights connected to $\\alpha$}\n \\State $\\textsc{UpdateGraph}(G_R,B)$\n \\State $p \\gets \\textsc{BellmanFord}(G_D,\\alpha)$ \\Comment{Earliest feasible assignment}\n \\State $q \\gets \\textsc{BellmanFord}(G_R,\\alpha)$ \\Comment{Latest feasible assignment}\n \\IfThen{$p = \\emptyset$ \\textbf{ or } $q = \\emptyset$}{\\textbf{return} $\\emptyset$}\n \\State $B'_i \\gets \\{[-p_i,q_i]\\},\\ \\forall\\ i \\in T$\n \\State $B \\gets B \\cap B'$ \\Comment{Tightens boundaries}\n \\DoWhile{$\\textsc{Change}(B) \\textbf{ and } L(B_{i}) \\leq U(B_{i})\\ \\forall\\ i \\in T$}\n \\State $s \\gets \\emptyset$\n \\IfThen{$L(B_{i}) \\leq U(B_{i})\\ \\forall\\ i \\in T$}{$s_i \\gets L(B_{i}),\\ \\forall\\ i \\in T$}\n \\State \\Return $s$\n \\end{algorithmic}\n\\end{algorithm}\n\nLines 6-7 compute the earliest feasible schedule $p$ and the latest feasible schedule $q$ over the updated graphs. If $p = \\emptyset$ or $q = \\emptyset$ then the instance is infeasible, because a negative cycle still exists even for the relaxed global boundaries of all time-points (line 8). Otherwise, line 9 constructs set $B'$ and line 10 computes the intersection of $B$ and $B'$. Operation $\\cap$ is the same used in ULT and defined by \\citet{art:ult}. Finally, lines 12-14 prepare solution $s$ to be returned. If boundaries in $B$ are feasible, line 13 assigns to every time-point its earliest feasible value. If the latest feasible solution is desired instead, we can assign $U(B_i)$ to $s_i$ in line 13.\n\nThe correctness of RULT follows directly from that of ULT \\cite{art:ult} in combination with the fact that $C_1$ constraints are fixed and the only intervals that must be considered are those in $C_2$. The asymptotic worst-case time complexity of RULT is similar to ULT's. Accounting for the efficiency gain in shortest path computations, which are performed with \\textsc{BellmanFord} instead of \\textsc{FloydWarshall}, RULT's complexity becomes: $O(|T|^2|C|K + |T|^2|K|^2)$. The space complexity of RULT is reduced to $O(|T|)$ given that we only have to allocate additional vectors of size $|T|$. Note that in our implementation, we do not explicitly maintain boundary set $B$.\n\n\n\n\\subsection{Bellman-Ford with Domain Check}\n\nAll of the algorithms described until now have employed \\textsc{BellmanFord} at some point during their execution. This should not be surprising since \\textsc{BellmanFord} can be implemented rather efficiently to detect negative cycles \\cite{art:sp-fp}, which is a core task when solving STPs, SDTPs and RDTPs. It seems only natural then to consider a variant of the original algorithm to solve SDTPs. Let us therefore define \\textit{Bellman-Ford with Domain Check} (BFDC), which incorporates small changes to \\textsc{BellmanFord} in order to address gaps of infeasible values in the shortest path computations. Our method draws inspiration from previous research on temporal problems \\cite{art:cra,art:stp-bf-inc,art:td-stp}.\n\nAlgorithm \\ref{alg:bfdc} describes the full BFDC procedure, which primarily works over graph $G_D$. Lines 1-5 involve the initialization of auxiliary variables. This includes the distance array $\\tau$, path length array $\\pi$ which calculates the number of nodes in the shortest path from $\\alpha$ up to $i \\in V$, the domain index array $z$ which holds the current domain index $z_i$ for each time-point $i \\in T$ and the first-in, first-out queue $Q$ used in \\textsc{BellmanFord}. After initialization, the main loop begins (lines 6-20). In line 7, an element $i$ is removed from the queue and its domain is checked in line 8. Procedure \\textsc{DomainCheck} is detailed in Algorithm \\ref{alg:check}. If \\textsc{DomainCheck} can prove the SDTP instance is infeasible, then it sets $\\tau = \\emptyset$. Otherwise the procedure updates assignments to $\\tau$, $\\pi$ and $z$ as necessary. The algorithm continues to line 9 where, if the instance has not been proven infeasible yet, all outgoing arcs from $i \\in V$ are relaxed and the shortest paths propagated (here \\textit{relax} refers to the nomenclature of \\citet{book:cormen}). \n\n\n\n\\begin{algorithm}\n \\caption{BFDC}\n \\label{alg:bfdc}\n \\footnotesize\n \\begin{algorithmic}[1]\n \\State $\\tau_i \\gets +\\infty,\\ \\forall\\ i \\in T\\cup \\{\\alpha\\}$\n \\State $\\pi_i \\gets 0,\\ \\forall\\ i \\in T\\cup \\{\\alpha\\}$\n \\State $z_i \\gets 1,\\ \\forall\\ i \\in T\\cup \\{\\alpha\\}$\n \\State $Q \\gets \\textsc{Push}(Q, \\alpha)$\n \\State $\\tau_\\alpha \\gets 0$\n \\While{$Q \\neq \\emptyset \\textbf{ and } \\tau \\neq \\emptyset$}\n \\State $i \\gets \\textsc{Pop}(Q)$\n \\State $\\textsc{DomainCheck}(i,\\tau,z,\\pi)$ \\Comment{Algorithm \\ref{alg:check}}\n \\If{$\\tau \\neq \\emptyset$}\n \\For{$(i,j,w_{ij}) \\in A_D$}\\Comment{Standard \\textsc{Relax} phase in \\textsc{BellmanFord}}\n \\If{$\\tau_j > \\tau_i + w_{ij}$}\n \\State $\\tau_j \\gets \\tau_i + w_{ij}$\n \\State $\\pi_j \\gets \\pi_i + 1$\n \\IfThen{$\\pi_j \\geq |T| \\textbf{ or } j = \\alpha$}{$\\tau \\gets \\emptyset$} \\Comment{Checks for negative cycle}\n \\IfThen{$j \\notin Q$}{$Q \\gets \\textsc{Push}(Q, j)$}\n \\EndIf\n \\IfThen{$\\tau = \\emptyset$}{\\textbf{break}}\n \\EndFor\n \\EndIf\n \\EndWhile\n \\IfThenElse{$\\tau \\neq \\emptyset$}{$s_i \\gets -\\tau_i,\\ \\forall\\ i \\in V$}{$s \\gets \\emptyset$}\n \\State \\Return $s$\n \\end{algorithmic}\n\\end{algorithm}\n\nFor each outgoing arc from $i$ in $A_D$ (recall graph $G_D=(V,A_D)$), line 10 checks whether the current shortest path up to $j$ adjacent to $i$ should be updated and, if so, then the algorithm also updates $\\pi_j$ and possibly queue $Q$. In line 14, if the path up to $j$ forms a cycle or the path leads back to $\\alpha$, the instance is determined to be infeasible. If $j$ is not yet in queue $Q$, we add it in line 15 (duplicated elements are not allowed). When the instance has been proven infeasible, line 17 aborts the for-loop (lines 10-18).\n\nThe main loop runs for as long as there are elements in $Q$ and $\\tau$ is not $\\emptyset$. Once one of these conditions is false, Algorithm \\ref{alg:bfdc} proceeds on to line 21. If the instance is feasible, assignment $s$ is created using the values of the shortest paths stored in $\\tau$, otherwise $\\emptyset$ is returned.\n\nProcedure \\textsc{DomainCheck} (Algorithm \\ref{alg:check}) represents the main difference between \\textsc{BellmanFord} and BFDC. In line 1, it verifies whether the current time $s_i=-\\tau_i$ assigned to $i$ belongs to its current domain indexed at $z_i$. If the assigned time does not exceed the domain's upper bound $u^{z_i}_i$ then \\textsc{DomainCheck} simply terminates. Otherwise, the algorithm searches for the first domain in increasing order to which $s_i=-\\tau_i$ belongs (lines 2-8). When a domain is found, lines 4-5 update the assignments for $\\tau_i$ and $\\pi_i$ accordingly. In case $s_i=-\\tau_i$ exceeds all domains in $D_i$ then we have a proof that the instance is infeasible (line 9).\n\n\\begin{lemma}\n Algorithm \\ref{alg:bfdc} is correct and returns either (i) the earliest feasible solution or (ii) proof that no solution exists. \\label{lem:bfdc-correct}\n\\end{lemma}\n\\begin{proof}\n First, note that $\\tau$ is always non-increasing in BFDC. This implies that the SDTP solution $s=-\\tau$ is non-decreasing. In every \\textsc{Relax} phase BFDC assigns the shortest path up to a subset of nodes in $G_D$ and therefore assigns the earliest feasible values to a subset of time-points. Whenever a \\textsc{DomainCheck} phase must increase the assignment to $z_i$ because $-\\tau_i > u^{z_i}_i$, it assigns the earliest feasible domain and either decreases $\\tau_i$ or leaves $\\tau_i$ unchanged (lines 4-5 in Algorithm \\ref{alg:check}). Value $\\tau_i$ is non-increasing and consequently decreasing the assignment of $z_i$ will never lead to a feasible solution. Therefore, $z$ is also non-decreasing in BFDC which implies domain assignment is a backtrack-free search.\n \n With these facts in mind, we can now show that there are two possibilities at the end of BFDC. If $\\tau \\neq \\emptyset$ then $s=-\\tau$ is the earliest feasible solution for the SDTP instance. This is true because $\\tau$ contains the shortest paths in $G_D$ from $\\alpha$ to every other node $i \\in V$, with this achieved by using the minimum feasible assignment of domains $z$. When $\\tau = \\emptyset$ then we have either exhausted the assignment $z_i$ to a time-point $i \\in T$ which implies that BFDC has run out of domains for $i$ ($z_i > |D_i|$), or there is a negative-cost cycle formed by $C_1$ constraints which has been detected during the \\textsc{Relax} phase (line 14 in Algorithm \\ref{alg:bfdc}).\n\\end{proof}\n\n\\begin{algorithm}\n \\caption{\\textsc{DomainCheck}}\n \\label{alg:check}\n \\footnotesize\n \\begin{algorithmic}[1]\n \\Require{Time-point $i$, distance array $\\tau$, domain index array $z$, path length array $\\pi$}\n \\If{$-\\tau_i > u^{z_i}_i$}\\Comment{Domain $[l^{z_i}_i,u^{z_i}_i] \\in D_i$}\n \\For{$z_i=z_i+1$ \\textbf{ until } $|D_i|$}\n \\If{$-\\tau_i \\leq u^{z_i}_i$}\n \\State $\\tau_i \\gets \\min\\{\\tau_i,-l^{z_i}_i\\}$\n \\IfThen{$\\tau_i = -l^{z_i}_i$}{$\\pi_i \\gets 1$}\n \\State \\textbf{break}\n \\EndIf\n \\EndFor\n \\IfThen{$-\\tau_i > U(D_i)$}{$\\tau \\gets \\emptyset$} \\Comment{No domain can accomodate current $\\tau_i$ assignment}\n \\EndIf\n \\end{algorithmic}\n\\end{algorithm}\n\nIt is possible to show that Lemma \\ref{lem:bfdc-correct} also holds for the reversed case: producing the \\textit{latest feasible solution}. For that, domains are sorted in descending order and computations occur over graph $G_R$ instead of $G_D$. This requires minor changes to how Algorithm \\ref{alg:check} works to account for the reversed order of domains., with the general reasoning concerning how the algorithm operates remaining the same. The latest feasible solution $s$, if it exists, can be retrieved directly via $s=\\tau$. Let us now turn to the asymptotic worst-case time complexity of BFDC which is established via Lemma \\ref{lem:bfdc-time}.\n\n\\begin{lemma}\n BFDC stops within a number of iterations proportional to $O(|T||C| + |T|\\omega)$. \\label{lem:bfdc-time}\n\\end{lemma}\n\\begin{proof}\n First, consider that the complexity of \\textsc{BellmanFord} is $O(|T||C|)$ over the same graph $G_D$. The addition of \\textsc{DomainCheck} does not change the size of queue $Q$ and therefore the overall number of iterations remains the same as standard \\textsc{BellmanFord}. The change lies in the computational overhead of each iteration individually.\n \n There are at most $O(|V|)$ phases in \\textsc{BellmanFord} with a first-in, first-out queue. In each phase, a node is extracted from $Q$ at most once \\cite{book:networks}. In other words: the operations taking place in lines 7-11 of Algorithm \\ref{alg:bfdc} are executed at most $|V|$ times per phase. These operations have a complexity equivalent to $O(\\textsc{OutDeg}(i) + |D_i|)$, where \\textsc{OutDeg}$(i)$ denotes the number of arcs in set $A_D$ which have $i$ as their source. Hence, each phase has complexity $O(\\sum_{i \\in V} \\textsc{OutDeg}(i) + |D_i|)$ which is equivalent to $O(|C| + \\omega)$. All together, we arrive at a complexity of $O(|V||C| + |V|\\omega)$ which is equivalent to $O(|T||C| + |T|\\omega)$ when solving SDTPs because $|V| = |T|$.\n\\end{proof}\n\nThe space complexity of BFDC is $O(|T|)$. The auxiliary arrays $\\tau$, $s$, $\\pi$, $z$ and queue $Q$ used in BFDC all require additional space proportional to $|T|$.\n\n\n\\subsection{Asymptotic worst-case complexities}\n\nLet us now assess the theoretical complexities of all the algorithms and draw some initial conclusions concerning what one should expect from empirical results. Table \\ref{tab:complexities} provides both the asymptotic worst-case time complexity and space complexity for each algorithm according to our implementation. Given that the ILP, CP and SCP models are often solved by means of general-purpose black-box solvers, we opted not to include their theoretical complexities in our analysis.\n\n \\begin{table}[!htb]\n \\centering\n \\caption{Asymptotic worst-case complexities for each algorithm.}\n \\label{tab:complexities}\n \\begin{tabular}{lrr}\n \\hline\n Algorithm & Time complexity & Space complexity\\\\\n \\hline\n ULT & $O(|T|^3|C|K + |C|^2K^2)$ & $O(|T|^2)$\\\\\n KAB & $O(|T|^2|C| + \\omega^\\frac{5}{2})$ & $O(|T|^2 + \\omega^2)$\\\\\n KAJ & $O(|T||C|\\log |T| + \\omega^\\frac{5}{2})$ & $O(|T|^2 + \\omega^2)$\\\\\n CRA & $O(|T||C|\\log |T| + |T|\\omega)$ & $O(|T|^2)$\\\\\n RULT & $O(|T|^2|C|K + |T|^2K^2)$ & $O(|T|)$\\\\\n BFDC & $O(|T||C| + |T|\\omega)$ & $O(|T|)$\\\\\n \\hline\n \\end{tabular}\n \\end{table}\n\n In terms of worst-case time complexity, BFDC clearly outperforms all other methods. CRA is the second fastest method. Meanwhile, it is difficult to rank KA and RULT because they have different terms which can dominate one another. Note that $\\omega = O(|T|K)$ and therefore whenever $\\omega \\leq (|T|K)^{\\frac{4}{5}}$ the second term (max-flow) in KA's complexity is never slower than RULT's second term. In this case, we can limit our comparison to the first term referring to shortest paths. Clearly, both KAB and KAJ are asymptotically faster than RULT in this regard. However, when $\\omega \\approx |T|K$ the time complexity of KA is lower than RULT's due to the max-flow phase. As previously mentioned, we can also see that the use of Johnson's Algorithm in KA reduces its time complexity, bringing KAJ closer to CRA. Finally, ULT is the slowest algorithm in Table \\ref{tab:complexities}, mainly due to its heavy utilization of \\textsc{FloydWarshall}. \n \n The time complexities documented in Table \\ref{tab:complexities} are indicative of the challenges faced when solving SDTPs. Despite their close ties to shortest path problems, the presence of negative cycles and disjunctive domains requires more complex and refined techniques. In particular, we wish to call attention to the increased space complexity in most of the established techniques in the literature. Only RULT and BFDC are able to solve SDTPs using linear space. Although this may appear unimportant given the availability of computational resources, a quadratic memory overhead can quickly become prohibitive in practice. This is often problematic given that SDTPs appear as subproblems of other more complex problems which require their own share of memory. We will discuss the impact of memory usage later in Section \\ref{sec:discussion}. \n \n A final remark concerns the relation between SDTPs and time-dependent STPs established in Section \\ref{sec:sdtp}. \\citet{art:td-stp} showed how the time-dependent STP can be solved in time $O(|T||C|)$. However, despite the relation between the two problems, Table \\ref{tab:complexities} shows that solving SDTPs requires asymptotically more time than time-dependent STPs in the worst case. This is primarily due to the discontinuity of time-point domains which renders certain assignments in SDTP solutions infeasible, thereby requiring additional procedures to correct the assignments and (re)check feasibility. When at most one domain exists per time-point ($K \\leq 1$), this correction is not necessary. \n \n\n\\section{Experiments} \\label{sec:exps}\n\nExperiments were carried out on a computer running Ubuntu 20.04 LTS equipped with two Intel Xeon E5-2660v3 processors at 2.60GHz, with a total of 160 GB RAM, 5 MB of L2 cache and 50 MB of L3 cache. Intel's Hyper-threading technology has been disabled at all times to avoid negatively influencing the experiments. All of the algorithms were implemented using \\texttt{C++} and compiled with GNU GCC 9.3 using optimization flag \\texttt{-O3}. The ILP, CP and SCP models were implemented using the \\texttt{C++} API of CPLEX 12.9. Methods were only allowed one thread during execution.\n\nOur experiments primarily focus on measuring observed computation times. In order to obtain accurate time measurements, we employ \\texttt{C++}'s \\texttt{std::steady\\_clock} to measure CPU time. To ensure as much fairness as possible when comparing methods that differ significantly with respect to the input representation, we decided to document only the computation time for solving an instance. This means our results do not include information concerning the time needed for input, output or preprocessing that is performed by some algorithms to transform data into a more suitable format. Similarly, the time to build ILP, CP and SCP models is not included in their results.\n\nWhile we understand that evaluating methods with respect to computation times is not always ideal \\cite{art:exp1,art:exp2}, it is difficult to obtain a single evaluation metric for algorithms that differ so much in terms of their basic operations and components. Additionally, \\citet{art:shapiro} argued that for tractable problems, the running time of algorithms is often a reasonable metric. \n\nSince we are proposing the first experimental study to evaluate algorithms for solving SDTPs, we introduce four datasets to assess the performance of the various methods and their implementations. The four datasets differ in terms of their problem structure and particularly with respect to the underlying distance graph. Instances are subdivided into \\textit{shortest path instances}, \\textit{negative cycle instances}, \\textit{vehicle routing instances} and \\textit{very large instances}. All of them include only integer values. It is therefore possible to accommodate all methods, including CP and SCP, without any changes. We will begin by first detailing the procedure by which we generated each instance set before presenting the computational results obtained from our experiments.\n\n\\subsection{Shortest path instances} \\label{sec:sp-instances}\n\nInstances are created with graph generators for Shortest Path (SP) problems. A graph $G=(V,A)$ is transformed into an SDTN $N=(T,C)$ by setting $T = V$, $C_1 = A$ and deriving $C_2$ constraints for the time-point domains from the shortest paths in $G$. Let us define the following parameters for an SDTP instance: number of time-points $|T|$, number of Type 1 constraints $|C_1|$, number of elements with more than one domain $|T_D|$, and number of domains $K > 1$ per $i \\in T_D$ such that $|D_i| = K$. There are four SP groups which differ in terms of how either graph $G$ or $C_2$ constraints are created. The generation process for each group is summarized below (for more details see Appendix \\ref{ap:instances}).\n\n\\begin{enumerate}\n \\item \\textsc{Rand}: generates graph $G$ using \\textsc{Sprand} introduced in the SPLib \\cite{art:splib}. Nodes and arcs are all created randomly. Constraints $C_2$ are generated based on the shortest path from a dummy node to every $i \\in V$. \n \\item \\textsc{Grid}: generates graph $G$ using \\textsc{Spgrid}, also introduced in the SPLib \\cite{art:splib}. Nodes are generated in a grid format with $X$ layers and $Y$ nodes per layer. Arcs connect nodes within the same layer and to those in subsequent layers. $C_2$ constraints are generated in the same way as for \\textsc{Rand}. \n \\item \\textsc{Seq}: generates graph $G$ using the tailored generator \\textsc{Spseq}. Nodes are generated at random similarly to \\textsc{Sprand}. A path connecting all nodes with $|V|-1$ arcs is created where the weight of all arcs is $w_{ij}=1$. Afterwards, the remaining $|A|-|V|+1$ arcs are created at random with greater weights. This creates a known shortest path which may be difficult for some methods to find. $C_2$ constraints are generated in the same way as for \\textsc{Rand} and \\textsc{Grid}.\n \\item \\textsc{Late}: generates graph $G$ using either \\textsc{Sprand} or \\textsc{Spseq}. $C_2$ constraints are created so that at least 60\\% of the earliest feasible solutions $s_i$ belong to the last domain of the respective time-point. \n\\end{enumerate}\n\nFor each of these four datasets, we also create four subsets to assess which key instance characteristics have the biggest impact on algorithmic performance. For each of these subsets we fix three of the parameters of an SDTP, and then vary the fourth. \n\n\\subsubsection{\\textsc{Nodes} dataset}\n\nThe number of time-points $|T|$ varies in the range \\{100,200,\\dots,12800,25600\\} for dataset \\textsc{Nodes}. Other parameters are fixed to $|C_1| = 6\\cdot|T|$, $|T_D|=0.8\\cdot|T|$ and $K=10$. Five instances were generated for each combination of dataset (1)-(4) and number of time-points $|T|$ (henceforth denoted a configuration): three feasible and two infeasible instances. For example, five instances have been generated for configuration (\\textsc{Rand}, \\textsc{Nodes}, $|T|=100$). \n\nFigure \\ref{fig:sp-nodes} provides the results for the \\textsc{Nodes} subset. Each graph reports the average computation times for each method according to the number of time-points for each dataset. The values reported are the average from 20 runs, so as to mitigate the impact of any outliers due to the short computation times needed to solve SDTPs \\cite{art:shapiro}. Additionally, methods are given a time limit of two seconds. If a method timed out for all instances of a given size, we omit these results for clarity. This explains the incomplete curves present in some of the graphs. However, if for some instance sizes a method could solve at least one instance (out of five), we report the averages including potential timeouts. KAJ rarely outperformed KAB. Based on these results, we decided to only show the results for KAB. In the experiments, we will comment on specific differences between the two methods whenever necessary.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/graph_sp_nodes.pdf}\n \\caption{Average computation times in microseconds ($\\mu$s) for the \\textsc{Nodes} subset. Both $x$- and $y$-axis are reported in log scale.}\n \\label{fig:sp-nodes}\n\\end{figure}\n\nThe general performance of the algorithms in graphs of Figure \\ref{fig:sp-nodes} is somewhat consistent. We can clearly see a cluster of curves towards the top of the graphs which include ULT, CP, SCP, ILP and KAB. It is also easy to distinguish a second cluster formed by RULT and BFDC at the bottom of the graphs. Meanwhile, CRA lies in-between these two clusters, typically starting at the bottom for small instances and trending towards the top for the largest ones. These differences are not surprising given that most methods in the top cluster are more general than those in the bottom cluster (including CRA). While these differences are to be expected, the question as to whether they hold in different scenarios must still be answered.\n\nULT demonstrates the fastest growth in the graphs and can rarely solve instances containing $|T| > 1000$. This behavior is easily explained by the use of \\textsc{FloydWarshall} in every iteration, which contributes to a $\\Theta(|T|^3)$ time complexity. CP and SCP are relatively consistent in execution time, with SCP able to solve slightly larger instances in \\textsc{Rand}, \\textsc{Seq} and \\textsc{Late}. For \\textsc{Late} instances, SCP outperformed CP in all scenarios. This showcases how CPLEX as a CP solver can benefit from SCP's model structure. Meanwhile, ILP is the quickest method in the topmost cluster of methods for \\textsc{Rand}, \\textsc{Grid} and \\textsc{SEQ}, while it performs similarly to SCP for \\textsc{Late} instances.\n\nOne can observe that KAB only outperforms other methods in the topmost cluster for small instances ($|T| \\leq 800$). The reason is that computing APSPs requires a significant amount of time and quickly becomes prohibitive for larger instances. For \\textsc{Late} instances, KAB was unable to solve those where $|T| > 3200$. This is because \\textsc{Late} instances tend to have larger interval sets $E$, which directly impact the creation of the conflict graph (line 3 in Algorithm \\ref{alg:kra}) thereby limiting KA's execution time.\n\nCP, SCP, ILP, ULT, and KAB always take longer than one millisecond to compute results. Meanwhile, CRA begins below or at this threshold in all cases and grows quickly, often reaching the one-second threshold. Nevertheless, we can see that CRA typically performs better than ILP. Even when CRA is slower, the differences are not significant. This is true except for the \\textsc{Late} instances, where CRA timed out for all five instances with $|T| = 25600$. In these cases, the combination of many time-points and late feasible schedules leads CRA to compute more entries of the distance matrix using \\textsc{Dijkstra}, causing major computational overhead for the method (line 8 in Algorithm~\\ref{alg:cra}). Both RULT and BFDC always remain below the 100 milliseconds threshold and are faster than CRA. This is despite the fact that RULT has a theoretical worst-case time complexity slower than CRA. We did not observe any timeout for either method in the bottom cluster, which contributes to their lower curves in Figure \\ref{fig:sp-nodes}. The relative position of both algorithms is also very consistent, with RULT only slightly slower than BFDC.\n\n\\subsubsection{\\textsc{Density} dataset}\nThe second subset is \\textsc{Density}, in which we vary the number of constraints $C_1$ in the range \\{20,25,\\dots,85,90\\}\\%, given as a percentage of the maximum number of constraints (maximum number of arcs in the base graph). Other parameters are fixed as follows: $|T|=1008$, $|T_D|=0.8\\cdot|T|$ and $K=10$. Similar to \\textsc{Nodes}, five instances are generated per configuration. Average computation times according to density growth are shown in Figure \\ref{fig:sp-density}.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/graph_sp_density.pdf}\n \\caption{Average computation times for the \\textsc{Density} subset. The $y$-axis is reported in log scale.}\n \\label{fig:sp-density}\n\\end{figure}\n\nOn the one hand, the performance of ULT barely changes with respect to varying densities due to \\textsc{FloydWarshall}'s phase which maintains ULT among the slowest methods. On the other hand, ULT was always able to solve at least one instance per configuration which is not true for all methods. Except for the \\textsc{Grid} instances, CP and SCP experience difficulties solving problems with a density greater than $40\\%$. KAB far outperforms ULT for \\textsc{Grid} instances, but generally performs similarly to ULT in all other cases. KAB is also never slower than CP or SCP. In terms of the topmost cluster, ILP was consistently the best performing method.\n\nHere we notice that CRA demonstrates far better performance than ULT, CP, SCP, ILP and KAB for the \\textsc{Rand}, \\textsc{Grid} and \\textsc{Seq} instances. It remains a sort of middling algorithm, but shows little variation in performance resulting from the network's density. However, for the \\textsc{Late} instances, CRA's performance compares to that of the ILP.\nSimilar to the \\textsc{Nodes} dataset, this is explained by the overhead incurred by \\textsc{Dijkstra} computations. For the \\textsc{Density} instances, however, the network is more connected, leading to more time-points being affected by changes made to others. This in turn also requires more assignment updates.\n\nBoth RULT and BFDC also show little variation with respect to the network's density. They remain the fastest algorithms, with no timeouts observed. \n\n\\subsubsection{\\textsc{NumDisj} dataset} \n\nIn the third subset, we vary the number of domains $K$ per time-point $i \\in T_D$ in the range \\{5,10,20,\\dots,90,100\\}. Other parameters are fixed as follows: $|T|=2000$, $|C_1| = 12000$ and $|T_D|=1600$. Figure \\ref{fig:sp-numd} shows the average computation times according to parameter $K$.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/graph_sp_numd.pdf}\n \\caption{Average computation times for the \\textsc{NumDisj} subset. The $y$-axis is reported in log scale.}\n \\label{fig:sp-numd}\n\\end{figure}\n\nIn these experiments, we can still differentiate the three clusters of methods from before, but their individual behaviors are now distinct. ULT suffers far more timeouts and often cannot solve a single instance in any configuration. However, this appears unrelated to parameter $K$ and more due to some other specific instance characteristic that was not captured in these experiments. CP and SCP can solve most instance sizes. Additionally, SCP is faster and can solve more instances as the value of $K$ increases compared to CP. This difference is pronounced for dataset \\textsc{Late}, where SCP not only outperforms CP but also CRA and ILP in all cases. Despite its slow performance for the \\textsc{Late} instances, ILP outperforms ULT, CP, SCP and KAB across all other configurations.\n\nKAB experiences the same difficulties solving \\textsc{Late} instances, where only the smallest ones with $K=5$ were solved. This is unsurprising since $\\omega$ is directly related to $K$ (recall Section \\ref{sec:sdtp}). KAJ performed slightly better than KAB and was able to solve \\textsc{Seq} instances with $K \\leq 90$. CRA again outperforms all methods in the topmost cluster except for the \\textsc{Late} instances, where SCP is faster. For \\textsc{Seq}, we notice the power of these \\textsc{Dijkstra} computations because they help CRA to easily find the hidden shortest path used during the instance's construction. This then leads to much shorter executions. RULT and BFDC remain the fastest methods. However, RULT is clearly impacted to a far greater extent by the growth in the number of time-point domains compared to BFDC for the \\textsc{Late} instances. This observation is aligned with their asymptotic worst-case time complexities.\n\nThe graphs in Figure \\ref{fig:sp-numd} suggest that the methods solved using CPLEX (CP, SCP and ILP) are those most impacted by increases in $K$. This may be due to the number of constraints created when more domains exist and an increase in the cardinality of sets $\\Phi_i$ in the SCP model.\n\n\\subsubsection{\\textsc{VarDisj} dataset} The fourth and final subset for SP instances is \\textsc{VarDisj}. This subset varies the size of $T_D$ in the range \\{10,20,\\dots,90,100\\}\\%, given as a percentage of the total number of time-points $|T|$ that have multiple domains. Other parameters are fixed as follows: $|T|=2000$, $|C_1|=6\\cdot|T|$ and $K=10$. Figure \\ref{fig:sp-vard} provides computation runtime results according to the size of set $T_D$.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/graph_sp_vard.pdf}\n \\caption{Average computation times for the \\textsc{VarDisj} subset. The $y$-axis is reported in log scale.}\n \\label{fig:sp-vard}\n\\end{figure}\n\nFigure \\ref{fig:sp-vard} shows how the results for this subset of instances differ significantly from the previous subsets. First, for instances \\textsc{Rand}, \\textsc{Grid} and \\textsc{Seq} the methods are now far more distinctly dispersed across different parts of the graphs. ULT is once again the slowest method, highlighting its difficulty in computing APSPs. KAB competes with CP and SCP in dataset \\textsc{Rand}, but is clearly slower than these methods for \\textsc{Grid} and \\textsc{Seq}. Similar to \\textsc{NumDisj}, KAJ demonstrated slightly better performance for \\textsc{Seq} instances than KAB, yet not enough to outperform CP or SCP. Indeed, these two methods always outperform KAB and ULT. Meanwhile, ILP is quicker than the previous four methods, maintaining its somewhat consistent behavior as the best general-purpose algorithm for solving SDTPs. CRA is clearly the method that suffers the most from an increasing number of time-points that have multiple domains. Nevertheless, for the first three datasets, CRA is always faster than ILP and for the \\textsc{Seq} instances even outperforms RULT for small ones. Finally, RULT and BFDC remain the fastest methods and as the size of set $T_D$ increases there is little noticeable impact on their performance.\n\nWhen we consider the \\textsc{Late} instances, which are arguably the most difficult to solve, the situation changes. ULT cannot solve a single instance in this set. KAB appears as the slowest method. Meanwhile, CP, SCP, ILP and CRA are all clustered below KAB. CP is clearly the slowest of the four methods. ILP and CRA exhibit some variations, but overall their growth trend is far less pronounced than for the other three sets (\\textsc{Rand}, \\textsc{Grid} and \\textsc{Seq}). SCP is very consistent across all experiments and for large $T_D$ sets outperforms the other three algorithms, albeit not by a very large margin. RULT is a whole order of magnitude slower than BFDC for almost all datasets except for \\textsc{Seq}. Nevertheless, neither RULT nor BFDC exhibit any significant variations in performance.\n\n\\subsection{Negative cycle instances}\n\nThe \\textsc{Negcycle} instances use the filter of same name proposed by \\citet{art:sp-fp}. This filter is applied to instances from Section \\ref{sec:sp-instances} by introducing a negative cycle into the underlying base graph. Only instances that are feasible before applying the filter are considered so that they become infeasible precisely due to the negative cycle. Similar to \\citet{art:sp-fp}, we consider four classes of negative cycles: (\\textsc{Nc02}) one cycle with three arcs; (\\textsc{Nc03}) $\\lfloor \\sqrt{|T|} \\rfloor$ cycles with three arcs each; (\\textsc{Nc04}) $\\lfloor \\sqrt[3]{|T|} \\rfloor$ cycles with $\\lfloor \\sqrt{|T|} \\rfloor$ arcs each; and (\\textsc{Nc05}) one Hamiltonian cycle. \n\nFor each one of these four classes, we vary $|T|$ in the range \\{100,200,\\dots,12800,25600\\} while fixing parameters $|C_1|=6\\cdot|T|$, $|T_d|=0.8\\cdot|T|$, $K=10$. For each configuration, we generate three instances. Figure \\ref{fig:nc} provides the computation results for each of the negative cycle classes, with the average time to prove infeasibility reported for each method over 20 runs.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/graph_all_nc.pdf}\n \\caption{Average computation times for the \\textsc{Negcycle} datasets. Both the $x$- and $y$-axis are reported in log scale.}\n \\label{fig:nc}\n\\end{figure}\n\nAn initial observation is that ULT experiences greater difficulty solving instances with fewer and smaller cycles (\\textsc{Nc02} and \\textsc{Nc03}). CP was unable to solve the largest instances in any of the cases, while SCP was able to prove infeasibility of the large instances when more than one cycle existed (\\textsc{Nc03} and \\textsc{Nc04}). Meanwhile, ILP demonstrated a very consistent performance despite the cycles.\n\nFor the first time, we notice KAB among the fastest algorithms at the bottom of the graphs. This is not surprising since for \\textsc{Negcycle} instances KAB is able to detect the cycle when computing distance matrix $\\delta$ (line 1 in Algorithm \\ref{alg:kra}). Also for the first time, BFDC demonstrates consistently the lowest execution time for \\textsc{Nc02} compared to CRA, RULT and KAB. For the same dataset, CRA is consistently the fastest method because it can also detect negative cycles in its first phase with standard \\textsc{BellmanFord} (line 1 in Algorithm \\ref{alg:cra}). This showcases the overhead incurred by \\textsc{DomainCheck} in BFDC (line 8 in Algorithm \\ref{alg:bfdc}).\n\nHowever, in datasets \\textsc{Nc03}-\\textsc{Nc05} the methods which feature in the cluster at the bottom are harder to differentiate. KAB typically appears to be the slowest, CRA the fastest, with BFDC and RULT lying somewhere in the middle. Nevertheless, the differences between KAB, RULT, BFDC and CRA for \\textsc{Negcycle} instances are negligible for most purposes.\n\n\\subsection{Vehicle routing instances}\n\nWe extract vehicle routing instances (\\textsc{Vrp}) from solutions to Vehicle Routing Problems with Multiple Synchronization (VRPMS) constraints \\cite{art:vrpms}. These VRPMS instances contain multiple routes for which departure times and service times must be assigned while complying with synchronization constraints between routes, in addition to maximum route duration constraints. We refer interested readers to Appendix \\ref{ap:instances} for more information concerning how exactly these instances have been generated.\n\n\n\\textsc{Vrp} instances primarily differ in terms of their number of time-points, which ranges from 10 to 1300. For each instance size, we again create five instances: three feasible and two infeasible. Figure \\ref{fig:vrp} presents the computation times per method according to the number of time-points in the instance, with the values reporting the average over 20 runs. Results are grouped according to instance feasibility given that some differences in performance can be observed depending on whether an instance is feasible.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/graph_all_vrp.pdf}\n \\caption{Average computation times for the \\textsc{Vrp} dataset. The $y$-axis is reported in log scale.}\n \\label{fig:vrp}\n\\end{figure}\n\nOne can quickly notice that ULT can easily prove infeasibility of instances, but it experiences difficulty producing solutions for feasible instances. Indeed, for feasible instances ULT cannot solve those where $|T| > 1000$. CP and SCP exhibit different performances, with SCP faster on average, particularly for infeasible instances. KAB consistently performs similarly to CP. The performance of ILP varies significantly for feasible instances and somewhat less significantly for the infeasible cases. It is difficult to conclude whether ILP is faster than SCP overall, although we can easily see that both methods are faster than ULT, CP and KAB. \n\nCRA, RULT and BFDC are once again clustered towards the bottom of the graph, signifying that they are the fastest methods. RULT is the slowest among the three, although CRA does vary a lot and is slower for certain cases. BFDC can be concluded to be the fastest method although, yet again, CRA does outperform BFDC in certain cases. Overall though, one can summarize the order from slowest to fastest as follows: RULT, CRA and BFDC.\n\nFinally, note that \\textsc{Vrp} instances all have an underlying distance graph which is very sparse. Vertices have at most two outgoing arcs and two incoming arcs, with the exception of those connected to the beginning of the time horizon $\\alpha$. Additionally, the graphs are almost acyclical and contain a very limited number of arcs that create cycles. While we do not exploit this structure when solving the \\textsc{Vrp} dataset, this would certainly be an interesting avenue for future research.\n\n\\subsection{Very large instances}\n\nAlthough the three previous datasets have diverse characteristics, they all fail to capture scenarios where the number of time-points is very large. These types of instances are important when it comes to truly verifying the scalability of methods which may have advantages for small-scale problems yet suffer when instance size grows significantly. To verify the extent to which our previous analysis holds for such instances, we generate five very large (\\textsc{Vl}) problems, all of which are feasible. Table \\ref{tab:vl} presents the characteristics of these instances in terms of their \\textit{Base} generation method, number of time-points, number of $C_1$ constraints, maximum number $K$ of domains per time-point and total number of domains $\\omega$. Base \\textsc{Tsp} means that graph $G$ was extracted from the Traveling Salesman Problem Library (TSPLib) \\cite{art:tsplib} instance \\texttt{pla85900} that contains 85900 nodes. For instance \\textsc{Vl}-1, a random subset of nodes is selected from \\texttt{pla85900}. Meanwhile, instances \\textsc{Vl}-3, \\textsc{Vl}-4 and \\textsc{Vl}-5 are generated with the SP procedures outlined in Section \\ref{sec:sp-instances}.\n\n\n\\begin{table}[!ht]\n\\caption{Very large instances.}\\label{tab:vl}\n\\centering\n\\begin{tabular}{llrrrr}\n\\hline\nInstance & Base & $|T|$ & $|C_1|$ & $K$ & $\\omega$\\\\\n\\hline\n\\textsc{Vl-1} & \\textsc{Tsp} & 50 000 & 500 000 & 20 & 905 000\\\\\n\\textsc{Vl-2} & \\textsc{Tsp} & 85 900 & 859 000 & 60 & 4 647 190\\\\\n\\textsc{Vl-3} & \\textsc{Seq} & 200 000 & 2 000 000 & 100 & 16 040 000\\\\\n\\textsc{Vl-4} & \\textsc{Late} & 400 000 & 4 000 000 & 180 & 57 680 000\\\\\n\\textsc{Vl-5} & \\textsc{Rand} & 1 000 000 & 10 000 000 & 500 & 400 200 000\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nFigure \\ref{fig:vl} provides a graph documenting the average execution times over 10 runs for a subset of the methods with the \\textsc{Vl} instances. We opted to test only the best performing methods: ILP, CRA, RULT and BFDC. For each run, the methods were given a one hour time limit.\n\nEven among the four best performing methods, it is obvious that CRA and ILP are limited with respect to the instance sizes they can solve. Indeed, they were unable to solve instances beyond \\textsc{Vl-3} within one hour of execution. While it is difficult to determine the precise reason for the behavior of the ILP solver, the reason for CRA lies in how large instances require more \\textsc{Dijkstra} computations (in particular for \\textsc{Vl-4}). The computation of APSPs is not only slow but also leads to far worse memory locality. This impacts cache usage which reduces the performance of CRA compared to RULT, even though the latter has a theoretically slower worst-case time complexity. Figure \\ref{fig:vl} also showcases the fact that \\textsc{Late} instances are much more difficult than other instances. This is true even when a \\textsc{Late} instance has less than half the number of time-points of a \\textsc{Rand} instance. \n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{images\/graph_all_vl.pdf}\n \\caption{Average computation times for the \\textsc{Vl} dataset.}\n \\label{fig:vl}\n\\end{figure}\n\n\\section{Discussion} \\label{sec:discussion}\n\n \\citet{art:exp2} noted that a lot of what is reported in experimental papers are observations about the \\textit{implementation} of an algorithm rather than the algorithm itself as a mathematical object. On the one hand, our study somewhat conforms to this trend. Despite being a well-known limitation of empirical studies, we hope to mitigate this by providing our code so that interested readers can inspect and even improve upon the implementations. On the other hand, some results documented in this paper are implementation-independent. For example: the reduced space complexity achieved by both RULT and BFDC compared to all other methods.\n\nLet us now turn our attention to a broad analysis of the computational study. Table \\ref{tab:summary} summarizes the results of the experiments from Section \\ref{sec:exps}, with the exception of \\textsc{Vl} instances. Columns \\textit{Max. time (ms)}, \\textit{Avg. time (ms)} and \\textit{Std. time (ms)} report the maximum, average and standard deviation of the recorded execution times per method in milliseconds. Column \\textit{Total time (s)} reports the total time required by each method to solve all of the instances, including eventual timeouts. Finally, column \\textit{Timeouts (\\%)} provides the percentage of runs for which the method timed out.\n\nWhen considering Table \\ref{tab:summary}, one must take into account the fact that ULT, CP, SCP and ILP are all more general than CRA, RULT, and BFDC. Hence, it should not come as a surprise that the latter algorithms outperform the former in almost all cases. Nevertheless, our experiments show that there are major differences in performance between algorithms when solving SDTPs and certain conclusions may appear counter-intuitive at first. For example, despite the polynomial worst-case asymptotic time complexity of ULT and KA, these algorithms exhibit poor general performance when solving SDTPs. Meanwhile, ILP demonstrated good performance for a problem that might have initially seemed more suitable for constraint programming.\n\n\\begin{table}[!ht]\n\\caption{Summary of results.}\\label{tab:summary}\n\\centering\n\\begin{tabular}{lrrrrr}\n\\hline\nMethod & Max. time (ms) & Avg. time (ms) & Std. time (ms) & Total time (s) & Timeouts (\\%)\\\\\n\\hline\nILP & 1947 & 350 & 462 & 7058 & 2.28\\\\\nCP & 2017 & 1062 & 985 & 21406 & 33.39\\\\\nSCP & 2002 & 875 & 954 & 17633 & 25.15\\\\\nULT & 1990 & 1476 & 812 & 29758 & 66.76\\\\\nKAB & 2000 & 868 & 873 & 17491 & 28.02\\\\\nKAJ & 2001 & 1065 & 1276 & 21478 & 30.26\\\\\nCRA & 1895 & 138 & 317 & 2772 & 1.19\\\\\nRULT & 152 & 6 & 13 & 115 & 0.00\\\\\nBFDC & 28 & 1 & 2 & 18 & 0.00\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nProfiling the implementations of both KAB and KAJ showed that $\\approx 90\\%$ of their execution time was consistently spent building distance matrix $\\delta$ and computing conflicts (lines 1-3 of Algorithm~\\ref{alg:kra}). Less than $5\\%$ of the total time was observed to be incurred by max-flow computations (line 6 of Algorithm~\\ref{alg:kra}). This showcases how the bottleneck is the computation of APSPs and conflicts rather than the theoretically slower max-flow step.\n\nSimilarly, profiling the implementation of CRA showed that \\textsc{Dijkstra} computations were responsible for up to 95\\% of the processing time. This observation includes our lazy evaluation implementation of the distance matrix. When the full matrix is precomputed as originally described by \\citet{art:cra}, the proportion of time spent on \\textsc{Dijkstra} could grow even more extreme. In many cases, precomputation of the distance matrix was not possible within the imposed time limit. Furthermore, precomputing the distance matrix for large instances is simply not possible due to insufficient memory. In spite of these drawbacks, one advantage of CRA is that some instances can be solved quickly during its first stage (\\textsc{BellmanFord}) when the initial solution is already feasible and there is therefore no time-point assignment $s_i$ which must undergo corrections.\n\nThe results detailed in Table \\ref{tab:summary} further confirm those observed in Section \\ref{sec:exps}. RULT and BFDC present the best performances overall, with computation times that are between two and three orders of magnitude shorter than all other methods. There is also no record of either of these algorithms timing out during our experiments. This performance can be explained by two factors. First, both RULT and BFDC focus on computing single-source shortest paths while ULT, KA and CRA consume a lot more computational resources solving APSPs. Second, and this comes as a direct consequence of the first factor, both RULT and BFDC have linear space complexity using only one-dimensional arrays of size $|T|$, thereby improving their cache locality and overall efficiency. Indeed, some instances could be solved almost entirely in cache by these two methods, while the quadratic space complexity of other methods made this far more unlikely.\n\nFigure \\ref{fig:cache} illustrates cache reference measurements for CRA, RULT and BFDC when solving the \\textsc{Vl} instances. We focus on this dataset because it required the most algorithmic effort. Recording of cache reference events was performed using the \\texttt{perf\\_events} package from the Linux kernel \\cite{web:perf}. The \\textit{Full scale} row in the top half of the figure demonstrates how difficult it can sometimes be to compare the behavior of different approaches. This is why we have also included the \\textit{Small scale} graphs below, which zoom into the \\textit{Full scale} graphs in order to reveal further details concerning behavior of each method. These graphs make it clear how both RULT and BFDC can solve the large instances much quicker simply by using the cache more efficiently. In all experiments both RULT and BFDC required fewer total cache references than the number of cache misses by CRA.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.95\\linewidth]{images\/graph_all_cache.pdf}\n \\caption{Cache references per method in the \\textsc{Vl} dataset.}\n \\label{fig:cache}\n\\end{figure}\n\nThe methods implemented in this paper also differ with respect to the type of solution produced. ULT, CRA, RULT and BFDC provide the earliest feasible solution at the end of their execution. However, ULT, RULT and BFDC can also return the latest feasible solution. \\citet{art:cra} did not comment on whether their method could return the latest feasible solution, although it appears possible when computing solutions over graph $G_R$ and with some minor changes to operations (e.g., \\textsc{UpdateAssignments}). By contrast, CP, SCP, ILP and KA are not guaranteed to return either the earliest or latest feasible solution. One could ensure finding either one of them by defining an appropriate objective function for the underlying model, but it is unclear how much this would impact their performance. For example, KA would require the solution of max-flows with arbitrary arc capacities rather than unitary capacities \\cite{art:kumar-estp}.\n\nWhile one could be tempted to conclude that BFDC and RULT should be the go-to methods when faced with SDTPs, this is not necessarily the conclusion we advocate for. Our advice is instead a little more nuanced. Given that BFDC performed the best for SDTPs on isolation, it represents the most sensible choice when evaluating, for example, the feasibility of interdependent vehicle routes. However, other problems which feature SDTPs may benefit from other algorithms to achieve the best performance. For restricted disjunctive temporal problems, \\citet{art:cra} introduced a method which exploits CRA's structure to obtain a low time complexity. In theory, it is also possible to employ BFDC, but this would increase the asymptotic worst-case time complexity of the algorithm for RDTPs. Similarly, \\citet{art:kumar-estp} showed that KA can be employed with minor changes to solve SDTPs where each domain is assigned an arbitrary preference weight and the goal is to find a feasible solution which maximizes the sum of the selected domains. In this problem context it is not clear how one could employ BFDC or RULT. On the other hand, we have shown empirically that KA experiences difficulty solving even medium-sized instances. Therefore, it may be worth considering further research on faster methods to solve these SDTPs with preferences.\n \n\\section{Conclusion}\n\nSimple disjunctive temporal problems generalize simple temporal problems. They have a wide range of real-world applications where they typically arise as subproblems. Some examples of application domains include robot planning, workforce scheduling, logistics and management systems. SDTPs can also be used in decomposition methods to solve more general temporal constraint satisfaction problems. It is therefore of interest for both researchers and practitioners to understand the empirical performance of algorithms for solving SDTPs in addition to their theoretical time bounds. Unfortunately, the literature previously understood very little about these methods in practice.\n\nTo bridge this gap and bring theory and experimentation in these temporal problems closer together, we provided a large exploratory and empirical study concerning new and established algorithms for solving SDTPs. Our results indicate that theoretical worst-case time complexities are not necessarily indicative of the observed computation times of these algorithms. Moreover, we showed that the quadratic space complexity of previous algorithms comes with several drawbacks that limit their use in practice, regardless of their asymptotic time complexity. Indeed, for very large datasets, some methods were unable to solve an otherwise simple problem due to memory limitations, even when executed on modern computers. By contrast, algorithms which possess a lower space complexity albeit a higher time complexity solved very large instances within only a few seconds.\n\nWe hope that the results of this paper provide useful evidence for future researchers that helps them make informed decisions concerning the best algorithm for their application, thereby reducing reimplementation efforts. The code we have made publicly available will also help future research test whether our conclusions hold for other datasets. Finally, our implementations also provide some common ground for benchmarking new algorithms or speedup techniques for simple disjunctive temporal problems and their special cases.\n\n In terms of future research, one could consider performing a similar computational experiment for the restricted disjunctive temporal problem \\cite{art:kra}. Instances could be derived from those introduced in this paper by adding Type 3 constraints. Another option is to investigate algorithms for variants of the SDTP with preferences associated with each domain of a time-point \\cite{art:kumar-estp}. For example, a certain time-point may have greater preference to be executed in the morning rather than in the evening. Another possibility is to extract SDTP instances from real-world applications and verify whether the conclusions from our research remain valid for other graph structures or if better performing methods exist.\n\n\n\n\\begin{acks}\n\n\nThis research was supported by Internal Funds KU Leuven (IMP\/20\/021) and by the Flemish Government, Belgium under \\textit{Onderzoeksprogramma Artifici\u00eble Intelligentie} (AI). Editorial consultation provided by Luke Connolly (KU Leuven).\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nThere is currently a great deal of interest in the rapidly evolving\nfield of observations of the polarization of the cosmic microwave\nbackground (CMB). This interest stems from the fact that such\nobservations have the potential to discriminate between inflation and\nother early-universe models through their ability to constrain an\nodd-parity $B$-mode polarization component induced by a stochastic\nbackground of gravitational waves at the time of last\nscattering~\\citep{kamionkowsky97,seljak97}.\n\nFrom an observational point of view, we are still a long way off from a\ndetection of this $B$-mode signature of inflation. However, much\nprogress has been made recently with the detection of the much\nstronger $E$-mode polarization signal on large scales by the {\\sevensize WMAP}\\,\nexperiment \\citep{page07,nolta08}.\nOn smaller scales, a growing number of\nballoon-borne and ground-based experiments have also measured $E$-mode\npolarization including {\\sevensize DASI}\\, \\citep{leitch05}, {\\sevensize CBI}\\,\n\\citep{sievers07}, {\\sevensize BOOMERANG}\\, \\citep{montroy06}, {\\sevensize MAXIPOL}\\,\n\\citep{wu07}, {\\sevensize CAPMAP}\\, \\citep{bischoff08} and {\\sevensize QUaD}\\,\n\\citep{pryke09}. Most recently, the high precision measurement of\nsmall-scale polarization by the {\\sevensize QUaD}\\, experiment has, for the first\ntime, revealed a characteristic series of acoustic peaks in the\n$E$-mode spectrum and put the strongest upper limits to date on the\nsmall-scale $B$-mode polarization signal expected from gravitational\nlensing by large-scale structure.\n\nBuilding on the experience gained from these pioneering experiments, a\nnew generation of experiments is now under\nconstruction with the ambitious goal of observing the primordial\n$B$-mode signal. Observing this signal is one of the most challenging goals\nof modern observational cosmology. There are a number of reasons why\nthese types of observations are so difficult. First and foremost, the\nsought-after signal is expected to be extremely small -- in terms of the \ntensor-to-scalar ratio\\footnote{%\nOur normalisation conventions follow those adopted in the\n{\\sevensize CAMB}\\, code~\\citep{lewis00}, so that $r$ is the ratio of primordial\npower spectra for gravitational waves and curvature perturbations.\nExplicitly, for slow-roll inflation in a potential $V(\\phi)$,\n$2 r\\approx M_{\\rm Pl}^2 [V'(\\phi)\/V(\\phi)]^2$ where\n$M_{\\rm Pl}= 2.436 \\times 10^{18}\\, \\mathrm{GeV}\/c^2$ is the reduced\nPlanck mass.}, the RMS polarization signal from primordial $B$-modes is\n$0.4 \\sqrt{r}\\, \\mu \\mathrm{K}$, and the current 95 per cent limit $r< 0.22$ from\n{\\sevensize WMAP}\\, temperature and $E$-mode polarization plus distance\nindicators~\\citep{komatsu09} implies an RMS $<180\\,\\mathrm{nK}$. Secondly,\npolarized emission from\nour own galaxy and from extra-Galactic objects act as a foreground\ncontaminant in observing the CMB polarized sky. Although our\nknowledge of such polarized foregrounds is currently limited,\nparticularly at the higher frequencies $\\mathrel{\\rlap{\\lower4pt\\hbox{\\hskip1pt$\\sim$} 100\\,\\mathrm{GHz}$\nof relevance to bolometer experiments, models suggest that such contamination\ncould be an order of magnitude larger than the sought-after signal on\nthe largest scales (e.g. \\citealt{amblard07}).\nThirdly, gravitational lensing by large-scale structure\nconverts $E$-modes into $B$-modes on small to medium scales\n(see~\\citealt{lewis06} for a review) and acts\nas a source of confusion in attempts to measure the primordial\n$B$-mode signal \\citep{knox02, kesden02}. Note however that the\nlensing $B$-mode signal is a valuable source of cosmological\ninformation in its own right and can be used to put unique constraints\non dark energy and massive neutrinos\n(e.g. \\citealt{kaplinghat03,smithchallinor06}).\nUnfortunately these latter two effects (foreground contamination and\nweak gravitational lensing) contrive in such a way as to render $B$-mode\npolarization observations subject to contamination on all angular\nscales (the primordial $B$-mode signal is dominated by foregrounds on\nlarge scales whilst on smaller scales it is swamped by the lensing\nsignal). Last, but not least, exquisite control of systematic and\ninstrumental effects will be required, to much better than $100\\, \\mathrm{nK}$,\nbefore any detection of $B$-modes can be claimed.\nThe sought-after signal is so small that\nsystematic and instrumental effects considered negligible for an\nexquisitely precise measurement of $E$-modes say, could potentially\nruin a detection of $B$-modes, if left uncorrected. One possible\napproach to mitigating some of these systematics in hardware is to\nmodulate the incoming polarization signal such that it is shifted to\nhigher frequency and thus away from low-frequency systematics which\nwould otherwise contaminate it. There are a number of\ntechniques for achieving this including the use of a rotating\nhalf-wave plate (HWP; e.g.~\\citealt{johnson07}),\nphase-switching, or Faraday rotation modulators~\\citep{keating03}.\n\nIn this paper, we investigate the ability of \nmodulation techniques that are either slow or fast \nwith respect to the temporal variation of the\nsignal\nto mitigate a range of possible systematic\neffects. Our analysis is based on simulations, and the\nsubsequent analysis, of data from a ground-based CMB $B$-mode\npolarization experiment. Previous investigations of the impact of\nsystematic effects on $B$-mode observations include the analytic works\nof \\citet{hu03}, \\citet{odea07} and~\\citet{shimon08},\nas well as the simulation-based\nanalysis of \\cite{mactavish08} who based their study on signal-only\nsimulations of the {\\sevensize SPIDER}\\, experiment. The simulation work presented\nhere is complementary to these previous analyses but we also take our\nanalysis further by including realistic noise in our simulations ---\nwe are thus able to quantify not only any bias found, but also any\ndegradation of performance due to the presence of systematic\neffects. Our work makes use of a detailed simulation pipeline which we\nhave created in the context of the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, experiment. Although the\nprecise details of our simulations are specific to {\\sevensize C}$_\\ell${\\sevensize OVER}\\,, our\ngeneral conclusions regarding the impact of modulation on a variety of\nsystematic effects are relevant to all upcoming ground-based $B$-mode\nexperiments and many of them are also relevant for both balloon-borne\nand space-based missions.\n\nThe paper is organised as follows. In Section \\ref{sec:cmb_expts}, we\nreview the relevant upcoming $B$-mode polarization\nexperiments. Section \\ref{sec:sims} describes our simulation technique\nand the systematic effects we have considered. In Section\n\\ref{sec:analysis}, we describe the map-making and power spectrum\nestimation techniques that we use to analyse the simulated\ndata. Section \\ref{sec:results} presents the results from our main\nanalysis of systematic effects. We discuss our results in Section\n\\ref{sec:discussion} where, for clarity, we group the possible\nsystematic effects considered into those that are, and those that are\nnot, mitigated by a modulation scheme. In this section, we also\ndemonstrate the importance of combining information from multiple\ndetectors during analysis and compare the simulated performance of\n{\\sevensize C}$_\\ell${\\sevensize OVER}\\, to the predicted performance from a Fisher\nanalysis. Our conclusions are summarised in Section\n\\ref{sec:conclusions}. Finally, Appendix~\\ref{app:pointing} develops\na simple model of the spurious $B$-mode power produced by\npointing jitter for experiments with a highly redundant scan strategy.\n\n\\section{CMB polarization experiments}\n\\label{sec:cmb_expts}\nA host of experiments are currently under construction (one is, in\nfact, already observing) with their primary goal being to constrain the\ntensor-to-scalar ratio, and thus the energy scale of inflation,\nthrough observations of the $B$-mode component of the CMB. Here, we\ngive a short summary of the planned experiments.\n\\vspace{-3mm}\n\\subsubsection*{(i) Ground-based experiments:}\n\\begin{itemize}\n \\item{{\\sevensize BICEP\/BICEP-2\/KECK} array:} The {\\sevensize BICEP}\\,\n experiment \\citep{yoon06} has recently completed its third and final season of\n operation from the South Pole. The experiment consisted of a total of 98 polarization-sensitive\n bolometers (PSBs) at 100 and 150~GHz. The optical design is\n very clean but with the downside of poor resolution -- 45 arcmin\n full-width at half-maximum (FWHM) at 150~GHz -- which limits the\n target multipole range to $\\ell <300$. The stated target sensitivity\n is $r = 0.1$. In the first observing season, three 100~GHz pixels and\n three 150~GHz pixels were equipped with Faraday rotation modulators\n and two pixels operated at 220~GHz. {\\sevensize\n BICEP-2} will consist of an upgrade to the {\\sevensize BICEP}\\, telescope with a\n 150~GHz 512-element array of antenna-coupled detectors. It will be\n deployed to the South Pole in November, 2009. The {\\sevensize KECK}\n array will consist of three {\\sevensize BICEP-2}-like telescopes\n (at 100, 150 and 220~GHz). It is hoped to be installed on the\n {\\sevensize DASI}\\, mount (previously occupied by {\\sevensize QUaD}) in November 2010. The\n nominal goal of this array is $r=0.01$.\n\n \\vspace{3mm}\\item{{{\\sevensize C}$_\\ell${\\sevensize OVER}}:} For an up-to-date overview\n see~\\citet{north08}. {\\sevensize C}$_\\ell${\\sevensize OVER}\\, is a\n three-frequency (97, 150 and 225~GHz) instrument to be sited at Pampa La\n Bola in the Atacama desert in Chile. It will have 576 single-polarization\n transition-edge sensors (TES), split equally among the three frequencies.\n The beam size of $\\sim$ 5.5 arcmin FWHM at 150~GHz will sample the \n multipole range $25 < \\ell < 2000$. The target sensitivity is\n $r \\sim 0.03$ and the polarization signal will be modulated with a\n HWP. The 97~GHz instrument is expected to be deployed to Chile in\n late 2009 with the combined 150\/225~GHz instrument to follow soon\n after in 2010. \n\n \\vspace{3mm}\\item{{{\\sevensize QUIET}}:} See~\\citet{samtleben08} for a recent overview.\n {\\sevensize QUIET}\\, is unique among\n planned $B$-mode experiments in that it uses pseudo-correlation\n HEMT-based receivers rather than bolometers. It will observe from Chile\n using the CBI mount -- a planned second phase will involve upgrading\n to $\\sim 1000$ element arrays and relocation of the 7-m Crawford Hill\n antenna from New Jersey to Chile. {\\sevensize QUIET}\\, will observe\n at 40 and 90~GHz. The target sensitivity for the second phase\n is $r \\sim 0.01$.\n\n \\vspace{3mm}\\item{{{\\sevensize POLARBEAR}}:} A three-frequency (90, 150 and\n 220~GHz) single-dish instrument to be sited in the Inyo Mountains, CA\n for its first year of operation, after which it will be relocated to\n the Atacama desert in Chile. It will use 1280 TES bolometers at each \n frequency with polarization modulation from a HWP. The planned beam\n size is 4 arcmin (FWHM) at 150~GHz. The target sensitivity is\n $r=0.015$ for the full instrument.\n\n \\vspace{3mm}\\item{{{\\sevensize BRAIN}}:} See~\\cite{charlassier08} for a recent\n review. {\\sevensize BRAIN}\\, is a unique\n bolometric interferometer project (c.f. {\\sevensize DASI}, {\\sevensize CBI}) to be sited on\n the Dome-C site in Antarctica. The final instrument will have $\\sim 1000$\n bolometers observing at 90, 150 and 220~GHz. {\\sevensize BRAIN}\\, will be primarily\n sensitive to multipoles $50 < \\ell < 200$. The full experiment is planned\n to be operational in 2011 and the stated target sensitivity is $r = 0.01$.\n\\end{itemize}\n\\vspace{-5mm}\n\\subsubsection*{(ii) Balloon-borne experiments}\n\\begin{itemize}\n \\item{{{\\sevensize EBEX}}:} See \\cite{oxley04} for a summary. {\\sevensize EBEX}\\, will\n observe at 150, 250 and 410~GHz and will fly a total of\n $1406$ TES with HWP modulation.\n The angular resolution is 8 arcmin and\n the target multipole range is $20 < \\ell < 2000$. The stated target\n sensitivity is $r = 0.02$. A test\n flight is planned for 2009 and a long-duration balloon (LDB)\n flight is expected soon after.\n\n \\vspace{3mm}\\item{{{\\sevensize SPIDER}}:} See~\\cite{crill08} for a recent description.\n {\\sevensize SPIDER}\\, will deploy $\\sim 3000$ antenna-coupled TES observing\n at 96, 145, 225 and 275~GHz, with a beam size of $\\sim 40$ arcmin at 145~GHz.\n The target multipole range is $10 < \\ell < 300$. A 2-6 day first\n flight is planned for 2010. The target sensitivity is\n $r=0.01$. Signal modulation will be provided by a (slow) stepped HWP\n and fast gondola rotation.\n\n \\vspace{3mm}\\item{{{\\sevensize PIPER}}:} This balloon experiment will deploy a\n focal plane of 5120 TES bolometers in a backshort-under-grid (BUG)\n configuration. Each flight of {\\sevensize PIPER}\\, will observe at a different\n frequency, covering 200, 270, 350 and 600~GHz after the four planned\n flights. The beam size is $\\sim15$ arcmin, corresponding to a target\n multipole range $\\ell < 800$. The first element of the optical system is\n a variable polarization modulator (VPM). The entire optical chain,\n including the modulators, are cooled to 1.5 K so that {\\sevensize PIPER}\\, observes\n at the background limit for balloon altitudes. Including removal of\n foregrounds, the experiment has the sensitivity to make a $2\\sigma$\n detection of $r = 0.007$. The first flight is scheduled for 2013.\n\n\\end{itemize}\n\\vspace{-5mm}\n\\subsubsection*{(iii) Space missions}\n\\begin{itemize}\n \\item{{{\\sevensize PLANCK}}:} See the publication of the~\\citet{planck06} for a detailed\n review of the science programme. {\\sevensize PLANCK}\\, will measure the temperature\n in nine frequency bands, seven of which will have (some) polarization\n sensitivity. The polarized channels (100, 143, 217 and 353~GHz)\n of the high-frequency instrument\n (HFI) use similar PSBs to those deployed on\n {\\sevensize BOOMERANG}\\, and have beam sizes 5--9.5 arcmin. For low $r$, sensitivity to\n primordial gravitational waves will\n mostly come from the large-angle reionisation\n signature~\\citep{zaldarriaga97b} and\n $r=0.05$ may be possible if foregrounds allow. {\\sevensize PLANCK}\\, will \n be sensitive to the multipole range $2 < \\ell <3000$ and is\n scheduled to launch in 2009. The HFI has no active or fast signal modulation\n (i.e.\\ other than scanning).\n\n \\vspace{3mm}\\item{{{\\sevensize CMBPOL\/BPOL}}:}\n Design studies have been conducted\n for satellite mission(s) dedicated to measuring primordial\n $B$-modes comprising $\\sim 2000$ detectors with the ability to measure\n $0.001 < r < 0.01$ if foregrounds allow it~\\citep{bock08,deBernardis08}.\n The timescale for launch of any selected mission is likely beyond 2020.\n\\end{itemize}\n\n\\section{Simulations}\n\\label{sec:sims}\n\nThe {\\sevensize C}$_\\ell${\\sevensize OVER}\\, experiment will consist of two telescopes -- a low frequency\ninstrument with a focal plane consisting of 192\nsingle-polarization 97~GHz TES detectors, and a high frequency instrument with\na combined focal plane of 150 and 225~GHz detectors (192 of\neach). Note that we have not included foreground contamination in our\nsimulations, so for this analysis, we consider only the 150~GHz\ndetector complement -- the corresponding reduction in sensitivity will\napproximate the effect of using the multi-frequency observations to\nremove foregrounds. Figure~\\ref{fig:hf_focal_plane} shows the\narrangement of the 150 and 225~GHz detectors on the high frequency\nfocal plane. The detectors are arranged in detector blocks consisting\nof eight pixels each. Each pixel consists of two TES detectors which are\nsensitive to orthogonal linear polarizations. The polarization\nsensitivity of the eight detector pairs within a block are along, and at\nright angles to, the major axis of their parent block. The detector\ncomplement, both at 150 and 225~GHz, therefore consists of three\n`flavours' of pixels with different polarization sensitivity\ndirections. The 97~GHz focal plane (not shown) has a similar mix of\ndetector orientations.\n\\begin{figure}\n \\centering\n \\resizebox{0.48\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig1.ps}}}\n \\caption{Layout of detectors on the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, high frequency focal\n plane. Each point indicates a pixel comprising two TES detectors\n sensitive to orthogonal linear polarizations. The polarization\n sensitivity directions of detectors within each block are along, and\n at right angles to, the major axis of the block. The outlined\n 150~GHz detector blocks at the centre of the array are used in the\n simulations described here. The field of view of the entire array is\n $\\sim 5 \\, {\\rm deg}^2$.}\n \\label{fig:hf_focal_plane}\n\\end{figure}\n\n\\subsection{Simulation parameters}\n\\label{sec:sims_params}\nBecause simulating the full {\\sevensize C}$_\\ell${\\sevensize OVER}\\, experiment is computationally\ndemanding (a single simulation of a two-year campaign\nat the full {\\sevensize C}$_\\ell${\\sevensize OVER}\\, data rate would require $\\sim 10^4$ CPU-hours),\nwe have scaled some of the simulation parameters in order to make our\nanalysis feasible.\n\\begin{enumerate}\n\\item We simulate only half the 150~GHz detector complement and have\n scaled the noise accordingly. We have verified with a restricted\n number of simulations using all the 150~GHz detectors that the\n marginally more even coverage obtained across our field using all detectors\n has little or no impact on our results or\n conclusions. The 150~GHz detector blocks used in our simulations are\n indicated in Fig.~\\ref{fig:hf_focal_plane} and include all three\n possible orientations of pixels on the focal plane.\n\\item The {\\sevensize C}$_\\ell${\\sevensize OVER}\\, detectors will have response times of $\\sim 200\\,\n \\mu$s and so the data will be sampled at $\\sim 1$~kHz in order to\n sample the detector response function adequately. Simulating at this\n rate is prohibitive so we simulate at a reduced data rate of\n $100$~Hz. For the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, beam size (FWHM = 5.5 arcmin at 150~GHz) and\n our chosen scan speed ($0.25^{\\circ}$\/s), this data rate is still fast\n enough to sample the sky signal adequately.\n\\item {\\sevensize C}$_\\ell${\\sevensize OVER}\\, will observe four widely separated fields on the sky,\n each covering an area of $\\sim 300 \\, {\\rm deg}^2$, over the course\n of two years. Two of the fields are in the southern sky and two\nlie along the equator. For our analysis, we observe each of the four fields\n for a single night only and, again, we have scaled the noise levels to those\n appropriate for the full two-year observing campaign.\n\\end{enumerate}\n\n\\subsection{Observing strategy}\n\\label{sec:obs_strategy}\nAlthough optimisation of the observing strategy is not the focus of\nthis work, a number of possible strategies have been investigated by\nthe {\\sevensize C}$_\\ell${\\sevensize OVER}\\, team. For our analysis, we use the most favoured scan\nstrategy at the time of writing. To minimise rapid variations in\natmospheric noise, the two telescopes will scan back and forth in\nazimuth at constant elevation, allowing the field to rise through the\nchosen observing elevation. Every few hours, the elevation angle of\nthe telescopes will be re-pointed to allow for\nfield-tracking. Although the precise details of the scan are likely to\nchange, the general characteristics of the scan and resulting field\ncoverage properties will remain approximately the same due to the\nlimitations imposed by constant-elevation scanning and observing from\nAtacama. The {\\sevensize C}$_\\ell${\\sevensize OVER}\\, telescopes are designed with the capability of\nscanning at up to $10^{\\circ}$\/s. However, for our analysis where we\nhave considered {\\sevensize C}$_\\ell${\\sevensize OVER}\\, operating with a rotating HWP (Section\n\\ref{sec:modulation}), we have chosen a relatively slow scan speed of\n$0.25^{\\circ}$\/s in light of the HWP rotation frequency which we have\nemployed ($f_\\lambda = 3$~Hz). Although the mode of operation of a HWP\non {\\sevensize C}$_\\ell${\\sevensize OVER}\\, is still under development, a continuously rotating HWP\nis likely to be restricted to rotation frequencies of $f_\\lambda <\n5$~Hz due to mechanical constraints (with current cryogenic rotation\ntechnologies, fast rotation, $\\mathrel{\\rlap{\\lower4pt\\hbox{\\hskip1pt$\\sim$} 5$~Hz, could possibly result in\nexcessive heat generation). Figure~\\ref{fig:hitmaps} shows the coverage maps for a\nsingle day's observing on one of the southern fields and on one of the\nequatorial fields. The corresponding maps for the other two fields are\nbroadly similar. Note that, in the real experiment, we expect that\nsomewhat more uniform field coverage than that shown in\nFig.~\\ref{fig:hitmaps} will be achievable by employing slightly\ndifferent scan patterns on different days.\n\\begin{figure*}\n \\centering\n \\resizebox{0.80\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig2.ps}}}\n \\caption{Hit-maps for one of the southern fields (left; RA\n 09:30 hrs , Dec -40.00$^{\\circ}$) and one of the equatorial fields\n (right; RA 04:00 hrs, Dec 0.00$^{\\circ}$) for a single day's\n observation with half the 150~GHz detector complement. The central\n part of the fields (shown in yellow and red) are roughly $20^{\\circ}$ in\n diameter. These maps have been constructed using a {\\sevensize HEALPIX}\\, resolution of\n $N_{\\rm side} = 1024$ corresponding to a pixel size $\\sim$3.4 arcmin.}\n \\label{fig:hitmaps}\n\\end{figure*}\n\n\\subsection{Signal simulations}\n\\label{sec:signal_sims}\nWe generate model $TT$, $EE$, $TE$ and $BB$ CMB power spectra using\n{\\sevensize CAMB}\\, \\citep{lewis00}. The input cosmology used consists\nof the best-fit standard $\\Lambda$CDM model to the 5-year {\\sevensize WMAP}\\, data\nset \\citep{hinshaw09}, but with a tensor-to-scalar ratio of $r=0.026$,\nchosen to match the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, `target' value. Realisations of CMB\nskies from these power spectra are then created using a modified\nversion of the {\\sevensize HEALPIX}\\footnote{%\nSee http:\/\/healpix.jpl.nasa.gov} \nsoftware \\citep{gorski05}. Our simulations include weak gravitational lensing\nbut ignore its non-Gaussian aspects. Using only Gaussian simulations\nmeans that we slightly mis-estimate the covariance matrices of our\npower spectrum estimates, particularly the $B$-mode\ncovariances~\\citep{smith04,smithchallinor06}. For the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, noise\nlevels, this is expected to have a negligible impact on the\nsignificance level of the total $B$-mode signal.\n\nAs part of the simulation process, the input CMB signal is convolved with a\nperfect Gaussian beam with FWHM $=5.5$ arcmin. Note that an important\nclass of systematic effects which we do not consider in this paper are\nthose caused by imperfect optics; see the discussion in\nSection~\\ref{sec:systematics}.\n\nFor our analysis of the simulated data sets (Section\n\\ref{sec:analysis}) we have chosen to reconstruct maps of the Stokes\nparameters with a map resolution of $3.4$ arcmin ({\\sevensize HEALPIX}\\, $N_{\\rm\nside} = 1024$). Note that this pixel size does not fully sample the\nbeam and it is likely that we will adopt $N_{\\rm side} = 2048$ for the\nanalysis of real data. In order to isolate the effects of the various\nsystematics we have considered, our simulated CMB skies have therefore\nbeen created at this same resolution --- we can then be sure that any\nbias found in the recovered CMB signals is due to the systematic\neffect under consideration rather than due to a poor choice of map\nresolution\\footnote\nWe have verified that spurious $B$-modes\ngenerated through the pixelisation are negligible for the {\\sevensize C}$_\\ell${\\sevensize OVER}\\,\nnoise levels.}. \nNote that our adopted procedure of simulating and map-making at\nidentical resolutions, although useful for the specific aims of this\npaper, is not a true representation of a CMB observation. For real\nobservations, pixelisation of the CMB maps will introduce a bias to\nthe measured signal on scales comparable to the pixel size adopted.\n \nUsing the pointing registers as provided by the scan strategy and\nafter applying the appropriate focal plane offsets for each detector,\nwe create simulated time-streams according to \n\\begin{equation} \nd_i = \\left[ T(\\theta) + Q(\\theta) \\cos(2\\phi_i) + U(\\theta) \\sin(2\\phi_i) \\right]\/2,\n\\label{eqn:signal_timestream}\n\\end{equation} \nwhere $\\theta$ denotes the pointing and $T, Q$ and $U$ are the sky\nsignals as interpolated from the\ninput CMB sky map. The polarization angle, $\\phi_i$ is, in general, a\ncombination of the polarization sensitivity direction of each detector,\nany rotation\nof the telescope around its boresight, the direction of travel of the\ntelescope in RA--Dec space and the orientation of the half-wave plate,\nif present.\n\n\\subsection{Noise simulations}\n\\label{sec:noise_sims}\nThe {\\sevensize C}$_\\ell${\\sevensize OVER}\\, data will be subject to several different noise\nsources. Firstly, photon loading from the telescope, the atmosphere\nand the CMB itself will subject the data to uncorrelated random\nGaussian noise. Secondly, the TES detectors used in {\\sevensize C}$_\\ell${\\sevensize OVER}\\, are\nsubject to their own sources of noise which will possibly\ninclude low-frequency $1\/f$ behaviour and correlations between\ndetectors. Thirdly, the atmosphere also has a very strong $1\/f$\ncomponent which will be heavily correlated across the detector\narray. Fortunately, the $1\/f$ component of the atmosphere is known to\nbe almost completely un-polarized and so can be removed from the\npolarization analysis by combining data from multiple detectors (see\nSection \\ref{sec:differencing}).\n\nThe white-noise levels due to loading from the instrument, atmosphere\nand CMB have been carefully modelled for the case of {\\sevensize C}$_\\ell${\\sevensize OVER}\\,\nobservations from Atacama. We will not present the details here, but\nfor realistic observing conditions and scanning elevations, we have\ncalculated the expected noise-equivalent temperature (NET) due to\nphoton noise alone to be $\\approx 146 \\, \\mu {\\rm K} \\sqrt{\\rm s}$. We\nadd this white noise component to our simulated signal time-streams\nfor each detector as\n\\begin{equation}\nd_i \\rightarrow d_i + \\frac{\\rm NET}{2} \\sqrt{f_{\\rm samp}} g_i,\n\\end{equation}\nwhere $f_{\\rm samp}$ is the sampling frequency and $g_i$ is a Gaussian\nrandom number with $\\mu = 0$ and $\\sigma = 1$. Note that the white-noise\nlevel in the detector time streams is ${\\rm NET} \/ 2$ since the\n{\\sevensize C}$_\\ell${\\sevensize OVER}\\, detectors are half-power detectors (equation\n\\ref{eqn:signal_timestream}). \n\nUsing instrument parameters appropriate for the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, detectors, \nwe use the small-signal TES model of \\cite{irwin05} to create a\nmodel noise power spectrum for the detector noise. This model includes\nboth a contribution from the super-conducting SQUIDs which will be used to\nrecord the detector signals (e.g. \\citealt{reintsema03}) and a\ncontribution from aliasing in the Multi Channel Electronics (MCE;\n\\citealt{battistelli08}) which will be used to read out the signals. \nFor the instrument parameters we have chosen, the effective\nNET of the detector noise in our simulations is approximately equal\nto the total combined photon noise contribution from the atmosphere, the\ninstrument and the CMB. Note however that for the final instrument, it\nis hoped that the detector NET can be reduced to half that of the\ntotal photon noise, thus making {\\sevensize C}$_\\ell${\\sevensize OVER}\\, limited by irreducible\nphoton loading.\nThe \\cite{irwin05} small-signal TES model does not \ninclude a $1\/f$ component to the detector noise so in order\nto investigate the impact of modulation on possible low-frequency\ndetector noise, we add a heuristically chosen $1\/f$ component to the\ndetector noise model with knee frequencies in the range, $0.01 <\nf_{\\rm knee} < 0.1$~Hz. \nThe MCE system which will be used to read out the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, data\nshould have low cross-talk between different channels. However,\ncorrelations will be present at some level and so we include\n10 per cent correlations between all of our simulated detector noise\ntime-streams. Generally, to simulate stationary noise that is correlated in time and\nacross the $N_{\\rm det}$ detectors we proceed as follows.\nLet the noise cross-power spectrum\nbetween detector $d$ and $d'$ be $P_{d,d'}(f)$. \nTaking the Cholesky decomposition of this matrix at each frequency,\n$L_{d,d'}(f)$, defined by\n\\begin{equation} \nP_{d,d'}(f) = \\sum_{d''} L_{d,d''}(f) L_{d',d''}(f), \n\\end{equation}\nwe apply $L$ to $N_{\\rm det}$ independent, white-noise time streams\n$g_{d}(f)$ in Fourier space,\n\\begin{equation}\ng_d(f) \\rightarrow \\sum_{d'} L_{d,d'}(f) g_{d'}(f) .\n\\end{equation}\nThe resulting time-streams, transformed to\nreal space then possess the desired correlations between detectors. Here,\nwe assume that the correlations are independent of frequency so that\nthe noise cross-power spectrum takes the form\n\\begin{equation}\nP_{d,d'}(f) = C_{d,d'} P(f) ,\n\\end{equation}\nwhere the correlation matrix $C_{d,d'} = 1$ for $d = d'$ and $C_{d,d'} = 0.1$\notherwise. In practice, we use discrete Fourier transforms to synthesise\nnoise with periodic boundary conditions (and hence circulant time-time\ncorrelations).\n\nWe use the same technique to simulate the correlated $1\/f$ component\nof the atmosphere. We have measured the noise properties of the atmosphere\nfrom data from the {\\sevensize QUaD}\\, experiment \\citep{hinderks09}. The 150~GHz\nfrequency channel of {\\sevensize QUaD}\\, is obviously well matched to the\n{\\sevensize C}$_\\ell${\\sevensize OVER}\\, 150~GHz channel although {\\sevensize QUaD}\\, observed the CMB from the\nSouth Pole rather than from Atacama. Although there are significant\ndifferences between the properties of the atmosphere at the South Pole\nand at Atacama (e.g. \\citealt{bussmann05}), the {\\sevensize QUaD}\\, observations\nstill represent the best estimate of the $1\/f$ noise properties of\nthe atmosphere available at present. A rough fit of the {\\sevensize QUaD}\\, data to the model, \n\\begin{equation}\nP(f) = {\\rm NET^2} \\left[1 + \\left( \\frac{f_{\\rm knee}}{f} \\right)^\\alpha \\right],\n\\label{eqn:pk_atms}\n\\end{equation}\nyields a knee frequency, $f_{\\rm knee} = 0.45$~Hz and spectral index,\n$\\alpha = 2.5$. Using this model power spectrum we simulated $1\/f$\natmospheric noise correlated across the array in exactly the same way\nas was used for the detector noise. Fortunately, the $1\/f$ component\nin the atmosphere is almost completely un-polarized.\nIf there were no instrumental polarization, detectors\nwithin the same pixel (which always look in\nthe same direction) would therefore be completely correlated\nwith one another and detectors from different pixels would also be\nheavily correlated. For the correlated atmosphere, we therefore use a\ncorrelation matrix given by $C_{d,d'} = 1.0$ for $|d-d'| \\le 1$\n(i.e. for detector pairs) and $C_{d,d'} = 0.5$ otherwise. In the\nfollowing sections, as one of the systematics we have investigated, we \nrelax the assumption that the atmosphere is un-polarized.\n\nFigure~\\ref{fig:noise_compare} compares the photon, atmospheric $1\/f$\nand detector noise contributions to our simulated data in frequency\nspace. At low frequencies, the noise is completely dominated by the\natmospheric $1\/f$ while the white-noise contributions from photon\nloading (including the uncorrelated component of the atmosphere) and\ndetector noise are approximately equal. Note that for observations\nwithout active modulation, and for the scan speed and observing\nelevations which we have adopted, the ``science band'' for the\nmultipole range $20 < \\ell < 2000$ corresponds roughly to $0.01 < f <\n1$~Hz in time-stream frequency. In contrast, for our simulations which\ninclude a continuously rotating HWP, the temperature signal remains within\nthe $0.01 < f < 1$~Hz frequency range but the polarized sky signal is\nmoved to a narrow band centred on $\\sim$12~Hz, well away from both the\ndetector and atmospheric $1\/f$ noise components (see\nSection~\\ref{sec:modulation} and Fig.~\\ref{fig:mod_frequency}). Note also that although the\natmospheric $1\/f$ dominates the detector $1\/f$ at low frequency, the\natmosphere is heavily correlated across detectors and can therefore be\nremoved by combining detectors (e.g. differencing detectors within a\npixel) but this is not true for the detector noise which is only\nweakly correlated between detectors. We demonstrate this in\nFig.~\\ref{fig:tod_noise} where we plot a five-minute sample of simulated\natmospheric and detector noise for the two constituent detectors\nwithin a pixel. Including the atmospheric $1\/f$ component, the\neffective total NET per detector measured from our simulated\ntime-streams is $293 \\, \\mu {\\rm K} \\sqrt{\\rm sec}$ whilst excluding\natmospheric $1\/f$, we measure $210 \\, \\mu {\\rm K} \\sqrt{\\rm s}$.\n\n\\begin{figure}\n \\centering\n \\resizebox{0.47\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig3.ps}}}\n \\caption{Frequency space comparison between the different noise\n sources in the simulations. The grey line shows the detector noise\n power spectrum (here with a $1\/f$ component with knee frequency,\n $f_{\\rm knee} = 0.1$~Hz). The correlated component to the\n atmosphere is shown as the dashed line and the total photon noise\n (including atmospheric loading) is shown as the dotted line. For\n the simulation parameters we have adopted, the temperature sky-signal from\n multipoles $20 < \\ell < 2000$ appears in the time-stream in the\n frequency range $0.01 < f < 1$~Hz. In the absence of fast modulation,\n the polarized sky signal also appears in this frequency range\n whereas in our simulations including fast modulation, the polarized\n sky signal appears in a narrow band centred on 12~Hz.}\n \\label{fig:noise_compare}\n\\end{figure}\n\\begin{figure}\n \\centering\n \\resizebox{0.48\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig4.ps}}}\n \\caption{A five-minute sample of simulated noise time-stream for the two\n detectors within a pixel (denoted ``A'' and ``B'') for both the\n atmospheric noise simulations (\\emph{top}) and for the detector\n noise simulations (\\emph{bottom}). The $1\/f$ component in the\n atmospheric noise time-streams is 100 per cent correlated between the\n A and B detectors and can be removed entirely by\n differencing. In contrast the $1\/f$ component in the detector noise\n (which is much weaker and not noticeable on this plot)\n is only weakly correlated between A and B and is not removed by\n differencing. For comparison, the signal-only time-streams are\n also plotted in the lower panels as the red curves.}\n \\label{fig:tod_noise}\n\\end{figure}\n\n\\subsection{Detector response}\n\\label{sec:detector_response}\nThe \\cite{irwin05} TES small-signal model mentioned above also\nprovides us with an estimate of the detector response (the conversion\nfrom incident power to resultant current in the detectors). In this\nmodel, the power-to-current responsivity, $s_I(\\omega)$, is given by\n\\begin{equation} \ns_I(\\omega) \\propto \\frac{1 - \\tau_+\/\\tau_I}{1 + i\\omega\\tau_+}\n\\frac{1 - \\tau_-\/\\tau_I}{1 + i\\omega\\tau_-}, \n\\label{eqn:response}\n\\end{equation} \nwhere $\\omega = 2 \\pi f$ is the angular frequency and $\\tau_+$ and\n$\\tau_-$ are the ``rise time'' and ``fall time'' (relaxation to steady\nstate) after a delta-function temperature impulse. Here, $\\tau_I$ is\nthe current-biased thermal time constant. The impulse-response in\nthe time domain is\n\\begin{equation}\ns_I(t) \\propto \\frac{e^{-t\/\\tau_+} - e^{-t\/\\tau_-}}{\\tau_+ - \\tau_-}\n\\Theta(t) ,\n\\end{equation}\nwhere $\\Theta(t)$ is the Heaviside step function.\nNote that the constant of\nproportionality in equation~(\\ref{eqn:response}) (which is also\npredicted by the model) is essentially the calibration (gain) of the\ndetectors. The simulated time-streams of\nequation~(\\ref{eqn:signal_timestream}) are converted to detector\ntime-streams through convolution with this response\nfunction. Note that the photon noise and correlated atmospheric noise\nare added to the time-stream before convolution (and so are also\nconvolved with the response function) while the detector noise is\nadded directly to the convolved time-streams. The {\\sevensize C}$_\\ell${\\sevensize OVER}\\, detectors\nare designed to be extremely fast with time-constants $\\tau_\\pm < 1 \\,\n{\\rm ms}$. For our simulations, we have used time-constants\npredicted by the small-signal TES model for {\\sevensize C}$_\\ell${\\sevensize OVER}\\, instrument\nparameters of $\\tau_+ = 300 \\, \\mu{\\rm s}$ and $\\tau_- = 322 \\, \\mu\n{\\rm s}$. \n\nNote that for our chosen scan speed of $0.25^{\\circ}$\/s, the effect of\nthe response function on the signal component in the simulations is\nsmall in the absence of fast modulation -- the {\\sevensize C}$_\\ell${\\sevensize OVER}\\,\ndetectors are so fast that signal attenuation\nand phase differences introduced by convolution with the detector\nresponse only become important at high frequencies, beyond the\nfrequency range of the sky signals. For our simulations without\nfast modulation, sky signals from multipoles $\\ell \\sim 2000$ will\nappear in the time-stream at $\\sim$ 1~Hz where the amplitude of the\nnormalised response function is effectively unity and the associated\nphase change is $-0.2^{\\circ}$. For our simulations\nincluding fast modulation, it becomes more important to correct the\ntime-stream data for the detector response. In this case, the polarized sky signals (from all\nmultipoles) appear within a narrow band centred on $12$~Hz in the\ntime-stream. Here, the amplitude of the response function is still very\nclose to unity but the phase change has grown to $-2.7^{\\circ}$. We\ninclude a deconvolution step in the analysis of all of our simulated\ndata to correct for this effect. Finally, we note that for the $\\sim 10$\nper cent errors in the time-constants which we have considered (see\nSection~\\ref{sec:systematics}), the resulting mis-estimation of the\npolarization signal will again be small, even for the case of\nfast modulation.\n\nIn principle we should also include the effect of sample integration: each\ndiscrete observation is an integral of a continuous signal over the sample period.\nOf course, in the case where down-sampled data is simulated\nthe sample integration should include the effect of the sample\naveraging. For integration over a down-sampled period $\\Delta$, there is an additional\nphase-preserving filtering by $\\mathrm{sinc}(\\omega \\Delta\/2)$ where\n$\\omega \\equiv 2\\pi f$. For our scan\nparameters, the filter is negligible (i.e.\\ unity) for unmodulated data\nwhile for modulated data the filter can be approximated at the frequency\n$4 \\omega_\\lambda$, where $\\omega_\\lambda$ is the angular rotation frequency\nof the HWP. This acts like a small decrease in the polarization efficiency,\nwith the discretised signal at the detector being\n\\begin{eqnarray}\nd_i &\\approx & \\frac{1}{2}\\big\\{T(\\theta) +\n\\mathrm{sinc}(2\\omega_\\lambda \\Delta) \\nonumber \\\\\n&&\\mbox{} \\times \\left[Q(\\theta)\\cos(2\\phi_i)\n+ U(\\theta) \\sin(2\\phi_i)\\right]\\big\\} .\n\\end{eqnarray}\nFor $\\omega_\\lambda = 2\\pi \\times 3\\,\\mathrm{Hz}$ and $\\Delta =\n10\\,\\mathrm{ms}$, the effective polarization efficiency is $0.98$ and\nthis has the effect of raising the noise level in the polarization maps by\n2 per cent. We do not include this small effect in our simulations, but could\neasily do so.\n\n\\subsection{Systematic effects}\n\\label{sec:systematics}\nIn the analysis that follows, we will investigate the impact of\nseveral systematic effects on the ability of a {\\sevensize C}$_\\ell${\\sevensize OVER}-like\nexperiment to recover an input $B$-mode signal. For a reference, we\nuse a suite of simulations which contain no systematics. This ideal simulation\ncontains the input signal, photon noise, $1\/f$ atmospheric noise\ncorrelated across the array (but un-polarized) and TES\ndetector noise with no additional correlated $1\/f$\ncomponent. Additionally, for our reference simulation all pointing\nregisters and detector polarization sensitivity angles use the nominal\nvalues and the signal is convolved with the detector response function\nusing the nominal time constants.\n\nWe then perform additional sets of simulations with the following\nsystematic effects included in isolation:\n\\begin{itemize}\n\\item $1\/f$ detector noise. We have considered an additional\n correlated $1\/f$ component to the detector noise with $1\/f$ knee \n frequencies of $0.1$, $0.05$ and $0.01$~Hz.\n\\item Polarized atmosphere. In addition to the un-polarized atmosphere\n present in the reference simulation, we consider a \\emph{polarized}\n $1\/f$ component in the atmosphere. To simulate polarized atmosphere,\n we proceed as described in Section~\\ref{sec:noise_sims} but now\n we add correlated $1\/f$ atmospheric noise to the $Q$ and $U$ sky signal\n time-streams such that equation~(\\ref{eqn:signal_timestream})\n becomes \n \\begin{eqnarray}\n d_i &=& \\frac{1}{2} \\left[T + \\left(Q + Q^{\\rm atms}_i\\right) \\cos(2 \\phi_i)\n \\nonumber \\right. \\\\\n && \\mbox{} \\phantom{xxxx} \\left. \n + \\left(U + U^{\\rm atms}_i\\right) \\sin(2 \\phi_i)\\right]. \n \\end{eqnarray}\n We take the $Q^{\\rm atms}$ and $U^{\\rm atms}$ atmospheric signals to\n have the same power spectrum as the common-mode atmosphere\n (equation~\\ref{eqn:pk_atms}) but a factor ten smaller in magnitude. \n\\item Detector gain errors. We consider three types of gain errors:\n (i) random errors in the gain that are constant in time, uncorrelated\n between detectors and have a 1 per cent RMS; (ii) gain drifts\n in each detector corresponding to a 1 per cent drift over the course of a\n two-hour observation -- the start and end\n gains for each detector are randomly distributed about the nominal\n gain value with an RMS of 1 per cent; (iii) systematic A\/B gain\n mis-matches (1 per cent mis-match) between the two detectors within each\n pixel. For this latter systematic, we have applied a constant 1 per\n cent A\/B mis-match to all pixels on the focal plane but the direction\n of the mis-match (that is, whether the gain of A is greater or smaller\n than B) is chosen randomly.\n \\item Mis-estimated polarization sensitivity directions. Random\n errors uncorrelated between detectors (including those with the\n same feedhorn) with RMS 0.5$^{\\circ}$ and which are constant in time,\n and a systematic mis-estimation of the\n instrument polarization coordinate reference system by 0.5$^{\\circ}$ are\n considered.\n\\item Mis-estimated half-wave plate angles. For the case where we\n consider an experiment which includes polarization modulation with a\n half-wave plate (see Section \\ref{sec:modulation}), we also\n consider random errors (with RMS 0.5$^{\\circ}$) in the recorded HWP angle \n which we apply to each 100~Hz time-sample. In addition, we consider\n a 0.5$^{\\circ}$ systematic offset in the half-wave plate angle measurements.\n\\item Mis-estimated time-constants. The analysis that follows\n includes a deconvolution step to undo the response function of the\n detectors and return the deconvolved sky signal. In all cases, we\n use the nominal time-constant values of $\\tau_+ = 300 \\, \\mu{\\rm s}$\n and $\\tau_- = 322 \\, \\mu {\\rm s}$ to perform the deconvolution. To\n simulate the effect of mis-estimated time-constants, we introduce\n both a random scatter (with RMS = 10 per cent across detectors) and a\n systematic offset ($\\tau_\\pm$ identically offset by 10 per cent for all\n detectors) in the time-constants when creating the simulated data.\n\\item Pointing errors. We simulate the effects of both a random jitter and a\n slow wander in the overall pointing of the telescope by introducing\n a random scatter uncorrelated between time samples (with RMS 30 arcsec)\n and an overall drift in the\n pointing (1 arcmin drift from true pointing over the course of a\n two-hour observation) when creating the simulated time-stream. Once\n again, the simulated data is subsequently analysed assuming perfect\n pointing registers.\n\\item Differential transmittance in the HWP. As a simple example of a\n HWP-induced systematic, we have considered a differential\n transmittance of the two linear polarizations by the\n HWP. Preliminary measurements of the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, HWPs suggest the level\n of differential transmittance will be in the region of\n $1\\!\\!\\!\\relbar\\!\\!\\!2$ per cent. For\n this work, we consider a 2 per cent differential transmittance in the HWP. \n\\end{itemize}\n\nThe range of systematic effects we include is not exhaustive. In\nparticular, we ignore effects in the HWP, when present, except for a\nmis-estimation of the rotation angle and a differential transmittance\nof the two linear polarizations. In addition, we ignore all optical\neffects. In practice, there are many possible HWP-related systematic\neffects which we have not yet considered. In general, a\nthorough analysis of HWP-related systematics require detailed physical\noptics modelling which is beyond the scope of our current analysis. We\ntherefore leave a detailed investigation of HWP-related systematics to future\nwork and simply urge the reader to bear in mind that where our\nanalysis has included a HWP, we have, in most cases, assumed a perfect one. For other\noptical effects, we note that \\citet{odea07} have already investigated\nsome relevant effects using analytic and numerical techniques and we\nare currently adapting their flat-sky numerical analysis to work with\nthe full-sky simulations described here. Our conclusions on the\nability of modulation to mitigate systematic effects associated with\nimperfect optics, which will be based on detailed physical optics\nsimulations of the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, beams (Johnson et al., in prep), will be\npresented in a future paper. Note that there are important\ninstrument-specific issues to consider in such a study to do with\nwhere the modulation is performed in the instrument. In {\\sevensize C}$_\\ell${\\sevensize OVER}, the\nmodulation will be provided by a HWP between the horns and mirrors and\nthis may lead to a difference in the relative rotation of the field\ndirections on the sky and the beam shapes as the HWP rotates compared\nto a set-up, as in {\\sevensize SPIDER}~\\citep{crill08}, where the HWP is after\n(thinking in emission) the beam-defining elements.\n\nFor a {\\sevensize C}$_\\ell${\\sevensize OVER}-like receiver, consisting of a HWP followed by a polarization\nanalyser (e.g.\\ an orthomode transducer) and detectors, we include\nessentially all\nrelevant systematic effects introduced \\emph{by the receiver}.\nTo see this, note that\nthe most general Jones matrix describing propagation of the two linear\n(i.e.\\ $x$ and $y$) polarization states through the polarization analyser\nis~\\citep{odea07}\n\\begin{equation}\n\\mathbfss{J} = \\left(\n\\begin{array}{cc}\n1+g_1 & \\epsilon_1 \\\\\n-\\epsilon_2 & (1+g_2)e^{i\\alpha}\n\\end{array}\n\\right) ,\n\\end{equation}\nwhere $g_1$, $g_2$ and $\\alpha$ are small real parameters and\n$\\epsilon_1$ and $\\epsilon_2$ are small and complex-valued. The detector\noutputs are proportional to the power in the\n$x$ and $y$-components of the transmitted field (after convolution\nwith the detector response function).\nTo first-order in small\nparameters, only $g_1$, $g_2$ and the real parts of $\\epsilon_1$ and\n$\\epsilon_2$ enter the detected power. In this limit, the perturbed Jones\nmatrix is therefore equivalent in terms of the detected power to\n\\begin{equation}\n\\mathbfss{J} \\sim \\left(\n\\begin{array}{cc}\n1+g_1 & 0 \\\\\n0 & 1+g_2\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\n\\cos \\alpha_1 & \\sin \\alpha_1 \\\\\n-\\sin\\alpha_2 & \\cos\\alpha_2\n\\end{array}\n\\right)\n\\end{equation}\nwhere the small angles $\\alpha_1 \\approx \\Re\\epsilon_1$ and $\\alpha_2 \\approx\n-\\Re\\epsilon_2$ denote the perturbations in the polarization-sensitivity\ndirections introduced above, and $g_1$ and $g_2$ are the gain errors.\nNote that instrument polarization (i.e.\\ leakage from $T$ to detected \n$Q$ or $U$) is only generated in the receiver through mismatches in the gain at first \norder, but also through $|\\epsilon_1|^2$ and\n$|\\epsilon_2|^2$ in an exact calculation. The latter effect is not\npresent in the simplified description in terms of offsets in the\npolarization-sensitivity directions. Note also that if we\ndifference the outputs of\nthe two detectors in the same pixel, in the presence of perturbations\n$\\alpha_1$ and $\\alpha_2$ to the polarization sensitivity directions we\nfind\n\\begin{eqnarray}\nd_1 - d_2 &=& \\cos(\\alpha_1-\\alpha_2)\\left(\nQ\\cos[2(\\phi-\\bar{\\alpha})] \\nonumber \\right. \\\\\n&&\\mbox{} \\phantom{xxxxxxxxxxxx} \\left. + U \\sin[ 2(\\phi-\\bar{\\alpha})] \\right) ,\n\\end{eqnarray}\nwhere $\\bar{\\alpha} = (\\alpha_1 + \\alpha_2)\/2$. This is equivalent\nto a common rotation of the pair by $\\bar{\\alpha}$ and a decrease in the\npolarization efficiency to $\\cos(\\alpha_1-\\alpha_2)$.\n\n\\subsection{Polarization modulation}\n\\label{sec:modulation}\n\nFor our reference simulation, and for each of the systematic effects\nlisted above, we simulate the experiment using three different\nstrategies for modulating the polarization signal. Firstly, we\nconsider the case where no explicit modulation of the polarization\nsignal is performed -- in this case, the only modulation achieved is\nvia telescope scanning and the relatively small amount of sky rotation\nprovided by the current {\\sevensize C}$_\\ell${\\sevensize OVER}\\, observing strategies. In addition,\nwe also consider the addition of a half-wave plate, either continuously\nrotating or ``stepped'', placed in front of the focal plane. A\nhalf-wave plate modulates the polarization signal such that the\noutput of a single detector (in the detector's local polarization\ncoordinate frame) is \n\\begin{equation} d_i = \\frac{1}{2}\\left[ T + Q \\cos ( 4\\phi_i )\n+ U \\sin ( 4\\phi_i ) \\right], \n\\label{eqn:pol_mod}\n\\end{equation} \nwhere, here, $\\phi_i$ is the angle between the detector's local\npolarization frame and the principal axes of the wave plate.\n\nFor a continuously rotating HWP, the polarized-sky signal is thus\nmodulated at $4 f_\\lambda$ where $f_\\lambda$ is the rotation frequency\nof the HWP. As well as allowing all three Stokes parameters to be\nmeasured from a single detector, modulation with a continuously\nrotating HWP (which we term ``fast'' modulation in this paper) moves\nthe polarization sky-signal to higher frequency and thus away from any\nlow-frequency $1\/f$ detector noise that may be\npresent; see Fig.~\\ref{fig:mod_frequency}.\n(Note that the temperature signal is not\nmodulated and one needs to rely on telescope scanning and analysis\ntechniques to mitigate $1\/f$ noise in $T$.) This ability to mitigate\nthe effect of $1\/f$ noise on the polarization signal is the prime\nmotivation for including a continuously rotating HWP in a CMB\npolarization experiment\\footnote{%\nThe ability to measure all three Stokes\nparameters from a single detector has also been suggested as\nmotivation for including a modulation scheme in CMB polarization\nexperiments. However, we will argue later in\nSection~\\ref{sec:differencing} that, at least for ground-based\nexperiments, an analysis based on extracting all three Stokes\nparameters from individual detectors in isolation using a real-space\ndemodulation technique may be a poor choice.}\nSystematic effects that generate an apparent polarization signal that is\nnot modulated at $4f_\\lambda$ can also be mitigated almost\ncompletely with fast modulation.\nMost notably, instrument polarization generated in the receiver will\nnot produce a spurious polarization signal in the recovered maps\nunless the gain and\ntime-constant mismatches vary sufficiently rapidly ($\\sim 4 f_\\lambda$)\nto move the \\emph{scan}-modulated temperature leakage up into the\npolarization signal band. \n\n\\begin{figure}\n \\centering\n \\resizebox{0.47\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig5.ps}}}\n \\caption{Frequency-space representation of polarization modulation\n with a continuously rotating HWP. The plotted power spectrum is that\n for a single azimuth scan from one of our signal-only simulations\n with the HWP continuously rotating at $f_\\lambda = 3$~Hz. The power\n in the frequency range $0.01 < k < 1$~Hz is the unmodulated\n temperature signal from sky multipoles in the range $20 < \\ell <\n 2000$. The power spike at $4 f_\\lambda = 12$~Hz is the modulated\n polarized signal. The dashed (dotted) line shows the expected\n temperature (polarization) signal band (with arbitrary\n normalisation) appropriate for the scan speed, modulation frequency\n and beam size we have used. The residual power between $1$ and\n $6$~Hz and on either side of the polarization power spike is due to\n pixelisation effects.}\n \\label{fig:mod_frequency}\n\\end{figure}\n\nAs mentioned in the previous section, as an example of a HWP-induced\nsystematic, we have considered the case of a 2 per cent differential\ntransmittance by the HWP of the two incoming linear polarizations. We\nmodel this effect using a non-ideal Jones matrix for the HWP of the\nform, \n\\begin{equation}\n\\mathbfss{J} = \\left(\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & -(1+\\delta)\n\\end{array}\n\\right) ,\n\\label{eqn:diff_trans_jones}\n\\end{equation}\nwhere $\\delta$ describes the level of differential\ntransmittance. Propagating through to detected power, for the\ndifference in output of the two detectors within a pixel, we find\n\\begin{eqnarray}\nd_1 - d_2 \\, &=& \\left[1 + \\delta + \\frac{\\delta^2}{4} \\right]\n(Q\\cos(4\\phi) + U\\sin(4\\phi) ) \\nonumber \\\\\n &-& \\left[\\delta + \\frac{\\delta^2}{2}\\right] I\\cos(2\\phi)\n + \\frac{\\delta^2}{4} Q.\n\\label{eqn:diff_trans_power}\n\\end{eqnarray}\nNote that, in this expression, both the HWP angle, $\\phi$, and the\nStokes parameters are defined in the pixel basis. The first term in\nequation (\\ref{eqn:diff_trans_power}) is the ideal\ndetector-differenced signal but mis-calibrated by a factor, $\\delta +\n\\delta^2\/4$. For reasonable values of $\\delta$, this mis-calibration\nis small ($\\mathrel{\\rlap{\\lower4pt\\hbox{\\hskip1pt$\\sim$} 2$ per cent) and, in any case, is easily dealt with during \na likelihood analysis of the power spectra. The\npotential problem term is the middle term which contains the\ntotal intensity signal modulated at $2 f_\\lambda$. Note that there\nwill be contributions to this term from the CMB monopole, dipole and\nthe atmosphere, which, for our simulations, we have taken to be 5 per cent\nemissive. Even for small values of $\\delta$ therefore, these\nHWP-synchronous signals will completely dominate the raw detector data\nand need to be removed from the data prior to the map-making step.\n\nWith a HWP operating in ``stepped'' mode where the angle of the HWP is\nchanged at regular time intervals (e.g. at the end of each scan), the\ngains are less clear. The polarization sky signal is not shifted to\nhigher frequencies so $1\/f$ detector noise can only be dealt with by\nfast scanning. What stepping the waveplate can potentially do is to increase\nthe range of polarization sensitivity directions with which a given pair of detectors\nsamples any pixel on the sky. This has two important effects: (i) \nit reduces the correlations between the errors in the reconstructed $Q$ and $U$\nStokes parameters in each sky pixel; and (ii) it can mitigate somewhat those\nsystematic effects that do not transform like a true polarization under \nrotation of the waveplate. Of course, one of the strongest motivations\nfor stepped, slow modulation is the avoidance of systematic effects\nassociated with the continuous rotation of the HWP. If these effects are\nsufficiently well understood, then the resulting spurious signals can\nbe rejected during analysis. However, if they are not well understood,\na stepped HWP, while not as effective in mitigating systematics, may\nwell be the preferred option.\n\nNote that for a perfect optical system, rotation of the waveplate is\nequivalent to rotation (by twice the angle) of the instrument.\nHowever, this need not hold with imperfect optics. For example,\nsuppose the beam patterns for the two polarizations of a given\nfeedhorn are purely co-polar (i.e.\\ the polarization sensitivity\ndirections are ``constant'' across the beams and orthogonal), but the\nbeam shapes are orthogonal ellipses. This set-up generates instrument\npolarization with the result that a temperature distribution that is\nlocally quadrupolar on the sky will generate spurious polarization\nthat transforms like true sky polarization under rotation of the\ninstrument~\\citep{hu03,odea07}. However, for an optical arrangement\nlike that in {\\sevensize SPIDER}, where the HWP is after (in emission) the\nbeam-defining optics, as the HWP rotates the polarization directions\nrotate on the sky but the beam shapes remain fixed. The spurious\npolarization from the mis-match of beam shapes is then constant as the\nHWP rotates for any temperature distribution on the sky, and so the\nquadrupolar temperature leakage can be reduced.\n\nIn our analysis, in addition to simulations without explicit\nmodulation we have also simulated an experiment with a HWP continuously\nrotating at $3$~Hz (thus modulating the polarization signal at\n$12$~Hz) and an experiment where a HWP is stepped (by 20$^{\\circ}$) at the end\nof each azimuth scan (for the scan strategy and scan speed we are\nusing, this corresponds to stepping the HWP roughly every $\\sim$ 90\ns).\n\nWe end this section with a comment on the ability of a continuously\nrotating HWP to mitigate un-polarized $1\/f$ atmospheric noise. The polarization\nsignal band is still, of course, moved to higher frequency and thus\naway from the $1\/f$ noise but the atmospheric $1\/f$ noise is so strong\nthat extremely rapid HWP rotation would be required to move the\npolarization band far enough into the tail of the $1\/f$ spectrum.\nSuch rapid rotation is not an option in practice as it would introduce its\nown systematic effects (e.g. excessive heat generation). This is the basis of our argument\nmentioned above that extracting all three Stokes parameters from a\nsingle detector may be a poor choice of analysis technique. However,\nsince the $1\/f$ atmospheric noise is un-polarized, it can be removed\n\\emph{completely} by combining data from multiple detectors. We\nrevisit this issue again with simulations in\nSection~\\ref{sec:differencing}. Finally, we note that if the atmosphere\ndoes contain a polarized $1\/f$ component, then we expect that this\nwill not be mitigated by modulation -- the polarized atmosphere would\nbe modulated in the same way as the sky signal and would\nshift up in frequency accordingly.\n\n\\section{Analysis of simulated data}\n\\label{sec:analysis}\nFor our reference simulation, and for each of the systematic effects\nand modulation strategies described in the previous section, we have\ncreated a suite of 50 signal-only, noise-only and signal-plus-noise\nsimulated datasets. Our analysis of the signal-only data will be used\nto investigate potential biases caused by the systematics while our\nsignal-plus-noise realisations are used together with the noise-only\nsimulations to investigate any degradation of the sensitivity of the\nexperiment due to the presence of the systematic effects.\n\nOur analysis of each dataset consists of processing the data through\nthe stages of deconvolution for the bolometer response function,\npolarization demodulation and map-making, and finally estimation of the\n$E$- and $B$-mode power spectra. For any given single realisation these\nprocesses are performed separately for each of the four observed\n{\\sevensize C}$_\\ell${\\sevensize OVER}\\, fields -- that is, we make maps and measure power spectra for\neach field separately. Since our fields are widely separated on the\nsky, we can treat them as independent and combine the power spectra\nmeasured from each using a simple weighted average to produce a single\nset of $E$- and $B$-mode power spectra for each realisation of the\nexperiment. Note that even if our fields were not widely separated,\nour procedure would still be unbiased (but sub-optimal) and \ncorrelations between the fields would be automatically taken\ninto account in our error analysis since, for any given realisation, \nthe input signal for all four fields is taken from the same simulated\nCMB sky. \n\n\\subsection{Time-stream processing and map-making}\n\\label{sec:map-making}\nWe first deconvolve the time-stream data for the detector response in\nfrequency space using the response function of\nequation~(\\ref{eqn:response}) and using the nominal time-constants in\nall cases. Once this is done, the data from detectors within each\npixel are differenced in order to remove both the CMB temperature signal and\nthe correlated $1\/f$ component of the atmospheric noise. For the case\nwhere the $1\/f$ atmosphere is completely correlated between the two\ndetectors and in the absence of instrumental polarization and\/or\ncalibration systematics, this process will remove the CMB $T$ signal and the\n$1\/f$ atmosphere completely. \n\nFor the case where we have simulated the effect of a differential\ntransmittance in the HWP, a further time-stream processing step is\nrequired at this point to fit for and remove the HWP-synchronous\nsignals from the time-stream. To do this, we have implemented a simple\niterative least-squares estimator to fit, in turn, for the amplitudes\nof both a cosine and sine term at the second harmonic of the\nHWP-rotation frequency, $f_\\lambda$. For our simulations containing\nboth signal and noise, the accuracy with which we are able to remove\nthe HWP-synchronous signals is determined by the noise level in the\ndata. A demonstration of the performance of this procedure is given\nin Fig.~\\ref{fig:hwp_sys_removal} where we plot the power spectra of\nsix minutes of simulated time-stream data (containing both signal\nand noise) before and after the removal of the HWP-synchronous signal.\n\n\\begin{figure}\n \\centering\n \\resizebox{0.455\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig6.ps}}}\n \\caption{Power spectra of a six minute segment of time-stream\n data before (upper panel) and after (lower panel) fitting for and\n removing the HWP-synchronous signal. A 2 per cent differential\n transmittance in the HWP was used to create the simulated data. The\n resulting spurious signal appears at $2 f_\\lambda = 6$~Hz in the upper\n panel and has a peak power of $\\sim 10^{10}$ in the units\n plotted. No obvious residuals are apparent in the lower panel after\n applying our procedure for removing the contamination.}\n \\label{fig:hwp_sys_removal}\n\\end{figure}\n\nAfter detector differencing, and the removal of the HWP-synchronous\nsignals if present, the resulting differenced time stream is then a\npure polarization signal:\n\\begin{equation} d_i = Q \\cos(2\\phi_i) + U \\sin(2\\phi_i),\n\\label{eqn:diff_timestream}\n\\end{equation} \nwhere again, the angle $\\phi_i$ is, in the most general case, a\ncombination of detector orientation, sky crossing angle, telescope\nboresight rotation and the orientation of the HWP if present. In\norder to construct maps of the $Q$ and $U$ Stokes parameters, these\nquantities need to be decorrelated from the differenced time-stream\nusing multiple observations of the same region of sky taken with\ndifferent values of $\\phi_i$. For an experiment which does not\ncontinuously modulate the polarization signal, the $Q$ and $U$ signals\nhave to be demodulated as part of the map-making step. Note however\nthat for an experiment where the polarization signal is continuously\nmodulated, there are a number of alternative techniques to demodulate\n$Q$ and $U$ at the time-stream level. In separate work, one of us has\ncompared the performance of a number of such demodulation schemes and\nour results will be presented in a forthcoming paper (Brown, in prep.).\nFor the purposes of our current analysis\nhowever, we have applied the same map-based demodulation scheme to all\nthree experiments which we have simulated. In this scheme, once the\ntime-streams from each detector pair have been differenced, $Q$ and\n$U$ maps are constructed as \n\\begin{eqnarray}\n\\left( \\begin{array}{c} Q \\\\ U \\end{array} \\right) &=& \\left(\n \\begin{array}{cc} \n \\langle\\cos^2(2\\phi_i)\\rangle & \\langle\\cos(2\\phi_i)\\sin(2\\phi_i)\\rangle \\\\\n \\langle\\cos(2\\phi_i)\\sin(2\\phi_i)\\rangle & \\langle\\sin^2(2\\phi_i)\\rangle \\\\ \n\\end{array} \\right)^{-1} \\nonumber \\\\\n&&\\mbox{} \\times \\left( \\begin{array}{c}\n \\langle\\cos(2\\phi_i)d_i\\rangle \\\\\n \\langle\\sin(2\\phi_i)d_i\\rangle \\\\ \\end{array} \\right),\n\\label{eqn:qu_mapmaking}\n\\end{eqnarray}\nwhere the angled brackets denote an average over all data falling within\neach map pixel. For the work presented here, this averaging is\nperformed using the data from all detector pairs in one operation. \nOne could alternatively make maps per detector pair which could then\nbe co-added later. For the case where the noise properties of each\ndetector pair are similar, the two approaches should be\nequivalent. Note that the map-maker we use for all of our analyses is\na na\\\"{\\i}ve one -- that is, we use simple binning to implement\nequation~(\\ref{eqn:qu_mapmaking}). There are, of course, more optimal\ntechniques available (e.g. \\citealt{sutton09} and references therein)\nwhich would out-perform a\nna\\\"{\\i}ve map-maker in the presence of non-white noise. However, for our\npurposes, where we wish to investigate the impact of modulation in\nisolation, it is more appropriate to apply the same na\\\"{\\i}ve map-making\ntechnique to all of our simulated data. We can then be sure that any\nimprovement we see in results from our simulations including\nmodulation are solely due to the modulation scheme employed. \n\n\\subsection{Power spectrum estimation}\n\\label{subsec:power}\nWe measure $E$- and $B$-mode power spectra from each of our\nreconstructed maps using the ``pure'' pseudo-$C_\\ell$ method of\n\\cite{smith06}. We will not describe the method in detail here and refer the\ninterested reader to \\cite{smith06} and \\cite{smith07} for\nfurther details. Here, we simply note that the ``pure''\npseudo-$C_\\ell$ framework satisfies most of the requirements of a power\nspectrum estimator for a mega-pixel CMB polarization experiment with\ncomplicated noise properties targeted at constraining $B$-modes: it\nis, just like normal pseudo-$C_\\ell$, a fast estimator scaling as\n$N_{\\rm pix}^{3\/2}$ where $N_{\\rm pix}$ is the number of map pixels\n(as opposed to a maximum likelihood estimator which scales as $N_{\\rm\npix}^3$); it is a Monte-Carlo based estimator relying on simulations \nof the noise properties of the experiment to remove the noise bias and \nestimate band-power errors and covariances -- it is thus naturally\nsuited to experiments with complicated noise properties for which \napproximations to the noise cannot be made; and it is near-optimal in the\nsense that it eliminates excess sample variance from $E \\rightarrow B$ mixing\ndue to ambiguous modes which result from incomplete sky observations\n\\citep{lewis02, bunn02}, and which renders simple pseudo-$C_\\ell$ techniques\nunsuitable for small survey areas~\\citep{challinor05}.\n\n\\subsubsection{Power spectrum weight functions}\n\nWith normal pseudo-$C_\\ell$ estimators, one multiplies the data with a function\n$W(\\hat{\\bmath{n}})$ that is chosen heuristically and apodizes the edge of the survey\n(to reduce mode-coupling effects). For example, if one\nis signal dominated, uniform weighting (plus apodization) is a reasonable\nchoice, whereas an inverse-variance weight is a good choice in the\nnoise-dominated regime. Similar reasoning applies for the\npure pseudo-$C_\\ell$ technique, but here one weights the spherical\nharmonic functions rather than the data themselves.\nTo see this, compare the\ndefinition of the ordinary and pure pseudo harmonic $B$-modes:\n\\begin{eqnarray} \n\\widetilde a_{\\ell m}^B = &-&\\frac{i}{2}\n\\sqrt{\\frac{(l-2)!}{(l+2)!}} \\int d^2\\hat{\\bmath{n}} \\bigg[ \\Pi_+(\\hat{\\bmath{n}})\nW(\\hat{\\bmath{n}}) \\bar{\\eth}\\bar{\\eth} Y_{\\ell m}^*(\\hat{\\bmath{n}}) \\nonumber \\\\ \n&&\\mbox{} - \\Pi_-(\\hat{\\bmath{n}}) W(\\hat{\\bmath{n}}) \\eth\\eth Y_{\\ell m}^*(\\hat{\\bmath{n}}) \\bigg]\n\\label{eq:almbdef} \\\\\n\\widetilde a_{\\ell m}^{B\\,{\\rm pure}} = &-&\\frac{i}{2}\n\\sqrt{\\frac{(l-2)!}{(l+2)!}} \\int d^2\\hat{\\bmath{n}} \\bigg[ \\Pi_+(\\hat{\\bmath{n}})\n\\bar{\\eth}\\bar{\\eth} \\big( W(\\hat{\\bmath{n}}) Y_{\\ell m}^*(\\hat{\\bmath{n}}) \\big)\n\\nonumber \\\\ \n&&\\mbox{} - \\Pi_-(\\hat{\\bmath{n}}) \\eth\\eth \\big(\nW(\\hat{\\bmath{n}}) Y_{\\ell m}^*(\\hat{\\bmath{n}}) \\big) \\bigg]. \n\\label{eq:almbpuredef} \n\\end{eqnarray}\nHere, $\\Pi_\\pm(\\hat{n}) = (Q\\pm\niU)(\\hat{\\bmath{n}})$ is the complex polarization and $\\eth, \\bar{\\eth}$ are\nthe spin raising and lowering operators defined\nin~\\citet{zaldarriaga97}. If $W(\\hat{\\bmath{n}})$ is chosen so that it vanishes\nalong with its first derivative on the survey boundary, then the\n$\\widetilde a_{\\ell m}^{B\\,{\\rm pure}}$ couple only to $B$-modes and\nthe excess sample variance due to $E$-$B$ mixing is eliminated.\nThe action of $\\eth, \\bar{\\eth}$ on the spin spherical\nharmonics is simply to convert between different spin-harmonics but\ntheir action on a general weight function is non-trivial for\n$W(\\hat{\\bmath{n}})$ defined on an irregular pixelisation such as {\\sevensize HEALPIX}\\footnote{%\nOne possibility that we have yet to explore is performing the derivatives\ndirectly in spherical-harmonic space. Since $W(\\hat{\\bmath{n}})$ is typically\nsmooth, its spherical transform will be band-limited and straightforward\nto handle.}. To\nget around this problem, we choose to calculate the derivatives of\n$W(\\hat{\\bmath{n}})$ in the flat-sky approximation where the differential\noperators reduce to\n\\begin{eqnarray}\n\\eth \\approx -(\\partial_x + i \\partial_y), \\\\\n\\bar{\\eth} \\approx -(\\partial_x - i \\partial_y).\n\\end{eqnarray}\nThe derivatives are then trivially calculated on a regular Cartesian \ngrid using finite differencing~\\citep{smith07}. \n\nThe most optimal weighting scheme for a pseudo-$C_\\ell$ analysis\ninvolves different weight functions for each $C_\\ell$ band-power\naccording to the signal-to-noise level expected in that band.\nHowever, this is a costly solution (requiring $3 N_{\\rm band}$\nspherical harmonic transforms) and the indications are, from some\nrestricted tests that we have carried out, that the improvement in\nresulting error-bars is small, at least for the specific noise\nproperties of our simulations. For the analysis presented here, we\nhave therefore chosen a simpler scheme whereby we have used a uniform\nweight, appropriately apodized at the boundaries for the entire $\\ell$\nrange for $E$-modes and for $\\ell \\le 200$ for $B$-modes. For $\\ell >\n200$ our simulated experiment is completely noise dominated for a\nmeasurement of $B$-modes and so here we use an inverse-variance\nweighting, again, appropriately apodized at the boundaries. For\nsimplicity, we have approximated the boundary of the map as a circle\nof radius $11^{\\circ}$. Note that restricting our power spectrum analysis\nto this central region of our maps means we are effectively using only\n$\\sim 70$ per cent of the available data. To calculate the derivatives of the\nweight functions, we project our weight maps (defined in {\\sevensize HEALPIX})\nonto a Cartesian grid using a gnomonic projection. Once the\nderivatives of the weight maps have been constructed on the grid using\nfinite differencing, they are transformed back to the original\n{\\sevensize HEALPIX}\\, grid. An example of the inverse variance weight maps we\nhave used and the resulting spin-1 and spin-2 weight functions, $\\eth\nW$ and $\\eth\\eth W$, for one of our fields are shown in\nFig.~\\ref{fig:ppcl_weights}.\n\\begin{figure*}\n \\centering\n \\resizebox{0.55\\textwidth}{!}{ \n \\rotatebox{0}{\\includegraphics{fig7.ps}}}\n \\caption{Inverse-variance weight functions used for power spectrum\n estimation for one of the southern fields. For the noise properties\n of our simulated data, the hit-map shown in the top left panel\n closely approximates the inverse-variance map. This map is heavily\n smoothed and apodized at the boundary of the map to produce the\n spin-0 weight function shown in the top right panel.\n The spin-1 and spin-2 weight functions, $\\eth W$ and $\\eth\\eth W$,\n are shown (as vector fields) in the bottom left and right-hand\n panels respectively.}\n \\label{fig:ppcl_weights}\n\\end{figure*}\n\n\\section{Results from simulations}\n\\label{sec:results}\nThe map-making and power spectrum estimation procedures described\nabove have been applied to each of our simulated datasets treating\neach of our four observing fields independently. For any given\nsimulation set, we have 50 Monte-Carlo simulations so we can estimate the\nuncertainties on the band-powers measured from each field. For each\nrealisation, we can then combine our measurements from the four fields\nusing inverse-variance weights to produce a final single estimate of\nthe power spectrum for each realisation. When presenting our results\nbelow, in all cases, we plot the mean of these final estimates. For\nour reference simulation, and for the $1\/f$ noise systematics, the\nerror-bars plotted are calculated from the scatter among the\nrealisations and are those appropriate for a single realisation. When\ninvestigating the $B$-mode bias from systematics, we plot the results\nfrom signal-only simulations and the error-bars plotted are the\nstandard error on the mean. For some of the noise-related systematics,\nwe will also examine the impact of modulation in the map domain where\nthe effects are already clear.\n\\subsection{Reference simulation}\n\\begin{figure*}\n \\centering\n \\resizebox{0.55\\textwidth}{!}{ \n \\includegraphics{fig8.ps}}\n \\caption{Sample maps constructed from simulated time-stream\n containing noise only (left panels) and both signal and noise\n (right panels). Temperature maps are shown in the top panel and\n $U$-polarization maps are shown in the bottom panels. These maps\n are for one of our reference simulations with no explicit\n modulation scheme and no systematics included. Note the striping\n in the noise-only $T$ map which is completely absent from the $U$\n maps due to differencing of detector pairs before map-making.}\n \\label{fig:maps_reference}\n\\end{figure*}\n\\begin{figure*}\n \\centering\n \\resizebox{0.70\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig9.ps}}}\n \\caption{Mean recovered $E$-mode (top) and $B$-mode (bottom) power\n spectra for the reference simulations without explicit\n modulation. The errors plotted are those appropriate for a single\n realisation. The input CMB power spectra used to create the signal\n component of the simulations are shown as the red curves. In the\n bottom panel the total $B$-mode input signal (including lensing) for\n a tensor-to-scalar ratio of $r=0.026$ is shown as the red curve and\n the $B$-mode signal due to lensing alone is shown as the dashed\n curve. The $\\ell < 200$ multipole range is shown in detail in the\n inset plots.}\n \\label{fig:cls_reference}\n\\end{figure*}\nTo provide a reference for the results which follow, in\nFigs.~\\ref{fig:maps_reference} and \\ref{fig:cls_reference} we show the results from our\nsuite of simulations with no explicit modulation and with no\nsystematics included. In Fig.~\\ref{fig:maps_reference} we plot examples of the\nreconstructed noise-only and signal-plus-noise $T$ and $U$ Stokes\nparameter maps (the reconstructed $Q$ maps -- not shown -- are\nqualitatively similar to $U$). The raw collecting power of an\nexperiment like {\\sevensize C}$_\\ell${\\sevensize OVER}\\, is apparent from the top two panels in this\nfigure. Although the noise $T$ map shown in the top-left panel is\nclearly dominated by striping due to the correlated noise from the\natmosphere, only signal is apparent in the signal-plus-noise map shown in\nthe top-right panel. (In fact, the noise contribution to the $T$\nsignal-plus-noise map is significant, particularly on large scales, and\nso would need to be accounted for when measuring the temperature power\nspectrum.) Conversely, the $U$ noise map is dominated by\nwhite noise; the correlated component of the atmosphere has been\nremoved completely from the polarization time-streams (as has the $T$\nsky signal) by differencing detector pairs before map-making.\n\nFigure~\\ref{fig:cls_reference} shows the mean recovered $E$- and $B$-mode power\nspectra from our reference simulations for the case of no explicit\nmodulation. Here, we see that our analysis is unbiased and recovers the input\npolarization power spectra correctly. For an input tensor-to-scalar\nratio of $r=0.026$, we recover a detection of $B$-modes \\emph{in\nexcess} of the lensing signal of $1.54\\sigma$.\n(We argue in Section~\\ref{sec:fisher} that this is an under-estimate of\nthe detection significance by around 10 per cent due to our ignoring small\nanti-correlations between the errors in adjacent band-powers.)\n\nThe corresponding plots for the stepped and continuously rotating HWP\nare very similar apart from the reconstructed polarization maps at the\nvery edges of the fields where a modulation scheme increases the\nability to decompose into the $Q$ and $U$ Stokes parameters. Since the\nedges of the field are excluded in our power spectrum analysis in any\ncase (see Fig.~\\ref{fig:ppcl_weights}), we find that the performance\n(in terms of $C_\\ell$ errors) of all three types of experiments which\nwe have considered is qualitatively the same in the absence of\nsystematic effects. Note that for all the systematics we have\nconsidered, the effects on the recovery of the $E$-mode spectrum is\nnegligible and so, in the following sections, we plot only the\nrecovered $B$-mode power spectra which are the main focus of this\npaper.\n\n\\subsection{$1\/f$ detector noise}\n\nFigure~\\ref{fig:maps_det_noise} shows the recovered noise-only maps from\na simulation containing a correlated $1\/f$ detector noise\ncomponent with a knee frequency of $f_{\\rm knee} = 0.1$~Hz. \nIn this figure, we have plotted the noise-only maps from\nthe three types of experiment we have considered: no modulation; a\nHWP which is stepped by 20$^{\\circ}$ at the end of each azimuth scan; and\na HWP continuously rotating at $3$~Hz. The impact of modulation on $1\/f$\ndetector noise is clear from this plot -- as described in\nSection~\\ref{sec:modulation}, the continuously rotating HWP shifts\nthe polarization band in frequency away from the $1\/f$ detector noise\nleaving only white noise in the resulting map. A stepped HWP, on the\nother hand, does not mitigate $1\/f$ detector noise in this way\nand so noise striping is apparent in the middle panel of\nFig.~\\ref{fig:maps_det_noise}. \n\\begin{figure*}\n \\centering\n \\resizebox{0.9\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig10.ps}}}\n \\caption{Sample noise-only $U$-maps from simulations containing a\n $1\/f$ component correlated to the detector noise}. For display\n purposes only, the maps have been smoothed with a Gaussian with a FWHM\n of 7 arcmin. In the case where explicit modulation is either absent (left\n panel) or slow (stepped HWP; middle panel), the $1\/f$ noise\n results in faint residual stripes in the polarization maps. In the case of\n a continuously rotating HWP, the polarization signal is modulated\n away from the low frequency $1\/f$ resulting in white-noise behaviour\n in the polarization map (right panel).\n \\label{fig:maps_det_noise}\n\\end{figure*}\n\nThe $B$-mode power spectra measured from our signal-plus-noise simulations\nincluding $1\/f$ detector noise are shown in\nFig.~\\ref{fig:cls_det_noise} (again for $f_{\\rm knee} = 0.1$~Hz) where\nwe show the results from all three types of experiment. \n\\begin{figure*}\n \\centering\n \\resizebox{0.9\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig11.ps}}}\n \\caption{Recovered $B$-mode power spectra for the simulations\n including a correlated $1\/f$ component to the detector noise with\n $f_{\\rm knee} = 0.1$~Hz for no modulation (top), a stepped HWP\n (middle) and a HWP continuously rotating at 3~Hz (bottom). See Table\n \\ref{tab:simsummary} for the significances with which each\n experiment detects the input signals.}\n \\label{fig:cls_det_noise}\n\\end{figure*}\nExamination of the figure suggests that the presence of a $1\/f$\ncomponent in the detector noise leads to a significant degradation in\nthe ability of the unmodulated and stepped-HWP experiments to recover\nthe input $B$-mode signal. This degradation happens at all multipoles\nbut is particularly acute on the largest scales ($\\ell < 200$) where\nthe primordial $B$-mode signal resides. The marginal detection of\nprimordial $B$-modes (for $r = 0.026$) which we saw in our reference\nsimulation (Fig.~\\ref{fig:cls_reference}) is now completely destroyed\nby the presence of the $1\/f$ correlated detector noise. Furthermore,\nnote that our analysis of the simulated data sets is optimistic in the\nsense that we have assumed that any correlated noise can be modelled\naccurately -- that is, our noise-only simulations, which we use to\nmeasure the noise bias, are generated from the same model noise power\nspectrum used to generate the noise component in our signal-plus-noise\nsimulations. (This is the reason that our recovered spectra are\nunbiased.) However, for the analysis of a real experiment, the noise\nproperties need to be measured from the real data and there are\nuncertainties and approximations inherent in this process. Any\ncorrelated detector noise encountered in a real experiment is unlikely\nto be understood to the level which we have assumed in our analysis\nand so will likely result not only in the increased uncertainties we\nhave demonstrated here but also in a biased result at low\nmultipoles. Estimating cross-spectra between maps made from subsets of\ndetectors for which the $1\/f$ detector noise is measured to be\nuncorrelated is a simple way to avoid this noise bias issue, at the\nexpense of a small increase in the error-bars~\\citep{hinshaw03}.\n\nThe results from the simulations where we have continuously modulated\nthe polarization signal recover the input $B$-mode signal to the same\nprecision that we saw with our reference simulation -- our marginal\ndetection of $r=0.026$ is retained even in the presence of the\ncorrelated detector noise. In Section \\ref{sec:significances} and\nTable \\ref{tab:simsummary} we show quantitatively that, for a detector\nknee frequency of $0.1$~Hz, the significance with which the\ncontinuously modulated experiment detects the primordial $B$-mode\nsignal is roughly twice that found for the un-modulated and\nstepped-HWP experiments. Also detailed in Table \\ref{tab:simsummary}\nare the results from our $1\/f$ noise simulations with knee frequencies\nof $0.05$ and $0.01$~Hz. We see, as expected, that the impact of fast\nmodulation is less for a lower knee frequency --- for $f_{\\rm knee} =\n0.05$, rapid modulation still significantly out-performs the\nun-modulated and stepped-HWP experiments while for $f_{\\rm knee} =\n0.01$~Hz, there is essentially no difference between the performance\nof the three types of experiment.\n\nNote that, in the case of rapid modulation, because the polarization\nsignal is moved completely away from the $1\/f$ frequency regime, the\nrecovered spectra should be immune to the issues of mis-estimation or\npoor knowledge of the noise power spectrum at low frequencies\nmentioned above. Although detector $1\/f$ noise can be mitigated by\nother methods (e.g. using a more sophisticated map-maker;\n\\citealt{sutton09}), these usually require accurate knowledge of the\nlow-frequency noise spectrum unlike the hardware approach of fast\nmodulation.\n\n\\subsection{Polarized atmospheric $1\/f$}\nIn contrast to the addition of $1\/f$ detector noise, which can be\nsuccessfully dealt with by rapid modulation, all three types of\nexperiment are degraded similarly by polarized low-frequency noise in\nthe atmosphere. In particular, the errors at low multipoles are\ninflated by a large factor since the large amount of polarized\natmosphere which we have input to the simulations swamps the input\n$B$-mode signal for $r=0.026$. We stress that the levels of polarized\natmosphere we have used in these simulations were deliberately chosen to\ndemonstrate the point that modulation does not help and the levels are\ncertainly pessimistic. \n\n\\subsection{Calibration errors}\nThe power spectra recovered from signal-only simulations where we\nintroduced random gain errors (constant in time) across the focal plane,\nor 1 per cent systematic A\/B gain mis-matches between the two detectors\nwithin each pixel are shown in Fig.~\\ref{fig:cls_gain_errors}. In both\ncases, we see a clear bias in the recovered $B$-mode signal in the\nabsence of fast modulation, but the bias is mitigated entirely by the\npresence of a HWP rotating at 3~Hz. The bias is also mitigated to some\ndegree by the stepped HWP but not completely. In our simulations, the bias is\ngenerally larger for the case of random gain errors since the\nvariance (across the focal plane) of the gain mismatches is twice as large\nin the former case. For our simulations where we allowed detector gains to drift\nover the course of a two-hour observation, we found a similarly\nbehaved $B$-mode bias to those shown in Fig.~\\ref{fig:cls_gain_errors}\nbut with a smaller magnitude (due to the two-hour drifts averaging down\nover the eight-hour observation).\n\n\\begin{figure*}\n \\centering\n \\resizebox{0.90\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig12.ps}}}\n \\caption{Mean recovered $B$-mode power spectra for the simulations\n including random gain errors across the focal plane (black points) or\n systematic 1 per cent A\/B gain mis-matches between detector pairs (blue points)\n for no modulation (top), a stepped\n HWP (middle) and a continuously rotating HWP (bottom). These spectra\n are measured from our suite of signal-only simulations. Our\n simulations containing both signal and noise exhibit the same biased\n recovery for the no-modulation and stepped-HWP cases. The presence\n of a fast modulation scheme (bottom panel) mitigates entirely the\n bias caused by these gain mis-matches. The standard errors in these\n mean recovered spectra are smaller than the plotted symbols.}\n \\label{fig:cls_gain_errors}\n\\end{figure*}\n\nA mis-match between the gains of the two detectors within a pixel\ncorresponds to a $T \\rightarrow Q$ leakage in the detector\nbasis. The projection of this instrumental polarization onto the sky\nwill therefore be suppressed if a wide range of sensitivity directions\n$\\phi_i$ contribute to each sky pixel, as is the case for fast\nmodulation. Note that in the case of a stepped HWP, one should be\ncareful to design the stepping strategy in such a way that it does not\nundo some of the effect of sky rotation. During our analysis, we have\nfound that the performance of a stepped HWP in mitigating systematic\neffects can depend critically on the direction, magnitude and\nfrequency of the HWP step applied. In fact, for some set-ups we have\ninvestigated, a stepped HWP actually worsened the performance in\ncomparison to the no modulation case due to interactions between the\nscan strategy and HWP stepping strategy. However, the results plotted in\nFig.~\\ref{fig:cls_gain_errors} for the stepped HWP case are for a\nHWP step of $20^{\\circ}$ between each azimuth scan which is large and\nfrequent enough to ensure that such interactions between the stepping strategy\nand the scan strategy are sub-dominant. \n\n\\subsection{Mis-estimated polarization angles}\nThe next set of systematics we have considered concern a\nmis-estimation of both the detector orientation angles (i.e.\\ the\ndirection of linear polarization to which each detector is sensitive\nto) and, for the case where a HWP is employed, a mis-estimation of the\nHWP orientation. We have performed simulations including both a random\nscatter (with an RMS of $0.5^{\\circ}$) and a systematic offset of\n$0.5^{\\circ}$ in the simulated detector and HWP angles. Note that for\nthe systematic offset in the detector angles, the same offset is\napplied to all detectors. For both the detector angles and the HWP,\nthe offset introduced corresponds to a systematic error in the\nestimation of the global polarization coordinate frame of the\nexperiment and the effects are therefore degenerate.\n\nFor the simulations which included a random scatter in the angles\n(both detector angles and HWP orientation), we found neither a bias in\nthe recovered $B$-mode power spectra, nor a degradation in the\nerror-bars from the simulations containing both signal and\nnoise. Following the discussion in Section~\\ref{sec:systematics},\ncommon errors in the detector angles for the pair of detectors in a single\nfocal-plane pixel give rise to a rotation of the polarization sensitivity\ndirection of the pixel, while differential errors reduce the polarization\nefficiency.\nFor a typical differential scatter of $\\sqrt{2}\\times 0.5^{\\circ}$,\nthe reduction in the polarization efficiency ($\\sim 10^{-4}$) is\nnegligible. For a given pixel on the sky, the impact of the polarization\nrotation is suppressed by $\\sqrt{N_\\mathrm{sample}}$, where\n$N_\\mathrm{sample}$ is the total number of samples contributing to that\npixel with independent errors in the angles. The combination of a\nlarge number of detectors and, in the case of random HWP angle errors,\ntheir assumed short correlation time in our simulations renders the\neffect of small and random scatter in the angles negligible.\n\nThe results from simulations which included a systematic error in the\nangles are shown in Fig.~\\ref{fig:cls_pol_angles}, where we plot the recovered\n$B$-mode power spectrum from our signal-only simulations. In contrast\nto the simulations with random scatter, there is a clear mixing between $E$\nand $B$ due to the systematic mis-calibration of the polarization\ncoordinate reference system of the instrument. A global mis-estimation\nof the polarization direction by an angle $\\psi$ in the reconstructed maps\nleads to spurious $B$-modes with\n\\begin{equation}\nC_\\ell^B = \\sin^2(2\\psi) C_\\ell^E \\approx 4 \\psi^2 C_\\ell^E .\n\\end{equation}\nNote that in\nFig.~\\ref{fig:cls_pol_angles}, we show the results from the detector angle\nsystematic only for the case of the non-modulated experiment but the\nplot is identical for both the stepped and continuously rotating HWP --\npolarization modulation cannot mitigate a mis-calibration of detector \nangles. The fact that the mixing apparent in Fig.~\\ref{fig:cls_pol_angles} is\ngreater for the HWP mis-calibration is simply because rotating the\nwaveplate by $\\psi$ rotates the polarization direction by $2\\psi$.\nAlthough the spurious $B$-mode power is most\nnoticeable at high multipoles, where $\\ell(\\ell+1)C_\\ell^E$ is largest,\nit is also present on large scales and,\nas is clear from the plot, would bias a measurement of the $B$-mode \nspectrum at all multipoles. \n\\begin{figure}\n \\centering\n \\resizebox{0.48\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig13.ps}}}\n \\caption{Mean recovered $B$-mode power spectra for the signal-only simulations\n including mis-estimated detector polarization sensitivity angles (black\n points) and mis-estimated HWP angles (blue points) where the angles have been\n systematically offset by 0.5$^{\\circ}$ in both cases. The standard errors in these\n mean recovered spectra are smaller than the plotted symbols.} \n \\label{fig:cls_pol_angles}\n\\end{figure}\n\n\\subsection{Mis-estimated time-constants}\n\nThe power spectra measured from simulations which included random and\nsystematic errors in the detector time-constants displayed neither a\nbias, nor a degradation in error-bars. This was to be expected for the\nslow scan speed and extremely fast time-constants we have considered\nin this analysis -- the response function of the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, detectors\nis effectively phase-preserving with zero attenuation in the frequency\nband which contains the sky-signal in our simulations. Note that this\nwould not necessarily have been the case had we considered a much\nfaster scan speed or more rapid polarization modulation.\n\n\\subsection{Pointing errors}\n\nOur analysis of simulations where we have introduced a jitter in the\npointing and\/or an overall wander in the pointing suggest that these\nsystematics have only a very small effect on the recovered $B$-mode\npower spectra, at least for the levels which we have considered\n(i.e. a $30$ arcsec random jitter in the pointing and\/or an overall\nwander of the pointing by $1$ arcmin over the course of a two-hour\nobservation). The only observed effect was a slight suppression of the\nrecovered $B$-mode signal at high multipoles consistent with a\nslight smearing of the effective beam. We note however that the effect \nwe observed was extremely small and was only noticeable in our\nsignal-only simulations. For our simulations containing noise, the\neffect was completely swamped by the errors due to random noise. \n\nIn principle, pointing errors can also lead to leakage\nfrom $E$ to $B$~\\citep{hu03,odea07}. In Appendix~\\ref{app:pointing}\nwe develop a toy-model for\nthe leakage expected from random pointing jitter in the case of\na scan\/modulation strategy that produces a uniform spread of polarization\nsensitivity directions in each sky pixel (such as by fast modulating\nwith a HWP). The result is a white-noise spectrum of $B$-modes but,\nfor the simulation parameters adopted here, the effect is very small --\nless than $1$ per cent of the $B$-mode power induced by weak gravitational lensing\non large scales.\n\n\\subsection{Differential transmittance in the HWP}\nThe power spectra reconstructed from our simulations which \nincluded a 2 per cent differential transmittance in the HWP exhibited\nno degradation in the accuracy of the recovered $B$-mode\nsignal --- even the relatively simple recipe which we have used to\nremove the HWP-synchronous signals from the time-stream \n(see Section \\ref{sec:map-making}) appears sufficient to recover the\n$B$-mode signal to the same accuracy as was seen in our reference\nsimulations. (We quantify this statement in the next section where we\nestimate the detection significances with which the different simulations\ndetect the $E$ and $B$-mode signals). As mentioned in\nSection~\\ref{sec:modulation}, the recovered polarization signal is\nmis-calibrated by $\\sim 2$ per cent in amplitude ($4$ per cent in\npower). Compared to the random noise however, this mis-calibration is\na small effect and is easily dealt with during, e.g. a cosmological\nparameter analysis by marginalising over it. \n\nNote that no prior information on the level of differential\ntransmittance was used during our analysis of the data. Our technique\nfor removing the HWP-synchronous signals is a blind one in this\nsense and should work equally well for other HWP-systematic effects\nthat result in spurious signals at harmonics of the HWP rotation\nfrequency, $f_\\lambda$.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\subsection{Controlling systematics with polarization modulation}\n\\label{sec:significances}\nThe main goal of the analysis presented in this paper is to\ndemonstrate and quantify with simulations the impact of two types of\npolarization modulation (slow modulation using a stepped HWP and rapid\nmodulation with a continuously rotating HWP) on the science return of\nupcoming CMB $B$-mode experiments in the presence of various\nsystematic effects. Although our list of included systematics is not\nan exhaustive one (in particular, we are still investigating the case\nof imperfect optics), we are nevertheless in a position to draw some\nrather general conclusions regarding the usefulness of modulation in\nmitigating systematics. It is, of course, important to bear in mind\nthat we have only considered two examples of a HWP-related systematic\neffect (imperfect HWP angles and differential transmittance in the\nplate). The are many more possible effects which will need to be\nwell understood and strictly controlled if fast polarization\nmodulation with HWPs is to realise its potential.\n\n\\subsubsection*{(i) Systematics mitigated by modulation}\n\\begin{itemize} \n\\item{\\bf Correlated $1\/f$ detector noise:} As expected by the general\n reasoning of Section~\\ref{sec:modulation}, and further borne out\n by our results from simulations, rapid polarization modulation is\n extremely powerful at mitigating a correlated $1\/f$ component in the\n detector noise. Any such $1\/f$ component is not mitigated by\n a HWP operating in stepped mode. \n\\item{\\bf Calibration errors:} Our results demonstrate that fast\n modulation is also useful for mitigating against possible\n calibration errors since it greatly increases the range of\n directions over which sky polarization is measured in a given pixel.\n For example, the clear bias introduced in our simulations by random\n gain drifts or systematic mis-calibrations between detectors was\n mitigated entirely by the HWP continuously rotating at $3$~Hz. This\n bias was also partly (but not completely) mitigated by stepping the\n HWP by 20$^{\\circ}$ between each of our azimuth scans. For some stepping\n strategies we have investigated, the bias actually increased --- a\n poor choice of stepping strategy can actually be worse than having\n no modulation because of interactions between the sky rotation and\n the HWP orientations.\n\\end{itemize}\n\n\\begin{table*}\n\\caption{Detection significances (in units of $\\sigma$) for our\n reference simulations, for our simulations with $1\/f$ noise\n systematics and for our simulations with a 2 per cent differential\n transmittance in the HWP. Also included for comparison are the predicted\n detection significances from a Fisher matrix analysis of the power\n spectrum errors (see Section \\ref{sec:fisher}) and from the\n simulations containing isotropic and uniform Gaussian noise (see\n text). The rightmost column displays the significance of the\n detection of the $B$-mode signal in excess of the lensing signal\n which corresponds directly to the significance with which each\n simulation detects the input tensor-to-scalar ratio of $r=0.026$.}\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c}\nSimulation & Modulation & $E$-mode & $B$-mode & $r = 0.026$ \\\\\n\\hline\nFisher predictions & --- & $128.6$ & $10.24$ & $1.90$ \\\\\n\\hline\nUniform noise & --- & $126.4$ & $10.30$ & $1.45$ \\\\\n\\hline\nReference simulation & None & $127.8$ & $10.41$ & $1.54$ \\\\\n & Stepped HWP & $130.1$ & $10.11$ & $1.41$ \\\\\n & Rotating HWP & $127.7$ & $10.72$ & $1.45$ \\\\\n\\hline\n$1\/f$ detector noise & None & $124.2$ & $7.29$ & $0.83$ \\\\\n($f_{\\rm knee} = 0.1$~Hz) & Stepped HWP & $124.6$ & $7.09$ & $0.79$ \\\\\n & Rotating HWP & $126.8$ & $9.96$ & $1.47$ \\\\\n\\hline\n$1\/f$ detector noise & None & $125.1$ & $8.61$ & $0.95$ \\\\\n($f_{\\rm knee} = 0.05$~Hz) & Stepped HWP & $127.5$ & $8.59$ & $1.14$ \\\\\n & Rotating HWP & $125.6$ & $10.31$ & $1.45$ \\\\\n\\hline\n$1\/f$ detector noise & None & $126.8$ & $9.78$ & $1.30$ \\\\\n($f_{\\rm knee} = 0.01$~Hz) & Stepped HWP & $128.0$ & $9.99$ & $1.44$ \\\\\n & Rotating HWP & $127.1$ & $10.28$ & $1.40$ \\\\\n\\hline\nPolarized atmosphere & None & $122.9$ & $6.86$ & $0.12$ \\\\\n & Stepped HWP & $124.9$ & $6.99$ & $0.10$ \\\\\n & Rotating HWP & $126.0$ & $8.24$ & $0.21$ \\\\\n\\hline\nDifferential transmittance & Rotating HWP & $127.6$ & $10.92$ & $1.59$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:simsummary}\n\\end{table*}\n\n\\vspace{-6mm}\n\\subsubsection*{(ii) Systematics not mitigated by modulation}\n\\begin{itemize}\n\\item{\\bf Polarized $1\/f$ atmosphere:} No amount of modulation (rapid or\n slow) will mitigate a polarized $1\/f$ component in the\n atmosphere. The results from our simulations containing polarized\n atmosphere are summarised in Table \\ref{tab:simsummary}. \n\\item{\\bf Pointing errors:} For our simulations which included pointing\n errors, the effect on the recovered $B$-mode power spectra was\n extremely small and was equivalent to a slight smoothing of the\n effective beam. Although the same amount of smoothing was observed\n in all the simulations (and so the effect is not mitigated by\n polarization modulation), the effect is negligible for the\n sensitivities and beam sizes considered here. The further leakage of\n $E$-mode power into $B$ modes due to pointing errors was, as expected\n (see Appendix~\\ref{app:pointing}) unobservably small in our simulations.\n\\item{\\bf Mis-calibration of polarization angles: } A polarization\n modulation scheme does not mitigate a systematic error in the\n calibration of the polarization sensitivity directions. Experiments\n using a HWP will require precise and accurate measurements of the\n HWP angle at any given time to avoid the $E \\rightarrow B$ mixing\n apparent in Fig.~\\ref{fig:cls_pol_angles}.\n\\end{itemize}\n\nOur results are in broad agreement with those of a similar study by\n\\cite{mactavish08} who based their analysis on signal-only simulations\nof the {\\sevensize SPIDER}\\, experiment. Both \\cite{mactavish08} and this study find\nthat polarization modulation with a continuously rotating HWP is\nextremely effective in mitigating the effects of $1\/f$ detector noise\nbut that, in the presence of significant $1\/f$ noise, the\nanalysis of an experiment where modulation is either absent or slow \nwill require near-optimal map-making techniques. In addition, both studies find that the effect\nof small and random pointing errors on the science return of upcoming\n$B$-mode experiments is negligible given the experimental\nsensitivities. The two analyses also find that the effect of random errors\n(with $\\sim 0.5^{\\circ}$ RMS) in the detector polarization sensitivity\nangles is negligible but that the global polarization coordinate frame\nof the experiment needs to be measured carefully --- \\cite{mactavish08}\nquote a required accuracy of $< 0.25^{\\circ}$ for {\\sevensize SPIDER}\\, which is\nconsistent with the requirement for an unbiased measurement of\nthe $B$-mode signal (for $r = 0.026$) at $\\ell < 300$ with {\\sevensize C}$_\\ell${\\sevensize OVER}. Finally, both\nstudies suggest that, in the absence of fast modulation, relative gain errors\nwill also need to be controlled to the $< 1$ per cent level (although, in\nthis paper, we have demonstrated that such gain errors are almost\nentirely mitigated using a fast modulation scheme; see\nFig.~\\ref{fig:cls_gain_errors}). \n\nIn Table~\\ref{tab:simsummary}, we quantify the impact of modulation on\nthe $1\/f$ noise systematics we have considered in this work by\nconsidering the significances with which we detect the $E$-mode and\n$B$-mode signals. For comparison, the detection significances in the\npresence of a 2 per cent differential transmittance in the HWP are also\npresented. To calculate the total significance of the detection\nwe compute the Fisher error on the amplitude of a fiducial\nspectrum,\n\\begin{equation}\n\\frac{S}{N} = \\left( \\sum_{bb'} P_b^{\\rm fid} {\\rm cov}^{-1}_{bb'}\nP_{b'}^{\\rm fid} \\right)^{1\/2},\n\\end{equation}\nwhere ${\\rm cov}^{-1}_{bb'}$ is the inverse of the band-power covariance\nmatrix for the given spectrum.\nFor the total significance of a detection of $E$ or $B$-modes, the\nfiducial band-powers are simply the binned input power spectra. In\norder to estimate the significance of a detection of primordial\n$B$-modes, we subtract the lensing contribution from the input\n$B$-mode power spectra. Because the primordial $B$-mode power spectrum\nis directly proportional to the tensor-to-scalar ratio, $r$, the\nsignificance with which we detect the $B$-mode signal in excess of the\nlensing signal translates directly to a significance for the detection\nof our input tensor-to-scalar ratio of $r=0.026$.\nWhen analysing the results of the\nsimulations, we approximate ${\\rm cov}^{-1}_{bb'} \\approx \\delta_{bb'}\/\n\\sigma_b^2$ since we are unable to estimate the off-diagonal elements\nfrom our small number of realisations (50) in each simulation set.\nWe know from a Fisher-based analysis (see Section~\\ref{sec:fisher}),\nthe results of which are also reported in Table~\\ref{tab:simsummary},\nthat neighbouring band-powers on the largest scales are, in fact,\n$\\sim 10$ per cent \nanti-correlated, and the diagonal approximation therefore\n\\emph{underestimates} the detection significance by $\\sim 10$ per cent.\nFor consistency, the numbers quoted for the Fisher analysis ignore the\noff-diagonal elements of the covariance matrix. Including the correlations\nincreases the $E$-mode significance to $144.6$ (from $128.6$),\nthe total $B$-mode significance to $11.57$ (from $10.24$) and\nthe primordial $B$-mode significance to $2.04$ (from $1.90$).\n\nIn comparing the entries in Table~\\ref{tab:simsummary} one should\nkeep in mind that the significances reported for the simulations are\nsubject to a Monte-Carlo error due to the finite number ($N_{\\mathrm{sim}}=50$)\nof simulations used to estimate the band-power errors. Approximating the\nband-powers as uncorrelated and Gaussian distributed, the\nsampling error in our estimates of the $S\/N$ is\n\\begin{equation}\n\\Delta (S\/N) \\approx \\frac{1}{S\/N} \\frac{1}{\\sqrt{2N_{\\mathrm{sim}}}}\n\\left[\\sum_b \\left(\\frac{P_b^{\\mathrm{fid}}}{\\sigma_b}\\right)^4\\right]^{1\/2} .\n\\end{equation}\nFor the reference simulation, this gives an error of $0.15$ in the\nsignificance of a detection of $r$ and $0.22$ in the significance of the\ntotal $B$-mode spectrum. The size of these errors likely explain the\napparent anomalies that rotating the HWP degrades the detection of $r$\nin the reference simulation, and that adding $1\/f$ detector noise\nimproves the detection of $r$ over the reference simulation for the case\nof a rotating HWP.\n\nAlso included in Table~\\ref{tab:simsummary} is the performance of our\nexperiment as estimated from a set\nof simple map-based simulations where we have injected uniform and\nisotropic white noise into signal-only $T$, $Q$ and $U$ maps\ndirectly. In these simulations, and also for the Fisher analysis,\nthe white-noise levels were chosen to\nmatch the noise levels in our main analysis and so they have identical\nraw sensitivity to the time-stream simulations but with perfectly\nbehaved noise properties. The broad agreement between our full\ntime-stream simulations and these simple map-based simulations\nsuggests that the anisotropic noise distribution introduced by the\n{\\sevensize C}$_\\ell${\\sevensize OVER}\\, scan strategy does not have a large impact on the\nperformance of the experiment. This agreement also suggests that the\nslightly poorer performance of the simulations in recovering the\n$r=0.026$ primordial $B$-mode signal as compared to the Fisher\npredictions is due to the sub-optimal performance on large scales of the\n(pure) pseudo-$C_\\ell$ estimator we have used.\n\n\\subsection{Importance of combining data from multiple detectors}\n\\label{sec:differencing}\nFor all of our analyses up to this point, in order to remove the\ncorrelated $1\/f$ atmospheric noise from the polarization\nanalysis, we have differenced detector pairs before\nmap-making. However, as mentioned in Section~\\ref{sec:modulation}, for\nthe case of a continuously modulated experiment, it is possible to\nmeasure all three Stokes parameters from a single detector in\nisolation. Here, we argue that this may be a poor choice of analysis\ntechnique in the presence of a highly correlated and common-mode\nsystematic such as atmospheric $1\/f$, at least when one employs\nreal-space demodulation techniques such as those that we have used in\nthis analysis. The key point to appreciate here is that, even with a\nrapid modulation scheme, and for an ideal experiment, it is impossible\nto separate completely the temperature and polarization signals in\nreal space using only a single detector.\\footnote{We note that this is\nnot necessarily true for the case of genuinely band-limited\ntemperature and polarization signals when a classical lock-in\ntechnique (such as that used in the analysis of the {\\sevensize MAXIPOL}\\, data;\n\\citealt{johnson07}) is used to perform the demodulation. We are\ncurrently working to integrate such a technique into our analysis.}\nIn contrast, the technique of detector differencing achieves this\ncomplete separation of the temperature and polarization signals (again\nfor the case of an ideal experiment). Note that this is true even in\nthe signal-only case. Consider again the modulated signal, in the\nabsence of noise, from a single detector sensitive to a single\npolarization: \\begin{equation} d_i = \\left[ T(\\theta) + Q(\\theta) \\cos(2\\phi_i) +\nU(\\theta)\\sin(2\\phi_i) \\right]\/2. \\end{equation} If for each observed data\npoint, $d_i$, the true sky signals, $T$, $Q$ and $U$ are different (as\nis the case for a scanning experiment), there is clearly no way to\nrecover the true values of $T$, $Q$ and $U$ at each point in time.\nThe approximation that one must make in order to demodulate the data\nin real space goes to the very heart of map-making -- that the true\ncontinuously varying sky signal can be approximated as a pixelised\ndistribution where $T$, $Q$ and $U$ are taken to be constant within\neach map-pixel. Armed with this assumption, all three Stokes\nparameters can be reconstructed from a single detector time-stream\nusing a generalisation of equation~(\\ref{eqn:qu_mapmaking}):\n\\begin{equation}\n\\left( \\begin{array}{c} T \\\\ Q \\\\ U \\end{array} \\right) = 2 \\, {\\mathsf M}^{-1} \\cdot\n\\left( \\begin{array}{c}\n \\langle d_i \\rangle \\\\\n \\langle\\cos(2\\phi_i)d_i\\rangle \\\\\n \\langle\\sin(2\\phi_i)d_i\\rangle \\\\ \\end{array} \\right).\n\\label{eqn:tqu_mapmaking}\n\\end{equation}\nwhere the decorrelation matrix, ${\\mathsf M}$ is now given by\n\\begin{equation} \n{\\mathsf M} = \\left( \\begin{array}{ccc} \n \\!\\!\\! 1 & \\!\\!\\! \\langle\\cos(2\\phi_i)\\rangle & \\!\\!\\!\\!\\! \\langle\\sin(2\\phi_i)\\rangle \\\\\n \\!\\!\\! \\langle\\cos(2\\phi_i)\\rangle & \\!\\!\\! \\langle\\cos^2(2\\phi_i)\\rangle &\n \\!\\!\\!\\!\\! \\langle\\cos(2\\phi_i)\\sin(2\\phi_i)\\rangle \\\\\n \\!\\!\\! \\langle\\sin(2\\phi_i)\\rangle & \\!\\!\\! \\langle\\cos(2\\phi_i)\\sin(2\\phi_i)\\rangle &\n \\!\\!\\!\\!\\! \\langle\\sin^2(2\\phi_i)\\rangle \\\\ \\end{array} \\!\\!\\!\\! \\right).\n\\end{equation}\n\\begin{figure}\n \\centering\n \\resizebox{0.48\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig14.ps}}}\n \\caption{Recovered noise only $U$-polarization maps from one of our\n reference simulations with continuous modulation. The map on the\n left is reconstructed from demodulated single detector ``pure'' $U$\n time streams and has not used information from multiple detectors to\n separate the $T$ and polarization signals. The map on the right is\n made from demodulated detector-pair ``pure'' $U$ time-streams and\n explicitly combines information from the two detectors within each\n pixel to separate $T$ from $Q\/U$. Although striping is absent from\n both maps, the white-noise level in the detector-differenced map is\n reduced compared to that made using the non-differencing\n analysis.}\n \\label{fig:maps_diff_nodiff}\n\\end{figure}\nIf, on the other hand, the data from different detectors are combined\n(e.g.\\ when detector differencing is used), the situation is different\n-- because the two detectors within a pixel observe exactly the same\nun-polarized component of the sky signal at exactly the same time,\ndifferencing the detectors removes\nthe $T$ signal completely without any assumptions regarding the scale\nover which the true sky signal is constant. In this case, the\ndecorrelation of $Q$ and $U$ using equation~(\\ref{eqn:qu_mapmaking})\nstill requires an assumption regarding the constancy of the $Q$ and\n$U$ signals over the scale of a map-pixel but now the much larger\ntemperature signal has been removed from the polarization analysis\ncompletely. \n\nNow, if in addition to the sky signal, we have a common-mode\ntime-varying systematic such as an un-polarized $1\/f$ component in the\natmosphere, this contaminant will again be removed entirely with the\ndetector differencing technique (as long as it is completely\ncorrelated between the two detectors) whilst it will introduce a\nfurther approximation into any attempt to decorrelate all three Stokes\nparameters from a single detector time-stream using\nequation~(\\ref{eqn:tqu_mapmaking}). \n\nTo illustrate this point, and to stress the importance of combining\ndata from multiple detectors, we have re-analysed the simulated data\nfrom our set of continuously modulated reference simulations but now\nwe perform the demodulation at the time-stream level for either single\ndetectors or single detector-pairs in isolation. Firstly, we\ndemodulate each detector time-stream individually using\nequation~(\\ref{eqn:tqu_mapmaking}) but now the averaging is performed\nover short segments in time rather than over all data falling within a\nmap-pixel. This procedure results in ``pure'' $T$, $Q$ and $U$\ntime-streams for each detector but at a reduced data rate determined\nby the number of data samples over which the averaging of\nequation~(\\ref{eqn:tqu_mapmaking}) is performed. For our second\nanalysis, we first difference each detector pair and then demodulate\nthe differenced time-streams using equation~(\\ref{eqn:qu_mapmaking}),\nagain applied over short segments of time, resulting in ``pure'' $Q$\nand $U$ time-streams for each detector pair, once again at a reduced\ndata rate. Maps of the $Q$ and $U$ Stokes parameters are then\nconstructed by simple binning of the demodulated $Q$ and $U$\ntime-streams from all detectors or detector pairs. In the case where\ndetector pairs are differenced, we are explicitly combining\ninformation from multiple detectors to separate the temperature and\npolarization signals whilst when we do not difference, we are\nattempting to separate the $T$ and $Q\/U$ signals present in detector\ntime-stream in isolation.\n\nThe results of these tests are shown in Figs.~\\ref{fig:maps_diff_nodiff} and\n\\ref{fig:cls_diff_nodiff}. Figure~\\ref{fig:maps_diff_nodiff} shows the noise-only\n$U$-polarization maps recovered using the two different analysis\ntechniques. Although striping from the atmospheric fluctuations is not\npresent in either of the maps, the extra uncertainty introduced when\none attempts to separate the $T$ and $Q\/U$ signals from individual\ndetectors in isolation clearly results in an increased white noise\nlevel in the polarization maps. This results in a degradation factor\nof $\\sim 2$ in the resulting measurements of the $B$-mode power\nspectrum on all angular scales (Fig.~\\ref{fig:cls_diff_nodiff}) with\na corresponding degradation in the detection significances for both\na measurement of the total $B$-mode signal and for a detection of $r=0.026$\n(Table~\\ref{tab:diff_nodiff}). These results are in excellent\nagreement with those of \\cite{sutton09} who found it necessary to\napply optimal mapping techniques to rapidly modulated single-detector\ntime-streams in the presence of $1\/f$ atmospheric noise. \n\nWe emphasise that the results presented in\nFig.~\\ref{fig:cls_diff_nodiff} and in Table~\\ref{tab:diff_nodiff} for\nthe case where we have analysed single detectors in isolation would\nlikely be improved if the Fourier domain filtering used in\n\\cite{johnson07} was implemented. We are currently working to\nintegrate this step into our algorithm, and we expect to report the\nsubsequent improvement in future publications.\n \n\\begin{figure}\n \\centering\n \\resizebox{0.48\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig15.ps}}}\n \\caption{Comparison between the $B$-mode power spectra recovered\n using an analysis based on differencing detector pairs (black\n points) and one based on demodulating each detector individually\n (blue points). In the presence of a time-varying common-mode\n systematic, such as the $1\/f$ atmospheric noise we have considered\n here, the analysis based on detector differencing is far superior to\n the analysis based on demodulating each detector individually.}\n \\label{fig:cls_diff_nodiff}\n\\end{figure}\n\n\\begin{table}\n\\caption{Detection significances (in units of $\\sigma$) from the\n analysis of identical simulated data with a HWP rotating\n continuously at $3$~Hz. The first analysis is based on detector\n differencing, the second based on demodulation of individual\n detectors in isolation.}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\nAnalysis & $E$-mode & $B$-mode & $r = 0.026$ \\\\\n\\hline\nDetector differencing & $132.9$ & $10.14$ & $1.43$ \\\\\nDemodulation & $123.4$ & $5.11$ & $0.68$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:diff_nodiff}\n\\end{table}\n\n\\subsection{Comparison of simulated and predicted {\\sevensize C}$_\\ell${\\sevensize OVER}\\, performance}\n\\label{sec:fisher}\nIt is common practice to make predictions of the performance of\nupcoming experiments using a Fisher-matrix analysis which\nattempts to predict the achievable errors on, for example, power\nspectra or cosmological parameters under some simplifying\nassumptions. Generally these assumptions will include uniform coverage\nof the observing fields and uncorrelated Gaussian noise resulting in\nan isotropic and uniform white-noise distribution across the observing\nfield. In contrast, the work described in this paper has made use of a\ndetailed simulation pipeline which we have created for the {\\sevensize C}$_\\ell${\\sevensize OVER}\\,\nexperiment. Our simulation pipeline includes the {\\sevensize C}$_\\ell${\\sevensize OVER}\\,\nfocal-plane designs as well as a realistic scan strategy appropriate\nfor observing the four chosen {\\sevensize C}$_\\ell${\\sevensize OVER}\\, fields from the telescope site\nin Chile. In addition we have employed a detailed model of the TES\ndetector noise properties and responsivity, and $1\/f$ atmospheric\nnoise correlated across the focal-plane array. Moreover, our errors\nare calculated using a Monte-Carlo analysis and so should\nautomatically include any effects due to correlations between map\npixels etc. An interesting exercise therefore is to compare the\nexpected errors from a Fisher-matrix analysis to those obtained\nfrom our simulation analysis. \n\nThe polarization band-power Fisher matrix is (e.g.~\\citealt{tegmark01})\n\\begin{equation}\n\\mathrm{cov}^{-1}_{(bP)(bP)'} = \\frac{1}{2}\\mathrm{tr}\\left(\n\\mathbfss{C}^{-1} \\frac{\\partial \\mathbfss{C}}{\\partial \\mathcal{C}_b^P}\n\\mathbfss{C}^{-1} \\frac{\\partial \\mathbfss{C}}{\\partial \\mathcal{C}_{b'}^{P'}}\n\\right)\n\\end{equation}\nwhere $P$ and $P'$ are $E$ or $B$, and $b$ labels the bandpower. Here,\n$\\mathbfss{C}$ is the covariance matrix of the noisy Stokes maps and\n$\\mathcal{C}_b^P$ are bandpowers of $\\ell (\\ell +1)C_\\ell^P\/(2\\pi)$.\nWe analyse a single circular field with area equal to that retained\nin the pseudo-$C_\\ell$ analysis described in Section~\\ref{subsec:power},\nand multiply the Fisher matrix by four to account for the\nnumber of fields observed (which are thus assumed to be fully independent).\nWe ignore inhomogeneity of the noise in the maps so that our problem\nhas azimuthal symmetry about the field centre. This allows us to\nwork in a basis where the data is Fourier transformed in azimuth and\nthe covariance matrix becomes block diagonal, thus speeding up the\ncomputation of the Fisher matrix considerably. The Fisher matrix\ntakes full account of band-power correlations (both between $b$ and\npolarization type) and the effect of ambiguities in isolating\n$E$ and $B$-modes given the survey geometry. We deal with power\non scales larger than the survey by including a junk band-power for\neach of $E$ and $B$ whose contribution to $\\mathrm{cov}_{(bP)(bP)'}$\nwe remove before computing detection significances.\n\nThe comparison between the predicted and simulated performance of\n{\\sevensize C}$_\\ell${\\sevensize OVER}\\, is shown in Fig.~\\ref{fig:clover_compare}. In this plot, we\nalso include the predicted $B$-mode errors from a na\\\"{\\i}ve mode-counting\nargument based on the fraction of sky observed, $f_{\\rm sky}$. For\nthese estimates, we assume independent measurements of the power\nspectrum in bands of width $\\Delta \\ell$ given by\n\\begin{equation}\n(\\Delta C_\\ell)^2 = \\frac{2}{(2\\ell + 1) f_{\\rm sky} \\Delta \\ell}\n(C_{\\ell} + N_{\\ell})^2,\n\\end{equation}\nwhere $C_\\ell$ is the band-averaged input signal and $N_\\ell$ is the\nband-averaged noise. For uncorrelated and isotropic Gaussian random\nnoise, the latter is given by $N_\\ell = w^{-1} B_\\ell^{-2}$\nwhere $B_\\ell = \\exp( - \\ell (\\ell + 1) \\sigma_B^2 \/ 2)$ is the\ntransform of the beam with $\\sigma_B = \\theta_B \/ \\sqrt{8 \\ln 2}$ for\na beam with FWHM of $\\theta_B$. The weight $w^{-1} =\n\\Omega_{\\rm pix} \\sigma^2_{\\rm pix}$, where the pixel noise in the $Q$\nand $U$ maps is \n\\begin{equation}\n\\sigma^2_{\\rm pix} = \\frac{ ({\\rm NET}\/\\sqrt{2})^2 \\Theta^2}{t_{\\rm obs}\n (N_{\\rm det}\/4) \\Omega_{\\rm pix}}. \n\\label{eqn:pixel_noise}\n\\end{equation}\nHere $\\Theta^2$ is the total observed area, $t_{\\rm obs}$ is\nthe total observation time and $\\Omega_{\\rm pix}$ is the pixel\nsize. In equation~(\\ref{eqn:pixel_noise}), we have used ${\\rm\n NET}\/\\sqrt{2}$ to account for the fact that a single measurement of\n$Q$ or $U$ requires a measurement from two detectors (or,\nalternatively, two measurements from a single detector) and we use\n$N_{\\rm det}\/4$ as the effective number of $Q$ and $U$ detectors.\n\nOver most of the $\\ell$ range, the agreement between the Fisher matrix\npredictions and the simulated performance is rather good -- the only\nsignificant discrepancy is for the lowest band-power where the\nsimulations fail to match the predicted Fisher error. This is almost\ncertainly due to the relatively poor performance of our power spectrum\nestimator on the very largest scales where pseudo-$C_\\ell$ techniques\nare known to be sub-optimal (compared to, for example, a maximum\nlikelihood analysis). In terms of a detection of the total $B$-mode\nsignal, the Fisher analysis predicts a detection for {\\sevensize C}$_\\ell${\\sevensize OVER}\\, of\n$\\sim 12.0\\sigma$. For comparison, the na\\\"{\\i}ve $f_{\\rm sky}$\nanalysis predicts a $12.4\\sigma$ detection. For our assumed\ntensor-to-scalar ratio, the Fisher matrix analysis predicts $r=0.026\n\\pm 0.013$ (a $2.04\\sigma$ detection) and the na\\\"{\\i}ve analysis\nyields $r=0.026 \\pm 0.011$ (a $2.39\\sigma$ detection). Comparing to\nthe detection significances quoted in Table~\\ref{tab:simsummary}, we\nsee that the detections recovered from the simulations fail to match\nthese numbers. For the case of the total $B$-mode amplitude, this\ndiscrepancy is entirely due to the fact that we are unable to measure\nand include in our analysis (anti-)correlations between the\nband-powers measured from our small number (50) of simulations --\nwhen we neglect the correlations in the Fisher matrix analysis, the\nFisher prediction drops to a $10.24\\sigma$ detection, in excellent\nagreement with our measured value from simulations. For the primordial\n$B$-mode signal only, the discrepancy found is also partly due to the\nsame effect (ignoring correlations in the Fisher analysis reduces the\nFisher prediction for primordial $B$-modes to $1.9\\sigma$). As\nmentioned above, we suspect that the additional decrease in sensitivity to\nprimordial $B$-modes seen in the simulations is due to the slightly\nsub-optimal performance of our implementation of the pure\npseudo-$C_\\ell$ estimator on the largest scales.\n\nWe should point out that in this work, we have made no attempt to optimise the\nsurvey strategy in light of recent instrument developments. In\nparticular, the survey size we have adopted for these simulations was\noptimised for a measurement of $r=0.01$ with {\\sevensize C}$_\\ell${\\sevensize OVER}\\, when the\nexperiment was expected to have twice the number of detectors now\nplanned. For the instrument parameters we have adopted in this analysis\n(which are a fair representation of the currently envisaged\nexperiment), the optimal survey area for a measurement of $r=0.026$ \nwould be significantly smaller than the $\\sim 1500$ deg$^2$ we have\nused here due to the increased noise levels from the reduced number of\ndetectors. Alternatively, if we had assumed a larger input value of\n$r$, the optimal survey size would increase. Optimisation of both the\nsurvey area and the scan strategy in light of these changes in the\ninstrument design is the subject of on-going work. \n\\begin{figure}\n \\centering\n \\resizebox{0.48\\textwidth}{!}{ \n \\rotatebox{-90}{\\includegraphics{fig16.ps}}}\n \\caption{Comparison between the predicted performance of {\\sevensize C}$_\\ell${\\sevensize OVER}\\,\n as calculated using a Fisher-matrix analysis and the simulated\n performance from our Monte-Carlo pipeline (for our reference\n simulation). Also shown for comparison are the errors predicted from\n a na\\\"{\\i}ve $f_{\\rm sky}$ analysis.}\n \\label{fig:clover_compare}\n\\end{figure}\nThere are, of course, many other sources of uncertainty which we have\nnot yet accounted for in our simulation pipeline and so both the\npredicted and simulated performance numbers should be taken only as\nguidelines at this time. However, it is encouraging that the extra\nsources of uncertainty which are included in our simulation pipeline\n(realistic instrument parameters, a realistic scan strategy,\ncorrelated noise), in addition to any uncertainties introduced as part\nof our subsequent analysis of the simulated data, do not degrade the\nexpected performance of {\\sevensize C}$_\\ell${\\sevensize OVER}\\, by a large amount. \n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe have performed a detailed investigation of the ability of both slow\nand fast polarization modulation schemes to mitigate possible\nsystematic effects in upcoming CMB polarization experiments, targeted\nat measuring the $B$-mode signature of gravitational waves in the\nearly universe. To do this we have used a simulation pipeline\ndeveloped in the context of the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, experiment, which includes\nrealistic instrument and observation parameters as well as $1\/f$\ndetector noise and $1\/f$ atmospheric noise correlated across the\n{\\sevensize C}$_\\ell${\\sevensize OVER}\\, focal-plane array. Using this simulation tool, we have\nperformed simulations of {\\sevensize C}$_\\ell${\\sevensize OVER}\\, operating with no explicit\nmodulation, with a stepped HWP and with a HWP rotating continuously at\n$3$~Hz. We have analysed the resulting time-stream simulations using\nthe technique of detector differencing coupled with a na\\\"{\\i}ve map-making\nscheme, and finally have reconstructed the $E$ and $B$-mode power\nspectra using an implementation of the near-optimal ``pure''\npseudo-C$_\\ell$ power spectrum estimator.\n\nAs expected, we find that fast modulation via a continuously rotating\nHWP is extremely powerful in mitigating a correlated $1\/f$ component\nin the detector noise but that a stepped HWP is not. In addition, we\nhave demonstrated that a polarized $1\/f$ component in the atmosphere\nis not mitigated by any amount of modulation and if present, would\nneed to be mitigated in the analysis using a sophisticated map-making\ntechnique. We have further verified with simulations that fast\nmodulation is very effective in mitigating instrumental polarization\nthat is fixed relative to the instrument basis, for example the $T\n\\rightarrow Q$ leakage caused by systematic gain errors and\nmis-matches between detectors, in agreement with the conclusions\nof~\\citet{odea07}. We have also demonstrated that modulation does not\nmitigate a systematic mis-calibration of polarization angles and that\nthese angles will need to be measured accurately in order to avoid a\nsystematic leakage between $E$ and $B$-modes. The other systematics\nwhich we have investigated (pointing errors, mis-estimated\ntime-constants) have a negligible impact on the recovered power\nspectra for the parameters adopted in our simulations.\n\nIn addition to our investigation of systematic effects, we have\nstressed the importance of combining data from multiple detectors and\nhave demonstrated the superior performance of a differencing technique\nas opposed to one based on measuring all three Stokes parameters from\nsingle detectors in isolation. We suggest that the latter technique,\nalthough possible in the presence of rapid modulation, is likely a\npoor choice of analysis technique, at least in the presence of a\ncommon-mode systematic effect such as atmospheric $1\/f$ noise.\n\nFinally, we have compared the simulated performance of the {\\sevensize C}$_\\ell${\\sevensize OVER}\\,\nexperiment with the expected performance from a simplified\nFisher-matrix analysis. For all but the very lowest multipoles, where\nthe simulations fail to match the Fisher predictions, we find\nexcellent agreement between the predicted and simulated\nperformance. In particular, despite the highly anisotropic noise\ndistribution present in our simulated maps, our measurement of the\ntotal $B$-mode signal matches closely with the Fisher matrix\nprediction (the latter assuming isotropic noise). On the other hand,\nthe measurement of the large scale $B$-mode signal (and thus of the\ntensor-to-scalar ratio, $r$) from the simulations is around 20 per\ncent worse\nthan the Fisher prediction. This is almost certainly due to the\nsub-optimality of our power spectrum analysis on large scales. It is\npossible that the Fisher matrix predictions could be recovered from\nthe simulations by using a more optimal weighting scheme in the pure\npseudo-$C_\\ell$ analysis, or, more likely, by using a\nmaximum-likelihood $C_\\ell$ estimator for the low multipoles.\n\nOne important class of systematic effects which we have not considered\nin this paper are those associated with imperfect optics. Additionally, \nwe have considered only two effects associated with an imperfect HWP.\nThe efficacy of fast modulation to mitigate systematic\neffects from imperfect optics, for example instrumental polarization\ndue to beam mis-match, is expected to depend critically on the optical\ndesign (such as the HWP location). We are currently working to\ninclude such optical systematics in our simulation pipeline, along\nwith a more detailed physical model of the atmosphere and models of\nthe expected polarized foreground emission. In future work, in\naddition to investigating further systematic effects, we will extend\nour simulations to multi-frequency observations and will use these to\ntest alternative foreground removal techniques. We will also apply the\n``destriping'' map-making technique of \\cite{sutton09} to our\nsimulations to assess the relative merits of destriping in analysis as\nopposed to a hardware based approach for mitigating $1\/f$ noise.\n \n\\section*{Acknowledgments}\nWe are grateful to the {\\sevensize C}$_\\ell${\\sevensize OVER}\\, collaboration for useful\ndiscussions. We thank Kendrick Smith for making his original pure\npseudo-$C_\\ell$ code available which we adapted to carry out some of\nthe analysis in this paper. We thank John Kovac and Jamie Hinderks for\nthe up-to-date descriptions of the {\\sevensize BICEP-2\/KECK} and {\\sevensize PIPER}\\,\nexperiments respectively. The simulation work described in this paper was carried\nout on the University of Cambridge's distributed computing facility,\n{\\sevensize CAMGRID}. We acknowledge the use of the {\\sevensize FFTW}\\, \\citep{frigo05},\n{\\sevensize CAMB}\\, \\citep{lewis00} and {\\sevensize HEALPIX}\\, \\citep{gorski05} packages.\n\n\\setlength{\\bibhang}{2.0em}\n\\vspace{-3mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThere is significant evidence that large glaciation events took place during the Proterozoic era (2500-540 million years ago). In particular, this evidence points to the existence of glacial formations at low latitudes, see the review articles \\cite{hoffmanschrag2002terranova, pierrehumbert2011climate} and the references therein. One theory on the exodus from such an extreme climate was put forward by Joseph Kirschvink \\cite{kirschvink1992protero}, who advocated that there was accumulation of greenhouse gases in the atmosphere, e.g. CO$_2$. His theory purports that during a large glaciation chemical weathering processes would be shut down, thus eliminating a CO$_2$ sink. Moreover, volcanic activity would continue during the glaciated state. The combination of these effects would \\textit{slowly} lead to enough build-up of atmospheric carbon dioxide to warm the planet and start the melting of the glaciers. Once a melt began, a deglaciation would follow \\textit{rapidly} due to ice-albedo feedback. \n\n\nThere has been a wealth of modelling work on ``snowball'' events ranging from computationally intensive global circulation models (GCMs) to low dimensional conceptual climate models (CCMs). In 1969, Mikhail Budyko and William Sellers independently proposed energy balance models (EBMs); these were CCMs capturing the evolution of the temperature profile of an idealized Earth \\cite{budyko2010effect, sellersglobal}. Many others, for example \\cite{ caldeira1992susceptibility, abbot2011, north1975theory}, have followed in the footsteps of Budyko and Sellers and used similar conceptual models capable of exhibiting snowball events. The low dimensionality of CCMs allows for a dynamical systems analysis, and hence a deeper investigation into some of the key feedbacks such as greenhouse gas and the ice-albedo effect. \n\nFrom the point of view of dynamical systems, many of these early works share a similar theme by focusing on a bifurcation analysis with respect to the \\textit{radiative forcing} parameter, one that depends on changes in atmospheric CO$_2$ and other greenhouse gas levels. The reader may find figures similar to those displayed in Figure \\ref{icf} in earlier works \\cite{abbot2011, hoffmanschrag2002terranova, pollard2005snowball}. These figures illustrate the glaciation state of an Earth with symmetric ice caps in terms of atmospheric CO$_2$ i.e. ice line latitude at $90^o$ means Earth is ice free while $0^o$ means Earth is fully glaciated. In both figures, the effect of the radiative forcing due to CO$_2$ and other greenhouse gases is treated statically as a parameter in a simple bifurcation analysis. Mathematically, the bifurcation analysis already extends beyond the realm of smooth dynamical systems. On one hand, the bifurcation diagrams are obtained conventionally, with the bold curves indicating the stable branches and the dotted curves showing the unstable ones. On the other hand, the extreme states (ice-free and ice-covered) are not true equilibria of the system, but treated as such in the literature as evidenced by the labeling of \"ice-free branch\" and \"ice-covered branch\". Indeed, the extreme states of the ice line are special because they serve as physical boundaries for the dynamics: the ice line cannot extend beyond the pole nor, because of the north-south symmetry assumed in the models, can it move downward of the equator. Therefore, mathematical models of snowball earth must reflect this physical imposition and be treated with a nonsmooth systems perspective.\n\n\nSince glacial extent varies over time and dynamic processes such as chemical weathering affect the level of atmospheric CO$_2$, the bifurcation diagrams in Figure \\ref{icf} can be viewed as phase planes with dynamic variables consisting of the glacier extent (ice line) and radiative forcing due to greenhouse gases. In particular, if a global glaciation did occur, and Kirschvink's argument about the accumulation of atmospheric CO$_2$ held, then we should expect orbits of this dynamical system to traverse the extreme ice latitudes, i.e. the equator and the pole. Following Kirschvink's idea further, we treat greenhouse gas effects on the energy balance by incorporating a slow CO$_2$ variable and treat the ice latitude as a relatively fast variable. The goal of the present article is to analyze such an interplay using a nonsmooth systems framework, thus providing mathematical support for Kirschvink's hypothesis. In this work, we treat the bifurcation diagram (similar to those in Figure \\ref{icf}) as a set of quasi-steady states of a slow-fast system and utilize the theory of Filippov.\n\n \\begin{figure}[ht]\n \\centering\n \\subfloat[Figure 10 from Abbot, Voigt, and Koll \\cite{abbot2011} ] {\\includegraphics[width=0.4\\textwidth]{fig1-abbot-2011-Aeta.pdf} {\\label{fig11}} } \\quad \\quad\n \\subfloat [Figure 6 from Hoffman and Schrag \\cite{hoffmanschrag2002terranova}] {\\includegraphics[width=0.4\\textwidth]{fig1-terranova.pdf} {\\label{fig12}}} \\quad\n \\caption{Bifurcation diagrams from energy balance models illustrating hysteresis in the climate system. In each figure, solid lines correspond to stable steady states while dashed lines correspond to unstable steady states. The positive horizontal axis can be thought of as increasing atmospheric carbon dioxide, and the vertical axis is the latitude of the ice line. Stability of snowball and ice-free states is \\textit{inferred}; these are physical boundaries and not true equilibria of the models.} \n\\label{icf}\n \\end{figure}\n\n \nWe present the topics as follows. In the next section, we motivate the model of interest. In Section \\ref{sect:fil}, we propose an extension of the model that is amenable to a nonsmooth dynamical systems treatment. In particular, we use Filippov's theory for differential equations with discontinuous right-hand sides to show that an ice line model based on a latitude-dependent EBM coupled with a simple equation for greenhouse gas evolution can be embedded in the plane to form a system that has unique forward-time solutions. We end Section \\ref{sect:fil} with an analysis of the system dynamics and conclude the paper with a discussion in Section \\ref{sect:discussion}.\n\n\n\\section{The Equations of Motion}\n\\subsection{The Budyko Energy Balance Model and the Ice Line Equation}\nThe Budyko EBM describes the evolution of the annual temperature profile $T=T(y,t)$, where $t$ denotes time and $y$ denotes the sine of the latitude. The governing equation may be written:\n\n\\begin{equation} \\label{bud}\nR \\frac{\\partial }{\\partial t}T(y,t) = Qs(y)(1-\\alpha(\\eta,y))-(A+BT(y,t))+C \\left( \\int_0^1 T(y,t)dy-T(y,t) \\right).\n\\end{equation}\n\nThe main idea is that the change in temperature is proportional to the imbalance in the energy received by the planet. The amount of short wave radiation entering the atmosphere is given by $Qs(y)(1-\\alpha(\\eta,y))$, where Q is the total solar radiation (treated as constant), $s(y)$ is the distribution of the solar radiation, and $\\alpha(\\eta,y)$ is the albedo at latitude $y$ given that the ice line is at latitude $\\eta$. The outgoing longwave radiation is the term $A+BT(y,t)$; it turns out that the highly complex nature of greenhouse gas effects on the Earth's atmosphere can be better approximated by a linear function of surface temperature than by the Stephan-Boltzmann law for blackbody radiation ($\\sigma T^4$), see the discussion in \\cite{graves1993new}. The parameter $A$ is particularly interesting here, because it is related to greenhouse gas effects on the climate system, which we will describe further in Section \\ref{sec-ghg}. The transport term $C ( \\int_0^1 T(y,t)dy - T )$ redistributes heat by a relaxation process to the global average temperature. Assuming symmetry of the hemispheres, one may take $y \\in [0,1]$ and $T(y,t)$ as a symmetric temperature profile over the interval $[-1,1]$, hence, an even function of $y$. A more detailed discussion of this model can be found in \\cite{tung2007topics}. For interested readers, Table \\ref{tab:ParValues} lists the polynomial expressions of some key functions in this model using standard parameter values. \n\nThe evolution of the ice-water boundary, or what is often called the \\textit{ice line}, $\\eta$, affects the albedo function $\\alpha(\\eta,y)$. In \\cite{tung2007topics} and \\cite{north1975theory}, a critical ice line annual average temperature, $T_c$, is specified. Above this temperature ice melts, causing the ice line to retreat toward the pole, and below it ice forms, allowing glaciers to advance toward the equator. One way to model this is described in \\cite{widiasih2013dynamics}, where the augmented ice line equation governing $\\eta$ is written as\n\n\\begin{equation} \\label{etadot}\n\\frac{d \\eta}{dt} = \\rho ( T(\\eta)-T_c ).\n\\end{equation}\n\nThe exact value of $\\rho$ is unspecified in our analysis, except that it is assumed to be large in comparison to the timescale governed by volcanism and weathering processes. The interested reader will find a detailed discussion of the value of $\\rho$ in \\cite{mcgwid2014simplification}.\nBecause \\eqref{bud} is an integro-differential equation, the phase space must contain the function space of the temperature profile. Therefore, the dynamics of this coupled system take place in some infinite-dimensional space. The work by Widiasih in \\cite{widiasih2013dynamics} treated the model in a discrete time framework and showed the existence of a one-dimensional attracting manifold. \n\nMcGehee and Widiasih \\cite{mcgwid2014simplification} imposed equatorial symmetry by using even Legendre polynomials with a discontinuity at the iceline $y=\\eta$, and showed the existence of a one-dimensional attracting invariant manifold, similar to that shown in \\cite{widiasih2013dynamics}. The invariant manifold is parametrized by the ice line, thereby reducing the dynamics to a single equation:\n\n\\begin{equation} \\label{etadot-reduced}\n\\frac{d\\eta}{dt}= h_0(\\eta; A)\n\\end{equation}\n\\noindent with\n\\begin{equation}\\label{h} \nh_0(\\eta;A):= \\rho\\left(\\frac{Q}{B+C} \\left(s(\\eta)(1-\\alpha(\\eta,\\eta))+\\frac{C}{B}(1-\\overline{\\alpha}(\\eta))\\right)-\\frac{A}{B}-T_c \\right)\n\\end{equation}\n\\noindent where $\\overline{\\alpha}(\\eta)=\\int_0^1 s(y) \\alpha(\\eta,y)dy$. The parameter $A$ appearing in \\eqref{etadot-reduced} is indeed the greenhouse gas parameter in \\cite{mcgwid2014simplification}, and is playing a similar role as that in the bifurcation diagram shown in Figure \\ref{fig11}. \n\nIn what follows, we apply the invariant manifold result of McGehee and Widiasih \\cite{mcgwid2014simplification} by coupling the ice line equation \\eqref{etadot-reduced} to an equation for atmospheric greenhouse gas evolution through the (former) parameter $A$. From here on, we highlight the explicit dependence of the equation for $\\eta$ on $A$ by writing \n\\begin{equation*}\n\\frac{d\\eta}{dt}=h(A,\\eta)\n\\end{equation*}\nwhere $h(A,\\eta)=h_0(\\eta;A)$ and $h(A,\\eta)$ is thought of as a real valued function over the plane $\\mathbb R \\times \\mathbb R$.\n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{|ccc||c|} \n\\hline\n\\textbf{Parameters} & Value & Units & \\textbf{Functions} \\\\ \\hline \\hline\n&&& \\\\\n$Q$ & 321 & $\\text{W}\\text{m}^{-2}$ & $s(y) = 1 - \\frac{0.482}{2} (3 y^2 - 1)$\\\\\n$s_1$ & 1 & dimensionless & \\\\\n$s_2$ & -0.482 & dimensionless & $h(A,\\eta)=\\rho\\left(112.88+56.91\\eta-24.31\\eta^2-11.05\\eta^3-\\frac{A}{1.5}\\right)$ \\\\\n$B$ & 1.5 & $\\text{W}\\text{m}^{-2}\\text{K}^{-1}$ & \\\\\n$C$ & 2.5B & $\\text{W}\\text{m}^{-2}\\text{K}^{-1}$ & $g(A,\\eta)=\\delta(\\eta-\\eta_c)$ \\\\\n$\\alpha_1$ & 0.32 & dimensionless & \\\\\n$\\alpha_2$ & 0.62 & dimensionless & \n$\\alpha(\\eta,y)=\\begin{cases} \n&\\alpha_1 \\text { when } y< \\eta \\\\\n& \\frac{\\alpha_1+\\alpha_2}{2} \\text{ when } y=\\eta\\\\\n&\\alpha_2 \\text{ when } y>\\eta \n \\end{cases}$ \\\\\n$T_c$ & $-10$ & ${}^\\circ\\text{C}$ & \\\\\n&&& \\\\\n\\hline\n\\end{tabular} \n\\caption{Parameter values as in \\cite{abbot2011} and functions as in \\cite{mcgwid2014simplification}.} \\label{tab:ParValues}\n\\end{table}\n\n\n\\subsection{Incorporating Greenhouse Gases} \\label{sec-ghg} \n\nWe now make an argument for a very simple form of an equation for greenhouse gas evolution. In the Budyko and Sellers models \\cite{ budyko2010effect,sellersglobal}, the parameter $A$ plays the important role of reradiation constant. The Earth absorbs shortwave radiation from the sun, and some of this is reradiated in the form of longwave radiation. The current value of $A$ is measured using satellite data to be approximately $202$ $W\/m^2$ \\cite{ graves1993new, tung2007topics}. \n\nHowever, throughout the span of millions of years $A$ is not constant, as the amount of heat reradiated to space depends crucially on greenhouse gases, especially carbon dioxide. In fact, it was posited in \\cite{caldeira1992susceptibility} that $A$ behaves like a function of the logarithm of atmospheric carbon dioxide, measured in parts per million. Intuitively, adding carbon dioxide to the atmosphere decreases its emissivity, allowing less energy to escape into space. Therefore, outgoing longwave radiation should vary inversely with CO$_2$.\n\nSince the land masses were concentrated in middle and low latitudes prior to the global glaciation period, Kirschvink postulated that the stage was set for an ice-covered Earth. Then \\lq On a snowball Earth, volcanoes would continue to pump CO$_2$ into the atmosphere (and ocean), but the sinks for CO2 \u2013 silicate weathering and photosynthesis would be largely eliminated \\rq \\cite{ hoffmanschrag2002terranova, kirschvink1992protero}. In more recent work, Hogg \\cite{hogg} put forth an elementary model for the evolution of greenhouse gases consistent with Kirschvink's theory. In short, he argued that the main sources for atmospheric CO$_2$ were due to an averaged volcanism rate and ocean outgassing. The main carbon dioxide sink was said to be due to the weathering of silicate rocks. For our purposes, it will be enough to work with volcanism and weathering. Let $V$ denote the rate of volcanism and $W$ the weathering rate, where the latter is assumed to depend on the location of the iceline, or more specifically the amount of available land or rock to be weathered. Putting this together with the assumption that the reradiation variable $A$ varies inversely with CO$_2$ gives\n\n\\begin{align} \n\\frac{dA}{dt}&=-(V-W\\eta)\\nonumber\\\\\n&:=\\delta(\\eta-\\eta_c). \\label{Adoteq}\n\\end{align} \n\nwhere $\\delta>0$ and $\\eta_c=\\frac{V}{W}$, the ratio of volcanism to weathering. Let $g(A,\\eta):=\\delta (\\eta -\\eta_c)$. \n\n\\begin{remark} In what follows, we often replace the right hand side of equation \\eqref{Adoteq} with the more general function $g(A,\\eta)$. While our analysis in Sections \\ref{sect:dynamics} and \\ref{sect:jorm} uses the linear function of ice line alone, it is conceivable that greenhouse gas feedbacks depend directly on their atmospheric concentration, see eg. \\cite{caldeira1992susceptibility, pierrehumbert2011climate}. We leave this avenue for future exploration and extension of the current work. \\end{remark}\n\nNext, allowing $A$ to vary in the ice line equation \\eqref{h}, we now have a system of equations for $A$ and $\\eta$:\n\\begin{equation} \\label{Aeta}\n\\begin{cases}\n&\\dot{A}=g(A,\\eta)\\\\\n&\\dot{\\eta}=h(A, \\eta).\\\\\n\\end{cases}\n\\end{equation}\n \n\nTo understand the timescale of $A$, we refer to the elucidation of the snowball scenario by Hoffman and Schrag in which they argue that weathering took place at a much slower rate than the ice-albedo feedback (see Fig. 7 on page 137 \\cite{hoffmanschrag2002terranova} ). \n\nThe idea of packaging the essence of the long term carbon cycle into one simple equation is not novel, and is consistent with earlier findings \\cite{ edmond1996fluvial, hogg, kump2000chemical}. The novelty comes from connecting such an equation to an energy balance model and analyzing the dynamics of the coupled system. Indeed, the parameter $A_{ex}$ on page 644 \\cite{kump2000chemical} is the variable $\\eta$ in equation \\eqref{Aeta} and it represents \\textit{the effective area of exposure of fresh minerals}. The parameter $\\eta_c$ can therefore be thought of as a \\textit{critical} area. The coupling of the ice line $\\eta$ and the greenhouse gas variable $A$ follows naturally. \n\nIn its current form, the model does not restrict dynamics to the physical region $0\\leq \\eta\\leq 1$. There are orbits that exit this interval on either end, and we must therefore create reasonable assumptions for projection of this motion onto the boundaries. Furthermore, $\\eta$ should evolve along the physical boundaries, as suggested by the bifurcation diagrams in Figure \\ref{icf}. This will be guaranteed by first carefully extending the vector field to the whole plane, and then utilizing Filippov's theory for the resulting nonsmooth system.\n\n\\section{A Filippov System for a Glaciated Planet\\label{sect:fil}}\n\nBefore we dive into the analysis of the coupled $(A, \\eta)$ system, we shall first build an intuition for the type of system of our focus and prove a general result for such a system. We then apply this result to an extended version of \\eqref{Aeta} and study its dynamics. We find that this framework both ensures that trajectories with initial conditions in the physical region remain in this region for all time and produces the expected sliding motion on the boundaries. \n\nVarious frameworks for analyzing non-smooth systems have been developed, eg. by Caratheodory, Rosenthal, Viktorovskii, Utkin \\cite{filippov1964, cortes2008}. Here, we choose to use the framework developed by A.F. Filippov \\cite{filippov1964}. \n\n\n\\subsection{Casting the Model in a Filippov framework}\n\nAs mentioned previously, the pole and the equator define physical constraints of the ice line; it must stay in the unit interval $[0,1]$. However, the model as it stands does not take this into account. A natural choice is to assume that $\\dot{\\eta}=0$ when $\\eta$ reaches a physical boundary. In addition to this, one needs to account for the stability\/instability of these states, thus allowing orbits to exit the boundary when it becomes unstable. More specifically, we should take\n\n\\begin{equation}\\label{AetaFillipov}\n\\begin{cases}\n\\dot{\\eta}&= \\begin{cases}\n 0 & \\{ \\eta=0 \\text{ and } h(\\eta,A)<0 \\}\\text{ \\text{or} }\\{ \\eta=1 \\text{ and } h(A, \\eta)>0\\}\\\\\n h(A, \\eta) & \\text{otherwise}\\end{cases} \\\\\n\\dot{A}&=g(A,\\eta)\n\\end{cases}\n\\end{equation}\n\nThe first equation forces $\\eta$ to stop at the physical boundary exactly when it is about to cross that boundary. When this happens, $\\eta$ should remain constant while $A$ continues to evolve. As $A$ evolves, so does the value of $h$ and this should result in the $\\dot{\\eta}$ equation ``turning back on'' as soon as $h$ becomes positive on the lower boundary, $\\eta=0$, or negative on the upper boundary, $\\eta=1$. This is analogy with Kirschvink's hypothesis, as we expect orbits that enter the snowball state to recover when enough carbon dioxide has built up in the atmosphere. However, it is not clear that classical results from dynamical systems apply because the system is nonsmooth as well as undefined beyond the physical boundary. Fundamentally, the existence and uniqueness of solutions needs to be verified. Our analysis takes the approach of \\textit{extending} the vector field to the nonphysical region in a way that makes the system amenable to the theory of Filippov. This mathematical extension is used to obtain exactly the expected dynamics of the system \\eqref{AetaFillipov} outlined above, including boundary motion. There are many possible modeling choices for the boundary, however, as will be discussed in Remark \\ref{remark}, our proposed extension ensures the consistency of the model dynamics with the existing snowball theory demonstrated by the literature e.g. in \\cite{abbot2011, caldeira1992susceptibility}. The type of extension we adopt is highlighted in Section \\ref{sect:ex-un} below. Much of the proof relies on the theory developed by Fillipov \\cite{filippov1964, filippov1988}.\n\n\\subsection{Existence and uniqueness of solutions \\label{sect:ex-un}}\n\nThe following general result will motivate and apply to an extended version of the system \\eqref{Aeta}, as well as guarantee the desired motion described in \\eqref{AetaFillipov}. \n\n\n\\begin{theorem} \\label{filippov-bud}\nLet $G(x,y), H(x,y): \\mathbb{R}^2 \\rightarrow \\mathbb{R}$ be $C^1$ functions. Define the vector field on $\\mathbb{R} \\times \\mathbb{R}$ as follows:\n\\begin{align} \\label{gen-pws} \n&\\dot{x} = G(x,y)\\\\\n&\\dot{y} = \\begin{cases} \n-|H| \\quad &\\text{when } y > 1\\\\\n\\frac{H-|H|}{2} \\quad &\\text{when } y=1 \\\\\nH \\quad &\\text{when } 0< y < 1\\\\\n\\frac{H+|H|}{2} \\quad &\\text{when } y = 0\\\\\\label{gen-pws1}\n|H| \\quad &\\text{when } y < 0. \\\\\n\\end{cases}\\\\\\nonumber\n\\end{align}\n\nThen, an initial value problem with time derivative \\eqref{gen-pws}-\\eqref{gen-pws1} has a unique forward-time Filippov solution. Furthermore, the strip $\\mathbb{R} \\times [0,1]$ is forward invariant. \n\\end{theorem}\n\n\\begin{proof}\n\n\nFirst, to simplify notation, let $F(x,y)$ denote the vector-valued function $[G(x,y),H(x,y)]$. To show the existence of a Filippov solution, one must show that $F$ is locally essentially bounded or equivalently, satisfies what Filippov called \\textit{Condition B} on p. 207 of \\cite{filippov1964}. This is straight forward to check since $F$ is Lipschitz continuous everywhere except at the boundaries $y=0,1$. \n\nTo show uniqueness of the Filippov solution, we must show the \\textit{one-sided Lipschitz condition}, ie. that for almost all $z_1$ and $z_2$ in some small neighborhood in $\\mathbb R^2$, the inequality\n\\begin{equation} \\label{esslip}\n \\left( F\\left( z_1 \\right)-F \\left( z_2 \\right)\\right)^T \\left( z_1-z_2 \\right) \\le K \\| z_1-z_2 \\|^2\n\\end{equation}\nholds for some constant $K$ (see inequality (51) p. 218 of \\cite{filippov1964}). Since $F$ is Lipschitz away from the discontinuity boundary, one easily checks that this condition is satisfied in this region. \n\nWe now consider initial value problems with initial conditions in the discontinuity boundary. There are two such discontinuity boundaries: the lines $y=0$ and $y=1$. First, let $z$ be a point on the line $y=1$. Let $\\delta$ be small enough that $B_{\\delta}(z)$ is a neighborhood contained within the set $\\left\\lbrace (x,y):y>0 \\right\\rbrace$. Suppose that $z_1=(x_1,y_1)$ satisfies $01$. We consider $\\left( F(z_2)-F(z_1) \\right)^T (z_2-z_1)$ for the following two cases:\n\\begin{enumerate}\n\\item If $H(z_1) \\ge 0$, then $\\left(F(z_2)-F(z_1)\\right)^T(z_2-z_1)=\\left(0, -|H(z_2)|-H(z_1) \\right)^T(x_2-x_1, y_2-y_1)<0$, since $-|H(z_2)|-H(z_1) < 0$ and $y_2-y_1>0$. Then, inequality \\eqref{esslip} is satisfied with $K=1$.\n\n\\item If $H(z_1)<0$, then $F(z_2)-F(z_1)=\\left(0, -|H(z_2)|+|H(z_1)| \\right)$. The second entry may be estimated further:\n$-|H(z_2)|+|H(z_1)| \\le |H(z_1)-H(z_2)| \\le k_H \\|z_1-z_2 \\|$ where $k_H$ is the Lipschitz constant of the function $H$. \nTherefore, the inequality \\eqref{esslip} is satisfied with $K=k_H$.\n\\end{enumerate}\n\nA similar argument holds for the other discontinuity boundary, and therefore, system \\eqref{gen-pws}-\\eqref{gen-pws1} satisfies the one-sided Lipschitz continuity condition \\eqref{esslip}, guaranteeing forward-time uniqueness of Filippov solutions. \n\nFilippov theory further guarantees that a sliding solution exists at the discontinuity boundary. With the normal vector taken in the direction of increasing $y$, one checks that system \\eqref{gen-pws}-\\eqref{gen-pws1} satisfies the conditions of Lemma 3 in \\cite{filippov1964}. This gives the sliding solution on the upper boundary $y=1$. If $z(t)$ denotes this solution, then, with $\\alpha=1\/2$,\n\\begin{align*} \n\\frac{dz}{dt}&=[G(x,1), \\alpha (-|H|)+(1-\\alpha) H] \\\\\n&=[G(x,1), 0].\n\\end{align*}\n\nFurthermore, above the upper discontinuity boundary the vector field points downward, while below the lower boundary, the vector fields points up. Therefore, since any solution can only cross into or slide along the boundary, the strip is forward invariant. \n\\end{proof}\n\nArmed with this result, we return to system \\eqref{Aeta}. Recall that $g$ and $h$ are both $C^1$ functions over the plane. Henceforth, we consider the initial value problem in the state space $(A,\\eta) \\in \\mathbb{R} \\times \\mathbb{R}$ endowed with the following piecewise Lipschitz vector field:\n\n\\begin{align}\\label{ig-pws}\n&\\dot{A} = g(A,\\eta)\\\\\n&\\dot{\\eta} = \\begin{cases} \n-|h| \\quad &\\text{when } \\eta > 1\\\\\n\\frac{h-|h|}{2} \\quad &\\text{when } \n\\eta=1 \\\\\nh \\quad &\\text{when } 0< \\eta < 1\\\\\\label{ig-pws1}\n\\frac{h+|h|}{2} \\quad &\\text{when } \\eta= 0\\\\\n|h| \\quad &\\text{when } \\eta < 0. \\\\\n\\end{cases}\\\\\\nonumber\n\\end{align}\nhaving an initial value $(A(0),\\eta(0)) \\in \\mathbb{R} \\times \\mathbb{R}$. \n\nAn application of Theorem \\ref{filippov-bud} shows that a unique forward-time Filippov solution to the initial value problem \\eqref{ig-pws}-\\eqref{ig-pws1} exists. Furthermore, the forward invariance of the strip $\\mathbb R \\times [0,1]$ ensures the physically relevant aspect of the ice line $\\eta$. We have arrived at the following conclusion:\n\n\\begin{corollary} \nThere exists a unique Filippov solution to \\eqref{ig-pws}-\\eqref{ig-pws1}. Furthermore, the strip $\\mathbb R \\times [0,1]$ is forward time invariant. \n\\end{corollary}\n\n\n\n\\begin{remark}\\label{remark}\nThere are many possible ways of extending system \\eqref{Aeta} beyond the strip $\\mathbb{R} \\times [0,1]$. The choice of extension as done in system \\eqref{ig-pws}-\\eqref{ig-pws1} assures the following points: \n\\begin{enumerate}\n\\item The attracting nature of the strip $\\mathbb{R} \\times [0,1]$.\n\\item The switching from sliding to crossing is entirely determined by zeroes of $h(A, \\eta)$. This point is especially important in the modeling of the physical system, since $h$ signals the ice line's advance or retreat and $h$ should be ``off\" for some positive amount of time at the extreme ice line locations (pole or equator) and should ``turn back on\" when the system has reduced or increased enough greenhouse gases.\n\\item The time derivative of the sliding mode is entirely determined by the governing function of the greenhouse gas effect, $g(A,\\eta)$. Again, this aspect is especially important in the modeling of the physical system, since at the extreme ice line location (pole or equator), the ice-albedo feedback shuts off and the greenhouse gas effect is the only driver of the system. \n\\end{enumerate}\n\\end{remark}\n\n\n\\subsection{Dynamics of the model \\label{sect:dynamics}}\n\nWe now focus our analysis on the system \\eqref{ig-pws}-\\eqref{ig-pws1}. We show that its sliding dynamics guarantee exit from the snowball scenario, and use the Filippov framework to simulate stable small ice cap states and large amplitude periodic orbits. \n\nThe $\\eta$-nullcline, given by $h(A,\\eta)=0$, is cubic in $\\eta$ and linear in $A$ (see Table \\ref{tab:ParValues}). It has a single fold in the region $0 \\leq \\eta \\leq 1$ at $\\eta =\\eta_f \\approx 0.77$, as can be seen in Figure \\ref{critman}. The $A$-nullcline is the horizontal line $\\eta=\\eta_c$. If $0<\\eta_c<1$, the system has a single fixed point. The eigenvalues associated with this fixed point are\n\n\\begin{equation*}\n\\lambda_{\\pm}=\\frac{1}{2}\\left(\\frac{\\partial h}{\\partial \\eta}\\pm \\sqrt{\\left(\\frac{\\partial h}{\\partial \\eta}\\right)^2-\\frac{4\\delta}{B}}\\right).\n\\end{equation*}\n\n From this and Figure \\ref{critman}, we see that if the fixed point lies above the fold and in the physical region, i.e. $\\eta_f<\\eta_c<1$, then it is stable, and it is unstable when $0<\\eta_c<\\eta_f$. This is reminiscent of the ``slope-stability theorem'' from the climate literature \\cite{cahalan1979stability}. In that work, the authors related changes in slopes of equilibrium curves to changes in stability of equilibria for a globally averaged energy balance model. They found that a small ice cap or ice-free solution could be stable, a large ice cap was unstable, and a snowball state was again stable. \n\n\\begin{remark} In the case that $\\eta_c=0,1$, one of the physical boundaries is entirely composed of equilibria, and stability is determined by the direction of the vector field near it. Physically, this degenerate scenario comes about when there is an absence of volcanism in the ice-covered state or a perfect balance between weathering and volcanism in the ice-free state. We do not discuss these cases further.\\end{remark}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{{phase.png}} \n\\caption{The physical region of the phase space and possible fixed points of the system given by the $\\eta$-nullcline, $h=0$. The location of the equilibrium is determined by the critical effective area of exposed land $0<\\eta_c<1$. Solid black portions of the curve represent stable equilibria while the dashed lines denote unstable equilibria. Solid black portions of the boundary are attractive sliding regions and dashed boundaries are crossing regions.}\\label{critman}\n\\end{figure} \n\n\\subsubsection{Sliding and escape from Snowball Earth}\n\nRecall that $\\eta_c$ was defined to be a ratio of volcanism to weathering. We now prove that the system always recovers from the ice-covered regime provided this parameter is positive, i.e. there is some contribution from volcanic outgassing to atmospheric greenhouse gas content.\n\n\\begin{theorem}\\label{exit} Suppose $\\eta_c>0$. Then any orbit of the system \\eqref{ig-pws}-\\eqref{ig-pws1} that enters the boundary $\\eta=0$ must exit in finite time.\\end{theorem}\n\n\\begin{proof}\nLet $A_*$ be the $A$-coordinate of the intersection of the $\\eta$-nullcline with the lower ice boundary $\\eta=0$.\nSuppose an orbit enters the boundary $\\eta=0$. At such an occurrence $A\\geq A_*$, since this is where $\\dot{\\eta}=h(A,\\eta)\\leq 0$. For $A\\geq A_*$, the confinement vector field in the non-physical region $\\eta<0$ is given by $\\dot{\\eta}=|h(A,\\eta)|\\geq 0$. $A$ is decreasing on either side of $\\eta=0$. The Filippov convention induces sliding according to \n\n\\begin{align*}\n\\dot{\\eta}&=\\frac{h(A,0)+|h(A,0)|}{2}=0\\\\\n\\dot{A}&=-\\delta \\eta_c.\n\\end{align*}\n\nAt the intersection point $A=A_*$, the vector field is tangent to the boundary and sliding terminates. For $A0$. Moreover, there is a single orbit of the smooth system that passes through the tangency point. Due to forward-time uniqueness from Theorem \\ref{filippov-bud}, the sliding solution must follow this orbit and reenter the physical region.\n\n\\end{proof}\n \n\\begin{remark} A similar argument shows that any orbit that enters an ice-free state must exit in finite time. The required condition is that there is \\textit{not} a perfect balance between volcanism and weathering, i.e. $\\eta_c\\neq 1$.\\end{remark}\n\n\n\\subsubsection{Numerical simulations\\label{po}}\n\nThe Filippov framework also allows for numerical simulation of the model. Let $(A_c,\\eta_c)$ be the location of the single fixed point. We find that small ice cap states corresponding to fixed points with $\\eta_f<\\eta_c<1$ are possible attractors of the system, as are periodic orbits born from the Hopf bifurcation at $\\eta_c=\\eta_f$. \n\nThe fold of the $\\eta$-nullcline is a \\textit{canard point}, and the mechanism that produces the large amplitude periodic orbit in Figure \\ref{dynamics} is a \\textit{canard explosion} \\cite{ benoit1981chasse, dumortier1996canard, krupa2001extending}. While mathematically interesting, canard orbits are unlikely observables for planar systems because they occur within an exponentially small parameter range. However, they can be robust in higher dimensions which makes more complex oscillatory behavior possible, see e.g. \\cite{benoit1983systemes,desroches2012mixed,szmolyan2001canards,wechselberger2005existence}. With this in mind, we remark that an interesting avenue for future research involves studying an extended version of the system \\eqref{ig-pws}-\\eqref{ig-pws1} with an additional variable ($w$ in \\cite{mcgwid2014simplification}) and foregoing their invariant manifold reduction.\n\n \\begin{figure}[ht]\n \\centering\n \\subfloat[Small ice cap equilibrium] {\\includegraphics[width=0.4\\textwidth]{sm_ice.png}} \\quad \\quad\n \\subfloat [Periodic orbit] {\\includegraphics[width=0.4\\textwidth]{relax_osc.png}} \\quad\n \\caption{Attractors of the system when (a) $\\eta_c=0.85$ and (b) $\\eta_c=0.6$. The $+$ symbol marks the intial condition and the horizontal long-dashed line is the $A$-nullcline. In (a), the orbit reaches the ice-free state and slides until it reaches the intersection of the folded curve $h(A,\\eta)=0$ with this boundary. It then enters the physical region and approaches the small ice cap equilibrium. In (b), the fixed point is unstable and the orbit oscillates between the ice-free and ice-covered boundaries. Simulations were performed using Mathematica 9.} \n\\label{dynamics}\n \\end{figure}\n\n\\subsection{Application to the Jormungand Model\\label{sect:jorm}}\n\nThe previous occurrence of a complete snowball event is still a highly contested topic in geology. In the model \\eqref{ig-pws}-\\eqref{ig-pws1}, the only possible stable fixed point is a small ice cap. In this section, we present a variant of the system, as introduced by Abbot, Voigt, and Koll \\cite{abbot2011}, and employ the Filippov framework as above. By modifying the albedo function $\\alpha(\\eta,y)$ so that it differentiates between the albedo of bare ice and snow-covered ice, the new system has an additional stable state that corresponds to a large ice cap. In \\cite{abbot2011}, this was called the \\textit{Jormungand} state because it allows for a snake-like band of open ocean at the equator. Moreover, there are again attracting periodic orbits. These can be seen in Figure \\ref{jorm}.\n\nIn this version of the model, the albedo is defined as follows:\n\n$$\\alpha_J(\\eta,y)=\\left\\{\\begin{array}{ll}\\alpha_w,& y<\\eta\\\\\n\\frac{1}{2}(\\alpha_w+\\alpha_2(\\eta)), & y=\\eta\\\\ \\alpha_2(y), & y>\\eta, \\end{array}\\right.\\eqno(14)$$\n\nwhere $\\alpha_2(y)=\\frac{1}{2}(\\alpha_s+\\alpha_i)+\\frac{1}{2}(\\alpha_s-\\alpha_i)\\tanh M(y-0.35) $.\n\nHere $\\alpha_w$ is the albedo of open water, $\\alpha_i$ is the albedo of {\\em bare} sea ice, and $\\alpha_s$ is the albedo of {\\em snow-covered} ice. The model assumes sea ice aquires a snow cover only for latitudes above $y=0.35$. We modify $h(A,\\eta)$ by replacing $\\alpha$ with $\\alpha_J$:\n\n\\begin{equation}\\label{hj}\nh_J(A,\\eta)=\\frac{Q}{B+C} \\left(s(\\eta)(1-\\alpha_J(\\eta,\\eta))+\\frac{C}{B}(1-\\overline{\\alpha_J}(\\eta))\\right)-\\frac{A}{B}-T_c \n\\end{equation}\nwhere we note that although $\\alpha(\\eta,y)$ has a discontinuity when $y=\\eta$, both $\\alpha(\\eta,\\eta)$ and $\\overline{\\alpha_J}(\\eta)=\\int_0^1 \\alpha(\\eta,y)s(y)dy$ are smooth functions of $\\eta$.\n\n\nFor this system, we can immediately apply Theorems \\ref{filippov-bud} and \\ref{exit} to obtain the following result.\n\n\\begin{corollary} The physical region is forward invariant with respect to the system \\eqref{ig-pws}-\\eqref{ig-pws1} where $h$ is replaced by $h_J$. Solutions exist and are unique in forward time, and any trajectory that enters an ice-covered state must exit in finite time provided $\\eta_c\\neq 0$.\\end{corollary}\n\n \\begin{figure}[ht]\n \\centering\n \\subfloat[] {\\includegraphics[width=0.4\\textwidth]{jorm1.png}} \\quad \\quad\n \\subfloat [] {\\includegraphics[width=0.4\\textwidth]{jorm2.png}} \\quad\n \\caption{Periodic orbits of the Jormungand system when (a) $\\eta_c=0.8$ and (b) $\\eta_c=0.15$. The folded curve is the $\\eta$-nullcline $h_J(A,\\eta)=0$ and dashing is as in Figures \\ref{critman} and \\ref{dynamics}.} \n\\label{jorm}\n \\end{figure}\n\n\\begin{table}[htb] \n\\centering\n\\begin{tabular}{|ccc||c|} \n\\hline\n\\textbf{Parameters} & Value & Units \\\\ \\hline \\hline\n&& \\\\\n$T_c$ & 0 & ${}^\\circ\\text{C}$ \\\\\n$M$ & 25 & dimensionless \\\\\n$\\alpha_w$ & $0.35$ & dimensionless \\\\\n$\\alpha_i$ & $0.45$ & dimensionless \\\\\n$\\alpha_s$ & $0.8$ & dimensionless \\\\\n&& \\\\\n\\hline\n\\end{tabular} \n\\caption{Parameter values as in Table \\ref{tab:ParValues} unless specified above. Additional values taken from \\cite{abbot2011}.} \\label{tab:ParJorm}\n\\end{table}\n\n\\section{Discussion\\label{sect:discussion}}\n\nIn this article we have extended a class of energy balance models to include a greenhouse gas component. The resulting system was nonsmooth due to physical constraints and did not immediately fit into an existing analytical framework. However, we defined an extended system that allowed us to utilize the theory of Filippov and obtain the expected motion, in agreement with physical arguments from climate literature. More specifically, we proved existence and uniqueness of forward-time solutions and showed that the system was confined to the physical region with sliding on the ice-covered and ice-free boundaries. We showed that the extended system always escapes from the ice-covered scenario, thus supporting Kirschvink's hypothesis about carbon dioxide accumulation in a snowball scenario due to a shut down of chemical weathering processes \\cite{kirschvink1992protero}. Using our extended system, we found that small ice-cap and large amplitude ice-cover oscillations were possible attractors of the system. We then applied our results to the case of Jormungand world and showed that it is possible to obtain further oscillatory dynamics between no ice cover, large ice-cover, and full ice-cover states. \n\nGeneral mathematical questions brought about by this work have to do with building a general framework for such models so that there is no need to appeal to artificial tools. A first step toward dealing with physical boundaries might be a general theory for semiflows generated by smooth vector fields on manifolds with boundary. In addition, the stable portions of the physical boundaries in our model should be thought of as equilibria of the fast subsystem, i.e. as part of the \\textit{critical manifold} from geometric singular perturbation theory \\cite{jones1995geometric}. In fact, they are much more \\textit{normally hyperbolic} than the curve $h=0$; they are reachable in finite time! However, the current theory cannot be directly applied to the full discontinuous system and we remark that developing an analogous Fenichel theory \\cite{fenichel1979geometric} for singularly perturbed discontinuous systems with sliding is an interesting avenue for future research.\n\nAnother interesting future direction is to study the explicit effect of the temperature (here we have considered it only through the equation for $\\eta$ based on the reduction by McGehee and Widiasih \\cite{mcgwid2014simplification}) on the dynamics of the system. In a nonsmooth system, the addition of a stable dimension may destroy an existing stable periodic orbit \\cite{sieber2010small}. Moreover, an additional dimension could result in more interesting glacial dynamics, such as mixed mode oscillations.\n \nFinally, it is natural to ask how the shape of the $\\eta$-nullcline changes with physical parameters and how this affects system dynamics, e.g. could the lower fold in Figure \\eqref{jorm} move to the left of the upper fold? We refer the reader to \\cite{widwal2014dynamics} for a number of examples. \\\\\n\n\n\\textbf{Acknowledgements:} This research was supported in part by the Mathematics and Climate Research Network and NSF grants DMS-0940366, DMS-0940363. AB was also supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. We thank the members of the MCRN Paleoclimate and Nonsmooth Systems seminar groups for many useful discussions, especially Mary Lou Zeeman and Emma Cutler. We also thank Andrew Roberts for his suggestions relating to the formulation of the problem.\n\n\n\n\\bibliographystyle{plain}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzmcle b/data_all_eng_slimpj/shuffled/split2/finalzzmcle new file mode 100644 index 0000000000000000000000000000000000000000..069f35b1c7a4a98ff4a337be6fa81cb8dce61635 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzmcle @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nWe know that the autoparallel curves in the Riemann geometry coincide with its geodesics. In this note my aim is to investigate if there is a such result in the symmteric teleparallel geometry (STPG).\n\n\\section{Mathematical Preliminaries}\n\nSpacetime, in general, is denoted by $\\{ M,g,\\nabla\\}$ where $M$\nis orientable and differentiable manifold, $g=g_{\\alpha \\beta}\ne^\\alpha \\otimes e^\\beta$ is the metric tensor written in terms of 1-forms\n$e^\\alpha$ and $\\nabla$ is connection associated with connection\n1-forms ${\\Lambda^\\alpha}_\\beta$. Cartan structure equations\ndefine nonmetricity 1-forms, torsion 2-forms and curvature\n2-forms, respectively\n \\ba\n Q_{\\alpha \\beta} &:=& - \\frac{1}{2} \\D g_{\\alpha \\beta}\n = \\frac{1}{2} (-\\d g_{\\alpha \\beta} + \\Lambda_{\\alpha \\beta}+\\Lambda_{\\alpha \\beta}) \\; , \\label{nonmet}\\\\\n T^\\alpha &:=& \\D e^\\alpha = \\d e^\\alpha + {\\Lambda^\\alpha}_\\beta \\wedge e^\\beta \\; , \\label{tors}\\\\\n {R^\\alpha}_\\beta &:=& \\D {\\Lambda^\\alpha}_\\beta := \\d {\\Lambda^\\alpha}_\\beta\n + {\\Lambda^\\alpha}_\\gamma \\wedge {\\Lambda^\\gamma}_\\beta \\label{curva}\n \\ea\nwhere $ \\d $ and $ \\D$ are exterior derivative and covariant\nexterior derivative. Geometrically, the nonmetricity\ntensor measures the deformation of length and angle standards\nduring parallel transport. For example, after parallel\ntransportation of a vector along a closed curve, the length of the\nfinal vector may be different from that of the initial vector. On\nthe other hand, torsion relates to the translational group.\nMoreover it is sometimes said that closed parallelograms do not\nexist in spacetime with torsion. Finally, curvature is related to\nthe linear group. That is, when a vector is parallel transported\nalong a closed loop, the vector undergos a rotation due to\ncurvature. These quantities satisfy Bianchi identities:\n \\ba\n \\D Q_{\\alpha \\beta} &=& \\frac{1}{2} ( R_{\\alpha \\beta} +R_{\\beta \\alpha}) \\; , \\label{bianc:0} \\\\\n \\D T^\\alpha &=& {R^\\alpha}_\\beta \\wedge e^\\beta \\; , \\label{bianc:1} \\\\\n \\D {R^\\alpha}_\\beta &=& 0 \\; . \\label{bianc:2}\n \\ea\n\n\n\nFull connection 1-forms can be decomposed uniquely as follows \\cite{fhehl1995},\\cite{rtucker1995},\\cite{madak2010}:\n \\ba\n {\\Lambda^\\alpha}_\\beta = \\underbrace{(g^{\\alpha \\gamma}\\d g_{\\gamma_\\beta} + {p^\\alpha}_\\beta)\/2 + {\\omega^\\alpha}_\\beta}_{Metric}\n + \\underbrace{{K^\\alpha}_\\beta}_{Torsion} + \\underbrace{{q^\\alpha}_\\beta + {Q^\\alpha}_\\beta}_{Nonmetricity} \\label{connect:dec}\n \\ea\nwhere ${\\omega^\\alpha}_\\beta$ Levi-Civita connection 1-forms\n \\ba\n {\\omega^\\alpha}_\\beta \\wedge e^\\beta = -\\d e^\\alpha \\; , \\label{LevCiv}\n \\ea\n${K^\\alpha}_\\beta$ contortion tensor 1-forms,\n \\ba\n {K^\\alpha}_\\beta \\wedge e^\\beta = T^\\alpha \\; , \\label{contort}\n \\ea\nand anti-symmetric 1-forms\n \\ba\n & & q_{\\alpha \\beta} = -( \\iota_\\alpha Q_{\\beta \\gamma } ) e^\\gamma + ( \\iota_\\beta Q_{\\alpha \\gamma})\n e^\\gamma \\; , \\label{q:ab} \\\\\n & & p_{\\alpha \\beta} = -( \\iota_\\alpha \\d g_{\\beta \\gamma } ) e^\\gamma + ( \\iota_\\beta \\d g_{\\alpha \\gamma})\n e^\\gamma \\label{p:ab}\\; .\n \\ea\nThis decomposition is self-consistent. To see that it is enough to\nmultiply (\\ref{connect:dec}) from right by $e^\\beta$ and to use\ndefinitions above. While moving indices vertically in front of\nboth $\\d$ and $\\D$, special attention is needed because $\\d\ng_{\\alpha \\beta} \\neq 0$ and $\\D g_{\\alpha \\beta} \\neq 0$.\nSymmetric part of the full connection comes from (\\ref{nonmet})\n \\ba\n \\Lambda_{(\\alpha \\beta)} = Q_{\\alpha \\beta } + \\frac{1}{2} \\d g_{\\alpha \\beta } \\label{connect:sym}\n \\ea\nand the remainder is anti-symmetric part\n \\ba\n \\Lambda_{[\\alpha \\beta]} = \\frac{1}{2} p_{\\alpha \\beta} + \\omega_{\\alpha \\beta} + K_{\\alpha \\beta} + q_{\\alpha \\beta} \\; . \\label{connect:ansym}\n \\ea\n\nIt is always possible to choose orthonormal basis 1-forms which I denote $\\{ e^a \\}$. Then the metric\n$g=\\eta_{ab}e^a \\otimes e^b$ where $\\eta_{ab} = \\mbox{diag}(\\pm 1, \\cdots , 1)$. In this case the splitting of the full connection (\\ref{connect:dec}) takes the form\n \\ba\n \\Lambda^a{}_b = \\omega^a{}_b + K^a{}_b + q^a{}_b + Q^a{}_b \\; . \\label{connect:decortonormal}\n \\ea\n\nIf only $Q_{\\alpha \\beta}=0$, connection is metric compatible;\nEinstein-Cartan geometry. If both $Q_{\\alpha \\beta}=0$ and\n$T^\\alpha =0$, connection is Levi-Civita; pseudo-Riemannian\ngeometry. If $R^\\alpha{}_\\beta=0$ and $Q_{\\alpha \\beta}=0$, it is\ncalled teleparallel (Weitzenb\\\"{o}ck) geometry. If $R^\\alpha{}_\\beta=0$ and\n$T^\\alpha =0$, it is called symmetric teleparallel geometry (STPG).\n\n\n\n\n\n \\section{STPG}\n\nIn STPG only nonmetricity tensor is nonzero:\n \\ba\n Q_{\\alpha \\beta} \\neq 0 \\quad , \\quad T^\\alpha =0 \\quad , \\quad {R^\\alpha}_\\beta\n =0 \\; . \\label{STPG1}\n \\ea\nHere I argue that in a well-chosen coordinate (or gauge) every metric has its own nonmetricity. This can be seen a kind of gauge fixing. Let us show this\nargument as follows. First one has to choose the natural frame $e^\\alpha =\n\\d x^\\alpha$ and the connection as $\\Lambda^\\alpha{}_\\beta =0$, then automatically\n$R^\\alpha{}_\\beta =0$ and $T^\\alpha =0$ and $Q_{\\alpha \\beta} = -\n\\frac{1}{2}\\d g_{\\alpha \\beta}$. This sequence of the operations corresponds to $\\omega_{ab} + q_{ab}$ in the orthonormal frame. Besides, in this geometry identities (\\ref{bianc:0})-(\\ref{bianc:2}) give one nontrivial identity, $\\D Q_{\\alpha \\beta} =0$. From now on let Greek indices denote natural (holonomic) ones.\n\n \\subsection{Autoparallel Curves}\n\nIn the Riemann geometry one requires that the tangent vector to the\nautoparallel curve points in the same direction as itself when\nparallel propagated, and demand that it maintains the same length,\n \\ba\n \\D T^\\alpha =0 \\; .\n \\ea\nOn the other hand, since intuitively autoparallel curves are\n\"those as straight as possible\" I do not demand the\nvector to keep the same length during parallel propagation in\nSTPG. It is known that nonmetricity is related to length and angle\nstandards at parallel transportation. Therefore I prescribe the\nparallel propagation of the tangent vector\n \\ba\n \\D T^\\mu = (a Q^\\mu{}_\\nu +bq^\\mu{}_\\nu ) T^\\nu + c Q T^\\mu\n \\label{eq:DTmu}\n \\ea\nwhere $T^\\mu = \\frac{d x^\\mu}{d \\tau}$ is the tangent vector to the curve\n$x^\\mu (\\tau)$ with an affine parameter $\\tau$ and $a,b,c$ are arbitrary constants. Here since I\nchoose $\\Lambda^\\alpha{}_\\beta =0$ I obtain $\\D T^\\mu = \\d T^\\mu\n= (\\partial_\\alpha T^\\mu) e^\\alpha$. Moreover, I write\n$Q^\\mu{}_\\nu=Q^\\mu{}_{\\nu \\alpha} e^\\alpha$ and $q^\\mu{}_\\nu =\nq^\\mu{}_{\\nu\\alpha} e^\\alpha$ and $Q= Q^\\nu{}_{\\nu\\alpha}\ne^\\alpha$. Then Eqn(\\ref{eq:DTmu}) turns out to be\n \\ba\n \\partial_\\alpha T^\\mu =(a Q^\\mu{}_{\\nu \\alpha} + b\n q^\\mu{}_{\\nu \\alpha}) T^\\nu + c Q^\\nu{}_{\\nu \\alpha} T^\\mu \\, .\n \\ea\nNow after multiplying this with $T^\\alpha$ I write $T^\\alpha\n\\partial_\\alpha := \\frac{d}{d\\tau}$. Thus I arrive at\n \\ba\n \\frac{d^2 x^\\mu}{d\\tau^2} = (a Q^\\mu{}_{\\nu \\alpha} + b\n q^\\mu{}_{\\nu \\alpha}) \\frac{d x^\\nu}{d \\tau} \\frac{d x^\\alpha}{d \\tau}\n + c Q^\\nu{}_{\\nu \\alpha} \\frac{d x^\\mu}{d \\tau} \\frac{d x^\\alpha}{d \\tau} \\, .\n \\ea\nHere by using $Q_{\\alpha \\beta} = -\\frac{1}{2} \\d g_{\\alpha \\beta}$ I get $Q^\\mu{}_{\\nu \\alpha} = -\\frac{1}{2}g^{\\mu\n\\beta}(\\partial_\\alpha g_{\\beta \\nu})$ and then $q^\\mu{}_{\\nu \\alpha} = -\\frac{1}{2}g^{\\mu \\beta}(\\partial_\\nu g_{\\alpha \\beta}\n- \\partial_\\beta g_{\\alpha \\nu})$. Consequently, I express\nthe autoparallel curve equation in terms of the metric as follows\n \\ba\n \\frac{d^2 x^\\mu}{d\\tau^2} &=& g^{\\mu \\beta} \\left[ - \\frac{a+b}{4} (\\partial_\\alpha g_{\\beta \\nu} + \\partial_\\nu g_{\\beta \\alpha})\n + \\frac{b}{2} (\\partial_\\beta g_{\\alpha \\nu}) \\right] \\frac{d x^\\nu}{d \\tau} \\frac{d x^\\alpha}{d \\tau} \\nonumber \\\\\n & & - \\frac{c}{2}g^{\\nu \\beta} (\\partial_\\alpha g_{\\beta \\nu}) \\frac{d x^\\mu}{d \\tau} \\frac{d x^\\alpha}{d \\tau}\n \\ea\nwhere I symmetrized the first term of the first line in $(\\nu \\alpha)= \\frac{1}{2}(\\nu \\alpha + \\alpha\n\\nu)$ indices. If I fix $a=b=1$ and $c=0$, then this equation\nbecomes the same as the autoparallel curve of the Riemann geometry. \n\n\n \\subsection{Geodesics}\n\nIntuitively, geodesics are \"curves as short as\npossible\". Interval between infinitesimal points is given by\nmetric\n \\ba\n ds^2 = g_{\\mu \\nu} dx^\\mu dx^\\nu\n \\ea\nLet me parameterize the curve between endpoints as $x^\\mu =\nx^\\mu(\\tau)$, then I obtain\n \\ba\n s=\\int_{\\tau_1}^{\\tau_2} \\left( -g_{\\mu \\nu} \\dot{x}^\\mu \\dot{x}^\\nu \\right)^{1\/2}d\\tau\n \\ea\nwhere dot denotes $\\tau$-derivative. I inserted a minus sign because of the Lorentz signature. I wish now to derive the\ncondition on a curve which makes it extremize the length between\nits endpoints, i.e., wish to find those curves whose length does\nnot change to first order under an arbitrary smooth deformation\nwhich keeps the endpoints fixed. This condition gives rise to the\nEuler-Lagrange equations\n \\ba\n \\frac{d}{d \\tau} \\left( \\frac{\\partial L}{\\partial \\dot{x}^\\alpha} \\right) - \\frac{\\partial L}{\\partial\n x^\\alpha}=0\n \\ea\nof the action integral $I=\\int_{\\tau_1}^{\\tau_2} L(x^\\mu ,\n\\dot{x}^\\mu , \\tau)$. Thus in my case the Lagrangian is\n \\ba\n L = \\left[ -g_{\\mu \\nu} (x) \\dot{x}^\\mu \\dot{x}^\\nu \\right]^{1\/2}\n \\ea\nwhere $x$ stands for coordinate functions $x^\\mu$. Now\nEuler-Lagrange equations yield\n \\ba\n \\ddot{x}^\\beta + \\frac{1}{2} g^{\\alpha \\beta} (\\partial_\\mu g_{\\alpha \\nu} + \\partial_\\nu g_{\\alpha \\mu}\n + \\partial_\\alpha g_{\\mu \\nu})\\dot{x}^\\mu \\dot{x}^\\nu =\n \\frac{\\dot{x}^\\beta}{2 ( g_{\\mu \\nu} \\dot{x}^\\mu \\dot{x}^\\nu )} \\frac{d(g_{\\mu \\nu} \\dot{x}^\\mu\n \\dot{x}^\\nu)}{d\\tau} \\label{eq:geodesic}\n \\ea\nAttention to the last term! Let me evaluate it. First, I write it\nas $ g_{\\mu \\nu} \\dot{x}^\\mu \\dot{x}^\\nu = g_{\\mu \\nu} T^\\mu T^\\nu\n= T_\\mu T^\\mu$. Now,\n \\ba\n \\d (T_\\mu T^\\mu) &=&(\\D g_{\\mu \\nu}) T^\\mu T^\\nu +2g_{\\mu \\nu} T^\\mu (\\D T^\\nu) \\nonumber \\\\\n &=& -2Q_{\\mu \\nu} T^\\mu T^\\nu + 2g_{\\mu \\nu} T^\\mu (\\D T^\\nu)\n \\ea\nHere the usage of Eqn(\\ref{eq:DTmu}) gives\n \\ba\n \\d (T_\\mu T^\\mu) = 2(a-1)Q_{\\mu \\nu} T^\\mu T^\\nu + 2c Q T_\\mu T^\\mu\n \\ea\nThis means that if I choose $a=1$ and $c=0$,\nEqn(\\ref{eq:geodesic}) becomes the same as the geodesic equation of the Riemann geometry. \n\n\n\\section{Result}\n\nThus if in STPG the parallel propagation of the tangent vector $T^\\mu =\n\\frac{dx^\\mu}{d\\tau}$ to a curve $x^\\mu(\\tau)$ is defined as\n \\ba\n \\D T^\\mu = ( Q^\\mu{}_\\nu + q^\\mu{}_\\nu ) T^\\nu\n \\ea\nthen the autoparallel curve equation is obtained as\n \\ba\n \\frac{d^2 x^\\mu}{d\\tau^2} + \\frac{1}{2} g^{\\mu \\beta} \\left( \\partial_\\alpha g_{\\beta \\nu} + \\partial_\\nu g_{\\beta \\alpha}\n - \\partial_\\beta g_{\\alpha \\nu} \\right) \\frac{d x^\\nu}{d \\tau} \\frac{d x^\\alpha}{d \\tau}=0 \\, . \\label{eq:autoparallel}\n \\ea\nwhich is the autoparallel curve equation of the Riemann geometry. Also it is shown that this is the geodesic equation of the STPG like in the Riemann geometry. \n\n\\section*{Acknowledgement}\n\nThe author thanks to F W Hehl for stimulating criticisms. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\label{SEC:IN}\n\\vspace{-2mm}\n\nWith the rise of big data analytics and cloud computing, cluster-based large-scale data processing has become a common paradigm in many applications and services. \nOnline companies of diverse sizes, ranging from technology giants to smaller startups, routinely store and process data generated by their users and applications on the cloud. \nData-parallel computing frameworks, such as Apache Spark~\\cite{zaharia2010spark,spark} and Hadoop~\\cite{hadoop}, are employed to perform such data processing at scale. \nJobs executed over such frameworks comprise hundreds or thousands of identical parallel subtasks, operating over massive datasets, and executed concurrently in a cluster environment.\n\n\n\\vspace{-1mm}\nThe time and resources necessary to process such massive jobs are immense. Nevertheless, jobs executed in such distributed environments often have significant computational overlaps: \ndifferent jobs processing the same data may involve common intermediate computations, as illustrated in Fig.~\\ref{FIG:JOBARRIVALS}.\nSuch computational overlaps arise naturally in practice. \nIndeed, computations performed by companies are often applied to the same data-pipeline: companies collect data generated by their applications and users, and store it in the cloud. \nSubsequent operations operate over the same pool of data, e.g., user data collected within the past few days or weeks. \nMore importantly, a variety of prominent data mining and machine learning operations involve common preprocessing steps. This includes database projection and selection~\\cite{maier1983theory}, preprocessing in supervised learning~\\cite{trevor2001elements}, and dimensionality reduction~\\cite{eldar2012compressed}, to name a few. \nRecent data traces from industry have reported $40\\sim60\\%$ recurring jobs in Microsoft production clusters~\\cite{jyothi2016morpheus}, and up to $78\\%$ jobs in Cloudera clusters involve data re-access~\\cite{chen2012interactive}.\n\n\n\\vspace{-1mm}\nExploiting such computational overlaps has a tremendous potential to drastically reduce job computation costs and lead to significant performance improvements. In data-parallel computing frameworks like Spark, computational overlaps inside each job are exploited through caching and \\emph{memoization}: the outcomes of computations are stored with the explicit purpose of significantly reducing the cost of subsequent jobs. \nOn the other hand, introducing caching also gives rise to novel challenges in resource management; \nto that end, the purpose of this paper is to design, implement and evaluate caching algorithms over data-parallel cluster computing environments.\n\n\\vspace{-1mm}\nExisting data-parallel computing frameworks, such as Spark, incorporate caching capabilities in their framework in a non-automated fashion. \nThe decision of which computation results to cache rests on the developer that submits jobs: the developer explicitly states which results are to be cached, while cache eviction is implemented with the simple policy (e.g., LRU or FIFO); neither caching decisions nor evictions are part of an optimized design. Crucially, determining which outcomes to cache is a hard problem when dealing with jobs that consist of operations with complex dependencies. \nIndeed, under the Directed Acyclic Graph (DAG) structures illustrated in Fig.~\\ref{FIG:JOBARRIVALS}, making caching decisions that minimize, e.g., total work is NP-hard~\\cite{ioannidis2016adaptive,shanmugam2013femtocaching}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.34\\textwidth]{Figures\/JOBARRIVALS.pdf}\\vspace{-3mm}\n\\caption{\\scriptsize {\\bf{Job arrivals with computational overlaps.}} Jobs to be executed over the cluster arrive at different times $t_1,\\ldots,t_5$. Each job is represented by a Directed Acyclic Graph (DAG), whose nodes correspond to operations, e.g., map, reduce, or join, while arrows represent order of precedence.\nCrucially, jobs have \\emph{computational overlaps}: their DAGs comprise common sets of operations executed over the same data, indicated as subgraphs colored identically across different jobs. Caching such results can significantly reduce computation time.\\vspace*{-1em}}\n\\label{FIG:JOBARRIVALS}\n\\end{figure}\n\n\\vspace{-1mm}\nIn this paper, we develop an adaptive algorithm for caching in a massively distributed data-parallel cluster computing environment, handling complex and massive data flows. Specifically, a mathematical model is proposed for determining caching decisions that minimize total work, i.e., the total computation cost of a job. \nUnder this mathematical model,\nwe have developed new {\\em adaptive} caching algorithms to make online caching decisions with optimality guarantees, e.g., minimizing total execution time. \nMoreover, we extensively validate the performance over several different databases, machine learning, and data mining patterns of traffic, both through simulations and through an implementation over Spark, comparing and assessing their performance with respect to existing popular caching and scheduling policies. \n\n\n\n\n\n\nThe remainder of this paper is organized as follows. \nSec.~\\ref{SEC:BM} introduces background and motivation. \nSec.~\\ref{SEC:AD} presents our model, problem formulation, and our proposed algorithms. \nTheir performance is evaluated in Sec.~\\ref{SEC:PE}. \nSec.~\\ref{SEC:RW} reviews related work, and we\nconclude in Sec.~\\ref{SEC:CO}.\n\n\n\n\n\\section{Background and Motivation}\n\\label{SEC:BM}\n\\vspace{-2mm}\n\n\\subsection{Resilient Distributed Datasets in Spark}\n\\vspace{-1mm}\n\nApache Spark has recently been gaining ground as an alternative for distributed data processing platforms. In contrast to Hadoop and MapReduce~\\cite{dean2008mapreduce},\nSpark is a memory-based general parallel computing framework. It provides {\\em resilient distributed datasets} (RDDs) as a primary abstraction: RDDs are distributed datasets stored in RAM across multiple nodes in the cluster.\nIn Spark, the decision of which RDDs to store in the RAM-based cache rests with the developer~\\cite{zaharia2012resilient}: the developer explicitly requests for certain results to persist in RAM. Once the RAM cache is full, RDDs are evicted using the LRU policy. Alternatively, developers are further given the option to store evicted RDDs on HDFS, at the additional cost of performing write operations on HDFS. RDDs cached in RAM are stored and retrieved faster; however, cache misses occur either because an RDD is not explicitly cached by the developer, or because it was cached and later evicted. In either case, Spark is resilient to misses at a significant computational overhead: if a requested RDD is neither in RAM nor stored in HDFS, Spark recomputes it from scratch. Overall, cache misses, therefore, incur additional latency, either by reading from HDFS or by fully recomputing the missing RDD.\n\n\n\\vspace{-2mm}\nAn example of a job in a data-parallel computing framework like Spark is given in Fig.~\\ref{FIG:DAG}. A job is represented as a DAG (sometimes referred to as the \\emph{dependency graph}). Each node of the DAG corresponds to a parallel operation, such as reading a text file and distributing it across the cluster, or performing a map, reduce, or join operation. Edges in the DAG indicate the order of precedence: an operation cannot be executed before all operations pointing towards it are completed, because their outputs are used as inputs for this operation. As in existing frameworks like Spark or Hadoop, the inputs and outputs of operations may be distributed across multiple machines: e.g., the input and output of a map would be an RDD in Spark, or a file partitioned across multiple disks in HDFS in Hadoop.\n\n\n\n\n\n\\vspace{-3mm}\n\\subsection{Computational Overlaps}\n\\vspace{-2mm}\n\n\nCaching an RDD resulting from a computation step in a job like the one appearing in Fig.~\\ref{FIG:DAG} can have significant computational benefits when jobs may exhibit \\emph{computational overlaps}: not only are jobs executed over the same data, but also consist of operations that are repeated across multiple jobs. This is illustrated in Fig.~\\ref{FIG:JOBARRIVALS}: jobs may be distinct, as they comprise different sets of operations, but certain subsets of operations (shown as identically colored subgraphs in the DAG of Fig.~\\ref{FIG:JOBARRIVALS}) are (a) the same, i.e., execute the same primitives (maps, joins, etc.) and (b) operate over the same data. \n\n\\vspace{-2mm}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.34\\textwidth]{Figures\/DAG.pdf}\\vspace{-3mm}\n\\caption{\\scriptsize \\textbf{Job DAG example}. An example of a parallel job represented as a DAG. Each node corresponds to an operation resulting RDD that can be executed over a parallel cluster (e.g., a map, reduce, or join operation). DAG edges indicate precedence. \nSimple, crunodes (in\/out) and cross nodes are represented with solid or lined textures.}\\vspace*{-1.3em}\n\\label{FIG:DAG}\n\\end{figure}\n\nComputational overlaps arise in practice for two reasons. The first is that operations performed by companies are often applied to the same data-pipeline: companies collect data generated by their applications and users, which they maintain in the cloud, either directly on a distributed file system like HDFS, or on NoSQL databases (like Google's Datastore~\\cite{barrett2008under} or Apache HBase~\\cite{hbase}). Operations are therefore performed on the same source of information: the latest data collected within a recent period of time.\nThe second reason for computational overlaps is the abundance of commonalities among computational tasks in data-parallel processing. Commonalities occur in several classic data-mining and machine learning operations heavily utilized in inference and prediction tasks (such as predictions of clickthrough rates and user profiling). \nWe give some illustrative examples below:\n\n\n\n\n\\vspace{-2mm}\n\\noindent\\textbf{Projection and Selection.} The simplest common preprocessing steps are \\emph{projection} and \\emph{selection}~\\cite{maier1983theory}. For example, computing the mean of a variable $\\mathtt{age}$ among tuples satisfying the predicate $\\mathtt{gender}=\\mathtt{female}$ and $\\mathtt{gender}=\\mathtt{female}\\land \\mathtt{income}\\geq 50$K might both first reduce a dataset by selecting rows in which $\\mathtt{gender}=\\mathtt{female}$. Even in the absence of a relational database, as in the settings we study here, projection (i.e., maintaining only certain feature columns) and selection (i.e., maintaining only rows that satisfy a predicate) are common. For example, building a classifier that predicts whether a user would click on an advertisement relies upon first restricting a dataset containing all users to the history of the user's past clicking behavior. This is the same irrespective of the advertisement for which the classifier is trained. \n\n\n\\vspace{-2mm}\n\\noindent\\textbf{Supervised Learning.} Supervised learning tasks such as regression and classification~\\cite{trevor2001elements}, i.e., training a model from features for the purpose of predicting a label (e.g., whether a user will click on an advertisement or image) often involve common operations that are label-independent. For example, performing ridge regression first requires computing the co-variance of the features~\\cite{trevor2001elements}, an identical task irrespective of the label to be regressed. Similarly, kernel-based methods like support vector machines require precomputing a kernel function across points, a task that again remains the same irrespective of the labels to be regressed~\\cite{scholkopf2001learning}. Using either method to, e.g., regress the click-through rate of an ad, would involve the same preprocessing steps, irrespectively of the labels (i.e., clicks pertaining to a specific ad) being regressed.\n\n\\vspace{-2mm}\n\\noindent\\textbf{Dimensionality Reduction.} Preprocessing also appears in the form of \\emph{dimensionality reduction}: this is a common preprocessing step in a broad array of machine learning and data mining tasks, including regression, classification, and clustering. Prior to any such tasks, data is first projected in a lower dimensional space that preserves, e.g., data distances. There are several approaches to doing this, including principal component analysis~\\cite{jolliffe2002principal}, compressive sensing~\\cite{eldar2012compressed}, and training autoregressive neural networks~\\cite{gregor2014deep}, to name a few. In all these examples, the same projection would be performed on the data prior to subsequent processing, and be reused in the different tasks described above.\n\n\\vspace{-2mm}\nTo sum up, the presence of computational overlaps across jobs gives rise to a tremendous opportunity of reducing computational costs. Such overlaps can be exploited precisely through the caching functionality of a data-parallel framework.\nIf a node in a job is cached (i.e., results are \\emph{memoized}), then neither itself nor any of its predecessors need to be recomputed.\n\n\n\\begin{comment}\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}[b]{0.4\\textwidth}\n\\includegraphics[width=\\textwidth]{job1}\n\\caption{Job1}\n\\label{fig:job1}\n\\end{subfigure}\n\\hspace{0.2in}\n\\begin{subfigure}[b]{0.4\\textwidth}\n\\includegraphics[width=\\textwidth]{job2}\n\\caption{Job2}\n\\label{fig:job2}\n\\end{subfigure}\n\\caption{\\small An Example of overlapping computation with Spark DAGs from two SQL queries. {\\bf too much space, may remove this figure}}\n\\end{figure}\n\\end{comment}\n\n\\vspace{-3mm}\n\\subsection{Problems and Challenges}\n\\vspace{-2mm}\n\nDesigning caching schemes poses several significant challenges. \nTo begin with, making caching decisions is an inherently combinatorial problem. Given (a) a storage capacity constraint, (b) a set of jobs to be executed, (c) the size of each computation result, and (d) a simple linear utility on each job, the problem is reduced to a knapsack problem, which is NP-hard. The more general objectives we discussed above also lead to NP-hard optimization problems~\\cite{fleischer2011tight}.\nBeyond this inherent problem complexity, even if jobs are selected from a pool of known jobs (e.g., classification, regression, querying), the sequence \nto submit jobs within a given time interval \\emph{is a priori unknown}. The same may be true about statistics about upcoming jobs, such as the frequency with which they are requested.\nTo that end, a practical caching algorithm must operate in an \\emph{adaptive} fashion: it needs to make online decisions on what to cache as new jobs arrive, and adapt to changes in job frequencies.\n\n\\vspace{-1mm}\nIn Spark, LRU is the default policy for evicting RDDs when the cache is full. There are some other conventional caching algorithms such as LRU variant~\\cite{LRU-K} that maintains the most recent accessed data for future reuse, and ARC~\\cite{NM-ARC} and LRFU~\\cite{LRFU} that consider both frequency and recency in the eviction decisions.\nWhen the objective is to minimize total work, these conventional caching algorithms are woefully inadequate, leading to arbitrarily suboptimal caching decisions~\\cite{ioannidis2016adaptive}. Recently, a heuristic policy~\\cite{geng2017lcs}, named ``Least Cost Strategy'' (LCS), was proposed to make eviction decisions based on the recovery temporal costs of RDDs. However, this is a heuristic approach and again comes with no guarantees.\nIn contrast, we intend to leverage Spark's internal caching mechanism to implement our caching algorithms and deploy and evaluate them over the Spark platform, while also attaining formal guarantees.\n\n\n\n\n\\section{Algorithm Design}\n\\label{SEC:AD}\n\\vspace{-2mm}\n\nIn this section, we introduce a formal mathematical model for making caching decisions that minimize the expected total work, i.e., the total expected computational cost for completing all jobs. \nThe corresponding caching problem is NP-hard, even in an offline setting where the popularity of jobs submitted to the cluster is \\emph{a priori known}. \nNevertheless, we show it is possible to pose this optimization problem as a submodular maximization problem subject to knapsack constraints. This allows us to produce a $1-1\/e$ approximation algorithm for its solution. Crucially, when job popularity is \\emph{not known}, we have devised an adaptive algorithm for determining caching decisions probabilistically, that makes caching decisions lie within $1-1\/e$ approximation from the offline optimal, in expectation. \n\n\\vspace{-2mm}\n\\subsection{DAG\/Job Terminology}\n\\vspace{-2mm}\n\nWe first introduce the terminology we use in describing caching algorithms.\nConsider a job represented as a DAG as shown in Fig.~\\ref{FIG:DAG}. Let $G(V,E)$ be the graph representing this DAG, whose nodes are denoted by $V$ and edges are denoted by $E$. Each node is associated with an operation to be performed on its inputs (e.g., map, reduce, join, etc.). \nThese operations come from a well-defined set of operation primitives (e.g., the operations defined in Spark). For each node $v$, we denote as $\\op(v)$ the operation that $v\\in V$ represents. The DAG $G$ as well as the labels $\\{\\op(v),v\\in V\\}$ fully determine the job.\nA node $v\\in V$ is a \\emph{source} if it contains no incoming edges, and a \\emph{sink} if it contains no outgoing edges. Source nodes naturally correspond to operations performed on ``inputs'' of a job (e.g., reading a file from the hard disk), while sinks correspond to ``outputs''.\nGiven two nodes $u,v\\in V$, we say that $u$ is a \\emph{parent} of $v$, and that $v$ is a \\emph{child} of $u$, if $(u,v)\\in E$. We similarly define \\emph{predecessor} and \\emph{successor} as the transitive closures of these relationships.\nFor $v\\in V$, we denote by $\\prd(v)\\subset V$, $\\scc(c)\\subset V$ the sets of predecessors and successors of $v$, respectively.\nNote that the parent\/child relationship is the opposite to usually encountered in trees, where edges are usually thought of as pointing away from the root\/sink towards the leaves\/sources.\nWe call a DAG a \\emph{directed tree} if (a) it contains a unique sink, and (b) its undirected version (i.e., ignoring directions) is acyclic.\n\n\\vspace{-2mm}\n\\subsection{Mathematical Model}\n\\vspace{-2mm}\n\nConsider a setting in which all jobs are applied to the same dataset; this is without loss of generality, as multiple datasets can be represented as a single dataset--namely, their union--and subsequently adding appropriate projection or selection operations as preprocessing to each job. Assume further that each DAG is a directed tree. Under these assumptions, let $\\ensuremath{\\mathcal{G}}$ be the set of all possible jobs that can operate on the dataset. We assume that jobs $G\\in\\ensuremath{\\mathcal{G}}$ arrive according to a stochastic stationary process with rate $\\lambda_G>0$. Recall that each job $G(V,E)$ comprises a set of nodes $V$, and that each node $v\\in V$ corresponds to an operation $\\op(v)$. We denote by as $c_v\\in \\reals_+$ the time that it takes to execute this operation given the outputs of its parents, and $s_v\\in \\reals_+$ be the size of the output of $\\op(v)$, e.g., in Kbytes. Without caching, the \\emph{total-work} of a job $G$ is then given by\n$W(G(V,E)) = \\sum_{v\\in V} c_v.$\nWe define the \\emph{expected total work} as:\n\\begin{align}\n\\bar{W} =\\sum_{G\\in \\mathcal{G}} \\lambda_G \\cdot W(G) = \\sum_{G(V,E)\\in \\mathcal{G}} \\lambda_{G(V,E)} \\sum_{v\\in V} c_v.\n\\end{align}\nWe say that two nodes $u,u'$ are \\emph{identical}, and write $u=u'$, if both these nodes and all their predecessors involve exactly the same operations.\nWe denote by $\\ensuremath{\\mathcal{V}}$\nthe union of all nodes of DAGs in $\\ensuremath{\\mathcal{G}}$.\nA \\emph{caching strategy} is a vector $x=[x_v]_{v\\in\\ensuremath{\\mathcal{V}}}\\in \\{0,1\\}^{|\\ensuremath{\\mathcal{V}}|}$, where $x_v\\in\\{0,1\\}$ is a binary variable indicating whether we have cached the outcome of node $v$ or not.\nAs jobs in \\ensuremath{\\mathcal{G}}{} are directed trees, when node $v$ is cached, \\emph{there is no need to compute that node or any predecessor of that node}.\nHence, under a caching strategy $x$, the total work of a job $G$ becomes:\n\\begin{align}\n\\textstyle W =\\sum_{v\\in V} c_v (1-x_v)\\prod_{u\\in \\scc(v)} (1-x_u).\n\\end{align}\nIntuitively, this states that the cost $c_v$ of computing $\\op(v)$ needs to be paid if and only if \\emph{neither} $v$ \\emph{nor} any of its successors have been cached.\n\n\\vspace{-3mm}\n\\subsection{Maximizing the Caching Gain: Offline Optimization} \n\nGiven a cache of size $K$ Kbytes, we aim to solve the following optimization problem:\n\n\n\\vspace{-2mm}\n\\begin{subequations}\\label{maxcachegain}\n\\small{{\\hspace*{\\stretch{1}} \\textsc{MaxCachingGain}\\hspace{\\stretch{1}} }}\n\\begin{align}\n\\text{Max:}& & & F(x) \\!=\\! \\bar{W} \\!-\\!\\! \\sum_{G\\in\\ensuremath{\\mathcal{G}}}\\!\\!\\lambda_G W(G,x)\\!\\label{obj}\\\\\n\\text{}&&& =\\!\\!\\! \\sum_{G(V,E)\\in\\ensuremath{\\mathcal{G}}}\\!\\!\\!\\!\\!\\! \\lambda_G\\!\\sum_{v\\in V}\\!c_v\\big[1- (1\\!-\\!x_v)\\!\\!\\!\\!\\!\\prod_{u\\in \\scc(v)}\\!\\!\\! (1\\!-\\!x_u)\\big] \\\\\n\\text{Sub.~to:}&&& \\textstyle\\sum_{v\\in \\ensuremath{\\mathcal{V}}} s_vx_v\\leq K, \\quad\nx_v\\in\\{0,1\\}, \\text{ for all } v\\in \\ensuremath{\\mathcal{V}}.\\label{intcont}\n\\end{align}\n\\end{subequations}\n\n\\vspace{-2mm}\nFollowing~\\cite{ioannidis2016adaptive}, we call function $F(x)$ the \\emph{caching gain}: this is the reduction on total work due to caching. This offline problem\nis NP-hard~\\cite{shanmugam2013femtocaching}. \nSeen as an objective over the set of nodes $v\\in \\ensuremath{\\mathcal{V}}$ cached, $F$ is a \\emph{monotone, submodular} function. Hence, \\eqref{maxcachegain} is a submodular maximization problem with a knapsack constraint. When all outputs have the same size, the classic greedy algorithm by Nemhauser et al.~\\cite{nemhauser} yields a $1-1\/e$ approximation. In the case of general knapsack constraints, there exist well-known modifications of the greedy algorithm that yields the same approximation ratio~\\cite{sviridenko-submodular,krause-submodular,kulik2009maximizing}.\n\n\n\n\n\\vspace{-2mm}\nBeyond the above generic approximation algorithms for maximizing submodular functions, \\eqref{maxcachegain} can be solved by \\emph{pipage rounding}~\\cite{ageev2004pipage}.\nIn particular, there exists a concave function $L:[0,1]^{|\\ensuremath{\\mathcal{V}}|}$ such that: \n\\begin{align}(1-1\/e)L(x)\\leq F(x) \\leq L(x), \\quad\\text{for all}~x\\in[0,1]^{|\\ensuremath{\\mathcal{V}}|}\\label{sandwitch}.\\end{align}\nThis \\emph{concave relaxation} of $F$ is given by:\n\\begin{align}\\label{relaxation}\nL(x) = \\sum_{G(V,E)\\in\\ensuremath{\\mathcal{G}}} \\lambda_G\\sum_{v\\in V}c_v\\min\\big\\{1, x_v+\\sum_{u\\in \\scc(v)}x_u\\big\\}.\n\\end{align}\nPipage rounding solves \\eqref{maxcachegain} by replacing objective $F(x)$ with its concave approximation $L(x)$ and relaxing the integrality constraints \\eqref{intcont} to the convex constraints $x\\in[0,1]^{|\\ensuremath{\\mathcal{V}}|}$. The resulting optimization problem is convex--in fact, it can be reduced to a linear program, and thus solved in linear time. Having solved this convex optimization problem, the resulting fractional solution is subsequently rounded to produce an integral solution. Several polynomial time rounding algorithms exist (see, e.g., \\cite{ageev2004pipage,swaprounding}, and \\cite{kulik2009maximizing} for knapsack constraints). Due to \\eqref{sandwitch} and the specific design of the rounding scheme, the resulting integral solution is guaranteed to be within a constant approximation of the optimal \\cite{ageev2004pipage,kulik2009maximizing}.\n\n\\vspace{-2mm}\n\\subsection{An Adaptive Algorithm with Optimality Guarantees}\n\\label{subsec:adaptive} \n\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=0.84\\textwidth]{Figures\/EXP-DAG.pdf}\\vspace{-3mm}\n\\caption{\\scriptsize {\\bf{An example of RDD dependency in synthetic jobs.}} Denote $J_x.S_y$ as stage $y$ in job $x$, then we have $J_0.S_0 = J_1.S_1=J_2.S_0=J_3.S_1$, \n$J_1.S_{0\\sim 5}=J_3.S_{0\\sim 5}$, and \n$J_0.S_{0\\sim 1}=J_2.S_{0\\sim 1}$. \nUnfortunately, even sharing the same computational overlap, by default these subgraphs will be assigned with different stage\/RDD IDs by Spark since they are from different jobs.\n\\vspace*{-1em}\n}\n\\label{FIG:EXP-DAG}\n\\end{figure*}\n\n\n\n\\begin{algorithm}[t]\n\\scriptsize\n\\SetAlFnt{\\footnotesize}\n\\SetKwFunction{FuncIterateJobs}{processJobs}\n\\SetKwProg{ProcIterateJobs}{Procedure}{}{}\n\n\\SetKwFunction{FuncIterate}{processJob}\n\\SetKwProg{ProcIterate}{Procedure}{}{}\n\n\\SetKwFunction{FuncCalCost}{estimateCost}\n\\SetKwProg{ProcCalCost}{Procedure}{}{}\n\n\\SetKwFunction{FuncSaveRDD}{updateCache}\n\\SetKwProg{ProcSaveRDD}{Procedure}{}{}\n\n\\SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{endif}\n\n\\ProcIterateJobs{\\FuncIterateJobs{\\ensuremath{\\mathcal{G}}}}\n{ \n $C_{\\ensuremath{\\mathcal{G}}}$ = Historical RDD access record\\;\n $C_G$ = Current job RDD access record\\;\n \\For{$G\\in\\ensuremath{\\mathcal{G}}$}\n {\n \n processJob($G(V,E)$, $C_G$)\\;\n updateCache($C_G$, $C_{\\ensuremath{\\mathcal{G}}}$)\\;\n }\n} \n\\ProcIterate{\\FuncIterate{$G(V,E)$, $C$}}\n{ \n $C_G$.clear()\\;\n \\For{v$\\in$V}\n {\n v.accessed=False\\;\n toAccess=set(DAG.sink())\\;\n \\While{toAccess$\\neq \\emptyset $}\n {\n v=toAccess.pop()\\; \n $C_G$[v]=estimateCost(v)\\;\n\n \\If{not v.cached}\n {\n \\For{u $\\in$ v.parents}\n {\n \\If{not u.accessed}\n {\n toAccess.add(u)\\;\n }\n }\n }\n access(v); \/* Iterate RDD $v$. *\/ \\\\\n v.accessed=True\\;\n }\n }\n \\KwRet\\;\n} \n\\ProcCalCost{\\FuncCalCost{v}}\n{\n cost=compCost[v]; \/* If all parents are ready. *\/\\\\\n toCompute=v.parents \/* Check each parent. *\/\\\\\n \\While{toCompute $\\neq \\emptyset$}\n {\n u=toCompute.pop()\\;\n \\If {not (u.cached or u.accessed or u.accessedInEstCost)}\n {\n cost+=compCost[u]\\;\n toCompute.appendList(u.parents)\\; \n u.accessedInEstCost=True\\;\n }\n }\n \\KwRet cost;\n}\n\\ProcSaveRDD{\\FuncSaveRDD{$C_G$, $C_{\\ensuremath{\\mathcal{G}}}$}}\n{\n \\For {$v \\in C_{\\ensuremath{\\mathcal{G}}}$}\n {\n \n \\If{$v \\in C_G$}\n {\n $C_{\\ensuremath{\\mathcal{G}}}[v] =(1-\\beta)\\times C_{\\ensuremath{\\mathcal{G}}}[v] + \\beta \\times C_G[v]$\\;\n }\n \\Else\n {\n $C_{\\ensuremath{\\mathcal{G}}}[v] =(1-\\beta)\\times C_{\\ensuremath{\\mathcal{G}}}[v] $\\;\n }\n updateCacheByScore($C_{\\ensuremath{\\mathcal{G}}}$)\\;\n \n }\n \\KwRet\\;\n}\n\\caption{\\small A Heuristic Caching Algorithm.}\n\\label{ALG:1}\n\\end{algorithm}\n\n\\vspace{-2mm}\nAs discussed above, if the arrival rates $\\lambda_G$, $G\\in\\ensuremath{\\mathcal{G}}$, are known, we can determine a caching policy within a constant approximation from the optimal solution to the (offline) problem \\textsc{MaxCachingGain} by solving a convex optimization problem.\nIn practice, however, the arrival rates $\\lambda_G$ may \\emph{not} be known. To that end, we are interested in an \\emph{adaptive} algorithm, that converges to caching decisions \\emph{without any prior knowledge of job arrival rates} $\\lambda_G$. \nBuilding on \\cite{ioannidis2016adaptive}, we propose an adaptive algorithm for precisely this purpose. We describe the details of this adaptive algorithm in {\\intechreport{the Appendix~\\ref{app:overview}.}{our technical report \\cite{techrep}.}}\nIn short, our adaptive algorithm performs \\emph{projected gradient ascent} over concave function $L$, given by \\eqref{relaxation}. \nThat is, our algorithm maintains at each time a fractional $y\\in[0,1]^{|\\ensuremath{\\mathcal{V}}|}$, capturing the probability with which each RDD should be placed in the cache. Our algorithm collects information from executed jobs; this information is used to produce an estimate of the gradient $\\nabla L(y)$. In turn, this is used to adapt the probabilities $y$ that we store different outcomes. Based on these adapted probabilities, we construct a randomized placement $x$ satisfying the capacity constraint \\eqref{intcont}. We can then show that the resulting randomized placement has the following property:\n\\begin{thm} \\label{mainthm}If $x(t)$ is the placement at time $t$, then \n$\\textstyle\\lim_{t\\to \\infty} \\mathbb{E}[F(x(t))] \\geq \\big(1 -{1}\/{e}\\big) F(x^*),$\nwhere $x^*$ is an optimal solution to the offline problem \\textsc{MaxCachingGain} (Eq.~\\eqref{maxcachegain}).\n\\end{thm}\nThe proof of Thm.~\\ref{mainthm} can be found in \\intechreport{Appendix~\\ref{app:proofofmainthm}.}{our technical report~\\cite{techrep}.}\n\n\n\n\n\\subsection{A Heuristic Adaptive Algorithm}\n\nBeyond attaining such guarantees, our adaptive algorithm gives us a great intuition to prioritize computational outcomes. Indeed, the algorithm prioritizes nodes $v$ that have a high gradient component $\\partial L\/\\partial x_v$ and a low size $s_v$. Given a present placement, RDDs should enter the cache if they have a high value w.r.t.~the following quantity \\intechreport{(Appendix~\\ref{app:proofofmainthm})}{~\\cite{techrep}}:\n\\begin{align}\\textstyle \\frac{\\partial L}{\\partial x_v}\/s_v \\simeq \\left(\\textstyle\\sum_{G\\in \\ensuremath{\\mathcal{G}}: v\\in G}\\lambda_G \\times \\Delta(w)\\right)\/s_v, \\label{approx}\\end{align}\nwhere $\\Delta(w)$ is the difference in total work if $v$ is not cached.\nThis intuition is invaluable in coming up with useful heuristic algorithms for determining what to place in a cache. In contrast to, e.g., LRU and LFU, that prioritize jobs with high request rate, Eq.~\\eqref{approx} suggests that a computation should be cached if (a) it is requested often, (b) caching it can lead to a significant reduction on the total work, and (c) it has small size. Note that (b) is \\emph{dependent on other caching decisions made by our algorithm}. Observations (a), (b), and (c) are intuitive, and the specific product form in \\eqref{approx} is directly motivated and justified by our formal analysis.\nThey give rise to the following simple heuristic adaptive algorithm: for each job submitted, maintain a moving average of (a) the request rate of individual nodes it comprises, and (b) the cost that one would experience if these nodes are not cached. Then, place in the cache only jobs that have a high such value, when scaled by the size $s_v$. \n\n\\vspace{-2mm}\nAlg.~\\ref{ALG:1} shows the main steps of our heuristic adaptive algorithm. \nIt updates the cache (i.e., storage memory pool) after the execution of each job (line 5) based on decisions made in the $updateCache$ function (line 6), which considers both the historical (i.e., $C_{\\ensuremath{\\mathcal{G}}}$) and current RDD (i.e., $C_G$) cost scores.\nParticularly, when iterating RDDs in each job following a recursive fashion, \nan auxiliary function $estimateCost$ is called to calculate and record the temporal and spatial cost of each RDD in that job (see line 14 and lines 22 to 31).\nNotice that $estimateCost$ does not actually access any RDDs, but conducts DAG-level analysis for cost estimation which will be used to determine cache contents in the $updateCache$ function.\nIn addition, a hash mapping table is also used to record and detect computational overlap cross jobs (details see in our implementation in Sec.~\\ref{SUBSEC:PE-SI}). \nAfter that, we iterate over each RDD's parent(s) (lines 16 to 18). Once all its parent(s) is(are) ready, we access (i.e., compute) the RDD (line 19). \nLastly, the $updateCache$ function first updates the costs of all accessed RDDs to decide the quantities cost collected above with a moving average window using a decay rate of $\\beta$, implementing an Exponentially Weighted Moving Average (EWMA).\nNext, $updateCache$ makes cross-job cache decisions based on the sorting results of the moving average window by calling the $updateCacheByScore$ function. \nThe implementation of this function can (1) refresh the entire RAM by top score RDDs; or (2) evict lower score old RDDs to insert higher score new RDDs. \n\n\n\n\\section{Performance Evaluation}\n\\label{SEC:PE}\n\\vspace{-2mm}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=0.85\\textwidth]{Figures\/EXP-SIM.pdf}\\vspace{-3mm}\n\\caption{\\small Hit ratio, access number and total work makespan results of large scale simulation experiments.\n\\vspace*{-2.3em}\n}\n\\label{FIG:EXP-SIM}\n\\end{figure*}\n\n\nIn this section, we first demonstrate the performance of our adaptive caching algorithm ( Alg.~\\ref{ALG:1}) on a simple illustrative example. \nWe then build a simulator to analyze the performance of large-scale synthetic traces with complex DAGs. \nLastly, we validate the effectiveness of our adaptive algorithm by conducting real experiments in Apache Spark with real-world machine learning workloads.\n\n\n\n\n\n\n\\vspace{-3mm}\n\\subsection{Numerical Analysis}\n\\label{SUBSEC:PE-NA}\n\\vspace{-2mm}\n\nWe use a simple example to illustrate how our adaptive algorithm (i.e., Alg.~\\ref{ALG:1}) performs w.r.t minimizing total work. \nThis example is specifically designed to illustrate that our algorithm significantly outperforms the default LRU policy used in Spark. \nAssume that we have 5 jobs ($J_0$ to $J_4$) each consisting of 3 RDDs, the first 2 of which are common across jobs. \nThat is, $J_0$'s DAG is $R_0$$\\rightarrow$$R_1$$\\rightarrow$$R_2$, \n$J_1$ is $R_0$$\\rightarrow$$R_1$$\\rightarrow$$R_3$, \n$J_2$ is $R_0$$\\rightarrow$$R_1$$\\rightarrow$$R_4$, etc. \nThe calculation time for $R_1$ is 100 seconds while the calculation time for other RDDs (e.g., $R_2$, $R_3$,...) is 10 seconds. \nWe submit this sequence of jobs twice, with the interarrival time of 10 seconds between jobs. Thus, we have 10 jobs in a sequence of \\{$J_0$, $J_1$, ..., $J_4$, $J_0$, $J_1$, ..., $J_4$\\}. We set the size of each RDD as 500MB and the cache capacity as 500MB as well. Hence, at most one RDD can be cached at any moment. \n\n\nTable~\\ref{TAB:sample} shows the experimental results of this simple example under LRU and our algorithm. Obviously, LRU cannot well utilize the cache because the recently cached RDD (e.g., $R_2$) is always evicted by the newly accessed RDD (e.g., $R_3$). As a result, none of the RDDs are hit under the LRU policy. By producing an estimation of the gradient on RDD computation costs, our algorithm instead places $R_1$ in the cache after the second job finishes and thus achieves a higher hit ratio of 36\\%, i.e., 8 out of 22 RDDs are hit. Total work (i.e., the total calculation time for finishing all jobs) is significantly reduced as well under our algorithm. \n\n\n\\begin{table}[h]\n\\scriptsize\n\\vspace{-0.1in}\n\\caption{Caching results of the simple case.}\n\\label{TAB:sample}\n\\centering\n\\begin{tabular}{|p{10mm}|p{2.5mm}|p{2.5mm}|p{2.5mm}|p{2.5mm}|p{2.5mm}|p{2.5mm}|p{9mm}|p{13mm}|}\n\\hline\nPolicy & $J_0$ & $J_1$ & $J_2$ & $J_3$ & ... & $J_4$ & hitRatio & totalWork \\\\ \\hline \nLRU & $R_2$ & $R_3$ & $R_4$ & $R_5$ & ... & $R_6$ &0.0\\% & 1100 \\\\ \\hline\nAdaptive & $R_2$ & $R_1$ & $R_1$ & $R_1$ & ... & $R_1$ &36.4\\% & 300 \\\\ \\hline\n\\end{tabular}\n\\vspace{-0.1in}\n\\end{table}\n\n\n\n\n\n\n\\vspace{-2mm}\n\\subsection{Simulation Analysis}\n\\label{SUBSEC:PE-SA}\n\\vspace{-2mm}\n\nTo further validate the effectiveness of our proposed algorithm, we scale up our synthetic trace by randomly generating a sequence of 1000 jobs to represent real data analysis applications with complex DAGs. Fig.~\\ref{FIG:EXP-DAG} shows an example of some jobs' DAGs from our synthetic trace, where some jobs include stages and RDDs with the same generating logic chain. For example, stage 0 in $J_0$ and stage 1 in $J_1$ are identical, but their RDD IDs are different and will be computed twice. \nOn average, each of these jobs consists of six stages and each stage has six RDDs. \nThe average RDD size is 50MB. \nWe use a decay rate of $\\beta=0.6$. \n\n\n\n\n\n\n\\vspace{-2mm}\nWe implement four caching algorithms for comparison: (1) NoCache: a baseline policy, which forces Spark to ignore all user-defined {\\em cache}\/{\\em persist} demands, and thus provides the lower bound of caching performance; (2) LRU: the default policy used in Spark, which evicts the least recent used RDDs; (3) FIFO: a traditional policy which evicts the earliest RDD in the RAM; and (4) LCS: a recently proposed policy, called ``Least Cost Strategy''~\\cite{geng2017lcs}, which uses a heuristic approach to calculate each RDD's recovery temporal cost to make eviction decisions. \nThe main metrics include \n(a) {\\em RDD hit ratio} that is calculated as the ratio between the number of RDDs hit in the cache and the total number of accessed RDDs, or the ratio between the size of RDDs hit in the cache and the total size of accessed RDDs;\n(b) {\\em Number of accessed RDDs} and {\\em total amount of accessed RDD data size} that need to be accessed through the experiment; \n(c) {\\em Total work} (i.e., makespan) that is the total calculation time for finishing all jobs; \nand \n(d) {\\em Average waiting time} for each job.\n\n\nFig.~\\ref{FIG:EXP-SIM} depicts the performance of the five caching algorithms.\nWe conduct a set of simulation experiments by configuring different cache sizes for storing RDDs. \nClearly, our algorithm (``Adaptive\") significantly improves the hit ratio (up to 70\\%) across different cache sizes, as seen Fig.~\\ref{FIG:EXP-SIM}(a) and (b). \nIn contrast, the other algorithms start to hit RDDs (with hit ratio up to 17\\%) only when the cache capacity becomes large. \nConsequently, our proposed algorithm reduces the number of RDDs that need to be accessed and calculated (see Fig.~\\ref{FIG:EXP-SIM}(c) and (d)), which further saves the overall computation costs, i.e., the total work in Fig.~\\ref{FIG:EXP-SIM}(e) and (f). \nWe also notice that such an improvement from ``Adaptive\" becomes more significant when we have a larger cache space for RDDs, which indicates that our adaptive algorithm is able to better detect and utilize those shareable and reusable RDDs across jobs. \n\n\n\n\n\\vspace{-3mm}\n\\subsection{Spark Implementation}\n\\label{SUBSEC:PE-SI}\n\\vspace{-2mm}\n\nWe further evaluate our cache algorithm by integrating our methodology into Apache Spark 2.2.1, hypervised by VMware Workstation 12.5.0. \nTable~\\ref{TAB:EV-SPEC} summarizes the details of our testbed configuration. \nIn Spark, the memory space is divided into four pools: storage memory, execution memory, unmanaged memory and reserved memory. \nOnly storage and execution memory pools (i.e., $UnifiedMemoryManager$) are used to store runtime data of Spark applications. \nOur implementation focuses on storage memory, \nwhich stores cached data (RDDs), internal data propagated through the cluster, and temporarily unrolled serialized data.\nFig.~\\ref{FIG:EXP-SP-ARC} further illustrates the main architecture of modules in our implementation. \nIn detail, different from Spark's built-in caching that responds to {\\em persist} and {\\em unpersist} APIs, we build an {\\em RDDCacheManager} module in the {\\em Spark Application Layer} to communicate with cache modules in the {\\em Worker Layer}. \nOur proposed module maintains statistical records (e.g., historical access, computation overhead, DAG dependency, etc.), and automatically decides which new RDDs to be cached and which existing RDDs to be evicted when the cache space is full. \n\n\\vspace{-1mm}\n\\small\n\\begin{table}[th]\n \\center\n \\caption{Testbed configuration.} \n \\label{TAB:EV-SPEC}\n \n \\begin{tabular}{|c|c|}\n \\hline\n \\textbf{Component} & \\textbf{Specs} \\\\ \\hline \n Host Server & Dell PowerEdge T310 \\\\ \\hline\n Host Processor & Intel Xeon CPU X3470 \\\\ \\hline\n Host Processor Speed & 2.93GHz \\\\ \\hline\n Host Processor Cores & 8 Cores \\\\ \\hline\n \n Host Memory Capacity & 16GB DIMM DDR3 \\\\ \\hline\n Host Memory Data Rate & 1333 MHz \\\\ \\hline\n \n Host Hypervisor & VMware Workstation 12.5.0 \\\\ \\hline\n Big Data Platform & Apache Spark 2.2.1 \\\\ \\hline\n Storage Device & Western Digital WD20EURS \\\\ \\hline\n Disk Size & 2TB \\\\ \\hline\n Disk Bandwidth & SATA 3.0Gbps \\\\ \\hline\n \n \n Memory Size Per Node & 1 GB \\\\ \\hline\n Disk Size Per Node & 50 GB\\\\ \\hline\n\n \\end{tabular} \n \\end{table}\n\\normalsize\n\\vspace{-1mm}\n\n\\begin{comment}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.31\\textwidth]{Figures\/EXP-SP-MEM.pdf}\\vspace{-3mm}\n\\caption{\\small Architecture of Apache Spark Unified Memory Manager.\\vspace*{-1.3em}}\\vspace*{-1.3em}\n\\label{FIG:EXP-SP-MEM}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.31\\textwidth]{Figures\/EXP-SP-ARC.pdf}\\vspace{-3mm}\n\\caption{\\small Module structure view of our Spark implementation, where our proposed {\\em RDDCacheManager} module cooperates with {\\em cache} module inside each worker node\n}\n\\label{FIG:EXP-SP-ARC}\n\\end{figure}\n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.29\\textwidth]{Figures\/EXP-SP-RES.pdf}\\vspace{-3mm}\n\\caption{\\small Hit ratio and normalized makespan results of a stress testing on cache-unfriendly {\\em Ridge Regression} benchmark with different cache sizes under four cache algorithms.\\vspace*{-1.3em}}\n\\label{FIG:EXP-SP-REC}\n\\end{figure}\n\nWe select {\\em Ridge Regression}~\\cite{hoerl1970ridge} as a benchmark because it is a ubiquitous technique, widely applied in machine learning and data mining applications~\\cite{li2013enhanced, huang2012extreme}.\nThe input database we use is a huge table containing thousands of entries (i.e., rows), and each entry has more than ten features (i.e., columns). \nMore than hundred Spark jobs are repeatedly generated with an exponential arrival rate. \nEach job's DAG contains at least one {\\em Ridge Regression}-related subgraph, which regresses a randomly selected feature column (i.e., target) by a randomly selected subset of the remaining feature columns (i.e., source), i.e., $f_t=\\Re (\\vec {f_s})$, where $f_t$ is the target feature, and $\\Re(\\vec {f_s})$ is the regressed correlation function with an input of source feature vector $\\vec {f_s}$.\nMoreover, different jobs may share the same selections of target and source features, and thus they may have some RDDs with exactly the same generating logic chain (i.e., a subset of DAGs). \nUnfortunately, the default Spark cannot identify RDDs with the same generating logic chain if they are in {\\em different} jobs.\nIn order to identify these reusable and identical RDDs, our proposed {\\em RDDCacheManager} uses a mapping table to records each RDD's generating logic chain {\\em across} jobs (by importing our customized header files into the benchmark), i.e., we denote $RDD_x$ by a hashing function $key\\leftarrow hash(G_x(V,E) )$, where $G_x(V,E)$ is the subgraph of $RDD_x$ ($V$ is the set of all ancestor RDDs and $E$ is the set of all operations along the subgraph).\nSince not all operations are deterministic~\\cite{determ} (e.g., {\\em shuffle} operation on the same input data may result in different RDDs), we only monitor those deterministic operations which guarantee the same output under the same input. \n\n\n\n\n\n\n\\vspace{-3mm}\nRather than scrutinizing the cache-friendly case where our adaptive algorithm appears to work well as shown in Sec.~\\ref{SUBSEC:PE-SA}, \nit will be more interesting to study the performance under the cache-unfriendly case (also called ``stress test''~\\cite{wagner2005stresstest}), where the space size of different combinations of source and target features is comparatively large, which causes the production of a large number of different RDDs across jobs. \nMoreover, the probability of RDDs reaccess is low (e.g., the trace we generated has less than 26\\% of RDDs are repeated across all jobs), and the temporal distances of RDDs reaccess are also relatively long~\\cite{vcacheshare}. Thus, it becomes more challenging for a caching algorithm to make good caching decisions to reduce the total work under such a cache-unfriendly case.\n\n\\vspace{-3mm}\nFig.~\\ref{FIG:EXP-SP-REC} shows the real experimental results under four different caching algorithms, i.e., FIFO, LRU, LCS, and Adaptive. To investigate the impact of cache size, we also change the size of storage memory pool to have different numbers of RDDs that can be cached in that pool. \nCompared to the other three algorithms, our adaptive algorithm achieves non-negligible improvements on both hit ratio (see in Fig.~\\ref{FIG:EXP-SP-REC}(a)) and makespan (see in Fig.~\\ref{FIG:EXP-SP-REC}(b)), especially when the cache size increases. \nSpecifically, the hit ratio can be improved by 13\\% and the makespan can be reduced by 12\\% at most, which are decent achievements for such a cache-unfriendly stress test with less room to improve performance. Furthermore, we observe that Adaptive significantly increases the hit ratio and reduces the makespan when we have more storage memory space, which again indicates that our caching algorithm has the ability to make good use of memory space. \nIn contrast, the other algorithms have less improvement on hit ratio and makespan, since they cannot conduct cross-job computational overlap detection. While, with a global overview of all accessed RDDs, our adaptive algorithm can effectively select proper RDDs from all jobs to be cached in the limited storage memory pool. \n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\\label{SEC:RW}\n\\vspace{-3mm}\n\n\n\n\n\\intechreport{In the era of big data, a large amount of data is needed to be analyzed and processed in a small amount of time. \nTo meet the requirement, two types of in-memory processing systems are proposed~\\cite{zhang2015memory}. %\nThe first type is data analytics system which is focusing on batch processing such as SINGA~\\cite{singa}, Giraph~\\cite{giraph}, and GridGain~\\cite{gridgain}.\nThe second type is real-time data processing systems such as Storm~\\cite{storm}, Spark Streaming~\\cite{sparkstream}, MapReduce Online~\\cite{condie2010mapreduce}. \n}{}\n\n\nMemory management is a well-studied topic across in-memory processing systems. \nMemcached~\\cite{Memcache} and Redis~\\cite{redis} are highly available distributed key-value stores.\nMegastore~\\cite{megastore} offers a distributed storage system with strong consistency guarantees and high availability for interactive online applications. \nEAD~\\cite{yang2018ead} and MemTune~\\cite{xu2016memtune} are dynamic memory managers based on workload memory demand and in-memory data cache needs. \n\\intechreport{\n\nA number of studies have also been done for modeling the multi-stage frameworks. \nStudy~\\cite{gu2013memory} compares the performance in both time and memory cost between Hadoop and Spark, and they observed that Spark is, in general, faster than Hadoop in iterative operations but Spark has to pay for more memory consumption. \nStudy~\\cite{wang2015performance} proposed a simulation driven prediction model that can predict job performance with high accuracy for Spark. \nA novel analytical model is designed in Study~\\cite{wang2016modeling}, which can estimate the effect of interference among multiple Apache Spark jobs running concurrently on job execution time.\n\n}{}\nThere are some heuristic approaches to evict intermediate data in big data platforms. \nLeast Cost Strategy (LCS)~\\cite{geng2017lcs} evicts the data which lead to minimum recovery cost in future. \nLeast Reference Count (LRC) \\cite{yu2017lrc} evicts the cached data blocks whose reference count is the smallest where the reference count dependent child blocks that have not been computed yet. \nWeight Replacement (WR)~\\cite{duan2016selection} is another heuristic approach to consider computation cost, dependency, and sizes of RDDs. \nASRW~\\cite{wang2015new} uses RDD reference value to improve the memory cache resource utilization rate and improve the running efficiency of the program.\nStudy~\\cite{kathpal2012analyzing} develops cost metrics to compare storage vs. compute costs and suggests when a transcoding on-the-fly solution can be cost-effective. \nWeighted-Rank Cache Replacement Policy (WRCP)~\\cite{ponnusamy2013cache} uses parameters as access frequency, aging, and mean access gap ratio and such functions as size and cost of retrieval. \nThese heuristic approaches do use optimization frameworks to solve the problem, but they are only focusing on one single job, and ignoring cross-job intermediate dataset reuse.\n\n\n\\section{Conclusion}\n\\label{SEC:CO}\n\\vspace{-3mm}\n\n\nThe big data multi-stage parallel computing framework, such as Apache Spark, has been widely used to perform data processing at scale. \nTo speed up the execution, Spark strives to absorb as much intermediate data as possible to the memory to avoid repeated computation. \nHowever, the default in-memory storage mechanism LRU does not choose reasonable RDDs to cache their partitions in memory, leading to arbitrarily sub-optimal caching decisions. \nIn this paper, we formulated the problem by proposing an optimization framework, and then developed an adaptive cache algorithm to store the most valuable intermediate datasets in the memory.\nAccording to our real implementation on Apache Spark, the proposed algorithm can improve the performance by reducing 12\\% of the total work to recompute RDDs.\nIn the future, we plan to extend our methodology to support more big data platforms.\n\\vspace{-5mm}\n\n\\subsection{Online Algorithm Overview}\\label{app:overview}\n\n We describe here our adaptive algorithm for solving {\\textsc{MaxCachingGain}} without a prior knowledge of the demands $\\lambda_G$, $G\\in \\mathcal{G}$. The algorithm is based on \\cite{ioannidis2016adaptive}, which solves a problem with a similar objective, but with matroid (rather than knapsack) constraints. We depart from \\cite{ioannidis2016adaptive} in both the objective studied--namely, \\eqref{obj}-- as well as in the rounding scheme used: the presence of knapsack constraints implies that a different methodology needs to be applied to round the fractional solution produced by the algorithm in each step. \n \n We partition time into periods of equal length $T>0$, during which we collect access statistics for different RDDs. In addition, we maintain as state information the\\emph{ marginals} $y_v\\in[0,1]^{|\\mathcal{V}|}$: intuitively each $y_{v}$ captures the probability that node $v\\in \\mathcal{V}$ is cached. \nWhen the period ends, we (a) adapt the state vector $y=[y_v]_{v\\in\\mathcal{V}}\\in [0,1]^{|\\mathcal{V}|}$, and (b) reshuffle the contents of the cache, in a manner we describe below. \n\n\\fussy\n\\noindent\\textbf{State Adaptation.} We use RDD access and cost measurements collected during a period to produce a random vector $z=[z_v]_{v\\in \\mathcal{V}}\\in \\reals_+^{|\\mathcal{V}|}$ that is an unbiased estimator of a subgradient of $L$ w.r.t.~to $y$. That is, if $y^{(k)}\\in[0,1]^{|\\mathcal{V}|}$ is the vector of marginals at the $k$-th measurement period, $z=z(y^{(k)})$ is a random variable satisfying:\n \\begin{align}\\label{estimateprop}\n\\mathbb{E}\\big[z(y^{(k)})\\big] \\in \\partial L(y^{(k)}\n\\end{align}\nwhere $\\partial L(y)$ is the set of subgradients of $L$ w.r.t $y$. \n We specify how to produce such estimates below, in Appendix~\\ref{app:distributedsub}. \n \n Having these estimates, we adapt the state vector $y$ as follows: at the conclusion of the $k$-th period, the new state is computed as\n \\begin{align}y^{(k+1)} \\leftarrow \\mathcal{P}_{\\ensuremath{\\mathcal{D}}} \\left( y^{(k)} + \\gamma^{(k)}\\cdot z(y^{(k)}) \\right),\\label{adapt}\\end{align}\nwhere $\\gamma^{(k)}>0$ is a gain factor and $\\mathcal{P}_{\\ensuremath{\\mathcal{D}}}$ is the projection to the set of relaxed constraints: \n$$\\ensuremath{\\mathcal{D}} = \\left\\{y\\in [0,1]^{|\\mathcal{V}|} : \\sum_{v\\in \\mathcal{V}}s_vy_{v}=K \\right\\}.$$\nNote that $\\mathcal{P}_{\\mathcal{D}}$ is a projection to a convex polytope, and can thus be computed in polynomial time.\n\n\n\\noindent\\textbf{State Smoothening.}\nUpon performing the state adaptation \\eqref{adapt}, each node $v\\in V$ computes the following\n``sliding average'' of its current and past states:\n \\begin{align}\\label{slide}\\bar{y}^{(k)} = \\textstyle \\sum_{\\ell = \\lfloor\\frac{k}{2} \\rfloor}^{k} \\gamma^{(\\ell)} y^{(\\ell)} \/\\left[\\sum_{\\ell=\\lfloor \\frac{k}{2}\\rfloor}^{k}\\gamma^{(\\ell)}\\right] .\\end{align}\n This ``state smoothening'' is necessary precisely because of the non-differentiability of $L$ \\cite{nemirovski2005efficient}. Note that $\\bar{y}^{(k)} \\in \\ensuremath{\\mathcal{D}}$, as a convex combination of points in $\\ensuremath{\\mathcal{D}}$.\n\n\\noindent\\textbf{Cache Placement.} Finally, at the conclusion of a timeslot, the smoothened marginals $\\bar{y}^{(k)}\\in [0,1]^{|\\mathcal{V}|}$ are \\emph{rounded}, to produce a new integral placement $x^{(k)}\\in\\{0,1\\}^{|\\ensuremath{\\mathcal{V}}|}$ that satisfies the knapsack constraint \\eqref{intcont}. There are several ways of producing such a rounding \\cite{ageev2004pipage,swaprounding,kulik2009maximizing}. We follow the probabilistic rounding of \\cite{kulik2009maximizing} (see also \\cite{horel2014budget}): starting from a fractional $y$ that maximizes $L$ over $\\ensuremath{\\mathcal{D}}$, the resulting (random) integral $x$ is guaranteed to be within $1-1\/e$ from the optimal, in expectation. \n\n\\begin{comment}\n Pipage rounding uses the following property of $F$: given a fractional solution $\\mathbf{y} \\in \\ensuremath{\\mathcal{D}}$, there are at least two fractional variables $y_{v}$ and $y_{v'}$, such that transferring mass from one to the other,(a) makes at least one of them 0 or 1, (b) the new $y'$ resulting from this mass transfer remains feasible (i.e., in $\\ensuremath{\\mathcal{D}}$), and (b) $F(y') \\geq F(y)$, that is, the caching gain at $y'$ is at least as good as $y$. \n This is a consequence of the fact, for any two $v,v'\\in \\mathcal{V}$, function $F(\\cdot,y_v,y_{v'})$ is convex w.r.t. the two variables $y_v$, $y_{v'}$; hence, maximizing it over the constraint set implied by $\\ensuremath{\\mathcal{D}}$, restricted to all other variables being fixed, will have a maximum at an extreme point (in which one of the two variables is either 0 or 1).\n \n This rounding process is repeated until $\\hat{\\mathbf{y}}$ has at most one fractional element left, at which point pipage rounding terminates, discards this fractional value, and returns $y'$.\n\\end{comment}\n\n\\subsection{Constructing an Unbiased Estimator of $\\partial L(y)$.}\\label{app:distributedsub}\nTo conclude our algorithm description, we outline here how to compute the unbiased estimates $z$ of the subgradients $\\partial L(y{(k)})$ during a measurement period. In the exposition below, drop the superscript $\\cdot^{(k)}$ for brevity. \n\nThe estimation proceeds as follows.\n\\begin{enumerate}\n\\item Every time a job $G(V,E)$ is submitted for computation, we compute\nthe quantity\n$$t_v=\\textstyle\\sum_{v\\in V}\\!c_v \\ensuremath{\\mathds{1}}\\big( x_v+\\sum_{u\\in \\scc(v)}x_u\\leq 1\\big),$$\nwhere \n$$\n\\ensuremath{\\mathds{1}}(A) =\\begin{cases}\n1, &\\text{if}~A~\\text{is true},\\\\\n0, &\\text{o.w.}\n\\end{cases}\n$$\n\\item Let $\\mathcal{T}_{v}$ be the set of quantities collected in this way at node $v$ regarding item $v\\in \\mathcal{V}$ during a measurement period of duration $T$. At the end of the measurement period, we produce the following estimates: \\begin{align}z_{v}= \\textstyle\\sum_{t\\in \\mathcal{T}_{v} }t\/T,\\quad v\\in\\mathcal{V}.\\label{estimation}\\end{align} \n\\end{enumerate}\nNote that, in practice, $z_v$ needs to be computed only for RDDs $v\\in \\mathcal{V}$ that have been involved in the computation of some job in the duration of the measurement period. \n\nIt is easy to show that the above estimate is an unbiased estimator of the subgradient:\n\\begin{lemma}\\label{subgradientlemma}\nFor $z=[z_{v}]_{v\\in \\mathcal{V}}\\in \\reals_+^{|\\mathcal{V}|}$ the vector constructed through coordinates \\eqref{estimation},\n$$\\mathbb{E}[z(y)] \\in \\partial L(y)\\text{ and }\\mathbb{E}[\\|z\\|_2^2] th_2\\\\\n uncertain \\quad \\quad \\quad \\hspace*{-1mm} \\textrm{otherwise}\n \\end{array}\n \\right.\n\\label{FixedTh}\n\\end{equation}\nThe two thresholds $th_1$ and $th_2$ have been fixed to $0.3$ and $0.7$, respectively. \nIf $prob(x, y) \\in (th_1,th_2)$, then $(x,y)$ is labeled as uncertain. To provide a significant pixel--level supervision, bounding--boxes that are not labeled as legible, machine printed and written in English have been added to the uncertainty region. This procedure has been used to extract the COCO\\_TS dataset.\nSome examples of the obtained supervisions are reported in Figure \\ref{supervision_example}.\n\\begin{figure*}[ht]\n\\begin{center}\n\\centerline{\n\\includegraphics[height=3.2cm,keepaspectratio]{generated_supervision\/COCO_train2014_000000005483.jpg}\n\\includegraphics[height=3.2cm,keepaspectratio]{generated_supervision\/COCO_train2014_000000011422.jpg}\n\\includegraphics[height=3.2cm,keepaspectratio]{generated_supervision\/COCO_train2014_000000014886.jpg}}\n\\vskip 0.08in\n\\centerline{\n\\includegraphics[height=3.2cm,keepaspectratio]{generated_supervision\/COCO_train2014_000000005483.png}\n\\includegraphics[height=3.2cm,keepaspectratio]{generated_supervision\/COCO_train2014_000000011422.png}\n\\includegraphics[height=3.2cm,keepaspectratio]{generated_supervision\/COCO_train2014_000000014886.png}}\n\\caption{The original images and the generated supervisions, on the top and at the bottom, respectively. The background is colored in black, the foreground in red, and the uncertainty region in yellow.}\n\\label{supervision_example}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Scene Text Segmentation} \\label{Scene_Text_Segmentation}\nThe COCO\\_TS dataset is used to train a deep segmentation network (bottom of Figure \\ref{training_scheme}) for scene text segmentation of both the ICDAR--2013 and Total--Text datasets. The effects obtained by the use of the COCO\\_TS dataset, as an alternative to synthetic data, will be described in the next section. \n\n\n\\section{Experiments} \\label{Experiments}\nIn the following, our experimental setup is shown. In particular, Section \\ref{PSP} and Section \\ref{training_details} introduce the segmentation network and define the implementation details used in our experimental setup. In Section \\ref{coco_eval}, the generated annotations for the COCO\\_TS dataset are evaluated, whereas Section \\ref{sts_eval} assesses the insertion of the COCO\\_TS dataset during the training of a scene text segmentation network.\n\n\\subsection{PSPNet}\\label{PSP}\nAll the experiments are carried out with the PSPNet architecture \\cite{PSP}, originally designed for semantic segmentation of natural images. This model, like most of the other semantic segmentation networks, takes an image as input and produces a per--pixel prediction. The PSPNet is a deep convolutional neural network, built on the ResNet model for image classification. To enlarge the receptive field of the neural network, a set of dilated convolutions replaces standard convolutions in the ResNet part of the network. The ResNet encoder produces a set of feature maps and a pyramid pooling module is used to gather context information. Finally, an upsample layer transforms, by bilinear interpolation, the low--dimension feature maps to the resolution of the original image. A convolutional layer produces the final per--pixel prediction. In this work, to better handle the presence of thin text and similarly to \\cite{tang2017scene}, we modified the network structure adding a two level convolutional decoder.\n\n\\subsection{Implementation Details}\\label{training_details}\nThe PSPNet architectures, used both for the background--foreground network and for scene text segmentation, are implemented in TensorFlow. Due to computational issues, in this work, the PSPNet based on the ResNet50 model is used as the CNN encoder. The experiments are realized based on the training procedure explained in the following.\nAs far as the background--foreground network is considered, the image crops are resized so that the min side dimension is equal to 185, while maintaining the original aspect--ratio. Random crops of $185\\times185$ are used during training. Instead, for the scene text segmentation network, the input images have not been resized, and random crops of $281\\times281$ are extracted for training. A multi--scale approach is employed during training and test. In the evaluation phase, a sliding window strategy is used for both the networks. The Adam optimizer \\cite{adam}, with a learning rate of $10^{-4}$, has been used to train the network. The experimentation was carried out in a Debian environment, with a single NVIDIA GeForce GTX 1080 Ti GPU.\n\n\\subsection{Evaluation of the Supervision Generation Procedure}\n\\label{coco_eval}\nThe quality of the generation procedure cannot be assessed on COCO--Text, \ndue to the absence of pixel--level targets. Therefore, we used the ICDAR--2013 dataset for which ground--truth labels are available.\nFollowing the procedure described in Section \\ref{sup_generation}, the segmentation annotations for the ICDAR--2013 test set have been extracted and compared to the original ground--truth. The results, measured using the pixel--level precision, recall and F1 score, are reported in Table \\ref{annotation_res}. For this analysis, the uncertainty region has been considered as text. \n\\begin{table*}[ht]\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lccc}\n\\toprule\n & Precision & Recall & F1 Score \\\\\n\\midrule\nProposed approach & 89.10\\% & 70.74\\% & 78.87\\% \\\\\n\\midrule\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Results of the annotation generation approach on the ICDAR--2013 test set.}\n\\label{annotation_res}\n\\end{table*}\n\\noindent\nA qualitative evaluation of the generated supervision for the COCO\\_TS dataset is reported in Figure \\ref{supervision_example}.\n\n\\subsection{Scene Text Segmentation evaluation}\n\\label{sts_eval}\nDue to the inherent difficulties in collecting large sets of pixel--level supervised images, only few public datasets are available for scene text segmentation. To face this problem, in \\cite{tang2017scene}, synthetic data generation has been employed. Nevertheless, due to the domain--shift, there is no guarantee that a network trained on synthetic data would generalize well also to real images. \nThe COCO\\_TS dataset actually contains real images and, therefore, we expect that, when used for network training, the domain--shift can be reduced. To test this hypothesis, the PSPNet is used for scene text segmentation and evaluated on the ICDAR--2013 and Total--Text test sets, that provides pixel--level annotations. In particular, the following experimental setups have been compared:\n \\begin{itemize} \n\\item \\textbf{Synth:} The training relies only on the synthetically generated images;\n\\item \\textbf{Synth + COCO\\_TS:} The network is pre--trained on the synthetic dataset and fine--tuned on the COCO\\_TS images;\n\\item \\textbf{COCO\\_TS:} The network is trained only on the COCO\\_TS dataset. \n \\end{itemize} \n\\noindent The influence of fine--tuning on the ICDAR--2013 and Total--Text datasets was also evaluated. The results, measured using the pixel--level precision, recall and F1 score, are reported in Table \\ref{ICDAR2013_Results} and Table \\ref{TotalText_Results}, respectively.\n\\begin{table*}[ht]\n\\label{icdar_results}\n\\begin{center}\n\\begin{small}\n\\subfloat[\\small{Results on the ICDAR--2013 test set} \\label{ICDAR2013_Results}]{%\n\\begin{tabular}{|lcccc|}\n\\hline\n & Precision & Recall & F1 Score & \\\\\n\\hline\nSynth & 73.19\\%& 55.67\\% & 63.23\\% & --\\\\\nSynth + COCO\\_TS & \\textbf{77.80\\%}& \\textbf{70.14\\%} & \\textbf{73.77\\%} & \\textbf{+10.54\\%} \\\\\nCOCO\\_TS & 78.86\\% & 68.66\\% & 73.40\\% & +10.17\\%\\\\\n\\hline\nSynth + ICDAR--2013 & 81.12\\%& 78.33\\% & 79.70\\% & --\\\\\nSynth + COCO\\_TS + ICDAR--2013 & 80.08\\%& 79.53\\% & 80.15\\% & +0.45\\%\\\\\nCOCO\\_TS + ICDAR--2013 & \\textbf{81.68\\%} & \\textbf{79.16\\%} & \\textbf{80.40\\%} & \\textbf{+0.70\\%}\\\\\n\\hline\n\\end{tabular}}\n\\end{small}\n\\begin{small}\n\\subfloat[\\small{Results on the Total--Text test set} \\label{TotalText_Results}]{%\n\\begin{tabular}{|lcccc|}\n\\hline\n & Precision & Recall & F1 Score & \\\\\n\\hline\nSynth & 55.76\\%& 22.87\\% & 32.43\\% & --\\\\\nSynth + COCO\\_TS & 72.71\\%& 54.49\\% & 62.29\\% & +29.86\\% \\\\\nCOCO\\_TS & \\textbf{72.83\\%} & \\textbf{56.81\\%} & \\textbf{63.83\\%} & \\textbf{+31.40\\%}\\\\\n\\hline\nSynth + Total Text & 84.97\\%& 65.52\\% & 73.98\\% & --\\\\\nSynth + COCO\\_TS + Total Text & 84.65\\%& 66.93\\% & 74.75\\% & +0.77\\%\\\\\nCOCO\\_TS + Total Text & \\textbf{84.31\\%} & \\textbf{68.03\\%} & \\textbf{75.30\\%} & \\textbf{+1.32\\%}\\\\\n\\hline\n\\end{tabular}}\n\\end{small}\n\\end{center}\n\\caption{Scene text segmentation performances using synthetic data and\/or the proposed COCO\\_TS dataset. The notation \"$+$ Dataset\" means that a fine--tune procedure has been carried out on \"Dataset\". The last column reports the relative increment, with and without fine--tuning, compared to the use of synthetic data only.}\n\\end{table*}\n\\noindent It is worth noting that training the network using the COCO\\_TS dataset is more effective than using synthetic images. Specifically, employing the proposed dataset, the F1 Score is improved of 10.17\\% and 31.40\\% on ICDAR--2013 and Total--Text, respectively. \nThese results are quite surprising and prove that the proposed dataset substantially increases the network performance, reducing the domain--shift from synthetic to real images. If the network is fine--tuned on ICDAR--2013 or Total--Text, the relative difference between the use of synthetic images and the COCO\\_TS dataset is reduced, but still remains significant. Specifically, the F1 Score is improved by 0.70\\% on ICDAR--2013 and 1.32\\% on Total--Text. \n\\noindent Furthermore, it can be observed that using only COCO\\_TS provides comparable results than training the network with both the synthetic and the proposed dataset. Therefore, the two datasets are not complementary and, in fact, the proposed COCO\\_TS is a valid alternative to synthetic data generation for scene text segmentation. Indeed, the use of real images increases the sample efficiency, allowing to substantially reduce the number of samples needed for training. In particular, the COCO\\_TS dataset contains 14690 samples that are less than 1\/50 of the synthetic dataset cardinality. \nSome qualitative output results of the scene text segmentation network are shown in Figure \\ref{segmentation_results_ICDAR} and Figure \\ref{segmentation_results_TotalText}.\n\\begin{figure}[!ht]\n\\begin{center}\n\\centerline{\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/src\/img_7.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/synth\/img_7.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/synth_coco\/img_7.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/coco\/img_7.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/tag\/img_7.png}}\n\\vskip 0.08in\n\\centerline{\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/src\/img_11.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/synth\/img_11.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/synth_coco\/img_11.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/coco\/img_11.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/tag\/img_11.png}}\n\\vskip -0.05in\n\\centerline{\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/src\/img_33.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/synth\/img_33.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/synth_coco\/img_33.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/coco\/img_33.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/icdar\/tag\/img_33.png}}}\n\\caption{Results on the ICDAR--2013 test set. In (a) the original image, in (b), (c) and (d) the segmentation obtained with Synth, Synth+COCO\\_TS and COCO\\_TS setups, respectively. The ground--truth supervision is reported in (e).}\n\\label{segmentation_results_ICDAR}\n\\end{center}\n\\begin{center}\n\\centerline{\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/src\/img2.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/synth\/img2.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/synth_coco\/img2.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/coco\/img2.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/tag\/img2.png}}\n\\vskip 0.08in\n\\centerline{\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/src\/img4.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/synth\/img4.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/synth_coco\/img4.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/coco\/img4.png}\n\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/tag\/img4.png}}\n\\vskip -0.05in\n\\centerline{\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/src\/img8.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/synth\/img8.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/synth_coco\/img8.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/coco\/img8.png}\\hskip 0.045in}\n\\subfloat[]{\\includegraphics[width=2.25cm,keepaspectratio]{img_segmentation\/total_text\/tag\/img8.png}}}\n\\caption{Results on the Total--Text test set. In (a) the original image, in (b), (c) and (d) the segmentation obtained with Synth, Synth+COCO\\_TS and COCO\\_TS setup, respectively. The ground--truth supervision is reported in (e).}\n\\label{segmentation_results_TotalText}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusions} \\label{Conclusions}\nIn this paper, a weakly supervised learning approach has been used to generate pixel--level supervisions for scene text segmentation. Exploiting the proposed approach, the COCO\\_TS dataset, which contains the segmentation ground--truth for a subset of the COCO--Text dataset, has been automatically generated. Unlike previous approaches based on synthetic images, a convolutional neural network is trained on real images from the COCO\\_TS dataset for scene text segmentation, showing a very significant improvement in the generalization on both the ICDAR--2013 and Total--Text datasets, although with only a fraction of the samples. To foster further research on scene text segmentation, the COCO\\_TS dataset has been released.\nInterestingly, our procedure for pixel--level supervision generation from bounding--box annotations is general and not limited to the COCO--Text dataset. It is a matter of future work to employ the same method to extract pixel--level supervisions for different text localization problems (f.i., on multilingual scene text datasets, such as MLT \\cite{MLT}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn the past few years, deep learning algorithms have successfully deployed in many areas of research, such as computer vision and pattern recognition, speech processing, text-to-speech and many other machine learning related areas \\cite{krizhevsky2012imagenet,bahdanau2014neural,amodei2016deep}. This has led the gradual performance improvement of many deep learning based statistical systems applied to various problems. These fundamental improvements also motivated researchers to explore problems of human-computer interaction (HCI), which have long been studied. One such problem involves understanding human emotions and reflecting them through machines, such as emotional dialogue models \\cite{zhou2018emotional,huang2018automatic}.\n\nIts natural for humans to identify the emotions and react accordingly. However, perception of emotion and affective response generation are still challenging problems for machines. In this study, we target to estimate facial animation representations from affective speech. For this purpose, we construct a two stage process, where the first stage uncovers the emotion in the affective speech signal, and the second stage defines a mapping from the affective speech signal to facial movements conditioned on the emotion. \n\n\n\nEmotion recognition from speech has been an active research area in HCI for various applications, specially for humanoid robots, avatars, chat bots, virtual agents etc. Recently, various deep learning approaches have been utilized to improve emotion recognition performance, which includes significant challenges in realistic HCI settings.\nBefore the deep learning era, classical machine learning models, such as hidden Markov\nmodels (HMMs), support vector machines (SVMs), and decision tree-based methods, have been used in speech emotion recognition \\cite{pan2012speech,schuller2003hidden,lee2011emotion,sadiq2017affect}.\nAs deep neural networks become widely available and computationally feasible, wide range of studies adapted complex deep neural network models for the speech emotion recognition problem. Among these models, \nConvolutional neural network (CNN) based models are successfully utilized for improved speech emotion recognition \\cite{bertero2017first,badshah2017speech}. In another setup, a recurrent neural network model based on text and speech inputs has been successfully used for emotion recognition \\cite{yoon2018multimodal}. \nEarly research on acoustic to visual mappings investigated different methods including Hidden Markov Models (HMM), Gaussian Mixture Models (GMM) and Dynamic Bayesian Networks (DBN). \nOne of the pioneers to produce speech driven facial animation used classical HMM \\cite{yamamoto1997speech}, where they successfully mapped states of the HMM to the lip parameters; moreover they also proposed to utilize visemes in mapping HMM states to visual parameters. Their idea of using visemes was later used by many researchers for synthesizing facial animations via speech signals \\cite{bozkurt2007comparison,verma2003using}. \n\nAnother early work on facial animation used classical machine learning approaches, where the 3-D facial movements were predicted from the LPC and RASTA-PLP acoustic features \\cite{brand1999voice}. Later, \\cite{kakumanu2001speech} proposed to consider context by tagging video frames to audio frames from past and future. \nMostly, the mapping from speech to visual representations is performed over offline data. As a real-time solution, \\cite{vougioukas2018end} proposed a generative adversarial network (GAN) based audio to visual mapping system that can synthesize the visual sequence at 150~frames\/sec. \n\nRecently, \\cite{taylor2017deep} focused on generating facial animations solely from the phoneme sequence. They use a DNN based sliding window predictor that learns arbitrary nonlinear mappings from phoneme label input sequences to facial lip movements. Contrary to the conventional methods of mapping phoneme sequence to fixed number of visemes in \\cite{verma2003using,bozkurt2007comparison}, they generate a sequence of output video frames for the input speech frame, and took the mean of that sequence to map the active speech frame to a single video frame.\n\nIn the literature, research on facial animation from affective speech is limited, this is mostly due to scarcity of labeled affective audio-visual data. \nIn a recent study, the problem of embedding emotions in speech driven facial animations has been modeled through an LSTM model \\cite{pham2017speech}. They employed the RAVDESS dataset in their study \\cite{livingstone2012ravdess}, which only includes two-sentence setup with limited phonetic variability.\nIn our earlier work, \\cite{asadiabadi2018multimodal}, we proposed a deep multi-modal framework, combining the work of \\cite{taylor2017deep} using phoneme sequence with spectral speech features to generate facial animations. Our work successfully demonstrated the good use of CNN models to capture emotional variability in mapping affective speech to facial representations.\n\nThe IEMOCAP dataset \\cite{busso2008iemocap} has been widely used in the literature for audio-visual emotion recognition task. Although the IEMOCAP delivers a rich set of affective audio-visual data, it mostly lacks frontal face videos and emotion categories are not all balanced. In this study, we choose to use the SAVEE \\cite{cooke2006audio} dataset. It delivers a balanced set of affective data as well as including clear frontal face videos, which help significantly to train better facial representation models.\n\nOur contributions in this paper are two fold. First we present a speech emotion recognition system which is trained with the SAVEE dataset to understand the emotional content in underlying speech signal. Secondly, we present a emotion dependent speech driven facial animation system which map the speech signal to facial domain according to emotional content.\n\nThe reminder of this paper is organized as follows. In Section~\\ref{sec:Method}, we describe the proposed methodology for the emotions based speech driven facial shape animations. we give experimental evaluations in Section~\\ref{sec:Res}. Finally, conclusions are discussed in Section~\\ref{sec:conc}.\n\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=90mm]{block2.png}\n \\caption{Block diagram of the proposed emotion dependent facial animation system.}\n \\label{fig:1}\n \n\\end{figure}\n\\section{Methodology}\n\\label{sec:Method}\nWe model the affective speech animation problem as a cascade of emotion classification followed by facial parameter regression as depicted in Figure~\\ref{fig:1}. The affective content of the input speech is first classified into one of the 7 emotion categories, then the corresponding emotion dependent deep model maps the spectral speech representation to the visual shape parameters. \n\n\\subsection{Dataset}\nIn this study, we use the Surrey Audio-Visual Expressed Emotion (SAVEE) dataset to evaluate the affective speech animation models \\cite{cooke2006audio}. The dataset consists of video clips from 4 British male actors with six basic emotions (disgust, anger, happy, sad, fear surprise) and the neutral state. A total of 480 phonetically balanced sentences are selected from the standard TIMIT corpus \\cite{garofolo1993darpa} for every emotional state. Audio is sampled at 44.1~kHz with video being recorded at a rate of 60~fps. Phonetic transcriptions are provided with the dataset. A total of approximately 102~K frames are available to train and validate models. Face recordings are all frontal and faces are painted with blue markers for tracking of facial movements.\n\n\\subsection{Feature extraction}\n\n\\subsubsection{Acoustic features}\nWe use the mel-frequency spectral coefficients, aka, MFSC to represent the speech acoustic features. For each speech frame, 40 dimensional MFSC features are extracted to define the acoustic energy distribution over 40 mel-frequency bands. Python's speech feature library is used for the feature extraction. The MFSC features are extracted from pre-emphasized overlapping Hamming windowed frames of speech at 100~Hz. The extracted feature set is z-score normalized to have zero mean and unit variance in each feature dimension. We represent the set of acoustic feature vectors as $\\left\\{f^a_j\\right\\}^N_{j=1}$, where $f^a_{j} \\in \\mathbb{R}^{40 \\times 1}$ and $N$ is the total number of frames. Figure~\\ref{fig:spectros} presents sample MFSC spectrograms from three different emotions.\n\\begin{figure}[tb]\n\\centering\n \\includegraphics[width=90mm]{speech_mfsc.png}\n \\caption{MFSC spectrograms of one sample sentence, \"she had your dark suit in greasy wash water all year,\" with the fear, neutral and surprise affective states.}\n \\label{fig:spectros}\n\\end{figure}\n\nA temporal sliding window of spectral image is defined to capture the spatio-temporal characterization of the input speech as $F^a_j = [f^a_{j-\\Delta_{a}},...,f^a_j,...,f^a_{j+\\Delta_{a}}]$, where $F^a_j$ is a $40 \\times K_{a}$ image, $K_{a}=2\\Delta_{a}+1$ is the temporal length of the spectral image at time frame $j$ and the sliding window moves with stride~$1$ for $j=1, \\dots, N$. \n\n\n\\subsubsection{Facial shape features}\nAs defined in our previous work \\cite{asadiabadi2018multimodal}, the facial shapes are described with a set of $M=36$ landmark points on the lower face region, along the jaw line, nose, inner and outer lips and represented as $S_{j} = \\left\\{(x^{j}_{i},y^{j}_{i})\\right\\}^M_{i=1}$, where $i$ is the landmark index and $j$ is the sample index. The landmarks used in our experiments are extracted using the Dlib face detector \\cite{king2009dlib} shown as red dots in Figure~\\ref{fig:marks}. To obtain a one-to-one correspondence between the acoustic and visual features, the videos in the dataset are re-sampled to 25~fps. From the re-sampled videos, the landmark points are extracted at a 25~Hz rate and later up-sampled to 100~Hz using cubic interpolation, thus yielding a one-to-one match to the acoustic feature sequence.\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=60mm]{markers.png}\n \\caption{Facial markers on the face: Blue markers are from the SAVEE dataset, red markers are the extracted landmarks using the Dlib face detector.}\n \\label{fig:marks}\n\\end{figure}\n\nThe extracted facial shape set \n$S=\\left\\{S_j\\right\\}^N_{j=1}$\nare aligned using the Generalized Procrustes Analysis (GPA) \\cite{ref:procrutes} to remove the possible rotation, scale and position differences across speakers. Then a statistical face shape model is generated utilizing Principal Component Analysis (PCA) algorithm. PCA projects the shapes in the facial space $S$ to a lower dimensional uncorrelated parameter space \n$P=\\left\\{P_j\\right\\}^N_{j=1}$.\nThe projection between the two spaces is carried out using the mean and truncated eigenvector matrix of the covariance of $S$ as defined in detail in \\cite{asadiabadi2018multimodal}. In this study, $18$ PCA parameters are used, which are covering around $99 \\%$ of the variation in the dataset. \n\nThe target output sequence of the DNN is obtained utilizing a sliding window of size $K_{v}$ and stride $1$ over the shape parameter space. The temporal shape feature sequence is represented as $\\left\\{F^v_j\\right\\}^N_{j=1}$ where $F^v_j = [f^v_{j-\\Delta_{v}},...,f^v_j,...,f^v_{j+\\Delta_{v}}] \\in \\mathbb{R}^{18K_{v} \\times 1}$ with $K_{v}=2\\Delta_v + 1$.\n\n\\subsection{Speech emotion recognition}\nEmotion recognition from speech has been widely studied in the literature and the speech spectrogram is known to discriminate emotions well. As Figure~\\ref{fig:spectros} demonstrates sample variations of the MFSC spectrograms on three different emotions, we set the $F^a_{j}$ spectral image as the acoustic feature to train and predict the emotion of the underlying speech. To this effect, we use a deep emotion recognition network (DERN), which includes 3 convolutional layers followed by a fully connected layer and a softmax layer at the output. The details of the DERN network are given in Table~\\ref{tab:emotion}. \n\nIn the emotion recognition phase, the DERN outputs an emotion label for each frame in a given test utterance as, $\\tilde{e}_j=DERN(F^a_{j})$, where $\\tilde{e}_j$ is the estimated emotion label for frame $j$. For an utterance consisting of $T$ frames, the estimated emotion sequence can be given as $\\{\\tilde{e}_j\\}_{j=1}^T$. Let's index the 7 emotions used in this study as $e^1, e^2, \\dots, e^7$. Utterance level emotion probability for the $i$-th emotion can be defined as\n\\begin{equation}\n p_i = \\frac{1}{T} \\sum_{j=1}^T 1(e^i = \\tilde{e}_j),\n\\end{equation}\nwhere ones function, $1()$, returns 1 when condition is true else a zero. Then the top two emotions for the utterance can be identified as\n\\begin{equation}\n i^* = \\arg\\max_i p_i \\;\\; \\text{and}\\;\\; i^{**} = \\arg\\max_{i - \\{i^*\\}} p_i.\n\\end{equation}\n\nThe top second emotion is utilized when the confidence to the top emotion is weak. Hence, probability of the top second emotion is updated as\n\\begin{equation}\n p_{i^{**}} =\n \\begin{cases}\n 0 & \\text{if $p_{i^{*}} > 0.65$} \\\\\n p_{i^{**}} & \\text{otherwise}.\n \\end{cases}\n\\end{equation}\nThen the normalized probabilities for the utterance level top two emotions are set as\n\\begin{equation}\n p^* = \\frac{p_{i^{*}}}{p_{i^{*}}+p_{i^{**}}} \\;\\; \\text{and}\\;\\; p^{**} = \\frac{p_{i^{**}}}{p_{i^{*}}+p_{i^{**}}},\n\\end{equation}\ncorresponding to the top two emotions $e^*$ and $e^{**}$, respectively.\n\\begin{table}[bht]\n\\caption{Network architecture for the DERN}\n\\label{tab:emotion}\n\\centering\n\\scalebox{0.95}{\n\\begin{tabular}{l c c c c c c}\n\\toprule[1pt]\\midrule[0.3pt]\n\\textbf{Layer} & \\textbf{Type} & \\textbf{Depth} & \\textbf{Filter Size} & \\textbf{Stride}\\\\\n\\midrule\n\\hline\n1 & CONV+ReLu & 32 & 5x5 & - \\\\\n\\hline\n2 & MaxPooling & 32 & 3x3 & 2 \\\\\n\\hline\n3 & CONV+ReLu & 64 & 5x5 & - \\\\\n\\hline\n4 & MaxPooling & 64 & 3x3 & 2 \\\\\n\\hline\n5 & CONV+ReLu & 128 & 5x5 & - \\\\\n\\hline\n6 & MaxPooling & 128 & 3x3 & 2 \\\\\n\\hline\n7 & FC+ReLu & 256 & - & - \\\\\n\\hline\n8 & Dropuout & - & - & - \\\\\n\\hline\n9 & FC+Softmax & 7 & - & - \\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\n\n\\subsection{Emotion dependent facial shape regression}\n\\label{ref: av2}\nWe train a deep shape regression network (DSRN) to estimate the facial shape features $F^v_j$ from the acoustic MFSC spectrogram images $F^a_j$ for each emotion category, separately. Note that each DSRN model is trained for each emotion category, hence the estimated facial shape feature can be defined as the fusion of two estimates extracted with the top two emotions,\n\\begin{equation}\n \\label{equ:two label}\n \\tilde{F}^v_j = p^* DSRN(F^a_j | {e}^*) + p^{**} DSRN(F^a_j | {e}^{**}).\n\\end{equation}\n\nIn this formalism, we have $K_v$ number of facial shape estimates for the frame $j$ that are extracted from the neighboring estimates. The final facial shape estimation is defined as the average of $K_v$ estimates,\n\\begin{equation}\n \\hat{f}^v_j = \\frac{1}{K_v} \\sum_{i=j-\\Delta_v}^{j+\\Delta_v} \\tilde{f}^v_i . \n\\end{equation}\n\n\nThe DSRN is constructed with 4 convolutional layers, which are followed by 2 fully connected layers to estimate the shape features. We use dropout regularization method with probability 50\\% in the fully connected layers only, to overcome the over-fitting. ReLu activation function is used at each layer with Adam optimizer for hyper learning rate optimizations. Mean square error (MSE) is chosen as the objective function to be minimized. During the training, we only apply convolutions and pooling over the frequency axis, in order to further prevent any over-fitting, and also to preserve the temporal nature of speech. Detailed specification of the DSRN network is given in Table~\\ref{tab:2}.\n\\begin{table}[bht]\n\\caption{Network architecture for the DSRN}\n\\label{tab:2}\n\\centering\n\\scalebox{0.95}{\n\\begin{tabular}{l c c c c c c}\n\\toprule[1pt]\\midrule[0.3pt]\n\\textbf{Layer} & \\textbf{Type} & \\textbf{Depth} & \\textbf{Filter Size} & \\textbf{Stride}\\\\\n\\midrule\n\\hline\n1 & CONV+ReLu & 32 & 5x1 & - \\\\\n\\hline\n2 & MaxPooling & 32 & 3x1 & 2x1 \\\\\n\\hline\n3 & CONV+ReLu & 64 & 5x1 & - \\\\\n\\hline\n4 & MaxPooling & 64 & 3x1 & 2x1 \\\\\n\\hline\n5 & CONV+ReLu & 128 & 5x1 & - \\\\\n\\hline\n6 & MaxPooling & 128 & 2x1 & 2x1 \\\\\n\\hline\n7 & CONV+ReLu & 128 & 3x1 & - \\\\\n\\hline\n8 & MaxPooling & 128 & 2x1 & 2x1 \\\\\n\\hline\n9 & FC+ReLu & 1024 & - & - \\\\\n\\hline\n10 & Dropuout & - & - & - \\\\\n\\hline\n11 & FC+ReLu & 500 & - & - \\\\\n\\hline\n12 & Dropuout & - & - & - \\\\\n\\hline\n13 & FC+Multi-Reg & $18 \\times K_v$ & - & - \\\\\n\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\n\\section{Experimental results}\n\\label{sec:Res}\n\nWe deploy 5-fold cross-validation training and testing scheme, where 10\\% of the dataset is hold for testing the trained networks. From the remain of the dataset 80\\% is used for training and the remaining 20\\% is used for validation in each fold. Both of the deep models, DERN and DSRN, are trained using the Keras\\footnote{\\url{https:\/\/keras.io\/}} with Tensorflow \\cite{abadi2016tensorflow} backend on a NVIDIA TITAN XP GPU. The temporal window size for acoustic and visual shape features is set as $K_a = 15$ and $K_v = 5$, respectively. \n\n\\subsection{Objective evaluations}\n\n\\subsubsection{Emotion recognition results}\n\nThe DERN model is trained over 200 epochs using the categorical cross-entropy loss function. Utterance level emotion recognition performances over the validation set are reported for each emotion category in Table~\\ref{tab:3}. All models sustain high average recall rates. The surprise emotion category model is observed to suffer the most, compared to other emotions.\n\n\\begin{table}[bht]\n\\caption{Utterance level accuracy (\\%) for the speech emotion recognition in each emotion category}\n\\label{tab:3}\n\\centering\n\\scalebox{0.85}{\n\\begin{tabular}{ccccccc}\n\\toprule[1pt]\\midrule[0.3pt]\nAngry & Disgust & Fear & Happy & Neutral & Sad & Surprise \\\\\n\\midrule\n\\hline\n 72.15 & 75.38 & 73.21 & 71.71 & 91.93 & 75.19 & 67.98 \\\\ \\hline\n\n\\end{tabular}}\n\\end{table}\n\n\n\n\\subsubsection{Facial shape regression results}\nThe mean squared error between predicted and original PCA coefficient values is used as the loss function for the training.\n\nThe DSRN model is trained for each emotion category separately, which defines the emotion dependent models. We as well train an emotion independent model using all combined training data, which sets a baseline for evaluations. Figure~\\ref{fig:loss} presents the MSE loss curve over the validation data through the learning process for emotion dependent and independent models. Note that the proposed emotion dependent regression attains significantly lower MSE loss values, which are more than 65\\% reduction in MSE compared to the emotion independent combined model. \n\nThe MSE loss performance of the cascaded trained DERN and DSRN models over the test set is given in Table \\ref{tab:4}. As obvious from the table, the proposed cascaded DERN and DSRN scheme performs better than the baseline model in terms of MSE in shape domain. In should be noted that given the true emotion labels of the utterances, the test loss of the DSRN is remarkably lower than the all combined model, which makes room to improve the performance of the DERN module in future work.\n\n\\begin{table}[b]\n\\caption{MSE loss on the test set for cascaded DERN and DSRN, true emotions and DSRN vs all combined model}\n\\label{tab:4}\n\\centering\n\\scalebox{0.85}{\n\\begin{tabular}{l c c c}\n\\toprule[1pt]\\midrule[0.3pt]\n\\textbf{Method} & \\textbf{DERN+DSRN}& \\textbf{Actual Emo+DSRN} & \\textbf{ALL Combined}\\\\\n\\midrule\n\\hline\nMSE & {5.57} & {3.23} & {6.73} \\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\\begin{figure}[bht]\n\\centering\n \\includegraphics[width=90mm]{all_loss_small.png}\n \\caption{The MSE loss over the validation data along the epochs for the emotion dependent (separate for each emotion) and independent (all combined) models.}\n \\label{fig:loss}\n\\end{figure}\n\n\n\n\n\\subsection{Subjective evaluations}\nThe resulting facial shape animations are also evaluated subjectively through visual user preference study. We use a mean opinion score (MOS) test to subjectively evaluate animations of the emotion dependent and independent models. The test is run with 15 participants using 7 conditions, which are the animations of utterances from the 7 emotion categories. Each test session for a participant contains 14 clips, where each condition is tested with the emotion dependent and independent models. During the test, all clips are shown in random order. In the test, each participant is asked to evaluate clips based on synchronization and emotional content of the animations using a five-point preference scale (1: Bad, 2: Poor, 3: Fair, 4: Good, and 5: Very Good).\n\n\n\\begin{table}[t]\n\\caption{Average preference scores for the emotion dependent and independent (all combined) model animation evaluations}\n\\label{tab:5}\n\\centering\n\\scalebox{0.95}{\n\\begin{tabular}{l c c}\n\\toprule[1pt]\\midrule[0.3pt]\n\\textbf{Method} & \\textbf{Mean} & \\textbf{Std} \\\\\n\\midrule\n\\hline\nEmotion Dependent & {3.03} & 0.96 \\\\\n\\hline\nAll Combined & 2.74 & 0.88\\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\\begin{figure}[bht]\n\\centering\n \\includegraphics[width=90mm]{subjective_emos1.png}\n \\caption{Emotion category based average preference scores for the emotion dependent and independent (all combined) model animation evaluations. }\n \\label{fig:subjec}\n\\end{figure}\n\nThe average preference scores are listed in Table~\\ref{tab:5}. The proposed emotion dependent facial shape animation scheme is preferred over the baseline emotion independent scheme. Furthermore, in each emotional category a similar preference tendency, except for happy and sad categories, is observed as presented in Figure~\\ref{fig:subjec}. \n\n\n\\section{Conclusions}\n\\label{sec:conc}\nIn this study we propose an emotion dependent speech driven facial animation system. A statistical shape model (PCA based) is trained to project the shape data into an uncorrelated lower dimensional parameter space, capturing 99\\% of variation in the training data. We observe that training separate models to map acoustic spectral features to visual shape parameters performs better than training a universal network with all the emotions combined. Our proposed emotion dependent facial shape animation model outperforms the emotion independent universal model in terms of the MSE loss and also in the subjective evaluations, facial animations of the proposed method preferred higher.\nAs a future work we will investigate the ways to improve the accuracy of the speech emotion recognition system.\n\n\\section{Acknowledgements}\nThis work was supported in part by the Scientific and Technological Research Council of Turkey (T\\\"{U}B\\.{I}TAK) under grant number 217E107.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOne of the longstanding problems in strong interaction is the\nunderstanding within QCD of the mechanism that is responsible for\nthe large single-spin asymmetries (SSA) observed in numerous\nhigh energy reactions with hadrons. Many different approaches were\nsuggested to solve this problem (see recent papers and review\n\\cite{Anselmino:2013rya,Anselmino:2012rq,reviews} and references\ntherein). Most of them are based on the assumption of the\nso-called transverse-momentum-dependent (TMD) factorization\n\\cite{Ji:2004xq,Ji:2004wu,Bacchetta:2008xw,collins}. The validity\nof this assumption is not clear so far \\cite{Rogers:2010dm}.\nFurthermore, in our paper we will show the existence of the\nnonperturbative QCD mechanism which violates explicitly the TMD\nfactorization for SSA.\n\nIt is well known that SSA arises from interference of different\ndiagrams and should include at least two ingredients. First off\nall, it should be a helicity-flip in the scattering amplitude and\nsecondly, the amplitude should have a nonzero imaginary part. The\nsmall current masses of quarks are only a source in perturbative\nQCD (pQCD) for helicity-flip. Furthermore, the imaginary part of\nthe scattering amplitude, which comes from loop diagrams, is\nexpected to be suppressed by extra power of the strong coupling\nconstant $\\alpha_s$. As a result, pQCD fails to describe large\nobserved SSA. On the other hand, it is known that QCD has a\ncomplicated structure of vacuum which leads to the phenomenon of\nspontaneous chiral symmetry breaking (SCSB) in strong interaction.\nTherefore, even in the case of a very small current mass of the\nquarks their dynamical masses arising from SCSB can be large.\nThe instanton liquid model of QCD vacuum \\cite{shuryak,diakonov}\nis one of the models in which the SCSB phenomenon arises in a\nvery natural way due to quark chirality flip in the field of\nstrong\n fluctuation of the vacuum gluon field called\ninstanton \\cite{Belavin:1975fg,'tHooft:1976fv}.\n The instanton is the well-known solution of QCD equation of motion in the Euclidian space-time which\nhas nonzero topological charge. In many papers (see reviews\n\\cite{shuryak,diakonov,Kochelev:2005xn}), it was shown that\ninstantons play a very important role in hadron physics.\nFurthermore, instantons lead to the anomalous quark-gluon\nchromomagnetic vertex with a large\n quark helicity-flip \\cite{kochelev1,diakonov}.\nTherefore, they can give the important contribution to SSA\n\\cite{kochelev1,Kochelev:1999nd,Cherednikov:2006zn,Dorokhov:2009mg,Ostrovsky:2004pd,diakonov,Qian:2011ya}.\n\n\nIn this paper, we will present the first consistent calculation\nof SSA in the quark-quark scattering based on the existence of the\nanomalous quark chromomagnetic moment (AQCM) induced by instantons\n\\cite{kochelev1} \\footnote{ The semi-classical mechanism for SSA\nbased on large AQCM has recently been discussed in papers\n\\cite{Abramov:2011zz,Abramov:2009tm}.}.\n\n\n\n\n\n\n\n\n\\section{ Quark-gluon interaction in non-perturbative QCD }\n\n\n\n\nIn the general case, the interaction vertex of a massive quark\nwith a gluon, Fig.1, can be written in the following form:\n\\begin{equation}\nV_\\mu(p_1^2,{p_1^\\prime}^2,q^2)t^a = -g_st^a[F_1(p_1^2,{p_1^\\prime}^2,q^2) \\gamma_\\mu\n +\n\\frac{\\sigma_{\\mu\\nu}q_\\nu}{2M_q}F_2(p_1^2,{p_1^\\prime}^2,q^2)],\n \\label{vertex}\n \\end{equation}\nwhere the first term is the conventional perturbative QCD\nquark-gluon vertex and the second term comes from the\nnonperturbative sector of QCD.\n\\begin{figure}[htb]\n\\vspace*{2.0cm} \\centering\n\\centerline{\\epsfig{file=vertices.eps,width=12cm,height=3.0cm,\nangle=0}} \\vskip 1cm \\caption{a) Perturbative helicity non-flip\nand b) nonperturbative helicity-flip quark-gluon vertices}\n\\label{vertexpicture}\n\\end{figure}\n\n\n\n\n\n\nIn Eq.\\ref{vertex} the form factors $F_{1,2}$ describe\nnonlocality of the interaction, $p_{1}, p_1^\\prime$ are\nthe momenta of incoming and outgoing quarks, respectively, $ q=p_1^\\prime-p_1$,\n $M_q$ is the quark mass, and $\\sigma_{\\mu\\nu}=(\\gamma_\\mu \\gamma_\\nu-\\gamma_\\nu \\gamma_\\mu)\/2$.\n\n\n\nThe form factor $F_2(p_1^2,{p_1^\\prime}^2,q^2)$\n suppresses the AQCM vertex\nat short distances when the respective virtualities are large. Within the\ninstanton model it is explicitly related to the Fourier-transformed quark\nzero-mode and instanton fields and reads\n\\begin{equation}\n F_2(p_1^2,{p_1^\\prime}^2,q^2) =\\mu_a\n\\Phi_q(\\mid p_1\\mid\\rho\/2)\\Phi_q(\\mid p_1^\\prime\\mid\\rho\/2)F_g(\\mid\nq\\mid\\rho) \\ , \\nonumber\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\Phi_q(z)&=&-z\\frac{d}{dz}(I_0(z)K_0(z)-I_1(z)K_1(z)), \\label{ffq}\\\\\nF_g(z)&=&\\frac{4}{z^2}-2K_2(z), \\label{ffg}\n\\end{eqnarray}\n$I_{\\nu}(z)$, $K_{\\nu}(z)$ are the modified Bessel functions and\n$\\rho$ is the instanton size.\n\nWe assume $F_1\\approx 1$ and $F_2(p_1^2,{p_1^\\prime}^2,q^2)\n \\approx \\mu_a F_g(q^2)$ since valence quarks in hadrons have small virtuality.\n\nWithin the instanton liquid model \\cite{shuryak,diakonov}, where\nall instantons have the same size $\\rho_c$, AQCM is\n\\cite{kochelev2}\n\\begin{equation}\n\\mu_a=-\\frac{3\\pi (M_q\\rho_c)^2}{4\\alpha_s}.\n\\label{AQCM}\n\\end{equation}\n In Eq.(\\ref{AQCM}), $M_q$ is the so-called dynamical quark mass.\nWe would like to point out two specific features of the formula\nfor AQCM. First, the strong coupling constant enters into the\ndenominator showing a clear nonperturbative origin of AQCM. The\nsecond feature is the negative sign of AQCM. As we will see below,\nthe sign of AQCM leads to the definite sign of SSA in the\nquark-quark scattering. The value of AQCM strongly depends on the\ndynamical quark mass which is $M_q=170$ MeV in the mean field\napproximation (MFA)\\cite{shuryak}\n and $M_q=350$ MeV in the Diakonov-Petrov model (DP) \\cite{diakonov}.\nTherefore, for fixed value of the strong coupling constant in\nthe instanton model, $\\alpha_s\\approx \\pi\/3\\approx 0.5$\n\\cite{diakonov}, we get\n\n\\begin{equation}\n{\\mu_a}^{MFA}=-0.4 \\ \\ \\ \\mu_a^{DP}= -1.6\n\\label{mu}\n\\end{equation}\n\nWe would like to mention that the Schwinger-type of the pQCD\ncontribution to AQCM\n\\begin{equation}\n \\mu_a^{pQCD}=-\\frac{\\alpha_s}{12\\pi}\\approx 1.3\\cdot 10^{-2}\n\\label{pQCD}\n\\end{equation}\nis by several orders of magnitude smaller in comparison with the\nnonperturbative contribution induced by instantons, Eq.\\ref{mu},\nand, therefore, it can give only a tiny contribution to\nspin-dependent cross sections\n \\cite{Ryskin:1987ya}\\footnote{Recently, a rather large AQCM has been obtained within the approach based on the\n Dyson-Schwinger equations (see review \\cite{Roberts:2012sv} and references therein).}.\n\n\n\n\n\n\n\\section{Single-spin asymmetry in high energy quark-quark scattering induced by AQCM}\n\nThe SSA for the process of transversely polarized quark scattering off unpolarized quark,\n $q^{\\uparrow}(p_1)+q(p_2) \\to q (p_1^\\prime)\n+ q(p_2^\\prime)$, is defined as\n\\begin{equation}\nA_{N}=\\frac{d \\sigma^{\\uparrow} - d \\sigma^{\\downarrow}}\n{d \\sigma^{\\uparrow}+d \\sigma^{\\downarrow}},\n\\end{equation}\nwhere ${\\uparrow} {\\downarrow}$ denote the initial quark spin\norientation perpendicular to the scattering plane and\n\n\\begin{equation}\nd \\sigma^{\\uparrow\\downarrow}=\\frac{|M(\\uparrow\\downarrow)|^2}{\\mathrm{2I}} d\\mathrm{PS_2(S,q_t)},\n\\end{equation}\nwhere $I$ is the initial flux, $S=(p_1+p_2)^2$,\n$M(\\uparrow\\downarrow)$ is the matrix element for the different\ninitial spin directions, $d\\mathrm{PS_2(S,q_t)}$ is the\ntwo-particle phase space and $q_t={{p_1}^\\prime}_t-{p_1}_t$ is the\ntransverse momentum transfer. In the high energy limit $S\\gg\nq_t^2,M_q^2$, we have $I\\approx S$ and\n$d\\mathrm{PS_2(S,q_t)}\\approx d^2q_t\/(8\\pi^2S)$.\n\n\n\nIn terms of the helicity amplitudes \\cite{Goldberger:1960md},\n\\cite{Buttimore:1978ry}\n\\begin{equation}\n\\Phi_1=M_{++;++},\\ \\ \\Phi_2=M_{++;--},\\ \\ \\Phi_3=M_{+-;+-},\\ \\ \\Phi_4=M_{+-;-+} ,\\ \\ \\Phi_5=M_{++;+-},\n\\nonumber\n\\end{equation}\nwhere the symbols $+$ or $-$ denote the helicity of quark in the\nc.m. frame, SSA is given by\n\\begin{equation}\nA_N=-\\frac{2Im[( \\Phi_1+\\Phi_2+\\Phi_3-\\Phi_4)\\Phi_5^*]} {|\\Phi_1|^2+|\\Phi_2|^2+|\\Phi_3|^2+|\\Phi_4|^2+4|\\Phi_5|^2)}.\n\\label{helicity}\n\\end{equation}\nIn Fig.2, we present the set of diagrams which give a\nsignificant contribution to $A_N$. Higher order terms in $\\mu_a$\nand $\\alpha_s$ are expected to be suppressed by a small instanton\ndensity in QCD vacuum \\cite{shuryak} and by a large power of the\nsmall strong coupling constant.\n\n\n\n\\begin{figure}[htb]\n\n \\vspace*{-0.0cm} \\centering\n \\centerline{\\epsfig{file=AN1grey.eps,width=14cm, angle=0}}\n \\caption{Contribution to SSA arising from different diagrams.}\n\n \\end{figure}\n\n\n\nFor estimation, we take the simple form for the gluon propagator in the Feynman gauge\n \\begin{equation}\n P_{\\mu\\nu}(k^2)=\\frac{g_{\\mu\\nu}}{k^2-m_g^2},\n \\nonumber\n \\end{equation} where $m_g$ can be treated as the infrared cut-off related to confinement \\cite{nikolaev},\nor as the dynamical gluon mass \\cite{RuizArriola:2004en},\n\\cite{aguilar}. Within the instanton model this parameter can be\nconsiderated as the effect of\n multiinstanton contribution to the gluon propagator.\n\n\n\nBy using in the high energy limit the Gribov decomposition for the\nmetric tensor into the transverse and longitudinal parts\n\\begin{equation}\n g_{\\mu\\nu}=g_{\\mu\\nu}^t+ \\frac{2(p_{2\\mu}p_{1\\nu}+p_{2\\nu}p_{1\\mu})}{S}\n \\approx \\frac{2(p_{2\\mu}p_{1\\nu}+p_{2\\nu}p_{1\\mu})}{S}\n \\nonumber\n\\end{equation}\nand the Sudakov parametrization of the four-momenta of particles \\cite{Arbuzov:2010zza},\n\\cite{Baier:1980kx}\n\\begin{equation}\nq_i=\\alpha_ip_2+\\beta_ip_1+q_{i,t}, \\ \\ q_{i,t}p_{1,2}=0, \\ \\ q_{i,t}^2=-\\vec{q_i^2}<0,\n\\nonumber\n\\end{equation}\nwe finally obtain\n \\begin{equation}\n A_N=-\\frac{5 \\alpha_s\\mu_a q_t(q_t^2+m_g^2)}{12 \\pi M_q} \\frac{ F_g(\\rho\n |q_t|)N(q_t)}{D(q_t)},\n \\label{SSA}\n \\end{equation}\n where\n\n \\begin{equation}\nN(q_t)=\\int \\!\\!d^2k_t \\frac{(1+\\mu_a^2(q_t \\cdotp \\! k_t + k_t^2)\n F_g(\\rho |k_t|) F_g(\\rho |q_t\\!+\\!k_t|)\/(2M_q^2) }\n {(k_t^2+m_g^2)((k_t+q_t)^2+m_g^2)}\\big)\n\\nonumber\n\\end{equation}\nand\n\\begin{equation}\n D(q_t)= \\Big(1+ (\\frac{\\mu_a q_t}{2M_q} F_g(\\rho |q_t|))^2\\Big)^2\n + \\frac{\\alpha_s^2 (q_t^2+m_g^2)^2}{12 \\pi^2}\n \\left( \\int \\!\\! \\frac{d^2k_t}{(k_t^2+m_g^2)((k_t+q_t)^2+m_g^2)}\\right)^2 \\nonumber\n\\end{equation}\n\n\\section{Results and discussion}\n\\begin{figure}[h]\n\\begin{minipage}[c]{8cm}\n\\vskip -0.5cm \\hspace*{-1.0cm}\n\\centerline{\\epsfig{file=Plot1grey.eps,width=10cm,height=6cm,angle=0}}\\\n\\end{minipage}\n\\begin{minipage}[c]{8cm}\n\\centerline{\\epsfig{file=Plot2grey.eps,width=10cm,height=6cm,angle=0}}\\\n\\hspace*{1.0cm} \\vskip -1cm\n\\end{minipage}\n\\caption{ Left panel: the $q_t$ dependence of SSA for different\nvalues of the infrared cut-off in the gluon propagator\n\\cite{nikolaev}, \\cite{RuizArriola:2004en}, \\cite{aguilar}. Right\npanel: the $q_t$ dependence of SSA for the different values of\nthe dynamical quark mass \\cite{shuryak}, \\cite{diakonov},\n\\cite{kochelev2}.}\n\\end{figure}\n In Fig.3, the result for $A_N$ as the function of transfer momentum transfer is presented\n for different values of the dynamical quark mass $M_q$ and the parameter infrared cut-off\n $m_g$.\nOur results show that SSA $A_N$ induced by AQCM is very large and\npractically independent of particular values of $M_q$ and $m_g$.\nWe would like to stress also that $A_N$ in our approach does not\ndepend on c.m. energy. The energy independence of SSA is in\nagreement with experimental data and in contradiction with naive\nexpectation that spin effects in strong interaction should vanish\nat high energy \\cite{Krisch:2010hr}.\n One can show that this property is directly related\nto the spin one t-channel gluon exchange. Another remarkable\nfeature of our approach is a flat dependence of SSA\n on transverse momentum of a final particle, Fig.3.\nIt comes from a rather soft power-like form factor in the\ngluon-quark vertex, Eq.\\ref{ffg}, and a small average size of\ninstanton, $\\rho_c\\approx 1\/3 fm$, in QCD vacuum \\cite{shuryak}.\nSuch a flat dependence has recently been observed by the STAR\ncollaboration in the inclusive $\\pi^0$ production in high energy\nproton-proton collision \\cite{STAR} and was not expected in the\nmodels based on TMD factorization and {\\it ad hoc} parametrization\nof Sivers and Collins functions \\cite{reviews}. Finally, the sign\nof the SSA is defined by the sign of AQCM and should be positive,\nEq.\\ref{SSA}. This sign is very important in explaining of the\nsigns of SSA observed for inclusive production of $\\pi^+,\\pi^-$\nand $\\pi^0 $ mesons in proton-proton and proton-antiproton high\nenergy collisions (see discussion and references in\n\\cite{reviews}, \\cite{Krisch:2010hr}).\n\n\nIt is evident that the instanton induced helicity-flip should also\ngive the contribution to SSA in the meson production in\nsemi-inclusive deep inelastic scattering (SIDIS) where large SSA\nin $\\pi$- and $K$-meson production was observed by HERMES\n\\cite{Airapetian:2010ds} and by COMPASS Collaborations\n\\cite{Martin:2013eja}. In the leading order in the instanton\ndensity the nonzero contribution to SSA in SIDIS is expected to\ncome from the interference of diagrams presented in Fig.4. Here,\nthe imaginary part arises from final state perturbative and\nnonperturbative interactions of the current quark with the\nspectator system. The real part of the amplitude presented by two\nfirst diagrams includes perturbative helicity-conserved\nphoton-quark vertex and the instanton induced helicity-flip\nvertex. The Pauli form factor corresponding to the last vertex was\ncalculated in \\cite{Kochelev:2003cp}.\n\n\\begin{figure}[htb]\n\\centering\n \\centerline{\\epsfig{file=SIDIS.eps,width=16cm,height=3cm,angle=0}}\n \\caption{The leading contributions to SSA in SIDIS.}\n\n \\end{figure}\n\n We should emphasize\nthe significant difference between our approach to SSA in SIDIS\nand perturbative final state interaction model presented in\n\\cite{Brodsky:2002cx}. In particular, one can expect that the\nmain contribution comes from the kinematical region where the\nvirtuality of gluon in Fig.4 is small. Therefore, soft gluon\ninteraction with quarks should be highly nonperturbative.\nFurthermore, the helicity flip in \\cite{Brodsky:2002cx} is related\nto the wave function of the nucleon. Due to that, SSA coming from\nthis mechanism, might be significant only in the region of small\ntransverse momentum of the final meson $k_t\\approx\n\\Lambda_{QCD}\\approx 250$ MeV. In our approach, we expect the\nlarge SSA at higher transverse momentum because the averaged\ninstanton size is much smaller than the confinement size\n$\\rho_c\\approx R_{conf}\/3$. This qualitative observation\ncorresponds to the experimental data presented by HERMES and\nCOMPASS where large SSA was observed only at rather large $k_t$.\nAdditionally, a significant $Q^2$ dependence of SSA found by\nCOMPASS Collaboration \\cite{Martin:2013eja} might be related to\nthe strong $Q^2$ dependence of the nonperturbative photon-quark\nvertex presented by second diagram in Fig.4.\n\n\n\n The additional contribution to SSA induced by instantons was\n suggested in the papers \\cite{Ostrovsky:2004pd} and \\cite{Qian:2011ya}.\nIt is based on the results from \\cite{Moch:1996bs},\n where the effects of instantons in\nthe nonpolarized deep inelastic scattering process were\ncalculated in a careful way \\footnote{This approach was applied to\nthe Drell-Yan process \\cite{Brandenburg:2006xu} as well.}. In\nthis case, the effect arises from phase shift in the quark\npropagator in the instanton field. This contribution might be\nconsidered as complementary to the AQCM effect.\n\n\nIn spite of the fact that our estimation is based mainly on\nsingle-instanton approximation (SIA) for AQCM \\cite{kochelev1},\nthe effects of the multiinstantons, which are hidden in the value\nof dynamical quark mass in Eq.\\ref{AQCM}, are also taken into\naccount in the effective way. The accuracy of such SIA was\ndiscussed in various aspects in \\cite{Faccioli:2001ug}. By\nanalyzing of several correlation functions the authors claimed\nthat dynamical quark mass can be different from the MFA value\n$M_q=170$ MeV. However, as it was discussed above, SSA induced by\nAQCM has rather a weak dependence on the value of dynamical mass,\nFig.3. Therefore, we believe that some effects beyond SIA can not\nlead to a significant change of our results.\n Furthermore, we would like to mention that the SSA\nmechanism based on AQCM is quite general and might happen in any\nnonperturbative QCD model with the spontaneous chiral symmetry\nbreaking. The attractive feature of the instanton liquid model is\nthat within this model this phenomenon comes from rather small\ndistances $\\rho_c\\approx 0.3$ fm. As the result, it allows to\nunderstand the origin of large observed SSA at large\ntransverse momentum.\n\n\n\n\nIn summary, we calculated the SSA in the quark-quark scattering\ninduced by AQCM and found that it was large. This phenomenon is\nrelated to the strong helicity-flip quark-gluon interaction\ninduced by the topologically nontrivial configuration of vacuum\ngluon fields called instantons. Our estimation shows that the\nsuggested mechanism can be responsible for anomalously large SSA\nobserved in different reactions at high energies. We would like to\nstress that quark-gluon and quark-photon nonperturbative\ninteractions violate the TMD factorization in inclusive meson\nproduction in both hadron-hadron and deep inelastic scatterings.\nTherefore, it cannot be treated as some additional contribution to\nthe Sivers distribution function or to the Collins fragmentation\nfunction. It is evident that the nonfactorizable mechanism for SSA\nbased on AQCM can be extended to other spin-dependent observables,\nincluding double-spin asymmetries in inclusive and exclusive\nreactions.\n\n\n\n\n\\section*{Acknowledgments}\nThe authors are very grateful to I.O. Cherednikov, A.E. Dorokhov,\nA.V. Efremov and E.~A.~Kuraev for discussion. The work of N.K. was\nsupported in part by Belarus-JINR grant, a visiting scientist\ngrant from the University of Valencia and by the MEST of the\nKorean Government (Brain Pool Program No. 121S-1-3-0318). We also\nacknowledge that this work was initiated through the series of\nAPCTP-BLTP JINR Joint Workshops.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzmwlb b/data_all_eng_slimpj/shuffled/split2/finalzzmwlb new file mode 100644 index 0000000000000000000000000000000000000000..2f42263c8f4b27f1835a189280070e6ab2194924 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzmwlb @@ -0,0 +1,5 @@ +{"text":"\\section{History}\n\\noindent\nThe \\emph{Julia set} $J(f)$ of $f$ is the closure of the set of repelling periodic points. It is also the smallest closed set containing at least three points which is completely invariant under $f^{-1}$. For the example $f(z)=z^2$, the Julia set of $f$ is the unit circle. The complement $F(f)=\\widehat\\C\\setminus J(f)$ of the Julia set, called the \\emph{Fatou set}, is the largest open set such that the iterates of $f$ restricted to it form a normal family. The Julia set and Fatou set are both invariant under $f$ and $f^{-1}$.\n\nThe \\emph{postcritical set} $\\post(f)$ of $f$ is the closure of the forward orbits of the critical points\n\\begin{eqnarray*}\n\\post(f)=\\overline{\\bigcup_{n\\geq 1}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}}.\n\\end{eqnarray*}\nThe postcritical set plays a crucial role in terms of understanding the expanding and contracting features of a rational map.\nIf the postcritical set $\\post(f)$ is finite, we say that the map $f$ is \\emph{postcritically finite}. In the postcritically finite case,\n\\begin{eqnarray*}\n\\post(f)=\\bigcup_{n\\geq 1}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}.\n\\end{eqnarray*}\n\nIn 1918, Samuel Latt\\`es described a special class of rational maps which have a simultaneous linearization for all of their periodic points (see \\cite{LatSur}). This class of maps is named after Latt\\`es, even though similar examples had been studied by Ernst Schr\\\"oder much earlier (see \\cite{SchUeber}). A \\emph{Latt\\`es map} $f\\:\\widehat\\C\\ra \\widehat\\C$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism, i.e., the map $f$ satisfies the following commutative diagram:\n\\begin{equation}\\label{lat}\n \\begin{CD}\n\\T @>\\bar{A}>> \\T\\\\\n@V\\Theta VV @VV\\Theta V\\\\\n\\widehat\\C @>f>> \\widehat\\C\n\\end{CD}\n\\end{equation}\nwhere $\\bar A$ is a map of a torus $\\T$ that is a quotient of an affine map of the complex plane, and $\\Theta$ is a finite-to-one holomorphic map. Latt\\`es maps were the first examples of rational maps whose Julia set is the whole sphere $\\widehat \\C$, and the postcritical set of a Latt\\`es map is finite. More importantly, Latt\\`es maps play a central role as exceptional examples in complex dynamics. We will discuss this further in the following section.\n}\n\n\\section{Introduction}\n\\noindent\nA \\emph{rational map} $f\\:\\widehat\\C \\ra \\widehat\\C$ is a map on the Riemann sphere $\\widehat\\C=\\C \\cup \\{\\infty\\}$ which can be written as a quotient of two relatively prime complex polynomials $p(z)$ and $q(z)$, with $q(z)\\not=0$,\n\\begin{eqnarray}\\label{rationalmap}\nf(z)=\\frac{p(z)}{q(z)}=\n\\frac{a_0z^{m}+\\ldots+a_{m}}{b_0z^l+\\ldots+b_l},\n\\end{eqnarray}\nwhere $a_i,b_j \\in \\C$ for $i=0,\\ldots,m$ and $j=0,\\ldots, l$.\nThe \\emph{postcritical set} $\\post(f)$ of $f$ is defined to be the forward orbits of the critical points\n\\begin{eqnarray*}\n\\post(f)=\\bigcup_{n\\geq 1}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}.\n\\end{eqnarray*}\nIf the postcritical set $\\post(f)$ is finite, we say that the map $f$ is \\emph{postcritically finite}.\n\nThurston introduced a topological analog of a postcritically finite rational map, now known as a \\emph{Thurston map} (see \\cite{DHThurston}). A \\emph{Thurston map} $f\\:\\S^2\\ra \\S^2$ is a branched covering map with finite postcritical set $\\post(f)$.\nThe notion of an expanding Thurston map was introduced in \\cite{BMExpanding} as a topological analog of a postcritically finite rational map whose Julia set is the whole sphere $\\widehat{\\C}$. Roughly speaking, a Thurston map is called \\emph{expanding} if all the connected components of the preimage under $f^{-n}$ of any open Jordan region disjoint from $\\post(f)$ become uniformly small as $n$ tends to infinity. We refer the reader to Definition~\\ref{expandingmap} for a more precise statement. A related and more general notion of expanding Thurston maps was introduced in \\cite{HPCoarse}. Latt\\`es maps are among the simplest examples of expanding Thurston maps.\n\n\nLet $f$ be an expanding Thurston map, and let $\\mathcal C$ be a Jordan curve containing $\\post(f)$.\nThe Jordan Curve Theorem implies that $\\S^2\\setminus\\mathcal C$ has precisely two connected components, whose closures we call \\emph{$0$-tiles}. We call the closure of each connected component of the preimage of $\\S^2\\setminus\\mathcal C$ under $f^n$ an \\emph{$n$-tile}. In Section 5 of \\cite{BMExpanding}, it is proved that the collection of all $n$-tiles gives a cell decomposition of $\\S^2$.\n\nEvery expanding Thurston map $f\\:\\S^2\\ra \\S^2$ induces a natural class of metrics on $\\S^2$, called \\emph{visual metrics} (see Definition \\ref{visual}), and each visual metric $d$ has an associated \\emph{expansion factor} $\\Lambda > 1$. This visual metric is essentially characterized by the geometric property that the diameter of an $n$-tile is about $\\Lambda^{-n}$, and the distance between two disjoint $n$-tiles is at least about $\\Lambda^{-n}$. The supremum of the expansion factors of all visual metrics is called the \\emph{combinatorial expansion factor} $\\Lambda_0$ (see \\cite[Theorem 1.5]{BMExpanding}). For Latt\\`es maps, the supremum is obtained. In general, the supremum is not obtained.\n\n\\excise{\n--------------------------\nThe points in $\\post(f)$ divide $\\mathcal C$ into several subarcs. Let $D_n=D_n(f,\\mathcal{C})$ be the minimum number of $n$-tiles needed to join two of these subarcs that are non-adjacent (see Definition \\ref{joinoppositesides} and \\eqref{defdn}). Even though $D_n=D_n(f,\\mathcal C)$ depends on the Jordan curve $\\mathcal C$, its growth rate is independent of the $\\mathcal C$. So the limit\n\\begin{equation}\\label{comb}\n \\Lambda_0(f)=\\lim_{n\\ra \\infty}\\big(D_n(f,\\mathcal C)\\big)^{1\/n}\n\\end{equation}\nexists and only depends on the map $f$ itself (see \\cite[Prop.~17.1]{BMExpanding}). We call this limit $\\Lambda_0(f)$ the \\emph{combinatorial expansion factor} of $f$. This quantity $\\Lambda_0(f)$ is invariant under topological conjugacy and multiplicative in the sense that $\\Lambda_0(f)^n$ is the combinatorial expansion factor of $f^n$.\n\nThe combinatorial expansion factor is closely related to the notion of \\emph{visual metrics and their expansion factors}. Every expanding Thurston map $f\\:\\S^2\\ra \\S^2$ induces a natural class of metrics on $\\S^2$, called \\emph{visual metrics} (see Definition \\ref{visual}), and each visual metric $d$ has an associated \\emph{expansion factor} $\\Lambda > 1$. This visual metric is essentially characterized by the geometric property that the diameter of an $n$-tile is about $\\Lambda^{-n}$, and the distance between two disjoint $n$-tiles is at least about $\\Lambda^{-n}$. The supremum of the expansion factors of all visual metrics is equal to the combinatorial expansion factor $\\Lambda_0$ (see \\cite[Theorem 1.5]{BMExpanding}). For Latt\\`es maps, the supremum is obtained. In general, the supremum is not obtained.\n----------------------------\n}\n\\bigskip\n\nA geodesic metric space $(X,d)$ is called a \\emph{Gromov hyperbolic} space if every geodesic triangle in it is ``very thin''. It can also defined in terms of \\emph{Gromov products}.\nFor any points $x,y,p\\in X$, the \\emph{Gromov product} $(x,y)_p$ of $x$ and $y$ with respect to the base point $p$ is defined as\n\\begin{eqnarray} \\label{gproduct}\n (x,y)_p \\:= \\frac12 \\left[d(x,p)+d(y,p)-d(x,y) \\right].\n\\end{eqnarray}\nThe space $X$ is called \\emph{$\\delta$-hyperbolic} (or \\emph{Gromov hyperbolic}) for some $\\delta\\geq 0$ if there exists a base point $p\\in X$ such that for all $x,y,z\\in X$, we have\n\\begin{eqnarray} \\label{trianglein}\n (x,y)_p\\geq \\min\\{(x,z)_p,(z,y)_p\\}-\\delta.\n\\end{eqnarray}\n\nWe construct a graph $\\G=\\G(f,\\mathcal C)$ by letting the tiles in the cell decompositions of $(f, \\mathcal C)$ be vertices of $\\G$.\nThere is an edge between the two vertices $X^n,Y^m\\in V$, denoted\n$X^n\\sim Y^m$ if as underlying tiles\n\\[|n-m|\\leq 1 \\mbox{ and } X^n\\cap Y^m\\not= \\emptyset.\\]\nIt turns out that the graph $\\G$ with the path metric is a Gromov hyperbolic space (see Theorem \\ref{gh}).\n\\begin{thm}\nLet $f\\: \\S^2 \\ra \\S^2$ be an expanding Thurston map\nand let $\\mathcal C \\subset \\S^2$ be a Jordan curve containing $\\post(f)$. Then the graph $\\G(f,\\mathcal C)$ equipped with the path metric $\\eta$ is a Gromov hyperbolic space.\n\\end{thm}\n\n\n\nThere is a natural boundary at infinity of a Gromov hyperbolic space. Roughly speaking, the \\emph{boundary at infinity} is the set of equivalence classes of geodesic rays in the Gromov hyperbolic space. It can also be equipped with a \\emph{Gromov product} by taking infimum of the infimum limit of the Gromov product along all the geodesic rays among the corresponding equivalence classes.\n A \\emph{visual metric} $\\rho$ on the boundary at infinity of a Gromov hyperbolic space is a metric that has a bounded ratio\n\\[ \\rho(\\xi,\\xi')\/ \\Lambda^{-(\\xi,\\xi')_p}\\]\nfor some fixed $\\Lambda>1$ and for all points $\\xi$ and $\\xi'$ on the boundary.\n\n\nIn Proposition \\ref{samevisual}, we show the following:\n\\begin{pro}\nFor an expanding Thurston map $f$ and a Jordan curve $\\mathcal C\\subset \\S^2$ containing $\\post (f)$, the boundary at infinity $\\partial_{\\infty}\\G$ of the graph tile $\\G(f,\\mathcal C)$ can be identified with $\\S^2$.\nUnder this identification, a metric $d$ is a visual metric on $\\S^2$ with respect\nto the expanding Thurston map $f$ if and only if $d$ is a visual metric on $\\partial_{\\infty}\\G$ (in the sense of Gromov hyperbolic spaces).\n\\end{pro}\n\nWe deduce that\nfor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, the classes of visual metrics on $\\partial_{\\infty}\\G(f,\\mathcal C)$ and $\\partial_{\\infty}\\G(f,\\mathcal C')$ can also be identified (see Corollary \\ref{corsamevisual}). A similar graph to $\\G(f,\\mathcal C)$ has also been studied by Kevin Pilgrim in \\cite{PilJulia}, from a somewhat different point of view. Our results overlap in some special cases. They consider the map $f$ being $C^1$ and $\\S^2\\setminus \\post{f}$ equipped with a special Riemannian metric, and prove that the Julia set of $f$ can be identified as the Gromov boundary of a certain Gromov hyperbolic one-complex.\n\n\nIn \\cite{BFAsymptotic}, the \\emph{asymptotic upper curvature} of a Gromov hyperbolic space is introduced. It is the analog of sectional curvature on Riemannian manifolds. Fix $\\kappa\\in [-\\infty,0)$. We call a metric space $X$ an \\emph{AC$_u(\\kappa)$-space} if there exists $p\\in X$ and a constant $c\\geq 0$ such that for all $x,x'\\in X$ and all finite sequences $x_0=x,x_1,\\ldots,x_n=x'$ in $X$,\n\\begin{eqnarray}\\label{acspaceeq}\n(x,x')_p\\geq \\min_{i=1,\\ldots,n}(x_{i-1},x_i)_p-\\frac1{\\sqrt{-\\kappa}}\\log n-c.\n\\end{eqnarray}\nHere we use the convention $1\/\\sqrt{\\infty}=0$.\nWe call\n\\begin{eqnarray*}\nK_u(X):=\\inf\\{\\kappa\\ \\in [-\\infty,0): X \\mbox{ is an AC$_u(\\kappa)$-space} \\}\n\\end{eqnarray*}\nthe \\emph{asymptotic upper curvature} of $X$. It is invariant under rough-isometry.\n\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, the metric spaces $\\G=\\G(f,\\mathcal C)$ and $\\G'=\\G(f,\\mathcal C')$ are rough-isometric (see Proposition~\\ref{roughgg}). Hence we may define the \\emph{asymptotic upper curvature} $K_u(f)$ of an expanding Thurston map $f$ as\n\\begin{eqnarray} \\label{asyf}\nK_u(f):= K_u(\\G(f,\\mathcal C)),\n\\end{eqnarray}\nwhere $\\mathcal C \\subseteq \\S^2$ is any Jordan curve containing $\\post(f)$. Using the notation above, we have the following theorem (see Theorem~\\ref{main2}).\n\n\\begin{thm} \\label{main3}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map. The asymptotic upper curvature of $f$ satisfies\n\\[K_u(f)\\geq-\\frac14\\log^2(\\deg f). \\]\nIf in addition, the map $f$ has no periodic critical points, then the tile graph $\\G=\\G(f)$ is an AC$_u(\\kappa)$-space with\n\\[ \\kappa= -\\frac14\\log^2(\\deg f),\\]\nif and only if the map $f$ is topologically conjugate to a Latt\\`es map.\n\\end{thm}\n\nRecall that in \\cite{HamEntropy}, the Hamenst{\\\"a}dt's entropy rigidity theorem establishes a connection between the curvature of a compact manifold $M$ and the topological entropy of the geodesic flow on the tangent space of $M$. Corollary 20.8 in \\cite{BMExpanding} shows that the topological entropy of an expanding Thurston map $f$ is $\\log(\\deg(f))$. Hence\nTheorem \\ref{main3} establishes a connection between the asymptotic upper curvature of an expanding Thurston map $f$ and the topological entropy of $f$, and provides a counterpart to Hamenst{\\\"a}dt's entropy rigidity theorem in the Sullivan dictionary.\n\n\\bigskip\n\\noindent\n\\textbf{Acknowledgements.} This paper is part of the author's PhD thesis under the supervision of Mario Bonk. The author would like to thank Mario Bonk for introducing her to and teaching her about the subject of Thurston maps and its related fields. The author is inspired by his enthusiasm and mathematical wisdom, and is especially grateful for his patience and encouragement.\nThe author would like to thank Dennis Sullivan for valuable conversations and sharing his mathematical insights.\nThe author also would like to thank Michael Zieve and Alan Stapledon for useful comments and feedback.\n\n\n\n\\section{Expanding Thurston maps and Cell Decompositions} \\label{expanding}\n\\noindent\nIn this section we review some definitions and facts on expanding Thurston maps. We refer the reader to Section 3 in \\cite{BMExpanding} for more details. We write $\\N$ for the set of positive integers, and $\\N_0$ for the set of non-negative integers. We denote the identity map on $\\S^2$ by ${\\rm id}_{\\S^2}$.\n\nLet $\\S^2$ be a topological 2-sphere with a fixed orientation. A continuous map $f\\:\\S^2\\ra \\S^2$ is called \\emph{a branched covering map} over $\\S^2$ if $f$ can be locally written as\n\\[z\\mapsto z^d\\]\nunder certain orientation-preserving coordinate changes of the domain and range. More precisely, we require that for any point $p\\in \\S^2$, there exists some integer $d>0$, an open neighborhood $U_p\\subseteq \\S^2$ of $p$, an open neighborhood $V_q\\subseteq \\S^2$ of $q=f(p)$, and orientation-preserving homeomorphism\n\\[\\phi\\: U_p\\ra U\\subseteq \\C\\]\nand\n\\[\\psi \\: V_p\\ra V\\subseteq \\C\\]\nwith $\\phi(p)=0$ and $\\psi(q)=0$ such that\n\\[(\\psi\\circ f \\circ \\phi^{-1} )(z)=z^d\\]\nfor all $z\\in U$. The positive integer $d=\\deg_f(p)$ is called the \\emph{local degree} of $f$ at $p$ and only depends on $f$ and $p$. A point $p\\in \\S^2$ is called a \\emph{critical point} of $f$ if $\\deg_f(p)\\geq 2$, and a point $q$ is called \\emph{critical value} of $f$ if there is a critical point in its preimage $f^{-1}(q)$. If $f$ is a branched covering map of $\\S^2$, $f$ is open and surjective. There are only finitely many critical points of $f$ and $f$ is \\emph{finite-to-one} due to the compactness of $\\S^2$. Hence, $f$ is a covering map away from the critical points in the domain and critical values in the range. The \\emph{degree $\\deg(f)$} of $f$ is the cardinality of the preimage over a non-critical value. In addition, we have\n\\[\\deg(f)=\\sum_{p\\in f^{-1}(q)}\\deg_f(p)\\]for every $q\\in \\S^2$.\n\nFor $n\\in \\N$, we denote the $n$-th iterate of $f$ as\n\\[f^n=\\underbrace{f\\circ f\\circ \\cdots \\circ f}_{\\textstyle{n} \\mbox{ factors}}.\\]\nWe also set $f^0={\\rm id}_{\\S^2}$.\n\nIf $f$ is a branched cover of $\\S^2$, so is $f^n$, and\n\\[\\deg(f^n)=\\deg(f)^n.\\]Let crit$(f)$ be the set of all the critical points of $f$. We define the set of \\emph{postcritical points} of $f$ as\n\\[\\post(f)=\\bigcup_{n\\in \\N}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}.\\]We call a map $f$ \\emph{postcritically-finite} if the cardinality of $\\post(f)$ is finite. Notice that $f$ is postcritically-finite if and only if there is some $n\\in \\N$ for which $f^n$ is postcritically-finite.\n\nLet $\\mathcal{C}\\subseteq \\S^2$ be a Jordan curve containing $\\post(f)$. We fix a metric $d$ on $\\S^2$ that induces the standard metric topology on $\\S^2$.\nDenote by \\emph{${\\rm mesh}(f,n,{\\mathcal C})$} the supremum of the diameters of all connected components of the set $f^{-n}(\\S^2\\setminus {\\mathcal C})$.\n\n\\begin{de} \\label{expandingmap}\nA branched covering map $f\\:\\S^2\\ra \\S^2$ is called a \\emph{Thurston map} if $\\deg(f)\\geq 2$ and $f$ is postcritically-finite. A Thurston map $f\\:\\S^2\\ra \\S^2$ is called \\emph{expanding} if there exists a Jordan curve $\\mathcal{C}\\subseteq \\S^2$ with $\\mathcal{C} \\supseteq \\post(f)$ and\n\\begin{equation} \\label{mesh}\n\\lim_{n\\ra \\infty}{\\rm mesh}(f,n,{\\mathcal C})=0.\n\\end{equation}\n\\end{de}\n\nThe relation \\eqref{mesh} is a topological property, as it is independent of the choice of the metric, as long as the metric induces the standard topology on $\\S^2$. Lemma 8.1 in \\cite{BMExpanding} shows that if the relation \\eqref{mesh} is satisfied for one Jordan curve ${\\mathcal C}$ containing $\\post(f)$, then it holds for every such curve. One can essentially show that a Thurston map is expanding if and only if all the connected components in the preimage under $f^{-n}$ of any open Jordan region not containing $\\post(f)$ become uniformly small as $n$ goes to infinity.\n\nThe following theorem (Theorem 1.2 in \\cite{BMExpanding}) says that there exists an invariant Jordan curve for some iterates of $f$.\n\\begin{thm} \\label{invariantJordancurve}\nIf $f \\: \\S^2\\ra \\S^2$ is an expanding Thurston map, then for some $n\\in \\N$ there exists a Jordan curve $\\mathcal{C}\\subseteq \\S^2$ containing $\\post(f)$ such that $\\mathcal C$ is invariant under $f^n$, i.e., $f^n({\\mathcal C})\\subseteq {\\mathcal C}$.\n\\end{thm}\n\nRecall that an \\emph{isotopy} $H$ between two homeomorphisms is a homotopy so that at each time $t\\in [0,1]$, the map $H_t$ is a homeomorphism. An \\emph{isotopy $H$ relative to a set $A$} is an isotopy satisfying\n\\[H_t(a)=H_0(a)=H_1(a)\\]\nfor all $a\\in A$ and $t\\in [0,1]$.\n\n\n\\begin{de}\nConsider two Thurston maps $f\\:\\S^2\\ra \\S^2$ and $g\\:\\S^2_1\\ra \\S^2_1$, where $\\S^2$ and $\\S^2_1$ are $2$-spheres. We call the maps $f$ and $g$ \\emph{(Thurston) equivalent} if there exist homeomorphisms $h_0,h_1\\:\\S^2\\ra \\S^2_1$ that are isotopic relative to $\\post(f)$ such that $h_0\\circ f=g\\circ h_1$.\nWe call the maps $f$ and $g$ \\emph{topologically conjugate} if there exists a homeomorphism $h\\:\\S^2\\ra \\S^2_1$ such that $h\\circ f=g\\circ h$.\n\\end{de}\nFor equivalent Thurston maps, we have the following commutative diagram\n\\[\\begin{CD}\n\\S^2 @>h_1>>\\S^2_1 \\\\\n@Vf VV @VVg V\\\\\n\\S^2 @>h_0>> \\S^2_1 .\n\\end{CD} \\]\n\n\n\nWe now consider the cardinality of the postcritical set of $f$. In Remark 5.5 in \\cite{BMExpanding}, it is proved that there are no Thurston maps with $\\#\\post(f)\\leq 1$. Proposition 6.2 in \\cite{BMExpanding} shows that all Thurston maps with $\\#\\post(f)=2$ are Thurston equivalent to a \\emph{power map} on the Riemann sphere,\n\\[z\\mapsto z^k, \\mbox{ for some }k\\in \\Z\\setminus\\{-1,0,1\\}.\\]\nCorollary 6.3 in \\cite{BMExpanding} states that if $f\\:\\S^2\\ra\\S^2$ is an expanding Thurston map, then $\\#\\post(f)\\geq 3$.\n\nLet $f\\:\\S^2\\ra\\S^2$ be a Thurston map, and let ${\\mathcal C}\\subseteq \\S^2$ be a Jordan curve containing $\\post(f)$. By the Sch\\\"onflies theorem, the set $\\S^2\\setminus {\\mathcal C}$ has two connected components, which are both homeomorphic to the open unit disk. Let $T_0$ and $T_0'$ denote the closures of these components. They are cells of dimension $2$, which we call \\emph{$0$-tiles}. The postcritical points of $f$ are called \\emph{$0$-vertices} of $T_0$ and $T_0'$, singletons of which are cells of dimension $0$. We call the closed arcs between vertices \\emph{$0$-edges} of $T_0$ and $T_0'$, which are cells of dimension $1$. These $0$-vertices, $0$-edges and $0$-tiles form a cell decomposition of $\\S^2$, denoted by $\\D^0=\\D^0(f,{\\mathcal C})$. We call the elements in $\\D^0$ $0$-cells. Let $\\D^1=\\D^1(f,{\\mathcal C})$ be the set of connected subsets $c\\subseteq \\S^2$ such that $f(c)$ is a cell in $\\D^0$ and $f|_c$ is a homeomorphism of $c$ onto $f(c)$. Call $c$ a $1$-tile if $f(c)$ is a $0$-tile, call $c$ a $1$-edge if $f(c)$ is a $0$-edge, and call $c$ a $1$-vertex if $f(c)$ is a $1$-vertex. Lemma 5.4 in \\cite{BMExpanding} states that $\\D^1$ is a cell decomposition of $\\S^2$. Continuing in this manner, let $\\D^n=\\D^n(f,{\\mathcal C})$ be the set of all connected subsets of $c\\subseteq \\S^2$ such that $f(c)$ is a cell in $\\D^{n-1}$ and $f|_c$ is a homeomorphism of $c$ onto $f(c)$, and call these connected subsets $n$-tiles, $n$-edges and $n$-vertices correspondingly, for $n\\in\\N_0$. By Lemma 5.4 in \\cite{BMExpanding}, $\\D^n$ is a cell decomposition of $\\S^2$, for each $n\\in \\N_0$, and we call the elements in $\\D^n$ $n$-cells. The following lemma lists some properties of these cell decompositions. For more details, we refer the reader to Proposition 6.1 in \\cite{BMExpanding}.\n\\begin{lem}\\label{tilenumber}\nLet $k,n\\in \\N_0$, let $f\\:\\S^2\\ra \\S^2$ be a Thurston map, let $\\mathcal C\\subset~ \\S^2$ be a Jordan curve with $\\mathcal C\\supseteq \\post(f)$, and let $m=\\#\\post(f)$.\n\\begin{enumerate}\n \\item\n If $\\tau$ is any $(n+k)$-cell, then $f^k(\\tau)$ is an $n$-cell, and $f^k|_{\\tau}$ is a homeomorphism of $\\tau$ onto $f^k(\\tau)$.\n \\item Let $\\sigma$ be an $n$-cell. Then $f^{-k}(\\sigma)$ is equal to the union of all $(n+k)$-cells $\\tau$ with $f^k(\\tau)=\\sigma$.\n \n \\item The number of $n$-vertices is less than or equal to $m\\deg(f)^n$, the number of $n$-edges is $m\\deg(f)^n$, and the number of $n$-tiles is $2\\deg (f)^n$.\n \\item The $n$-edges are precisely the closures of the connected components of\n $f^{-n}(\\mathcal C)\\setminus f^{-n}(\\post(f))$. The $n$-tiles are precisely the closures of the connected components of $\\S^2\\setminus f^{-n}(\\mathcal C)$.\n \\item Every $n$-tile is an $m$-gon, i.e., the number of $n$-edges and $n$-vertices contained in its boundary is equal to $m$.\n\\end{enumerate}\n\\end{lem}\n\n\\excise{\n------------------------\nLet $\\sigma$ be an $n$-cell. Let $W^n(\\sigma)$ be the union of the interiors of all $n$-cells intersecting with $\\sigma$, and call $W^n(\\sigma)$ the \\emph{$n$-flower} of $\\sigma$. In general, $W^n(\\sigma)$ is not necessarily simply connected. The following lemma (from Lemma 7.2 in \\cite{BMExpanding}) says that if $\\sigma$ consists of a single $n$-vertex, then $W^n(\\sigma)$ is simply connected.\n\\begin{lem} \\label{flower}\nLet $f\\: \\S^2\\ra \\S^2$ be a Thurston map, and let $\\mathcal C$ be a Jordan curve containing $\\post(f)$. If $\\sigma$ is an $n$-vertex, then $W^n(\\sigma)$ is simply connected. In addition, the closure of $W^n(\\sigma)$ is the union of all $n$-tiles containing the vertex $\\sigma$.\n\\end{lem}\nOne of the most important properties of $n$-flowers is that they build a connection between $n$-tiles of different Jordan curves due to the following lemma in \\cite[Lemma 7.12]{BMExpanding}.\n\\begin{lem}\\label{flowertile}\nLet $\\mathcal C$ and $\\mathcal C'$ be Jordan curves in $\\S^2$ both containing\n$\\post(f)$. Then there exists a number $M$ such that each $n$-tile for $(f, \\mathcal C)$\nis covered by $M$ $n$-flowers for $(f, \\mathcal C')$.\n\\end{lem}\n\\begin{re}\nThe exact same proof for this lemma shows that for $n'\\geq n$, there exists a number $M$ such that each $n'$-tile $(f, \\mathcal C)$ is covered by $M$ $n$-flowers for $(f, \\mathcal C')$.\n\\end{re}\n-------------------------\n}\n\nWe obtain a sequence of cell decompositions of $\\S^2$ from a Thurston map and a Jordan curve on $\\S^2$. It would be nice if the local degrees of the map $f$ at all the vertices were bounded, and this can be obtained by the assumption of no periodic critical points (see \\cite[Lemma 16.1]{BMExpanding}).\n\\begin{lem} \\label{noperiodic}\nLet $f : \\S^2\\ra \\S^2$ be a branched covering map. Then f has no\nperiodic critical points if and only if there exists $N\\in \\N$ such that\n\\[\\deg_{f^n}(p)\\leq N,\\]\nfor all $p\\in \\S^2$ and all $n\\in \\N$.\n\\end{lem}\nHenceforth we assume that \\emph{all Thurston maps have no periodic critical points}.\n\nLet $f\\:\\S^2\\ra \\S^2$ be an expanding Thurston map and let $\\mathcal{C}$ be a Jordan curve containing $\\post(f)$.\n\\begin{de}\\label{joinoppositesides}\nA set $K\\subseteq \\S^2$ \\emph{joins opposite sides} of $\\mathcal C$ if $\\#$$\\post(f)\\geq 4$\nand $K$ meets two disjoint $0$-edges, or if $\\#$$\\post(f) = 3$ and $K$ meets all\nthree $0$-edges.\n\\end{de}\n\nLet $D_n=D_n(f,\\mathcal C)$ be the minimum number of $n$-tiles needed to join opposite sides of a Jordan curve $\\mathcal{C}$. More precisely,\n\\begin{eqnarray}\\label{defdn}\nD_n =\\min\\{N\\in \\N: \\mbox{ there exist $n$-tiles } X_1, . . . ,X_N \\mbox{ such that } \\\\\n\\bigcup_{j=1}^N X_j\\mbox{ is connected and joins opposite sides of }\\mathcal C\\}. \\nonumber\n\\end{eqnarray}\nOf course, $D_n$ depends on $f$ and $\\mathcal C$.\n\n\n\nLet $f$ be an expanding Thurston map. For any two Jordan curves $\\mathcal{C}$ and $\\mathcal{C'}$ with $\\post(f)\\subset \\mathcal C,\\mathcal C'$, inequality (17.1) in \\cite{BMExpanding} states that there exists a constant $c>0$ such that for all $n>0$,\n\\[ \\frac1{c}D_n(f,\\mathcal C)\\leq D_n(f,\\mathcal C')\\leq c D_n(f,\\mathcal C).\\]\nProposition 17.1 in \\cite{BMExpanding} says that:\n\\begin{pro} \\label{expansionfactor}\nFor an expanding Thurston map $f\\: \\S^2\\ra \\S^2$, and a Jordan curve $\\mathcal C$ containing $\\post(f)$,\nthe limit\n\\[\\Lambda_0=\\Lambda_0(f):=\\lim_{n\\ra\\infty}D_n(f,\\mathcal{C})^{1\/n}\\] exists and is independent of $\\mathcal C$.\n\\end{pro}\n\nWe call $\\Lambda_0(f)$ the \\emph{combinatorial expansion factor} of $f$.\n\nProposition 17.2 in \\cite{BMExpanding} states that:\n\\begin{pro}\nIf $f\\:\\S^2\\ra \\S^2$ and $g\\:\\S^2_1\\ra \\S^2_1$ are expanding Thurston maps that are topologically conjugate, then $\\Lambda_0(f)=\\Lambda_0(g)$.\n\\end{pro}\n\n\n\n\\begin{de}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map, and let\n${\\mathcal C}\\subseteq \\S^2$ be a Jordan curve containing $\\post(f) $. Let $x, y \\in \\S^2$.\nFor $x \\not= y$ we define\n\\begin{eqnarray*}\nm_{f,\\mathcal C}(x, y) = \\min\\{n\\in \\N_0 :\\mbox{ there exist disjoint $n$-tiles }X \\mbox{ and } Y \\\\\n\\mbox{ for }(f, \\mathcal C) \\mbox{ with } x\\in X \\mbox{ and } y\\in Y \\}.\n\\end{eqnarray*}\nIf $x = y$, we define $m_{f,\\mathcal C}(x, x)= \\infty$.\n\\end{de}\n\nThe minimum in the definition above is always obtained since the\ndiameters of $n$-tiles go to $0$ as $n\\ra \\infty$. We usually drop one or both\nsubscripts in $m_{f,\\mathcal C}(x, y)$ if $f$ or $\\mathcal C$ is clear from the context. If\nwe define for $x,y\\in \\S^2$ and $x \\not= y$,\n\\begin{eqnarray*}\nm'_{f,\\mathcal C}(x, y) = \\max\\{n\\in \\N_0 : \\mbox{ there exist nondisjoint $n$-tiles }X \\mbox{ and } Y \\\\\n\\mbox{ for } (f, \\mathcal C) \\mbox{ with } x\\in X \\mbox{ and } y\\in Y \\},\n\\end{eqnarray*}\nthen $m_{f,\\mathcal C}$ and $m'_{f,\\mathcal C}$ are essentially the same up to a constant (see Lemma 8.6 (v) in \\cite{BMExpanding}).\n\\begin{lem} \\label{twom}\nLet $m_{f,\\mathcal C}$ and $m'_{f,\\mathcal C}$ as defined above. There exists a constant $k>0$, such that for any $x,y\\in \\S^2$ and $x\\not=y$,\n\\[ m_{f,\\mathcal C}(x,y)-k\\leq m'_{f,\\mathcal C}(x,y)\\leq m_{f,\\mathcal C}(x,y)+1.\\]\n\\end{lem}\n\n\\begin{de}\\label{visual}\nLet $f \\: \\S^2\\ra \\S^2$ be an expanding Thurston map and\n$d$ be a metric on $\\S^2$. The metric $d$ is called a \\emph{visual metric} for $f$ if there\nexists a Jordan curve $\\mathcal C\\subseteq \\S^2$ containing $\\post(f) $, constants $\\Lambda > 1$ and $C \\geq 1$ such that\n\\[\\frac1{C}\\Lambda^{-m_{f,\\mathcal C}(x, y)} \\leq d(x, y) \\leq C\\Lambda^{-m_{f,\\mathcal C}(x, y)}\\]\nfor all $x, y \\in \\S^2$.\n\\end{de}\n\nProposition 8.9 in \\cite{BMExpanding} states that for any expanding Thurston map $f\\:\\S^2\\ra \\S^2$, there exists a visual metric for $f$, which induces the standard topology on $\\S^2$. Lemma 8.10 in the same paper gives the following characterization of visual metrics.\n\\begin{lem} \\label{charvisual}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map. Let $\\mathcal C\\subseteq \\S^2$\nbe a Jordan curve containing $\\post(f)$, and $d$ be a visual metric for $f$ with\nexpansion factor $\\Lambda > 1$. Then there exists a constant $C > 1$ such that\n\\begin{enumerate}\n \\item $d(\\sigma,\\tau)\\geq (1\/C)\\Lambda^{-n}$ whenever $\\sigma$ and $\\tau$ are disjoint $n$-cells,\n \\item $(1\/C)\\Lambda^{-n}\\leq \\diam (\\tau)\\leq C\\Lambda^{-n}$ for $\\tau$ as any $n$-edge or $n$-tile.\n\\end{enumerate}\nConversely, if $d$ is a metric on $\\S^2$ satisfying conditions $(1)$ and $(2)$\nfor some constant $C>1$, then $d$ is a visual metric with expansion\nfactor $\\Lambda > 1$.\n\\end{lem}\n\n\n\\section{Gromov Hyperbolic Spaces}\n\\noindent\nIn this section, we review the definitions of Gromov hyperbolic spaces and the asymptotic upper curvature for Gromov hyperbolic spaces.\n\nLet us first review some basic facts about Gromov hyperbolic spaces. We refer the reader to \\cite{BSElements} as a general source on Gromov hyperbolic spaces. Let $(X,d)$ be a geodesic metric space. For any points $x,y,p\\in X$, the \\emph{Gromov product} $(x,y)_p$ of $x$ and $y$ with respect to base point $p$ is defined as\n\\begin{eqnarray} \\label{gproduct}\n (x,y)_p := \\frac12 \\left[d(x,p)+d(y,p)-d(x,y) \\right].\n\\end{eqnarray}\nThe space $X$ is called \\emph{$\\delta$-hyperbolic} (or Gromov hyperbolic) for some $\\delta\\geq 0$ if there exists a base point $p\\in X$, such that for all $x,y,z\\in X$ we have\n\\begin{eqnarray} \\label{trianglein}\n (x,y)_p\\geq \\min\\{(x,z)_p,(z,y)_p\\}-\\delta.\n\\end{eqnarray}\nIf this inequality holds for some base point $p\\in X$, then it also holds for any other $p'\\in X$ with $\\delta$ being replaced by $2\\delta$.\n\nLet $(X,d)$ be a Gromov hyperbolic metric space with a fixed base point $p\\in X$. A sequence of points $\\{x_i\\}\\subseteq X$ \\emph{converges to infinity} if\n\\begin{eqnarray*}\n\\lim_{i,j\\ra\\infty} (x_i,x_j)_p=\\infty.\n\\end{eqnarray*}\nThis property of a sequence $\\{x_i\\}$ does not depend on the base point $p\\in X$. We say two sequences converging to infinity $\\{x_i\\}$ and $\\{x_i'\\}$ are \\emph{equivalent} if\n\\begin{eqnarray*}\n\\lim_{i\\ra\\infty} (x_i,x_i')_p=\\infty.\n\\end{eqnarray*}\nThe \\emph{boundary at infinity} $\\partial_{\\infty} X$ of $X$ is defined to be the set of equivalence classes of sequences of points converging to infinity. One can also define the \\emph{Gromov product} for points $\\xi,\\xi'\\in \\partial_{\\infty}X$ and $p\\in X$ as\n\\begin{eqnarray*}\n(\\xi,\\xi')_p:=\\inf \\liminf_{i\\ra\\infty}(x_i,x_i')_p\n\\end{eqnarray*}\nwhere the infimum is taken over all sequences $\\{x_i\\}\\in \\xi$ and $\\{x_i'\\}\\in \\xi'$. Here $(\\xi,\\xi')_p=\\infty$ if and only if $\\xi=\\xi'$.\n\nA metric $\\rho$ on the boundary at infinity $\\partial_{\\infty} X$ of a Gromov hyperbolic space $X$ is called \\emph{visual} if there exist $p\\in X$, $\\Lambda>1$ and $k\\geq 1$ such that for all $\\xi,\\xi'\\in \\partial_{\\infty}X$, we have that\n\\begin{eqnarray} \\label{visualg}\n\\frac1{k}\\Lambda^{-(\\xi,\\xi')_p}\\leq \\rho(\\xi,\\xi') \\leq k \\Lambda^{-(\\xi,\\xi')_p}.\n\\end{eqnarray}\nWe call the constant $\\Lambda$ in this inequality the \\emph{expansion factor} of the visual metric $\\rho$.\nRecall that we also defined a visual metric for an expanding Thurston map (see Definition 2.11).\nWhen it is not clear from context, we will refer to the visual metric defined in \\eqref{visualg} as a `visual metric in the Gromov hyperbolic sense'.\n\n\nGiven two metric spaces $(X,d_X)$ and $(Y,d_Y)$, a map $f\\:X\\ra Y $ is called a \\emph{quasi-isometry} if there are constants $\\lambda\\geq 1$ and $k\\geq 0$ such that for all $x,x'\\in X$\n\\[\\frac1{\\lambda}d_X(x,x')-k\\leq d_Y(f(x),f(x'))\\leq \\lambda d_X(x,x')+k \\] and for all $y\\in Y$,\n\\[\\inf_{x\\in X} d_Y(f(x),y)\\leq k. \\]\nIf $\\lambda=1$, we call the map $f$ a \\emph{rough-isometry}. We say that the spaces $X$ and $Y$ are \\emph{quasi-isometric (rough-isometric)} if there is a quasi-isometry (rough-isometry) between them.\n\n\nIn \\cite{BFAsymptotic}, Bonk and Foertsch introduced the notion of upper curvature bounds for Gromov hyperbolic spaces up to rough-isometry (see \\cite[Definition 1.1 and 1.2]{BFAsymptotic}).\n\\begin{de}\nLet $\\kappa\\in [-\\infty,0)$. We call a metric space $X$ an \\emph{AC$_u(\\kappa)$-space} if there exists $p\\in X$ and a constant $c\\geq 0$ such that for all $x,x'\\in X$ and all finite sequences $x_0=x,x_1,\\ldots,x_n=x'$ in $X$ with $n>0$,\n\\begin{eqnarray}\\label{acspaceeq}\n(x,x')_p\\geq \\min_{i=1,\\ldots,n}(x_{i-1},x_i)_p-\\frac1{\\sqrt{-\\kappa}}\\log n-c.\n\\end{eqnarray}\nHere we use the convention $1\/\\sqrt{\\infty}=0$.\nWe call\n\\begin{eqnarray*}\nK_u(X):=\\inf\\{\\kappa\\: X \\mbox{ is an AC$_u(\\kappa)$-space}\\in [-\\infty,0) \\}\n\\end{eqnarray*}\nthe \\emph{asymptotic upper curvature} of $X$.\n\\end{de}\nRough-isometric Gromov hyperbolic spaces have the same asymptotic upper curvature since under rough-isometries, Gromov products only change by a fixed additive amount, which can be absorbed in the constant $c$ in \\eqref{acspaceeq}.\n\nThe asymptotic upper curvature is related to the expansion factors of visual metrics in Gromov hyperbolic spaces, due to the following theorem \\cite[Theorem 1.5]{BFAsymptotic}.\n\\begin{thm} \\label{acku}\nLet $X$ be a Gromov hyperbolic metric space. If there exists a visual metric on $\\partial_{\\infty}X$ with expansion factor $\\Lambda>1$, then $X$ is an AC$_u(\\kappa)$-space with $\\kappa=-\\log^2\\Lambda$. Conversely, if $X$ is an AC$_u(\\kappa)$-space, then for every $1<\\Lambda1$.\n\nWe define a graph by the cell decompositions of $(f, \\mathcal C)$ as follows.\nLet\n\\[V=V(f,\\mathcal C)\\]\nbe the set of all tiles in the cell decompositions $\\D^n(f,\\mathcal C)$ of $(f, \\mathcal C)$ for $n\\geq -1$, where $\\D^{-1}(f,\\mathcal C)$ contains a single $(-1)$-tile $\\S^2$.\nLet $V$ be the set of vertices of the graph. Define the edge set $E$ as follows:\nthere is an edge between the two vertices $X^n,Y^m\\in V$, which we indicate by the notation\n$X^n\\sim Y^m$ if for the underlying tiles we have\n\\[|n-m|\\leq 1 \\mbox{ and } X^n\\cap Y^m\\not= \\emptyset.\\]\nWe call the graph\n\\[\\G(f,\\mathcal C):=G(V,E)\\]\nthe \\emph{tile graph} of $(f, \\mathcal C)$. We usually drop one or both\nparameters in $\\G(f,\\mathcal C)$ if $f$ or $\\mathcal C$ are clear from the context.\nWe call \\[\\ell\\: V \\ra \\Z\\]\nthe \\emph{level function}, where\nfor an $n$-tile $X^n$, we have $\\ell(X^n)=n$.\n\nIf $X\\cap Y=\\emptyset$, let\n\\begin{align*}\n\\bar m_{f,\\mathcal C}(X,Y) :=\\max\\{m\\in \\N_{-1}\\:&\\mbox{\nthere exist non-disjoint $m$-tiles $X^m$ and} \\\\\n&\\mbox{$Y^m$, such that } X\\cap X^m\\not= \\emptyset,\nY\\cap Y^m\\not= \\emptyset\\};\n\\end{align*}\nif $X\\cap Y\\not=\\emptyset$, let\n\\begin{align*}\n\\bar m_{f,\\mathcal C}(X,Y) :=\\infty.\n\\end{align*}\nHere we assume that the $\\infty$-tile is the empty set.\nFor $X,Y\\in \\G$, define\n\\begin{eqnarray} \\label{mG}\nm(X,Y)=m_{f,\\mathcal C}(X,Y)=\\min\\{\\ell(X),\\ell(Y), \\bar m_{f,\\mathcal C}(X,Y)\\}.\n\\end{eqnarray}\nThe tile graph $\\G$ is path connected since any tile can be connected to the $(-1)$-tile $\\S^2$.\nWe give $\\G$ the path metric $\\eta$. Notice that $\\G$ is a\ngeodesic space under this metric. The distance of $X\\in V$ to the base point $\\S^2$ is\n\\[\\eta(X,\\S^2)= \\ell(X)+1.\\]\nFor $X,Y\\in \\G$, we let\n\\begin{eqnarray} \\label{gp}\n(X,Y)&:=& (X,Y)_{X^{-1}} = (X,Y)_{\\S^2} \\nonumber \\\\\n&=& 1\/2[\\eta(X,\\S^2)+\\eta(Y,\\S^2)-\\eta(X,Y)]\\\\\n&=&1\/2[\\ell(X)+\\ell(Y)-\\eta(X,Y)]+1, \\nonumber\n\\end{eqnarray} be the\n\\emph{Gromov product} of $X$ and $Y$ with respect to $X^{-1}=\\S^2$.\n\nIn the following, we are going to prove that the tile graph $\\G$ equipped with the path metric $\\eta$ is a Gromov\nhyperbolic space.\n\n\\excise{\n\\begin{eg}\nWe define a map $f$ as follows (see the picture below): we glue along the boundary of two unit squares $[0,1]^2$, and get a pillow-like space which is homeomorphic to $\\widehat{\\C}$; we color one of the squares black and the other white; we divide each of the squares into 4 smaller squares of half the side length, and color them with black and white in checkerboard fashion; we map one of the small black pillows to the bigger black pillows by Euclidean similarity, and extend the map to the whole pillow-like space by reflection. In fact, the map $f$ is an expanding Thurston map (see Example 4.13 in \\cite{YinLattes}), and the postcritical set $\\post (f)$ consists of the four common corner points of the two big squares.\n\\begin{center}\n\\mbox{ \\scalebox{0.7}{\\includegraphics{2by2.eps}}}\n\\end{center}\nLet $\\mathcal C$ be the common boundary of the two big squares, then $\\mathcal C$ contains $\\post (f)$.\nOn the $n$-th level, the set of $n$-tiles corresponding to\n\\end{eg}\n}\n\n\n\n\\begin{lem} \\label{mdiam}\nThere exists a constant $C>1$ such that for any tiles $X,Y\\in \\G$,\n\\[\\frac1{C}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq C \\Lambda^{-m(X,Y)}.\\]\n\\end{lem}\nHere and in the following, the diameter function $\\diam(\\cdot)$ is with respect to the visual metric $d$ on $\\S^2$.\n\n\\begin{proof}\nLet $m=m(X,Y)$, and let $X^m, Y^m$ be non-disjoint $m$-tiles such that\n\\[X\\cap X^m\\not=\\emptyset \\mbox{ and } Y\\cap Y^m\\not=\\emptyset.\\]\nWe have that\n\\begin{eqnarray*}\n\\diam (X\\cup Y) &\\leq &\\diam (X)+\\diam (X^m)+\\diam (Y^m)+\\diam (Y)\\\\\n&\\leq& 4 C' \\Lambda^{-m},\n\\end{eqnarray*}\nwhere $C'>1$ is the same as the constant in Lemma \\ref{charvisual}, which only depends on $f$.\nLet $\\bar m= \\bar m_{f,\\mathcal C}(X,Y)$, and let $X^{\\bar m+1}, Y^{\\bar m+1}$ be disjoint $(\\bar m+1)$-tiles such that\n\\[X\\cap X^{\\bar m+1}\\not=\\emptyset,\\quad Y\\cap Y^{\\bar m+1}\\not=\\emptyset.\\] Then\n\\begin{eqnarray*}\n\\diam (X\\cup Y) &\\geq& \\max\\{ \\diam (X), \\diam (Y), d(X^{\\bar m+1}, Y^{\\bar m+1}) \\} \\\\\n&\\geq & \\frac1{C'}\\max\\{\\Lambda^{-\\ell(X)}, \\Lambda^{-\\ell(Y)}, \\Lambda^{-\\bar m}\\} \\\\\n&\\geq &\\frac1{C'}\\Lambda^{-\\min\\{\\ell(X),\\ell(Y),\\bar m\\}}\\\\\n&= &\\frac1{C'}\\Lambda^{-m},\n\\end{eqnarray*}\nwhere $C'>1$ is the same $C$ as in Lemma \\ref{charvisual}, which only depends on $f$. Let $C=4C'$, and the lemmas follows.\n\\end{proof}\n\n\\begin{lem} \\label{pdiam}\nThere exists a constant $k\\geq 1$ such that for any tiles $X,Y\\in \\G$,\n\\[\\diam (X\\cup Y) \\leq k\\Lambda^{-(X,Y)}.\\]\n\\end{lem}\n\n\\begin{proof}\nLet $\\eta=\\eta(X,Y)$. Pick any path $X_0=X, X_1,\\ldots, X_{\\eta}=Y$. Then\n\\begin{eqnarray*}\n\\diam (X\\cup Y)& \\leq & \\sum_{i=0}^{\\eta}\\diam(X_i)\\\\\n&\\leq & C \\sum_{i=0}^{\\eta}\\Lambda^{-\\ell(X_i)} \\\\\n&\\leq & C \\min_{0\\leq l\\leq \\eta}\\left\\{\\sum_{i=0}^{l}\\Lambda^{-\\ell(X)+i}+ \\sum_{i=l+1}^{\\eta}\\Lambda^{-\\ell(Y)+(\\eta-i)}\\right\\}\\\\\n&\\leq & \\frac{C\\Lambda}{\\Lambda-1} \\min_{0\\leq l\\leq \\eta}\\left\\{\\Lambda^{-\\ell(X)+l}+ \\Lambda^{-\\ell(Y)+(\\eta-l)}\\right\\}.\n\\end{eqnarray*}\nNotice that on the right hand-side the minimum is obtained when the two exponents of $\\Lambda$ are the same:\n\\[ -\\ell(X)+l= -\\ell(Y)+(\\eta-l),\\]\nso we let\n\\[l=\\left[\\frac12(\\ell(X)-\\ell(Y)+\\eta)\\right]\\]\nbe the integer part of $\\frac12(\\ell(X)-\\ell(Y)+\\eta)$. Hence, we have\n\\begin{eqnarray*}\n\\diam (X\\cup Y) &\\leq & \\frac{2C\\Lambda}{\\Lambda-1} \\Lambda^{-[1\/2(\\ell(X)+\\ell(Y)-\\eta)]} \\\\\n&\\leq &k\\Lambda^{-(X,Y)},\n\\end{eqnarray*}\nwhere $C>1$ is the same $C$ as in Lemma \\ref{charvisual}, and\n\\[k=\\frac{2C\\Lambda^3}{\\Lambda-1}\\] also only depends on $f$.\n\\end{proof}\n\n\n\\begin{pro} \\label{mp}\nThere exists a constant $C'>0$, such that for any tiles $X,Y\\in \\G$,\n\\begin{eqnarray*}\nm(X,Y)-1\\leq (X,Y)\\leq m(X,Y)+C'.\n\\end{eqnarray*}\n\\end{pro}\n\n\\begin{proof}\nBy Lemma \\ref{mdiam} and Lemma \\ref{pdiam}, we have that\n\\[ \\frac1{C}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq k\\Lambda^{-(X,Y)}\\] for some constants $C,k>1$ which only depend on $f$. Hence, there exists a constant $C'>0$, such that for any tiles $X,Y\\in \\G$,\n\\[(X,Y)\\leq m(X,Y)+C'.\\]\n\nFor the other inequality, let $m=m(X,Y)$, and let $X^m, Y^m$ be non-disjoint $m$-tiles such that\n\\[X\\cap X^m\\not=\\emptyset,\\quad Y\\cap Y^m\\not=\\emptyset.\\]\nSo \\[\\eta(X,X^m)\\leq \\ell(X)-m+1 ,\\]and\n\\[\\eta(Y,Y^m)\\leq \\ell(Y)-m+1 .\\]\nBy the triangle inequality, we have that\n\\begin{eqnarray*}\n\\eta(X,Y)&\\leq& \\eta(X,X^m)+\\eta(X^m,Y)\\\\\n&\\leq & \\eta(X,X^m)+\\eta(Y^m,Y)+1\\\\\n&\\leq & (\\ell(X)-m)+(\\ell(Y)-m)+3.\n\\end{eqnarray*}\nHence, we obtain that\n\\begin{eqnarray*}\n(X,Y)&= &(X,Y)_{X^{-1}} =1\/2[\\ell(X)+\\ell(Y)-\\eta(X,Y)]+1\\\\\n&\\geq & 1\/2[\\ell(X)+\\ell(Y)-(\\ell(X)-m)-(\\ell(Y)-m)-3]+1\\\\\n&\\geq &m(X,Y)-1.\n\\end{eqnarray*}\n\\end{proof}\n\n\\excise{\n----------------------------------\n\\begin{cor}\nThere exists a constant $C'>0$, such that for any tiles $X,Y\\in \\G$, we have\n\\[ (X,Y)\\leq m(X,Y)+C'.\\]\n\\end{cor}\n\\begin{proof}\nBy the Lemma \\ref{mdiam} and Lemma \\ref{pdiam}, we have\n\\[ \\frac1{k}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq k'\\Lambda^{-(X,Y)},\\] for some constant $k,k'>1$ which only depends on $f$. The corollary follows easily.\n\\end{proof}\n\n\\begin{lem}\n\\[ m(X,Y)\\leq (X,Y).\\]\n\\end{lem}\n\n\\begin{proof}\nLet $m=m(X,Y)$, and let $X^m, Y^m$ be non-disjoint $m$-tiles such that\n\\[X\\cap X^m\\not=\\emptyset,\\quad Y\\cap Y^m\\not=\\emptyset.\\]\nBy triangle inequality, we have\n\\begin{eqnarray*}\n\\eta(X,Y)&\\leq& \\eta(X,X^m)+\\eta(X^m,Y)\\\\\n&\\leq & \\eta(X,X^m)+\\eta(Y^m,Y)+1\\\\\n&\\leq & (\\ell(X)-m)+(\\ell(Y)-m)+1.\n\\end{eqnarray*}\nHence,\n\\begin{eqnarray*}\n(X,Y)&= &(X,Y)_{X^{-1}} =1\/2[\\ell(X)+\\ell(Y)-\\eta(X,Y)]+1\\\\\n&\\geq & 1\/2[\\ell(X)+\\ell(Y)-(\\ell(X)-m)-(\\ell(Y)-m)-1]+1\\\\\n&\\geq &m(X,Y).\n\\end{eqnarray*}\n\\end{proof}\n-------------------------------}\n\n\n\\begin{lem} \\label{triangleineq}\nThere exists a number $c\\geq 0$ such that for any tiles $X,Y,Z\\in \\G$,\n\\[m(X,Y)\\geq \\min\\{m(X,Z),m(Y,Z)\\} -c .\\]\n\\end{lem}\n\n\\begin{proof}\nFor any $X,Y,Z\\in \\G$,\n\\begin{eqnarray*}\n\\diam(X\\cup Y)&= &\\max\\{d(x,y),d(x,x'),d(y,y')\\: x,x'\\in X, y,y'\\in Y\\}\\\\\n&\\leq & \\max\\{d(x,z)+d(z,y),d(x,x'),d(y,y')\\: \\\\\n&& \\hspace{1.5cm} x,x'\\in X, y,y'\\in Y, z\\in Z\\}\\\\\n&\\leq & \\max\\{d(x,z),d(x,x')\\: x\\in X,z\\in Z\\}\\\\\n&& +\\max\\{d(z,y),d(y,y')\\: y\\in Y,z\\in Z\\}\\\\\n&\\leq & \\diam(X\\cup Z)+\\diam(Z \\cup Y),\n\\end{eqnarray*}\nand so\n\\begin{eqnarray} \\label{diamtri}\n\\diam(X\\cup Y)&\\leq & 2\\max\\{ \\diam(X\\cup Z),\\diam(Z \\cup Y)\\}.\n\\end{eqnarray}\n\nBy Lemma \\ref{mdiam}, there exists a constant $k>1$, such that for any $X,Y\\in \\G$,\n\\begin{eqnarray*}\n\\frac1{k}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq k \\Lambda^{-m(X,Y)}.\n\\end{eqnarray*}\nHence, by the inequalities above and inequality \\eqref{diamtri}, we have that\n\\begin{eqnarray*}\nm(X,Y)&\\geq & -\\log_{\\Lambda}\\big(k\\diam(X\\cup Y)\\big) \\\\\n&\\geq & -\\log_{\\Lambda}\\Big(2k\\max\\big\\{ \\diam(X\\cup Z),\\diam(Z \\cup Y) \\big\\}\\Big)\\\\\n&\\geq & \\min\\Big\\{ -\\log_{\\Lambda}\\big(2k\\diam(X\\cup Z)\\big),-\\log_{\\Lambda}\\big(2k\\diam(Z \\cup Y)\\big) \\Big\\}\\\\\n&\\geq & \\min\\{m(X,Z),m(Y,Z)\\} -c\n\\end{eqnarray*}\nfor some $c\\geq 0$ that only depends on $f$.\n\\end{proof}\n\n\\begin{thm} \\label{gh}\nLet $f\\: \\S^2 \\ra S^2$ be an expanding Thurston map\nand let $\\mathcal C \\subset \\S^2$ be a Jordan curve containing $\\post(f)$. Then the tile graph $\\G(f,\\mathcal C)$ equipped with the path metric $\\eta$ is a Gromov hyperbolic space.\n\\end{thm}\n\n\\begin{proof}\nFor any tiles $X,Y\\in \\G$, by Proposition \\ref{mp}, the Gromov product $(X,Y)$ defined in equation \\eqref{gp} is equal to $m(X,Y)$ up to a constant which only depends on $f$. So by Lemma \\ref{triangleineq}, there exists a constant $c'>0$, such that for any tiles $X,Y,Z\\in \\G$,\n\\[(X,Y)\\geq \\min\\{(X,Z), (Y,Z)\\} -c' .\\] Therefore, the graph $G(f,\\mathcal C)$ equipped with the path metric $\\eta$ is a Gromov hyperbolic space.\n\\end{proof}\n\n\\begin{re}\nIn the proofs of Proposition \\ref{mp} and Lemma \\ref{triangleineq}, we used visual metrics as a bridge to connect $m(\\cdot,\\cdot)$ and the Gromov product $(\\cdot,\\cdot)$. This idea is contained in \\cite{BPCohomologie}. Theorem \\ref{gh}, Proposition \\ref{mp} and Lemma \\ref{triangleineq} can also be proved combinatorially without using visual metrics.\n\\end{re}\n\n\\begin{pro} \\label{roughgg}\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, the tile graphs $\\G=\\G(f,\\mathcal C)$ and $\\G'=\\G(f,\\mathcal C')$ equipped with path metric respectively are rough-isometric.\n\\end{pro}\n\n\\begin{proof}\nBy equation \\eqref{gp}, for any $X,Y\\in \\G(f,\\mathcal C)$, we have\n\\begin{eqnarray} \\label{dp}\n \\eta(X,Y) &=& \\ell(X)+\\ell(Y)+2-2(X,Y).\n\\end{eqnarray}\nWe have similar relations for the path metric $\\eta'$ of $\\G'$. Let $m=m_{f,\\mathcal C}$ and $m'=m_{f,\\mathcal C'}$ as defined in equation \\eqref{mG}.\nWe know that $m(X,Y)$ and $(X,Y)$ are equal up to a constant that only depends on $f$ by Proposition \\ref{mp}. So\nif we can show that there exists a level-preserving bijection $g\\:\\G\\ra \\G'$ and a constant $\\lambda\\geq 0$, such that for any $X,Y\\in \\G$,\n\\begin{eqnarray*}\n m(X,Y)-\\lambda\\leq m'(g(X),g(Y))\\leq m(X,Y)+\\lambda,\n\\end{eqnarray*}\nthen by equation \\eqref{dp}, the map $g$ will be a rough isometry between the path metrics of $\\G$ and $\\G'$.\n\nFix $p\\in {\\rm post}(f)$. We will define\n\\[g\\: \\G(f,\\mathcal C)\\ra \\G(f,\\mathcal C')\\]\nby specifying a bijection between $n$-tiles of $(f,\\mathcal C)$ and $(f,\\mathcal C')$ for all $n\\geq -1$. For $n=-1$, let\n\\[g(\\S^2)=\\S^2.\\]\nFor $n\\geq 0$, and for any $q\\in f^{-n}(p)$, we claim that there exists a bijection $g_{n,q}$ between $n$-tiles of $(f,\\mathcal C)$ containing $q$ and $n$-tiles of $(f,\\mathcal C')$ containing $q$,\n\\begin{equation*}\n g_{n,q}\\: \\Big\\{n\\mbox{-tile } X\\in \\D^n(f,\\mathcal C): q\\in X \\Big\\}\\ra \\Big\\{n\\mbox{-tile } X'\\in \\D^n(f,\\mathcal C'): q\\in X' \\Big\\}.\n\\end{equation*}\nIndeed, the number of tiles containing $q$ is equal to the degree of $f^n$ at $q$, and this justifies the existence of the bijection $g_{n,q}$. Since every $n$-tile contains exactly one point in $f^{-n}(p)$, we get a bijection of all $n$-tiles by $g_{n,q}$ for $q\\in f^{-n}(p)$.\n\nFor any $X, Y\\in \\G$, let $X',Y' \\in \\G'$ be their images under $g$. It follows from the definition of $g$ that\n\\[X\\cap X'\\not=\\emptyset \\mbox{ and } Y\\cap Y'\\not=\\emptyset.\\]\nNow we are going to show that there exists $k\\geq 1$, such that for any $X,Y\\in \\G$,\n\\begin{eqnarray*}\n\\frac1{k}\\diam (X'\\cup Y') \\leq \\diam (X\\cup Y)\\leq k\\diam (X'\\cup Y').\n\\end{eqnarray*}\nLet $m=m(X,Y)$. We have that\n\\begin{eqnarray*}\n\\diam (X\\cup Y)&\\leq &\\diam(X) + \\diam (X'\\cup Y') + \\diam (Y)\\\\\n&\\leq & C^2 \\diam (X')+C^2\\diam(Y') + \\diam(X'\\cup Y')\\\\\n&\\leq & (C^2+1) \\diam (X'\\cup Y'),\n\\end{eqnarray*}\nwhere $C>1$ is the same $C$ as in Lemma 2.12, which only depends on $f$.\nThis implies that\n\\[\\diam (X\\cup Y)\\leq k\\diam (X'\\cup Y'), \\]for some $k>1$ only depending on $f$. Similarly, we get that\n\\[\\diam (X'\\cup Y')\\leq k\\diam (X\\cup Y). \\]\n\nSince $\\diam (X\\cup Y)$ and $\\Lambda^{-m(X,Y)}$ are the same up to a scaling by Lemma \\ref{mdiam}, there exists a constant $\\lambda>0$, such that\n\\[m(X,Y)-\\lambda\\leq m'(g(X),g(Y))\\leq m(X,Y)+\\lambda \\] for all $X,Y\\in \\G(f,\\mathcal C)$.\n\\end{proof}\n\n\\begin{re}\nIn the proof of Proposition \\ref{roughgg}, the bijective rough-isometry $g$ between tile graphs of two different Jordan curves induces a bijection $g_{\\infty}$ on the boundary at infinity of these two tile graphs.\n\\end{re}\n\n\\begin{pro} \\label{samevisual}\nThe boundary at infinity $\\partial_{\\infty}\\G$ of a graph tile $\\G(f,\\mathcal C)$ can be identified with $\\S^2$.\nUnder this identification, a metric $d$ is a visual metric on $\\S^2$ with respect\nto the expanding Thurston map $f$ if and only if $d$ is a visual metric on $\\partial_{\\infty}\\G$ (in the sense of Gromov hyperbolic spaces).\n\\end{pro}\n\nHere the metric $d$ on $\\partial_{\\infty}\\G$ means the pull-pack metric of $d$ under the identification.\n\n\\begin{proof}\nLet $d$ be a visual metric with expansion factor $\\Lambda$ of $\\S^2$ with respect to $f$.\n\nFor any sequence $\\{X_n\\}$ converging to $\\infty$\n\\[\\lim_{i,j\\ra \\infty} (X_i,X_j)=\\infty,\\] we have a filtration\n\\begin{eqnarray*}\n\\bigcup_{i={1}}^{\\infty} X_i\\supset \\bigcup_{i={2}}^{\\infty} X_i\\supset\\ldots \\supset\\bigcup_{i={n}}^{\\infty} X_i \\supset \\bigcup_{i=n+1}^{\\infty} X_i\\supset\\ldots\n\\end{eqnarray*}\nwith\n\\[ \\diam \\left(\\bigcup_{i={n}}^{\\infty} X_i\\right)\\ra 0\\mbox{ as } n\\ra \\infty.\\]\nHence, there exists a limit point $x\\in \\S^2$ such that for any $\\epsilon>0$, there exists $N>0$ such that for all $n>N$,\n\\begin{eqnarray} \\label{inepsilonn}\n\\bigcup_{i={n}}^{\\infty} X_i\\subset N_{\\epsilon}(x),\n\\end{eqnarray}\nwhere $N_{\\epsilon}(x)$ is an $\\epsilon$-neighborhood of $x$ in $\\S^2$, i.e.,\n\\[ X_n\\subset N_{\\epsilon}(x),\\] or\n\\begin{eqnarray*}\nd(x,X_n)< \\epsilon.\n\\end{eqnarray*}\nWe claim that the limit point is unique. Indeed, if there exists $y\\in \\S^2$ also satisfying \\eqref{inepsilonn}, then\n\\[d(x,y)\\leq d(x,\\diam(X_n))+d(y,\\diam(X_n))\\ra 0\\mbox{ as } n\\ra\\infty.\\]\nHence, $x=y$.\nLet $\\{Y_n\\}$ be an sequence converging to infinity equivalent to $\\{X_i\\}$, i.e.,\n\\[\\lim_{i\\ra \\infty} (X_i,Y_i)=\\infty.\\]\nWe claim that the limit point of $\\{Y_n\\}$ is $x$. Indeed, by Lemma \\ref{pdiam}, we have\n\\[d(x,Y_n)\\leq d(x,X_n)+d(Y_n,X_n)\\leq d(x,X_n)+k{\\Lambda}^{-(Y_n,X_n)}\\ra 0 \\]\nas $n$ goes to infinity since $(Y_n,X_n)\\ra \\infty$. Hence, any two equivalent sequences converging to infinity have the same limit point, and we can assign a limit point to an equivalence class of sequences converging to infinity.\n\nWe define\n\\[h\\: \\partial_{\\infty}\\G\\ra \\S^2\\]\nby mapping any equivalence class of sequences converging to infinity to its limit point. For any $x\\in \\S^2$, there exists $X_i$ with $\\ell(X_i)=i$ containing $x$, for any $i\\geq -1$. Then by Lemma \\ref{pdiam}, we have that\n\\begin{eqnarray*}\n(X_i,X_j)&\\geq& -\\log_{\\Lambda}\\diam(X_i\\cup X_j)+\\log k\\\\\n&\\geq&-\\log_{\\Lambda} \\big(\\min\\{ \\diam(X_i),\\diam(X_j)\\} \\big) +\\log k \\ra \\infty\n\\end{eqnarray*}\nas $i,j\\ra\\infty$, where $k\\geq 1$ is a constant as in Lemma \\ref{pdiam}.\nSo $\\{X_i\\}$ is a converging sequence with limit point $x$. Hence, the map $h$ is surjective.\nIn order to prove the injectivity, for any two\nsequences converging to infinity $\\{X_i\\}$ and $\\{Y_i\\}$, we let $x$ and $y$ be their limit points respectively. If $x=y$, then\n\\[\\diam(X_n\\cup Y_n) \\ra 0 \\mbox{ as } n\\ra \\infty.\\]\nSo by Lemma \\ref{mdiam} and \\ref{mp},\n\\[(X_n,Y_n) \\geq m(X_n,Y_n)-1 \\geq -\\log_{\\Lambda}\\big(C\\diam(X_n\\cup Y_n)\\big) -1\\ra \\infty \\]\nas $n$ goes to infinity, which implies that $\\{X_i\\}$ and $\\{Y_i\\}$ are equivalent.\nHence,\n$h$ is injective.\n\nWe only need to show that that there exists a constant $C>0$ such that for any $\\xi,\\xi'\\in \\partial_{\\infty}\\G$, $x=h(\\xi)$ and $y=h(\\xi')$,\n\\[\\frac1{C} \\Lambda^{-(\\xi,\\xi')}\\leq d(x,y)\\leq C\\Lambda^{-(\\xi,\\xi')}. \\]\nPick any $\\{X_n\\}\\in \\xi$ and $\\{Y_n\\}\\in \\xi'$. By Lemma \\ref{mdiam}\n\\begin{eqnarray*}\n\\frac1{C}\\Lambda^{-(X_n,Y_n)}\\leq \\diam(X_n\\cup Y_n) \\leq C \\Lambda^{-(X_n,Y_n)}.\n\\end{eqnarray*}\nTaking the limit superior, we get\n\\begin{eqnarray*}\n\\frac1{C}\\limsup_{n\\ra\\infty} \\Lambda^{-(X_n,Y_n)}\\leq \\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n)\n\\leq C\\limsup_{n\\ra\\infty} \\Lambda^{-(X_n,Y_n)}.\n\\end{eqnarray*}\nHence, we have\n\\begin{eqnarray} \\label{infdiam}\n\\frac1{C}\\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}&\\leq& \\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n)\\\\\n &&\\hspace{2cm}\\leq C \\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}. \\nonumber\n\\end{eqnarray}\nSince\n\\[d(x,y)=\\lim_{n\\ra\\infty}\\diam(X_n,Y_n)\\] and\n\\[(\\xi,\\xi')\\leq \\liminf_{n\\ra\\infty} (X_n,Y_n),\\]\nby inequality \\eqref{infdiam}\n\\begin{eqnarray} \\label{dleq}\nd(x,y)&=&\\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n) \\nonumber\\\\\n &\\leq& {C} \\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}\\\\\n &\\leq& {C} \\Lambda^{- (\\xi,\\xi')}.\\nonumber\n\\end{eqnarray}\nSince \\[(\\xi,\\xi')= \\inf \\liminf_{n\\ra\\infty} (X_n,Y_n)\\] where infimum is taken for all $\\{X_n\\}\\in \\xi$ and $\\{Y_n\\}\\in \\xi'$, by inequality \\eqref{infdiam},\n\\begin{eqnarray*}\n\\frac1{C}\\sup\\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}\\leq \\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n) =d(x,y)\n\\end{eqnarray*}\nwhere supremum is taken for all $\\{X_n\\}\\in \\xi$ and $\\{Y_n\\}\\in \\xi'$. Hence,\n\\begin{eqnarray} \\label{dgeq}\n\\frac1{C}\\Lambda^{-(\\xi,\\xi')}=\\frac1{C}\\Lambda^{-\\inf\\liminf_{n\\ra\\infty} (X_n,Y_n)}\\leq d(x,y).\n\\end{eqnarray}\nCombining equations \\eqref{dleq} and \\eqref{dgeq}, we get that\n\\begin{eqnarray} \\label{dleqgeq}\n\\frac1{C}\\Lambda^{-(\\xi,\\xi')}\\leq d(x,y)\\leq 4C\\Lambda^{-(\\xi,\\xi')},\n\\end{eqnarray}\nso\n\\begin{eqnarray*}\n\\frac1{C}\\Lambda^{-(\\xi,\\xi')}\\leq d(h(\\xi),h(\\xi'))\\leq C\\Lambda^{-(\\xi,\\xi')}\n\\end{eqnarray*}\nfor all $\\xi,\\xi'\\in \\partial_{\\infty}\\G$.\nTherefore, the pull-back of the metric $d$ on $\\S^2$ under $h$ is a visual metric on $\\partial_{\\infty}\\G$.\n\nSince $d$ is a visual metric with respect to $f$, equation \\eqref{dleqgeq} implies that there exists a constant $c\\geq 0$ such that for all $x,y\\in \\S^2$, and $\\xi=h^{-1}(x)$, $\\xi=h^{-1}(y)$,\n\\begin{eqnarray} \\label{mxi}\n(\\xi,\\xi')-c\\leq m(x,y)\\leq (\\xi,\\xi')+c.\n\\end{eqnarray}\nLet $\\rho$ be a visual metric on $\\partial_{\\infty}\\G$ on the Gromov hyperbolic space, so there exists constant $k\\geq 1$, such that for any $\\xi,\\xi'\\in \\partial_{\\infty}\\G$,\n\\[ \\frac1{k}\\Lambda^{-(\\xi,\\xi')}\\leq \\rho(\\xi,\\xi' ) \\leq k\\Lambda^{-(\\xi,\\xi')}.\\]\nBy equation \\eqref{mxi}, there exists a constant $k'\\geq 1$, such that\n\\[\\frac1{k'}\\Lambda^{-(x,y)}\\leq \\frac1{k}\\Lambda^{-(\\xi,\\xi')}\\leq \\rho(h^{-1}(x),h^{-1}(y) ) \\leq k\\Lambda^{-(\\xi,\\xi')}\\leq k'\\Lambda^{-m(x,y)},\\]\nwhere $x,y\\in \\S^2$, $\\xi=h^{-1}(x)$ and $\\xi=h^{-1}(y)$.\nTherefore, the pull-back of the metric $\\rho$ on $\\partial_{\\infty}\\G$ under $h^{-1}$ is a visual metric on $\\S^2$.\n\\end{proof}\n\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, let $\\partial_{\\infty}\\G=\\partial_{\\infty}\\G(f,\\mathcal C)$ and $\\partial_{\\infty}\\G'=\\partial_{\\infty}\\G(f,\\mathcal C')$ be the boundary at infinity of the tile graphs $\\G(f,\\mathcal C)$ and $\\G(f,\\mathcal C')$ respectively. By the proposition above, there exist identifications\n\\[h\\: \\partial_{\\infty}\\G \\ra \\S^2\\]\nand\n\\[h'\\: \\partial_{\\infty}\\G' \\ra \\S^2.\\]\nSo we have the following diagram\n\\[\n\\xymatrix@R=0.5cm{\n \\partial_{\\infty}\\G \\ar[dd]_{g_{\\infty}} \\ar[dr]^{h} \\\\\n & \\S^2\\ar[dl]^{(h')^{-1}} \\\\\n \\partial_{\\infty}\\G' }\n\\]\nThis induced bijection $g_{\\infty}=(h')^{-1}\\circ h$ should be the same as $g_{\\infty}$ as in the remark after Proposition \\ref{roughgg}. In addition, under this identification, visual metrics on $\\partial_{\\infty}\\G$ and $\\partial_{\\infty}\\G'$ are also identified. This is the following corollary.\n\n\\begin{cor} \\label{corsamevisual}\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, there exists a natural identification between $\\G=\\partial_{\\infty}\\G(f,\\mathcal C)$ and $\\G'=\\partial_{\\infty}\\G(f,\\mathcal C')$. Under this identification, a metric $\\rho$ is a visual metric on $\\partial_{\\infty}\\G$ if and only if it is a visual metric on $\\partial_{\\infty}\\G'$.\n\\end{cor}\n\n\n\n\n\\section{Asymptotic Upper Curvature}\n\\noindent\nIn this section, we define the asymptotic upper curvature for an expanding Thurston map. After review the definition of Latt\\`es maps, we give a curvature characterization of Latt\\`es maps.\n\nLet $f \\: \\S^2 \\ra S^2$ be an expanding Thurston map.\nWe define the \\emph{asymptotic upper curvature} of $f$ as\n\\begin{eqnarray} \\label{asyf}\nK_u(f)\\:=K_u(\\G(f,\\mathcal C)),\n\\end{eqnarray}\nwhere $\\mathcal C \\subset \\S^2$ is any Jordan curve containing $\\post(f)$ and $\\G=\\G(f,\\mathcal C)$ denotes\nthe Gromov hyperbolic graph constructed from the cell decompositions of $(f, \\mathcal C)$.\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$, the Gromov hyperbolic space $\\G(f,\\mathcal C)$ and $\\G(f,\\mathcal C')$ are rough-isomeric by Proposition \\ref{roughgg}, and the asymptotic upper curvature is invariant under rough-isometry, so\n\\[K_u(\\G(f,\\mathcal C))=K_u(\\G(f,\\mathcal C')). \\]\nTherefore, the asymptotic upper curvature $K(f)$ is well-defined in equation \\eqref{asyf}.\n\nA \\emph{Latt\\`es map} $f\\:\\widehat\\C\\ra \\widehat\\C$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism, i.e., the map $f$ satisfies the following commutative diagram:\n\\begin{equation}\\label{lat}\n \\begin{CD}\n\\T @>\\bar{A}>> \\T\\\\\n@V\\Theta VV @VV\\Theta V\\\\\n\\widehat\\C @>f>> \\widehat\\C\n\\end{CD}\n\\end{equation}\nwhere $\\bar A$ is a map of a torus $\\T$ that is a quotient of an affine map of the complex plane, and $\\Theta$ is a finite-to-one holomorphic map. Latt\\`es maps were the first examples of rational maps whose Julia set is the whole sphere $\\widehat \\C$, and a Latt\\`es map is an expanding Thurston map. In \\cite{YinLattes}, we have the following combinatorial characterization of Latt\\`es maps:\n\\begin{thm}[Yin, 2011] \\label{main0}\nA map $f\\:\\S^2\\ra \\S^2$ is topologically conjugate to a Latt\\`es map if and only if the following conditions hold:\n\\begin{itemize}\n \\item $f$ is an expanding Thurston map;\n \\item $f$ has no periodic critical points;\n \\item there exists $c>0$ such that $D_n\\geq c(\\deg f)^{n\/2}$ for all $n>0$.\n\\end{itemize}\n\\end{thm}\n\nThis leads to an curvature characterization of Latt\\`es maps as follows.\n\n\\begin{thm} \\label{main2}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map\nThe asymptotic upper curvature of $f$ satisfies\n\\[K_u(f)\\geq-\\frac14\\log^2(\\deg f). \\]\nIf in addition, the map $f$ has no periodic critical points, then the tile graph $\\G=\\G(f)$ is an AC$_u(\\kappa)$-space with\n\\[ \\kappa= -\\frac14\\log^2(\\deg f),\\]\nif and only if the map $f$ is topologically conjugate to a Latt\\`es map.\n\\end{thm}\n\n\\begin{proof}\nThe first part follows directly from the definition of asymptotic upper curvature of $f$ and from Theorem \\ref{acku}.\n\nIf $f$ is topologically conjugate to a Latt\\`es map, then by Corollary \\ref{cormain}, there exists a visual metric on $\\S^2$ with respect to $f$ with expansion factor $\\Lambda=\\deg(f)^{1\/2}$. By Proposition \\ref{samevisual}, there exists a visual metric on $\\partial_{\\infty}\\G$ in the sense of Gromov hyperbolic spaces with expansion factor $\\Lambda=\\deg(f)^{1\/2}$. By Theorem \\ref{acku}, the Gromov hyperbolic space $\\G$ is an AC$_u(\\kappa)$-space with \\[\\kappa=-\\frac14\\log^2(\\deg f).\\]\n\nIf $\\G$ is an AC$_u(\\kappa)$-space with\n\\[ \\kappa= -\\frac14\\log^2(\\deg f),\\]\nthen for all $X,X'\\in \\G$ and all finite sequences $X_0=X,X_1,\\ldots,X_n=X'$ in $X$,\n\\begin{eqnarray}\\label{acspaceeq1}\n(X,X')\\geq \\min_{i=1,2,\\ldots,n}(X_{i-1},X_i)-\\frac{\\log n}{\\log(\\deg f)^{1\/2}}-c.\n\\end{eqnarray}\n\nLet $D_n$ be the minimum number of $n$-tiles needed to join opposite sides of Jordan curve $\\mathcal C$ as defined in \\label{defdn}, for $n>0$. Let $P_n=X_1\\ldots X_{D_n}$ be an $n$-tile chain joining opposite sides of $\\mathcal C$.\nBy the equation \\eqref{acspaceeq1}, we have\n\\begin{eqnarray*}\n(X_1,X_{D_n})\\geq \\min_{i=1,2,\\ldots,D_n}(X_{i-1},X_i)-\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}-c,\n\\end{eqnarray*}\nso\n\\begin{eqnarray} \\label{logdndegf}\n\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}\\geq \\min_{i=1,2,\\ldots,D_n}(X_{i-1},X_i)-(X_1,X_{D_n})-c.\n\\end{eqnarray}\n\nBy equation \\eqref{gp}, we have\n\\begin{eqnarray} \\label{XiXi}\n(X_{i-1},X_i) &=&\\frac12[\\ell(X_{i-1})+\\ell(X_{i})-\\eta(X_{i-1},X_i)]+1 \\nonumber\\\\\n&=& \\frac12[2n-\\eta(X_{i-1},X_i)]+1\\\\\n&\\geq & n+1-\\frac14N\n\\end{eqnarray}\nwhere $N>0$ is a constant only depending on $f$ as in Lemma \\ref{noperiodic}, i.e. the constant $N$ is the upper bound of the degree of $f^n$ at any point in $\\S^2$.\nApplying equation \\eqref{XiXi} and Lemma \\ref{pdiam} to equation \\eqref{logdndegf}, we have\n\\begin{eqnarray*}\n\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}&\\geq &\\min_{i=1,2,\\ldots,D_n}(X_{i-1},X_i)-(X_1,X_{D_n})-c\\\\\n&\\geq& n+1-\\frac14N- \\log_{\\Lambda}\\big(\\diam(X_1\\cup X_{D_n})\\big)+\\log_{\\Lambda}k-c, \\\\\n\\end{eqnarray*}\nwhere $k\\geq 1$ only depends on $f$ and $\\mathcal C$ as in Lemma \\ref{pdiam}, and $N>0$ only depends on $f$.\nLet $D$ be the maximum of diameters of the two $0$-tiles, then\n\\[\\log_{\\Lambda}\\big(\\diam(X_1\\cup X_{D_n})\\big)\\leq D. \\]\nSo\n\\begin{eqnarray*}\n\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}\n&\\geq& n+1-\\frac14N- \\log_{\\Lambda}(D)+\\log_{\\Lambda}k-c \\\\\n&=& n+C.\n\\end{eqnarray*}\nHere the constant\n\\[C=1-\\frac14N- \\log_{\\Lambda}(D)+\\log_{\\Lambda}k-c \\]\nonly depends on $f$ and $\\mathcal C$.\nHence, we have\n\\begin{eqnarray*}\n{\\log D_n}&\\geq & (n+C) {\\log(\\deg f)^{1\/2}}.\n\\end{eqnarray*}\nTherefore, we have\n\\begin{eqnarray*}\n{D_n}&\\geq & C'(\\deg f)^{n\/2},\n\\end{eqnarray*}\nwhere $C'=(\\deg f)^{C\/2}$ only depends on $f$ and $\\mathcal C$.\nBy Theorem \\ref{main0}, the map $f$ is topologically conjugate to a Latt\\`es map.\n\\end{proof}\n\n\\newpage\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe circum-galactic medium (CGM) is the interface between cold flows from the intergalactic medium onto a galaxy, and hosts hot halo gas and material ejected from galaxies \\citep[for reviews, see][]{Putman12,Tumlinson17}. With various processes in galaxy evolution consuming (e.g. star formation) and removing (e.g. winds) gas, the CGM is shaped by the processes internal to the galaxy. Early progress in the study of the CGM came from connecting absorption lines in quasar (QSO) spectra with galaxies imaged in the foreground, tracing the extent and properties of the CGM gas as a function of the host galaxy's properties \\citep[e.g.][]{Bergeron86,Bowen95,Lanzetta95,Adelberger05,Chen10,Steidel10,Bordoloi11,Prochaska11,Turner14}. Building on these foundations, our understanding of the CGM has been significantly improved through surveys with the Hubble Space Telescope (HST) Cosmic Origins Spectrograph \\citep[COS;][]{Green12}. The first of several surveys of the CGM surrounding low redshift galaxies was the COS-Halos survey \\citep{Tumlinson13} which targetted the CGM around 44 $\\sim$L$^{\\star}$ galaxies, demonstrating that the properties of the CGM differ depending on whether the central galaxy is passive or star-forming \\citep[defined using a specific star formation rate cut of sSFR$\\rm{=10^{-11} yr^{-1}}$;][]{Tumlinson11,Werk13,Borthakur16}. The COS-Halos team found a distinct lack of O\\ion{vi} around passive galaxies, while H\\ion{i}{} was found at the same strength around all galaxies \\citep{Tumlinson11,Thom12}. Additionally, connections have been made between the CGM and properties of the host galaxy, including: increased H\\ion{i}{} content of the CGM with larger interstellar medium (ISM) gas masses \\citep[COS-GASS;][]{Borthakur15}, the presence of extended gas reservoirs around galaxies of all stellar mass \\citep[COS-Dwarfs;][]{Bordoloi14}, and enhanced metal content around starbursting hosts \\citep[COS-Burst;][]{Borthakur13,Heckman17}.\n\n\nAn important stage in the evolution of galaxies is when their central supermassive black holes are actively accreting material. This active galactic nucleus (AGN) phase may be responsible for the removal of gas from star forming reservoirs within galaxies via winds and outflows \\citep{Veilleux05,Tremonti07,Sturm11,Woo17}, and has been associated with the evolution of galaxies off the star-forming main sequence to passive galaxies \\citep{Springel05,Schawinski07,Fabian12,Bluck14,Bluck16}. In addition, radio-mode feedback and radiation from the AGN keeps the CGM hot, buoyant, and consistently ionized \\citep[][]{McNamara07,Bower17,Hani17}, as well as preventing gas from returning to the host galaxy. Such processes have been proposed to be responsible for O\\ion{vi} bimodality seen in the CGM by COS-Halos without an active AGN \\citep{Oppenheimer17}.\n\nObservationally linking the environmental and feedback effects of AGN hosts with their CGM has primarily been done through the use of projected QSO-QSO pairs at higher redshifts. This technique has the added benefit of looking at the role of a stronger and weaker QSO radiation fields located in the respective transverse (background QSO) and line-of-sight (foreground QSO; i.e.~along the outflow) CGM \\citep{Bowen06,Farina13,Johnson15}. Cool gas traced by H\\ion{i}{} and Mg\\ion{ii} is anisotropically distributed about the QSO, with larger column densities of H\\ion{i}{} preferentially found along the transverse direction \\citep{QPQ1,Farina13}; suggesting that radiation from the QSO does not affect the transverse medium \\citep[][]{QPQ2,QPQ7,Farina14}. An excess of cool gas (relative to the intergalactic medium) has been found all the way out to one Mpc, with a stronger enhancement at smaller impact parameters \\citep{QPQ6}. When the QSO-QSO pairs are split by the bolometric luminosity of the QSO host, the Mg\\ion{ii} covering fraction is larger for high-luminosity QSOs (covering fraction of $\\approx 60$\\% for luminosities of ${\\rm L_{Bol}\\geq45.5}$ erg s$^{-1}$) compared to low luminosity QSOs \\citep[$\\approx20$\\%, ${\\rm L_{Bol}\\leq45.5}$ erg s$^{-1}$;][]{Johnson15}. All of these observations of excess cool gas around luminous QSOs is suggestive of either a viewing angle effect with the ionizing radiation exciting cool gas along the line of sight to the QSO, or an environmental effect of haloes hosting massive QSOs such as debris from galaxy interactions fuelling QSO activity \\citep[][]{QPQ6,Farina14,Johnson15}.\n\n\n\n\n\n\n\nMost of the work described above has focused on high luminosity quasars. However, there has been little focus on how the less luminous but more common Seyfert-like AGN shape their surrounding CGM. In the only observational study of the CGM surrounding Seyfert galaxies, \\cite{Kacprzak15} found a low (10\\%) Mg\\ion{ii} $\\lambda$ 2796 \\AA{} covering fraction around 14 AGN (between 100 and 200 kpc) in the transverse direction relative to field and QSO host galaxies, but a reservoir of cool gas still exists along the line of sight to the AGN. They suggest that AGN-driven outflows are destroying the cool gas in the transverse direction (i.e.~along the outflow), suggesting that the difference between their observations of the CGM of AGN-dominated galaxies with previous observations of QSOs \\citep[e.g.][]{QPQ6} is caused by the viewing angle of the AGN.\n\nPredictions from zoom-in simulations of galaxies taken from the EAGLE cosmological simulation \\citep{Schaye15} suggest that radiative feedback from the AGN should ionize the gas out to a distance of two virial radii \\citep{Oppenheimer13,Segers17}. After implementing non-equilibrium ionization into their models, \\cite{Oppenheimer13} have predicted that AGN proximity fossil zones exist around galaxies that host (or have hosted) bright AGN, with the metals remaining in an over ionized state for several megayears \\citep[depending on the luminosity duty cycle and lifetime of the AGN;][]{Segers17,Oppenheimer18}. However, the detailed CGM properties in simulations can be quite sensitive to the size of the CGM clouds, implementation of feedback, and different recipes between codes \\citep{Stinson12,Gutcke17,Nelson17}.\n\n\n\n\n\nIn this paper, we investigate the observational properties of the CGM around galaxies hosting Type II Seyfert AGN (which we will henceforth simply refer to as AGN). We measure the rest-frame equivalent widths (EWs) of a range of ionization species present in the CGM material probed by QSO sightlines near 20 AGN-host galaxies. We provide a systematic comparison to non-AGN galaxies observed in the literature to quantify whether the CGM around AGN host galaxies is different from their counterparts. Throughout the paper, we assume a flat $\\Lambda$CDM Universe with $H_{0}=67.8~{\\rm km~s^{-1}~Mpc^{-1}}$ and $\\Omega_{M}=0.308$ \\citep{Planck15}.\n\n\n\n\n\\section{Data}\n\\subsection{Sample selection and properties}\n\\label{sec:SampProps}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{tb_BPT_Lbol.pdf}\n\\caption{The BPT diagram of all SDSS galaxies with spectroscopic observations (blue shaded region; only showing SDSS galaxies with $>5\\sigma$ detections of diagnostic emission lines). The solid circles show the COS-AGN galaxies, and are coloured based on their AGN luminosity. The dashed pink and green lines denote the~\\citet[K01]{Kewley01} and~\\citet[K03]{Kauffmann03} cuts typically used to select AGN and composite galaxies. LINERS classified using the \\citet{Kewley06} emission line metrics (see Table \\ref{tab:SightProps}) are denoted with a white dot on top of the datapoint.}\n\\label{fig:BPT}\n\\end{center}\n\\end{figure}\n\n\n\nThe QSO sightlines through the CGM of AGN galaxies were selected by cross-matching coordinates of Sloan Digital Sky Survey \\citep[SDSS;][]{Abazajian09} galaxies hosting AGN with the locations of UV-bright QSOs ($17300$m\\AA{}) between 100 and 200 kpc of the AGN. The Mg\\ion{ii} EW threshold adopted by \\cite{Kacprzak15} is typical of that used to select strong H\\ion{i}{} absorbers at low redshifts \\citep[log(N(H\\ion{i}{})\/cm$^{-2}$)$\\gtrsim18.5$;][]{Rao06}. Translating this Mg\\ion{ii} EW threshold into a corresponding Ly$\\alpha$ EW theshold for these column densities of gas yields a much higher than the threshold used in this work (EW $\\gtrsim 1300$ m\\AA{} --- assuming a minimum broadening parameter of 5 km~s$^{-1}${} --- compared to EW $>124$ m\\AA{} for COS-AGN). Using this larger threshold, only one of the COS-AGN sightlines has an EW $\\gtrsim 1300$ m\\AA{}, giving a consistent result with the observations from \\cite{Kacprzak15}. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_covering_fractions_R.pdf}\n\\caption{Covering fractions (with 1$\\sigma$ errors) of gas within $\\pm500$ km~s$^{-1}${} of the host galaxy for a variety of species measured in COS-AGN and control-matched samples. Top panel: The global covering fractions measured across all impact parameters from Table \\ref{tab:covfracs} are shown for the COS-AGN galaxies (green circles), and the control-matched passive (red squares) and star-forming (blue diamonds) galaxies. Bottom panels: The covering fractions of an individual species as a function of impact parameter ($\\rho_{\\rm imp}${}), split into two bins by the median $\\rho_{\\rm imp}${} of the COS-AGN sample (164 kpc). The EW thresholds (EW$_{\\rm thrsh}$) used to determine the covering fraction for each species (including the measurements presented in the top panel) are given above the corresponding species panel in the bottom two rows. The horizontal error bars represent the entire range of $\\rho_{\\rm imp}${} probed by each sample within the respective bin. The three points are offset from the centre of each bin by a small amount for clarity. No points are shown for a species that do not have spectral coverage of the corresponding absorption line. }\n\\label{fig:CovFracR}\n\\end{center}\n\\end{figure*}\n\n\n\n\n\n\\subsubsection{Relative EW analysis}\n\nFigure \\ref{fig:EWRho} shows the raw EW values for a variety of ionic species as a function of $\\rho_{\\rm imp}${}. The COS-AGN points are colour-coded by their L$_{\\rm AGN}${}. The control-matched sample is shown as black points, while the grey points are the remaining un-matched literature sightlines. The bold COS-AGN points are CGM sightlines that have spectroscopic companions (see Section \\ref{sec:SampProps}). The top left panel demonstrates that the Ly$\\alpha$ EWs for the COS-AGN follow the general trend of decreasing EW as a function of $\\rho_{\\rm imp}${} seen in previous low redshift studies \\citep[][]{Chen10, Werk14,Borthakur16}. For metal species, the lack of detections in both the COS-AGN and control sample makes a comparative analysis difficult. For the remainder of this section, we focus only on the Ly$\\alpha$ EWs.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_EW_rho_all.pdf}\n\\caption{The rest-frame equivalent widths (within $\\pm500$ km~s$^{-1}${} of the host galaxy) as a function of impact parameter. The coloured circles (EW detections) and triangles (EW upper limits) represent the COS-AGN sample, and are colour coded by the bolometric luminosity of the AGN (L$_{\\rm AGN}${}). The median error bar on the COS-AGN EW measurements is given by the black error bar in the top right region of each panel. The black squares show the EW values of the control-matched galaxies, while the grey squares denote the rest of the literature comparison sample. Data outlined with a thick black line represent COS-AGN systems with nearby galaxies. COS-AGN galaxies flagged as LINERs are indicated by small white dots on top of the respective data points.}\n\\label{fig:EWRho}\n\\end{center}\n\\end{figure*}\n\nIn order to quantify any difference between the COS-AGN and control samples, we calculate $\\Delta$log(EW\/m\\AA), which is defined as ${\\rm \\Delta log(EW\/m\\AA) = log[EW_{AGN}\/median(EW_{Controls})]}$, such that a positive ${\\rm \\Delta log(EW\/m\\AA)}${} would imply that the CGM surrounding the AGN has a larger EW than the median of its control-matched galaxies. The left panel of Figure \\ref{fig:deltaEWRho} shows $\\Delta$log(EW) for Ly$\\alpha$ as a function of $\\rho_{\\rm imp}${} for the COS-AGN galaxies. For reference, the grey band represents the interquartile range of $\\Delta$log(EW) for the entire literature sample matched to itself. The right panel shows the distributions of $\\Delta$log(EW) for the COS-AGN (orange) and literature (grey) galaxies, with medians of the distributions indicated by the arrows. \n\nTo include the non-detections of the controls in the analysis, we calculate the median EW of the controls twice: once including limits as if they were detections, and once setting the non-detected EWs to 0~m\\AA{}. These median EWs span the range of true median EW if the absorption lines were actually detected. For this calculation, we only include non-detections when the upper limits are more sensitive (i.e. smaller) than the largest detected EW as these limits are constraining enough to affect the median value. The corresponding $\\Delta$log(EW) range is shown on Figure \\ref{fig:deltaEWRho} as the thick grey errorbars. The 1$\\sigma$ jackknife errors on $\\Delta$log(EW) are typically smaller than the size of the points.\n\nThe median $\\Delta$log(EW) of the COS-AGN sample is enhanced by $+0.10\\pm0.13$ dex relative to the controls. Repeating this control-matching experiment for the literature sample yields a median $\\Delta$log(EW) of $0.00\\pm0.28$. Note that the errors on these median $\\Delta$log(EW) represent the median absolute deviation (MAD) of the distribution. A KS test rejects the null hypothesis that the distributions of $\\Delta$log(EW) for the COS-AGN and control samples are same at 20\\% confidence. When the LINER galaxies are removed from the COS-AGN sample, the median $\\Delta$log(EW) changes to $+0.10\\pm0.15$, and the KS test yields a rejection of the null hypothesis at 14\\% confidence.\n\n\n\n\n\n\n\n\nTo check if there is any effect from splitting the control sample into star-forming and passive galaxies, we measured the $\\Delta$log(EW) for all star-forming galaxies in the non-AGN literature sample to their control-matched passive counterparts. The median $\\Delta$log(EW) obtained for star-forming galaxies relative to the matched passive controls is $+0.03\\pm0.26$, while a KS-test reveals the null hypothesis of the $\\Delta$log(EW) distributions for the star-forming and passive galaxies are the same is rejected at 38\\% confidence. Therefore there is no significant difference in the offset of the AGN hosts from the controls that would be caused by the star formation rate.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_dEW_rho_HI.pdf}\n\\caption{The difference in the Ly$\\alpha$ equivalent width of the COS-AGN sightlines relative to their control matched counterparts ($\\Delta$log(EW\/m\\AA{})) as a function of impact parameter ($\\rho_{\\rm imp}${}). The EW measure material within $\\pm500$ km~s$^{-1}${} of the host galaxy. The points are colour-coded by the bolometric luminosity of the AGN (L$_{\\rm AGN}${}). The errorbars denote how the maximal shift on including control matched EW upper limits in the calculation of $\\Delta$log(EW\/m\\AA{}). For reference, the horizontal grey band represents the interquartile range of $\\Delta$EW of the literature sample control matched with itself. The normalized distributions of $\\Delta$EW for both the COS-AGN and control galaxies are shown in the right panel, with the median of each histogram given by an arrow. Data outlined by a thick black line are COS-AGN sightlines flagged as having nearby galaxies. COS-AGN galaxies flagged as LINERs are indicated by small white dots on top of the respective data points.}\n\\label{fig:deltaEWRho}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Stacked Spectra}\n\n\nThe results in the previous sub-section indicate a possible (but not significant) difference in the Ly$\\alpha$ absorption properties of the COS-AGN sample, and possibly in some of the metal species as well. However, that analysis is limited by the modest S\/N of the data, the small sample size and lack of detections of metal species. Therefore, in this section we consolidate the data by stacking all of the COS-AGN spectra and comparing it to a stack of the control sample. In brief, all spectra are shifted to the rest-frame (using the redshift of the strongest component of the H\\ion{i}{} absorption profile) and rebinned to a linear dispersion of 0.064 \\AA{} pixel$^{-1}$ (similar to the resolution of the COS-AGN spectra). These rebinned spectra are mean combined without any weighting. We note that either using the systemic redshifts of the galaxies or using different weighting schemes does not significantly change the results.\n\nTable \\ref{tab:EWStack} gives the measured EWs of the various absorption lines of interest from the final stacked COS-AGN spectrum. We require that the absorption line be detected at $>3\\sigma$, otherwise the EW is set to a $3\\sigma$ upper limit. The stacked EW errors are calculated using a standard jackknife approach by removing each COS-AGN from the stacked spectrum and recalculating the EW. The numbers in brackets in each column give the EW offsets from removing the `strongest' (${jack,min}$) and `weakest' (${jack,max}$) absorber sightline from the stack in a jackknife fashion (i.e. these are the maximal variations in the EW from the jackknife), as well as the number of sightlines that contributed to the stack (N$_{spec}$). The four columns on the right are the measured EWs when the stacking process is only applied to COS-AGN sightlines split into two bins of log(L$_{\\rm AGN}${}) and $\\rho_{\\rm imp}${} at the median value of the COS-AGN sample (log(L$_{\\rm AGN}${}\/erg s$^{-1}$)$=42.9$ and $\\rho_{\\rm imp}${}$=164$ kpc; respectively).\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Measured EW of stacked spectra}\n\\label{tab:EWStack}\n\\begin{tabular}{lc|c|cc|cc|}\n\\hline\nIon & Line & \\multicolumn{5}{c}{Stacked EW ($_{jack,min}^{jack,max}$; N$_{spec}$) [m\\AA{}]} \\\\\n & [\\AA] & All sight-lines&\tlog(L$_{\\rm AGN}${}\/erg s$^{-1}$)$\\leq$42.9 & log(L$_{\\rm AGN}${}\/erg s$^{-1}$)$>$42.9& $\\rho_{\\rm imp}${}$\\leq$164 kpc& $\\rho_{\\rm imp}$$>$164 kpc \\\\\n\\hline\nH\\ion{i}{}\t & 1215\t & 739$\\pm$25 ($_{-69}^{+44}$; 14)\t & 937$\\pm$53 ($_{-116}^{+77}$; 7)\t & 539$\\pm$38 ($_{-66}^{+70}$; 7)\t & 799$\\pm$41 ($_{-68}^{+73}$; 7)\t & 680$\\pm$66 ($_{-176}^{+87}$; 7)\t\\\\\nC\\ion{ii}\t & 1036\t & $<$80 (2)\t & $<$80 (2)\t & . . . & . . . & $<$80 (2)\t\\\\\nC\\ion{ii}\t &\t1334& $<$37 (11)\t & $<$53 (6)\t & $<$50 (5)\t & $<$54 (5)\t & $<$51 (6)\t\\\\\nC\\ion{iv}\t & 1548\t & 194$\\pm$13 ($_{-40}^{+20}$; 11)\t & 236$\\pm$38 ($_{-89}^{+61}$; 5)\t & $<$194 (6)\t & 185$\\pm$17 ($_{-38}^{+29}$; 6)\t & 205$\\pm$39 ($_{-96}^{+53}$; 5)\t\\\\\nC\\ion{iv}\t & 1550\t & 131$\\pm$9 ($_{-24}^{+15}$; 13)\t & 132$\\pm$21 ($_{-56}^{+21}$; 6)\t & $<$167 (7)\t & $<$132 (7)\t & 159$\\pm$24 ($_{-52}^{+33}$; 6)\t\\\\\nN\\ion{v}\t & 1238\t & $<$32 (13)\t & $<$42 (6)\t & $<$47 (7)\t & $<$47 (7)\t & $<$43 (6)\t\\\\\nN\\ion{v}\t & 1242\t & $<$38 (10)\t & $<$46 (5)\t & $<$60 (5)\t & $<$51 (6)\t & $<$54 (4)\t\\\\\nO\\ion{i}\t &\t1302& $<$31 (12)\t & $<$45 (5)\t & $<$42 (7)\t & $<$47 (6)\t & $<$39 (6)\t\\\\\nO\\ion{vi}\t &\t1037& $<$379 (1)\t & $<$379 (1)\t & . . . & . . . & $<$379 (1)\t\\\\\nSiII\t & 1190\t & $<$29 (10)\t & $<$36 (5)\t & $<$46 (5)\t & $<$61 (3)\t & $<$32 (7)\t\\\\\nSi\\ion{ii}\t & 1190\t & $<$32 (9)\t & $<$44 (4)\t & $<$46 (5)\t & $<$61 (3)\t & $<$37 (6)\t\\\\\nSi\\ion{ii}\t & 1260\t & 88$\\pm$8 ($_{-28}^{+11}$; 15)\t & 94$\\pm$28 ($_{-75}^{+30}$; 6)\t & 84$\\pm$10 ($_{-17}^{+15}$; 9)\t & $<$101 (6)\t & 103$\\pm$18 ($_{-47}^{+21}$; 9)\t\\\\\nSi\\ion{iii}\t & 1206\t & 152$\\pm$16 ($_{-62}^{+18}$; 12)\t & 211$\\pm$37 ($_{-98}^{+33}$; 8)\t & $<$120 (4)\t & $<$98 (6)\t & 238$\\pm$40 ($_{-112}^{+32}$; 6)\t\\\\\nSi\\ion{iv}\t & 1393\t & $<$44 (11)\t & $<$69 (4)\t & $<$57 (7)\t & $<$59 (6)\t & $<$66 (5)\t\\\\\nFe\\ion{ii}\t & 1144\t & $<$29 (12)\t & $<$34 (7)\t & $<$52 (5)\t & $<$49 (5)\t & $<$35 (7)\t\\\\\nFe\\ion{ii}\t & 1608& $<$102 (8)\t & $<$94 (2)\t & $<$134 (6)\t & $<$129 (5)\t & $<$168 (3)\t\\\\\n\\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Measured EW of stacked control spectra}\n\\label{tab:EWStackControl}\n\\begin{tabular}{lc|c|cc|cc|}\n\\hline\nIon & Line & \\multicolumn{5}{c}{Stacked EW ($_{jack,min}^{jack,max}$; N$_{spec}$) [m\\AA{}]} \\\\\n & [\\AA] & All sight-lines&\t log(sSFR\/yr$^{-1}$)$<-11$ & log(sSFR\/yr$^{-1}$)$\\geq-11$ & $\\rho_{\\rm imp}${}$\\leq$164 kpc& $\\rho_{\\rm imp}$$>$164 kpc \\\\\n\\hline\nH\\ion{i}{}\t & 1215\t & 577$\\pm$12 ($_{-39}^{+24}$; 43)\t & 485$\\pm$30 ($_{-84}^{+52}$; 19)\t & 649$\\pm$21 ($_{-65}^{+40}$; 24)\t & 781$\\pm$20 ($_{-50}^{+50}$; 25)\t & 239$\\pm$15 ($_{-37}^{+30}$; 18)\t\\\\\nC\\ion{ii}\t & 1334\t & 36$\\pm$3 ($_{-11}^{+6}$; 43)\t & $<$29 (19)\t & 46$\\pm$5 ($_{-20}^{+9}$; 24)\t & 62$\\pm$5 ($_{-18}^{+10}$; 25)\t & $<$31 (18)\t\\\\\nSi\\ion{ii}\t & 1190\t & $<$22 (15)\t & $<$39 (5)\t & $<$26 (10)\t & $<$22 (15)\t & . . . \\\\\nSi\\ion{ii}\t & 1260\t & 56$\\pm$4 ($_{-14}^{+7}$; 33)\t & 85$\\pm$10 ($_{-28}^{+17}$; 16)\t & $<$42 (17)\t & 103$\\pm$6 ($_{-20}^{+8}$; 23)\t & $<$55 (10)\t\\\\\nSi\\ion{iii}\t & 1206\t & 98$\\pm$4 ($_{-15}^{+9}$; 39)\t & 77$\\pm$9 ($_{-28}^{+14}$; 21)\t & 126$\\pm$8 ($_{-24}^{+15}$; 18)\t & 156$\\pm$7 ($_{-20}^{+14}$; 25)\t & $<$47 (14)\t\\\\\nSi\\ion{iv}\t & 1393\t & $<$35 (37)\t & $<$44 (17)\t & $<$55 (20)\t & 58$\\pm$5 ($_{-15}^{+14}$; 23)\t & $<$65 (14)\t\\\\\nSi\\ion{iv}\t & 1402\t & $<$27 (37)\t & $<$34 (17)\t & $<$41 (20)\t & $<$33 (21)\t & $<$44 (16)\t\\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\nA similar procedure was completed for all the spectra in the literature sample that were used in the control sample. Each control sightline in the stacked spectrum was weighted by the number of times the sightline was matched to a unique AGN host (such that the most frequently matched control sightline was given a higher weighting). Although there is very little difference in the measured EW without such a weighting, we elect to use this weighting scheme such that the derived jackknife errors represent the true range in EWs when a given sightline is excluded. The EWs measured from the stacked spectrum of the control galaxies is given in Table \\ref{tab:EWStackControl}. We repeat the stacked EW calculations of splitting the controls by the median $\\rho_{\\rm imp}${} of the COS-AGN sample and log(sSFR\/yr$^{-1}$)$=-11$ (i.e. whether the controls are star-forming or passive; see Table \\ref{tab:EWStackControl}). Note that the COS-GASS results \\citep{Borthakur15,Borthakur16} focus on a smaller subset of species (H\\ion{i}{}, C\\ion{ii}, Si\\ion{ii}, Si\\ion{iii}, and Si\\ion{iv}), thus we are only able to provide a stacked spectrum for these species. We point out that the Si\\ion{iv} EW of the control stacked spectrum is poorly constrained due to an uncertain continuum in some of the literature sample spectra.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_dEW_stack_full.pdf}\n\\caption{$\\delta$log(EW) measured from the stacked spectra for a variety of detected species. Top panel: The $\\delta$log(EW) is shown for the COS-AGN sightlines relative to: all controls (black circles), AGN and controls for small ($\\rho_{\\rm imp}$$<164$ kpc; red circles) and large ($\\rho_{\\rm imp}${}$\\geq164$ kpc; blue circles) impact parameters. Errorbars on all points represent the possible range in $\\delta$log(EW) spanned by the maximal jackknife errors (i.e.~$_{-jack,min}^{+jack,max}$). Upper and lower limits are plotted when an absorption line is detected in either the COS-AGN or control stack (respectively), but not the other. We note that the lower errorbar on the $>164$ kpc $\\delta$log(EW) for Si\\ion{iii} 1206~\\AA{} line represents how shallow the lower limit becomes upon including the jackknife errors. The bottom ten panels show the absorption profiles of the stacked spectra (the COS-AGN spectrum in the middle row; the control spectrum in the bottom row) for each element. The colour coding of which sightlines are included is the same as in the top panel. The absorption lines are offset vertically by 0.5 in relative flux for clarity.}\n\\label{fig:StackdEW}\n\\end{center}\n\\end{figure*}\n\n\nWe repeat a similar differential EW analysis as above, where we calculate $\\delta$log(EW)$={\\rm log(EW_{AGN}) - log(EW_{Control})}$ using the EWs derived from the respective stacked spectra (Tables \\ref{tab:EWStack} and \\ref{tab:EWStackControl}). The top panel of Figure \\ref{fig:StackdEW} provides a comparison of EW between all the AGN sightlines and all controls (black points). The errorbars represent the combination of the maximal jackknife errors, providing the entire range of possible $\\delta$log(EW) from removing a single sightline from each sample. The stacking confirms the results from Figure \\ref{fig:deltaEWRho}, where the COS-AGN sightlines have an enhanced Ly$\\alpha$ relative to the controls, with an enhancement of $\\delta$log(EW)$\\approx0.17$ dex. However, there is a negligible difference in $\\delta$log(EW) for the metal species detected at all impact parameters. Removing the absorption surrounding the LINER galaxies does not change the qualitative picture presented in Figure \\ref{fig:deltaEWRho}; the measured $\\delta$EW from the LINER-free stacks shift $\\leq0.08$ dex (in either direction). \n\nThe additional red and blue points in the top panel of Figure \\ref{fig:StackdEW} show the values of $\\delta$log(EW) calculated only when including only sightlines with $\\rho_{\\rm imp}${} smaller or larger than the median value of the COS-AGN sample (164 kpc; respectively). Splitting the stacking into the `inner' and `outer' CGM around AGN hosts uncovers that the enhancement seen in the Ly$\\alpha$ $\\delta$log(EW) is driven by the the COS-AGN sightlines that probe $\\rho_{\\rm imp}${}$\\geq164$ kpc ($\\delta$log(EW)$=+0.45\\pm0.05$~dex; using standard jackknife EW errors). This enhancement in the outer CGM gas is also seen by the detections of the cool gas tracers Si\\ion{ii} ($\\delta$log(EW)$>0.27$~dex) and Si\\ion{iii} ($\\delta$log(EW)$>0.75$~dex) in COS-AGN, even after removing the strongest metal absorber (towards QSO J0852+0313; see Figure \\ref{fig:J0852}), whilst these species are not detected in the stacked spectrum of the control galaxies. We remind the reader that this enhancement was suggested in our covering fraction analysis of Si\\ion{iii} (Figure \\ref{fig:CovFracR}). For the inner CGM, the stacked spectrum hints that there is a deficit of metal species around AGN galaxies relative to the control sample. The combination of the excess Ly$\\alpha$ EW and tentative Si\\ion{ii} and Si\\ion{iii} EW enhancements at high $\\rho_{\\rm imp}${} is suggestive that these EW enhancement are tracing the cool gas phase of the CGM, rather than just the H\\ion{i}{} gas kinematics. We note that the inner CGM distribution has a slightly higher median M$_{\\star}${} than the outer CGM bin (log(M$_{\\star}$\/M$_{\\odot}$){} of 10.8 dex relative to 10.5 dex), and the inner and outer CGM bins contain approximately the same ratio of passive and star-forming galaxies. We remind the reader that this discrepancy in the median M$_{\\star}${} of each bins is on the order of the size of our control matching tolerance.\n\n\n\n\n\n\\section{Discussion}\n\nIn the previous section, we demonstrated that the CGM around AGN hosts is not much different than the control-matched non-AGN hosts. Statistically significant differences are found in the analysis of the stacked spectrum; the COS-AGN systems have a higher EW of Ly$\\alpha$ (and potentially Si\\ion{ii} and Si\\ion{iii} as well) relative to their non-AGN host counterparts at high impact parameters ($\\rho_{\\rm imp}${} $\\geq 164$ kpc; $\\delta$log(EW\/m\\AA{})$=+0.45\\pm0.05$~dex). The kinematics of the gas traced by the absorption show the gas is likely bound to the halo, whilst no strong kinematic offsets relative to their host are present. We now consider whether these observations are a result of the AGN directly influencing the CGM of the host galaxy, or an effect of the environments (either internal or external to the host galaxy) in which AGN are typically found.\n\n\n\n\\subsection{Are we seeing the effects of AGN feedback?}\n\n\\subsubsection*{Mock COS-AGN simulation}\n\\label{sec:Sims}\nThe radiation field of the AGN may be expected to have a profound effect on the ionization structure of the CGM, by enhancing the ionizing radiation field to which the surrounding gas is subjected, and in the case of metals even long after the AGN has turned off \\citep{Oppenheimer13,Segers17,Oppenheimer18}. To quantify the expected effects of turning on and off the AGN ionizing radiation spectrum on the CGM of a COS-AGN galaxy, we created a mock COS-AGN survey using previously run cosmological zoom-in hydrodynamical simulations with an additional AGN ionizing source following previous work on non-equilibrium ionization effects \\citep{Oppenheimer13,Oppenheimer16}. The purpose of these simulations is to isolate the effect of adding a constant AGN ionizing radiation source versus a control sample without AGN ionizing radiation to study the effect of the changed ionization structure of the CGM. This exploration is not meant to model or consider the effect of AGN feedback mechanically transforming the CGM via superwinds as the inclusion of the AGN radiation is not tied to the accretion onto the central super massive black hole. The AGN radiation is added to these simulations at the position of the central super massive black hole and only alters the ionization states of the CGM.\n\nFor our simulation suite we selected three representative galaxy haloes from \\cite{Oppenheimer16} that were chosen from the EAGLE volume \\citep{Crain15,Schaye15}. These three halos are representative of the properties of the COS-AGN sample at $z=0.075$ [log(M$_{\\star}$\/M$_{\\odot}$)$=10.3$, $10.9$, $11.0$, log(sSFR \/ yr$^{-1}$)$=-10.3$, $-10.7$, $-11.0$, residing in haloes log(M$_{200}$\/M$_{\\odot}$)$=12.1$, $12.8$, $13.3$; respectively]. The haloes were ran at EAGLE HiRes resolution (gas particle mass of $2.3\\times10^{5}~{\\rm M_{\\odot}}$, dark matter particle mass of $1.2\\times10^{6}~{\\rm M_{\\odot}}$, and softening length of 350 proper pc) using the Recal feedback prescription \\citep{Schaye15} and are zooms Gal001, Grp003, and Grp008 listed in Table 1 of \\citet{Oppenheimer16} from the initial conditions ($z=127$).\n\nTo include the effects of the AGN ionizing spectrum on the CGM at different luminosities, we inserted an additional AGN ionizing source \\citep[with an ionizing spectrum from][]{Sazonov04} placed at the centre of the galaxy, instantaneously affecting the radiation field from $z=0.1$ onwards and reaches equilibrium at $z=0.075$. We note that this AGN radiation model has no dynamical effect on the gas accretion or outflows from the AGN, and is not tied to the accretion of material onto the central super-massive black hole. A range of AGN luminosities was used to match the COS-AGN luminosities (log(L$_{\\rm AGN}$\/erg s$^{-1}$){}=42--44 dex, in increments of 0.5 dex), and a control run with no AGN radiation was included for creating the control sample. The time-dependent ionization from the AGN in addition to the \\cite{Haardt01}\\footnote{The \\cite{Haardt01} background is adopted as it better reproduces the statistics of the Ly$\\alpha$ forest in the EAGLE volumes and other simulations \\citep{Rahmati15,Kollmeier14}.} ultra-violet background is followed using the \\cite{Richings14a} ionization network. \n\n\nA mock COS-AGN sample was then generated from this simulation suite to match the properties of the COS-AGN sample. For each observed galaxy in COS-AGN, a simulated galaxy was selected by matching the stellar mass (with a log(M$_{\\star}$\/M$_{\\odot}$){}$=\\pm0.5$~dex tolerance), star formation rate (log(sSFR \/ yr$^{-1}$){}$=\\pm0.5$~dex tolerance), and AGN luminosity (log(L$_{\\rm AGN}$\/erg s$^{-1}$){}$=\\pm0.25$~dex) of the simulated haloes to each COS-AGN galaxy. If multiple matches were identified, a random simulated halo was selected such that a mock sample contains the same number of galaxies as the actual COS-AGN sample (20 galaxies). A mock spectrum was generated for each galaxy using the \\textsc{SpecWizard} package \\citep[][]{Theuns98,Schaye03} by placing a randomly oriented quasar sightline through the CGM at the corresponding impact parameter of the matched COS-AGN galaxy sightline. A control sample was generated in a similar fashion, but all AGN ionizing sources in the simulated galaxies were set to L$_{\\rm AGN}${}$=0$~erg~s$^{-1}$. The above procedure was repeated 200 times (including choosing another random halo if multiple haloes were matched), in order to produce 200 mock survey samples for both the AGN and control samples. These 200 realizations of the COS-AGN survey were needed to reproduce the range in measured EW variations for the metal species between the mock samples, and is used to quantify the expected spread in EWs from different orientations and from halo to halo variations. Further details on how these mock spectra were generated will be presented in a forthcoming paper (Horton et al., in prep.).\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_sim_dEW_stack_full.pdf}\n\\caption{The top panel shows the measured $\\delta$EW using a stacked spectrum from the zoom-in simulations of AGN hosts, split into bins of $\\rho_{\\rm imp}${} (all $\\rho_{\\rm imp}${}, $<164$ kpc and $\\geq164$ kpc; black, red and blue points, respectively). The respective open symbols show the $\\delta$EW measured in the observations (i.e.~identical points presented in the top panel of Figure \\ref{fig:StackdEW}). Errorbars were calculated using the jackknife approach identical to that used for the observations. The bottom panels show the simulated stacked velocity profiles for each species for the AGN (middle row) and non-AGN control run (bottom row). The dashed lines represent the continuum levels of the absorption profiles. The Ly$\\alpha$ are staggered by 0.5 in relative flux for clarity. The relative flux scale has been enlarged by a factor of four for the metal line profiles (indicated by a $\\times4$ in the bottom right of each panel), with the dashed continuum lines separated by a relative flux of 0.125. For the outer CGM, the simulations predict little change in the EW of H\\ion{i}{} and Si\\ion{iii} due to the ionizing radiation from the AGN, which is in stark contrast to the COS-AGN observations similarly presented in Figure \\ref{fig:StackdEW}.}\n\\label{fig:Sims}\n\\end{center}\n\\end{figure*}\n\n\\subsubsection*{Stacked mock COS-AGN spectra}\nWe assessed the relative impact of the AGN ionizing radiation on the measured simulated EWs by repeating the stacking procedure above, but with the simulated data. We computed the relative EW ratio $\\delta$log(EW) as with the observations. Note that this relative EW analysis removes any systematic differences between the observed and simulated EWs caused by assumptions in the physics models (e.g. feedback) or ability to resolve small clouds in the simulations \\citep{Schaye07,Stinson12,Crighton15,Gutcke17,Nelson17}. A quantitative comparison between the simulated and observed EWs will be presented in Horton et al.~(in prep.), although we note that the simulations produce a systematically weaker H\\ion{i}{} EW from the observations, while metal lines show better agreement with the observed data \\citep[see][]{Oppenheimer16,Oppenheimer17}. \n\nThe simulated stacked sightline spectra were created from the mocks by centering the velocity profile on the redshift of the simulated galaxy, rebinning the mock spectra to 15 km~s$^{-1}${}, and mean stacking these rebinned spectra. Three simulated stacked spectra were created for the same three bins of impact parameters that were previously used for the observed data: all $\\rho_{\\rm imp}${}, $\\rho_{\\rm imp}${}$<164$ kpc, and $\\rho_{\\rm imp}${}$\\geq164$ kpc. Jackknife errors on the measured EWs were computed by removing all 200 mock spectra associated with the matched COS-AGN sightline which contributed the most and least to the derived EW. This approach for calculating the simulated jackknife error is identical to that used for the observations. Using the same analysis for our COS-AGN sightlines, Figure \\ref{fig:Sims} presents the $\\delta$log(EW) analysis for these simulated stacked spectra (solid points in the top panel) to quantify the effect of including AGN radiation on the CGM relative to the control simulations. The inclusion of an AGN ionizing spectrum results in a negligible change in the EW as a function of impact parameter for Ly$\\alpha$. At high impact parameters ($\\rho_{\\rm imp}${}$\\geq 164$~kpc), the simulated results are in contrast to the observations (open points on Figure \\ref{fig:Sims}) where the COS-AGN sample shows a significant enhancement in the Ly$\\alpha$ EW relative to the control sample, as well as a potential enhancement for Si\\ion{ii} and Si\\ion{iii}. We do note that for the inner impact parameter bin, the results from the simulation are consistent with the lack of metal species detected of our COS-AGN stacked spectrum ($\\rho_{\\rm imp}${}$<164$ kpc; Figure \\ref{fig:StackdEW}). \n\n\nCompounded by the fact that the H\\ion{i}{} ionizing radiation from a \\cite{Sazonov04} AGN at the observed $\\rho_{\\rm imp}${} of the COS-AGN sample is weaker ($\\lesssim25\\%$ ) than the ionizing radiation from the UV background \\citep{Haardt01} at most of the impact parameters probed by COS-AGN (see the bottom right panel of Figure \\ref{fig:SampDists}), it is unlikely that photoionizing radiation from the AGN is responsible for the observed difference in EWs between the COS-AGN and the control samples. We note that at the impact parameters of COS-AGN, the light travel time of radiation from the AGN ($\\sim5\\times10^{5}$ yr) is comparable to the AGN's lifetime \\citep[$\\lesssim10^{6}$ yr;][]{Schirber04,Goncalves08,Furlanetto11,Keel12}, implying that photoionization events caused by an AGN likely require previous AGN cycle(s) to have ionized the CGM gas, and have remained in the predicted long-lived fossil zone around the AGN \\citep{Oppenheimer13,Segers17} provided an AGN has been active within the last several Myrs. Unfortunately, our simulations predict that low ionization species cannot distinguish the presence of fossil zones in COS-AGN due to the low intensity of the photo-ionizing radiation.\n\n\nDespite being unable to probe the effects of the lower ionization species, ions such as C\\ion{iv}, N\\ion{v}, and O\\ion{vi} are better indicators of these proximity zone fossil. We highlight that in our simulated COS-AGN haloes, C\\ion{iv} and N\\ion{v} are still sensitive to the ionizing radiation of the harder AGN ionizing spectrum relative to the \\cite{Haardt01} UV background \\citep[see figure 3 in][]{Segers17}, as demonstrated by the enhanced EWs for these ionization species out to 300 kpc. However, to observe such an excess with N\\ion{v} 1238~\\AA{} would require spectra with S\/N of $\\sim30$ to detect an absorption line of the predicted strength displayed in Figure \\ref{fig:Sims}. The excess EW of C\\ion{iv} 1548~\\AA{} would be an excellent test of the effects of the AGN ionizing field as we have already detected absorption in our stacked COS-AGN spectra (Table \\ref{tab:EWStack}). Such a test would required observing the C\\ion{iv} 1548~\\AA{} covering fraction in the CGM of non-AGN galaxies from the control-matched sample.\n\n\\subsubsection*{AGN-driven winds and outflows}\nAn alternative form of feedback is AGN-driven winds or outflows \\citep[][]{Concas17, Fiore17,Woo17}. As the typical lifetime of an AGN ($\\lesssim10^6$ yr) is much smaller than the expected travel time of winds out to the impact parameters probed by COS-AGN \\citep[$\\sim10^8$ yr, assuming a constant, maximum velocity of $1000$ km~s$^{-1}${}; e.g.][]{Tremonti07,Veilleux13,Ishibashi15}, any signatures of winds or outflowing material would likely trace material expelled from a previous cycles of AGN activity or star-formation in the host galaxy \\citep[e.g.][]{Nedelchev17,Kauffmann17,Woo17}. The typically observed signatures of outflowing winds from a galaxy manifest as kinematic offsets of ionized emission or absorption lines from the host galaxy \\citep[e.g.][]{Bordoloi14b,Rubin14,Woo16,Heckman17,Perna17}, but as demonstrated in Figure \\ref{fig:Kinematics}, the Ly$\\alpha$ gas that we are probing in the COS-AGN sightlines does not have any strong bulk motion away from the host galaxy relative to the control matched sample. If such winds or outflows were driven by previous AGN or star-forming activity, the lack of kinematic offsets from the host galaxy suggests that these winds have dissipated over time, and should have deposited metals into the CGM \\citep[e.g.][]{Muzahid15,Turner15}. The low metal covering fraction at low impact parameter ($\\leq164$ kpc; Figure \\ref{fig:CovFracR}) suggests an absence of outflowing material polluting the CGM with metals, rejecting the notion that any recent (within 160~Myr) AGN-driven winds have enhanced the CGM. We note that the Ly$\\alpha$ EW at these column densities is more sensitive to the kinematics of the gas than to the amount of gas. Although we have rejected the possibility that AGN-driven winds are responsible for our results, it is possible that the gas is more turbulent in the CGM of AGN hosts compared to their control matches. To estimate the effects of turbulence, we would require observations of unsaturated low ionization metal absorption lines to verify if the EW enhancements are from enhanced column densities or kinematic broadening.\n\nGiven that AGN-driven winds can affect the absorption line profile by up to $\\approx \\pm 1000$ km~s$^{-1}${}, our adopted search window of $\\pm500$ km~s$^{-1}${} (Section \\ref{sec:DataEW}) could potentially miss gas present in a wind. We searched for the Ly$\\alpha$ profiles within $\\pm1000$ km~s$^{-1}${} and found two systems (J1214+0825 and J2133$-$0712) with minimal absorption outside the original window not associated with other absorption systems or the Galaxy. These two additional absorption components do not show any associated metal line absorption. As these missed components are small and narrow, the calculated flux-weighted velocity centroids presented in Figure \\ref{fig:Kinematics} would still be contained within the already identified absorption component. However, we remind the reader that our adopted search window of $\\pm500$ km~s$^{-1}${} is adopted to be consistent with methods used in the literature and control samples.\n\nRather than AGN feedback, it is possible that the effects we are seeing are from a different process co-eval or prior to the onset of AGN accretion. Several works have pointed out that AGN activity coincide with a recent starburst; with the AGN having significant accretion events at least $\\sim200$ Myr after the starburst has occurred \\citep{Wild07,Davies07,Wild10,Yesuf14} giving the neutral material time to propagate out to the impact parameters probed by COS-AGN \\citep{Heckman17}. With a sample of QSO sightlines probing the CGM around 17 low-redshift starburst and post-starburst galaxies, \\cite{Heckman17} have observed a similar signature of enhanced EWs of Ly$\\alpha$, Si\\ion{iii}, and C\\ion{iv} (the latter of which is not measured in our control sample) relative to a control-matched sample (matched in stellar mass and impact parameter). In the range of impact parameters and stellar masses probed by COS-AGN, the strength of our enhanced EW signature is consistent with the values probed by \\cite{Heckman17}. However, the results of \\cite{Heckman17} show strong offsets in the kinematics of the gas from the host galaxy \\citep[$\\approx 100$ km~s$^{-1}${}; see figure 5 from][]{Heckman17}, whereas the COS-AGN sightlines do not (bottom panel of Figure \\ref{fig:Kinematics}). Assuming the AGN activity was triggered by the starburst, a minimum delay time of 200 Myr could allow for any starburst-driven winds to dissipate and kinematic offsets to no longer be present at the impact parameters of the COS-AGN sample. Although this starburst picture provides a possible explanation of our observations, we caution that starbursts are not the only astrophysical event linked to AGN accretion activity. For example, mergers that trigger the AGN \\citep{Ellison11,Ellison13,Satyapal14,Silverman14,Goulding17} could potentially affect the surrounding CGM gas. Past and future work focussing on the CGM of galaxy mergers can further test this result \\citep[][Bordoloi et al. in prep.]{Johnson14,Hani17}.\n\n\n\n\\subsection{Are we seeing the effects from environment or other galaxy properties?}\nIf the AGN (or host galaxy) is not responsible for the observed differences in the CGM, an alternative is that the circumgalactic environment in which an AGN host is found is different. Results from \\emph{Quasars probing Quasars} \\citep{QPQ6}, \\cite{Farina14}, and \\cite{Johnson15} have all suggested that the excess of cool gas seen in the CGM out to 1 Mpc around $z\\approx1$ quasars is a result of residing in group environments. Although the excess of cool gas around quasars goes in the same direction as the 0.1 dex enhancement we find in the outer CGM of $z\\approx0.1$ COS-AGN galaxies (though we \\emph{do not} see an excess in the Ly$\\alpha$ EW in the inner CGM of AGN hosts, as seen for QSOs), the dark matter haloes of the AGN in our sample are typically an order of magnitude smaller than group dark matter haloes that host quasars. As stated in Section \\ref{sec:ControlMatching}, the $\\delta_{5}$ parameter provides an estimate of the environment. Given that differences in the distributions of $\\delta_{5}$ between AGN and star-forming galaxies in the SDSS vary on the order of a percent for a given stellar mass, it is likely that the enhanced EW of cool gas is not due to the contribution from a difference in the galaxy environment.\n\n\n\n\nGiven that we find an excess in the H\\ion{i}{} content of the outer CGM around AGN hosts, the results from \\cite{Borthakur13} which find a connection between the ISM and CGM gas properties would imply that the ISM would also host a large reservoir of H\\ion{i}{} gas. Such an enhancement in the ISM gas mass (relative to non-AGN galaxies) has been previously seen in AGN hosts \\citep[e.g.][]{Vito14}, where an excess of $\\sim0.2$ dex in gas mass is seen for AGN hosts similar to those probed in our sample. If such large gas reservoirs do exist in the ISM of AGN hosts, the AGN could be fuelled by the excess cool gas in the ISM, which in turn is fed by the cool CGM gas surrounding the AGN host. We note that other works such as \\cite{Fabello11} have found that the ISM of optically-selected AGN hosts contain the same H\\ion{i}{} gas mass as their star-forming counterparts. However, $\\sim$50\\% of the \\cite{Fabello11} AGN sample are so-called `composite' AGN (galaxies whose emission lines contain contributions from both star formation and AGN activity), which are not representative of the AGN hosts selected in our COS-AGN sample. \n\n\nA further test of this accretion picture would be to look for any orientation effects. If we are probing the gas reservoirs that are fuelling the AGN, we are likely to find the accreting gas along the major axis of the galaxy \\citep[e.g.][]{Kacprzak12,Nielsen15,Ho17}. Unfortunately, we do not have the ability to measure robust inclinations for many of our COS-AGN galaxies from SDSS imaging (see Figure \\ref{fig:postage}), and have too small of a sample size to produce a significant statistic. In addition, the \\cite{Ho17} sample are at much closer impact parameters ($\\lesssim 50$ kpc) whereas \\cite{Borthakur15} showed that at higher impact parameters (such as those probed by COS-AGN) there is no evidence of orientation effects for galaxies in COS-GASS. However, we do note that for the 4--5 sightlines that are probing along the edge of the disc relative to the 2--3 that are perpendicular to the disc, there is no significant difference in the median $\\Delta$log(EW). A larger sample would be required to test this explicitly.\n\n\n\n\\section{Summary}\nUsing a sample of 19 quasar sightlines through the circumgalactic medium (CGM) of 20 Type II Seyfert AGN and LINERs, we have demonstrated that there are mild differences in the rest-frame equivalent widths (EWs) of cool CGM gas around AGN hosts relative to their non-AGN counterparts. After matching in stellar mass and impact parameter, we find:\n\\begin{itemize}\n\n\\item[1.] The covering fraction of Ly$\\alpha$ gas for the AGN is 94$^{+6}_{-23}$\\%, which is comparable to the star-forming control galaxies (100$^{+0}_{-21}$\\%) and consistent with passive galaxies (75$^{+25}_{-21}$\\%). The covering fractions of metal species (C\\ion{ii}, Si\\ion{ii}, Si\\ion{iii}, C\\ion{iv}, Si\\ion{iv}, and N\\ion{v}) are consistent with the control-matched galaxies (Figure \\ref{fig:CovFracR}).\n\n\\item[2.] An insignificant increase in the Ly$\\alpha$ EW for AGN relative to control-matched galaxies on a sightline by sightline basis. The measured median EW offset between these two population is $+0.10\\pm0.13$ dex (Figure \\ref{fig:deltaEWRho}). \n\n\\item[3.] After stacking the spectra, the observed EW enhancement of low ionization species for AGN is seen only at high impact parameters ($\\rho_{\\rm imp}${}$\\geq164$ kpc; the median impact parameter of COS-AGN) for both Ly$\\alpha$ ($\\delta$log(EW)$=+0.45\\pm0.05$) and cool metal line tracers (Si\\ion{ii} 1260~\\AA{} [$\\delta$log(EW)$>0.27$~dex] and Si\\ion{iii} 1206~\\AA{} [$\\delta$log(EW)$>0.75$~dex]; Figure \\ref{fig:StackdEW}). These results are inconsistent with the expected effects from AGN feedback seen in our zoom-in simulations at high impact parameters (Figure \\ref{fig:Sims}). At lower impact parameters ($\\rho_{\\rm imp}${}$<164$ kpc), our results are consistent with the simulations.\n\n\\item[4.] The Ly$\\alpha$ line kinematics for the COS-AGN sightlines does not differ significantly from what is observed around the control-matched sample, suggesting there is no strong bulk motion in the CGM due to the presence of an AGN (Figure \\ref{fig:Kinematics}). As all but one system show gas within the escape velocity of the host halo, the probed material is likely bound to the AGN host.\n\n\\end{itemize}\n\nThese results suggest that the circumgalactic environments that host AGN show little difference than their non-AGN hosts on a sightline-by-sightline basis, likely attributed to our small sample size. We only detect a significant difference in the amount of cool gas in our stacked spectrum at high impact parameters ($\\rho_{\\rm imp}${}$\\geq164$~kpc), which we use to interpret our results. Given the lack of signatures (in both EW and kinematic diagnostics) of recent AGN feedback on the CGM from winds and ionizing radiation, we speculate that possible causes for these stacked spectrum results could be from the accretion of cool gas to feed the AGN, or a remnant effect from previous evolutionary activity of the host galaxy (such as starbursts) prior to the AGN accretion phase.\n\n\n\\section*{Acknowledgments}\nWe thank the anonymous referee for their comments to improve the clarity of this manuscript. We are grateful for Sanchayeeta Borthakur providing the reduced spectra and all Si species equivalent widths from the COS-GASS survey. This manuscript benefited greatly from discussions with H.W. Chen, T. Heckman, C. Martin, J.X. Prochaska, and J. Werk. BDO's contribution was made possible by the HST observing grant HST-GO-13774. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{section-introduction}\n\\noindent Special functions play a prominent role in mathematics and physics. They are solutions of important equations and come in various forms, in terms of series or integrals. We are interested in the way special functions connect to representation theory. That such a connection exists is well known. Special functions appear in the study of spherical functions (see e.g. \\cite{Helgason1984}, Chapter IV), of matrix coefficients of representations (see e.g. \\cite{Vilenkin1993-1}, Section 4.1) and of symmetry breaking operators (see \\cite{Kobayashi-Kubo-Pevzner2016} and \\cite{Kobayashi-Pevzner2016}). In this work, we focus on the $K$-types of degenerate principle series of $\\spg(n,\\C)$, looking for $K$-finite vectors that have explicit formulas in terms of special functions. Here, $K$ refers to $\\spg(n)$.\n\\vskip 8pt\n\\noindent The reason why we choose to work with degenerate principle series of $\\spg(n,\\C)$ is twofold.\n\\vskip 8pt\n\\noindent First, the geometric setting we choose is motivated by the following sequence of groups of isometries of vector spaces over different number fields:\n$$\\xymatrix{\n\\spg(n) \\ar@{~>}[d] & \\subset & \\sug(2n) \\ar@{~>}[d] & \\subset & \\sog(4n) \\ar@{~>}[d] \\\\\n\\quat^n & \\simeq & \\C^{2n} & \\simeq & \\R^{4n}.}$$\nThis sequence enables us to use a refined version of the classical theory of spherical harmonics. Moreover, the non-commutativity of the skew field $\\quat$ of quaternions adds a rich ingredient to the underlying representation theory and harmonic analysis: see for example \\cite{Howe-Tan1993}, \\cite{Kobayashi1992}, \\cite{Pasquale1999} and the very recent paper \\cite{Schlichtkrull-Trapa-Vogan2018}. \n\\vskip 8pt\n\\noindent Second, the degenerate principal series representations of the complex symplectic Lie group $\\spg(n,\\C)$ are \"small\" in the sense of the Kirillov-Gelfand dimension(see e.g. \\cite{Vogan2017}) amongst infinite-dimensional unitary representations of $\\spg(n,\\C)$. According to the guiding principle \"small representation of a group $=$ large symmetries in a representation space\" suggested by T. Kobayashi in \\cite{Kobayashi2013}, explicit models of such representations are a natural source of information on special functions that arise in this framework as specific vectors. This philosophy has been applied to the analysis of minimal representations of $\\org(p,q)$ (see \\cite{Kobayashi-Mano2011} and \\cite{Kobayashi-Orsted2003}) and small principal series representations of the real symplectic group $\\spg(n,\\R)$ (see \\cite{Kobayashi-Orsted-Pevzner2011}).\n\\vskip 8pt\n\\noindent Fix an integer $n \\geq 2$, set $m=n-1$ and write $G=\\spg(n,\\C)$ and $K=\\spg(n)$. The representations we work with are defined by parabolic induction with respect to a maximal parabolic subgroup $Q$ of $G$ whose Langlands decomposition is $Q=MAN$(see Section \\ref{subsection-non_comp._pict._and_heisenberg_group} for explicit description) with:\n$$M \\simeq \\ung(1) \\times \\spg(m,\\C), \\; A \\simeq \\R^{\\times}_{+} \\; {\\rm and} \\; N \\simeq {\\rm H}_{\\C}^{2m+1}.$$\nHere, $\\mathrm{H}_{\\C}^{2m+1}$ refers to the $(2m+1)$-dimensional complex Heisenberg group. We denote the $\\ung(1)$ component of an element $m$ of $M$ by $e^{i \\theta(m)}$ and the positive real scalar that corresponds to an element $a$ of $A$ by $\\alpha(a)$. Now consider for $(\\lambda,\\delta) \\in \\R \\times \\Z$ the character $\\chi_{i\\lambda,\\delta}$ defined on $Q$ by:\n$$\\chi_{i\\lambda,\\delta}(man)= \\big( e^{i \\theta(m)}\\big)^{\\delta} \\, \\big( \\alpha(a)\\big)^{i\\lambda}.$$\nThe corresponding induced representation $\\pild={\\rm Ind}_Q^G \\, \\chi_{i\\lambda,\\delta}$ can be realised on the completion $V_{i\\lambda,\\delta}$ of the complex vector space\n{\\small $$V_{i\\lambda,\\delta}^0 = \\left \\lbrace f \\in C^0 \\big( \\C^{2n} \\setminus \\{0\\} \\big) \\; \\Big\/ \\; \\forall c \\in \\C \\setminus \\{0\\}: \\; f(c \\, \\cdot)=\\left( \\frac{c}{\\vert c \\vert} \\right)^{-\\delta} \\vert c \\vert^{-i\\lambda-2n} f(\\cdot) \\right \\rbrace$$}with respect to the $L^2$-norm on $S^{4n-1}$.\n\\vskip 8pt\n\\noindent Representations $\\pild$ form a degenerate principal series of $G$. It is proved in \\cite{Gross1971} that $\\pild$ is irreducible if $(\\lambda,\\delta) \\neq (0,0)$ and in \\cite{Clare2012} that $\\pi_{0,0}$ decomposes into a sum of two irreducible subrepresentations. The isotypic decomposition of $\\pild \\big|_K$ is multiplicity free and given by (see \\cite{Clare2012} and \\cite{Howe-Tan1993}):\n\\begin{equation} \\label{equation-general_isot._dec.}\n\\pild \\big|_K \\simeq \\sideset{}{^\\oplus}\\sum_{ \\substack{l-l' \\geq |\\delta| \\\\\nl-l' \\equiv \\delta [2]}} \\, \\pi^{l,l'}.\n\\end{equation}\nIn this sum, $l$ and $l'$ are integers such that $l \\geq l' \\geq 0$ and $\\pi^{l,l'}$ is the irreducible representation of $K$ whose highest weight is $(l,l',0,\\cdots,0)$; we denote the irreducible invariant subspace of $V_{i\\lambda,\\delta}$ (with respect to the left action of $K$) that corresponds to $\\pi^{l,l'}$ by $V^{l,l'}$, calling it a \\textit{component} of $V_{i\\lambda,\\delta}$.\n\\vskip 8pt\n\\noindent Our aim is to describe specific elements of components $V^{l,l'}$ in terms of special functions. This will require suitable changes of the carrying space $V_{i\\lambda,\\delta}$ (together with the action of $G$) so as to have a clearer view of $\\pild$, depending on the kind of special functions we have in mind; each point of view is called a \\textit{picture} of $\\pild$ (above definition is the \\textit{induced picture}).\n\\vskip 8pt\n\\noindent This paper is organised as follows:\n\\begin{itemize}\n\\item Section 2: in the compact picture, we study the $K$-type structure, by which we mean the detailed description of the components of $V_{i\\lambda,\\delta}$ and connections that exist between them (Propositions \\ref{proposition-left_action_decomposition}, \\ref{proposition-hwv_for_right_action} and \\ref{proposition-isotypic_decomposition_for_sp(1)}).\n\\item Section 3: we consider the case $l=l'$ and use, again in the compact picture, invariance properties with respect to $\\spg(1)$ and $1 \\times \\spg(n-1)$ to exhibit in components $V^{l,l}$ elements which can be seen as solutions of hypergeometric differential equations (Theorem \\ref{theorem-hypergeometric_equation}).\n\\item Section 4: we define the non-standard picture of $\\pild$ (which was introduced in \\cite{Kobayashi-Orsted-Pevzner2011} and followed in \\cite{Clare2012}) by applying a certain partial Fourier transform $\\mathcal{F}$ to the non-compact picture. For a wide class of components, namely components $V^{l,0}$ (said otherwise, those components such that $l'=0$), this leads to elements that can be expressed in terms of modified Bessel functions (Theorem \\ref{theorem-final_formula}).\n\\end{itemize}\nLet us put together our most important results:\n\\begin{nnthm}[Main results] \\label{thm-quat._sph._harm._thm} \\\n\n\\noindent Let $n \\in \\N$ be such that $n \\geq 2$ and set $m=n-1$.\\\\\n\\noindent Consider the group $G=\\spg(n,\\C)$, its maximal compact subgroup $\\spg(n)$ and the parabolic subgroup $Q=MAN \\simeq \\ung(1) \\times \\spg(m,\\C) \\times \\R^{\\times}_{+} \\times {\\rm H}_{\\C}^{2m+1}$.\\\\\n\\noindent Consider a pair $(\\lambda,\\delta) \\in \\R \\times \\Z$, together with the character $\\chi_{i\\lambda,\\delta}$ defined by $\\chi_{i\\lambda,\\delta}(man)= \\big( e^{i \\theta(m)}\\big)^{\\delta} \\, \\big( \\alpha(a)\\big)^{i\\lambda}$\nand the degenerate principal series representations $\\pi_{i\\lambda,\\delta} = {\\rm Ind}_Q^G \\, \\chi_{i\\lambda,\\delta}$ of $G$. \n\\begin{enumerate}\n\\item For $l \\in \\N$, consider the $K$-type $\\pi^{l,l}$. The corresponding subspace $V^{l,l}$ of $V_{i\\lambda,\\delta}$ contains an element which can be seen as a function of a single variable $\\tau \\in [0,1]$ and the restriction $\\varphi$ of this function to $]0,1[$ satisfies the following hypergeometric equation:\n$$\\quad \\; \\tau(1-\\tau) \\varphi''(\\tau) \\, + \\, 2(1-n\\tau) \\, \\varphi'(\\tau) \\, + \\, l(l+2n-1) \\, \\varphi(\\tau) \\, = \\, 0.$$\n\\item For $(l,\\alpha,\\beta) \\in \\N^3$ such that $l=\\alpha+\\beta$ and $\\delta=\\beta-\\alpha$, consider the $K$-type $\\pi^{l,0}$ of $\\pild$. Then, in the non-standard picture, highest weight vectors of $\\pi^{l,0}$ are proportionnal to a function\n$$\\psi \\, : \\, \\C \\times \\C^m \\times \\C^m \\, \\longrightarrow \\C$$\nwhose expression for $s \\neq 0$ and $v \\neq 0$ is\n$$\\quad \\psi(s,u,v) \\, = \\, R(s,u,v) \\, K_{\\frac{i\\lambda+\\delta}{2}} \\left( \\pi \\sqrt{ 1 + \\Vert u \\Vert^2 } \\sqrt{ \\vert s \\vert^2+4\\Vert v \\Vert^2 } \\right)$$\nwhere we set\n$$\\; R(s,u,v) \\, = \\, \\frac{(-i \\overline{s})^{\\alpha} \\, \\pi^{i\\lambda+\\beta+n}}{2^{\\frac{i\\lambda+l}{2}+1} \\, \\Gamma \\left( \\frac{i\\lambda+l}{2}+n \\right)} \\, \\left( \\frac{ \\sqrt{ |s|^2+4\\Vert v \\Vert^2}}{\\pi \\sqrt{1+\\Vert u \\Vert^2}} \\right)^{\\frac{i\\lambda+\\delta}{2}}$$\nand where $K_{\\frac{i\\lambda+\\delta}{2}}$ denotes a modified Bessel function of the third kind (see appendix for definition).\n\\end{enumerate}\n\\end{nnthm}\n\\noindent Before we enter the details, let us motivate the use of partial Fourier transforms. For one thing, they have proved useful in the study of Knapp-Stein operators (see \\cite{Clare2012}, \\cite{Kobayashi-Orsted-Pevzner2011}, \\cite{Pevzner-Unterberger2007} and \\cite{Unterberger2003}). For another, partial Fourier transforms with respect to appropriate Lagrangian subspaces modify the nature of constraints imposed on specific vectors in representation spaces and have been used to find explicit formulas for $K$-finite vectors in \\cite{Kobayashi-Mano2011}, \\cite{Kobayashi-Orsted2003}, and \\cite{Kobayashi-Orsted-Pevzner2011}. These works establish formulas that involve Bessel functions. This has lead us to apply similar techniques to degenerate principal series of $\\spg(n,\\C)$.\n\\section{$K$-type structure} \\label{section-k_type_structure}\n\\noindent We investiqate the $K$-type structure of the degenerate principal series $\\pild={\\rm Ind}_Q^G \\, \\chi_{i\\lambda,\\delta}$ of $\\spg(n,\\C)$. \n\\subsection{Compact picture and general facts} \\label{subsection-compact_picture_and_general_facts} \n\\\n\\vskip 8pt\n\\noindent Let us fix $(\\lambda,\\delta) \\in \\R \\times \\Z$. In the compact picture (see \\cite{Knapp1986}, Chapter VII), the carrying space of $\\pild$ is a subspace of $L^2(G)$. The natural action of $G$ on $\\C^{2n}$ enables one to identify it with the following Hilbert space:\n$$L^2_{\\delta}(S^{4n-1})=\\left \\lbrace f \\in L^2(S^{4n-1}) \\; \\big\/ \\; \\forall \\theta \\in \\R: \\; f(e^{i\\theta} \\, \\cdot)=e^{-i\\delta \\theta} f(\\cdot) \\right \\rbrace.$$\n\\vskip 8pt\n\\noindent The compact picture is the ideal setting to look for irreducible invariant subspaces with respect to the left action of $K$, because one can benefit from decompositions given by standard harmonic analysis (see e.g. Chapter 9 of \\cite{Faraut2008}, Chapters IV and V of \\cite{Knapp2002} and Chapters 9 and 11 of \\cite{Vilenkin1993-2}).\n\\vskip 8pt\n\\noindent Denote by $\\harm^k$ the complex vector space of polynomial functions $f$ on $\\R^{4n}$ that are harmonic and homogeneous of degree $k$. Let us write $\\sph^k=\\harm^k \\big|_{S^{4n-1}}$ (elements of $\\sph^k$ are called spherical harmonics). Then:\n\\begin{itemize}\n\\item each $\\sph^k$ is invariant under the left action of $\\sog(4n)$ (this is also true for each $\\harm^k$);\n\\item the representations defined by the left action of $\\sog(4n)$ on the various spaces $\\sph^k$ are irreducible and pairwise inequivalent;\n\\item $\\displaystyle L^2\\left( S^{4n-1} \\right) = \\widehat{\\bigoplus_{k \\in \\N}} \\; \\sph^k$.\n\\end{itemize}\nConsider the identification $(x,y) \\in \\R^{2n}\\times \\R^{2n} \\longleftrightarrow z=x+iy \\in \\C^{2n}$. Then:\n\\begin{itemize}\n\\item functions of the variables $x$ and $y$ (in particular elements of the spaces $\\harm^k$) can be regarded as functions of the variables $z$ and $\\bar{z}$; \n\\item matrices $A+iB$ of $\\glg(2n,\\C)$ (resp. $\\sug(2n)$), where $A$ and $B$ denote $2n \\times 2n$ real matrices, can be regarded as matrices $\\left( \\begin{array}{cc}\n A & -B \\\\\n B & A \\\\\n \\end{array}\n\\right)$ of $\\glg(4n,\\R)$ (resp. $\\sog(4n)$);\n\\item accordingly, the left action $L$ of $\\sug(2n)$ on functions of $z$ and $\\bar{z}$ is defined by $L(u)f(z,\\bar{z}) = f(u^{-1}z, \\, \\overline{u^{-1} z}) = f(u^{-1}z,\\, {^t u}\\bar{z})$.\n\\end{itemize}\n\\vskip 8pt\n\\noindent The Laplace operator can be written $\\displaystyle \\Delta =4\\sum_{i=1}^{2n} \\frac{\\partial^2}{\\partial z_i \\partial \\bar{z_i}}$. For $\\alpha,\\beta \\in \\N$, consider the space $\\harm^{\\alpha,\\beta}$\nof polynomial functions $f$ of the variables $z$ and $\\bar{z}$ such that $f$ is homogeneous of degree $\\alpha$ in $z$ and degree $\\beta$ in $\\bar{z}$ and such that $\\Delta f=0$. Let us write $\\sph^{\\alpha,\\beta}=\\harm^{\\alpha,\\beta} \\big|_{S^{4n-1}}$. Then:\n\\begin{itemize}\n\\item each $\\sph^{\\alpha,\\beta}$ is invariant under the left action of $\\sug(2n)$ (this is also true for each $\\harm^{\\alpha,\\beta}$);\n\\item the representations defined by the left action of $\\sug(2n)$ on the various spaces $\\sph^{\\alpha,\\beta}$ are irreducible and pairwise inequivalent;\n\\item $\\displaystyle \\sph^k = \\bigoplus_{\n\\substack{\n(\\alpha,\\beta) \\in \\N^2 \\\\\n\\alpha+\\beta=k}} \\sph^{\\alpha,\\beta}$.\n\\end{itemize}\nThis leads to the Hilbert sum $\\displaystyle L^2_{\\delta}(S^{4n-1})=\\widehat{\\bigoplus_{\n\\substack{(\\alpha,\\beta) \\in \\N^2 \\\\\n\\delta=\\beta-\\alpha}}} \\sph^{\\alpha,\\beta}$. We see that, in order to describe the isotypic decomposition of $\\pild \\big|_K$, we need to understand how each $\\sph^{\\alpha,\\beta}$ breaks into irreducible invariant subspaces under the left action of $K$.\n\\vskip 8pt\n\\noindent From now on, we consider $3$-tuples $(k,\\alpha,\\beta) \\in \\N^3$ such that $\\alpha+\\beta=k$ and denote by $L$ the left action of $K$, be it on $\\harm^k$, $\\sph^k$, $\\harm^{\\alpha,\\beta}$ or $\\sph^{\\alpha,\\beta}$.\n\\subsection{Left action of $\\spg(n)$} \\label{subsection-left_action}\n\\\n\\vskip 8pt\n\\noindent Recall that the Lie algebra $\\spa(n,\\C)$ of $G$ is\n$$\\mathfrak{g}=\\left\\lbrace X=\\left(\n\\begin{array}{cc}\nA & C \\\\\nB & -{}^t\\!{A} \\\\\n\\end{array}\n\\right) \\in \\m(2n,\\C)\\ \/ \\ B\\ {\\rm and}\\ C\\ {\\rm are\\ symmetric} \\right\\rbrace$$ \nand that the Lie algebra $\\spa(n)$ of $K$ is\n$$\\mathfrak{k}=\\left\\lbrace X=\\left(\n\\begin{array}{cc}\nA & -\\overline{B} \\\\\nB & \\overline{A} \\\\\n\\end{array}\n\\right) \\in \\m(2n,\\C)\\ \/ \\ A\\ {\\rm is\\ skew \\; and}\\ B\\ {\\rm is\\ symmetric} \\right\\rbrace.$$ \n\\noindent Consider the complexification $\\mathfrak{g}=\\mathfrak{k}\\oplus i\\mathfrak{k}$ of $\\mathfrak{k}$. Let $\\mathfrak{h}$ be the usual Cartan subalgebra of $\\mathfrak{g}$ consisting of diagonal elements of $\\mathfrak{g}$. If $r$ belongs to $\\{1,...,n\\}$, denote by $L_r$ the linear form that assigns to an element of $\\mathfrak{h}$ its $r^{\\rm th}$ diagonal term.\n\\vskip 8pt\n\\noindent The set $\\Delta$ of roots of $\\mathfrak{g}$ (with respect to $\\mathfrak{h}$) consists of the following linear forms:\n\\begin{itemize}\n\\item $L_r-L_s$ and $-L_r+L_s$ ($1 \\leq r 0$ ($I_m$ denotes the $m$-dimensional identity matrix);\n\\item $N$ consists of matrices {\\small $n=\\left(\n\\begin{array}{cccc}\n1 & {}^t\\!{u} & 2s & {}^t\\!{v} \\\\\n0 & I_m & v & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & -u & I_m \\\\\n\\end{array}\n\\right)$} such that $s\\in \\C$ and $u$ and $v$ both belong to $\\C^m$.\n\\end{itemize}\nThen $Q=MAN$ is indeed a parabolic subgroup of $G$.\n\\vskip 8pt\n\\noindent We regard the $(2m+1)$-dimensional complex Heisenberg group $\\mathrm{H}_{\\C}^{2m+1}$ as $\\C \\times \\C^m \\times \\C^m$ equipped with the following product: \n$$(s,u,v)(s',u',v')=\\left( s+s'+\\frac{\\langle v,u' \\rangle - \\langle u,v' \\rangle}{2},u+u',v+v' \\right).$$\nHere, $\\langle \\cdot,\\cdot \\rangle$ is defined for elements $(x,y) \\in \\C^m \\times \\C^m$ by: \n$$\\langle x,y \\rangle = \\sum_{i=1}^m x_i y_i.$$\nWe point out that, later on, this sum will be denoted by $x \\cdot y$ instead of $\\langle x,y \\rangle$ when we consider $(x,y) \\in \\R^m \\times \\R^m$.\n\\vskip 8pt\n\\noindent We use the following sequence of identifications (with $n$ as above):\n$$\\begin{array}{ccccc}\n\\mathrm{H}_{\\C}^{2m+1} & \\simeq & \\overline{N} & \\simeq & {\\rm H}\\\\\n(s,u,v) & \\longmapsto & {}^t\\!{n} & \\longmapsto & (1,u,2s,v). \\\\\n\\end{array}$$\nwhere ${\\rm H}=\\{1\\} \\times \\C^m \\times \\C \\times \\C^m$ is regarded as a complex hyperplane of $\\C^{2n}$. The first identification map is a group isomorphism (one can also prove that $\\mathrm{H}_{\\C}^{2m+1} \\simeq N$) and the second identication results from the natural action of $\\overline{N}$ on $\\C^{2n}$ (applied to the element $(1,0,\\cdots,0)$ of $\\C^{2n}$).\n\\vskip 8pt\n\\noindent One can now identify $L^2(\\overline{N})$ with $L^2(\\mathrm{H}_{\\C}^{2m+1})$ and, given $f \\in V_{i\\lambda,\\delta}^0$ (in the induced picture), regard the restriction $f|_{\\rm H}$ as an element of $L^2(\\mathrm{H}_{\\C}^{2m+1})$.\n\\vskip 8pt\n\\noindent From now on, we write $\\C^{2m+1}$ instead of $\\mathrm{H}_{\\C}^{2m+1}$.\n\n\\subsection{Definition of the non-standard picture} \\label{subsection-definition_of_the_non_stand._pict.}\n\\\n\\vskip 8pt\n\\noindent We first define two partial Fourier transforms\n\\begin{align*}\n\\mathcal{F}_{\\tau} & : L^2(\\C^{2m+1}) \\longrightarrow L^2(\\C^{2m+1})\\\\\n\\mathcal{F}_{\\xi} &: L^2(\\C^{2m+1}) \\longrightarrow L^2(\\C^{2m+1})\n\\end{align*}\nby setting for functions $g \\in L^2(\\C^{2m+1})$ that fulfill integrability conditions: \n$$\\mathcal{F}_{\\tau}(g)(s,u,v)=\\int_{\\C} g(\\tau,u,v) \\, e^{-2i\\pi {\\rm Re}(s\\tau) } \\, d\\tau$$\n$$\\mathcal{F}_{\\xi}(g)(s,u,v)=\\int_{\\C^m} g(s,u,\\xi)\\,e^{-2i\\pi {\\rm Re} \\langle v,\\xi \\rangle}\\,d\\xi.$$\n\\noindent We then define the partial Fourier transform on which is based the non-standard picture:\n$$\\mathcal{F}= \\mathcal{F}_{\\tau} \\circ \\mathcal{F}_{\\xi}.$$\n\\begin{nndef}\n\\noindent The \\textit{non-standard picture} of $\\pi_{i\\lambda,\\delta}$ has $L^2(\\C^{2m+1})$ as carrying space. The action of $G$ is then the conjugate under $\\mathcal{F}$ of the action of $G$ in the non-compact picture; in other words, $\\mathcal{F}$ intertwines the action of $G$ in the non-compact picture and the action of $G$ in the non-standard picture.\n\\end{nndef}\n\\subsection{Connection with modified Bessel functions} \\label{subsection-connection_with_modified_bessel_functions}\n\\subsubsection{Selected components} \\label{subsubsection-selected_components}\n\\\n\\vskip 8pt\n\\noindent \nBecause we intend to use the right action of $\\spg(1)$, we consider an entire space $\\harm^k$ of harmonic polynomials. We restrict our attention to those components of Proposition \\ref{proposition-left_action_decomposition} that are labelled by $\\gamma=0$ and are subspaces of $\\harm^k$, namely the components $V_0^{\\alpha,\\beta}$ with $\\alpha + \\beta=k$; the corresponding highest weight vectors are the polynomials \n$$P_0^{\\alpha,\\beta}(z,w,\\bar{z},\\bar{w})=w_1^{\\alpha}\\bar{z_1}^{\\beta}.$$\n\\noindent We denote by $g_{\\alpha,\\beta}$ the restriction of $P_0^{\\alpha,\\beta}$ to the unit sphere $S^{4n-1}$. Let us call $g$ the function in the induced picture that corresponds to $g_{\\alpha,\\beta}$. It extends $g_{\\alpha,\\beta}$, meaning that $g_{|_{S^{4n-1}}}=g_{\\alpha,\\beta}$. By definition of the induced picture, $g$ must satisfy for all non-zero complex numbers $c$:\n$$g(c\\,\\cdot)=\\left( \\frac{c}{|c|}\\right)^{-\\delta} \\, \\vert c \\vert^{-i\\lambda-2n} \\, g.$$\n\\noindent We define\n$$a(s,u)=\\sqrt{1+4\\vert s \\vert ^2 + \\Vert u \\Vert^2}$$ \n$$r(s,u,v)= \\sqrt{ a^2(s,u) + \\Vert v \\Vert^2}.$$\n\\noindent By restricting $g$ to the complex hyperplane ${\\rm H} \\simeq \\C^{2m+1}$, we get a function $G_{\\alpha,\\beta}$ defined on $\\C^{2m+1}$ by:\n\\begin{align*}\n& G_{\\alpha,\\beta}(s,u,v) = g(1,u,2s,v)\\\\\n& = \\left( \\frac{1}{r(s,u,v)} \\right)^{i\\lambda+2n} g_{\\alpha,\\beta} \\left( \\frac{1}{r(s,u,v)},\\frac{u}{r(s,u,v)},\\frac{2s}{r(s,u,v)},\\frac{v}{r(s,u,v)} \\right).\n\\end{align*}\n\\noindent Finally, due to the total homogeneity degree $k$ of $P_0^{\\alpha,\\beta}$:\n$$G_{\\alpha,\\beta}(s,u,v)=\\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + \\Vert v \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\, .$$\nWe call $G_{\\alpha,\\beta}$ the \\textit{non-compact form} of $P_0^{\\alpha,\\beta}$. The aim of the rest of Section \\ref{section-non_standard_picture_and_mod._bessel_functions} is to determine the \\textit{non-standard form} of $P_0^{\\alpha,\\beta}$, that is, $\\mathcal{F}(G_{\\alpha,\\beta})$. We apply $\\mathcal{F}_{\\xi}$ in Section \\ref{subsubsection-first_transform} and $\\mathcal{F}_{\\tau}$ in Section \\ref{subsubsection-second_transform}. Calculations involve Bessel functions and various formulas that we have gathered in the appendix. \n\\subsubsection{First transform} \\label{subsubsection-first_transform}\n\\\n\\vskip 8pt\n\\noindent \nBy definition (integrability condition is fulfilled):\n\\begin{equation} \\label{equation-step_1-definition}\n\\displaystyle \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(s,u,v)=\\int_{\\C^m} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + \\Vert \\xi \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\,e^{-2i\\pi {\\rm Re} \\langle v,\\xi \\rangle}\\,d\\xi.\n\\end{equation}\n\\noindent In real coordinates, writing $\\xi=x+iy$ and $v=a+ib$ (elements $x,y,a,b$ each belong to $\\R^m$) and identifying $\\xi$ and $v$ with the elements $(x,y)$ and $(a,b)$ of $\\R^m \\times \\R^m$, formula (\\ref{equation-step_1-definition}) reads:\n\\begin{multline} \\label{equation-step_2-real_coordinates}\n\\displaystyle \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(s,u,v)=\\\\\n\\int_{\\R^m \\times \\R^m} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + \\Vert x \\Vert^2 + \\Vert y \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\,e^{-2i\\pi (a \\cdot x-b \\cdot y)}\\,dxdy.\n\\end{multline}\n\\noindent Switch to polar coordinates by writing $(x,y)=rM$ and $(a,-b)=r'M'$, with $M$ and $M'$ in $S^{2m-1} \\subset \\R^{2m}$, $r =\\sqrt{\\Vert x \\Vert^2+\\Vert y \\Vert^2} =\\Vert \\xi \\Vert$ and $r'=\\sqrt{a^2+(-b)^2}=\\Vert v \\Vert$. Then the integral in Formula (\\ref{equation-step_2-real_coordinates}) becomes\n\\begin{equation} \\label{equation-step_3-polar_coordinates}\n\\int_0^{\\infty} \\left( \\int_{S^{2m-1}} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} e^{-2i\\pi rr'M \\cdot M'} d\\sigma(M) \\right) r^{2m-1} dr\n\\end{equation}\nwhere $M \\cdot M'$ now denotes the Euclidean scalar product of $\\R^{2m}$ applied to the points $M$ and $M'$ of the sphere $S^{2m-1}$ seen as vectors of $\\R^{2m}$. Integral (\\ref{equation-step_3-polar_coordinates}) can be written:\n\\begin{equation} \\label{equation-step_4-polar_coordinates-2nd_version}\n\\displaystyle \n\\int_{0}^{\\infty} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\left( \\int_{S^{2m-1}} e^{-2i\\pi rr'M \\cdot M'} d\\sigma(M) \\right) r^{2m-1} dr.\n\\end{equation}\n\\noindent Proposition \\ref{proposition-bochner} then changes (\\ref{equation-step_4-polar_coordinates-2nd_version}) into:\n\\begin{equation} \\label{equation-step_5-bessel_expression_1}\n\\int_{0}^{\\infty} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} 2 \\pi (rr')^{1-m} J_{m-1}(2 \\pi rr') r^{2m-1} dr.\n\\end{equation}\n\\noindent Because $r'=\\Vert v \\Vert$, (\\ref{equation-step_5-bessel_expression_1}) becomes:\n\\begin{equation} \\label{equation-step_6-bessel_expression_2}\n\\displaystyle \n2^{\\alpha+1} \\pi s^{\\alpha} \\Vert v \\Vert^{1-m} \\int_{0}^{\\infty} \\frac{r^m}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} J_{m-1}(2 \\pi \\Vert v \\Vert r) dr.\n\\end{equation}\n\\noindent We now want to apply Proposition \\ref{proposition-integral_formulas-erdelyi}. But it uses another notation system than ours. To understand how to switch from one system of notation to the other, let us define new variables $x,y,\\mu$: \n$$x=r\\ ;\\ y=2 \\pi \\Vert v \\Vert\\ ;\\ \\mu=\\frac{i\\lambda+k}{2}+n-1\\ ;\\ \\nu=m-1.$$\n\\noindent Then (\\ref{equation-step_6-bessel_expression_2}) becomes:\n$$2^{\\alpha+1} \\pi s^{\\alpha} \\Vert v \\Vert^{1-m} y^{-\\frac{1}{2}} \\int_{0}^{\\infty} \\frac{x^{\\nu+\\frac{1}{2}}}{ \\left( a^2(s,u) + r^2 \\right)^{\\mu+1}} J_{m-1}(xy) \\sqrt{xy} dx.$$\n\\noindent Proposition \\ref{proposition-integral_formulas-erdelyi} (first formula) now gives (as long as $\\Vert v \\Vert > 0$)\n$$2^{\\alpha+1} \\pi s^{\\alpha} \\Vert v \\Vert^{1-m} y^{-\\frac{1}{2}} \\frac{a^{\\nu-\\mu} y^{\\mu+\\frac{1}{2}} K_{\\nu-\\mu}(ay)}{2^{\\mu} \\Gamma(\\mu+1)}$$\nwhich, back to our own notation choices, is equal to\n$$\\frac{2^{\\alpha+1} s^{\\alpha} \\pi^{\\frac{i\\lambda+k}{2}+n}}{\\Gamma\\left( \\frac{i\\lambda+k}{2}+n \\right)} \\left( \\frac{\\Vert v \\Vert}{a(s,u)} \\right)^{\\frac{i\\lambda+k}{2}+1} K_{-\\left( \\frac{i\\lambda+k}{2}+1 \\right)}(2\\pi a(s,u) \\Vert v \\Vert).$$\n\\noindent To make formulas lighter, from now on we will write:\n{\\small $$\\Lambda=\\frac{i\\lambda+k}{2}.$$}\nBecause $K_{-(\\Lambda+1)}=K_{\\Lambda+1}$, we have proved:\n\\begin{prop} \\label{proposition-first_transform-k_bessel_expression}\nGiven any $\\lambda \\in \\R$ and any $(k,\\alpha,\\beta) \\in \\N^3$ such that $\\alpha+\\beta=k$, consider the non-compact form $G_{\\alpha,\\beta}$ of the highest weight vector $P_0^{\\alpha,\\beta}$. Then for all $(s,u,v)$ in $\\C \\times \\C^m \\times \\C^m$ such that $v \\neq 0$:\n$$\\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(s,u,v)=2^{\\alpha+1} s^{\\alpha} \\frac{\\pi^{\\Lambda+n}}{\\Gamma\\left( \\Lambda + n \\right)} \\left( \\frac{\\Vert v \\Vert}{a(s,u)} \\right)^{\\Lambda+1} K_{ \\Lambda+1 }(2\\pi a(s,u) \\Vert v \\Vert).$$\n\\end{prop}\n\\subsubsection{Second transform} \\label{subsubsection-second_transform}\n\\\n\\vskip 8pt\n\\noindent Consider any $(s,u,v) \\in \\C \\times \\C^m \\times \\C^m$ such that $s \\neq 0$ and $v \\neq 0$. We want to compute:\n\\begin{equation} \\label{equation-second_transform_integral}\n\\mathcal{F}_{\\tau} \\Big( \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta}) \\Big) \\, (s,u,v) = \\int_{\\C} \\ \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(\\tau,u,v) \\,e^{-2i\\pi {\\rm Re}(s\\tau)} \\ d\\tau.\n\\end{equation}\n\\noindent Using propositions \\ref{proposition-first_transform-k_bessel_expression} and \\ref{proposition-bessel_asymptotics-erdelyi}, one shows that $\\tau \\longmapsto \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(\\tau,u,v)$ is indeed integrable. Let us use letters $a,b,x,y$ again, this time taking them to refer to real numbers: $s=a+ib$ and $\\tau=x+iy$. Then ${\\rm Re}(s\\tau)=ax-by$ and (\\ref{equation-second_transform_integral}) becomes\n$$\\int_{\\R^2} \\frac{2^{\\alpha+1} \\tau^{\\alpha} \\pi^{\\Lambda+n}}{\\Gamma\\left( \\Lambda + n \\right)} \\left( \\frac{\\Vert v \\Vert}{a(\\tau,u)} \\right)^{\\Lambda+1} K_{ \\Lambda+1 }(2\\pi a(\\tau,u) \\Vert v \\Vert) \\,e^{-2i\\pi (ax-by)}\\,dxdy$$\nwhich can be re-organised as\n\\begin{equation} \\label{equation-ax_by_integral}\n\\frac{2^{\\alpha+1} \\pi^{\\Lambda+n} \\Vert v \\Vert^{\\Lambda+1}}{\\Gamma\\left( \\Lambda + n \\right)} \\int_{\\R^2} \\frac{\\tau^{\\alpha} K_{ \\Lambda+1 }(2\\pi a(\\tau,u) \\Vert v \\Vert) }{\\Big( a(\\tau,u) \\Big)^{\\Lambda+1}} \\,e^{-2i\\pi (ax-by)}\\,dxdy.\n\\end{equation}\n\\noindent Let us again use polar coordinates (outside the origin):\n\\begin{itemize}\n\\item $(x,y)= r v_{\\theta}$ with $r>0$, $\\theta \\in \\R$ and $v_{\\theta}=(\\cos \\theta, \\sin \\theta)$; thus, $\\tau =r e^{i\\theta}$.\n\\item $(a,-b)=r' v_{\\theta'}$ with $r'>0$, $\\theta' \\in \\R$ and $v_{\\theta'}=(\\cos \\theta', \\sin \\theta')$; thus, $\\overline{s} =r' e^{i\\theta'}$.\n\\end{itemize}\n\\noindent Let us write $a(r,u)$ instead of $a(\\tau,u)$:\n$$a(r,u)=\\sqrt{1+4r^2+\\Vert u \\Vert^2}.$$\n\\noindent Integral (\\ref{equation-ax_by_integral}) can now be written:\n\\begin{multline} \\label{equation-cos_sin_integral}\n\\frac{2^{\\alpha+1} \\pi^{\\Lambda+n} \\Vert v \\Vert^{\\Lambda+1}}{\\Gamma\\left( \\Lambda + n \\right)} \\int_0^{\\infty} \\frac{r^{\\alpha} K_{ \\Lambda+1 }(2\\pi a(r,u) \\Vert v \\Vert) }{\\Big( a(r,u) \\Big)^{\\Lambda+1}} \\\\\n\\left( \\int_0^{2\\pi} e^{i\\alpha\\theta}\\,e^{-2i\\pi rr' (\\cos \\theta \\cos \\theta'+\\sin \\theta \\sin \\theta')} d\\theta \\right)\\,rdr.\n\\end{multline}\n\\noindent Following Proposition \\ref{proposition-bessel_integral_formula}, the inner integral\n$$\\int_0^{2\\pi} e^{i\\alpha \\theta} \\, e^{-2i\\pi r r' (\\cos \\theta \\, \\cos \\theta' \\, + \\, \\sin \\theta \\, \\sin \\theta')} \\, d\\theta$$\nis equal to:\n\\begin{equation} \\label{equation-inner_integral_computation}\n2 \\pi e^{i\\alpha\\left( \\theta'-\\frac{\\pi}{2} \\right) } \\, J_{\\alpha}(2\\pi r r').\n\\end{equation}\n\\noindent Because $r'=\\vert s \\vert$ and $\\theta'={\\rm Arg}(\\overline{s})$, (\\ref{equation-inner_integral_computation}) is equal to:\n$$2 \\pi e^{i\\alpha \\left( {\\rm Arg}(\\overline{s}) - \\frac{\\pi}{2} \\right) } \\, J_{\\alpha}(2\\pi r \\vert s \\vert ).$$\n\\noindent This turns (\\ref{equation-cos_sin_integral}) into:\n\\begin{multline} \\label{equation-non_standard_version_of_hwv_gamma=0}\n\\frac{2^{\\alpha+2} \\pi^{\\Lambda+n+1} \\Vert v \\Vert^{\\Lambda+1} e^{i\\alpha \\left( {\\rm Arg}(\\overline{s}) - \\frac{\\pi}{2} \\right) } }{\\Gamma\\left( \\Lambda + n \\right)} \\\\\n\\int_0^{\\infty} \\frac{r^{\\alpha + 1} \\, K_{ \\Lambda+1 }(2\\pi a(r,u) \\Vert v \\Vert) }{\\Big( a(r,u) \\Big)^{\\Lambda+1}} \\, J_{\\alpha}(2\\pi r |s|) \\, dr.\n\\end{multline}\n\\noindent We can now apply the second formula of proposition \\ref{proposition-integral_formulas-erdelyi}. To help follow notation choices made in this proposition, we set:\n\\begin{itemize}\n\\item $x=2r$, $dx=2dr$ and $\\beta=\\sqrt{1+\\Vert u \\Vert^2 } > 0$;\n\\item $a=2\\pi\\Vert v \\Vert > 0$ (careful: this variable $a$ is not what we have denoted $a(r,u)$);\n\\item $y=\\pi\\vert s\\vert > 0$, $\\nu=\\alpha$ and $\\mu=\\Lambda+1$.\n\\end{itemize}\n\\noindent Plugging these expressions in (\\ref{equation-non_standard_version_of_hwv_gamma=0}) and using Proposition \\ref{proposition-integral_formulas-erdelyi}, we finally obtain:\n\\begin{thm} \\label{theorem-final_formula}\nGiven any $\\lambda \\in \\R$ and any $(k,\\alpha,\\beta) \\in \\N^3$ such that $\\alpha+\\beta=k$, consider the non-compact form $G_{\\alpha,\\beta}$ of the highest weight vector $P_0^{\\alpha,\\beta}$. Then for all $(s,u,v)$ in $\\C \\times \\C^m \\times \\C^m$ such that $s \\neq 0$ and $v \\neq 0$:\n\\begin{multline*}\n\\mathcal{F}(G_{\\alpha,\\beta})(s,u,v)\\, = \\\\\n\\int_{\\C \\times \\C^m} \\, \\frac{(2\\tau)^{\\alpha}}{ \\left(1+4\\vert \\tau \\vert ^2 + \\Vert u \\Vert^2 + \\Vert \\xi \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\, e^{-2i\\pi {\\rm Re} \\left( s \\tau + \\langle v,\\xi \\rangle \\right) } \\,d\\tau d\\xi \\, = \\\\\nR(s,u,v) \\, K_{\\frac{i\\lambda+\\delta}{2}} \\left( \\pi \\sqrt{ 1 + \\Vert u \\Vert^2 } \\sqrt{ \\vert s \\vert^2+4\\Vert v \\Vert^2 } \\right)\n\\end{multline*}\nwhere we set\n$$R(s,u,v) \\, = \\, \\frac{(-i \\, \\overline{s})^{\\alpha} \\, \\pi^{i\\lambda+\\beta+n}}{2^{\\frac{i\\lambda+k}{2}+1} \\, \\Gamma \\left( \\frac{i\\lambda+k}{2}+n \\right)} \\left( \\frac{ \\sqrt{ |s|^2+4\\Vert v \\Vert^2}}{\\pi \\sqrt{1+\\Vert u \\Vert^2}} \\right)^{\\frac{i\\lambda+\\delta}{2}}.$$\n\\end{thm}\n\\noindent This theorem establishes the second part of our \"Main results\" theorem (stated in the introduction).\n\\begin{nnsrem} \\\n\\begin{itemize}\n\\item The partial Fourier transform $\\mathcal{F}$ changes, up to a constant $\\frac{-1}{i\\pi}$, multiplication by the $s$ coordinate into differentiation with respect to $s$. This implies $\\mathcal{F}(G_{\\alpha+1,\\beta-1})=\\frac{2}{-i\\pi}\\frac{\\partial}{\\partial s} \\Big( \\mathcal{F}(G_{\\alpha,\\beta}) \\Big)$, which is in fact proved in \\cite{Mendousse2017} for $\\beta \\geq 2$ (due to integrability issues).\n\\item One inevitably notices in the formula of Theorem \\ref{theorem-final_formula} a structure in two variables. Indeed, the particular value $\\alpha=0$ and the square root terms lead us to study the functions\n$$\\begin{array} {cccc}\n\\psi_{\\nu} : & ]0,+\\infty[ \\times ]0,+\\infty[ & \\longrightarrow & \\C \\\\\n & (x,y) & \\longmapsto & \\left( \\frac{x}{y} \\right)^{\\nu} K_{\\nu} (xy) \\\\\n\\end{array}$$\nwhere the parameter $\\nu$ is a complex number. Fix $\\nu \\in \\C$, $x_0 > 0$ and define the function\n$$\\begin{array} {cccc}\n\\varphi_{x_0,\\nu} : & ]0,+\\infty[ & \\longrightarrow & \\C \\\\\n & y & \\longmapsto & \\psi_{\\nu}(x_0,y). \\\\\n\\end{array}$$\n\\noindent One can show:\n$$ \\varphi_{x_0,\\nu}''(y) \\, + \\, \\frac{(1+2\\nu)}{y} \\, \\varphi_{x_0,\\nu}'(y) \\, - \\, x_0^2 \\, \\varphi_{x_0,\\nu}(y) \\, = \\, 0.$$\n\\noindent This equation belongs to the family of \\textit{Emden-Fowler equations} (or \\textit{Lane-Emden equations}) and its solutions can be written as the following combinations of Bessel functions of the first and second kind:\n$$u(t) \\, = \\, C_1 \\, t^{-\\nu} \\, J_{\\nu} \\left( -itx_0 \\right) \\, + \\, C_2 \\, t^{-\\nu} \\, Y_{\\nu} \\left( -itx_0 \\right).$$\nSimilar conclusions hold if one fixes $y_0$ instead of $x_0$.\n\\end{itemize}\n\\end{nnsrem}\n\\section{Appendix: Bessel functions} \\label{section-appendix}\n\\noindent In these definitions, following for instance \\cite{Lebedev1972} (sections 5.3 and 5.7), we take $\\nu \\in \\C$ and $z \\in \\C \\, \\setminus \\{0\\}$ such that $-\\pi < {\\rm Arg}(z) < \\pi$:\n\\begin{enumerate}\n\\item The \\textit{Bessel function of the first kind} is the function $J_{\\nu}$ defined by:\n$$J_{\\nu}(z)=\\sum_{k=0}^{\\infty} \\frac{(-1)^k}{\\Gamma(k+1) \\Gamma(k+\\nu+1)} \\left( \\frac{z}{2} \\right)^{\\nu+2k}.$$\n\\item The \\textit{Bessel function of the second kind} is the function $Y_{\\nu}$ defined by:\n$$Y_{\\nu}(z)=\\frac{J_{\\nu}(z) \\cos(\\nu \\pi) - J_{-\\nu}(z)}{\\sin (\\nu \\pi)}$$\nwhen $\\nu \\notin \\Z $ and, when $\\nu \\in \\Z$, by\n$$Y_{\\nu}(z)=\\lim_{\\substack{\n\\epsilon \\rightarrow \\nu \\\\\n0< |\\epsilon-\\nu| < 1}\n} Y_{\\epsilon}(z).$$\n\\item The \\textit{modified Bessel function of the first kind} is the function $I_{\\nu}$ defined by: \n$$I_{\\nu}(z)=\\sum_{k=0}^{\\infty} \\frac{1}{\\Gamma(k+1) \\Gamma(k+\\nu+1)} \\left( \\frac{z}{2} \\right)^{\\nu+2k}.$$\n\\item The \\textit{modified Bessel function of the third kind} is the function $K_{\\nu}$ defined by \n$$K_{\\nu}(z)=\\frac{\\pi}{2} \\frac{I_{-\\nu}(z) - I_{\\nu}(z)}{\\sin (\\nu \\pi)}$$\nwhen $\\nu \\notin \\Z $ and, when $\\nu \\in \\Z$, by\n$$K_{\\nu}(z)=\\lim_{\\substack{\n\\epsilon \\rightarrow \\nu \\\\\n0 < |\\epsilon-\\nu| < 1}\n} K_{\\epsilon}(z).$$\n\\end{enumerate}\n\\noindent As a consequence of Formula (2) of Section 7.3.1 in Chapter VII of \\cite{Erdelyi-higher_etc.-2}:\n\\begin{prop}[An integral representation of Bessel functions] \\label{proposition-bessel_integral_formula} Given $\\nu \\in \\N$, $\\rho >0$ and $a > 0$:\n$$J_{\\nu}(\\rho) \\, = \\, \\frac{1}{2\\pi e^{i\\nu\\left( a-\\frac{\\pi}{2} \\right)}} \\, \\int_0^{2\\pi} e^{i\\nu \\theta} \\, e^{-i\\rho(\\cos a \\, \\cos \\theta \\, + \\, \\sin a \\, \\sin \\theta)} \\, d\\theta.$$\n\\end{prop}\n\\noindent Formulas in the next proposition are stated in \\cite{Erdelyi-tables_etc.-2} (Chapter VIII: Formula (20) of Section 8.5 and Formula (35) of Section 8.14):\n\\begin{prop}[Two integral formulas involving Bessel functions] \\label{proposition-integral_formulas-erdelyi}\\\n\\begin{itemize}\n\\item For any real number $y > 0$ and any complex numbers $a,\\nu,\\mu$ such that ${\\rm Re}(a) > 0$ and $-1 < {\\rm Re}(\\nu) < 2 {\\rm Re}(\\mu) + \\frac{3}{2}$, one has:\n$$\\int_0^{\\infty} x^{\\nu + \\frac{1}{2}} \\left( x^2 + a^2 \\right)^{-\\mu-1} J_{\\nu}(xy) \\sqrt{xy} \\ dx =$$\n$$ \\frac{a^{\\nu-\\mu} y^{\\mu+\\frac{1}{2}} K_{\\nu-\\mu}(ay)}{2^{\\mu} \\Gamma(\\mu+1)}.$$\n\\item For any real number $y > 0$ and any complex numbers $a,\\beta,\\nu,\\mu$ such that ${\\rm Re}(a) > 0$, ${\\rm Re}(\\beta) > 0$ and ${\\rm Re}(\\nu) > -1$, one has:\n$$\\int_0^{\\infty} x^{\\nu + \\frac{1}{2}} \\left( x^2 + \\beta^2 \\right)^{-\\frac{\\mu}{2}} K_{\\mu} \\left( a(x^2+\\beta^2)^{\\frac{1}{2}} \\right) J_{\\nu}(xy) \\sqrt{xy}\\ dx =$$\n$$a^{-\\mu} \\beta^{\\nu+1-\\mu} y^{\\nu + \\frac{1}{2}} (a^2+y^2)^{\\frac{\\mu}{2}-\\frac{\\nu}{2}-\\frac{1}{2}} K_{\\mu-\\nu-1} \\left( \\beta (a^2+y^2)^{\\frac{1}{2}} \\right).$$\n\\end{itemize}\n\\end{prop}\n\\noindent One can find the next formula in \\cite{Erdelyi-higher_etc.-2} (Section 7.4.1, Formula (4)):\n\\begin{prop}[Asymptotic expansion for modified Bessel functions] \\label{proposition-bessel_asymptotics-erdelyi} \nFor any fixed $P \\in \\N \\setminus \\{0\\}$ and $\\nu \\in \\C$:\n$$K_{\\nu}(z)=\\left( \\frac{\\pi}{2z} \\right)^{\\frac{1}{2}} e^{-z} \\left( \\left[ \\sum_{p=0}^{P-1} \\frac{ \\Gamma \\left( \\frac{1}{2} + \\nu + p\\right) }{p! \\ \\Gamma \\left( \\frac{1}{2} + \\nu - p\\right) } \\ (2z)^{-p} \\right] + {\\rm O} \\left( |z|^{-P} \\right) \\right).$$\n\\end{prop}\n\\noindent The following proposition can be derived from Section 9.6 in \\cite{Faraut2008}:\n\\begin{prop}[Bochner formula] \\label{proposition-bochner} \nConsider any integer $p \\geq 2$. For $\\xi'\\in S^{p-1}$ and $s > 0$:\n$$\\int_{S^{p-1}}e^{-2i\\pi s \\xi \\cdot \\xi'} d\\sigma(\\xi)\n=2 \\pi s^{1-\\frac{p}{2}} J_{\\frac{p}{2}-1}(2\\pi s)$$\nwhere $d\\sigma$ denotes the Euclidean measure of $S^{p-1}$ and $\\xi \\cdot \\xi'$ denotes the Euclidean scalar product of $\\R^p$ applied to the elements of the sphere $\\xi$ and $\\xi'$ seen as elements of $\\R^p$.\n\\end{prop}\n\n\\footnotesize{ \n\\noindent \\textbf{Contact information.}\\\\\n\\noindent Universit{\\'e} de Reims Champagne-Ardenne\\\\\n\\noindent Laboratoire de Math{\\'e}matiques FRE 2011 CNRS\\\\\n\\noindent UFR Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039\\\\\n\\noindent 51687 REIMS Cedex 2, FRANCE\\\\\n\\noindent \\texttt{gregory.mendousse@univ-reims.fr}\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nBlazars, active galactic nuclei (AGN) with strong nonthermal\nemission from an aligned\nrelativistic jet \\citep[][]{br78,up95},\nare the most luminous persistent objects in the universe.\nThese sources emit photons\nacross the whole electromagnetic spectrum from the radio to gamma-ray bands.\nTheir spectral energy distributions (SEDs) are well characterized with a\ndouble-hump structure where the low-energy hump, peaking in the IR\/optical\/UV\/X-ray\nband, is thought to be produced by synchrotron emission of the jet electrons.\nTheir high-energy peak in the gamma-ray band is produced by\nsynchrotron self-Compton (SSC) and external Compton (EC) scattering, or possibly\nby hadronic processes \\citep[e.g.,][]{mb92,bms97,gtf+10}.\n\nBlazars are heuristically classified into flat spectrum radio quasars (FSRQs) and\nBL Lacertae objects (BL Lacs). The former show broad optical emission lines associated\nwith clouds surrounding or in the accretion disk. The latter lack such lines and have a jet\ncontinuum strong enough to obscure spectral features of the host galaxy\n\\citep[][]{mbi+96,lpp+04}. \\citet{pg95} further divided BL Lacs\nbased on the synchrotron peak frequency ($\\nu^{\\rm sy}_{\\rm pk}$) into low synchrotron peak\n(LSP, $\\nu^{\\rm sy}_{\\rm pk}<10^{14}\\rm Hz$), intermediate peak\n(ISP, $10^{14}\\rm Hz<\\nu^{\\rm sy}_{\\rm pk}<10^{15}\\rm Hz$), and high peak\n(HSP, $10^{15}\\rm Hz<\\nu^{\\rm sy}_{\\rm pk}$) subclasses. FSRQs are\nalmost all classified as LSP \\citep[][]{fermiblazar10}.\n\n \\citet{fmc+98} found that 5\\,GHz luminosity,\nthe synchrotron peak luminosity ($L^{\\rm sy}_{\\rm pk}$), and\nthe gamma-ray dominance (ratio of the peak\ngamma-ray to peak synchrotron $\\nu F_{\\nu}$ luminosity)\nare correlated with $\\nu^{\\rm sy}_{\\rm pk}$.\nThey characterize this as\na ``blazar sequence'' trend from low-peaked powerful sources (i.e., FSRQs) to high-peaked\nless powerful sources (HSPs). A plausible physical explanation for this\nsequence is provided by \\citet{gcf+98}; more luminous sources tend to have stronger disk\naccretion, and the external photons from the broad line region (BLR) or the disk in these\nsources provide additional seeds for Compton upscattering which cools the jet\nelectrons, lowering $\\nu^{\\rm sy}_{\\rm pk}$, while increasing the Compton luminosity.\nIndeed, as the typical accretion state evolves over cosmic time, this picture\nmay provide an explanation of evolution in the FSRQ\/BL Lac blazar populations\n\\citep[][]{bd02,cd02}. Quantitatively, this may explain the apparent ``negative evolution''\n(increase at low redshift) observed for HSP BL Lacs \\citep[][]{rsp+00,beb+03,arg+14}.\n\n\\newcommand{\\tablenotemark{a}}{\\tablenotemark{a}}\n\\newcommand{\\tablenotemark{b}}{\\tablenotemark{b}}\n\\newcommand{\\tablenotemark{c}}{\\tablenotemark{c}}\n\\begin{table*}[t]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Summary of observations used in this work\n\\label{ta:ta1}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{cccccccc} \\hline\\hline\nSource & R.A. & Decl. & Redshift & Observatory & Start date & Obs. ID & Exposure \\\\\n & & & & & (MJD) & & (ks) \\\\ \\hline\n\\multirow{4}{*}{J0022} & \\multirow{4}{*}{0$^{\\rm h}$22$^{\\rm m}$09.25$^{\\rm s}$} & \\multirow{4}{*}{$-$18$^\\circ$53$'$34.9$''$} & \\multirow{4}{*}{0.774} & GROND & 57031.1 & $\\cdots$ & 0.25\/0.24\\tablenotemark{a} \\\\\n & & & & {\\em Swift} & 57031.7 & 00080777001 & 1.9\\tablenotemark{b} \\\\\n & & & & {\\em XMM} & 57026.8 & 0740820501 & 15\/9\\tablenotemark{c} \\\\\n & & & & {\\em NuSTAR} & 57026.7 & 60001141002--4 & 110 \\\\ \\hline\n\\multirow{4}{*}{J0630} & \\multirow{4}{*}{6$^{\\rm h}$30$^{\\rm m}$59.515$^{\\rm s}$} &\\multirow{4}{*}{$-$24$^\\circ$06$'$46.09$''$} &\\multirow{4}{*}{$>$1.239} & GROND & 56949.2 & $\\cdots$ & 0.25\/0.24\\tablenotemark{a} \\\\\n & & & & {\\em Swift} & 56948.5 & 00080776001 & 0.27\\tablenotemark{b} \\\\\n & & & & {\\em XMM} & 56948.2 & 0740820401 & 8\/4\\tablenotemark{c} \\\\\n & & & & {\\em NuSTAR} & 56947.7 & 60001140002 & 67 \\\\ \\hline\n\\multirow{4}{*}{J0811} & \\multirow{4}{*}{8$^{\\rm h}$11$^{\\rm m}$03.214$^{\\rm s}$} & \\multirow{4}{*}{$-$75$^\\circ$30$'$27.85$''$} & \\multirow{4}{*}{0.689} & GROND & 56903.3 & $\\cdots$ & 0.25\/0.24\\tablenotemark{a} \\\\\n & & & & {\\em SWIFT} & 56908.2 & 00091903001 & 0.39\\tablenotemark{b} \\\\\n & & & & {\\em XMM} & 56901.2 & 0740820601 & 9\/6\\tablenotemark{c} \\\\\n & & & & {\\em NuSTAR} & 56901.2 & 60001142002 & 113 \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\hspace{-2.0 mm}\n$^{\\rm a}${ For {\\it g$'$r$'$i$'$z$'$\/JHK} bands.}\\\\\n$^{\\rm b}${ For the UW1 band. Exposures in the other UVOT bands may differ from this value.}\\\\\n$^{\\rm c}${ For MOS1,2\/PN.}\\\\\n\\vspace{-1.0 mm}\n\\end{table*}\n\n On the other hand, \\citet{gpp+12} used Monte Carlo simulations to argue that the\n$L^{\\rm sy}_{\\rm pk}$ and $\\nu^{\\rm sy}_{\\rm pk}$ anti-correlation may be primarily a\nselection effect. \\citet{pgr12} discuss four sources with high $\\nu^{\\rm sy}_{\\rm pk}$ and\nhigh peak (synchrotron + SSC) power as examples well off of the blazar sequence.\nSuch sources might be FSRQs with unusually strong jet emission along the Earth line-of-sight\nmasking the underlying host components. Thus simultaneous observations and careful SED\nmodeling of such (generally higher-redshift) BL Lac sources is interesting as it can\nhelp us understand the underlying emission zone physics and whether it is truly different\nfrom the bulk of the blazar population. Characterization via less redshift-dependent parameters\n\\citep[e.g. gamma-ray dominance or Compton dominance;\nsee][for example]{fmc+98,f13} may also help clarify\ntheir place in the population. Also, comparing robust SED model fits with\ngamma-ray spectra of high-$z$ blazars can reveal the effect of absorption\nby the extragalactic background light (EBL), which provides important constraints\non evolution of cosmic star formation \\citep[e.g.,][]{fermiEBL, HESSEBL}.\nBL Lacs are believed to have higher Compton dominance and less sensitivity\nto local soft photon fields and so are particularly useful for such study.\n\n Appropriate high-redshift HSP BL Lac objects are rare because they\nare faint especially in the gamma-ray band, and HSPs appear to exhibit negative\nevolution \\citep[][]{arg+14}. We select three\n{\\it Fermi}-detected \\citep[][]{fermi2fgl,fermi2lac} sources,\n3FGL~J0022.1$-$1855 (J0022, $z=0.774$),\n3FGL~J0630.9$-$2406 (J0630, $z>1.239$),\nand 3FGL~J0811.2$-$7529 (J0811, $z=0.689$), whose optical spectra\nare unusual, showing no emission lines but a set of strong low excitation (Mg~I,\nFe~II, Al~II etc) absorption lines on a blue, power-law\ncontinuum. These indicate that the AGN is viewed through the disk of an intervening\nabsorber. In \\citet{src+13}, this was taken to be the host galaxy; indeed for\nJ0630 the photometry of \\citet{rsg+12} supports this as the host redshift. With\nestimated redshifts of 0.774, $>$1.239, and 0.689 \\citep[][]{rsg+12,src+13} for\nJ0022, J0630 and J0811, respectively,\nthese are thus luminous high-peak sources suitable for studying the extreme of the BL Lac\npopulation. At these redshifts, we may also see the effects of\nextragalactic background light absorption at\nthe high end of the {\\it Fermi} band. To probe this absorption, and the high end\nof the jet particle population most sensitive to Compton cooling, we require particularly\ngood characterization of the peak and high-energy cutoff (near-IR to hard X-ray)\nof the synchrotron component. Under classic SSC modeling,\nthis allows us to characterize the high-energy Compton component, as well, thus\nproviding inferences about the Compton cooling at the source and EBL absorption of the\nGeV photons as they propagate to Earth.\n\nIn this paper, we present broadband SEDs of the three high-redshift BL Lacs which are\nsimultaneous across the critical $\\nu >\\nu^{\\rm sy}_{\\rm pk}$ range (Section~\\ref{sec:sec2}).\nJ0630 has been previously discussed as a high-$\\nu^{\\rm sy}_{\\rm pk}$,\nhigh-power source \\citep{pgr12}; our improved data allow more refined modeling,\nwhich is discussed in Section~\\ref{sec:sec3}, including EBL constraints. The implications\nof our inferred model parameters are discussed in Section~\\ref{sec:sec4}.\nWe use $H_0=70\\rm \\ km\\ s^{-1}\\ Mpc^{-1}$, $\\Omega_m=0.3$, $\\Omega_{\\Lambda}=0.7$ \\citep[e.g.,][]{ksd+11},\nand redshift values given in Table~\\ref{ta:ta1} ($z=1.239$ for J0630) throughout.\n\n\\section{Observations and Data Reduction}\n\\label{sec:sec2}\n\nBL Lac objects can be variable on all timescales from minutes to years \\citep[][]{HESSvariable},\nso coordinated broad-band coverage is important for characterizing the instantaneous SED.\nWe therefore carried out nearly contemporaneous observations of the sources using the\nGamma-Ray burst Optical\/Near-Infrared Detector (GROND)\ninstrument at the 2.2-m MPG telescope at the ESO La Silla Observatory \\citep{gbc+08}\nas well as the {\\it Swift} \\citep{gcg+04}, {\\it XMM-Newton} \\citep{jla+01}\nand {\\it NuSTAR} \\citep{hcc+13} satellites, covering the upper range of the synchrotron component.\nOur sources showed relatively modest variability in the {\\it Fermi} \\citep[][]{fermimission} band\nand so we average over 6 years of Large Area Telescope (LAT) data to best characterize the\nmean Compton component of these relatively faint (but luminous, for BL Lacs) sources.\nArchival radio, optical, and near-IR observations are provided for comparison\nalthough we do not use them in the SED fitting.\n\n\\subsection{Contemporaneous observations: GROND, Swift, XMM-Newton, and NuSTAR}\n\\label{sec:sec2_1}\n\nThe GROND data were reduced and analyzed with the standard tools and methods\ndescribed in \\citet{kkg+08}. The photometric data were obtained using\nFWHM-matched PSF ($g^\\prime r^\\prime i^\\prime z^\\prime$) or aperture photometry ({\\it JHK}).\nThe $g^\\prime$, $r^\\prime$, $i^\\prime$, and $z^\\prime$ photometric\ncalibration was obtained via standard star fields observed on the\nsame nights as the target integrations. The {\\it J}, {\\it H}, and {\\it Ks} photometry was\ncalibrated against selected in-field 2MASS stars \\citep[][]{scs+06}.\n\nFor {\\it Swift} UVOT data, we performed aperture photometry for the six\n{\\it Swift} filters \\citep[][]{pbp+08} using the {\\tt uvotsource} tool\nin HEASOFT 6.16\\footnote{http:\/\/heasarc.nasa.gov\/lheasoft\/}. We measured photometric magnitude of the sources using\na $R=5^{\\prime\\prime}$ aperture. Backgrounds were estimated using a $R=20^{\\prime\\prime}$ circle\nnear the source.\n\nX-ray SEDs of the sources were measured with {\\it XMM-Newton} and {\\it NuSTAR}.\nThe sources were detected with very high significance ($>20\\sigma$)\nwith {\\it XMM-Newton} but with relatively low significance ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 6\\sigma$)\nwith {\\it NuSTAR}.\nFor the {\\em XMM-Newton} data, we processed the observation data files\nwith {\\ttfamily epproc} and {\\ttfamily emproc} of Science Analysis System (SAS)\nversion 14.0.0\\footnote{http:\/\/xmm.esac.esa.int\/sas\/} and then applied\nstandard filters. The {\\it NuSTAR} data were processed with the standard pipeline\ntool {\\tt nupipeline} of {\\tt nustardas}\n1.4.1 integrated in the HEASOFT 6.16.\nWe used {\\it NuSTAR} CALDB version 20140414 and applied the standard\nfilters.\\footnote{See http:\/\/heasarc.gsfc.nasa.gov\/docs\/nustar\/analysis\/nustar\\\\\\_swguide.pdf for more details}\nWe then extracted source events using circular regions with $R=20^{\\prime\\prime}$ and $R=30^{\\prime\\prime}$\nfor the {\\it XMM-Newton} and the {\\it NuSTAR} data, respectively. Backgrounds were\nextracted from nearby source-free regions.\n\n\\subsection{Gamma-ray observations}\n\\label{sec:sec2_2}\nFor the gamma-ray data, we used the {\\it Fermi} observations taken between\n2008 August 4 and 2015 January 31. The Pass 8 data \\citep[][]{fermiP8}, based on a complete\nand improved revision of entire LAT event-level analysis, were downloaded\nfrom Fermi Science Support Center\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/},\nand we analyzed the data using the {\\it Fermi}\nScience tool 10-00-04 along with the instrument response functions (IRFs) P8R2\\_SOURCE\\_V6.\nWe extracted source class events in the 100\\,MeV--500\\,GeV band in a\n$R=5^\\circ$ region of interest (ROIs)\nand $<80^\\circ$ zenith angle and $<52^\\circ$ rocking angle cuts.\nThese events were analyzed using the background models ({\\tt gll\\_iem\\_v06} and\n{\\tt iso\\_P8R2\\_SOURCE\\_V6\\_v06}) and all 3FGL sources within $15^\\circ$.\nWe first modeled fluxes on a one-month cadence to check for strong source\nvariability using the standard {\\it Fermi} likelihood analysis with {\\tt gtlike}\n(see Figure~\\ref{fig:fig1} and Section~\\ref{sec:sec3_1}).\nNo strong flares were seen and so we combined all the LAT\ndata, modeling the mission-averaged spectrum.\nIn Figure~\\ref{fig:fig1}, we mark the epochs of the\ncontemporaneous campaign and historical spectra. For J0630 we also have access\nto optical monitoring from the KAIT program \\citep[][]{crf+14}, shown on the top panel.\nVariability is clearly seen in the optical band.\n\n\\begin{figure}\n\\includegraphics[width=3.65 in]{fig1.eps}\n\\vspace{-3mm}\n\\figcaption{Optical ({\\it R} band) and gamma-ray (100\\,MeV--500\\,GeV) light curves.\nThe top panel shows KAIT (right scale)\nand {\\it Fermi} (left scale) fluxes for J0630. Our contemporaneous observation\nepoch and the optical spectrum epochs are marked. The lower panel shows the LAT light curves and\nmultiwavelength epochs for J0022 and J0811. The modest LAT variability justifies\nthe use of mission-averaged spectra.\n\\label{fig:fig1}\n}\n\\vspace{0mm}\n\\end{figure}\n\n\\subsection{Archival observations}\n\\label{sec:sec2_3}\nFor comparison, we also collected archival data in the radio to UV band. We assembled data\nfrom various catalogs (e.g., {\\it WISE} and 2MASS for IR data) or reanalyze the\narchival data (e.g., VLT\/Keck spectra and {\\it Swift} UVOT). For the catalog data, we convert the\nmagnitude to flux appropriately.\nThe VLT\/Keck data reduction and calibration were described in \\citet{src+13}.\nThe archival UVOT data are processed as described above (Section~\\ref{sec:sec2_1}).\nThe measurements are corrected for Galactic extinction in constructing\nthe SED (Section~\\ref{sec:sec3_2}).\nArchival measurements are used only in flux variability studies.\n\n\\subsection{Discovery of a serendipitous source}\n\\label{sec:sec2_4}\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ccc}\n\\hspace{-3.0 mm}\n\\includegraphics[width=2.1 in]{fig2a.eps} &\n\\hspace{1.0 mm}\n\\includegraphics[width=2.05 in]{fig2b.eps} &\n\\hspace{-3.0 mm}\n\\includegraphics[width=2.7 in]{fig2c.eps} \\\\\n\\end{tabular}\n\\figcaption{{\\it Left}: {\\it NuSTAR} image\nof the field containing J0811. The color scale is arbitrarily adjusted\nfor better visibility. {\\it Fermi}\/LAT 3FGL ellipse (95\\%, white) and the best-fit circle (95\\%, magenta)\nare shown, and a $R=30''$ circle is drawn around the serendipitous source (denoted as J0810.0$-$7527).\n{\\it Middle}: Location of the sources we are studying in the {\\it WISE}\n[3.4]$-$[4.6]$-$[12]$\\mu m$ color-color diagram \\citep[Figure taken from][]{dma+12}.\nThe four sources, including J0810, are located in the middle of the BZB\n(naming convention for BL Lac in the ROMA-BZCAT catalog) distribution.\nSee \\citet{dma+12} for more detail.\n{\\it Right}: Observed SED of the serendipitous source. Note that we used\n$N_{\\rm H}=6.9\\times 10^{20}\\rm \\ cm^{-2}$, the optical extinction inferred value,\nfor constructing the SED. Notice that this new source is quite\nhard, emitting more strongly in the {\\it NuSTAR} band than in the {\\it XMM-Newton} band.\n\\label{fig:fig2}\n}\n\\vspace{0mm}\n\\end{figure*}\n\nWe discovered a serendipitous X-ray source (J0810) in the field of J0811 (Figure~\\ref{fig:fig2}).\nThe X-ray ({\\it XMM-Newton}) position of the source is\nR.A. = 08\\textsuperscript{h}10\\textsuperscript{m}03\\textsuperscript{s}\nand decl. = $-$75$^\\circ$27$'$21$''$ (J2000, $\\delta_{\\rm R.A.,\\ decl.} =2''$ statistical only),\nonly 6$'$ from J0811 (Figure~\\ref{fig:fig2} left).\nWe find that the spectrum cannot be described with a simple absorbed power law\n($\\chi^2$\/dof=185\/118, $p=7\\times10^{-5}$).\nA broken power-law model\\footnote{http:\/\/heasarc.gsfc.nasa.gov\/docs\/xanadu\/xspec\/manual\/XS\\\\modelBknpower.html}\nexplains the data ($\\chi^2$\/dof=116\/116, $p=0.47$) and\nthe best-fit parameters are $N_{\\rm H}=1.4\\pm0.3\\times10^{21}\\rm \\ cm^{-2}$,\nlow-energy photon index $\\Gamma_{\\rm 1}=3.4\\pm0.3$, high-energy photon index\n$\\Gamma_{\\rm 2}=1.74\\pm0.07$, break energy $E_{\\rm break}=1.46\\pm0.08$\\,keV\nand 3--10\\,keV flux $F_{\\rm 3-10 keV}=2.7\\pm0.2\\times 10^{-13}\\rm \\ erg \\ s^{-1}\\ cm^{-2}$.\n\nTogether with archival radio, optical, and {\\it Swift} UV data, we\nconstruct the SED of the source (Figure~\\ref{fig:fig2} right). If we use the best-fit X-ray\n$N_{\\rm H}$, the extrapolated spectrum matches poorly to the optical. Instead\nwe de-absorb using the value from the optical\/UV extinction $N_{\\rm H}=6.9\\times10^{20}\\rm \\ cm^{-2}$.\nX-ray fits with absorption fixed at this value are statistically acceptable\n(null hypothesis probability $p=0.3$).\nThe SED of this source suggests a blazar with $\\nu^{\\rm sy}_{\\rm pk}$ in the optical range,\nand a rise to a Compton component in the hard X-ray band. Its location in the\n{\\it WISE} color-color diagrams \\citep[Figure~\\ref{fig:fig2} middle; see also][]{dma+12}\nsuggests that the source should be a BL Lac.\nIf the Compton component peaks at $>100$\\,MeV, this source may contribute to the J0811 SED,\nsince the source is within the aperture we used for J0811. If we free the position of J0811 in the\n{\\it Fermi} analysis, we find a maximum likelihood coincident with J0811\n(magenta circle in Figure~\\ref{fig:fig2} left). Also, a second source\nat the J0810 position does not significantly increase the model test statistic (TS).\n\nWe then increased the zenith angle cut to $<100^\\circ$ to have more events\nand used a small spatial bin size (0.05$^\\circ$) to see if J0810 is detected\nin the {\\it Fermi} band. We performed binned likelihood analysis with\nthe new data. In this case, a gamma-ray counterpart\nof J0810 is detected significantly (TS=56); the model without J0810 is only 0.03\\% as\nprobable as the one with J0810.\nIn the 0.1--500\\,GeV band, J0810 has $\\sim$20\\% of the flux (with 40\\% flux uncertainty)\nof J0811 with a similar power-law index ($\\Gamma_{\\gamma}=1.8\\pm0.1$).\nThese spectral parameters for J0810 may not be very accurate because of mixing from\nthe brighter source, J0811.\nSince J0811 is brighter than J0810 in the gamma-ray band,\nwe attribute all of the LAT flux to J0811 in SED modeling\nand discuss implication of J0810 contamination on the model (see Section~\\ref{sec:sec3_3}).\n\n\\begin{figure*}\n\\centering\n\\vspace{-75.0 mm}\n\\hspace{-12.0 mm}\n\\includegraphics[width=5.7 in,angle=90]{fig3.eps} \\\\\n\\vspace{-5.0 mm}\n\\figcaption{Observed broadband SED and best-fit models for\n({\\it a}) J0022, ({\\it b}) J0630, and ({\\it c}) J0811.\nData points with an error bar are taken from the contemporaneous\nobservations (Sections~\\ref{sec:sec2_1} and \\ref{sec:sec2_2})\nand diamonds are from the archival observations (Section~\\ref{sec:sec2_3}).\nThe dashed lines are the best-fit SSC SED models\nof \\citet{bms97} with (black) and without (red) EBL absorption \\citep[][]{frd10}.\nNote that the archival data are not taken contemporaneously even if they\nare plotted in the same color and symbol. The insets plot the VLT\/Keck spectra of\n\\citet{src+13}, showing the lack of emission lines and the low excitation\nabsorption complexes placing lower limits on the redshift. These observations\nappear to have been in a brighter, harder optical state.\n\\vspace{2.0 mm}\n\\label{fig:fig3}\n}\n\\vspace{0mm}\n\\end{figure*}\n\n\\section{Data Analysis and Results}\n\\label{sec:sec3}\n\n\\subsection{Variability}\n\\label{sec:sec3_1}\n\n We have examined the collected data for variability, since short\ntimescales can give useful constraints on the characteristic size of the emission zone\nin the various wavebands. We first examined our contemporaneous data sets for\nshort-timescale variations. For the {\\it XMM-Newton} and {\\it NuSTAR} data, spanning\n$\\sim$10--100\\,ks, we constructed exposure-weighted light curves using various time bin sizes\n($\\sim$100--20,000~s), ensuring $> 20$ counts in each time bin, and calculated $\\chi^2$\nfor a constant flux. The probability for constancy was always high ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi$10\\%),\nimplying no significant short-term variability for any of the three sources at this epoch.\nSimilarly, the optical\/UV data from the contemporaneous epoch did not show sub-day variability.\n\n However, on longer time scales, the optical synchrotron peak flux does show\nsubstantial variability, as can be seen by comparing the contemporaneous and archival points\nin Figures~\\ref{fig:fig1} and \\ref{fig:fig3}. J0022, for example varies by $\\sim 6\\times$.\nAs noted, the VLT\/Keck spectra also appear to represent brighter epochs, although slit losses limit\nthe precision of the flux calibration. In general, the brighter epochs appear to have harder\nnear-IR to UV spectra, suggesting increased electron energy (or increased bulk Doppler factor)\nin flaring events. A much better characterization of J0630's optical variability is available from\nthe KAIT {\\it Fermi} AGN monitoring data \\citep[][]{crf+14}.\\footnote{http:\/\/brando.astro.berkeley.edu\/kait\/agn\/}\nThe dominant modulation is slow on $\\sim$year timescales; this is of modest amplitude compared\nto other BL Lacs ($\\sim$50\\%). KAIT resolves times as short as the $\\sim$3d cadence and\nwe do see statistically significant ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi$6$\\sigma$) changes between consecutive observations.\nThis suggests that at least some of the jet flux arises in compact $r<10^{16}$cm structures.\n\n We can use the LAT band to probe variability in the Compton peak emission. Since these\nsources are not very bright, we were able to only probe $\\sim$month timescales. To\nthis end we generated lightcurves by fitting source fluxes to 100\\,MeV--500\\,GeV\nphotons from a 5 degree ROI about each source using the {\\tt gtlike} tool for each time bin.\nFor this we fixed the background model\nnormalization and the background source spectral parameters at the mission-averaged values (see below),\nallowing only the source flux to vary with the spectral index held fixed at the values given\nin Table~\\ref{ta:ta2} .\nFigure~\\ref{fig:fig1} shows the corresponding light curves. The variability\nis not strong ($\\chi^2$\/dof values for a constant light curve of \n5\/8, 92\/72, and 28\/33 for J0022, J0630 and J0811, respectively).\nWe confirm the results of the 3FGL catalog \\citep[][]{fermi3fgl};\nour sources are not flagged as variable in the 3FGL catalog at a 99\\% confidence.\nFinally, examination of light curves assembled by the\nAgenzia Spaziale Italiana science data center\\footnote{http:\/\/www.asdc.asi.it\/fermi3fgl\/}\nalso shows no significant variability in any source. We conclude that the three sources have\nbeen relatively quiescent for BL Lacs -- this gives us confidence that the mission-averaged LAT\nspectrum may be usefully compared with our contemporaneous campaign fluxes for SED fitting.\n\n\\subsection{Constructing broadband SEDs}\n\\label{sec:sec3_2}\n\n\\newcommand{\\tablenotemark{a}}{\\tablenotemark{a}}\n\\newcommand{\\tablenotemark{b}}{\\tablenotemark{b}}\n\\newcommand{\\tablenotemark{c}}{\\tablenotemark{c}}\n\\begin{table}[t]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Galactic foreground reddening values and X-ray\/gamma-ray fit results\n\\label{ta:ta2}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{cccccccc} \\hline\\hline\nSource\t\t\t&\t\t\t\t& J0022 & J0630 & J0811 \\\\ \\hline\n$E(B-V)$\t\t& (mag)\t\t\t\t& 0.024 & 0.056 & 0.125 \\\\\n$N_{\\rm H,\\ Dust}$\\tablenotemark{a}\t& ($10^{20}\\rm \\ cm^{-2}$)\t& 1.3 & 3.1 & 7 \\\\\n$N_{\\rm H}$\t\t& ($10^{20}\\rm \\ cm^{-2}$)\t& 4(1) & 13(1) & 7(1) \\\\\n$\\Gamma_{\\rm X}$\t& $\\cdots$\t\t\t& 2.55(6) & 2.98(7) & 2.45(7) \\\\\n$F_{\\rm X}$\\tablenotemark{b}\t& $\\cdots$\t\t\t& 0.93(8) & 1.6(1) & 1.4(1) \\\\\n$\\Gamma_{\\rm \\gamma}$\t& $\\cdots$\t\t\t& 1.86(6) & 1.83(3) & 1.93(4) \\\\\n$F_{\\rm \\gamma}$\\tablenotemark{c}\t& $\\cdots$\t\t\t& 6.3(9) & 25(2) & 23(2) \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\vspace{-0.5 mm}\n$^{\\rm a}$ Dust-extinction equivalent $N_H$, converted with\n$N_{\\rm H}=1.8\\times 10^{21} A(V)\\rm \\ cm^{-2}\\ mag^{-1}$ and $R_V=3.1$ \\citep{ps95}.\\\\\n$^{\\rm b}$ 3--10\\,keV flux in units of $10^{-13}\\rm \\ erg\\ s^{-1}\\ cm^{-2}$.\\\\\n$^{\\rm c}$ 0.1--500\\,GeV flux in units of $10^{-9}\\rm \\ photons \\ s^{-1}\\ cm^{-2}$.\n\\end{table}\n\nNext we assembled broadband SEDs for the sources using the data described in Section~\\ref{sec:sec2}.\nThe optical\/UV magnitudes were corrected for the dust map extinction in these directions\n(Table~\\ref{ta:ta2}) obtained from the NASA\/IPAC extragalactic database, using\nthe \\citet{sf11} calibration. We show the SEDs in Figure~\\ref{fig:fig3}.\nNote that Lyman-$\\alpha$ forest absorption was visible in J0630 at frequencies above\n$\\sim10^{15}$\\,Hz in the UVOT data, as expected from its large redshift;\nwe do not use the high-frequency UVOT data $\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 10^{15}$\\,Hz in the J0630 SED modeling.\n\nThe X-ray response files are produced with the standard tools in SAS and in {\\tt nustardas}\nfor the {\\it XMM-Newton} and {\\it NuSTAR} spectra, respectively.\nWe fit the spectra in the 0.3--79\\,keV band with an absorbed power-law model\nin {\\tt XSPEC} 12.8.2 and found that the model describes the data well, having\n$\\chi^2$\/dof$\\ifmmode\\stackrel{<}{_{\\sim}}\\else$\\stackrel{<}{_{\\sim}}$\\fi$1 for all three sources. The fact that all X-ray spectra are\nwell modeled by a single absorbed power law is important to the modeling below.\nThe absorption corrections for the X-ray data were obtained from the $N_{\\rm H}$\nin the power-law fits. The fit results are presented in Table~\\ref{ta:ta2}.\n\n While the X-ray fit and extinction map values for the absorption agree well for J0811,\nJ0022 and especially J0630 show stronger X-ray absorption. Given the modest dust map resolution,\nand the $\\sim$50\\% conversion uncertainties \\citep[e.g.,][]{g75,w11,fgos15}, the discrepancy\nfor J0022 may be reconciled. However the large value for J0630 seems difficult to accommodate\nand we have no clear explanation. The Galactic H{\\scriptsize I} column\ndensity\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/cgi-bin\/Tools\/w3nh\/w3nh.pl}\ntoward J0630 is 7--12$\\times 10^{20}\\rm cm^{-2}$, consistent with the X-ray inferred value.\nIf we assume the X-ray value for de-extinction of the optical, we find an unnatural UV flux rise\n(similarly, using the optical value makes an unnatural cutoff in the low energy X-ray spectrum).\nThus we can only accommodate the X-ray fit value if the optical\/UV flux has an extra\nblue, narrow-band component. This seems unnatural. Alternatively the dust map extinction might\nbe correct and the X-ray component may be spatially separated from the optical emission,\nexperiencing extra local (host) absorption.\nMeasuring the J0630 VLT absorption line strengths indicated that the intervening\/host\ngalaxy supplies negligible extinction $E(B-V)<0.01$ to the optical component, which is\nconsistent with the low effective $E(B-V)$.\nAcknowledging this inconsistency, we use the two values in Table~\\ref{ta:ta2}\nwhen constructing the SED.\n\nFor the {\\it Fermi} SED, we performed binned likelihood analysis\nusing the same configuration as described in Section~\\ref{sec:sec2_2} with the 6.5-yr data.\nIn doing so, we fit spectra for all bright sources (detected with $\\gapp5\\sigma$) in the ROI\nand the background amplitudes. Spectral parameters for faint sources or those outside the ROI\nare held fixed at the 3FGL values.\nThe results are shown in Table~\\ref{ta:ta2}.\nThe highest-energy bands in which a significant detection (TS$>15$)\nwas made are 29--75\\,GeV, 75--194\\,GeV, and 75--194\\,GeV for J0022,\nJ0630 and J0811, respectively (see Figure~\\ref{fig:fig3}).\nWe then derive the SEDs using the best-fit power-law model, and show the inferred spectrum in\nFigure~\\ref{fig:fig3}, where the TS is greater than 15\nfor each data point. We performed the analysis using different ROI sizes,\nfinding consistent results.\nIn Figure~\\ref{fig:fig3} we show the results obtained for the 5$^\\circ$ extraction\nas it gives the highest TS value.\n\nWe show the broadband SEDs in Figure~\\ref{fig:fig3}.\nA non-contemporaneous broadband SED for J0630 with sparser X-ray and gamma-ray data\nhas been previously reported \\citep[][]{gtf+12, pgr12}; the results are broadly similar to\nour measurements.\n\n\\subsection{SED modeling}\n\\label{sec:sec3_3}\n\n\\newcommand{\\tablenotemark{a}}{\\tablenotemark{a}}\n\\newcommand{\\tablenotemark{b}}{\\tablenotemark{b}}\n\\begin{table*}[]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Best-fit parameters for the SSC model of B97 with single power-law injection\n\\label{ta:ta3}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{ccccc} \\hline\\hline\nParameter & Symbol & 3FGL~J0022.1$-$1855 & 3FGL~J0630.9$-$2406 & 3FGL~J0811.2$-$7529 \\\\ \\hline\nRedshift & $z$ & 0.774 & $>1.239$ & 0.689 \\\\\nDoppler factor & $\\delta_{\\rm D}$ & 19 & 71 & 33 \\\\\nBulk Lorentz factor & $\\Gamma$ & $>9.6$ & $>35.3$ & $>16.5$ \\\\\nViewing angle (deg.) & $\\theta_v$ & $<3.0$ & $<0.81$ & $<1.74$ \\\\\nMagnetic field (mG) & $B$ & 60 & 1016 & 7 \\\\\nComoving radius of blob (cm) & $R'_b$ & $1.12\\times10^{14}$ & $1.78\\times10^{13}$ & $1.52\\times10^{14}$ \\\\ \nEffective radius of the blob ($cm$) & $R'_{\\rm E}=(3 R'^{2}_b t_{\\rm evol} c\/4)^{1\/3}$ & $1.4\\times10^{15}$ & $1.9\\times10^{14}$ & $1.7\\times10^{15}$ \\\\ \\hline\nInitial electron spectral index & $p_{\\rm 1}$ & 3.14 & 4.26 & 3.19 \\\\ \nInitial minimum electron Lorentz factor & $\\gamma'_{\\rm min}$ & $2.88\\times 10^{4}$ & $1.41\\times 10^4$ & $1.18\\times 10^{4}$ \\\\\nInitial maximum electron Lorentz factor & $\\gamma'_{\\rm max}$ & $1.5\\times10^{6}$ & $2.7\\times10^7$ & $3\\times10^{7}$ \\\\\nInjected particle luminosity (erg s$^{-1}$)\\tablenotemark{a} & $L_{\\rm inj}$ & $9\\times10^{42}$ & $7\\times10^{41}$ & $8\\times10^{42}$ \\\\ \n$\\chi^2$\/dof & $\\cdots$ & 151.1\/122 & 186\/140 & 128.5\/94 \\\\ \\hline\nSynchrotron peak frequency (Hz)\\tablenotemark{b} & $\\nu^{\\rm sy}_{\\rm pk}$ & $5.6\\times10^{14}$ & $1.5\\times10^{15}$ & $5.8\\times10^{14}$ \\\\\nSynchrotron peak luminosity($\\rm erg\\ s^{-1}$)\\tablenotemark{b} & $L^{\\rm sy}_{\\rm pk}$ & $4.6\\times10^{45}$ & $6.7\\times10^{46}$ & $5.1\\times10^{45}$ \\\\\nCompton dominance & CD & 1.2 & 1.4 & 2.1 \\\\ \\hline \\hline\n\\end{tabular}}\n\\end{center}\n\\hspace{-2.0 mm}\n$^{\\rm a}${Energy injected into the jet in the jet rest frame \\citep[see][]{bc02}.}\\\\\n$^{\\rm b}${Quantities in the observer frame.\nThe luminosity quoted is that inferred assuming isotropic emission.}\\\\\n\\end{table*}\n\n We use the one-zone synchro-Compton model of \\citet[][hereafter B97]{bms97} to model\nthe SEDs of the sources. The code evolves a spherical blob of electron\/positron\nplasma with a power-law injected energy distribution,\nfollowing the e$^+$\/e$^-$ population over $10^7$\\,s ($t_{\\rm evol}$)\nassuming that the particle energy loss is dominated by radiative cooling\nas the blob zone flows along a jet axis.\nAs blobs are continuously injected, the emission zone forms\na cylindrical shape (i.e., jet)\nelongated along the jet axis ($l=ct_{\\rm evol}=3\\times10^{17}$\\,cm)\nand the time-integrated spectrum determines the jet emission.\nThe effect of pair-absorption is calculated and included in the model.\nThe full model has 16 parameters including those for disk and BLR emission;\nto simplify we start with standard BL Lac assumption\nthat self-Compton emission dominates so that the seed photons from BLR\nand disk are negligible. The seven remaining parameters we adjust are\nthe low-energy and high-energy cutoffs ($\\gamma'_{\\rm min,max}$)\nand spectral index of the power-law electron distribution ($p_{\\rm 1}$),\nthe magnetic field strength ($B$), the bulk Lorentz factor of the jet ($\\Gamma$)\n(this is done for a fixed viewing angle $\\theta_{\\rm v}$,\nhence equivalent to adjusting the Doppler factor $\\delta_{\\rm D}$)\nand the blob rest frame size ($R'_b$) and electron density ($n_{e}$),\nwhich serve to normalize the total flux.\nThis model has also been used for modeling SED of other blazars \\citep[e.g.,][]{hba+01,r06}.\n\n We use the following steps to find best-fit SED parameters:\n(1) adjust the parameters to visually match the SED for initial values,\n(2) vary each individual parameter over a range\n(a factor of $\\sim$2 initially and decreased with iterations) with\nten grid points while holding the other parameters fixed,\n(3) find the parameter value that provides the minimum $\\chi^2$,\n(4) update the parameter found in step (3) with the best-fit value,\n(5) repeat (2)--(4) until the fit does not improve any more.\nBecause the X-ray spectra are so well described by a simple power law,\nwe initially identify their spectra with synchrotron emission of a cooled electron population,\nstrongly constraining the fit parameter set.\nWe do not include the highest energies ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 40$\\,GeV) LAT points in the initial\nfits, as we will use them later for EBL constraints as done by \\citet{dfp+13}. We update\nonly one parameter each iteration although we vary all seven parameters.\nWe present the best-fit parameters in Table~\\ref{ta:ta3}.\nWe also measured $\\nu^{\\rm sy}_{\\rm pk}$, $L^{\\rm sy}_{\\rm pk}$ and CD\nusing the best-fit SED model, and present them in Table~\\ref{ta:ta3}.\n\n In the model $\\Gamma$ and $\\theta_{\\rm v}$\nappear only in combination through the Doppler factor $\\delta_{\\rm D} = [\\Gamma(1-\\beta\\mu)]^{-1}$,\nwhere $\\beta=\\sqrt{1-1\/\\Gamma^2}$ and $\\mu=\\mathrm{cos}(\\theta_{\\rm v})$. Hence, the model determines\nonly $\\delta_{\\rm D}$ unless one has external constraints on one of $\\Gamma$ or $\\theta_{\\rm v}$.\nTherefore, for a given $\\delta_{\\rm D}$, only lower and upper limit for $\\Gamma$ and $\\theta_{\\rm v}$ can\nbe inferred, also given in Table~\\ref{ta:ta3}.\n\n While the procedure above converges well to a local minimum, there is always a risk\nthat quite distinct solutions could provide better fits. The high dimensionality of the fit space,\nplus the incomplete SED coverage makes it difficult to locate such minima. To aide our exploration of\nparameter space, we used the initial scans to define the covariance between the various\nquantities. We find that simple power-law co-dependencies capture most of the covariance\ntrend around the fit minimum. We fit an amplitude and slope for each parameter pair. Thus,\nby varying one\ncontrol parameter, say $B$, and then setting the others to the covariance-predicted values, we can\ntake larger steps without wandering too far from the $\\chi^2$ minimum surface. For each\nsuch trial solution, we then compute small test grids to rapidly converge to the local\nminimum (with the control parameter held fixed). In this way we explored the minima\nconnected to the `best fit' solution tabulated above. This gave us larger ranges for\n`acceptable' (i.e. null hypothesis probability $p>0.01$) solutions. For example for J0630\nacceptable solutions were found for $0.3$\\,G$40$\\,GeV here.\nNot unexpectedly, EBL absorption provides no significant improvement to the fits\nof the lower redshift sources J0022 and J0811. However, we see clear\nimprovements ($\\Delta\\chi^2\\sim10$ corresponding to $\\sim5\\sigma$)\nfor J0630. Only the high UV model provides no improvement.\nThe $\\chi^2$ decrease is similar for the more conventional models.\n\n\\begin{table}\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Best-fit $\\chi^2$ values for the EBL models tested in this work\n\\label{ta:ta4}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{cccccccc} \\hline\\hline\nModel & J0022 & J0630 & J0811 & reference \\\\ \\hline\nNo EBL & 151.1 & 197.4 & 128.9 & $\\cdots$ \\\\\nDom{\\'{\\i}}nguez & 151.1 & 186.2 & 129.6 & [1] \\\\\nFranceshini & 151.1 & 186.2 & 129.6 & [2] \\\\\nGilmore Fiducial & 151.0 & 189.2 & 130.0 & [3] \\\\\nGilmore Fixed & 151.1 & 186.5 & 129.6 & [3] \\\\\nHelgason & 151.1 & 186.3 & 129.5 & [4] \\\\\nKneiske04 best fit & 151.1 & 191.4 & 130.6 & [5] \\\\\nKneiske \\& Dole & 151.1 & 187.4 & 129.8 & [6] \\\\\nKneiske high UV & 150.3 & 205.1 & 132.8 & [5] \\\\\nStecker high opac. & 151.0 & 194.0 & 131.6 & [7] \\\\\nStecker low opac. & 151.0 & 187.4 & 130.2 & [7] \\\\\nFinke `C' & 151.1 & 187.0 & 129.7 & [8] \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\footnotesize{\nReferences: [1] \\citet{dpr+11}\n[2] \\citet{frv08}\n[3] \\citet{gsp+12}\n[4] \\citet{hk12}\n[5] \\citet{kbm+04}\n[6] \\citet{kd10}\n[7] \\citet{sms12}\n[8] \\citet{frd10}\n}\\\\\n\\end{table}\n\n Since the redshift measurement for J0630 is only a lower limit,\nwe attempted to fit $z$ in the EBL model fits.\nAllowing one more free parameter (holding the other parameters fixed)\nimproves the fit in general but the improvement\nis small except for the case of the disfavored models.\nFor all models the best-fit $z$ is less than the spectroscopic lower limit, although\nthis is within errors for the best-fit models. Accordingly, we hold $z$ fixed at 1.239.\n\n Although the LAT observations continue, unless there is a strong flare,\nwe are unlikely to greatly improve the J0630 EBL constraints without going to higher\nenergy. This will be challenging with present and future generation air Cerenkov telescopes;\nwe predict an absorbed 200\\,GeV energy flux\nof $\\nu F_{\\nu} \\sim4\\times10^{-14}\\rm \\ erg\\ cm^{-2}\\ s^{-1}$ which is\nan order of magnitude lower than the 5-$\\sigma$ sensitivity of the Cherenkov\nTelescope Array\\footnote{https:\/\/portal.cta-observatory.org\/Pages\/Home.aspx}.\nFurther LAT study of other high-redshift BL Lacs\ncan certainly probe the EBL evolution at $z>1.5$.\n\n\\subsection{Alternative Fits}\n\\label{sec:sec3_5}\n\n\\begin{figure}\n\\centering\n\\hspace{-5.0 mm}\n\\includegraphics[width=3.5 in]{fig4.eps} \\\\\n\\figcaption{An SED model fit with (black dotted line) and without (red dotted line)\nthe EBL absorption model (Finke `C' in Table~\\ref{ta:ta4}) for the J0630 data\nwith a hard injection spectrum. The parameters\nfor this model are:\n$\\delta_{\\rm D}=73$,\n$\\theta_{\\rm v}=0.74^\\circ$, $B=10$\\,mG,\n$R'_{\\rm b}=2\\times10^{14}\\rm cm$, $p_{\\rm 1}=2.35$,\n$\\gamma_{\\rm min}=5\\times 10^{3}$,\nand $\\gamma_{\\rm max}=2\\times10^5$.\n\\label{fig:fig4}\n\\hspace{-10mm}\n}\n\\end{figure}\n\nThe best-fit parameters for our BL Lacs are unusual with steep $p_{\\rm 1}>3$ injection spectra.\nJ0630 is the most extreme, with $p_{\\rm 1}\\approx 4.3$ and a strong $\\sim 1$\\,G magnetic field.\nThe excellent power-law fits to the {\\it XMM-NuSTAR} X-ray data drive these values.\nWe have attempted to fit J0630 with more conventional $24\\times10^{4}$, $\\gamma_{\\rm max}'=5\\times10^{6}$, $p_{\\rm 1}=4.1$\nand a small electron density $\\sim 10^{-1}\\rm cm^{-3}$ in order not to\noverproduce the optical and the Compton emission.\n\n We are focused on the LAT band fit, so it is interesting to see that\nthis model has a very similar cutoff to that of Figure~\\ref{fig:fig3}b, requiring a similar EBL absorption.\nThe $\\chi^2$ values (18 data points ignoring the X-ray data) are 62 and 86 with and without the\nEBL absorption, respectively.\nEvidently inverse Compton emission from the X-ray component, if any, is in the highly absorbed\nTeV band. We can speculate that the soft X-ray component rises in\na different zone of the jet \\citep[e.g.,][]{m14},\narguably with large $B$ and a steep, highly cooled spectrum. Whether this connects to the apparently\ndifferent absorption for this component is unclear.\n\n\\begin{table*}[t]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Best-fit parameters of the FDB08 model\n\\label{ta:ta6}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{ccccc} \\hline\\hline\nParameter & Symbol & J0022.1$-$1855 & 3FGL~J0630.9$-$2406 & 3FGL~J0811.2$-$7529 \\\\ \\hline\nRedshift & $z$ & 0.774 & $>$1.239 & 0.689 \\\\\nDoppler factor & $\\delta_{\\rm D}$ & 29 & 110 & 49 \\\\\nMagnetic field (mG) & $B$ & 37 & 4.7 & 7.9 \\\\\nVariability timescale (s) & $t_v$ & $10^5$ & $10^5$ & $10^5$ \\\\\nComoving radius of blob (cm) & $R'_b$ & $4.9\\times10^{16}$ & $1.5\\times10^{17}$ & $8.7\\times10^{16}$ \\\\ \\hline\nLower-energy electron spectral index & $p_{\\rm 1}$ & 2.5 & 2.4 & 2.6 \\\\\nHigh-energy electron spectral index & $p_{\\rm 2}$ & 4.0 & 4.5 & 4.0 \\\\\nMinimum electron Lorentz factor & $\\gamma'_{\\rm min}$ & $6\\times10^{3}$ & $10^3$ & $3\\times10^{3}$ \\\\\nBreak electron Lorentz factor & $\\gamma'_{brk}$ & $3.9\\times10^{4}$ & $6.9\\times10^4$ & $4.9\\times10^{4}$ \\\\\nMaximum electron Lorentz factor & $\\gamma'_{\\rm max}$ & $3.0\\times10^{6}$ & $3.0\\times10^6$ & $6\\times10^{6}$ \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\hspace{-2.0 mm}\n\\vspace{2.0 mm}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n\\vspace{-80.0 mm}\n\\hspace{-12.0 mm}\n\\includegraphics[width=5.7 in,angle=90]{fig5.eps} \\\\\n\\vspace{-4.0 mm}\n\\figcaption{SED models with the disk component for J0630.\n({\\it a}): A model with the disk component added to the baseline synchrotron+SSC model\nin Figure~\\ref{fig:fig3}b.\n({\\it b}): Similar to (a), but baseline model is that in Figure~\\ref{fig:fig4}.\n({\\it c}): Same as (b) with larger $B$ and lower $\\gamma_{\\rm max}$.\nThe model parameters are further adjusted from the baseline ones to match the SED.\nThe EBL model we used for the plot is the ``Finke C'' model in Table~\\ref{ta:ta4}.\nSee text for more details.\n\\label{fig:fig5}\n}\n\\end{figure*}\n\nIf we allow an additional X-ray emitting component, we might also consider a more\ncomplex injection model \\citep[][hereafter FDB08]{fdb08}.\nWe try an electron distribution that is a broken power law or a log parabola.\nTo compare parameters, we fit to this model by first choosing\na variability timescale and then adjusting the other parameters ($\\delta_{\\rm D}$, $B$, \nand the electron distribution) until a good fit was obtained.\nWe assumed $t_v=10^{5}$\\,s which is consistent with the timescale for the optical flux variability\nin J0630 ($t_v\\lapp3$\\,days). The broken power-law model is always more satisfactory than\nthe log-parabola version and we show the best-fit parameters for our three BL Lacs in Table~\\ref{ta:ta6}.\nIt is interesting to compare to our cooling model fits. In particular, the power law breaks strongly\nto large $p_{\\rm 2}$ values. This is imposed by fiat here, but the drive to such large break\nis difficult to accommodate in self-consistent cooling and can require large\nmagnetic field strengths (Table~\\ref{ta:ta3}).\nWe conclude that if conventional $p_{\\rm 1}\\sim2-3$ electron injection spectra\nare adopted, we will always require an additional steep component not easily achieved by\nradiative cooling.\n\n We have noted that the $>$GeV LAT spectrum is not affected by this extra electron component\n(and thus our EBL conclusions for J0630 are robust). However this is in the context of SSC models.\n\\citet{gtf+12} and \\citet{pgr12} noted that HSP BL Lacs\ncan also have low level disk\/BLR emission, overwhelmed by\n(and invisible behind) the jet synchrotron component along the Earth line-of-sight, yet\nproviding substantial seed photons for Compton up-scatter. These may have significant impact on\nthe high-energy hump of the SED \\citep[blue FSRQ model;][]{gtf+12, pgr12}.\nThus, we explore B97 model for J0630\nwith a disk component (orders of magnitude fainter than the baseline synchrotron emission)\nwhich can produce additional Compton emission at $\\sim 10^{24}-10^{26}$\\,Hz (Figure~\\ref{fig:fig5}).\nWe assume a small BL covering fraction given the strong limits\non broad line equivalent widths \\citep[][]{src+13}.\n\n In Figure~\\ref{fig:fig5}a, we add disk EC emission to the model of\nFigure~\\ref{fig:fig3} with a soft ($p_{\\rm 1}=4.26$) injection spectrum. The strong constraint\nof the X-ray data preclude any large change in the SSC component. We find that the additional\nEC emission contributes primarily at high LAT energies. The net effect is to under-produce\nthe low energy gamma-rays leading to an excessively hard LAT spectrum, while not significantly\nchanging the high-energy spectral shape. Thus the EC is not statistically demanded by this model,\nbut even if EC is added, significant EBL absorption should be present;\nimprovement of the fit when the EBL models in Table~\\ref{ta:ta4} are included is typically\n$\\Delta \\chi^2\\sim20$.\n\nAddition of the disk\/EC component to the model in Figure~\\ref{fig:fig4}\n(hard injection spectrum) provides more flexibility since we do not need to match the\nX-ray spectrum, having assumed above that the X-ray emission in this model\nis from a different region than the peak jet emission.\nIn this case, the shape of the SSC component can be adjusted to match the low-energy\ngamma-ray data and the EC emission accounts for the higher energy data (Figure~\\ref{fig:fig5}b);\nthis model reproduces\nthe optical\/UV and gamma-ray data better than the baseline model (Figure~\\ref{fig:fig4})\ndoes. Nevertheless, the effect of EBL absorption is clearly visible in Figure~\\ref{fig:fig5}b,\nand including the EBL models improves the fit by $\\Delta\\chi^2\\sim40$.\n\nIt may be imagined that the sharp drop above $10^{25}$\\,Hz in the unabsorbed model\n(dashed magenta line in Figure~\\ref{fig:fig5}b) may be able to reproduce\nthe sharp drop in the SED without a visible effect of the EBL absorption\nif the peak frequency of the EC component can be lowered.\nThis can be done by lowering $\\gamma_{\\rm max}'$, but merely adjusting\n$\\gamma_{\\rm max}'$ will damage the goodness of fit in the optical-UV band.\nHowever, by adjusting $B$, $\\gamma'_{\\rm max}$, and $\\Gamma$ ($\\delta_{\\rm D}$),\nlowering only $\\nu^{\\rm IC}_{\\rm pk}$ without affecting\n$\\nu^{\\rm sy}_{\\rm pk}$ is possible since the latter is $\\propto \\Gamma B\\gamma_{\\rm max}'^2$\nwhile the former is $\\propto \\Gamma^2 \\gamma_{\\rm max}'^2$.\nWe first adjust $B$ (decrease) and $\\gamma_{\\rm max}'$ (increase), and find that\n$\\nu^{\\rm sy}_{\\rm pk}$ is also lowered in this case owing to stronger cooling caused\nby the stronger magnetic field strength. So we lower $\\Gamma$, and adjusted $B$ and $\\gamma_{\\rm max}'$.\nIn this way, we were able to match the steep fall in the SED at $\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 10^{25}$\\,GHz\nwithout invoking EBL absorption (Figure~\\ref{fig:fig5}c). For this model, we use\n$B=15$\\,mG, $\\gamma_{\\rm max}'=8\\times 10^{4}$ and $\\delta_{\\rm D}=27$\n(corresponding to $\\Gamma>14$ and $\\theta_v<2.1^\\circ$).\nIn this case, as we intended, the fit\nis better when the EBL absorption is not considered; the EBL effect makes\nthe model underpredict the data, and including the EBL models increases\n$\\chi^2$ by $\\sim3$ typically.\nNote that for models in Figures~\\ref{fig:fig5}b and c,\nwe assumed that there is a sharp high-energy cutoff in the synchrotron emission.\nHowever, if such a sharp cutoff does not exist,\nthe high-frequency SSC\/EC component should be enhanced, perhaps similar to that\nin figure~\\ref{fig:fig5}a, requiring the EBL absorption.\n\nNote that we can also add BLR-reflected disk photons to this\nmodel \\citep[see][for example]{r06}. The EC emission of the reflected photons\nonly appears at higher frequencies than the direct disk component and thus\nsuffers from severe EBL absorption. Therefore, we do not consider this component here.\n\n\\section{Discussion and Conclusions}\n\\label{sec:sec4}\nWe constructed broadband SEDs for three high-redshift BL Lac objects, J0022,\nJ0630, and J0811, using nearly contemporaneous observations in the\noptical to X-ray band. Studying the LAT data, we conclude that the variability\non day to year timescales is fairly low for these three systems. This allows us to use the\n6-year (mission averaged) LAT spectrum in forming our SED.\nWe fit the SEDs with a synchrotron\/Compton model to infer physical properties of the sources.\n\nInterestingly, Figure~\\ref{fig:fig3} shows that there is a\ntrend for high-flux optical states to be spectrally harder. Similar trends have\nbeen seen in other blazars \\citep[e.g.,][]{zlz+12}.\nOur contemporaneous data (and SED modeling) are for the low, relatively quiescent state. We\nlack the broad-band high state coverage to study the physical properties imposing this variation\nvia separate SED fits.\nStill, if the variation (increase in $L^{\\rm sy}_{\\rm pk}$ and $\\nu^{\\rm sy}_{\\rm pk}$)\nwere produced by an increase in the external photon field, one expects $\\nu^{\\rm sy}_{\\rm pk}$\nto decrease as the jet particles should cool more efficiently. This is not observed and so\nwe infer that the variation is likely produced in the injection particle spectrum or in the\njet blob flow (e.g., increase in $\\delta_{\\rm D}$) and $B$ field. This suggests correlated optical\nGeV variability, which may be too weak for the LAT to detect.\n\n The basic B97 modeling constrains the emission parameters well under\nthe assumptions of pure SSC emission and radiative cooling of the injected\nelectrons (Figure~\\ref{fig:fig3}). The SED fits assuming only the assigned statistical\nerrors is adequate (probabilities $pr = 10^{-2}$--$10^{-3}$)\nHowever there are almost certainly additional\nsystematic errors including extinction uncertainty and inter-instrument calibrations.\nFor example, increasing the measurement uncertainties by 5\\% (all the SED data points)\nmakes the fit acceptable, with $pr\\sim$10\\%.\n\n The SED parameters are, however, somewhat unusual, giving\nparticularly soft injection spectra, with $p_{\\rm 1}$ well above that expected for relativistic\nshock acceleration, $p_{\\rm 1}\\sim 2-2.5$. For J0022 and J0811,\nhigher $p_{\\rm 1}$ are required because of\nthe flat SED ($\\alpha=0$ in $\\nu F_{\\rm \\nu}\\propto \\nu^{\\alpha}$)\nin the optical band, which requires $p_{\\rm 1}\\sim3$. If we identify this with the cooled\nspectrum, allowing harder injection, then we cannot accommodate the steeper X-ray spectrum\nsince radiative cooling produces only a $\\Delta \\alpha=0.5$ break\n(if the electrons were in the Klein-Nishina regime the break would be even weaker). Similarly,\nmatching the J0630 optical spectrum ($\\alpha \\sim 0.2$) and X-ray spectrum ($\\alpha \\sim -1$)\nis not possible if we let the electrons cool with the break between the optical and\nthe X-ray bands (Figure~\\ref{fig:fig4}). Thus we are forced to very steep injection spectra\nif the X-rays are produced by the same population as the optical emission.\nThis conclusion is supported by fitting with more complex heuristic electron spectra (FDB08 model).\nWith such models we can avoid the very high magnetic field strength\nrequired for J0630 to implement the rapid X-ray cooling and use lower\n10\\,mG fields.\n\n The minimum electron energies for the sources are rather high. While these values\nare not unusual when compared to those in other works \\citep[e.g.,][]{tgg+10},\nit is not clear what environments\/conditions are required in the acceleration site\nto achieve such high minimum electron energies and\nfurther investigations are needed to tell whether or not\nsuch values are realistic. Note that we do not\nuse the equipartition magnetic-field strength in our modeling,\nand the particle energy is much larger than the\nmagnetic energy in our models. In particular, the inferred magnetic field strength for J0811\nis very low compared to those for previously studied BL~Lacs\n\\citep[see][for example]{fdb08, tgg+10, zlz+12},\nalthough there are several objects in the literature with lower inferred $B$\n(and lower magnetic-to-particle-energy ratio). As we already noted (Section~\\ref{sec:sec3_3}),\nit may be possible to find another solution with lower $\\gamma_{\\rm min}$ and higher $B$.\nCovering the SED more completely will help to infer the parameters\nmore precisely. Nevertheless, the SED at the high-energy end is primarily\ndetermined by the X-ray spectrum in our model, and thus our conclusion\non the EBL would not change.\n\n By excluding the X-rays from the SED fit we can indeed accommodate lower injection $p_{\\rm 1}$,\nbut the cost is that the X-ray must be an independent, steep spectrum component. Heuristic\nmodeling with inferred stationary e$^+$e$^-$ spectra confirm that a very steep population is\nneeded to model the X-ray component. Thus a simple, single-zone SSC model with typical\nparticle acceleration spectra is inadequate. The additional ingredient may be a separate, steep\ncooled jet population for the X-ray emission. There is some indication for separate X-ray\/optical\ncomponents seen in the different absorption columns inferred from the two bands for J0630.\nHowever other effects (e.g. adiabatic expansion cooling) may also be relevant.\n\n We find that the $\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi$100\\,GeV LAT points for our highest redshift source J0630 are generally\nsignificantly over-predicted by our SED models and take this to be strong evidence of the\neffect of EBL absorption. Standard EBL models do a good job of producing the observed spectral\ncutoff, but high UV models are not satisfactory\n\\citep[see also][]{fermiEBL, HESSEBL}.\nThis conclusion is fairly robust, and\nEBL absorption is still required if we allow the observed X-ray emission to be a separate jet\ncomponent. Introduction of EC components from faint (unobserved) disk emission\naffects the shape of the LAT spectrum. In general the harder EC spectrum does not match the LAT data\nand it is difficult to arrange components to mimic the high-energy cutoff; EBL\nabsorption is still preferred unless the synchrotron cutoff is extraordinarily\nsharp. We can approximate this with\nan abrupt cut-off in the electron energy distribution (Figure~\\ref{fig:fig5}c), but\nsuch a sharp feature is unlikely to be realized in physical\nacceleration models.\nNote that the effects of EBL absorption are not clearly visible\nin the low redshift sources as expected in EBL models; optical depth at 50\\,GeV for $z=0.7$\nis only 0.08 estimated with the Dom{\\'{\\i}}nguez model in Table~\\ref{ta:ta4}.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{cc}\n\\hspace{-0.0 mm}\n\\includegraphics[width=3.2 in]{fig6a.eps} &\n\\hspace{4 mm}\n\\includegraphics[width=3.1 in]{fig6b.eps} \\\\\n\\end{tabular}\n\\figcaption{{\\it Left}: Synchrotron peak luminosity vs. synchrotron peak frequency.\n{\\it Right}: Compton dominance vs. synchrotron peak frequency.\nWe use black filled circles for FSRQs, red empty circles for BL Lacs, and green\nsquares for sources which are not clearly classified.\nThe three BL Lacs we study are shown as purple diamonds. Note that the red circle\nfor J0630 shows the position of the source reported in a previous study \\citep[][]{f13}.\n\\label{fig:fig6}\n}\n\\vspace{0mm}\n\\end{figure*}\n\nWe conclude with a few comments about the place of our sources in the BL Lac population.\nOur objects are luminous with high $\\nu_{\\rm pk}^{\\rm sy}$ so it is natural to consider\ntheir relation to the `blazar sequence'. In Figure~\\ref{fig:fig6}, we plot $L^{\\rm sy}_{\\rm pk}$\nand CD \\citep[][]{f13} vs. $\\nu_{\\rm pk}^{\\rm sy}$ (in the source rest frame) for\nblazars from the 3LAC sample, including our three sources.\nThe general trend is commonly attributed to the effect of an increased external photon field\n(e.g., from the BLR or disk) for blazars with lower $\\nu^{\\rm sy}_{\\rm pk}$ and magnetic field strength\n\\citep[e.g.,][]{gcf+98, f13}.\nOur three sources are HSPs\/ISPs,\nbut are relatively close to the ISP border. They show higher $L_{\\rm pk}^{\\rm sy}$\nand higher CD than the general population, but only J0630 is a true outlier, in the\n$L_{\\rm pk}^{\\rm sy}$ plot. In fact with the quiescent state SED assembled here, it is\nsomewhat less extreme than in previous studies. Still, as one of the four high-redshift BL Lacs\ncalled out by \\citet[][]{pgr12} it does present some challenges to the simple blazar sequence.\nA more complete study of the high-redshift LAT BL Lacs is needed to see if such sources are\na robust population and thus conflict with the blazar sequence correlation. If so, sources\nsuch as J0630 may be FSRQs viewed very close to the jet axis ($\\theta_{\\rm v}<0.81$\\,deg;\nTable~\\ref{ta:ta3}) so that the disk\/BLR emission is overwhelmed by the beamed jet emission.\nA detailed study along the lines of the blue FSRQ model \\citep[][]{gtf+12}\nusing our high-quality contemporaneous SEDs\nwould be quite interesting.\n\n Since $L_{\\rm pk}^{\\rm sy}$ is redshift-dependent, it is more subject to\nselection effects in a survey study. Thus it is argued \\citep[e.g.,][]{f13} that CD is a more\nrobust classifier of the blazar status, being redshift independent (although still\nsensitive to viewing angle effects, if EC components contribute). In\nFigure~\\ref{fig:fig6} right \\citep[see][for more details]{f13}, we see that our three sources lie\nnear the upper edge of the HSP population. These are highly Compton-dominated sources but\nnot really distinct from the rest of the HSP population. Since our three sources, and the other\nhigh-peak\/high-power BL Lacs, still follow a general correlation in this plot, it suggests that\nthe blazar sequence scenario may still be robust to inclusion of high-power, high-redshift BL Lacs.\n\nNonetheless, the Doppler factors ($\\delta_{\\rm D}$) of these three sources are fairly large.\nFollowing the cosmic evolution, \\citet{arg+14} inferred the distribution of\nthe Lorentz factor ($\\Gamma$) and the viewing angle ($\\theta_{\\rm v}$) for the LAT blazar\npopulation. We note that the distribution for $\\theta_{\\rm v}$\nderived by \\citet{arg+14} (their Figure~9) is broad and the values we inferred with the\nmodels (Tables~\\ref{ta:ta3}) are not exceptional. However, the\nbest-fit Lorentz factors are very high considering the power-law distribution\nwith the slope $k=-2.03\\pm0.70$ for BL Lacs \\citep[][]{arg+14}. In order for\nthe chance probability of having $\\Gamma>35.3$ (for J0630) to be greater than 1\\%,\n$k$ should be greater than $-2.49$. So perhaps our sources do represent a high velocity,\ntightly beamed wing of the BL Lac population and their unusual properties are due to beaming\neffects.\n\n Whether or not BL Lacs at $z>1$ contradict our present picture of the source evolution,\nour SED measurements, particularly that for J0630, show that these sources can be a powerful\nprobe of the EBL and its evolution. We anticipate more striking EBL constraints, pushing to\nthe peak of cosmic star formation via further study of high-redshift {\\it Fermi}-detected BL Lacs.\n\n\\bigskip\n\nThis work was supported under NASA Contract No. NNG08FD60C,\nand made use of data from the {\\it NuSTAR} mission,\na project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory,\nand funded by the National Aeronautics and Space Administration. We thank the {\\it NuSTAR} Operations,\nSoftware and Calibration teams for support with the execution and analysis of these observations.\nThis research has made use of the {\\it NuSTAR} Data Analysis Software (NuSTARDAS) jointly developed by\nthe ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA).\n\nThe \\textit{Fermi} LAT Collaboration acknowledges generous ongoing support\nfrom a number of agencies and institutes that have supported both the\ndevelopment and the operation of the LAT as well as scientific data analysis.\nThese include the National Aeronautics and Space Administration and the\nDepartment of Energy in the United States, the Commissariat \\`a l'Energie Atomique\nand the Centre National de la Recherche Scientifique \/ Institut National de Physique\nNucl\\'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana\nand the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education,\nCulture, Sports, Science and Technology (MEXT), High Energy Accelerator Research\nOrganization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and\nthe K.~A.~Wallenberg Foundation, the Swedish Research Council and the\nSwedish National Space Board in Sweden.\n\nAdditional support for science analysis during the operations phase is gratefully\nacknowledged from the Istituto Nazionale di Astrofisica in Italy and\nthe Centre National d'\\'Etudes Spatiales in France.\n\nH.A. acknowledges supports provided by the NASA sponsored {\\it Fermi}\nContract NAS5-00147 and by\nKavli Institute for Particle Astrophysics and Cosmology (KIPAC).\nPart of the funding for GROND (both hardware as well as personnel)\nwas generously granted from the Leibniz-Prize to Prof. G. Hasinger\n(DFG grant HA 1850\/28-1).\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{intro}\nEffective field theories (EFTs) have become a standard tool in nuclear few-body\nphysics to construct the interactions between the considered degrees of\nfreedom~\\cite{Epelbaum:2008ga,Hammer:2012id}. For example, chiral effective\ntheory is a low-energy expansion of the nucleon-nucleon ($NN$) interaction that\nemploys only nucleons and pions as degrees of freedom and that uses the pion\nmass $m_\\pi$ (or a small momentum) over a large scale $\\Lambda$ that can be\nassociated with the lightest degree of freedom not included in the EFT (e.g.~the\n$\\rho$-meson). This framework is then used to derive the nuclear Hamiltonian in\na systematic low-energy expansion. The resulting potential has been used\nextensively in few-nucleon studies and ab initio nuclear structure calculations.\nIt was pointed out that the most singular piece of the one-pion exchange (OPE)\nin the deuteron channel is an inverse cube\npotential~\\cite{Sprung:1994,PavonValderrama:2005gu}. The renormalization of\nthis leading order (LO) potential has been studied repeatedly in the two- and\nthree-nucleon\nsector~\\cite{PavonValderrama:2004nb,Nogga:2005hy,Birse:2005um,Long:2011xw,Song:2016ale}.\nHere, we study the renormalization of the finite range inverse cube\npotential (FRIC) in the much simpler three-boson system thereby\nremoving the complications due to\nthe spin-dependent tensor force. In particular, we examine whether the\nthree-body system with pairwise inverse cube interactions requires a three-body\ncounterterm for renormalization, and whether residual cutoff corrections can be\nused as a reliable tool to build a power counting scheme as suggested in\nRef.~\\cite{Griesshammer:2015osb}. We note that there is also interest in atomic\nphysics regarding the inverse cube interaction. However, most attention is\nfocused on the low-energy properties in the \\textit{infinite} range\nlimit~\\cite{Mueller:2013,Gao:1999}.\n\nSince the residual cutoff dependence to some extent can be influenced\nby the chosen regularization scheme, we carry out this analysis\nfor various schemes that are currently used by the\ncommunity. Specifically, we consider a \\textit{local}\nregularization scheme~\\cite{Gezerlis:2013ipa} that cuts off the\npotential in coordinate space at a small distance $R$, a non-local\nregularization scheme~\\cite{Epelbaum:2008ga} that cuts off the high momenta\nin the momentum space form of the two-body interaction $V(p,p')$\nseparately, and a semi-local regularization scheme~\\cite{Epelbaum2015}\nthat applies these strategies separately to the long-range inverse\ncube part of the interaction and the short-distance regulator.\n\nThese different regularization schemes have different advantages for\ndifferent methods that are used to diagonalize the nuclear\nHamiltonian. For example, local interactions are commonly used in\nquantum Monte Carlo calculations, though progress has been made\nincluding nonlocal interactions\n(e.g.~\\cite{Roggero:2014kea,Lynn:2012}). However, while these have\nbeen used extensively in the literature, a detailed comparison of\nthese approaches is missing.\n\nWe find that the regularization schemes analyzed can be used to obtain\nregulator-independent results at large cutoffs. We find however that the regulator\ndependence of the short-distance counterterm is different for the\nregulation schemes we apply. In agreement with findings in the\nthree-nucleon sector\\cite{Nogga:2005hy, Song:2016ale}, we find that\nthree-body observables are completely renormalized without the\ninclusion of an additional three-body counterterm. However, an\nanalysis of the cutoff dependence of three-body observables shows also\nthat observables converge more slowly than expected from previous\nstudies of the three-nucleon sector~\\cite{Song:2016ale}.\n\nIn Sec.~\\ref{sec:theory}, we discuss the regularization schemes as well as the\nrenormalization and calculation of observables. In\nSec.~\\ref{sec:results}, we present the results obtained for the two- and\nthree-boson system as well as quantitative analyses of the remaining cutoff\ncorrections. We conclude with a summary and an outlook.\n\n\\section{Theory}\n\\label{sec:theory}\n\nIn the following subsections, we describe the interaction that is used\nin this work, how it is regulated, and how it is renormalized. We\ncomment also briefly on technical details such as the normalization of\nstates and the calculation of observables through the Schr\\\"odinger,\nLippmann-Schwinger, and Faddeev equations.\n\nThe non-regulated and singular potential $V_S$ that we consider is a FRIC\npotential of the form\n\\begin{equation}\n\t\\label{eq:fric_pot}\n V_{\\rm{S}}(r) = -C_3 \\frac{e^{-m_\\pi r}}{r^3}~.\n\\end{equation}\nWe choose $m_\\pi = 138$ MeV and $C_3 = 0.8$ fm$^2$ such that a\ndeuteron-like state ($B_2=2.2$ MeV) exists when we regulate the\npotential at $\\sim 1~\\textrm{fm}$. This potential has to be regulated\nat short distances and observables will depend strongly on the\nregularization scale as the interaction is too singular~\\cite{Frank:1971xx}.\nBelow we display how a (\\textit{smeared out}) short-distance counterterm can be\nintroduced to address this problem.\n\nWe perform our calculations in momentum space, and we Fourier transform the\ninteraction $V$ and carry out a partial-wave projection\n\\begin{equation}\n \\label{eq:ft_pwp}\n \\tilde{V}_{l}(p,k) \\equiv FT\\left[V(r)\\right] =\n \\frac{2}{\\pi}\\int_0^\\infty drr^2 j_l(pr) V(r) j_{l}(kr)~,\n\\end{equation}\nwhere $j_l(z)$ are the spherical Bessel functions of order $l$.\n\n\n\\subsection{Regulator Formulations}\\label{sec:regs}\n\n\\subsubsection{Local Regulation}\\label{sec:local_reg}\n\nFor a local, singular potential, $V_S(r)$, we have\nimplemented three different forms of regulation: local, semi-local, and\nnonlocal. The locally regulated potential has the form\n\\begin{equation}\n \\label{eq:local}\n V(r) = \\rho(r;R)V_S(r) + g(R)\\chi(r;R)~,\n\\end{equation}\nwhere $\\rho(r;R)$ is an arbitrary function that minimally fulfills two requirements.\nFirst, it must overcome $V_S(r)$ in the $r\\rightarrow 0$ limit such that the\nproduct $\\rho(r;R)V_S(r)$ is finite.\nSecond, in the limit of $r\\rightarrow\\infty$, $\\rho(r;R)$ must go to one.\nFor the locally regulated case we use\n\\begin{equation}\n \\label{eq:local_reg}\n \\rho(r;R) = {\\left(1-e^{-{(r\/R)}^2}\\right)}^4~,\n\\end{equation}\nwhere $R$ is the range at which the characteristic behavior of $V_S(r)$ is cut\noff. The counterterm\n\\begin{equation}\n \\label{eq:local_cterm}\n g(R)\\chi(r;R)~,\n\\end{equation}\nhas two components. The first, $g(R)$ is an $R$-dependent coupling\nstrength. We tune this parameter to match some low-energy, two-body\nobservable such as the two-body binding energy. The second,\n$\\chi(r;R)$, is a contact-like interaction or a \\textit{smeared} $\\delta$\nfunction such that\n\\begin{equation}\n\t\\lim_{R\\rightarrow 0}\\chi(r;R) \\sim \\delta(r)~.\n\\end{equation}\nFor the locally regulated case we use\n\\begin{equation}\n \\label{eq:locally_xterm}\n \\chi(r;R) = e^{-{(r\/R)}^3}~.\n\\end{equation}\nWe discuss below that the RG flow of the locally-regulated\ncounterterm strength, $g(R)$, contains multiple\nbranches~\\cite{Beane:2000wh}. To ensure consistency between our\nresults and others', we also implement a semi-local regulation\nscheme.\n\n\\subsubsection{Semi-Local Regulation}\\label{sec:semi_local_reg}\n\nThe difference between local regulation and semi-local regulation\nlies in the definition of the counterterm. In Eq.~\\eqref{eq:local}\nwe defined the counterterm in coordinate space. This counterterm,\nthat regulates the relative distance in the two-body system and\nthereby the momentum exchange, has multiple solutions (provided the\nshort-distance cutoff is small enough) for which the two-body binding\nenergy $B_2$ is reproduced.\n\nIf we instead define the counterterm in momentum space as\n\\begin{equation}\n \\label{eq:momspace_cterm}\n g(R)\\tilde{\\chi}(p;R)\\tilde{\\chi}(k;R)~,\n\\end{equation}\nsuch that, by itself, only permits one state, we obtain a unique RG flow.\nThe full potential in momentum space is then\n\\begin{equation}\n \\label{eq:momspace_int_sl}\n \\tilde{V}(p,k) = FT\\left[\\rho(r;R)V_S(r)\\right] +\n g(R)\\tilde{\\chi}(p;R)\\tilde{\\chi}(k;R)~,\n\\end{equation}\nwhere $FT$ represents the Fourier transform and partial-wave projection\nshown in Eq.~\\eqref{eq:ft_pwp}.\n\nFor the semi-locally regulated case, similar to~\\cite{Epelbaum2015}, we use\n\\begin{equation}\n \\label{eq:semilocal_reg}\n \\rho(r;R) = {\\left[1-e^{-{(r\/R)}^2}\\right]}^4~,\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:mom_reg}\n \\tilde{\\chi}(p;R) = e^{-{(pR\/2)}^2} = e^{-{(p\/\\Lambda)}^2}~,\n\\end{equation}\nwhere $\\Lambda\\equiv 2\/R$. For a brief discussion on the different $\\rho(r;R)$\nfunctions used for the locally and semi-locally regulated cases,\nsee Appendix~\\ref{sec:rho_choice}.\n\n\\subsubsection{Nonlocal Regulation}\\label{sec:nonlocal_reg}\n\nFor the fully nonlocal interaction, we take the semi-local\ninteraction Eq.~\\eqref{eq:momspace_int_sl}, including the forms of $\\rho(r;R)$ and\n$\\tilde{\\chi}(p;R)$, and modify the first term as follows\n\\begin{equation}\n \\label{eq:momspace_int_nl}\n \\tilde{V}(p,k) = \\tilde{\\chi}(p;R) FT\\left[\\rho(r;R_<)V_S(r)\\right]\n\\tilde{\\chi}(k;R) + g(R)\\tilde{\\chi}(p;R)\\tilde{\\chi}(k;R)~.\n\\end{equation}\nThe momentum-space regulators multiplying the first term suppress the diagonal\nmatrix elements where the incoming and outgoing momenta are large but similar,\nremoving some sensitivity to the choice of $\\rho(r;R)$ that we discuss\nin~\\ref{sec:rho_choice}. The short-distance cutoff used before we take the\nFourier transform, $R_<$, is chosen to be much less than $R$. This allows us to\nensure that the resulting cutoff dependence in the observables is attributable\nto the regulator function, $\\tilde{\\chi}(p;R)$, rather than the Fourier\ntransform.\n\n\n\\subsection{Two-Body Bound States}\\label{sec:two_body_bound_states}\nWe calculate two-body binding energies by solving the Schr\\\"odinger equation\n\\begin{equation}\n \\label{eq:shroedinger}\n (\\hat{H}_0 + \\hat{V})\\ket{\\psi} = E\\ket{\\psi}~,\n\\end{equation}\nin coordinate and momentum space. In coordinate space, we tune the counterterm\nsuch that for a desired value $E$, the radial equation\n\\begin{equation}\n \\label{eq:shroedinger_radial}\n -\\frac{1}{m}\\frac{d^2u}{dr^2} + V(r) u(r) = E\\,u(r)~,\n\\end{equation}\nis solved where $u(r)\\equiv rR_0(r)$. We have dismissed the centrifugal term as\nonly s-waves are considered. In momentum space, we\nrearrange Eq.~\\eqref{eq:shroedinger} such that we have\n\\begin{equation}\n \\label{eq:determinant}\n \n \\hat{G}_0(E)\\hat{V}\\ket{\\psi} = \\ket{\\psi}~,\n\\end{equation}\nwhere $G_0(z)\\equiv 1\/(z-\\hat{H}_0)$. After discretization with the basis states\n$|p_i\\rangle$, Eq.~\\eqref{eq:determinant} becomes an eigenvalue problem that is\neasily solved by finding the energies that fulfill\n\\begin{equation}\n \\label{eq:determinant1}\n \\det\\left[\\hat{1}-K_{ij}(E)\\right] = 0~,\n\\end{equation}\nwhere $K_{ij}(E) = \\braket{p_i|\\hat{G}_0(E)\\hat{V}|p_j}$ and we tune the counterterm such\nthat the requirement Eq.~\\eqref{eq:determinant1} is satisfied.\n\n\\subsection{Lippmann-Schwinger Equation}\\label{sec:lse}\nTo obtain two-body phase shifts, we solve numerically the\nLippmann-Schwinger Equation for the two-body $t$-matrix\n\\begin{equation}\n\\label{eq:lse}\n\\hat{t} = \\hat{V} + \\hat{V}\\hat{G}_0\\hat{t}~.\n\\end{equation}\nIn the partial-wave projected momentum basis, considering bosons interacting in\n$s$-waves only, we have\n\\begin{align}\n\\nonumber\n \\braket{p\\,|\\hat{t}|p^\\prime }\n &=\\braket{p\\,|\\hat{V}|p^\\prime } + \\braket{p\\,|\\hat{V}\\,\\hat{G}_0(E+i\\epsilon)\\,t|p^\\prime}~,\\\\\nt(p, p^\\prime; E) & = \\tilde{V}(p,p^\\prime) + \\int_0^\\infty\n dq\\,q^2\\,\\frac{\\tilde{V}(p,q) \\,t(q, p^\\prime; E)} {E+i\\epsilon-q^2\/m}\n\\end{align}\nwhere $m$ is the nucleon mass and $\\epsilon \\to +0$. From the on-shell matrix element $t(p,p;E=p^2\/m)$\nwe extract the phase shift via\n\\begin{equation}\n t(p,p;E=p^2\/m) = -\\frac{2}{m\\pi} \\frac{1}{p\\cot{\\delta}-ip}~.\n\\end{equation}\nThe scattering length is defined by the effective range expansion\n\\begin{equation}\n \\label{eq:ere}\n p\\cot\\delta \\approx = -\\frac{1}{a} + \\frac{r_s}{2}p^2~,\n\\end{equation}\nwhich allows us to calculate it exactly from the on-shell $t$-matrix amplitude\nat $p=0$.\n\\begin{equation}\n a = \\frac{m\\pi}{2} t(0,0;0)~.\n\\end{equation}\n\n\n\\subsection{Three-Body Bound States}\\label{sec:3b_bound_states}\nTo calculate three-body binding energies, we start with the equation for a\nsingle Faddeev component of a system containing three identical particles\n\\begin{equation}\n\t\\label{eq:Faddeev_bound_state}\n \\ket{\\psi} = \\hat{G}_0(E)\\hat{t}\\hat{P}\\ket{\\psi}~,\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\hat{P} = \\hat{P}_{12}\\hat{P}_{23} + \\hat{P}_{13}\\hat{P}_{23}~,\n\\end{equation}\nis the permutation operator with $\\hat{P}_{ij}$ interchanging particles\n$i$ and $j$~\\cite{Gloeckle:99109}.\nAfter projecting onto the partial-wave, momentum basis for three identical\nbosons described by two Jacobi momenta $p$ (the relative momentum between\nparticles 1 and 2) and $q$ (the relative momentum between particle 3 and the\ncenter of mass of the 1--2 subsystem), we discretize the equation and solve for\nthe bound state energy $E$ using the same techniques as in the two-body case,\nas long as $E$ remains below the deepest state in the two-body\nspectrum. However, this limitation is in conflict with our goal of\nstudying the cutoff dependence of two- and three-body observables. As\nwe go to higher momentum-space cutoffs (smaller $R$ values), spurious\nbound states enter the two-body spectrum. Three-body states quickly\nbecome resonances in this regime, bounded above and below by two-body\nbound states. There are two ways that we deal with this.\n\nThe first method follows~\\cite{Nogga:2005hy} and is repeated here. It involves\n\\textit{removing} the spurious two-body state from the spectrum by transforming\nthe potential\n\\begin{equation}\n \\label{eq:state_removal}\n \\hat{V} \\rightarrow \\hat{V} + \\ket{\\phi}\\lambda\\bra{\\phi}~,\n\\end{equation}\nwhich takes the eigenvalue of the state $\\phi$ and modifies it by an amount\n$\\lambda$.\nUsing this transformed potential in the Lippmann-Schwinger equation and taking\nthe limit\nof $\\lambda\\rightarrow\\infty$ (removing the state from the spectrum), we have\n\\begin{equation}\n \\lim_{\\lambda\\rightarrow\\infty} \\hat{t}(\\lambda) =\n \\hat{t} - \\ket{\\eta}\\frac{1}{\\braket{\\phi|\\hat{G}_0|\\eta}}\\bra{\\eta}~,\n\\end{equation}\nas our modified $t$-matrix where\n\\begin{equation}\n\t\\label{eq:eta}\n \\ket{\\eta} = \\ket{\\phi} + \\hat{t}\\hat{G}_0\\ket{\\phi}~.\n\\end{equation}\nThis only requires that we have the wave function $\\braket{p|\\phi}$ to\ncalculate the modified $t$-matrix where that state no longer\ncontributes a pole. \nIn practical calculations using a large, finite $\\lambda$ value in\n(\\ref{eq:state_removal}) is sufficient.\nIf there are several spurious two-body states, the procedure is repeated\nfor each of them.\n\nThe second method we employ to study the cutoff dependence of\nthree-body resonances is to look for the resonances in the three-body\nphase shifts.\n\n\\subsection{Three-Body Phase Shifts}\\label{sec:3b_phase_shifts}\nIn the cutoff regime where spurious two-body bound states exist, we\ncan scatter a third particle off the spurious deep two-body state\nand scan the phase shifts in the energy range between the two-body\nstates for a resonance. To do this, we calculate the three-body\n$T$-matrix using~\\cite{Gloeckle:1995jg}\n\\begin{equation}\n\\label{eq:T}\n \\hat{T} = \\hat{t}\\hat{P} + \\hat{t}\\hat{G}_0\\hat{P}\\hat{T}~,\n\\end{equation}\nwhich relates to the elastic scattering operator $\\hat{U}$ by\n\\begin{equation}\n \\label{eq:U}\n \\hat{U} = \\hat{P}\\hat{G}_0^{-1} + \\hat{P}\\hat{T}~.\n\\end{equation}\nIn the partial-wave-projected, momentum basis, considering bosons interacting\nonly via $s$-waves, we have\n\\begin{equation}\n\t\\begin{split}\n \\braket{pq|\\hat{T}|\\phi} & = \\braket{pq|\\hat{t}\\hat{P}|\\phi} + \\\\\n & \\int_0^\\infty dq^\\prime {(q^\\prime)}^2 \\int_{-1}^1 dx\\,\n \\frac{t(p,\\pi_1,E-3q^2\/4m)\\,G(q,q^\\prime, x)}\n {E+i\\epsilon-q^2\/m-{(q^\\prime)}^2\/m-qq^\\prime x\/m}\n \\braket{\\pi_2q^\\prime|\\hat{T}|\\phi}~,\n\t\\end{split}\n\\end{equation}\nwhere the incoming state $\\ket{\\phi}=\\ket{\\varphi k}$ contains the\nwave function $\\varphi(p)$ of the two-body bound state and the relative\nmomentum $k$ between the third particle and the center of mass of the\ntwo-body subsystem,\n$G(q,q^\\prime,x)$ is a geometrical factor introduced by the\npermutation operator, $\\pi_1 = \\sqrt{q^2\/4+{(q^\\prime)}^2+qq^\\prime x}$, and\n$\\pi_2 = \\sqrt{q^2+{(q^\\prime)}^2\/4+qq^\\prime x}$.\n\nThe elastic scattering amplitude $M$ is related to the $U$ operator by\n\\begin{equation}\n M = -\\frac{2m\\pi}{3}\\braket{\\phi|\\hat{U}|\\phi}~,\n\\end{equation}\nand the phase shift by\n \n \n \n\\begin{equation}\n \\label{eq:three-body-pw-amp}\n M = \\frac{1}{k\\cot\\delta-ik}~.\n\\end{equation}\nIn the three-body sector, we have a similar effective range expansion\n\\begin{equation}\n \\label{eq:ere-atom-dimer}\n k\\cot\\delta \\approx -\\frac{1}{a_{AD}} + \\frac{r_{s,AD}}{2}k^2~,\n\\end{equation}\nwhich defines the atom-dimer scattering length $a_{AD}$ and atom-dimer effective\nrange $r_{s,AD}$.\nWe also study the inelasticity parameter given in terms of the $S$-matrix by\n\\begin{equation}\n \\eta = e^{-2\\delta_i}~,\n \\label{eq:inelasticity}\n\\end{equation}\nwhere the phase shift is complex and the usual decomposition\n\\begin{equation}\n \\delta = \\delta_r + i\\delta_i~,\n\\end{equation}\nis taken.\n\n\\subsection{Quantitative Uncertainty Analysis}\nTo analyze the uncertainties induced by short-distance physics of our\nregularization procedure, we study in this section the regulator\ndependence of observables. Similar to the analysis done by Song\n{\\textit{et al.}}~\\cite{Song:2016ale}, our uncertainty analysis is based on a\nsimple power series expansion of observables quantities $\\mathcal{O}$\nof the form\n\\begin{equation}\n \\label{eq:power_series_uncertainties}\n \\mathcal{O}(\\Lambda) \\approx \\mathcal{O}_\\infty \\left[1+\\sum_i^\\infty\n c_i{\\left(\\frac{q}{\\Lambda}\\right)}^i\\right]~,\n\\end{equation}\nwhere $q$ is associated with the low-momentum scale relevant to the calculation;\nhowever, $i$ is \\textit{not} assumed to be an integer. For the purposes of this\nproject, we truncate the summation over $i$ after the first term $i=n$, leaving\n\\begin{equation}\n \\label{eq:lo_correction}\n \\mathcal{O}(\\Lambda) \\approx \\mathcal{O}_\\infty \\left[1 + c_n\n {\\left(\\frac{q}{\\Lambda}\\right)}^n\\right]~,\n\\end{equation}\nWe seek to establish the value of $n$. In Ref.~\\cite{Song:2016ale},\n$n$ was found by fitting the first few terms in the above expansion\nwith integer $n$ to the cutoff dependence of observables. Here, we\nstudy the cutoff dependence at very large cutoffs, focus on the\ndominant term in the expansion, and fit $n$ itself to data and \nallow for non-integer values.\n\nTo extract the power of the leading cutoff correction, we examine both the\n$\\Lambda$ and the $q$ dependence. The first approach we take to investigate the\n$\\Lambda$ dependence is to calculate observable $\\mathcal{O}$ over a range of\n$\\Lambda$ values, and fit the results to Eq.~\\eqref{eq:lo_correction} for a\nrange of $n$ values. For each $n$, we evaluate a penalty function that we define\nas\n\\begin{equation}\n \\label{eq:penalty_function}\n p_n = \\sum_i {\\left(\\frac{\\mathcal{O}_{calc}(\\Lambda_i) -\n \\mathcal{O}_{fit}(\\Lambda_i)}{\\mathcal{O}_{calc}(\\Lambda_i)}\\right)}^2~,\n\\end{equation}\nwhere $\\mathcal{O}_{calc}(\\Lambda)$ is the observable calculated for a specific\nvalue of $\\Lambda$ and $\\mathcal{O}_{fit}(\\Lambda)$ is the value of the\nobservable as it is ``reproduced'' by Eq.~\\eqref{eq:lo_correction} and the fit\nparameters $\\mathcal{O}_\\infty$ and $c_n$. Once we have $p_n$ for a range of $n$\nvalues, we search for a minimum $p_n$ where $n$ is optimal.\n\nGriesshammer has shown~\\cite{Griesshammer:2015osb} that the $q$ dependence of\nobservables provides a necessary though insufficient window into the order of\ncutoff-dependent corrections. To isolate the $q$ dependence, we have to restrict\nthe observables we study to those whose $q$ dependence is well understood. Doing\nso allows us to calculate the observable at two different cutoffs and study the\nrelative difference\n\\begin{equation}\n \\label{eq:gh_diff}\n 1 - \\frac{\\mathcal{O}(\\Lambda_1)}{\\mathcal{O}(\\Lambda_2)}\n \\approx\n q^n c_n \\left[\\frac{1}{\\Lambda_2^n} - \\frac{1}{\\Lambda_1^n}\\right]~.\n\\end{equation}\nTaking the logarithm, we get\n\\begin{equation}\n \\label{eq:gh_diff_log}\n \\ln\\left[1 - \\frac{\\mathcal{O}(\\Lambda_1)}{\\mathcal{O}(\\Lambda_2)}\\right] =\n n\\ln q + b~,\n\\end{equation}\nwhere $n$ and $b$ are the slope and intercept that we fit, respectively.\n\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Renormalization Group Flow}\\label{sec:rgflow}\nThe first thing we compare between the regulation schemes is the RG\nflow. We choose to fix the shallowest two-body state at\n$B_2 = 2.2$ MeV. Figure~\\ref{fig:rg_flows} shows the stark difference\nbetween the RG flow found using a local counterterm and the RG flows\nfound with nonlocal counterterms. The main difference is the issue of\nuniqueness. For the locally regulated potential, as pointed out\nby~\\cite{Beane:2000wh}, $g(R)$ has multiple solutions that give a two-body bound\nstate at the desired binding energy. There is one branch where there exists one\nstate in the two-body system. Each branch below that branch contains\nsuccessively one additional state. The RG flow shown for the locally regulated\ninteraction connects four of those branches, ``hopping'' downward when it is\neasier to add an additional state than to continue to maintain the shallowness\nof the fixed state. Only two of the ``hops'' are visible in the plot due to the\nscale and the relative difference between the magnitudes of $g$ between the\ndifferent branches. Note also the difference in the units of the upper and\nlowers plots if Fig.~\\ref{fig:rg_flows}. There is a factor of $R^3$ that comes\nfrom the Fourier transform and partial-wave projection of $\\chi(r;R)$.\n\nThe other two functions shown in the lower plot of Fig.~\\ref{fig:rg_flows}\nare qualitatively very\nsimilar. They correspond to the semi-local and nonlocal regulation schemes.\nWhile the same $\\rho(r;R)$ is used in both, the prescription is somewhat\ndifferent as one can see from Eq.~\\eqref{eq:momspace_int_sl}\nand Eq.~\\eqref{eq:momspace_int_nl}. The\nsemi-local regulation scheme brings in spurious bound states faster than the\nnonlocal regulation scheme, but as mentioned before, nonlocal regulation cuts\noff the potential at large incoming and outgoing momenta, suppressing\nhigh-momentum contributions. Still, they are very similar interactions, thus\nthey provide very similar RG flows.\n\n\\begin{figure}\n\t\\includegraphics[width=\\linewidth]{rg_flows.pdf}\n \\caption{RG flows of the counterterm coupling $g$. The yellow circles in the upper\n plot represent $g(R)$ values calculated with a local regulator and local\n counterterm. The red, solid line in the upper plot are the $g(R)$ values\n used to calculate the phase shifts in\n Fig.~\\ref{fig:two_body_phase_shifts}. The blue, dashed line in the lower\n plot corresponds to the semi-locally regulated interaction. The orange,\n dashed line corresponds to the nonlocally regulated interaction.\n }\\label{fig:rg_flows}\n\\end{figure}\n\n\\subsection{Two-Body Scattering}\\label{sec:two-body-scattering}\nAs the different regulation schemes are tuned to reproduce the same\nshallow state at $B_2=2.2$ MeV, we expect that differences in\nlow-energy scattering observables are highly suppressed when large\ncutoffs are employed. We calculate the phase shifts using all three\nregulation schemes and show the results in\nFig.~\\ref{fig:two_body_phase_shifts}. The left plot contains the phase\nshifts of an non-renormalized, nonlocally regulated potential with\n$g(R) = 0$, demonstrating the strong cutoff dependence of low-energy\nobservables and the need for a counterterm. The most important feature\nof the right plot is the agreement between the different regulation\nschemes. \nIt is also worth mentioning the ``turning point'' $\\Lambda$ value at which phase\nshifts clearly begin to flatten out.\nAt low energies, the point is near 2 GeV.\nAs the scattering energy increases that point increases as well.\nImportantly, this behavior agrees with studies of the OPE\npotential~\\cite{Song:2016ale,Nogga:2005hy} where similar convergence behavior is\nfound across a range of partial-wave channels.\nOur $C_3$ value is chosen to mimic the OPE in the bosonic sector such that we\ncan expect similar renormalization behavior.\nObserving this similarity is consistent with the known result that the\none-pion-exchange potential goes like an inverse cube potential at short\ndistances (high cutoffs)~\\cite{Sprung:1994,PavonValderrama:2005gu}.\n\n\\begin{figure}\n\t\\includegraphics[width=\\linewidth]{two_body_phase_shifts.pdf}\n \\caption{[Left] Cutoff dependence of the s-wave phase shifts at $E = 1$ (red,\n dashed), $10$ (green, dotted), and $100$ MeV (blue, dot-dashed) calculated\n via a nonlocally regulated potential without a counterterm. [Right] Cutoff\n dependence of the s-wave phase shifts at (from top to bottom) $E = 1$, $10$,\n and $100$ MeV in the center-of-mass frame. The solid, red lines are the phase\n shifts calculated from a locally regulated potential. The green, dashed\n lines are the phase shifts at the same energies calculated with a\n semi-locally regulated interaction. The blue, dot-dashed lines are the phase\n shifts using a nonlocally regulated interaction. All three schemes include a\n contact-like counterterm.\n }\\label{fig:two_body_phase_shifts}\n\\end{figure}\nIt is clear from Fig.~\\ref{fig:two_body_phase_shifts} that a two-body\ncontact interaction is sufficient to renormalize the two-body phase\nshifts. The corresponding result for the two-body scattering length is\nshown in Fig.~\\ref{fig:two_body_scattering_length}.\n\n\\begin{figure}\n \\includegraphics[width=0.9\\linewidth]{two_body_scattering_length.pdf}\n \\caption{The scattering length is shown as a function of the high-momentum\n (short-distance) cutoff. The blue circles are the numerical results.\n }\\label{fig:two_body_scattering_length}\n\\end{figure}\n\nOne of the advertised, key advantages of EFT is quantifiable\nuncertainty which in turn requires a power counting that orders\ncontributions in the Hamiltonian according to their\nimportance. These uncertainties have usually two sources: (i) the\ntruncation of the low-energy expansion and (ii) uncertainties that are\nintroduced when low-energy counterterms are fitted to data. Here we\nfocus on the first source of uncertainties and some information on\nthis truncation error is contained in the convergence behavior of\nobservables as the short-distance cutoff is increased. To study this\nproblem, we first chose a range of cutoffs over which to fit the\nscattering length to Eq.~\\eqref{eq:lo_correction}. However, as the window of\ncutoffs over which the fit was carried out was narrowed to include only the highest\nvalues of $\\Lambda$, the resulting $n$ was found to be unstable.\nAs a result, we plotted $\\Lambda(da\/d\\Lambda)$, shown in\nFig.~\\ref{fig:a_analysis}.\nThe solid, red line in the left-hand plot of Fig.~\\ref{fig:a_analysis} is the\nexpected $\\Lambda(da\/d\\Lambda)$ dependence based on a fit to\nEq.~\\eqref{eq:lo_correction} with $n=1.5$.\nClearly, there is behavior in the cutoff dependence of the scattering length\nthat is not captured by the simple form assumed in\nEq.~\\eqref{eq:lo_correction}.\n\nEmpirically, we model the residual cutoff dependence by\n\\begin{equation}\n \\label{eq:rgi_correction}\n \\Lambda\\frac{da}{d\\Lambda} \\approx \\frac{1}{\\Lambda^n}\\left[A +\n B\\cos \\left({h\\Lambda^{1\/3}+f}\\right)\\right]~,\n\\end{equation}\nwhere $A, B, h$ and $f$ are treated as fit parameters.\nWe choose a range of $n$ values over which we carry out the fit and evaluate the\nquality of the fit with Eq.~\\eqref{eq:penalty_function} at each value.\nThe right-hand plot of Fig.~\\ref{fig:a_analysis}\nshows $\\Lambda(da\/d\\Lambda)$ in blue circles with $n_{\\min}=1.7$.\nThe red, dashed line in the left-hand plot of Fig.~\\ref{fig:a_analysis}\nrepresents Eq.~\\eqref{eq:rgi_correction} with the fit parameters found when\nusing $n_{min}$.\nThe agreement between the data and the empirical formula is excellent.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{a_analysis.pdf}\n \\caption{[Left] RG analysis of the two-body scattering length as a function of\n the cutoff. Blue circles represent the data. The red, dashed line represents\n a fit to Eq.~\\eqref{eq:lo_correction} with $n=1.5$. [Right] The blue circles\n represent the same data as the left-hand plot. The red, dashed line\n represents a fit to Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}=1.7$.\n }\\label{fig:a_analysis}\n\\end{figure}\n\nWe expect that all low-energy, two-body observables come with similar cutoff\ndependence.\nIn keeping with our study of the cutoff dependence of the scattering length, we\napplied the same analysis to the phase shifts and cross sections.\nIn Fig.~\\ref{fig:delta_sigma_analysis} we plot the results.\nIn both cases, the calculation was performed at a relative, center-of-mass\nmomentum of 106~MeV.\nThe analyses produced minima of the penalty functions\n(Eq.~\\eqref{eq:penalty_function}) near $n_{\\min}=1.7$.\nSimilar analyses performed at different energies produced similar results.\nThe only trend worth mentioning is the slight decrease of $n_{\\min}$ to\napproximately 1.5 as the scattering energy increases.\nOverall, the agreement between the data and Eq.~\\eqref{eq:rgi_correction} found\nfor the scattering length is found for the phase shift and cross section as\nwell.\nThe $n_{\\min}$ values are collected in Table~\\ref{tab:nmins}.\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{delta_sigma_analysis.pdf}\n \\caption{[Left] RG analysis of the phase shift at a center-of-mass momentum of\n 106 MeV as a function of the cube root of the cutoff. Blue circles\n represent the data. The red, dashed line represents a fit to\n Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}\\approx 1.7$. [Right] The same analysis\n of the cross section at a center-of-mass momentum of 106 MeV as a function\n of the cube root of the cutoff. The legend is the same as in the left-hand\n plot.}\\label{fig:delta_sigma_analysis}\n\\end{figure}\n\nInterestingly, the $h$ values vary by less than a few percent around 1.5\nMeV$^{-1\/3}$ between the observables.\nThis fairly constant oscillation frequency matches up with the frequency of new\nbound states in the RG flow.\nAs shown below, this correspondence carries over to the three-body sector as\nwell.\n\nThe order of corrections is independent of the method used to obtain it.\nIn that spirit, we apply in addition to our modified power series expansion the\nmethod proposed by Griesshammer~\\cite{Griesshammer:2015osb}.\nFig.~\\ref{fig:delta_gh_analysis} shows the comparison of the phase shifts at\n$\\Lambda=3408$ and $6704$ MeV.\nBy Eq.~\\eqref{eq:gh_diff_log}, we expect the behavior to be linear.\nIn fact, we are able to extract a reliable slope of $n=1.5$ by fitting the data\nto Eq.~\\ref{eq:gh_diff_log}.\nUnfortunately, we found that other observables such as the cross section and\n$k\\cot\\delta$ provide unreliable results.\nSpecifically, zeros and unpredictable crossings precluded the extraction of\nlinear behavior.\nSelecting the phase shifts as the quantities of interest follows naturally from\nthese unfortunate conditions as discussed by\nGriesshammer~\\cite{Griesshammer:2015osb}.\n\\begin{figure}[ht]\n \\includegraphics[width=\\linewidth]{delta_gh_analysis.pdf}\n \\caption{Residual cutoff corrections to the two-body phase shifts as a\n function of the relative momentum.\n The blue circles represent the numerical calculation.\n The red line represents a fit to Eq.~\\eqref{eq:gh_diff_log}, resulting in\n $n=1.5$.\n The pink, shaded region represents the range of $k$ over which the fit was\n performed.\n The vertical, green line is the binding momentum $\\gamma$.\n }\\label{fig:delta_gh_analysis}\n\\end{figure}\n\n\n\\subsection{Three-Body Scattering}\n\n\nThe first observable in the three-body sector that we study is the atom-dimer\nscattering length.\nFigure~\\ref{fig:a_AD_cutoff_dependence} shows the convergence of $a_{\\textrm{AD}}$\nwith respect to the momentum cutoff $\\Lambda$, clearly demonstrating that a\ntwo-body contact term is sufficient to renormalize three-body observables.\n\\begin{figure}[ht]\n\t\\includegraphics[width=0.9\\linewidth]{a_AD_cutoff_dependence.pdf}\n \\caption{The cutoff dependence of the atom-dimer scattering length.\n }\\label{fig:a_AD_cutoff_dependence}\n\\end{figure}\n\nAgain, we apply the analysis based on Eq.~\\eqref{eq:rgi_correction} to the\natom-dimer scattering length.\nThe results are shown in Fig.~\\ref{fig:a_AD_analysis}.\nAs in the two-body sector, Eq.~\\eqref{eq:rgi_correction} is able to accurately\ndescribe the oscillatory convergence behavior occuring on top of the typically\nexpected $\\Lambda$-dependence.\nThe fit was performed over a range of cutoffs --- from\n$\\Lambda_\\textrm{lower}=3.1$~GeV to $\\Lambda_\\textrm{upper}=8.1$~GeV.\nFor the atom-dimer scattering length, the best fit to\nEq.~\\eqref{eq:rgi_correction} occurs at $n_{\\min}=1.3$.\nBecause this analysis involves the derivative of the observable with respect to\n$\\Lambda$ and three-body observables are particularly difficult to obtain to\narbitrary accuracy, we are often forced to constrain our fit window.\nThe atom-dimer scattering length, as well as the other three-body observables\npresented below, are selected because they provide stable results over a\nsignificant range of cutoffs.\n\\begin{figure}[ht]\n \\includegraphics[width=0.9\\linewidth]{a_AD_analysis.pdf}\n \\caption{$\\Lambda (da_{AD}\/d\\Lambda)$ as a function of the momentum-space\n cutoff. The blue circles are the calculation. The solid, red line is the\n fit to Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}=1.3$.\n }\\label{fig:a_AD_analysis}\n\\end{figure}\n\nIn addition to the atom-dimer scattering length, we also conduct analyses of\nthree-body phase shifts and inelasticities at center-of-mass, kinetic energies\nof 10, 50, and 100~MeV.\nThe results are shown in Fig.~\\ref{fig:3b_scatter_analysis}.\nThe $n_{\\min}$ values, ranging from 1.1 to 1.3, used to plot the solid, red\nlines corresponding to Eq.~\\ref{eq:rgi_correction} are tabulated in\nTable~\\ref{tab:nmins}.\nThe bounds of the cutoff range are included as well to assure the reader that\nthe behavior represents a significant and relevant portion of the cutoff\ndependence.\n\\begin{figure}[ht]\n \\includegraphics[width=\\linewidth]{3b_scatter_analysis.pdf}\n \\caption{[Upper Left] Eq.~\\ref{eq:rgi_correction}-based analysis of the 2+1\n phase shift at $E=10$~MeV. The blue circles are the calculation. The solid,\n red line is the fit to Eq.~\\ref{eq:rgi_correction}. [Upper Right] Same\n analysis and legend applied to the 2+1 phase shift at $E=50$~MeV. [Lower\n Left] Inelasticity at $E=50$~MeV. [Lower Right] Inelasticity at\n $E=100$~MeV.\n }\\label{fig:3b_scatter_analysis}\n\\end{figure}\n\n\\subsection{Three-Body Bound States}\\label{sec:three-body-bound}\n\nOne of the main goals of these efforts has been to examine the\nsufficiency of a two-body counterterm to renormalize three-body\nobservables.\nIn Fig.~\\ref{fig:three_body_states} we plot the cutoff dependence of the\nthree-body binding energies associated with two three-body states that appear in\nthe system defined by the nonlocally regulated interaction\nEq.~\\eqref{eq:momspace_int_nl}.\nThe results shown come from the solution of Eq.~\\eqref{eq:Faddeev_bound_state},\nthough equivalent results were found by calculating the three-body phase shifts\ndefined by Eq.~\\eqref{eq:three-body-pw-amp} and scanning for resonances.\nThe ground state and excited state binding energies at $\\Lambda=10$~GeV are\n-18.086~MeV and -2.2379~MeV, respectively.\nThe primary feature of Fig.~\\ref{fig:three_body_states} is the convergence of\nthe binding energies in the infinite $\\Lambda$ limit.\nAt $\\approx 2$~GeV, the binding energies (or rather, the resonant energies)\nbegin to flatten out, just as in the two-body phase shifts.\n\\begin{figure}[ht]\n\t\\includegraphics[width=\\linewidth]{e3_cutoff_dependence.pdf}\n \\caption{[Left] Three-body ground state\/resonance energy as a function of the\n short-distance cutoff. [Right] Three-body excited state\/resonance energy as\n a function of the short-distance cutoff.\n }\\label{fig:three_body_states}\n\\end{figure}\n\nUnfortunately, small inaccuracies in the three-body binding energies left only\nsmall windows of cutoffs over which a fit to Eq.~\\eqref{eq:rgi_correction} could\nbe performed when all four fit parameters were treated as such.\nUsing the values of $h$ and $f$ from the fit of the atom-dimer scattering length to\nEq.~\\eqref{eq:rgi_correction}, we fit only $A$ and $B$ for the ground state\nbinding energy and show the results in Fig.~\\ref{fig:3b_gs_cutoff_dep}.\nAn $n_{\\min}$ value of 1.4 is found to minimize the penalty function, and \nthe form of Eq.~\\eqref{eq:rgi_correction} is further validated.\n\\begin{figure}[ht]\n\t\\includegraphics[width=\\linewidth]{3b_gs_cutoff_dep.pdf}\n \\caption{RG analysis of the three-body, ground-state binding energy. The blue\n circles are the calculation. The red line represents a fit to\n Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}=1.4$ and the values of $h$ and\n $f$ taken from the same fit of the atom-dimer scattering length.\n }\\label{fig:3b_gs_cutoff_dep}\n\\end{figure}\n\nThroughout all of the three-body observables, we see a consistency among the\n$h$ values.\nNotably, it is enforced manually for the three-body ground state.\nThey range from 1.4 to 1.5 MeV$^{-1\/3}$ which is also consistent with the $h$\nvalues found by fitting the two-body observables.\nThis consistency between the two- and three-body sectors can be seen in\nTable~\\ref{tab:nmins} which establishes the pervasive nature of these oscillations.\n\n\\setlength\\tabcolsep{12pt}\n\\begin{center}\n \\begin{table}\n \\begin{tabular} { c c c c c c }\n \\hline\\hline\n Observable & $n_{\\min}$ & $\\Lambda_{\\textrm{lower}}$ (GeV) &\n $\\Lambda_{\\textrm{upper}}$ (GeV) & h (MeV$^{-1\/3}$) \\\\\n \\hline\n $a(\\Lambda)$ & 1.7 & 3.6 & 10.0 & 1.5 \\\\\n $\\delta(\\Lambda;E=12\\textrm{MeV})$ & 1.7 & 2.6 & 10.0 & 1.5 \\\\\n $\\sigma(\\Lambda;E=12\\textrm{MeV})$ & 1.7 & 2.4 & 10.0 & 1.5 \\\\\n $\\delta(k)$ & 1.5 & 3.4 & 6.7 & --- \\\\\n $a_{AD}(\\Lambda)$ & 1.3 & 3.1 & 8.1 & 1.5 \\\\\n $\\delta_{2+1}(\\Lambda;E=10\\textrm{MeV})$ & 1.3 & 3.7 & 7.7 & 1.4 \\\\\n \n $\\delta_{2+1}(\\Lambda;E=50\\textrm{MeV})$ & 1.2 & 3.7 & 7.0 & 1.4 \\\\\n $\\eta_{2+1}(\\Lambda;E=50\\textrm{MeV})$ & 1.3 & 3.7 & 7.0 & 1.5 \\\\\n $\\eta_{2+1}(\\Lambda;E=100\\textrm{MeV})$ & 1.1 & 3.7 & 7.1 & 1.4 \\\\\n $E_3^{(0)}$ & 1.4 & 3.5 & 7.8 & 1.5* \\\\\n \\hline\n \\end{tabular}\n \\caption{$n_{\\min}$ values for various two- and three-body observables\n alongside the bounds of cutoffs over which the fit to\n Eq.~\\eqref{eq:rgi_correction} was performed as well as the frequency that\n optimizes the fit. * The $h$ value for $E_3^{(0)}$ was taken from the fit of\n $a_{AD}$.}\\label{tab:nmins}\n\\end{table}\n\\end{center}\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this manuscript, we have set out to understand the renormalization\nproperties of the FRIC potential in the two- and\nthree-body sector.\nIn particular, we have studied the regulator dependence of observables such as\ntwo-body phase shifts, three-body binding energies, the atom-dimer scattering\nlength, phase shifts, and inelasticity parameter.\nMotivated by a recent development in the nuclear theory community, we did these\ncalculations using different, frequently used regulator functions.\n\nOur results in the two-body sector confirm that the two-body sector is\nproperly renormalized. One input parameter is required (at leading\norder) to renormalize one low-energy counterterm and thereby the\ntwo-body sector. In the three-body sector, we have demonstrated that a\nthree-body force is not needed at leading order to renormalize\nthree-body observables for the inverse cube interaction.\n\nIn both the two- and three-body sectors, we have observed significant\noscillatory behavior in the cutoff dependence of observables. These\noscillations are not captured by a simple power series expansion.\n\nInstead, we have empirically found that a generalized oscillatory dependence of\nthe form presented in Eq.~\\eqref{eq:rgi_correction} allows accurate fits of the\ndata to be made and a much clearer picture of the power of the cutoff dependence\nto be revealed.\n\nOur analysis strongly indicates that $n$ is smaller in the three-body sector\nthan in the two-body sector.\nThis would suggest that a three-body force is needed at next-to-leading order.\n\nOur analysis also indicates that $n$ is consistent with approximately 1.5\nfor two-body observables and approximately 1 for three-body\nobservables. It is an interesting question whether this has any\nsignificance for the counting of two- and three-body counterterms in\nan EFT for the inverse cube potential. For example, the singular\n$1\/r^2$ has been considered previously as the starting point for an\nEFT expansion in Ref.~\\cite{Long:2007vp}, however the inverse cube and\nall other singular coordinate space potentials need their own\nindependent analysis.\n\nHaving tested several different local, semi-local, and nonlocal regulators and\nhaving found no significant differences above $\\approx$2~GeV, we conclude that\nthese oscillations are most likely attributable to the singular nature of the\ninverse cube potential in coordinate space.\n\nIn the future, we plan to carry out an analysis of higher order corrections in\nthe three-boson and three-nucleon sector.\nHowever, we plan to also extend our work to the infinite range inverse cube\npotential that is of relevance to the atomic dipole interaction.\nThis will let us combine the results obtained by M\\\"uller~\\cite{Mueller:2013}\nwith three-body observables and study the dependence of three-body observables\non the boundary condition employed in the two-body sector.\nA more detailed analysis of the short-distance behaviour of the three-nucleon\nwave function might also provide novel insights into the power counting of\nelectroweak currents~\\cite{Valderrama:2014vra}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzngxl b/data_all_eng_slimpj/shuffled/split2/finalzzngxl new file mode 100644 index 0000000000000000000000000000000000000000..fccbfd33a88ff76163a4fb47f2a19410811d7ddc --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzngxl @@ -0,0 +1,5 @@ +{"text":"\\section*{\\centering{#1}}}\n\\newcommand{\\smat}[4]{\\left(\\begin{smallmatrix} #1 & #2\\\\ #3 & #4\\end{smallmatrix}\\right)}\n\\newcommand{\\mat}[4]{\\begin{pmatrix} #1 & #2\\\\ #3 & #4\\end{pmatrix}}\n\\newcommand{\\vet}[2]{\\left(\\begin{smallmatrix} #1 \\\\ #2 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Gr}\n\n\\newcommand\\scalemath[2]{\\scalebox{#1}{\\mbox{\\ensuremath{\\displaystyle\n #2}}}}\n\n\\setcounter{MaxMatrixCols}{200} \n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{corollary}[thm]{Corollary}\n\\newtheorem{question}[thm]{Question}\n\\newtheorem{claim}[thm]{Claim}\n\\newtheorem{lemma}[thm]{Lemma}\n\\newtheorem{proposition}[thm]{Proposition}\n\\newtheorem{numerology}[thm]{Numerology}\n\\newtheorem{criterion}[thm]{Criterion}\n\\newtheorem{conj}[thm]{Conjecture}\n\\newtheorem{NB}{NB:}\n\\newtheorem*{aim*}{Aim of this paper}\n\\newtheorem*{problem}{Problem}\n\n\\theoremstyle{definition}\n\\newtheorem{definition}[thm]{Definition}\n\\newtheorem{rmk}[thm]{Remark}\n\\newtheorem{ex}[thm]{Example}\n\\newtheorem{caveat}[thm]{Caveat}\n\\newtheorem{notz}{Notazione}[section] \n\n\\setlength{\\topmargin}{-1.2cm}\n\\setlength{\\textheight}{23.4cm}\n\\setlength{\\textwidth}{17cm}\n\\setlength{\\oddsidemargin}{-0.2cm}\n\\setlength{\\evensidemargin}{-0.2cm}\n\n\\newenvironment{sistema}\n{\\left\\lbrace\\begin{array}{@{}l@{}}}\n{\\end{array}\\right.}\n\n\\makeatletter \n\\def\\l@subsection{\\@tocline{1}{0,2pt}{2pc}{8mm}{\\ \\ }} \n\\makeatletter \n\\def\\l@section{\\@tocline{1}{0,2pt}{2pc}{8mm}{\\ \\ }} \n\n\\author{Lorenzo De Biase}\n\\address{School of Mathematics - Cardiff University \\\\ CF24 4AG, UK}\n\\email[L.~De Biase]{debiase.lorenzo7@gmail.com}\n\n\\author{Enrico Fatighenti}\n\\address{Department of Mathematical Sciences\\\\\nLoughborough University\\\\\n LE11 3TU, UK}\n\\email[E.~Fatighenti]{enricofatighenti6@gmail.com}\n\n\\author{Fabio Tanturri}\n\\address{Dipartimento di Matematica \\\\\nUniversit\\`a di Genova\\\\\nVia Dodecaneso 35\\\\\n16146 Genova, Italy}\n\\email[F.~Tanturri]{tanturri@dima.unige.it}\n\n\\@namedef{subjclassname@2020}{\\textup{2020} Mathematics Subject Classification}\n\\subjclass[2020]{Primary 14J30, 14J45; Secondary 14M15, 14E30}\n \n\\title[Fano 3-folds from homogeneous vector bundles over Grassmannians]{Fano 3-folds from homogeneous vector bundles over Grassmannians}\n\n\\begin{document}\n\\begin{abstract}\nWe rework the Mori--Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.\n\\end{abstract}\n\\maketitle\n\\section{Introduction}\n\n\\thispagestyle{empty}\n\nThe classification of Fano 3-folds is one of the most influential results in birational geometry. Out of the 105 families, 17 have Picard rank $\\rho=1$. They are usually called \\emph{prime}. Their classification was completed first by Iskovskikh \\cite{isk}, using the birational technique of the \\emph{double projection from a line}. The classification was reworked by Mukai \\cite{mukai}, using the biregular \\emph{vector bundle method}. Mukai was able to describe most of the prime Fano varieties as complete intersections in certain homogeneous or quasi-homogeneous varieties. The latter in turn can be embedded in Grassmannians as zero loci of sections of homogeneous vector bundles.\n\nMori and Mukai \\cite{morimukai} classified as well the 88 remaining families of Fano 3-folds with Picard rank $\\rho \\geq 2$. However, the proof has little in common with the vector bundle strategy, relying on the powerful birational Mori's theory of extremal rays.\n\nOne of the aims of this paper is to rewrite the entire classification of 3-folds in a biregular fashion, finding models for the non-prime Fano 3-folds which are akin to the Mukai's vector bundle ones.\nIn particular, for each of the 105 Fano $X$ we will look for a suitable embedding $X \\subset \\prod \\Gr(k_i, n_i)$, such that $X$ can be described as the zero locus of a general global section of a homogeneous vector bundle $\\mathcal{F}$ over $\\prod \\Gr(k_i, n_i)$.\n\nIn \\cite{corti} Coates, Corti, Galkin, and Kasprzyk carried out a similar program. In particular, they were able to write down each of the 105 Fano 3-folds as zero loci of sections of vector bundles over GIT quotients. In some cases, their \\emph{key varieties} are products of Grassmannians, and we decided to adopt their models. However, in many cases, their model of choice is a complete intersection in a toric variety, which was particularly suitable for their purpose of computing the quantum periods, with the aim of using ideas from mirror symmetry for further classification results.\n\nOur motivating purpose is instead to attack the classification of Fano varieties in higher dimension from a representation-theoretical angle. In \\cite{kuchle} K\\\"uchle classified Fano 4-folds of index 1 that can be obtained from completely reducible, homogeneous vector bundles over a single Grassmannian $\\Gr(k,n)$. The resulting 20 families are therefore a sort of higher dimensional analogue of the Mukai models for 3-folds obtained via the vector bundle method. One of the main advantages of K\\\"uchle's method is that it relies only on very simple combinatorial data as input, such as the weight of the representation corresponding to the bundle involved. Moreover, this description allows an efficient computation of the invariants of the Fano, such as the Hodge numbers, for example using in combination the Koszul complex and Borel--Weil--Bott Theorem on the ambient variety. Such methods can be easily automatised via computer algebra, and extended to the case of products of Grassmannians $\\prod \\Gr(k_i, n_i)$, to say the least. This is exactly what we did. This paper originated from the construction of 3-folds via these methods; in a series of subsequent projects, we plan to work on more classification-type results, in dimension 4 and above.\n\nAs an initial benchmark for our strategy, we wanted to check how many of the 105 3-folds could be described using our methods. We found out that all 105 of them are. Although we do not believe that the same will be true in dimension 4 and higher, we hope to be able to find out many new and interesting examples of non-prime Fano 4-folds.\n\n\n\\subsection*{Main results}\n\nThe results of the paper are partially summarised in the following theorem. In what follows and throughout the whole paper, the notation $\\mathscr{Z}(\\mathcal{F}) \\subset G$ will denote the zero locus of a general global section of the vector bundle $\\mathcal{F}$ in the variety $G$.\n\\begin{thm}\\label{mainthm}\nLet $X$ be a smooth Fano 3-fold. Then there exist an ambient variety $G=\\prod \\Gr(k_i,n_i)$, product of (possibly weighted) Grassmannians, and a homogeneous vector bundle $\\mathcal{F}$ on $G$ such that $X= \\mathscr{Z}(\\mathcal{F}) \\subset G.$\n\\end{thm}\nThe only Fano varieties requiring weighted Grassmannians (actually, a unique weighted projective space) in their description without any alternative description are 1--11, 2--1, and 10--1, the others involving only classical Grassmannians. The weighted projective space in question is $\\mathbb{P}(1^3,2,3)$. The Fano 1--11 is a section of $\\mathcal{O}(6)$ on the latter, 2--1 is a blow up of 1--11 and 10--1 is a linear section (multiplied with a $\\mathbb{P}^1$). Notice that for the Fano 1--11 (which was present in this form in Mukai's classification as well), $-K_X$ is not very ample. A few other weighted projective spaces appear, but for all of them we provide alternative descriptions.\n\nIn the statement of Theorem \\ref{mainthm} we have not specified any hypothesis on the vector bundle $\\mathcal{F}$. Our \\emph{gold standard} for a homogeneous vector bundle $\\mathcal{F}$ is to be completely reducible and globally generated. Bundles with these properties are particularly suitable when facing classification problems. \nFor 85 out of the 105 families, we managed to find a vector bundle of this form; for the remaining ones, we used homogeneous bundles which are extensions of some other homogeneous completely reducible ones, so that the description is slightly more complicated but still well within our range of techniques. Out of these 20 families, for 5 of them the vector bundle is particular: it\nis of the form $\\mathcal{F}= \\mathcal{F}' \\oplus \\mathcal{G}$ where $\\mathcal{G}$ is a line bundle with no global sections on the total space, but with sections on $\\mathscr{Z}(\\mathcal{F}')$. This happens when we need to blow up along a subvariety involving an exceptional divisor coming from a previous blow up. We deal with this phenomenon in Caveat \\ref{caveatBundle}.\n\nWe partially collect these refinements in the following theorem.\n\\begin{thm} \\label{thm:refined} Let $X$ be a Fano as in Theorem \\ref{mainthm}. Then\n\\begin{itemize}\n \\item For 102\/105 families of Fano there exists a description without weighted factors in $G$.\n \\item For 85\/105 families of Fano there exists a description such that the bundle $\\mathcal{F}$ is completely reducible.\n\\end{itemize}\n\\end{thm}\n\nThe two theorems are proven in Section \\ref{Fano3folds}, which we devote to the construction of the aforementioned families, except for those which are already known in the literature. We collect all the models in Section \\ref{tables}; we include models for Del Pezzo surfaces as well. All models are general in moduli.\n\nWe draw the reader's attention to Section \\ref{identifications} as well. This is mainly a collection of technical lemmas and results, and we believe that most of them are well-known to experts. \nNonetheless, some of them are of independent interest, as they provide a dictionary between zero loci of sections of vector bundles and birational geometry. They were quite useful for translating Mori--Mukai models into our descriptions, and we believe that they can and will be useful for higher dimensional analyses. In this line of thought, we also present a few results involving flag varieties, even if they play only a small role in what follows.\n\n\\subsection*{Our models}\n\nMori--Mukai characterisation of the 88 non-prime 3-folds often involves intricate birational descriptions. The typical situation consists in blowing up a simpler 3-fold along a curve. Whenever the curve is a complete intersection in the base 3-fold, finding a suitable model in a product of Grassmannians is almost algorithmic; when the curve is not, then we perform a delicate analysis to understand how the curve can be cut in the ambient Fano. Subsequently, Lemma \\ref{lem:blowup}, Corollary \\ref{cor:cayleycrit}, and Lemma \\ref{lem:blowDegeneracyLocus} allow us to describe the resulting 3-fold as a complete intersection in a suitable projective bundle. We then need to describe the latter as a zero locus of some vector bundle over a product of Grassmannians. In many cases, this is a straightforward procedure and the proof takes few lines. However, some projective bundles turn out to be particularly tricky, and we have to deal with them case-by-case. \n\nFor other Fano we need to blow up a variety along a subvariety of codimension at least 3. To handle these cases, we collect and develop a few results which allow us to characterise these blow ups in term of zero loci of sections.\n\nWe want to give here an introductory example of a Fano 3-fold whose description is not immediate, yet admits a quite simple description in our model. We compare the original Mori--Mukai approach and the Coates--Corti--Galkin--Kasprzyk one with ours.\n\nLet us consider the Fano of rank 2, number 16 in the Mori--Mukai list. Following the notation which will be adopted in our paper, we will call it 2--16.\n\n\\begin{description}[leftmargin=0pt]\n\\item[2--16, Mori--Mukai] Blow up of the complete intersection of two quadrics in $\\mathbb{P}^5$ in a conic $C$. Notice that $C$ is not a complete intersection in the ambient variety $\\mathbb{Q}_1 \\cap \\mathbb{Q}_2 \\subset \\mathbb{P}^5$.\n\\item[2--16, Coates--Corti--Galkin--Kasprzyk] A codimension-2 complete intersection\n$\\mathscr{Z}(L+M, 2M) \\subset F$ where $F$ has weight data\n\\[\n\\begin{array}{rrrrrrrl} \n\t\\multicolumn{1}{c}{s_0} & \n\t\\multicolumn{1}{c}{s_1} & \n\t\\multicolumn{1}{c}{s_2} & \n\t\\multicolumn{1}{c}{x} & \n\t\\multicolumn{1}{c}{x_3} & \n\t\\multicolumn{1}{c}{x_4} & \n\t\\multicolumn{1}{c}{x_5} & \\\\ \n\t\\cmidrule{1-7}\n\t1 & 1 & 1 & -1 & 0 & 0 & 0& \\hspace{1.5ex} L\\\\ \n\t0 & 0 & 0 & 1 & 1 & 1 & 1 & \\hspace{1.5ex} M \\\\\n\\end{array}\n\\]\n\\end{description}\n\nFinally, our description realises this Fano as the zero locus of a general section of a globally generated homogeneous vector bundle over a (non-toric) product of Grassmannians.\n\\begin{description}[leftmargin=0pt]\n\\item[2--16, our description] The zero locus \\[\\mathscr{Z}(\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4),\\] where $\\mathcal{U}$ is the rank 2 tautological subbundle.\n\\end{description}\n\nOur construction methods often allow for multiple models. For instance, the above Fano 2--16 can be realised as well as \\[ \\mathscr{Z}(\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,4). \\]\nIn order to preserve the compactness of this paper we usually decided to present only one model for each Fano variety, with notable exceptions whenever we found an alternative description too elegant not to include it, or whenever they were important intermediate steps in the identification of the model. Our choice of model depends on our personal taste. The criterion for an $X= \\mathscr{Z}(\\mathcal{F}) \\subset \\prod \\Gr(k_i, n_i)$ was to pick the model with either the smallest number of factors or with the rank of $\\mathcal{F}$ as low as possible. To mention an example in lower dimension, the Del Pezzo surface of degree 5 can be equivalently described as $\\mathscr{Z}(\\mathcal{O}(1,0,0,0,1)\\oplus \\mathcal{O}(0,1,0,0,1) \\oplus \\mathcal{O}(0,0,1,0,1) \\oplus \\mathcal{O}(0,0,0,1,1)) \\subset (\\mathbb{P}^1)^4 \\times \\mathbb{P}^2 $ or as $\\mathscr{Z}(\\mathcal{O}(1)^{\\oplus 4}) \\subset \\Gr(2,5)$. We will prefer the latter description to the former. \n\n\n\n\\subsection*{Further directions}\n\\label{futureDirections}\n\nAs mentioned in the first part of the introduction, our methods are built with the explicit intention of being applied in higher dimension. Over a single Grassmannian $\\Gr(k,n)$ homogeneous, completely reducible vector bundles can be written as direct sums of $\\Sigma_{\\alpha} \\mathcal{Q} \\otimes \\Sigma_{\\beta}\\mathcal{U}$, where $\\Sigma_{\\alpha}$ (resp.\\ $\\Sigma_{\\beta}$) denotes the Schur functor indexed by the non-increasing sequence $\\alpha$ (resp.\\ $\\beta$); a similar expression holds for flag varieties and their products. This makes to some extent possible a methodical search for varieties which are zero loci of sections of bundles of this form.\n\nWhat we plan to do in a series of subsequent works is to classify all Fano in dimension 4 that can be obtained in this way, comparing our results with the already existing known classes of Fano 4-folds (\\cite{batyrev, coates, kalashnikov}, to cite a few). We are confident that many new and interesting examples can be found in this way, and the results of this paper are for sure strong motivations. We are particularly interested in the case of 4-folds of index 1 with Picard rank as high as possible and which are not a product. The \\emph{champion} at the moment is the Fano of Picard rank 9 constructed by Casagrande, Codogni, and Fanelli in \\cite{casagrande}; see, e.g., \\cite{CasagrandeSurvey} for a survey of results on the topic.\n\nAnother case of interest are Fano varieties in higher dimension with special Hodge-theoretical properties. In particular, Fano varieties in any dimension of K3 type (in the sense of \\cite{eg2}) have recently been studied due to their possible links with hyperk\\\"ahler manifolds. Finally, we remark that zero loci are particular cases of degeneracy loci of morphisms between vector bundles. It is certainly possible to further extend the above program to this framework, which has already been explored from many points of view (see, e.g., \\cite{TanturriHilbert}), or even to the so-called orbital degeneracy loci, a recently introduced wider class of varieties \\cite{BFMT2,BFMT}.\n\n\n\n\\subsection*{Plan of the paper} \nSection \\ref{identifications} is where we establish our toolbox, and state or prove several lemmas, useful to translate the Mori--Mukai birational language into our biregular one, and vice versa. Section \\ref{computeinvariants} is devoted to explaining how we are able to compute the invariants for all the models we present. Section \\ref{Fano3folds} is the core of the paper. A detailed description of all the families which are not provided in the literature is given. Section \\ref{tables} contains the tables and recap all the results in a schematic and handy fashion.\n\n\n\\subsection*{Notation and conventions}Throughout the whole paper, the notation $\\mathscr{Z}(\\mathcal{F}) \\subset X$ denotes the zero locus of a general global section of the vector bundle $\\mathcal{F}$ in the variety $X$. We will denote by $X_d$ a general hypersurface of degree $d$ inside $X$.\n\nIf $E$ is a rank $r$ vector bundle over a variety $X$, we denote by $\\mathbb{P}_X(E)$ (or simply by $\\mathbb{P}(E)$ when no confusion can arise) the projective bundle $\\pi:\\mathrm{Proj}(\\Sym E^{\\vee}) \\rightarrow X$; we remark that we adopt the subspace notation, as in \\cite[Chapter 9]{EisenbudHarris3264}. If we denote by $\\mathcal{O}_{\\mathbb{P}(E)}(1)$ (or simply $\\mathcal{O}(1)$) the relatively ample line bundle, this yields $H^0(\\mathbb{P}(E),\\mathcal{O}_{\\mathbb{P}(E)}(1)) \\cong H^0(X, E^{\\vee})$. Moreover $\n\\omega_{\\mathbb{P}(E)} \\cong \\mathcal{O}_{\\mathbb{P}(E)}(-r) \\otimes \\pi^*(\\omega_X \\otimes \\det(E^{\\vee})\n$\nand, for any line bundle $L$, the isomorphism $\\mathbb{P}(E)\\cong \\mathbb{P}(E\\otimes L)$ induces $\\mathcal{O}_{\\mathbb{P} (E)}(1) \\otimes L^\\vee = \\mathcal{O}_{\\mathbb{P}(E\\otimes L)}(1)$.\n\nFor products of varieties $X_1 \\times X_2$, the expression $\\mathcal{F}_1 \\boxtimes \\mathcal{F}_2$ will denote the tensor product between the pullbacks of $\\mathcal{F}_i$ via the natural projections. For products of Grassmannians $\\Gr(k_1,n_1) \\times \\Gr(k_2,n_2)$, we will almost always adopt the short form $\\mathcal{O}(a,b):=\\mathcal{O}(a) \\boxtimes \\mathcal{O}(b)$; we will often omit the pullbacks when no confusion can arise, so that, e.g., $\\mathcal{Q}_{\\Gr(k_1,n_1)}(1,2)=\\mathcal{Q}_{\\Gr(k_1,n_1)}(1) \\boxtimes \\mathcal{O}_{\\Gr(k_2,n_2)}(2)$. \n\nBy $\\Fl(k_1, \\ldots, k_r, n)$ we will denote the flag variety of subspaces $V_{k_1} \\subset V_{k_2} \\subset \\ldots \\subset V_{k_r} \\subset \\mathbb{C}^n$. We will denote by $\\pi_i$ the projection to the $i$-th Grassmannian $\\Gr(k_i,n)$. $\\mathcal{U}_i$ and $\\mathcal{Q}_i$ will denote the pullback of the tautological bundles via $\\pi_i$. For short, we will write $\\mathcal{O}(a,b)=\\pi_1^*(\\mathcal{O}(a)) \\otimes \\pi_2^*(\\mathcal{O}(b))$. In the rare cases where a flag is involved in a product of varieties, the different Picard groups will be separated by a semicolon, i.e., $\\mathcal{O}(a,b;c)=\\mathcal{O}(a,b) \\boxtimes \\mathcal{O}(c)$ on $\\Fl(k_1,k_2,n) \\times \\Gr(k',n')$.\n\nMany data for Table \\ref{tab:3folds} (and for the paper overall) are taken from \\cite{fanography}. They rely on the tables from \\cite{isp5,pcs,kps,cfst}. Many other alternative descriptions are taken from \\cite{corti}. We include the relevant citation to the alternative description in the table whenever appropriate. The notation X--Y for a Fano means a Fano of Picard rank $X$ which is the number $Y$ in the Mori--Mukai list.\nFinally, $\\mathbb{Q}_3$ denotes the 3-dimensional quadric hypersurface (Fano 1--16) and $\\mathbb{V}_5$ denotes the index 2, degree 5 linear section of $\\Gr(2,5)$ (Fano 1--15).\n\\subsection*{Acknowledgements} \nWe are indebted to Daniele Faenzi for many enlightening suggestions. Thanks to Vladimiro Benedetti, Marcello Bernardara, Giovanni Mongardi, and Miles Reid for useful discussions. EF and FT were partially supported by a ``Research in Paris'' grant held at Institut Henri Poincar\\'e. We thank the institute for the excellent working conditions. We acknowledge the Laboratoire Paul Painlev\\'e -- Universit\\'e de Lille, the Dipartimento di Matematica ``Giuseppe Peano'' -- Universit\\`a di Torino and INdAM for partial support as well. All three authors are members of INdAM-GNSAGA.\n\n\\section{Identifications}\n\\label{identifications}\n\nMost of the Fano 3-folds with Picard rank $\\rho\\geq 2$ arise as blow up of other Fano 3-folds with centre in distinguished subvarieties. Sometimes other standard birational descriptions are involved. The purpose of this subsection is therefore to establish a toolbox that allow us to translate the Mori--Mukai birational language into models suitable for our type of descriptions. Most of the lemmas appearing in this section are probably well-known to the experts: however for some of them we have not been able to locate clear proofs in the literature. \n\nThe most basic result is the description of the blow up of a projective space in a linear subspace. We will use the following lemma:\n\n\n\\begin{lemma} \\label{lem:blow}\nLet $\\mathcal{Q}$ be the tautological quotient bundle on $ \\mathbb{P}^{n-r}$. We have\n\\[\n\\Bl_{\\mathbb{P}^{r-1}}\\mathbb{P}^n \\cong \\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^{n-r}}(0,1)) \\subset \\mathbb{P}^{n-r} \\times \\mathbb{P}^n.\n\\]\n\n\\begin{proof}\nLet $V$ be a $n+1$-dimensional vector space such that $\\mathbb{P}^n \\cong \\mathbb{P}(V)$. By \\cite[Proposition 9.11]{EisenbudHarris3264},\n$\\Bl_{\\mathbb{P}^{r-1}}\\mathbb{P}^n$ is isomorphic to the projectivization of the vector bundle\n\\[\nE = \\mathcal{O}_{\\mathbb{P}^{n-r}}(-1) \\oplus (V' \\otimes \\mathcal{O}_{\\mathbb{P}^{n-r}}),\n\\]\nwhere $V' \\subset V$ has dimension $r$ and $\\mathbb{P}^{n-r}$ is identified with $\\mathbb{P}(V\/V')$. The bundle $E$ fits into the short exact sequence\n\\[\n0 \\rightarrow E \\rightarrow (V\/V' \\oplus V') \\otimes \\mathcal{O}_{\\mathbb{P}^{n-r}} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^{n-r}} \\rightarrow 0,\n\\]\nhence $\\mathbb{P}(E)$ can be also expressed as the zero locus of $\\mathcal{Q} \\boxtimes \\mathcal{O}(1)$ inside $\\mathbb{P}^{n-r} \\times (\\mathbb{P}^n\\cong \\mathbb{P}(V\/V' \\oplus V'))$, as claimed.\n\\end{proof}\n\\end{lemma}\n\nIn the above lemma we used the fact that, as soon as we have a short exact sequence on $X$ of vector bundles\n$0 \\rightarrow\nE \\rightarrow\nF \\rightarrow\nG \\rightarrow\n0,$\nthen $\\mathbb{P}(E)$ can be obtained as the zero locus of a section of $t$ of $\\pi^*(G) \\otimes \\mathcal{O}_{\\mathbb{P}(F)}(1)$ over $\\pi:\\mathbb{P}(F)\\rightarrow X$. If $H^1(E)=0$, then $t$ can be chosen to be general; a particularly favourable situation will occur when $F\\cong \\mathcal{O}^{\\oplus r}$, so that $\\mathbb{P}(E)$ embeds into $X \\times \\mathbb{P}^{r-1}$.\n\nLemma \\ref{lem:blow} can be generalised for the Grassmannians context.\n\\begin{lemma}\\label{lem:blowInGrass}\nWe have\n\\[\n\\Bl_{\\Gr(k-1, n-1)}\\Gr(k,n) \\cong \\mathscr{Z}(\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee}) \\subset \\Gr(k,n-1) \\times \\Gr(k,n),\n\\]\nwhere the centre of the blow up $\\Gr(k-1, n-1)$ is identified with $\\mathscr{Z}(\\mathcal{Q})\\subset \\Gr(k, n)$.\n\\begin{proof}\nLet $V_n, V_{n-1}$ be complex vector spaces of dimension $n, n-1$ respectively. A section of $\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee}$ over $\\Gr(k,V_{n-1}) \\times \\Gr(k,V_n)$ can be regarded as a section of $\\mathcal{U}^{\\vee} \\boxtimes \\mathcal{U}^{\\vee}$ over $\\Gr(n-k-1,V_{n-1}^\\vee) \\times \\Gr(k,V_n)$, and the corresponding zero loci are canonically isomorphic.\n\nA section of the latter vector bundle is of the form\n\\[\ns=f_1 x_1 + \\ldots + f_{n-1} x_{n-1}\n\\]\nfor some $f_i \\in V_{n-1}$, $x_i \\in V_n^\\vee$. Let us fix bases for $V_{n-1}^{\\vee}, V_n$ accordingly. Up to the action of $\\GL_k$, a point in $\\Gr(k,V_n)$ is represented by a $k \\times n$ matrix\n\\[\nA=\\left( \\begin{array}{ccc}\na_{1,1} & \\cdots & a_{1,n}\\\\\n\\vdots & & \\vdots \\\\\na_{k,1} & \\cdots & a_{k,n}\n\\end{array}\n\\right)\n\\]\nand a section $x_i \\in V_n^\\vee$ evaluates as $x_i (A)=(a_{1,i} \\cdots a_{k,i})^t$. Analogously, a section $f_i \\in V_n$ evaluates on a point $B \\in \\Gr(n-k-1,V_{n-1}^\\vee)$, seen as a $n-k-1 \\times n-1$ matrix, as $f_i(B)=(b_{1,i} \\cdots b_{n-k-1,i})^t$.\n\nThe evaluation of a section $f_i x_i$ on a point $(A,B)$ is given by the $k \\times n-k-1$ matrix $x_i(A) \\cdot (f_i(B))^t$. It is straightforward to check that\n\\[\ns(A,B) = 0 \\qquad \\mbox{if and only if} \\qquad A \\cdot \\left( \n\\begin{array}{c}\nB^t \\\\\n\\hline 0 \\cdots 0\n\\end{array} \\right)=0.\n\\]\n\nLet $Y$ be the zero locus of $s$. We want to study the fibres of the (restriction of the) natural projection $Y \\rightarrow \\Gr(k,V_n)$. This amounts to solving a linear system with $b_{1,1}, b_{1,2}, \\ldots, b_{1,n-1}, b_{2,1}, \\ldots, b_{n-k-1,n-1}$ as variables. With this choice of coordinates, the matrix associated to the linear system is the $(n-1)(n-k-1) \\times k(n-k-1)$ matrix\n\\[\n\\left(\n\\begin{array}{cccc}\n\\tilde{A} \\\\\n& \\tilde{A} \\\\\n& & \\ddots \\\\\n& & & \\tilde{A}\n\\end{array}\n\\right), \\qquad \\mbox{where } A=\\left(\n\\begin{array}{c|c}\n\\tilde{A} & \\begin{array}{c}\na_{1,n}\\\\\n\\vdots\\\\\na_{k,n}\n\\end{array}\n\\end{array}\n\\right).\n\\]\nThe fiber over a general point $A$, i.e., whenever $\\tilde{A}$ has maximal rank, is a single point $\\in \\Gr(n-k-1,V_{n-1}^\\vee)$, hence $Y \\rightarrow \\Gr(k,V_n)$ is birational. The fiber over $A$ is positive-dimensional if and only if $\\tilde{A}$ has rank at most $k-1$, i.e., if and only if\n\\[\n\\rank \\left(\n\\begin{array}{c}\nA \\\\\n\\hline 0 \\cdots 0 \\, 1\n\\end{array}\n\\right)0$, which implies that $H^0(\\Fl, E)$ surjects onto $H^0(Y, E|_Y)$.\n\n\nWe do a recap in the following handy corollary. This is will be useful to deal with the case of complete intersection curves of different degrees.\n\\begin{corollary}\\label{cor:cayleycrit}\nAssume that we have a bundle $F=E \\oplus G$ on $\\Fl(1,2,n)$, with $G=\\pi_2^*\\widetilde{G}$ for a bundle $\\widetilde{G}$ on $\\Gr(2,n)$ and $X=\\mathscr{Z}(F) \\subset Y= \\mathscr{Z}(G) \\subset \\Fl(1,2,n)$. Denote by $\\widetilde{Y}$ the zero locus $\\widetilde{Y}=\\mathscr{Z}(\\widetilde{G}) \\subset \\Gr(2,n)$. Assume that\n$H^0(Y, E|_Y) \\cong H^0(\\widetilde{Y},\\mathcal{U}|_{\\widetilde{Y}} \\otimes L)$ for some line bundle $L$ on $\\widetilde{Y}$. Denote by $\\widetilde{X}=\\mathscr{Z}(\\mathcal{U}|_{\\widetilde{Y}} \\otimes L)\\subset \\widetilde{Y}$. Then $X \\cong \\Bl_{\\widetilde{X}} \\widetilde{Y}$.\n\\end{corollary}\n\nThere is a further generalisation of the Cayley trick, that applies to degeneracy loci as well, which we recall for completeness.\n\n\\begin{lemma}[{\\cite[Lemma 2.1]{kuznetsovKuchle}}]\n\\label{lem:blowDegeneracyLocus}\nLet $\\varphi:E \\rightarrow F$ be a morphism of vector bundles of ranks $r+1$, $r$ respectively on a Cohen--Macaulay variety $X$.\nDenote by $D_k(\\varphi)$ the $k$-th degeneracy locus of $\\varphi$, i.e., the locus where the morphism has corank at least $k$.\nConsider the projectivization $\\pi:\\mathbb{P}(E) \\rightarrow X$, then $\\varphi$ gives a global section of the vector bundle $\\pi^*F \\otimes \\mathcal{O}(1)$.\nIf $\\codim D_k(\\varphi) \\ge k + 1$ for all $k \\ge 1$ then the zero locus of $\\varphi$ on $\\mathbb{P}(E)$ is isomorphic to the blow up of $X$ along $D_1(\\varphi)$.\n\\end{lemma}\n\nIn practice, we will often need to find some projective bundle $\\mathbb{P}(\\mathcal{O}(-d_1,\\dotsc,-d_m) \\oplus \\mathcal{O}^{\\oplus r})$ as the zero locus of a suitable vector bundle over a product of Grassmannians. The following remark will be very helpful for this sake; an instance of its application will be Lemma \\ref{projBundle1-12}. \n\n\\begin{rmk}\n\\label{rem:principalParts} Let $L$ be a line bundle on $X$. For any $k$ one can define $\\mathcal{P}^k(L)$, the bundle of $k$-principal parts of $L$, of rank $\\binom{k+\\ddim X}{k}$. One has $\\mathcal{P}^0(L)=L$; by \\cite[Exp II, Appendix II 1.2.4.]{SGA} there exist natural short exact sequences\n\\begin{equation}\\label{seq:principalparts} 0 \\rightarrow \\Sym^k (\\Omega_X)(L) \\rightarrow \\mathcal{P}^k(L) \\rightarrow \\mathcal{P}^{k-1}(L) \\rightarrow 0. \\end{equation}\nIf $X \\cong \\mathbb{P}^n$ these bundles of principal parts are homogeneous, and in \\cite[Thm 1.1]{re} their splitting type is determined. The situation is particularly simple when we consider $L=\\mathcal{O}_{\\mathbb{P}^n}(d)$ with $d \\geq k$: in this case one has $\\mathcal{P}^k(\\mathcal{O}_{\\mathbb{P}^n}(d)) \\cong \\Sym^k V_{n+1} \\otimes \\mathcal{O}_{\\mathbb{P}^n}(d-k)$. The sequence above for $k=1$ coincides with the dualised twisted Euler sequence.\n\\end{rmk}\n\n\n\n\nWe finish this section with a classical remark on how to characterise double covers as hypersurfaces in projective bundles. A detailed proof can be found for example in \\cite[Lemma 1.2]{lyu}. The formula below can be easily generalised to the case of $k$-cyclic covers, using $\\mathcal{O}_P(k)$ instead.\n\n\\begin{rmk}\\label{lem:doublecovers}\nLet $X$ be a 2-fold cyclic covering of $Y$, ramified along a smooth divisor $D$, and $L$ a line bundle with $L^{\\otimes 2}=\\mathcal{O}_Y(D)$, which is assumed to be $2$-divisible in $\\mathrm{Pic}(Y)$. Then $X$ can be identified with $\\mathscr{Z}(\\mathcal{O}_P(2))$ in $P:=\\mathbb{P}_Y(\\mathcal{O} \\oplus L^{\\vee})$. \n\\end{rmk}\n\n\n\\section{How to compute invariants}\n\\label{computeinvariants}\nIn this section we explain and show with a concrete example how we can compute the invariants of a zero locus of a general section of a given homogeneous vector bundle on a product of flag varieties.\n\nAs a matter of fact, such computations are not strictly necessary for the identification of the models we found for the Fano 3-folds in the next section. However, we want to stress out the importance of having such a tool for two reasons. On the one hand, one could start producing in an automatised way many examples coming from homogeneous vector bundles on products of flag varieties and later try to identify them using the existing classifications. This was exactly the starting point of this project and what made us able to characterise, along the process, many zero loci of sections from a geometric point of view. Several results of Section \\ref{identifications} have been found by trying to generalise the evidences coming from all the examples we had. On the other hand, it goes without saying that these methods will certainly be very useful when a similar search will be performed for varieties which have not yet been classified.\n\n\\subsection{The invariants \\texorpdfstring{$h^0(-K)$ and $(-K)^3$}{h\\^{}0(-K) and (-K)\\^{}3}}\n\nThese invariants can be computed via intersection theory. If $X$ is a product of flag varieties, then we know its graded intersection ring of algebraic cycle classes modulo numerical equivalence. We know how to integrate, so that Hirzebruch--Riemann--Roch Theorem yields a way to compute $\\chi(E)$ for any vector bundle $E$ with assigned Chern classes.\n\nThe situation does not change much when we consider a subvariety $\\mathscr{Z}(\\mathcal{F}) \\subset X$ given as the zero locus of a general section of some vector bundle $\\mathcal{F}$ on $X$. If we know the Chern classes of $\\mathcal{F}$, we can write down the graded intersection ring of $\\mathscr{Z}(\\mathcal{F})$, as well as count points on $0$-dimensional cycles.\n\nIn concrete examples, instead of doing computations by hand, it is of course convenient to use some computer algebra software. Our choice fell on \\cite{M2}, for which an already developed package \\cite{Schubert2} implementing the methods we need is available. This allows us to compute $(-K_{\\mathscr{Z}(\\mathcal{F})})^3$, as the Chern classes of the canonical sheaf of $\\mathscr{Z}(\\mathcal{F})$ are easy to express. As for $h^0(-K_{\\mathscr{Z}(\\mathcal{F})})$, we certainly know how to compute $\\chi(-K_{\\mathscr{Z}(\\mathcal{F})})$. But $-K=K-2K$ and $-2K$ is ample, so the Kodaira Vanishing Theorem implies $h^{i>0}(-K_{\\mathscr{Z}(\\mathcal{F})})=0$.\n\n\\subsection{Hodge numbers and tangent cohomology}\n\nBeside the aforementioned invariants, one of the most important data one would like to know about a Fano variety is its Picard rank. More in general, it is rather important to compute $h^{i,j}$ for a given variety. In our setting this is perfectly doable using classical tools as the Koszul complex and a bit of representation theory, even though the computations may quickly become cumbersome if the involved vector bundles have high rank or several summands.\n\nWe briefly recall the strategy. Let us suppose that $Y=\\mathscr{Z}(\\mathcal{F})\\subset X$. Assume that $\\rank(\\mathcal{F})=r$. For each $j \\in \\mathbb{N}$, we have the $j$-th exterior power of the conormal sequence\n\\begin{equation}\n\\label{wedgeKConormal}\n0\\rightarrow\n\\Sym^j \\mathcal{F}^\\vee|_Y \\rightarrow\n(\\Sym^{j-1} \\mathcal{F}^\\vee \\otimes \\Omega_X)|_Y \\rightarrow\n\\dotso \\rightarrow\n(\\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X)|_Y \\rightarrow\n\\dotso \\rightarrow\n\\Omega^j_X|_Y \\rightarrow\n\\Omega^j_Y \\rightarrow\n0.\n\\end{equation}\nAs our goal is $h^i(\\Omega^j_Y)$, we can compute the dimensions of the cohomology groups of all the other terms in \\eqref{wedgeKConormal}, split it into short exact sequences and use the induced long exact sequences in cohomology to get the result.\n\nEach term $(\\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X)|_Y$ is in turn resolved by an exact Koszul complex\n\\begin{equation*}\n0\\rightarrow\n\\bigwedge^r \\mathcal{F}^\\vee \\otimes \\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X \\rightarrow\n\\dotso \\rightarrow\n\\mathcal{F}^\\vee \\otimes \\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X \\rightarrow\n\\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X,\n\\end{equation*}\nso that we are led to compute the cohomology groups of the terms above. If $X$ is a product of Grassmannians and $F$ is completely reducible, then those terms are completely reducible as well: a decomposition can be found via suitable plethysms. The cohomology groups can be then obtained via the usual Borel--Weil--Bott Theorem. \n\nThings get worse if $X$ has some genuine flag variety as a factor, in which case $\\Omega_X$ is an extension of completely reducible vector bundles, or if $F$ itself is an extension thereof. In these cases, one needs to deal carefully with the exterior\/symmetric power of an extension (which is an extension itself) and the tensor product of extensions; in the end, each term of the Koszul complex above is again an extension of completely reducible vector bundles, whose cohomology groups can be easily computed and arranged to get the result.\n\nIt may happen that several cohomology groups do not vanish, so that in the induced long exact sequences in cohomology there are boundary homomorphisms whose rank is a priori not known. This leads to some ambiguity in the final results, and can be partially solved by considering the additional relations involving $h^{i,j}$ such as the symmetries in the Hodge diamond and the computation of $\\chi(\\Omega^j_Y)$ as done above.\n\nAdditionally, suppose that we want to get some information on the automorphism group and the space of deformations of $Y$. One way is to compute $h^0(T_Y)$ and $h^1(T_Y)$ via the normal sequence\n\\[\n0 \\rightarrow\nT_Y \\rightarrow\nT_X|_Y \\rightarrow\n\\mathcal{F}|_Y \\rightarrow\n0.\n\\]\nAs before, one can compute the cohomology groups of the terms on the right via the usual Koszul complex and get some information on $h^i(T_Y)$.\n\nA rather easy example of application of the whole routine is provided in Section \\ref{aworkedexample}. It is evident that such computations cannot be done by hand for more complicated examples, especially for a significant number of cases. A Macaulay2 \\cite{M2} package which was developed to implement and automatise the procedure just described will be presented in \\cite{FatighentiTanturriPackage}.\n\n\\subsection{A worked example}\n\\label{aworkedexample}\n\nLet us show how to concretely compute the Hodge numbers of $Y:= \\mathscr{Z}(\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4)=:X$, which we will prove to be a model for \\hyperlink{Fano2--16}{2--16}.\n\nOur vector bundle $\\mathcal{F}:=\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)$ has rank $3$. For $j=0$, \\eqref{wedgeKConormal} simply becomes $\\mathcal{O}_Y \\rightarrow \\mathcal{O}_Y$. The Koszul complex resolving $\\mathcal{O}_Y$ is\n\\[\n0 \\rightarrow\n\\mathcal{O}(-2,-3) \\rightarrow\n\\mathcal{U}_{\\Gr(2,4)}(-1,-2) \\oplus \\mathcal{O}(-2,-1) \\rightarrow\n\\mathcal{O}(0,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,0) \\rightarrow\n\\mathcal{O};\n\\]\nthe only non-zero cohomology group is $H^0(\\mathcal{O})\\cong \\mathbb{C}$, which gives $h^{0,0}=1$ and $h^{0,j}=0$ for $j>0$.\n\nFor $j=1$, \\eqref{wedgeKConormal} yields the usual conormal short exact sequence. The term on the left is $\\mathcal{F}^\\vee|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee & = &\\mathcal{O}(-2,-5) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-3,-3) \\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee & = & \\mathcal{U}_{\\Gr(2,4)}(-1,-4) \\oplus \\Sym^2\\mathcal{U}_{\\Gr(2,4)}(-2,-2)\\oplus \\mathcal{O}(-2,-3)^{\\oplus 2} \\oplus\\mathcal{U}_{\\Gr(2,4)}(-3,-1) \\\\\n\\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee & = & \\mathcal{O}(0,-4) \\oplus \n\\mathcal{U}_{\\Gr(2,4)}(-1,-2)^{\\oplus 2} \\oplus \n\\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(-2,0) \\oplus \n\\mathcal{O}(-2,-1)\\\\\n\\mathcal{F}^\\vee & = & \\mathcal{O}(0,-2) \\oplus \n\\mathcal{U}_{\\Gr(2,4)}(-1,0);\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology group is $h^4(\\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee)=1$, which yields $h^3(\\mathcal{F}^\\vee|_Y)=1$.\n\nThe middle term is $\\Omega_X|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes \\Omega_X &= & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\mathcal{U}_{\\Gr(2,4)}(-2,-4) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(-4,-3)\n\\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\Omega_X & = & \n\\mathcal{Q}_{\\Gr(2,4)}\\otimes\\left(\\Sym^2\\mathcal{U}_{\\Gr(2,4)}(-1,-3)\n\\oplus \\mathcal{O}(-1,-4) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-2,-2)\n\\right) \\oplus \\\\ & & {}\\oplus\n \\mathcal{Q}_{\\mathbb{P}^2}(-4,-1) \\oplus \\mathcal{U}_{\\Gr(2,4)} \\otimes \\mathcal{Q}_{\\mathbb{P}^2}(-3,-2)\n\\\\\n\\mathcal{F}^\\vee \\otimes \\Omega_X & = & \n\\mathcal{Q}_{\\Gr(2,4)}\\otimes\\left(\\mathcal{U}_{\\Gr(2,4)}(0,-3)\n\\oplus \\Sym^2\\mathcal{U}_{\\Gr(2,4)}(-1,-1)\n\\oplus \\mathcal{O}(-1,-2)\n\\right) \\oplus \\\\ & & {}\\oplus\n \\mathcal{Q}_{\\mathbb{P}^2}(-2,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}\\otimes \\mathcal{Q}_{\\mathbb{P}^2}(-3,0)\\\\\n\\Omega_X & = & \\mathcal{Q}_{\\Gr(2,4)}\\otimes \\mathcal{U}_{\\Gr(2,4)}(0,-1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(-2,0);\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology groups are $h^3(\\mathcal{F}^\\vee \\otimes \\Omega_X)=1$ and $h^1(\\Omega_X)=2$, which yield $h^1(\\Omega_X|_Y)=2$ and $h^2(\\Omega_X|_Y)=1$.\n\nThe long exact sequence in cohomology induced by the conormal sequence then gives $h^{1,1}=2$ and $h^{1,2}=2$, while the other $h^{1,j}$ are zero.\n\n\nSimilar computations can be performed to compute $h^i(T_Y)$, by considering the normal sequence. The middle term is $T_X|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes T_X &= & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\mathcal{U}_{\\Gr(2,4)}(-2,-2) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(-1,-3) \n\\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes T_X & = & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\left( \n\\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(-1,-1) \\oplus \\mathcal{O}(-1,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-2,0)\n\\right) \\oplus \\\\ & & {}\\oplus\n\\mathcal{Q}_{\\mathbb{P}^2} \\otimes \\left( \n\\mathcal{U}_{\\Gr(2,4)}(0,-2) \\oplus \\mathcal{O}(-1,-1) \n\\right)\n\\\\\n\\mathcal{F}^\\vee \\otimes T_X & = & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\left(\n\\mathcal{U}_{\\Gr(2,4)}(0,-1) \\oplus\n\\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(-1,1) \\oplus\n\\mathcal{O}(-1,0)\n\\right) \\oplus \\\\ & & {}\\oplus\n\\mathcal{Q}_{\\mathbb{P}^2} \\otimes \\left( \n\\mathcal{O}(1,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}\n\\right)\n\\\\\nT_X & = & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\mathcal{U}_{\\Gr(2,4)}(0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(1,0)\n;\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology groups are \n$h^1(\\mathcal{F}^\\vee \\otimes T_X)=1$ and $h^0(T_X)=23$, which yield $h^0(T_X|_Y)=24$. Similarly, the term on the right is $\\mathcal{F}|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes \\mathcal{F} &= & \n\\mathcal{O}(-2,-1) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,-2)\n\\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\mathcal{F} & = & \n\\mathcal{U}_{\\Gr(2,4)}(-1,0) \\oplus \\mathcal{O}(-2,1) \\oplus \\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(0,-1) \\oplus \\mathcal{O}(0,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,0)\n\\\\\n\\mathcal{F}^\\vee \\otimes \\mathcal{F} & = & \n\\mathcal{O}^{\\oplus 2} \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(1,-1) \\oplus \\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(0,1)\n\\\\\n\\mathcal{F} & = & \n\\mathcal{O}(0,2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(1,1)\n;\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology groups are \n$h^2(\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\mathcal{F})=1$, $h^0(\\mathcal{F}^\\vee \\otimes \\mathcal{F})=2$, and $h^0(\\mathcal{F})=32$, which yield $h^0(\\mathcal{F}|_Y)=31$.\n\nThus, $h^1(T_Y)-h^0(T_Y)=31-24=7$, and indeed Fano \\hyperlink{Fano2--16}{2--16} is known to have a $7$-dimensional moduli space.\n\n\n\\section{Fano 3-folds as zero loci of sections}\n\\label{Fano3folds}\n\nIn this section a model for each Fano 3-fold as the zero locus of a general section of a vector bundle over a product of Grassmannians is given, provided that such a description is not available in the literature. For each model we prove the identification with the corresponding Fano; all the examples have been checked to have the right Hodge diamond and invariants as described in Section \\ref{computeinvariants}.\n\n\n\\hypertarget{Fano1--1}{\\subsection*{Fano 1--1}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^3$ with branch locus a divisor of degree 6.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(6)) \\subset \\mathbb{P}(1,1,1,1,3)$.\n\\subsubsection*{Identification} The obvious description as weighted hypersurface is classical. We want to give however another description embedded in a product of non-weighted Grassmannians.\n\nBy Lemma \\ref{lem:doublecovers}, we can express our Fano as the zero locus of $\\mathcal{O}(2)$ over $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-3) \\oplus \\mathcal{O})$. We thus need to express such projective bundle as the zero locus of a section of a suitable vector bundle. To do that, we adopt a general strategy which will be explained in more details for \\hyperlink{Fano1--12}{1--12} or \\hyperlink{Fano2--2}{2--2}: we start from the short exact sequences provided by Remark \\ref{rem:principalParts}\n\\begin{equation}\n\\begin{gathered}\n0 \\rightarrow\n\\mathcal{O}(-3) \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 4} \\rightarrow\n\\mathcal{Q}(-2) \\rightarrow\n0,\\\\\n0 \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 4} \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\Sym^2\\mathcal{Q}(-1) \\rightarrow\n0,\\\\\n\\label{thirdseq}\n0 \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\mathcal{O}^{\\oplus 20} \\rightarrow\n\\Sym^3\\mathcal{Q} \\rightarrow\n0.\n\\end{gathered}\n\\end{equation}\nWe can arrange the first two using the snake lemma as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-3) \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\Lambda \\rightarrow\n0\n\\]\nfor a uniquely defined extension $\\Lambda \\in \\Ext^1(\\Sym^2\\mathcal{Q}(-1),\\mathcal{Q}(-2))$. The latter sequence can be again arranged with the third one in \\eqref{thirdseq}, to get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-3) \\rightarrow\n\\mathcal{O}^{\\oplus 20} \\rightarrow\nK \\rightarrow\n0\n\\]\nfor another uniquely defined extension $K \\in \\Ext^1(\\Sym^3\\mathcal{Q},\\Lambda)$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to the above sequence, we get that our Fano can be expressed as\n\\[\n\\mathscr{Z}(K(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^{3} \\times \\mathbb{P}^{20}.\n\\]\n\n\\hypertarget{Fano1--12}{\\subsection*{Fano 1--12}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^3$ with branch locus a smooth quartic surface.\n\\subsubsection*{Our description} $\\mathscr{Z} (\\mathcal{O}(4)) \\subset \\mathbb{P}(1,1,1,1,2)$.\n\\subsubsection*{Identification}\nThe obvious description as weighted hypersurface is classical. We want to give however a rather simple description as a subvariety in a product of projective spaces. We notice that our Fano is, by Lemma \\ref{lem:doublecovers}, the zero locus of $\\mathcal{O}(2)$ on $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-2) \\oplus \\mathcal{O})$.\n\n\\begin{lemma}\n\\label{projBundle1-12}\nThe projective bundle $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-2) \\oplus \\mathcal{O})$ can be obtained as the zero locus of $\\Lambda(0,1)$ over $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^3$ fitting into sequence \\eqref{Lambda1--12} below.\n\\begin{proof}\nOur goal is to write $\\mathcal{O}_{\\mathbb{P}^3}(-2) \\oplus \\mathcal{O}_{\\mathbb{P}^3}$ as a subbundle of $\\mathcal{O}_{\\mathbb{P}^3}^{\\oplus 11}$. By Remark \\ref{rem:principalParts}, we have two (dual) canonical short exact sequences on $\\mathbb{P}^3$\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow \\mathcal{O}(-1)^{\\oplus 4} \\rightarrow \\mathcal{Q}(-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(-1)^{\\oplus 4} \\rightarrow \\mathcal{O}^{\\oplus 10} \\rightarrow \\Sym^2 \\mathcal{Q} \\rightarrow 0.\n\\end{gather*}\nThese fit as the first row and middle column of the exact diagram on $\\mathbb{P}^3$ here below, which can be completed by the snake lemma as\n\\begin{equation}\n\\label{snake1-12}\n\\begin{gathered}\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d]\\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d]^-= \\ar[r] & \\mathcal{O}(-1)^{\\oplus 4} \\ar[d] \\ar[r] & \\mathcal{Q}(-1) \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d] \\ar[r] & \\mathcal{O}^{\\oplus 10} \\ar[d] \\ar[r] & \\Lambda \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & 0 \\ar[r]\\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]^-= \\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]\\ar[d] & 0 \\\\\n& 0 & 0 & 0 \\\\\n}\n\\end{gathered}\n\\end{equation}\nfor a uniquely determined homogeneous vector bundle $\\Lambda$ of rank $9$. The last column describes $\\Lambda$ as a non-split extension\n\\begin{equation}\n\\label{Lambda1--12}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}(-1) \\rightarrow \\Lambda \\rightarrow \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^3} \\rightarrow 0.\n\\end{equation}\n\nThe rank $9$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^3, \\Sym^2 \\mathcal{Q})\\cong \\Sym^2 V_4$. Adding the middle row of \\eqref{snake1-12} to $\\mathcal{O} \\rightarrow \\mathcal{O}$, we get\n\\[\n0 \\rightarrow \\mathcal{O}(-2)\\oplus \\mathcal{O} \\rightarrow \\mathcal{O}^{\\oplus 11} \\rightarrow \\Lambda \\rightarrow 0,\n\\]\nwhence the conclusion of the lemma.\n\\end{proof}\n\\end{lemma}\n\nThe previous lemma yields that a model for \\hyperlink{Fano1--12}{1--12} is $\\mathscr{Z}(\\Lambda(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10}$.\n\n\n\\hypertarget{Fano2--2}{\\subsection*{Fano 2--2}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ with branch locus a $(2,4)$ divisor.\n\\subsubsection*{Our description} $\\mathscr{Z} (\\mathcal{O}(0,0,2) \\oplus K(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$, where $K \\in \\Ext^2(\\mathcal{O}(1,0)^{\\oplus 6},\\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ fits into sequences \\eqref{K2-2}.\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ over $\\mathbb{P}(\\mathcal{O}(-1,-2) \\oplus \\mathcal{O})$, the latter being a projective bundle on $\\mathbb{P}^1 \\times \\mathbb{P}^2$. We need to express such projective bundle as the zero locus of a suitable vector bundle. \n\n\\begin{lemma}\n\\label{projBundle2-2}\nThe projective bundle $\\mathbb{P}(\\mathcal{O}(-1,-2) \\oplus \\mathcal{O})$ can be obtained as the zero locus of $K(0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$, being $K \\in \\Ext^2(\\mathcal{O}(1,0)^{\\oplus 6},\\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequences \\eqref{K2-2} below.\n\\begin{proof}\nOur goal is to write $\\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(-1,-2) \\oplus \\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}$ as a subbundle of $\\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}^{\\oplus 13}$. By Remark \\ref{rem:principalParts}, we have two (dual) canonical short exact sequences on $\\mathbb{P}^2$\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow \\mathcal{O}(-1)^{\\oplus 3} \\rightarrow \\mathcal{Q}(-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(-1)^{\\oplus 3} \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\Sym^2 \\mathcal{Q} \\rightarrow 0.\n\\end{gather*}\nThese fit as the first row and middle column of the exact diagram on $\\mathbb{P}^2$ here below, which can be completed by the snake lemma as\n\\begin{equation}\n\\label{snake2-2}\n\\begin{gathered}\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d]\\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d]^-= \\ar[r] & \\mathcal{O}(-1)^{\\oplus 3} \\ar[d] \\ar[r] & \\mathcal{Q}(-1) \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d] \\ar[r] & \\mathcal{O}^{\\oplus 6} \\ar[d] \\ar[r] & \\Lambda \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & 0 \\ar[r]\\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]^-= \\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]\\ar[d] & 0\\\\\n& 0 & 0 & 0 \n}\n\\end{gathered}\n\\end{equation}\nfor a uniquely determined homogeneous vector bundle $\\Lambda$ of rank $5$. The last column describes $\\Lambda$ as a non-split extension\n\\begin{equation}\n\\label{LambdaProvv}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(-1) \\rightarrow \\Lambda \\rightarrow \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^2} \\rightarrow 0.\n\\end{equation}\n\nWe can pull back the middle row of \\eqref{snake2-2} on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ and twist it by $\\mathcal{O}(-1,0)$. This and the standard (pulled back) Euler sequence on $\\mathbb{P}^1$ can be inserted as the first row and second column in the exact diagram below, which can be again completed by the snake lemma as\n\\begin{equation}\n\\label{KSnake}\n\\begin{gathered}\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d]\\\\\n0 \\ar[r] & \\mathcal{O}(-1,-2) \\ar[d]^-= \\ar[r] & \\mathcal{O}(-1,0)^{\\oplus 6} \\ar[d] \\ar[r] & \\Lambda(-1,0) \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{O}(-1,-2) \\ar[d] \\ar[r] & \\mathcal{O}^{\\oplus 12} \\ar[d] \\ar[r] & K \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & 0 \\ar[r]\\ar[d] & \\mathcal{O}(1,0)^{\\oplus 6} \\ar[r]^-= \\ar[d] & \\mathcal{O}(1,0)^{\\oplus 6} \\ar[r]\\ar[d] & 0 \\\\\n& 0 & 0 & 0 \\\\\n}\n\\end{gathered}\n\\end{equation}\nfor a uniquely determined homogeneous vector bundle $K$ of rank $11$. We can further describe $K$ as an element of $\\Ext^2(\\mathcal{O}(1,0)^{\\oplus 6},\\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ obtained by combining the short exact sequences\n\\begin{equation}\n\\begin{gathered}\n\\label{K2-2}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(-1,-1) \\rightarrow \\Lambda(-1,0) \\rightarrow \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^2}(-1,0) \\rightarrow 0,\n\\\\\n0 \\rightarrow \\Lambda(-1,0) \\rightarrow K \\rightarrow \\mathcal{O}(1,0)^{\\oplus 6} \\rightarrow 0.\n\\end{gathered}\n\\end{equation}\n\nThe bundle $K$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^1 \\times \\mathbb{P}^2, \\mathcal{O}(1,0)^{\\oplus 6})$. Adding the middle row of \\eqref{KSnake} to $\\mathcal{O} \\rightarrow \\mathcal{O}$, we get\n\\[\n0 \\rightarrow \\mathcal{O}(-1,-2)\\oplus \\mathcal{O} \\rightarrow \\mathcal{O}^{\\oplus 13} \\rightarrow K \\rightarrow 0,\n\\]\nwhence the conclusion of the lemma.\n\\end{proof}\n\\end{lemma}\n\nBy construction, the bundle $\\mathcal{O}(2)$ on $\\mathbb{P}(\\mathcal{O}(-1,-2) \\oplus \\mathcal{O})$ is identified with $\\mathcal{O}(0,0,2)$ over the zero locus of $K$ on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$, whence the conclusion. \n\n\n\\hypertarget{Fano2--3}{\\subsection*{Fano 2--3}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano1--12}{1--12} in an elliptic curve which is the intersection of two divisors from $|-\\frac{1}{2}K|$.\n\\subsubsection*{Our description} $\\mathscr{Z} (\\mathcal{O}(4,0) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}(1,1,1,1,2) \\times \\mathbb{P}^1$.\n\n\\subsubsection*{Identification} The first bundle on $\\mathbb{P}(1,1,1,1,2)$ gives \\hyperlink{Fano1--12}{1--12}. We can conclude by Lemma \\ref{lem:blowup}.\n\nIt is possible to provide a rather simple description involving only projective spaces. To do this, recall that a model for \\hyperlink{Fano1--12}{1--12} is $\\mathscr{Z}(\\Lambda(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10}$. The adjunction formula shows that the canonical divisor is the restriction of $\\mathcal{O}(-2,0)$; by Lemma \\ref{lem:blowup}, a model for \\hyperlink{Fano2--3}{2--3} is therefore given by\n\\[\n\\mathscr{Z}(\\Lambda(0,1,0) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^1.\n\\]\n\n\\hypertarget{Fano2--5}{\\subsection*{Fano 2--5}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--13 in a plane cubic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,3) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification} The first bundle on $\\mathbb{P}^4$ gives 1--13. We conclude by Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano2--8}{\\subsection*{Fano 2--8}}\n\\subsubsection*{Mori-Mukai} Double cover of \\hyperlink{Fano2--35}{2--35} with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth.\n\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{12}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3},\\mathcal{Q}_{\\mathbb{P}^2}(0,-1))$ a uniquely defined extension on $\\mathbb{P}^2 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda2-8} below.\n\\subsubsection*{Identification}\nAs shown below, $Y:={}$ \\hyperlink{Fano2--35}{2--35} can be obtained as $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$. By Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ on $\\mathbb{P}_Y(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$.\n\nAs it turns out, the projective bundle $\\mathbb{P}_{\\mathbb{P}^2 \\times \\mathbb{P}^3}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ can be obtained as the zero locus of $\\Lambda(0,0,1)$ over $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{12}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3},\\mathcal{Q}_{\\mathbb{P}^2}(0,-1))$ a uniquely defined extension on $\\mathbb{P}^2 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda2-8} below. To see it, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) two (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 3} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(0,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 3} \\rightarrow \\mathcal{O}^{\\oplus 12} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{gather}\n\\label{inclusion2-8}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 12} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda2-8}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(0,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3} \\rightarrow 0,\n\\end{gather}\nwhere the rank $11$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^3, \n\\mathcal{Q}^{\\oplus 3}) \\cong (V_4)^{\\oplus 3}$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion2-8} we get the desired description for $\\mathbb{P}_{\\mathbb{P}^2 \\times \\mathbb{P}^3}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ and the conclusion.\n\n\n\\hypertarget{Fano2--10}{\\subsection*{Fano 2--10}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--14 in an elliptic curve which is an intersection of 2 hyperplanes.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1))\\subset \\Gr(2,4) \\times \\mathbb{P}^1$.\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano2--11}{\\subsection*{Fano 2--11}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--13 in a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1) \\oplus \\mathcal{O}(1,2)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow}, the zero locus of the first summand gives $\\Bl_{\\mathbb{P}^1}(\\mathbb{P}^4)$. Let $\\mathbb{P}^4=\\mathbb{P}(V_5)$ with dual coordinates $x_0, \\dotsc, x_4 \\in V_5^\\vee$. Assume that $\\mathbb{P}^1=\\mathbb{P}(V_2)$ is given by the vanishing of $x_2, \\dotsc, x_4$. A general section in $H^0(\\mathbb{P}^2 \\times \\mathbb{P}^4,\\mathcal{O}(1,2))$ is identified with a cubic in $\\Sym^3(V_5^\\vee)\/\\Sym^3(V_2^\\vee)$, i.e., a cubic without terms in $x_0^3, x_0^2x_1, x_0x_1^2,x_1^3$. Such cubic contains $\\mathbb{P}(V_2)$, hence the claim.\nNotice that, using the equivalent Corollary \\ref{cor:blowupflag}, we can describe \\hyperlink{Fano2--11}{2--11} as well as the zero locus $\\mathscr{Z}(\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(2,1)) \\subset \\Fl(1,3,5)$.\n\n\n\n\n\\hypertarget{Fano2--15}{\\subsection*{Fano 2--15}}\n\\subsubsection*{Mori-Mukai}Blow up of $\\mathbb{P}^3$ in the intersection of a quadric and a cubic where the quadric is smooth.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification}\n\nBy Lemma \\ref{lem:blowDegeneracyLocus}, our Fano is the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(2)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(-1)\\oplus \\mathcal{O}) \\rightarrow \\mathbb{P}^3$.\n\nAdding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to the standard Euler sequence on $\\mathbb{P}^3$ we get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-1) \\oplus \\mathcal{O} \\rightarrow\n\\mathcal{O}^{\\oplus 5} \\rightarrow\n\\mathcal{Q} \\rightarrow\n0,\n\\]\nwhence the result.\n\nAnother simple description of our Fano is\n\\begin{equation}\n\\label{anotherDescription}\n\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,2)) \\subset \\Fl(1,2,5);\n\\end{equation}\nthe two descriptions are equivalent thanks to the correspondence between subvarieties of flags and of products of Grassmannians given by Lemma \\ref{lem:identificationsOnFlags}, Remark \\ref{rmk:wisniewski}, and Lemma \\ref{lem:blow}. From these one can immediately identify $(\\mathscr{Z}(\\mathcal{Q}_2) \\subset \\Fl(1,2,V_5)) \\cong \\mathbb{P}_{\\mathbb{P}(V_5\/v_0)}(\\mathcal{O}(-1) \\oplus (v_0 \\otimes\\mathcal{O})) \\cong \\Bl_{\\mathbb{P}(v_0)} \\mathbb{P}(V_5)$ as $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}\\boxtimes \\mathcal{O}_{\\mathbb{P}^4}(1)) \\subset \\mathbb{P}(V_5\/v_0) \\times \\mathbb{P}((V\/v_0) \\oplus v_0)$. On the latter we have that $\\mathcal{O}(1,0) \\cong p^*\\mathcal{O}_{\\mathbb{P}^3}(1)$ and $\\mathcal{O}(0,1) \\cong \\pi^*\\mathcal{O}_{\\mathbb{P}^4}(1)$, where $p$ is the projective bundle map and $\\pi$ the blow up map.\n\nWe want to provide a direct way to describe our Fano as \\eqref{anotherDescription}, in order to show how the Cayley trick can be effectively used. First note that by Corollary \\ref{cor:onF12n} $X=\\mathscr{Z}(\\mathcal{Q}_2) \\subset F:=\\Fl(1,2,5)$ is identified with $\\mathbb{P}(E)=\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-1) \\oplus \\mathcal{O}) \\cong \\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-3) \\oplus \\mathcal{O}(-2)) $. To use Corollary \\ref{cor:cayleycrit} we want to show that \n\\[\nH^0(X, \\mathcal{O}_X(1,2)) \\cong H^0(\\mathbb{P}(E), \\mathcal{O}_{\\mathbb{P}(E)}(1)) \\cong H^0(\\mathbb{P}^3, \\mathcal{O}_{\\mathbb{P}^3}(2) \\oplus \\mathcal{O}_{\\mathbb{P}^3}(3)).\n\\]\nIn order to compute $H^0(X, \\mathcal{O}_X(1,2))$ we use the Koszul complex for $X \\subset F$ twisted by $\\mathcal{O}_{F}(1,2)$. The only non-zero cohomology groups are\n\\[\n\\begin{array}{ll}\nH^0(F, \\bigwedge^3 \\mathcal{Q}^{\\vee}_2 \\otimes \\mathcal{O}_F(1,2)) \\cong \\Sigma_{2,2,2,1}V_5 \\cong \\mathbb{C}^{40},&\nH^0(F, \\bigwedge^2 \\mathcal{Q}^{\\vee}_2 \\otimes \\mathcal{O}_F(1,2)) \\cong \\Sigma_{3,2,2,1}V_5 \\cong \\mathbb{C}^{175}\\\\\n\\rule{0pt}{12pt}\nH^0(F, \\mathcal{Q}^{\\vee}_2 \\otimes \\mathcal{O}_F(1,2)) \\cong \\Sigma_{3,3,2,1}V_5 \\cong \\mathbb{C}^{280},&\nH^0(F, \\mathcal{O}_F(1,2)) \\cong \\Sigma_{3,3,3,1}V_5 \\cong \\mathbb{C}^{175}.\n\\end{array}\n\\]\nAs in Lemma \\ref{lem:identificationsOnFlags}, $\\mathcal{U}_2|_X= \\overline{\\mathcal{U}_1} \\oplus \\mathcal{O}$: this is therefore equivalent to split $V_5 = V_4 \\oplus \\mathbb{C} v_0$, and apply the above Schur functors to a such decomposed $V_5$ to get $\\SL(4)\\times \\mathbb{C}^*$ representations, with the $\\mathbb{C}^*$ component being the trivial representation. As it turns out,\n\\[\n\\begin{array}{rcl}\n\\Sigma_{2,2,2,1}(V_4 \\oplus \\mathbb{C}) & = & \\Sigma_{2,2,1}V_4\\oplus\\Sigma_{2,2,2}V_4\\oplus\\Sigma_{2,2,1,1}V_4\\oplus\\Sigma_{2,2,2,1}V_4, \\\\\n\\rule{0pt}{12pt} \\Sigma_{3,2,2,1}(V_4 \\oplus \\mathbb{C}) & = & \\Sigma_{3,2,2,1}V_4\\oplus\\Sigma_{3,2,2}V_4\\oplus\\Sigma_{3,2,1,1}V_4\\oplus\\Sigma_{3,2,1}V_4\\oplus\\Sigma_{2,2,2,1}V_4\\oplus\\Sigma_{2,2,2}V_4\\oplus\\\\ & & {}\\oplus \\Sigma_{2,2,1,1}V_4\\oplus\\Sigma_{2,2,1}V_4,\\\\\n\\rule{0pt}{12pt} \\Sigma_{3,3,2,1}(V_4 \\oplus \\mathbb{C})& = &\\Sigma_{3,3,2,1}V_4\\oplus\\Sigma_{3,3,2}V_4\\oplus\\Sigma_{3,3,1,1}V_4\\oplus\\Sigma_{3,3,1}V_4\\oplus\\Sigma_{3,2,2,1}V_4\\oplus \\Sigma_{3\n ,2,2}V_4\\oplus\\\\ & &{}\\oplus\\Sigma_{3,2,1,1}V_4\\oplus\\Sigma_{3,2,1}V_4,\\\\\n\\rule{0pt}{12pt} \\Sigma_{3,3,3,1}(V_4 \\oplus \\mathbb{C})& = &\\Sigma_{3,3,3,1}V_4\\oplus\\Sigma_{3,3,3}V_4\\oplus\\Sigma_{3,3,2,1}V_4\\oplus\\Sigma_{3,3,2}V_4\\oplus\\Sigma_{3,3,1,1}V_4\\oplus\\Sigma_{3\n ,3,1}V_4.\n\\end{array}\n\\]\n\n\nTherefore, splitting the Koszul complex in short exact sequences, we get the natural isomorphism (of vector spaces)\n\\[H^0(X, \\mathcal{O}_X(1,2)) \\cong \\Sigma_{2,2,2}V_4 \\oplus \\Sigma_{3,3,3}V_4 \\cong \\Sym^2 V_4^{\\vee} \\oplus \\Sym^3 V_4^{\\vee} \\cong H^0(\\mathbb{P}^3, \\mathcal{O}_{\\mathbb{P}^3}(2) \\oplus \\mathcal{O}_{\\mathbb{P}^3}(3)),\\]\nas claimed. It suffices to use Corollary \\ref{cor:cayleycrit} to show that $X$ coincides with the Mori--Mukai description as the blow up of $\\mathbb{P}^3$ in the complete intersection of a quadric and a cubic surfaces.\n\n\n\\hypertarget{Fano2--16}{\\subsection*{Fano 2--16}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--14 in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,4)$.\n\\subsubsection*{Identification} Let $Y=\\mathscr{Z}(\\mathcal{O}_F(0,2)) \\subset \\Fl(1,2,4)$ and $\\widetilde{Y}= \\mathscr{Z}(\\mathcal{O}_G(2)) \\subset \\Gr(2,4)$. One directly checks that \\[H^0(Y, \\mathcal{O}_Y(1,0)) \\cong H^0(\\widetilde{Y}, \\overline{\\mathcal{U}}^{\\vee}|_{\\widetilde{Y}}).\\]\nIn fact, both spaces can be naturally identified with $V_4^{\\vee}$, as in \\hyperlink{Fano2--15}{2--15}. Then it suffices to apply Corollary \\ref{cor:cayleycrit} to get that $X=\\mathscr{Z}(\\mathcal{O}_Y(1,0)) \\subset Y \\cong \\Bl_{\\mathscr{Z}( \\overline{\\mathcal{U}}^{\\vee}|_{\\widetilde{Y}})}\\widetilde{Y}$, where we used that by duality on $\\Gr(2,4)$, $\\mathcal{U}(1) \\cong \\mathcal{U}^{\\vee} \\cong (\\pi_2)_{*} \\mathcal{O}_F(1,0)$. We conclude the proof by noting that $\\widetilde{Y}$ is a complete intersection of two quadrics in $\\mathbb{P}^5$, and $(\\mathscr{Z}( \\overline{\\mathcal{U}}^{\\vee}|_{\\widetilde{Y}}) \\subset \\widetilde{Y}) = \\mathscr{Z}(\\mathcal{U}^{\\vee} \\oplus \\mathcal{O}_G(2)) \\subset \\Gr(2,4)$ which is a plane conic.\n\nWe want to give an alternative description of this Fano in the product of two Grassmannians. For this, let us start by the Mori--Mukai description. Lemma \\ref{lem:blowInGrass} enables us to describe $\\Bl_{\\mathbb{P}^2} \\Gr(2,4)$ in the product $(\\mathbb{P}^2)^{\\vee} \\times \\Gr(2,4)$. We then need to cut with an extra quadric intersecting the blown up $\\mathbb{P}^2$. As we are going to see in full details for \\hyperlink{Fano2--26}{2--26}, for this it suffices to take a section of $\\mathcal{O}(0,2)$. Summarising, we can describe our \\hyperlink{Fano2--16}{2--16} as \n\\[ \\mathscr{Z}(\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4).\n\\]\n\\hypertarget{Fano2--17}{\\subsection*{Fano 2--17}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in an elliptic curve of degree 5.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1) \\oplus \\mathcal{O}(1,1)) \\subset \\Fl(1,2,4)$.\n\\subsubsection*{Identification}\nA model for this 3-fold in $\\Gr(2,4) \\times \\mathbb{P}^3$ can be found in \\cite{corti}. As an exception to our self-imposed rule, we want to give here an alternative description in a flag variety, since we find it particularly nice. Let us show that our Fano can be written as\n\\[\n\\mathscr{Z}(\\mathcal{O}(1,1) \\oplus \\mathcal{O}(0,1)) \\subset \\Fl(1,2,4).\n\\]\n\n As before, we check that \\[H^0(Y, \\mathcal{O}_Y(1,1)) \\cong H^0(\\widetilde{Y}, \\overline{\\mathcal{U}}^{\\vee}(1)|_{\\widetilde{Y}}),\\]\nwhere we are using the same notation as above. These spaces are both $16$-dimensional and isomorphic as vector spaces to $\\Sigma_{2,1}V_4^{\\vee}\/V_4^{\\vee}$, where we can interpret $\\Sigma_{2,1}V_4$ as the kernel of the natural contraction map $\\lrcorner: V_4 \\otimes \\bigwedge^2 V_4^{\\vee} \\rightarrow V_4^{\\vee}$. These spaces of sections are not $\\SL(V_4)$-representations: in fact $\\widetilde{Y}$ (and similarly for the section on the flag) is not homogeneous for the whole group, but rather for $\\SO(V_3)$, and one could write a more elegant expression for the spaces of section as in \\hyperlink{Fano2--15}{2--15}. To conclude we apply Corollary \\ref{cor:cayleycrit}: we have that $X=\\mathscr{Z}(\\mathcal{O}_Y(1,1)) \\cong \\Bl_{\\widetilde{Z}} \\widetilde{Y}$ where $\\widetilde{Y}$ is a quadric 3-fold, and the centre of the blow up is $\\widetilde{Z}= \\mathscr{Z}(\\mathcal{U}^{\\vee}(1) \\oplus \\mathcal{O}(1)) \\subset \\Gr(2,4)$, which can be easily checked to be an elliptic curve of degree 5.\n\n\\hypertarget{Fano2--18}{\\subsection*{Fano 2--18}}\n\\subsubsection*{Mori-Mukai} Double cover of 2-34 with branch locus a divisor of degree $(2,2)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18} below.\n\n\\subsubsection*{Identification}\n\nBy Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ on $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$. As it turns out, the latter projective bundle can be obtained as the zero locus of $\\Lambda(0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18} below. To see it, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) two (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{gather}\n\\label{inclusion2-18}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda2-18}\n0 \\rightarrow \\mathcal{O}(1,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2} \\rightarrow 0,\n\\end{gather}\nwhere the rank $5$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^2, \n\\mathcal{Q}^{\\oplus 2}) \\cong (V_3)^{\\oplus 2}$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion2-18} we get the desired description for $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ and the conclusion.\n\n\\hypertarget{Fano2--19}{\\subsection*{Fano 2--19}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--14 in a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)^{\\oplus 2} ) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^5$. \n\n\\subsubsection*{Identification} It suffices to apply Lemma \\ref{lem:blow} and argue as done for \\hyperlink{Fano2--11}{2--11}. The zero locus of the first factor identifies $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1))$ with the blow up $\\Bl_{\\mathbb{P}^1}(\\mathbb{P}^5)$. Let $\\mathbb{P}^5=\\mathbb{P}(V_6)$ with dual coordinates $x_0, \\dotsc, x_5 \\in V_6^\\vee$. Assume that $\\mathbb{P}^1=\\mathbb{P}(V_2)$ is given by the vanishing of $x_2, \\dotsc, x_5$. A general section in $H^0(\\mathbb{P}^3 \\times \\mathbb{P}^5,\\mathcal{O}(1,1)^2)$ is identified with two quadrics in $\\Sym^2(V_6^\\vee)\/\\Sym^2(V_2^\\vee)$, i.e., quadrics without terms in $x_0^2, x_1^2, x_0x_1$. Such quadrics have generically maximal rank, so their intersection is smooth and contains $\\mathbb{P}(V_2)$, hence the claim.\nNotice that, using the equivalent Corollary \\ref{cor:blowupflag}, we can describe \\hyperlink{Fano2--19}{2--19} as the zero locus of \n$\\mathscr{Z}(\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(1,1)^{\\oplus 2}) \\subset \\Fl(1,3,6)$ as well.\n\n\n\n\\hypertarget{Fano2--22}{\\subsection*{Fano 2--22}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{V}_5$ in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\Gr(2,5)}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}) \\subset \\mathbb{P}^3 \\times \\Gr(2,5).$\n\\subsubsection*{Identification} In \\cite{corti} this variety is described as $\\mathscr{Z}(\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}) \\subset \\Fl(1,2,5)$. This description is equivalent to the one given here simply applying Lemma \\ref{lem:blowInGrass} with $k=3$ (where we identify $\\Gr(3,4)$ and $\\Gr(3,5)$ with $\\mathbb{P}^3$ and $\\Gr(2,5)$). The three residual sections of $\\mathcal{O}(0,1)$ cut both $\\Gr(2,5)$ (in $\\mathbb{V}_5$) and $\\Gr(2,4)$ (in a conic).\n\n\n\\hypertarget{Fano2--23}{\\subsection*{Fano 2--23}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in an intersection of a hyperplane and a quadric.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1) \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,6)$.\n\n\\subsubsection*{Identification} We apply Corollary \\ref{cor:cayleycrit}. In the notation of the corollary, we denote by $Y \\subset \\Fl(1,2,6)$ the zero locus of $\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2)$. We identify $\\widetilde{Y}$ with a three dimensional quadric $Q \\subset \\mathbb{P}^4$. What we need to check is \\[H^0(Y, \\mathcal{O}_Y(1,1)) \\cong H^0(Q, \\mathcal{O}_Q(1)) \\oplus H^0(Q, \\mathcal{O}_Q(2)). \\]\n\nTo verify this, one can argue as for \\hyperlink{Fano2--15}{2--15}: one can compute the $\\SL(V_6)$-representations arising from the Koszul complex resolving $\\mathcal{O}_Y(1,1)$. These representations, when seen as $\\SL(V_5)\\times \\mathbb{C}^*$-representations under the splitting $V_6 = V_5 \\oplus \\mathbb{C} v_0$, sum up to $\\Sigma_{1,1,1,1}V_5 \\oplus \\Sigma_{2,2,2,2}V_5\/ \\mathbb{C}$, which is clearly isomorphic to the right hand side.\n\nTherefore $X \\cong \\Bl_{\\widetilde{Z}} Q$, where $\\widetilde{Z}$ is given by the intersection of a quadratic and linear forms in $Q$.\n\nWe provide the following alternative description for this Fano:\n\\[ \\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^4}(0,1) \\oplus \\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^4 \\times \\mathbb{P}^5,\n\\]\nwhich can be shown to be equivalent to the previous one following the same lines of \\hyperlink{Fano2--15}{2--15}.\n\n\n\\hypertarget{Fano2--26}{\\subsection*{Fano 2--26}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{V}_5$ in a curve of genus 0.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{2,4} \\boxtimes \\mathcal{U}^{\\vee}_{2,5} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}) \\subset \\Gr(2,4) \\times \\Gr(2,5).$\n\n\\subsubsection*{Identification} \nBy Lemma \\ref{lem:blowInGrass} we can identify $\\mathscr{Z}(\\mathcal{Q}_{2,4} \\boxtimes \\mathcal{U}^{\\vee}_{2,5}) \\subset \\Gr(2,4) \\times \\Gr(2,5)$ as the blow up $\\Bl_{\\mathbb{P}^3} \\Gr(2,5)=\\Gr(2,V_5)$, where $\\mathbb{P}^3$ is identified with $\\mathscr{Z}(\\mathcal{Q}) \\subset \\Gr(2,V_5)$, given by a vector $w \\in V_5^\\vee$.\n\nWithout loss of generality, we can assume $w=x_0$. We have a splitting $V_5^{\\vee}= x_0 \\oplus W_4$ that induces a splitting $\\bigwedge^2 V_5^{\\vee} = \\bigwedge^2 W_4 \\oplus x_0 \\wedge W_4$. For simplicity, let us fix a basis $x_0,\\dotsc,x_4$ of $V_5^{\\vee}$ and the corresponding dual basis $e_0,\\dotsc,e_4$ of $V_5$. The above $\\mathbb{P}^3$ is by definition described by the points in $\\Gr(2,5)$ of the form $e_0 \\wedge \\alpha$, where $\\alpha \\in \\langle e_1,\\dotsc,e_4 \\rangle$.\n\nBy construction, any $f \\in \\bigwedge^2 W_4=|\\mathcal{O}(1,0)|$ does not contain any summand of the form $x_0 \\wedge \\beta$, so that $f(e_0 \\wedge \\alpha)=0$. In other words, $f \\in \\Ann(\\mathbb{P}^3)$, hence its zero locus in $\\Bl_{\\mathbb{P}^3} \\Gr(2,5)$ contains the whole exceptional divisor and does not cut it. The two extra sections of $\\mathcal{O}(0,1)$ cut the exceptional divisor in a codimension two linear subspace. Therefore our zero locus can be seen as the blow up of $\\mathscr{Z}(\\mathcal{O}_{\\Gr(2,V_5)}^{\\oplus 3}(1))\\subset \\Gr(2,V_5)$ along $\\mathbb{P}^1 \\cong \\mathscr{Z}(\\mathcal{O}_{\\mathbb{P}^3}^{\\oplus 2}(1))\\subset \\mathbb{P}^3$.\n\n\n\n\n\nAnother description of this Fano is as $\\mathscr{Z}(\\mathcal{U}_1^{\\vee} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}) \\subset \\Fl(2,3,5)$. By Lemma \\ref{lem:identificationsOnFlags} this can be easily identified with the alternative description of this Fano given in \\cite{corti}.\n\n\n\\hypertarget{Fano2--28}{\\subsection*{Fano 2--28}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in a plane cubic.\n\\subsubsection*{Our description} \n$\\mathscr{Z}(\\Lambda(0,1) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10}$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^3$ fitting into sequence \\eqref{Lambda1--12} above.\n\n\\subsubsection*{Identification}\nOur Fano variety is the blow up of $\\mathbb{P}^3$ along the intersection of two divisors of degree $1$ and $3$. By Lemma \\ref{lem:blowDegeneracyLocus}, it corresponds to the zero locus of $\\pi^*\\mathcal{O}_{\\mathbb{P}^3}(1) \\otimes \\mathcal{O}(1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-2)\\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^3$. We conclude by Lemma \\ref{projBundle1-12}.\n\n\n\\hypertarget{Fano2--29}{\\subsection*{Fano 2--29}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano2--30}{\\subsection*{Fano 2--30}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1)) \\subset \\Fl(1,2,5)$.\n\n\\subsubsection*{Identification} We apply Corollary \\ref{cor:cayleycrit}. Following the notation of the corollary, we denote by $Y \\subset \\Fl(1,2,5)$ the zero locus of $\\mathcal{Q}_2$ and we identify $\\widetilde{Y}$ with a $\\mathbb{P}^3$. What we need to check is \\[H^0(Y, \\mathcal{O}_Y(1,1)) \\cong H^0(\\mathbb{P}^3, \\mathcal{O}_{\\mathbb{P}^3}(1)\\oplus \\mathcal{O}_{\\mathbb{P}^3}(2)). \\]\n\nTo verify this, one can argue as for \\hyperlink{Fano2--15}{2--15} or \\hyperlink{Fano2--23}{2--23}: the representations arising from the Koszul complex resolving $\\mathcal{O}_Y(1,1)$, when seen as $\\SL(V_4)\\times \\mathbb{C}^*$-representations, sum up to $\\Sigma_{1,1,1}V_4 \\oplus \\Sigma_{2,2,2}V_4$, which is clearly isomorphic to the right hand side.\n\nNotice that as an alternative description we can follow the same lines of \\hyperlink{Fano2--15}{2--15} and describe the Fano \\hyperlink{Fano2--30}{2--30} as \n\\[\n\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4.\n\\]\n\\hypertarget{Fano2--31}{\\subsection*{Fano 2--31}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in a line.\n\n\\subsubsection*{Our description} $\\mathscr{Z}( \\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,1)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4) $.\n\\subsubsection*{Identification}\nWe may regard $\\mathbb{P}^2$ as $\\Gr(2,3)$, so that our Fano is given as $\\mathscr{Z}(\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee} \\oplus \\mathcal{O}(0,1))$. Then we argue as for \\hyperlink{Fano2--26}{2--26}. By Lemma \\ref{lem:blowInGrass} we can identify $\\mathscr{Z}(\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee}) \\subset \\Gr(2,3) \\times \\Gr(2,4)$ as $\\Bl_{\\mathbb{P}^2} \\Gr(2,4)$, where $\\mathbb{P}^2$ is identified with $\\mathscr{Z}(\\mathcal{Q}) \\subset \\Gr(2,4)$. As shown for \\hyperlink{Fano2--26}{2--26}, the remaining section of $\\mathcal{O}(0,1)$ cuts such $\\mathbb{P}^2$ in a codimension one linear subspace and the ambient $\\Gr(2,4)$ in a three-dimensional quadric, hence the conclusion.\n\n\n\\hypertarget{Fano2--33}{\\subsection*{Fano 2--33}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano2--35}{\\subsection*{Fano 2--35}}\n\\subsubsection*{Mori-Mukai} $\\Bl_p \\mathbb{P}^3$ or $\\mathbb{P}_{\\mathbb{P}^2}(\\mathcal{O} \\oplus \\mathcal{O}(-1))$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification}\nThis is a straightforward application of Lemma \\ref{lem:blow}. Notice that equivalently we could describe \\hyperlink{Fano2--35}{2--35} as $\\mathscr{Z}(\\mathcal{Q}_2) \\subset \\Fl(1,2,4)$.\n\n\\hypertarget{Fano2--36}{\\subsection*{Fano 2--36}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}_{\\mathbb{P}^2}(\\mathcal{O} \\oplus \\mathcal{O}(-2))$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}.\n\n\\subsubsection*{Identification}\nWe argue as in Lemma \\ref{projBundle1-12}, with the appropriate changes. By Remark \\ref{rem:principalParts}, we have two (dual) canonical short exact sequences on $\\mathbb{P}^2$\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow 3\\mathcal{O}(-1) \\rightarrow \\mathcal{Q}(-1) \\rightarrow 0,\\\\\n0 \\rightarrow 3\\mathcal{O}(-1) \\rightarrow 6\\mathcal{O} \\rightarrow \\Sym^2 \\mathcal{Q} \\rightarrow 0.\n\\end{gather*}\nWe combine them and get\n\\begin{gather}\n\\label{inclusion2-36}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda2-36}\n0 \\rightarrow \\mathcal{Q}(-1) \\rightarrow \\Lambda \\rightarrow \\Sym^2 \\mathcal{Q}\\rightarrow 0,\n\\end{gather}\nwhere the rank $5$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^2, \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^2}) \\cong \\Sym^2 V_3$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion2-36} we get the desired description for $\\mathbb{P}(\\mathcal{O}(-2) \\oplus \\mathcal{O})$.\n\n\n\\hypertarget{Fano3--1}{\\subsection*{Fano 3--1}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ with branch locus a divisor of degree $(2,2,2)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(K(0,0,0,1) \\oplus \\mathcal{O}(0,0,0,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$, where the bundle $K$ is a uniquely defined extension in $\\Ext^2(\\mathcal{O}(0,0,1)^{\\oplus 4},\\mathcal{O}(1,-1,-1))$ on $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into the chain of extensions \\eqref{K3-1} below.\n\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ on the projective bundle $\\mathbb{P}_{\\mathbb{P}^1\\times\\mathbb{P}^1\\times \\mathbb{P}^1}(\\mathcal{O}(-1,-1,-1) \\oplus \\mathcal{O})$. As it turns out, the latter projective bundle can be obtained as the zero locus of $K(0,0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$, being $K \\in \\Ext^2(\\mathcal{O}(0,0,1)^{\\oplus 4},\\mathcal{O}(1,-1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into \\eqref{K3-1} below. \n\nTo see it, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) three (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1,-1) \\rightarrow \\mathcal{O}(0,-1,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(0,0,-1)^{\\oplus 4} \\rightarrow \\mathcal{O}(0,1,-1)^{\\oplus 2} \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,0,-1)^{\\oplus 4} \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\mathcal{O}(0,0,1)^{\\oplus 4} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{equation}\n\\label{Inclusion3-1}\n0 \\rightarrow \\mathcal{O}(-1,-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow K \\rightarrow 0,\n\\end{equation}\nwhere the rank $7$ bundle $K$, fitting into the chain of extension \\eqref{K3-1}, is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^1\\times\\mathbb{P}^1 \\times \\mathbb{P}^1, \n\\mathcal{O}(0,0,1)^{\\oplus 4}) \\cong (V_2)^{\\oplus 4}$. \n\\begin{equation}\n\\begin{gathered}\n\\label{K3-1}\n0 \\rightarrow \\mathcal{O}(1,-1,-1) \n\\rightarrow \\Lambda \\rightarrow \\mathcal{O}(0,1,-1)^{\\oplus 2} \\rightarrow 0,\n\\\\\n0 \\rightarrow \\Lambda \\rightarrow K \\rightarrow \\mathcal{O}(0,0,1)^{\\oplus 4} \\rightarrow 0.\n\\end{gathered}\n\\end{equation}\n\nAdding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{Inclusion3-1} and from the previous considerations we get the conclusion.\n\n\n\\hypertarget{Fano3--2}{\\subsection*{Fano 3--2}}\n\\subsubsection*{Mori-Mukai} A divisor from $|\\mathcal{O}(2) \\otimes \\pi^*\\mathcal{O}(0,1)|$ on the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2} ) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^1$ such that $X \\cap Y$ is irreducible, where $X$ is the Fano itself and $Y \\in |\\mathcal{O}(1)|$.\n\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^5$, being $\\Lambda \\in \\Ext^1(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into \\eqref{Lambda3-2} below.\n\n\n\\subsubsection*{Identification} \nWe need to find $\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2} )$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1$. To do that, we argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the two Euler exact sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 4} \\rightarrow \\mathcal{O}(0,1)^{\\oplus 2} \\rightarrow 0,\n\\end{gather*}\nand get\n\n\n\n\\begin{gather}\n\\label{inclusion3-2}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 4} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda3-2}\n0 \\rightarrow \\mathcal{O}(1,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{O}(0,1)^{\\oplus 2} \\rightarrow 0.\n\\end{gather}\nwhere the rank $3$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^1 \\times \\mathbb{P}^1, \\mathcal{O}(0,1)^{\\oplus 2}) \\cong V_2^{\\oplus 2}$. Adding $\\mathcal{O}^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 2}$ to \\eqref{inclusion3-2} we get the desired description for $\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2})$ and the conclusion.\n\n\n\n\\hypertarget{Fano3--4}{\\subsection*{Fano 3--4}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--18}{2--18} in a smooth fiber of the composition of the double cover projection to $\\mathbb{P}^1 \\times \\mathbb{P}^2$ with the projection to $\\mathbb{P}^2$.\n\\subsubsection*{Our description}$\\mathscr{Z}(\\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,0,2,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1$, where the bundle $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ is a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18}.\n\\subsubsection*{Identification} The first two bundles define $Y \\times \\mathbb{P}^1$, being $Y \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$ the Fano \\hyperlink{Fano2--18}{2--18}. The curve on $Y$ we need to blow up is a complete intersection of two $(0,1,0)$ divisors, which cut in $Y$ the preimage of a $\\mathbb{P}^1$-fiber of the projection $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\rightarrow \\mathbb{P}^2$. We therefore conclude by Lemma \\ref{lem:blowup}.\n\n\n\n\n\n\\hypertarget{Fano3--5}{\\subsection*{Fano 3--5}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in a curve $C$ of degree $(5,2)$ such that $C \\hookrightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2 \\rightarrow \\mathbb{P}^2$ is an embedding.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,1)^{\\oplus 2}) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^7$, with $\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ fitting into \\eqref{Lambda2-18}.\n\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:blowDegeneracyLocus} and Lemma \\ref{lem:expectedRes} below, our Fano is the zero locus of $\\pi^* (\\mathcal{O}(0,1)^{\\oplus 2}) \\otimes \\mathcal{O}(1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2}) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2$. We thus need to find the latter projective bundle as the zero locus of a suitable vector bundle.\n\nA straightforward modification of the argument used for \\hyperlink{Fano2--18}{2--18} provides the desired description: adding $\\mathcal{O}^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 2}$ to \\eqref{inclusion2-18} we get\n\\[\n0 \\rightarrow \\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\Lambda \\rightarrow 0,\n\\]\nwhere $\\Lambda$ fits into \\eqref{Lambda2-18}. The conclusion follows as soon as we have proved the following lemma.\n\n\\begin{lemma}\\label{lem:expectedRes}\nThe ideal sheaf of a general rational curve $C$ of bidegree $(5,2)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^2$ admits a locally free resolution of the form\n\\begin{equation}\n\\label{shaperesolution}\n0 \\rightarrow\n\\mathcal{O}(-1,-4)^{\\oplus 2} \\rightarrow\n\\mathcal{O}(0,-2) \\oplus \\mathcal{O}(-1,-3)^{\\oplus 2} \\rightarrow\n\\mathcal{I}_C \\rightarrow 0\n\\end{equation}\nand, conversely, a general $3 \\times 2$ matrix as above yields a presentation for the ideal sheaf of a general curve $C$.\n\\begin{proof}\nThe aim is to show, on the one hand, that the above resolution is the simplest (in terms of Betti numbers) such a curve is expected to have. On the other hand, if we manage to show that a curve having that resolution exists, a semicontinuity argument yields that a general curve shares the same behaviour.\n\nThe first task requires a bit of commutative algebra, which we specialise to our setting $\\mathbb{P}:=\\mathbb{P}^1 \\times \\mathbb{P}^2$. Let $R:=\\oplus_{(a,b) \\in \\mathbb{Z}^2}H^0(\\mathbb{P},\\mathcal{O}(a,b))$ be the Cox ring of $\\mathbb{P}$. If $I_C$ denotes the ideal of $C$, which can be seens as a finitely generated $R$-module, we have a multigraded minimal free resolution\n\\[\n0 \\rightarrow F_r \\rightarrow \\dotso \\rightarrow F_0 \\rightarrow I_C \\rightarrow 0,\n\\]\nwhere the $F_i$ are finitely generated free modules $F_i = \\oplus_{(a,b) \\in \\mathbb{Z}^2}R(-a,-b)^{\\oplus \\beta_{i,(a,b)}}$, being $\\beta_{i,(a,b)}$ the so-called multigraded Betti numbers, which are independent of the chosen resolution.\n\nThe so-called multigraded Hilbert series of $I_C$ is the formal Laurent series\n\\[\nH_{I_C}:=\\sum_{(a,b) \\in \\mathbb{Z}^2} \\dim_{\\mathbb{C}}(I_C)_{(a,b)}\\cdot s^a t^b,\n\\]\nwhich is well-known to encode the Betti numbers $\\beta_{i,(a,b)}$ in the following way: it factors as\n\\[\nH_{I_C}=\\frac{\n\\sum_{(a,b) \\in \\mathbb{Z}^2}\\left( \\sum_{i=0}^r (-1)^{i} \\beta_{i,(a,b)}\\right) \\cdot s^a t^b\n}{(1-s)^2(1-t)^3}.\n\\]\n\nBy Riemann--Roch we can compute $H^0(C,\\mathcal{O}_{C}(a,b))$ for any $(a,b) \\in \\mathbb{Z}^2$; if we assume that $C$ has maximal rank, i.e., that $H^0(\\mathbb{P},\\mathcal{O}_{\\mathbb{P}}(a,b)) \\rightarrow H^0(C,\\mathcal{O}_{C}(a,b))$ has maximal rank for all $(a,b) \\in \\mathbb{Z}^2$, then we explicitly have $\\dim_{\\mathbb{C}}(I_C)_{(a,b)}$ and $H_{I_C}$. Straightforward computations then show that the numerator of $H_{S\/I_C}$ is $t^2 + 2st^3 -2st^4$; thus, the expected resolution of $I_C$ has the shape \\eqref{shaperesolution}.\n\nTo conclude, it suffices to show the existence of a curve with the right genus and degree having the desired resolution. This task can be rather difficult, depending on the given invariants: on $\\mathbb{P}^1 \\times \\mathbb{P}^2$, different approaches can be adopted, such as liaison theory or the construction of the Hartshorne--Rao module of the curve, see, e.g., \\cite{KeneshlouTanturri1,KeneshlouTanturri2}. Our situation, however, is favourable, as the minors of a general matrix\n\\[\n\\mathcal{O}(-1,-4)^{\\oplus 2} \\rightarrow\n\\mathcal{O}(0,-2) \\oplus \\mathcal{O}(-1,-3)^{\\oplus 2}\n\\]\ngenerate the ideal of a smooth curve of maximal rank with the desired invariants. This can be checked via any computer algebra software like \\cite{M2}.\n\\end{proof}\n\\end{lemma}\n\n\nIf we consider the normal sequence for $Y=\\mathscr{Z}(\\mathcal{F}) \\subset \\mathbb{P}:=\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^7$, a few cohomology computations via the Koszul complex as described in Section \\ref{computeinvariants} provide that $h^0(T_{\\mathbb{P}}|_Y)=74, h^0(\\mathcal{F}|_Y)=79$ and the higher cohomology groups vanish. In \\cite[Corollary 8.8]{pcs} it is shown that the family of Fano \\hyperlink{Fano3--5}{3--5} has a unique member with infinite automorphism group. This means that a general model $Y$ admits a $(79-74=5)$-dimensional family of deformations, which is the dimension of the moduli of Fano \\hyperlink{Fano3--5}{3--5}, hence $Y$ is general in moduli.\n\n\\hypertarget{Fano3--6}{\\subsection*{Fano 3--6}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of a line and an elliptic curve of degree 4.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,2) \\oplus \\mathcal{O}(0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification}\nA quartic elliptic curve is a given by a complete intersections of two quadrics in $\\mathbb{P}^3$. It then suffices to apply twice Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--8}{\\subsection*{Fano 3--8}}\n\\subsubsection*{Mori-Mukai} Divisor from the linear system $|(\\alpha \\circ \\pi)^* (\\mathcal{O}_{\\mathbb{P}^2}(1)) \\boxtimes \\mathcal{O}_{\\mathbb{P}^2}(2)|$ on $\\Bl_p \\mathbb{P}^2 \\times \\mathbb{P}^2$, where $\\pi: \\Bl_p \\mathbb{P}^2 \\times \\mathbb{P}^2 \\rightarrow \\Bl_p \\mathbb{P}^2$ is the first projection and $\\alpha: \\Bl_p \\mathbb{P}^2 \\rightarrow \\mathbb{P}^2$ is the blow up map. \n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,2) \\oplus \\mathcal{O}(1,1,0)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$.\n\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--9}{\\subsection*{Fano 3--9}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}_{\\mathbb{P}^2}(\\mathcal{O} \\oplus \\mathcal{O}(-2))$ in a quartic curve on $\\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^6}(0,0,1) \\oplus K(0,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^{20}$, where the bundle $\\Lambda\\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ is a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36} and $K \\in \\Ext^3(\\Sym^4 \\mathcal{Q},\\mathcal{Q}(-3))$ is a uniquely defined extension on $\\mathbb{P}^2$ fitting into the chain of extensions \\eqref{Kappa3--9}.\n\n\\subsubsection*{Identification}\nWe need to blow up \\hyperlink{Fano2--36}{2--36} in a quartic curve $C$ on the base $\\mathbb{P}^2$. The first bundle defines $Y:=$ \\hyperlink{Fano2--36}{2--36} inside $\\mathbb{P}^2\\times \\mathbb{P}^6$; since $C$ is the zero locus of a map\n\\[\n\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-4,0) \\rightarrow \\mathcal{O}\n\\]\non $Y$, by Lemma \\ref{lem:blowDegeneracyLocus} our Fano will be the zero locus of $\\mathcal{O}(1)$ over $\\mathbb{P}_Y(\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-4,0))$.\n\nFor the first bundle $\\mathcal{O}(0,-1)$, we have the standard (pulled back) Euler sequence\n\\begin{equation}\n\\label{of0-1}\n0 \\rightarrow\n\\mathcal{O}(0,-1) \\rightarrow\n\\mathcal{O}^{\\oplus 7} \\rightarrow\n\\mathcal{Q}_{\\mathbb{P}^6} \\rightarrow\n0;\n\\end{equation}\nthe second bundle $\\mathcal{O}(-4,0)$ requires a cumbersome though straightforward merging of the following (dualised) short exact sequences on $\\mathbb{P}^2$ given by Remark \\ref{rem:principalParts}:\n\\begin{gather*}\n0 \\rightarrow\n\\mathcal{O}(-4) \\rightarrow\n\\mathcal{O}(-3)^{\\oplus 3} \\rightarrow\n\\mathcal{Q}(-3) \\rightarrow\n0\\\\\n0 \\rightarrow\n\\mathcal{O}(-3)^{\\oplus 3} \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 6} \\rightarrow\n\\Sym^2\\mathcal{Q}(-2) \\rightarrow\n0\\\\\n0 \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 6} \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\Sym^3\\mathcal{Q}(-1) \\rightarrow\n0\\\\\n0 \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\mathcal{O}^{\\oplus 15} \\rightarrow\n\\Sym^4\\mathcal{Q} \\rightarrow\n0.\n\\end{gather*}\nArranging them repeatedly as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}, we get to a uniquely defined homogeneous rank $14$ vector bundle $K$ on $\\mathbb{P}^2$ which fits into\n\\begin{equation}\n\\label{of-40}\n0 \\rightarrow\n\\mathcal{O}(-4) \\rightarrow\n\\mathcal{O}^{\\oplus 15} \\rightarrow\nK \\rightarrow\n0\n\\end{equation}\nand into the following chain of extensions\n\\begin{equation}\n\\begin{gathered}\n\\label{Kappa3--9}\n0 \\rightarrow\n\\mathcal{Q}(-3) \\rightarrow\nK_1 \\rightarrow\n\\Sym^2\\mathcal{Q}(-2) \\rightarrow\n0\\\\\n0 \\rightarrow\nK_1 \\rightarrow\nK_2 \\rightarrow\n\\Sym^3\\mathcal{Q}(-1) \\rightarrow\n0\\\\\n0 \\rightarrow\nK_2 \\rightarrow\nK\\rightarrow\n\\Sym^4\\mathcal{Q} \\rightarrow\n0.\n\\end{gathered}\n\\end{equation}\nOne can directly check using \\eqref{of-40} and \\eqref{Kappa3--9} that $H^0(K) \\cong \\Sym^4 V_3$ and $H^1(K) \\cong V_3$.\nThe conclusion follows by considering the direct sum of \\eqref{of-40} and \\eqref{of0-1}.\n\n\n\n\\hypertarget{Fano3--10}{\\subsection*{Fano 3--10}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in the disjoint union of 2 conics.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(0,0,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification} It suffices to apply twice Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--11}{\\subsection*{Fano 3--11}}\n\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in an elliptic curve which is the intersection of two divisors from $|-\\frac{1}{2}K|.$\n\\subsubsection*{Our description} \n$\\mathscr{Z}(\\mathcal{O}(1,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification} We recall first that \\hyperlink{Fano2--35}{2--35} is the blow up of $\\mathbb{P}^3$ at a point, which we have already identified as $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$. As such, its anticanonical class is $\\mathcal{O}(2,2)$ by adjunction. It then suffices to apply Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--12}{\\subsection*{Fano 3--12}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of $\\mathbb{P}^3$ in the disjoint union of a line and a twisted cubic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1)\\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O} (1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3.$\n\n\\subsubsection*{Identification} The variety $\\mathscr{Z}(\\mathcal{O}(1,1)\\oplus \\mathcal{O}(1,1) ) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$ is the Fano 3-fold 2--27, the blow up of $\\mathbb{P}^3$ in a twisted cubic. The result then follows by Lemma \\ref{lem:blowup}, with the two extra $(0,1)$ divisors cutting a line in space which by construction is disjoint from the twisted cubic. To make this explicit, take coordinates $[z_0,z_1], \\ [y_0, y_1, y_2], \\ [x_0,\\ldots, x_3]$. The divisor of degree $(1,0,1)$ is therefore given by an expression of type $\\sum z_i f_i(x_i)$. Say for simplicity $z_0 x_0 +z_1 x_3$. The line $L$ in $\\mathbb{P}^3$ which we are blowing up is therefore given by $x_0=x_3=0$. On the other hand the two divisors of degree $(0,1,1)$ define the twisted cubic as follows: they are given by the solutions of, e.g., \\[ \n {\\begin{pmatrix}\n x_0 & x_1 & x_2 \\\\\n x_1 & x_2 & x_3 \\\\\n \\end{pmatrix} }\n {\\begin{pmatrix} y_0 \\\\\n y_1 \\\\\n y_2\n \\end{pmatrix}}=0.\n \\]\nIn particular this locus is trivially identified with the blow up of $\\mathbb{P}^3$ where the matrix drops rank, that is $ {\\rank \\begin{pmatrix}\n x_0 & x_1 & x_2 \\\\\n x_1 & x_2 & x_3 \\\\\n \\end{pmatrix} } <2$. The latter are the equations of the twisted cubic in $\\mathbb{P}^3$, which we can easily check to be disjoint from the line $L$.\n\n\\hypertarget{Fano3--14}{\\subsection*{Fano 3--14}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of $\\mathbb{P}^3$ in the disjoint union of a plane cubic curve and a point outside the plane.\n\\subsubsection*{Our description} $\\mathscr{Z}( \\Lambda(0,1,0) \\oplus \\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(1,0,0)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^2$, where the bundle $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ is a uniquely defined extension on $\\mathbb{P}^3$ fitting into sequence \\eqref{Lambda1--12} above.\n\n\n\n\n\\subsubsection*{Identification} The first two bundles on $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$ determine \\hyperlink{Fano2--28}{2--28}, i.e., the blow up of $\\mathbb{P}^3$ in a plane cubic curve. To blow it up in a point, we can apply Lemma \\ref{lem:blow} for the base $\\mathbb{P}^3$, adding a $\\mathbb{P}^2$ factor and the corresponding bundle. The extra point will in general be outside the plane.\n\n\\hypertarget{Fano3--15}{\\subsection*{Fano 3--15}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of $\\mathbb{Q}_3$ in the disjoint union of a line and a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the zero locus of the last two bundles on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$ gives us $\\mathbb{P}^1 \\times \\Bl_{\\mathbb{P}^1}\\mathbb{Q}_3$. We still have to cut with a section of $\\mathcal{O}(1,0,1)$. By Lemma \\ref{lem:blowup} this is the blow up of $\\Bl_{\\mathbb{P}^1} \\mathbb{Q}_3$ in the locus cut by two linear sections, which is in general disjoint from the $\\mathbb{P}^1$. The result follows.\n\n\\hypertarget{Fano3--16}{\\subsection*{Fano 3--16}} \n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in the proper transform of a twisted cubic containing the centre of the blow up.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,1,0)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^2$.\n\n\\subsubsection*{Identification} We first fix the system of coordinates\n$\\mathbb{P}^2_{[y_0\\ldots y_2]} \\times \\mathbb{P}^3_{[x_0\\ldots x_3]} \\times \\mathbb{P}^2_{[w_0\\ldots w_2]}.$ As a first step we use Lemma \\ref{lem:blow} to identify $\\mathcal{Q}_{\\mathbb{P}^2} (0,1) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 $ as \\hyperlink{Fano2--35}{2--35}, i.e., $\\Bl_p \\mathbb{P}^3$. The two remaining divisors are, on $\\mathbb{P}^2 \\times \\mathbb{P}^3$, of degree $(0,1)$ and $(1,0)$ and are both trivially identified with linear forms on $\\mathbb{P}^3$, but with a distinction. Without loss of generality, assume that $p$ is the point $[1,0,0,0]$. We have $(f \\in |\\mathcal{O}(1,0)|) \\in \\mathrm{Ann}(p)$, while $(g \\in |\\mathcal{O}(0,1)|)$ gives a non-zero element of $V_4^{\\vee}\/\\mathrm{Ann}(p)$. In other words, $f=f(x_1,x_2,x_3)$ does not contain the coordinate $x_0$, while the converse holds for $g$. Both the divisors were twisted by $\\mathcal{O}_{\\mathbb{P}^2}(1)$, giving rise to two divisors of degree $(1,1)$ on $\\Bl_p\\mathbb{P}^3 \\times \\mathbb{P}^2_{[w_0\\ldots w_2]}$. As in \\hyperlink{Fano3--12}{3--12}, these lead to the blow up of $\\Bl_p\\mathbb{P}^3$ in a twisted cubic, that (since $f \\in \\mathrm{Ann}(p)$) passes through the point $p \\in \\mathbb{P}^3$. The result follows.\n\n\n\\hypertarget{Fano3--18}{\\subsection*{Fano 3--18}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of a line and a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2(0;0,0) \\oplus \\mathcal{O}(0;1,1) \\oplus \\mathcal{O}(1;0,1)) \\subset \\mathbb{P}^1 \\times \\Fl(1,2,5)$.\n\n\\subsubsection*{Identification}\nThis Fano can be evidently identified with the blow up of \\hyperlink{Fano2--30}{2--30} in a line disjoint from the conic. Recall that we described \\hyperlink{Fano2--30}{2--30} as $(\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1)) \\subset \\Fl(1,2,5)) \\cong \\mathscr{Z}(\\mathcal{O}(1)) \\subset \\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-1)\\oplus \\mathcal{O})$. The result then follows from Lemma \\ref{lem:blowup}, since two divisors of degree $(0,1)$ cut a line in the base $\\mathbb{P}^3$.\n\nWe can write an alternative description for this Fano, based on the alternative description already given for \\hyperlink{Fano2--30}{2--30}. Using Lemma \\ref{lem:blowup} the Fano \\hyperlink{Fano3--18}{3--18} will be\n\n\\[\n\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)) \\subset \\mathbb{P}^1\\times \\mathbb{P}^3 \\times \\mathbb{P}^4.\n\\]\n\n\\hypertarget{Fano3--19}{\\subsection*{Fano 3--19}}\n\\subsubsection*{Mori-Mukai}Blow up of $\\mathbb{Q}_3$ in two non-collinear points.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:blow}, the first divisor yields the blow up of $\\mathbb{P}^4$ along a line. The second divisor is identified with a general quadric in $\\mathbb{P}^4$, hence it cuts out a quadric hypersurface in $\\mathbb{P}^4$ blown up along two points. The general quadric does not contain the line, so the blown up points are in general non-collinear.\n\n\\hypertarget{Fano3--20}{\\subsection*{Fano 3--20}} \n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in the disjoint union of two lines.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_2}(0,1,0) ) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^5 \\times \\mathbb{P}^2$.\n\n\\subsubsection*{Identification} We remark that, by Lemma \\ref{lem:blow}, another model for \\hyperlink{Fano2--31}{2--31} (the blow up of $\\mathbb{Q}_3$ in one line) is given by $\\mathscr{Z}(\\mathcal{O}(1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2} (0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^5$. Our model for \\hyperlink{Fano3--20}{3--20} is just the iteration of the blow up process, where the second and the third bundles give the blow up of $\\mathbb{P}^4$ along two disjoint lines $L_1, L_2$ and the first bundle gives a quadric which contains both the lines. Notice that in fact a section $\\sum_k f_{1,k} f_{2,k}$ of the bundle $\\mathcal{O}(1,0,1)$ identifies a quadric in $\\mathbb{P}^4$ and $f_{i,k} \\in \\Ann(L_i)$ for $i=1,2$ (see also the arguments used for \\hyperlink{Fano2--19}{2--19} and \\hyperlink{Fano3--16}{3--16}).\n\n\\hypertarget{Fano3--21}{\\subsection*{Fano 3--21}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in a curve of degree $(2,1)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1) \\oplus \\Lambda(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18}.\n\n\\subsubsection*{Identification}\nOn $\\mathbb{P}^1 \\times \\mathbb{P}^2$, a general complete intersection of a $(0,1)$ and a $(1,2)$ divisors is a smooth curve of degree $(2,1)$. In order to blow it up, we can use Lemma \\ref{lem:blowDegeneracyLocus}, according to which our Fano will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*(0,1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2$.\n\nThe above projective bundle has already been found when dealing with \\hyperlink{Fano2--18}{2--18}: it is the zero locus of $\\Lambda(0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, with $\\Lambda$ fitting into \\eqref{Lambda2-18}.\n\n\\hypertarget{Fano3--22}{\\subsection*{Fano 3--22}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in a conic on $\\lbrace x \\rbrace \\times \\mathbb{P}^2 $, $\\lbrace x \\rbrace \\in \\mathbb{P}^1$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\Lambda(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}.\n\\subsubsection*{Identification}\nWe need to blow up on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ a complete intersection curve given by two divisors of degree $(1,0)$ and $(0,2)$. To do that, we use Lemma \\ref{lem:blowDegeneracyLocus}: our Fano will then be the zero locus of $\\mathcal{O}(1)$ over the projective bundle $\\mathbb{P}(\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2))$.\n\nTo find the above projective bundle, we can add the standard (pulled back) Euler sequence on $\\mathbb{P}^1$ to \\eqref{inclusion2-36} and get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2) \\rightarrow\n\\mathcal{O}^{\\oplus 8} \\rightarrow\n\\mathcal{O}(1,0) \\oplus \\Lambda \\rightarrow\n0,\n\\]\nbeing $\\Lambda$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}. The conclusion follows.\n\n\\hypertarget{Fano3--23}{\\subsection*{Fano 3--23}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in the proper transform of a conic containing the centre of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0) \\oplus \\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the first bundle (when seen on the first two factors) gives $X:=$ \\hyperlink{Fano2--35}{2--35}, the blow up of $\\mathbb{P}^3$ in one point $p$. We need to blow up $X$ along the proper transform of a conic $Q$ containing $p$. Note that $Q$ is cut out by a hyperplane and a quadric in $\\mathbb{P}^3$ both containing $p$, so that $Q$ is the degeneracy locus of a map $\\mathcal{O}_X(-1,-1) \\oplus \\mathcal{O}_X(-1,0) \\rightarrow \\mathcal{O}_X$ (see, e.g., the arguments used for \\hyperlink{Fano2--19}{2--19} and \\hyperlink{Fano3--16}{3--16}). Lemma \\ref{lem:blowDegeneracyLocus} yields that our Fano will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*(\\mathcal{O}(1,0))$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(0,-1) \\oplus \\mathcal{O}) \\rightarrow \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\nSuch projective bundle can be found in $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$, as the sequence on $\\mathbb{P}^2 \\times \\mathbb{P}^3$\n\\[\n0 \\rightarrow\n\\mathcal{O}(0,-1) \\oplus \\mathcal{O} \\rightarrow\n\\mathcal{O}^{\\oplus 5} \\rightarrow\n\\mathcal{Q}_{\\mathbb{P}^3} \\rightarrow\n0\n\\]\nshows. The conclusion follows.\n\n\\hypertarget{Fano3--24}{\\subsection*{Fano 3--24}}\n\\subsubsection*{Mori-Mukai} The fiber product of 2--32 with $\\Bl_p\\mathbb{P}^2$ over $\\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2.$\n\\subsubsection*{Identification} See \\cite[\\textsection 77]{corti}.\n\n\\hypertarget{Fano3--25}{\\subsection*{Fano 3--25}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-1,0))$, or the blow up of $\\mathbb{P}^3$ in two disjoint lines.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1)^{\\oplus 2}) \\subset \\Fl(1,2,4)$.\n\n\\subsubsection*{Identification} We can identify $\\Fl(1,2,4)$ with $\\mathbb{P}_{\\Gr(2,4)}(\\mathcal{U})$. Let $Z:= \\mathbb{P}^1 \\times \\mathbb{P}^1$. The two $(0,1)$ sections give us $\\mathbb{P}_Z(\\mathcal{U}|_Z)$. By \\cite[Theorem 1.4]{ottaviani} the restriction of $\\mathcal{U}$ to $Z$ coincides with the direct sum of $\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-1,0)$. The result follows. \n\n\nAn alternative description is $\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$, by simply apply twice Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--26}{\\subsection*{Fano 3--26}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of a point and a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\n\\subsubsection*{Identification} The bundle $\\mathcal{Q}_{\\mathbb{P}^2} (0,1)$ on $ \\mathbb{P}^2 \\times \\mathbb{P}^3$ gives the Fano \\hyperlink{Fano2--35}{2--35} by Lemma \\ref{lem:blow}. Two extra sections of $\\mathcal{O}(0,1)$ on this space cut a line that does not intersect the exceptional divisor (equivalently, a line in $\\mathbb{P}^3$ that does not pass through the blown up point). The identification therefore follows by Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--28}{\\subsection*{Fano 3--28}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}^1 \\times \\Bl_p \\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$.\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--29}{\\subsection*{Fano 3--29}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in a line on the exceptional divisor.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(1,0,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1) \\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(0,-1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^2 \\times \\mathbb{P}^9$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the first bundle gives, on the first two factors, the blow up $Y$ of $\\mathbb{P}^3$ along a point. We then need to blow up a line in the exceptional divisor. By \\cite[Corollary 9.12]{EisenbudHarris3264}, the exceptional divisor is a $(1,-1)$ divisor in $Y$; in order to cut out a line on it, we have to intersect it with the strict transform of a hyperplane in $\\mathbb{P}^3$ passing through the point, which is a $(0,1)$ divisor (see, e.g., the argument used for \\hyperlink{Fano3--16}{3--16}).\n\nSummarising, we need to blow $Y$ up along the intersection of the two aforementioned divisors. By Lemma \\ref{lem:blowDegeneracyLocus}, this yields that our Fano variety is the zero locus of $\\pi^* \\mathcal{O}(0,-1) \\otimes \\mathcal{O}(1)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2)) \\rightarrow Y$. \n\nTo express the above projective bundle, we can add the standard (pulled back) Euler sequence on $\\mathbb{P}^3$ to \\eqref{inclusion2-36} and get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2) \\rightarrow\n\\mathcal{O}^{\\oplus 10} \\rightarrow\n\\mathcal{Q}_{\\mathbb{P}^3} \\oplus \\Lambda \\rightarrow\n0,\n\\]\nbeing $\\Lambda$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}. The conclusion follows.\n\n\\begin{caveat}\n\\label{caveatBundle}\nThe above bundle $\\mathcal{O}(0,-1,1)$ has clearly no sections on $\\mathbb{P}^3 \\times \\mathbb{P}^2 \\times \\mathbb{P}^9$, so our notation seems misleading. In fact, this bundle acquires a $4$-dimensional space of global sections once it is restricted to the zero locus of the previous ones, so that the direct sum above should be taken with a pinch of salt.\n\nThis phenomenon naturally occurs when we need to consider the exceptional divisor of a blow up obtained via Lemma \\ref{lem:blow}: as already remarked, if $Y=\\Bl_{\\mathbb{P}^{n-m-1}}\\mathbb{P}^n=\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^m}(0,1))\\subset \\mathbb{P}^m \\times \\mathbb{P}^n$, then the exceptional divisor is a $(-1,1)$ divisor in $Y$. Notice that $\\mathcal{O}_{\\mathbb{P}^n \\times \\mathbb{P}^m}(-1,1)|_Y \\cong \\mathcal{O}_Y(-1,1)$ indeed has global sections.\n\\end{caveat}\n\n\n\\hypertarget{Fano3--30}{\\subsection*{Fano 3--30}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in the proper transform of a line containing the centre of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_2(0,1,0)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^1$.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the first bundle (when seen on the first two factors) gives a Fano $X$ which is \\hyperlink{Fano2--35}{2--35}, the blow up of $\\mathbb{P}^3$ in one point $p$. We need to blow up $X$ along the proper transform of a line containing $p$, which is the complete intersection of two divisors of degree $(1,0)$ on $\\mathbb{P}^2 \\times \\mathbb{P}^3$ (see, e.g., the argument used for \\hyperlink{Fano3--16}{3--16}). We conclude by Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--31}{\\subsection*{Fano 3--31}}\n\\subsubsection*{Mori-Mukai} Blow up of the cone over a smooth quadric in $\\mathbb{P}^3$ in the vertex, or $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$.\n\\subsubsection*{Our description} of $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,5)$.\n\n\\subsubsection*{Identification} By Corollary \\ref{cor:blowupflag} we have that $\\mathscr{Z}(\\mathcal{Q}_2) \\subset \\Fl(1,2,5)$ is isomorphic to $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-1) \\oplus \\mathcal{O})$. The extra quadric cuts only the base $\\mathbb{P}^3$, and yields the identification. \n\nWe want to give an alternative description as \\[\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,0)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4.\\] \nBy Lemma \\ref{lem:blow}, $\\mathcal{Q}_{\\mathbb{P}^3} (0,1)$ gives the blow up of $\\mathbb{P}^4$ at a point $p_0$, with dual coordinate $x_0$. A section of $\\mathcal{O}(2,0)$ gives a quadric in the space $\\Sym^2(V_5^{\\vee}\/\\langle x_0 \\rangle)$. This gives the equation of a cone over a smooth, degenerate quadric in $\\mathbb{P}^3_{[x_1, \\ldots, x_4]}$. The result follows.\n\n\n\\hypertarget{Fano4--2}{\\subsection*{Fano 4--2}}\n\\subsubsection*{Mori-Mukai} \nBlow up of the cone over a smooth quadric in $\\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric. \n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1,0) \\oplus \\mathcal{O}(2,0,0) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^4}(0,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4 \\times \\mathbb{P}^5$.\n\n\\subsubsection*{Identification} \nWe use the alternative description of \\hyperlink{Fano3--31}{3--31}. In fact to blow up the requested elliptic curve it suffices to blow up $Y:=$ \\hyperlink{Fano3--31}{3--31}, in its intersection with a hyperplane not passing through the vertex of the cone and a general quadric, i.e., in the intersection of a $(0,1)$ and a $(0,2)$ divisors. Lemma \\ref{lem:blowDegeneracyLocus} yields that our Fano variety is the zero locus of $\\pi^* \\mathcal{O}(0,1) \\otimes \\mathcal{O}(1)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(0,-1)\\oplus \\mathcal{O}) \\rightarrow Y$. Such projective bundle can be obtained as the zero locus of the remaining bundle by considering the Euler sequence on $\\mathbb{P}^4$, which yields and embedding of $\\mathcal{O}(0,-1)\\oplus \\mathcal{O}$ inside $\\mathbb{P}(5 \\mathcal{O} \\oplus \\mathcal{O})$.\n\n\\hypertarget{Fano4--3}{\\subsection*{Fano 4--3}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ in a curve of degree $(1,1,2)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0,1) \\oplus \\mathcal{O} (0,0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2.$\n\n\\subsubsection*{Identification} A complete intersection of divisors of degree $(1,1,0)$, $(1,1,1)$ is a curve of degree $(1,1,2)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. In order to blow it up, we use Lemma \\ref{lem:blowDegeneracyLocus}: our Fano is then the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(1,1,0)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(0,0,-1) \\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. From the standard Euler sequence on $\\mathbb{P}^1$ we get\n\\[\n0 \\rightarrow\n\\mathcal{O}(0,0,-1) \\oplus \\mathcal{O} \\rightarrow\n\\mathcal{O}^{\\oplus 3} \\rightarrow\n\\mathcal{O}(0,0,1) \\rightarrow\n0,\n\\]\nhence the conclusion.\n\n\n\\hypertarget{Fano4--4}{\\subsection*{Fano 4--4}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--19}{3--19} in the proper transform of a conic through the points.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^4 \\times \\mathbb{P}^1. $\n\n\\subsubsection*{Identification} The first two bundles on $\\mathbb{P}^2 \\times \\mathbb{P}^4$ give the Fano \\hyperlink{Fano3--19}{3--19}. We then just need to use Lemma \\ref{lem:blowup}, since two sections of $\\mathcal{O}(1,0)$ cut the three dimensional quadric in a conic passing through the two points (see also the argument used for \\hyperlink{Fano3--16}{3--16}).\n\n\\hypertarget{Fano4--5}{\\subsection*{Fano 4--5}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1 $, where the bundle $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ is a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18}.\n\n\\subsubsection*{Identification} The first two bundles describe, on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, the variety \\hyperlink{Fano3--21}{3--21}. We need to blow it up along a curve of degree $(1,0)$, which is the complete intersection of two divisors of degree $(0,1)$ on $\\mathbb{P}^1 \\times \\mathbb{P}^2$. The result follows from Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano4--6}{\\subsection*{Fano 4--6}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of 3 lines.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(1,0,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1. $\n\n\\subsubsection*{Identification} It suffices to apply three times Lemma \\ref{lem:blowup}. By dimension reasons the three lines on $\\mathbb{P}^3$ which are cut each times are disjoint.\n\n\\hypertarget{Fano4--7}{\\subsection*{Fano 4--7}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of 2--32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0; 1,0) \\oplus \\mathcal{O}(0,1;0,1)) \\subset \\Fl(1,2,3) \\times \\mathbb{P}^1 \\times \\mathbb{P}^1. $\n\n\\subsubsection*{Identification} The flag variety $F:=\\Fl(1,2,3)$ can be identified with 2--32, that is a $(1,1)$ section of $\\mathbb{P}^2 \\times (\\mathbb{P}^2)^{\\vee}.$ Notice that under this identification the generators of the Picard group of the flag are the restriction of the canonical ones on $\\mathbb{P}^2 \\times (\\mathbb{P}^2)^{\\vee}.$ In particular $H^0(F, \\mathcal{O}_{F}(1,0)) \\cong V_3^{\\vee}$ and $H^0(F, \\mathcal{O}_{F}(0,1)) \\cong V_3$.\nThe zero locus of two sections of $\\mathcal{O}_{F}(1,0)$ is a $(0,1)$ curve, and the opposite holds for $\\mathcal{O}_{F}(0,1)$. We then apply twice Lemma \\ref{lem:blowup} to conclude.\n\nOf course thanks to the above identification and Lemma \\ref{lem:blowup} this Fano can be described as well as \n\\[\n\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1.\n\\]\n\n\\hypertarget{Fano4--8}{\\subsection*{Fano 4--8}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--31}{3--31} (i.e., $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$) in a $(1,1)$-section of the base $\\mathbb{P}^1 \\times \\mathbb{P}^1$, or blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ in a curve of degree $(0,1,1)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2;0) \\oplus \\mathcal{O}(1,0;1) ) \\subset \\Fl(1,2,5) \\times \\mathbb{P}^1.$\n\n\\subsubsection*{Identification} We use the first description by Mori--Mukai, together with our description of \\hyperlink{Fano3--31}{3--31}. Given this, it suffices to apply Lemma \\ref{lem:blowup}, since the zero locus of two extra copies of $\\mathcal{O}_{F}(1,0)$ on $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}_{F}(0,2)) \\subset F:=\\Fl(1,2,5)$ is such a curve. In fact $Z:=\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}_{F}(0,2) \\oplus \\mathcal{O}_{F}(1,0)) \\subset F$ corresponds to the base $\\mathbb{P}^1 \\times \\mathbb{P}^1$; on $Z$, both the restrictions $\\mathcal{O}_{F}(1,0)|_Z \\cong \\mathcal{O}_{F}(0,1)|_Z $ coincide with $\\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(1,1)$, as can be easily checked via a Chern classes computation.\n\nAlternatively, we can use Lemma \\ref{lem:blowup} to give another description of this Fano, given the alternative one for \\hyperlink{Fano3--31}{3--31}. In particular \\hyperlink{Fano4--8}{4--8} will be given as\n\\[\n\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1,0) \\oplus \\mathcal{O}(2,0,0) \\oplus \\mathcal{O}(0,1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4 \\times \\mathbb{P}^1.\n\\]\n\\hypertarget{Fano4--9}{\\subsection*{Fano 4--9}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--25}{3--25} in an exceptional rational curve $E$ of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0,0) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1.$\n\n\\subsubsection*{Identification} First we use that the bundle $\\mathcal{Q}_{\\mathbb{P}^2} (0,1) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$ gives the blow up $\\Bl_p \\mathbb{P}^3$ by Lemma \\ref{lem:blow}. Lemma \\ref{lem:blowup} yields that the other two bundles yield the blow up along two other lines $L, L'$ in $\\mathbb{P}^3$: $L$ (corresponding to $\\mathcal{O}(1,0,1,0)$) passing through $p$, $L'$ avoiding it (see, e.g., the argument used for \\hyperlink{Fano3--16}{3--16}). Therefore we identify the above variety with $\\Bl_{\\Sigma} \\mathbb{P}^3$, where $\\Sigma:= L \\cup L' \\cup p$, and $p \\in L$. This is the same as $\\Bl_{E}(\\Bl_{L \\cup L'} \\mathbb{P}^3)$. Since the exceptional divisor of the second blow up $\\pi_2$ is a $\\mathbb{P}^1$-bundle over the union of the two lines, (with $E=\\pi_2^{-1}(p)$) the result follows.\n\n\\hypertarget{Fano4--10}{\\subsection*{Fano 4--10}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}^1 \\times \\Bl_2 \\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\n\n\n\\subsubsection*{Identification} Lemma \\ref{lem:blow} identifies the zero locus of a general section of the second bundle with $\\mathbb{P}^1 \\times \\Bl_p \\mathbb{P}^3$. A section of the remaining bundle gives a quadric in $\\mathbb{P}^3$ containing $p$ (see, e.g., the argument used for \\hyperlink{Fano2--19}{2--19}), which identifies our model with $\\Bl_p (\\mathbb{P}^1 \\times \\mathbb{P}^1)$. We remark that Lemma \\ref{lem:blowup} provides another simple model, i.e., the zero locus of $\\mathcal{O}(1,0,1,0)\\oplus \\mathcal{O}(0,1,1,0)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$.\n\n\\hypertarget{Fano4--11}{\\subsection*{Fano 4--11}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--28}{3--28} in $\\lbrace x\\rbrace \\times E$, $x \\in \\mathbb{P}^1$ and $E$ the $(-1)$-curve.\n\\subsubsection*{Our description}. $\\mathscr{Z}(\\mathcal{O}(0,1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,0,1) \\oplus \\Lambda(0,0,0,1) \\oplus \\mathcal{O}(0,0,-1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into \\eqref{Lambda3-2}.\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:blow} the first bundle defines, on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$, the Fano \\hyperlink{Fano3--28}{3--28}. By \\cite[Corollary 9.12]{EisenbudHarris3264}, we need to blow up the intersection of a $(1,0,0)$ and a $(0,1,-1)$ divisors. Using Lemma \\ref{lem:blowDegeneracyLocus}, our Fano will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(0,0,-1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,0,-1) \\otimes \\mathcal{O}(0,-1,0)) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$.\n\nFor $\\mathcal{O}(-1,0,-1)$ we can pull back sequence \\eqref{inclusion3-2} and get\n\\begin{equation}\n0 \\rightarrow \\mathcal{O}(-1,0,-1) \\rightarrow \\mathcal{O}^{\\oplus 4} \\rightarrow \\Lambda \\rightarrow 0,\n\\end{equation}\nwhere $\\Lambda$ fits into \\eqref{Lambda3-2}. Adding it with the standard Euler sequence on $\\mathbb{P}^2$, we get\n\\[\n0 \\rightarrow \\mathcal{O}(-1,0,-1) \\oplus \\mathcal{O}(0,-1,0) \\rightarrow \\mathcal{O}^{\\oplus 7} \\rightarrow \\Lambda \\oplus \\mathcal{Q}_{\\mathbb{P}^2} \\rightarrow 0,\n\\]\nwhich gives the conclusion.\n\nWe remark that the last bundle in the description should be taken with a caveat, as it has no global sections on the ambient space, but acquires some when restricted to the zero locus of the previous bundles. See Caveat \\ref{caveatBundle}.\n\n\\hypertarget{Fano4--12}{\\subsection*{Fano 4--12}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--33}{2--33} in the disjoint union of two exceptional lines of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(-1,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda4-12} below.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} (or Lemma \\ref{lem:blowup}) the first bundle gives, on the first two factors, the blow up $Y$ of $\\mathbb{P}^3$ along a line. We then need to blow up two disjoint lines in the exceptional divisor. By \\cite[Corollary 9.12]{EisenbudHarris3264}, the exceptional divisor is a $(-1,1)$ divisor in $Y$; in order to cut out two lines on it, we have to intersect it with the strict transform of a general quadric hypersurface in $\\mathbb{P}^3$, which is a $(0,2)$ divisor cutting the blown up line in two points.\n\nSummarising, we need to blow $Y$ up along the intersection of the two aforementioned divisors. By Lemma \\ref{lem:blowDegeneracyLocus}, this yields that our Fano variety is the zero locus of $\\pi^* \\mathcal{O}(-1,1) \\otimes \\mathcal{O}(1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^3$.\n\nTo describe this projective bundle, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) two (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{gather}\n\\label{inclusion4-12}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda4-12}\n0 \\rightarrow \\mathcal{O}(1,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2} \\rightarrow 0,\n\\end{gather}\nwhere the rank $7$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^3, \n\\mathcal{Q}^{\\oplus 2}) \\cong (V_4)^{\\oplus 2}$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion4-12} we get that $\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ is the zero locus of $\\Lambda(0,0,1)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^8$, whence the conclusion.\n\nWe remark that the last bundle in the description should be taken with a caveat, as it has no global sections on the ambient space, but acquires some when restricted to the zero locus of the previous bundles. See Caveat \\ref{caveatBundle}.\n\n\n\\hypertarget{Fano4--13}{\\subsection*{Fano 4--13}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ in a curve of degree $(1,3,1)$.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\Lambda(0,0,0,1) \\oplus \\mathcal{O}(1,0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$, being $\\Lambda \\in \\Ext^1(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ (the first two copies) fitting into \\eqref{Lambda3-2}.\n\\subsubsection*{Identification}\nThe complete intersection between a $(2,1,1)$ and a $(1,0,1)$ divisors is a curve of degree $(1,3,1)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. In order to blow it up, we use Lemma \\ref{lem:blowDegeneracyLocus}: our Fano $Y$ will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(1,0,1)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1,0) \\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. From \\eqref{inclusion3-2} we get that this projective bundle is the zero locus of $\\Lambda(0,0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$, where $\\Lambda$ is a bundle on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ (the first two copies) fitting into \\eqref{Lambda3-2}. The conclusion follows.\n\nAnalogously, we could have used the complete intersection of a $(3,1,0)$ and a $(1,1,0)$ divisors, which is again a curve of degree $(1,3,1)$. A similar argument requires the projective bundle $\\mathbb{P}(\\mathcal{O}(-2,-1,0) \\oplus \\mathcal{O}(0,0,-1))$ and produces a model $Y'$ in $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$. If we consider the normal sequence for $Y=\\mathscr{Z}(\\mathcal{F}) \\subset \\mathbb{P}:=\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$,\na few cohomology computations via the Koszul complex as described in Section \\ref{computeinvariants} provide that $h^0(T_{\\mathbb{P}}|_Y)=33, h^0(\\mathcal{F}|_Y)=34$ and the higher cohomology groups vanish. In \\cite[Lemma 8.11]{pcs} it is shown that the family of curves of degree $(1,1,3)$ on $(\\mathbb{P}^1)^3$ has dimension one (up to the action of $\\Aut((\\mathbb{P}^1)^3)$), and that for all but one curve the automorphism group is finite. This means that a general model $Y$ admits a $(34-33=1)$-dimensional family of deformations, which is the dimension of the moduli of Fano \\hyperlink{Fano4--13}{4--13}, hence $Y$ is general in moduli. The corresponding computations for $Y'\\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$ give, analogously, $73-72=1$, so that the models $Y'$ are also general in the moduli space of Fano \\hyperlink{Fano4--13}{4--13}.\n\n\n\\hypertarget{Fano5--1}{\\subsection*{Fano 5--1}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--29}{2--29} in the disjoint union of three exceptional lines of the blow up.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(-1,1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^8}(0,0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^{11}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda4-12}.\n\n\\subsubsection*{Identification} This Fano variety is the blow up of \\hyperlink{Fano4--12}{4--12} along a rational curve. If we consider the model for \\hyperlink{Fano4--12}{4--12} in $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^8$ (given by the zero locus of the first three bundles), we can check that the intersection of a $(0,1,0)$ divisor and a $(0,0,1)$ divisor is indeed a rational curve $C$, and the corresponding blow up $Y$ can be checked to have the right Hodge diamond and invariants. To ensure that $Y$ is Fano (hence, it is \\hyperlink{Fano5--1}{5--1}) we can check that a fiber $F$ of the exceptional divisor has $F.K_Y=-1$, so that $-K_Y$ is ample by \\cite[Thm 1.4.3]{isp5}.\n\nAs usual, we blow up $C$ via Lemma \\ref{lem:blowDegeneracyLocus}: our Fano will be the zero locus of $\\mathcal{O}(1)$ over $\\mathbb{P}(\\mathcal{O}(0,-1,0) \\oplus \\mathcal{O}(0,0,-1))$. This projective bundle can be easily described by considering the direct sum of the two Euler sequences, which yields\n\\[\n0 \\rightarrow \\mathcal{O}(0,-1,0) \\oplus \\mathcal{O}(0,0,-1) \\rightarrow \\mathcal{O}^{\\oplus 13} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3} \\oplus \\mathcal{Q}_{\\mathbb{P}^8} \\rightarrow 0.\n\\]\nThe conclusion follows.\n\n\\hypertarget{Fano5--2}{\\subsection*{Fano 5--2}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--25}{3--25} in the disjoint union of two exceptional lines on the same irreducible component.\n\n\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(-1,1,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^1$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda4-12}.\n\n\\subsubsection*{Identification} \nRecall that \\hyperlink{Fano4--12}{4--12} is the blow up of $\\mathbb{P}^3$ in a line and then in the disjoint union of two exceptional lines of the blow up, and is given by the zero locus of the first three bundles. To get \\hyperlink{Fano5--2}{5--2} we need to blow it up along the strict transform of a line not intersecting any of the other three. The previously found model for \\hyperlink{Fano4--12}{4--12} was in $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8}$, and such a line is the complete intersection of two $(0,1,0)$ divisors. Lemma \\ref{lem:blowup} yields the conclusion.\n\n\n\\section{Tables}\n\\label{tables}\nIn this last section we collect in an exhaustive table all the models for Fano 3-folds we exhibited in Section \\ref{Fano3folds}, together with the models already existing in the literature. In Table \\ref{tab:3folds}, MM stands for the Mori--Mukai numeration; the Picard rank $\\rho$ is the first number. In the column ``Inv'' an entry $(a,b,c)$ means the invariants $(h^0(-K), K^3, h^{2,1})$ of the corresponding Fano. The column $X$ refers to the ambient variety, whereas $\\mathcal{F}$ is the bundle whose zero locus produces the 3-fold. In some cases alternative descriptions (marked by ``\\emph{alt.}'') are given, whenever we find them equally interesting. In the column ``Notes'' we put either the reference for the chosen model, when it was not provided by us, or a further explanation of the bundles appearing in the previous column.\n\nWe include a second table, Table \\ref{tab:delpezzo}, for Del Pezzo surfaces, whose models can be easily figured out from Table \\ref{tab:3folds}. Each family in the table (except 2--1) will correspond to the blow up of $\\mathbb{P}^2$ in $9-K^2$ points in sufficiently general position. All models (for 3-folds and Del Pezzo surfaces) are general.\n\n\\begin{centering}\n\\begin{scriptsize}\n\\setlength\\tabcolsep{4pt}\n\\begin{longtable}{ccccc}\n\n\\caption{Fano 3-folds.}\\label{tab:3folds}\\\\\n\\toprule\nMM&\nInv& \n$X$ & \n$\\mathcal{F}$ & \nNotes \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n\\endfirsthead\n\\multicolumn{5}{l}{\\vspace{-0.25em}\\scriptsize\\emph{\\tablename\\ \\thetable{} continued from previous page}}\\\\\n\\toprule\nMM&\nInv& \n$X$ & \n$\\mathcal{F}$ & \nNotes \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n\\endhead\n\\multicolumn{5}{r}{\\scriptsize\\emph{Continued on next page}}\\\\\n\\endfoot\n\\bottomrule\n\\endlastfoot\n\n\\hyperlink{Fano1--1}{1--1} & $(4,2,52)$ &\\ $\\mathbb{P}(1^4,3)$\\ & $\\mathcal{O}(6)$&\\cite{isp5}\\\\\n\\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}}& $\\mathbb{P}^3 \\times \\mathbb{P}^{20}$&$\\mathcal{O}(0,2) \\oplus K(0,1)$&$K \\in \\Ext^2_{\\mathbb{P}^3}(\\Sym^3 \\mathcal{Q}, \\mathcal{Q}(-2))$\\\\\n\\rowcolor[gray]{0.95} 1--2 & $(5,4,30)$& $\\mathbb{P}^4$& $\\mathcal{O}(4)$& \\cite{isp5} \\\\\n1--3 & $(6,6,20)$& $\\mathbb{P}^5$& $\\mathcal{O}(2) \\oplus \\mathcal{O}(3)$& \\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--4 & $(7,8,14)$& $\\mathbb{P}^6$& $\\mathcal{O}(2)^{\\oplus 3}$&\\cite{isp5} \\\\\n1--5 & $(8,10,10)$& $\\Gr(2,5)$& $\\mathcal{O}(2) \\oplus \\mathcal{O}(1)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--6 & $(9,12,7)$& $\\OGr^+(5,10)$& $\\mathcal{O}(\\frac{1}{2})^{7}$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Gr(2,5)$& $\\mathcal{U}^{\\vee}(1)\\oplus \\mathcal{O}(1)$&\\cite{corti} \\\\\n1--7 & $(10,14,5)$& $\\Gr(2,6)$&\n$\\mathcal{O}(1)^5$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--8 & $(11,16,3)$& $\\Gr(3,6)$& $\\bigwedge^2\\mathcal{U}^{\\vee} \\oplus \\mathcal{O}(1)^{3}$&\\cite{isp5} \\\\\n1--9 & $(12,18,2)$& $\\Gr(2,7)$& $\\mathcal{Q}^{\\vee}(1) \\oplus \\mathcal{O}(1)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--10 & $(14,22,0)$& $\\Gr(3,7)$& $(\\bigwedge^2\\mathcal{U}^{\\vee})^{\\oplus 3}$&\\cite{isp5} \\\\\n1--11 & $(7,8,21)$& $\\mathbb{P}(1^3,2,3)$& $\\mathcal{O}(6)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano1--12}{1--12} & $(11,16,10)$& $\\mathbb{P}(1^4,2)$& $\\mathcal{O}(4)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$& $\\Lambda(0,1) \\oplus \\mathcal{O}(0,2)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2 \\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n1--13 & $(15,24,5)$& $\\mathbb{P}^4$& $\\mathcal{O}(3)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--14 & $(19,32,2)$& $\\mathbb{P}^5$& $\\mathcal{O}(2)^{\\oplus 2}$&\\cite{isp5} \\\\\n1--15 & $(23,40,0)$& $\\Gr(2,5)$& $\\mathcal{O}(1)^{\\oplus 3}$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--16 & $(30,54,0)$& $\\mathbb{P}^4$& $\\mathcal{O}(2)$&\\cite{isp5} \\\\\n1--17 & $(35,64,0)$& $\\mathbb{P}^3$& & \\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 2--1 & $(5,4,22)$& $\\mathbb{P}(1^3,2,3) \\times \\mathbb{P}^1$& $\\mathcal{O}(6,0) \\oplus \\mathcal{O}(1,1)$&\\cite{corti} \\\\\n\\hyperlink{Fano2--2}{2--2} & $(6,6,20)$& $\\mathbb{P}^1\\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$& $\\mathcal{O}(0,0,2) \\oplus K(0,0,1)$&$K \\in \\Ext_{\\mathbb{P}^1 \\times \\mathbb{P}^2}^2(\\mathcal{O}(1,0)^{\\oplus 6}, \\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--3}{2--3} & $(7,8,11)$& $\\mathbb{P}(1^4,2) \\times \\mathbb{P}^1$& $\\mathcal{O}(4,0) \\oplus \\mathcal{O}(1,1)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^1$& $\\Lambda(0,1,0) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{O}(1,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2 \\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n2--4 & $(8,10,10)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,3)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--5}{2--5} & $(9,12,6)$& $\\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,3) \\oplus \\mathcal{O}(1,1)$& \\\\\n2--6 & $(9,12,9)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(2,2)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} 2--7 & $(10,14,5)$& $\\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,2)$&\\cite{corti} \\\\\n\\hyperlink{Fano2--8}{2--8} & $(10,14,9)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{12}$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3}, \\mathcal{Q}_{\\mathbb{P}^2}(0, -1))$ \\\\\n\\rowcolor[gray]{0.95} 2--9 & $(11,16,5)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1) \\oplus \\mathcal{O}(1,2)$& \\cite{corti} \\\\\n\\hyperlink{Fano2--10}{2--10} & $(11,16,3)$& $\\Gr(2,4) \\times \\mathbb{P}^1$& $\\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1)$& \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--11}{2--11} & $(12,18,5)$& $\\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^2}(0,1) \\oplus \\mathcal{O}(1,2)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,3,5)$& $\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(2,1)$& \\\\\n2--12 & $(13,20,3)$& $\\mathbb{P}^3 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1)^{\\oplus 3}$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} 2--13 & $(13,20,2)$& $\\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,1)^{\\oplus 2} \\oplus \\mathcal{O}(0,2)$&\\cite{corti} \\\\\n2--14 & $(13,20,1)$& $\\Gr(2,5) \\times \\mathbb{P}^1$& $\\mathcal{O}(1,0)^{\\oplus 3} \\oplus \\mathcal{O}(1,1)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--15}{2--15} & $(14,22,4)$& $\\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,1)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,2)$& \\\\\n \\hyperlink{Fano2--16}{2--16} & $(14,22,2)$& $\\mathbb{P}^2 \\times \\Gr(2,4)$& $\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,2)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--17}{2--17} & $(15,24,1)$& $\\Gr(2,4) \\times \\mathbb{P}^3$& $\\mathcal{U}_{\\Gr(2,4)}^{\\vee}(0,1) \\oplus \\mathcal{O}(1,1) \\oplus \\mathcal{O}(1,0)$&\\cite{corti} \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{O}(0,1) \\oplus \\mathcal{O}(1,1)$& \\\\\n \\hyperlink{Fano2--18}{2--18} & $(15,24,2)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--19}{2--19} & $(16,26,2)$& $\\mathbb{P}^3 \\times \\mathbb{P}^5$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)^{\\oplus 2}$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,3,6)$& $\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(1,1)^{\\oplus 2}$& \\\\\n 2--20 & $(16,26,0)$& $\\Gr(2,5) \\times \\mathbb{P}^2$& $\\mathcal{U}^{\\vee}_{\\Gr(2,5)}(0,1) \\oplus \\mathcal{O}(1,0)^{\\oplus 3}$&\\cite{corti} \\\\\n \\rowcolor[gray]{0.95} 2--21 & $(17,28,0)$& $\\Gr(2,4) \\times \\mathbb{P}^2$& $\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(0,1)^{\\oplus 2} \\oplus \\mathcal{O}(1,0)$&\\cite{corti} \\\\\n \\hyperlink{Fano2--22}{2--22} & $(18,30,0)$& $\\mathbb{P}^3 \\times \\Gr(2,5)$& $\\mathcal{Q}_{\\Gr(2,5)}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}$&\\cite{corti} \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--23}{2--23} & $(18,30,1)$& $\\mathbb{P}^4 \\times \\mathbb{P}^5$& $\\mathcal{Q}_{\\mathbb{P}^4}(0,1) \\oplus \\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1)$& \\\\*\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,6)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,1)$& \\\\\n 2--24 & $(18,30,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,2)$& \\cite{isp5} \\\\ \n \\rowcolor[gray]{0.95} 2--25 & $(19,32,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,2)$&\\cite{corti} \\\\\n \\hyperlink{Fano2--26}{2--26} & $(20,34,0)$& $\\Gr(2,4) \\times \\Gr(2,5)$& $\\mathcal{Q}_{\\Gr(2,4)} \\boxtimes \\mathcal{U}_{\\Gr(2,5)}^{\\vee} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(2,3,5)$& $\\mathcal{U}_1^{\\vee} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}$& \\\\\n \\rowcolor[gray]{0.95} 2--27 & $(22,38,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1)^{\\oplus 2}$&\\cite{corti} \\\\\n \\hyperlink{Fano2--28}{2--28} & $(23,40,1)$& $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$& $\\Lambda(0,1) \\oplus \\mathcal{O}(1,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--29}{2--29} & $(23,40,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,1)$& \\\\\n \\hyperlink{Fano2--30}{2--30} & $(26,46,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--31}{2--31} & $(26,46,0)$& $\\mathbb{P}^2 \\times \\Gr(2,4)$& $\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,1)$& \\\\\n 2--32 & $(27,48,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1)$&\\cite{isp5} \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}}& $\\Fl(1,2,3)$& &\\cite{isp5} \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--33}{2--33} & $(30,54,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1)$& \\\\\n 2--34 & $(30,54,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2$& &\\cite{isp5} \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--35}{2--35} & $(31,56,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{Q}_{\\mathbb{P}^2}(0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{Q}_2$& \\\\\n \\hyperlink{Fano2--36}{2--36} & $(34,62,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\Lambda(0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n\n \n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--1}{3--1} & $(9,12,8)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$& $K(0,0,0,1) \\oplus \\mathcal{O}(0,0,0,2)$&$K \\in \\Ext^2_{(\\mathbb{P}^1)^3}(\\mathcal{O}(0,0,1)^{\\oplus 4},\\mathcal{O}(1,-1,-1))$ \\\\\n \\hyperlink{Fano3--2}{3--2} & $(10,14,3)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^5$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,2)$&$\\Lambda \\in \\Ext^1_{(\\mathbb{P}^1)^2}(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ \\\\\n \\rowcolor[gray]{0.95} 3--3 & $(12,18,3)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,2)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--4}{3--4} & $(12,18,2)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1$& $\\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,0,2,0) \\oplus \\mathcal{O}(0,1,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--5}{3--5} & $(13,20,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^7$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,1)^{\\oplus 2}$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ \\\\\n \\hyperlink{Fano3--6}{3--6} & $(14,22,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,2) \\oplus \\mathcal{O}(0,1,1)$& \\\\\n \\rowcolor[gray]{0.95} 3--7 & $(15,24,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,1,1)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--8}{3--8} & $(15,24,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(0,1,2) \\oplus \\mathcal{O}(1,1,0)$& \\\\\n \\rowcolor[gray]{0.95} & & & &\\multicolumn{1}{c}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1)),$} \\\\\n \\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano3--9}{3--9}}& \\multirow{-2}{*}{$(16,26,3)$} & \\multirow{-2}{*}{$\\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^{20}$} & \\multirow{-2}{*}{$\\Lambda(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^6}(0,0,1) \\oplus K(0,0,1)$}& \\multicolumn{1}{c}{$K \\in \\Ext^3_{\\mathbb{P}^2}(\\Sym^4\\mathcal{Q}, \\mathcal{Q}(-3))$}\\\\\n \\hyperlink{Fano3--10}{3--10} & $(16,26,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(0,0,2)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--11}{3--11} & $(17,28,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n \\hyperlink{Fano3--12}{3--12} & $(17,28,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,0,1)$& \\\\\n \\rowcolor[gray]{0.95} 3--13 & $(18,30,0)$& $(\\mathbb{P}^2)^3$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(1,0,1) \\oplus(0,1,1)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--14}{3--14} & $(19,32,1)$& $\\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^2$& $\\Lambda(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(1,0,0) \\oplus \\mathcal{O}(1,1,0)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1)) $ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--15}{3--15} & $(19,32,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n \\hyperlink{Fano3--16}{3--16} & $(20,34,0)$& $\\mathbb{P}^2_1 \\times \\mathbb{P}^2_2 \\times \\mathbb{P}^3$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} 3--17 & $(21,36,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,1)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--18}{3--18} & $(21,36,0)$& $\\mathbb{P}^1\\times \\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\Fl(1,2,5)$& $\\mathcal{Q}_2(0;0,0) \\oplus \\mathcal{O}(0;1,1) \\oplus \\mathcal{O}(1;0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--19}{3--19} & $(22,38,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,2) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,1)$& \\\\\n \\hyperlink{Fano3--20}{3--20} & $(22,38,0)$& $\\mathbb{P}^2_1 \\times \\mathbb{P}^2_2 \\times \\mathbb{P}^5$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_2}(0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--21}{3--21} & $(22,38,0)$& $\n \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\mathcal{O}(0,1,1) \\oplus \\Lambda(0,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1)) $ \\\\\n \\hyperlink{Fano3--22}{3--22} & $(23,40,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\mathcal{O}(1,0,1) \\oplus \\Lambda(0,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2 \\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--23}{3--23} & $(24,42,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0) \\oplus \\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)$& \\\\\n \\hyperlink{Fano3--24}{3--24} & $(24,42,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano3--25}{3--25} & $(25,44,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{O}(0,1)^{\\oplus 2}$& \\\\\n\\hyperlink{Fano3--26}{3--26} & $(26,46,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n\\rowcolor[gray]{0.95} 3--27 & $(27,48,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$& &\\cite{isp5} \\\\\n\\hyperlink{Fano3--28}{3--28} & $(27,48,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,0,1)$& \\\\\n\\rowcolor[gray]{0.95} & & & \\multicolumn{1}{c}{$\\mathcal{Q}_{\\mathbb{P}^2}(1,0,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1) \\oplus$} &\n\\\\\n\\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano3--29}{3--29}} & \\multirow{-2}{*}{$(28,50,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^3 \\times \\mathbb{P}^2 \\times \\mathbb{P}^9$}& \\multicolumn{1}{c}{$\\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(0,-1,1)$}&\\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1))$} \\\\\n\\hyperlink{Fano3--30}{3--30} & $(28,50,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano3--31}{3--31} & $(29,52,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,0)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2)$& \\\\\n4--1 & $(15,24,1)$& $(\\mathbb{P}^1)^4$& $\\mathcal{O}(1,1,1,1)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} & & & $\\mathcal{Q}_{\\mathbb{P}^3}(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^4}(0,0,1) \\oplus $ & \\\\\n\\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano4--2}{4--2}} & \\multirow{-2}{*}{$(17,28,1)$}& \\multirow{-2}{*}{$\\mathbb{P}^3 \\times \\mathbb{P}^4 \\times \\mathbb{P}^5$}& $\\oplus \\mathcal{O}(2,0,0) \\oplus \\mathcal{O}(0,1,1) $ &\\\\\n\n \\hyperlink{Fano4--3}{4--3} & $(18,30,0)$& $(\\mathbb{P}^1)^3 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,0,1) \\oplus \\mathcal{O}(0,0,1,1)$& \\\\\n \n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--4}{4--4} & $(19,32,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,0,2) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n \\hyperlink{Fano4--5}{4--5} & $(19,32,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1$& $\\mathcal{O}(0,1,1,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--6}{4--6} & $(20,34,0)$& $(\\mathbb{P}^1)^3 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,0,1) \\oplus \\mathcal{O}(0,1,0,1) \\oplus \\mathcal{O}(0,0,1,1)$& \\\\\n \\hyperlink{Fano4--7}{4--7} & $(21,36,0)$& $(\\mathbb{P}^1)^2 \\times (\\mathbb{P}^2)^2$& $\\mathcal{O}(0,0,1,1) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)$&\\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $ (\\mathbb{P}^1)^2 \\times \\Fl(1,2,3)$& $\\mathcal{O}(1,0; 1,0) \\oplus \\mathcal{O}(0,1;0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--8}{4--8} & $(22,38,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\Fl(1,2,5)$& $\\mathcal{O}(1;1,0) \\oplus \\mathcal{O}(0;0,2) \\oplus \\mathcal{Q}_2$& \\\\\n \\hyperlink{Fano4--9}{4--9} & $(23,40,0)$& $(\\mathbb{P}^1)^2 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--10}{4--10} & $(24,42,0)$& $(\\mathbb{P}^1)^3 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,0,0,1) \\oplus \\mathcal{O}(0,1,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1\\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n\n & & & \\multicolumn{1}{c}{$\\mathcal{O}(0,1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,0,1) \\oplus$}\\\\\n \\multirow{-2}{*}{\\hyperlink{Fano4--11}{4--11}}\n & \\multirow{-2}{*}{$(25,44,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1 \\times \\mathbb{P}^6$}& \\multicolumn{1}{c}{$\\oplus \\Lambda(0,0,0,1) \\oplus \\mathcal{O}(0,0,-1,1)$}&\\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{(\\mathbb{P}^1)^2}(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$} \\\\\n \n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--12}{4--12} & $(26,46,0)$& $ \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8}$& $\\mathcal{O}(1,1,0) \\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(-1,1,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ \\\\ \n \\hyperlink{Fano4--13}{4--13} & $(16,26,0)$& $ \\mathbb{P}^1_1 \\times \\mathbb{P}^1_2 \\times \\mathbb{P}^1_3 \\times \\mathbb{P}^4$& $\\mathscr{Z}(\\Lambda(0,0,0,1) \\oplus \\mathcal{O}(1,0,1,1)) $&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1_1 \\times \\mathbb{P}^1_2}(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ \\\\\n \\rowcolor[gray]{0.95} & & & $\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(-1,1,1,0) \\oplus $& \\\\\n \\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano5--1}{5--1}} & \\multirow{-2}{*}{$(17,28,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^{11}$}& $\\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^8}(0,0,0,1)$& \\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$}\\\\\n \n\n& & & $\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus$\n\\\\\n\\multirow{-2}{*}{\\hyperlink{Fano5--2}{5--2}} & \\multirow{-2}{*}{$(21,36,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^1$}& $\\oplus \\mathcal{O}(-1,1,1,0) \\oplus \\mathcal{O}(0,1,0,1)$&\\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$} \\\\\n\\rowcolor[gray]{0.95} 5--3 & $(21,36,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$& $\\mathcal{O}(1,1,0)^{\\oplus 2}$& \\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $(\\mathbb{P}^1)^4$& $\\mathcal{O}(1,1,1,0)$& \\cite{isp5} \\\\\n6--1 & $(18,30,0)$& $\\Gr(2,5) \\times \\mathbb{P}^1$& $\\mathcal{O}(1,0)^{\\oplus 4}$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 7--1 & $(15,24,0)$& $\\mathbb{P}^4 \\times \\mathbb{P}^1$& $\\mathcal{O}(2,0)^{\\oplus 2}$&\\cite{isp5} \\\\\n8--1 & $(12,18,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^1$& $\\mathcal{O}(3,0)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 9--1 & $(9,12,0)$& $\\mathbb{P}(1^3,2) \\times \\mathbb{P}^1$& $\\mathcal{O}(4,0)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$& $\\mathcal{O}(2,2,0)$&\\cite{isp5} \\\\\n10--1 & $(6,10,0)$& $\\mathbb{P}(1^2,2,3) \\times \\mathbb{P}^1$& $\\mathcal{O}(6,0)$&\\cite{isp5} \\\\\n\\end{longtable}\n\\end{scriptsize}\n\\end{centering}\n\n\n\\begin{longtable}{cccccrc}\n\\caption{Del Pezzo surfaces.}\\label{tab:delpezzo}\\\\\n\\toprule\n\\multicolumn{1}{c}{DP}&\\multicolumn{1}{c}{$K^2$}& \\multicolumn{1}{c}{X} & \\multicolumn{1}{c}{$\\mathcal{F}$} \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4}\n\\endfirsthead\n\\multicolumn{5}{l}{\\vspace{-0.25em}\\scriptsize\\emph{\\tablename\\ \\thetable{} continued from previous page}}\\\\\n\\midrule\n\\endhead\n\\multicolumn{5}{r}{\\scriptsize\\emph{Continued on next page}}\\\\\n\\endfoot\n\\bottomrule\n\\endlastfoot\n\\rowcolor[gray]{0.95} 1--1 & $9$& $\\mathbb{P}^2$& \\\\\n 2--1 & $8$& $\\mathbb{P}^1 \\times \\mathbb{P}^1$& \\\\\n\\rowcolor[gray]{0.95} 2--2 & $8$& $\\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1) $\\\\\n\n3--1 & $7$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1)$\\\\\n\\rowcolor[gray]{0.95} 4--1 & $6$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1)^{\\oplus 2}$\\\\\n\\rowcolor[gray]{0.95} \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $(\\mathbb{P}^1)^3$& $\\mathcal{O}(1,1,1)$\\\\\n5--1 & $5$& $\\Gr(2,5)$& $\\mathcal{O}(1)^{\\oplus 4}$\\\\\n\\rowcolor[gray]{0.95} 6--1 & $4$& $\\mathbb{P}^4 $& $\\mathcal{O}(2)^{\\oplus 2}$\\\\\n7--1 & $3$& $\\mathbb{P}^3 $& $\\mathcal{O}(3)$\\\\\n\\rowcolor[gray]{0.95} 8--1 & $2$& $\\mathbb{P}(1^3,2)$& $\\mathcal{O}(4)$\\\\\n\\rowcolor[gray]{0.95} \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(2,2)$\\\\\n9--1 & $1$& $\\mathbb{P}(1^2,2,3)$& $\\mathcal{O}(6)$\\\\\n\n\\end{longtable}\n\n\\frenchspacing\n\n\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\\pagestyle{myheadings}\n\\markboth{\\sc Churchill \\& Vogt \\hfill The Halo: QSO Absorption~~}\n {\\sc Churchill \\& Vogt \\hfill The Halo: QSO Absorption~~}\n\nAs Sidney van den Bergh stated in his closing comments of the meeting,\nthere are two general approaches to studying the history and\nevolution of the Galactic Halo. \nThe first is to study the kinematic, chemical, and structural\ncomponents of the Halo itself, and the second is to study the \nhalos of external galaxies with the partial aim of placing the Halo in\nthe broader context offered by various morphologies and environments\n(cf.~Morrison, this volume).\nGenerally, the tracers of Halo formation (e.g.~globular clusters,\nRR~Lyrae stars, Blue Horizontal Branch stars, star counts, etc.) are\ntreated as frozen relics of past formation processes.\nThe hope is that these processes have left signatures, such as\nkinematic trends or metallicity gradients, which can be used to\nclearly discern between competing formation scenarios.\n\nOne important tracer, which to a certain degree has not captured the\nattention of those studying Milky Way evolution {\\it per se},\nis low density gas in the Halo.\nUsing the absorption lines seen in the spectra of background quasars\n(QSO Absorption Lines, or QALs), several groups have carefully mapped\nout the sky locations of large cloud complexes (cf.~van~Woerden, this\nvolume), including high--velocity clouds (HVCs), which by themselves\nhave a sky covering factor of 38\\%.\nThe connection between HVCs and Halo tracers is not known, though\nMajewski (this volume) has reported stellar moving groups which\ncorrelate well with the locations and velocities of HVCs toward\nthe North Galactic Pole.\nFrom QAL surveys of galaxies at redshifts $0.4 \\leq z \\leq\n2.2$, gaseous halos of normal galaxies are known to extend to $\\sim\n70$~kpc and to contain an estimated $10^{9}$--$10^{10}$~M$_{\\sun}$ of\ngas (Steidel 1993; Steidel \\& Sargent 1992).\nHalo gas is the reservoir for star formation and chemical evolution, \nand plays a central role in the formation of the tracers used to study \nHalo formation. \nIts study in early epoch galaxies promises to yield important\nclues to the processes that regulated galaxy formation\n(cf.~Bechtold, this volume).\n\n\\section*{Learning about the Milky Way from QALs}\n\nQAL studies are unique in that they directly probe galactic gas over\nthe entire history of galaxy evolution.\nThus, they provide a powerful method from which to indirectly\n``view'' Milky Way Halo formation over a multi--billion year period\nand offer a very broad context within which studies using Halo tracers\ncan be placed.\nIn tandem with high spatial resolution imaging (Hubble Space Telescope)\nand high quality spectra of the absorbing galaxies themselves, the\npotential provided by QAL studies can be fully realized.\n\nFrom high resolution QAL spectra (see Fig.~1), we can measure the\nnumber of clouds intercepted in a galaxy, their column densities,\nbroadening mechanisms, and line--of--sight velocities. \nHST images provide the absorbing galaxies' environments, luminosities,\ncolors, impact parameters to the QSO (projected galactocentric\ndistance of the absorbing clouds), and orientations relative to the\nline of sight.\nSpectra of the galaxies can be used to estimate star formation rates,\nand (with LRIS on Keck) obtain rotation curves out to $z \\sim 1$.\nCase by case, we can study the physical details of absorbing gas and\nits relationship to the host galaxy, and then piece together a\ncomprehensive picture of halo evolution directly from the large range of\nepochs the galaxies sample.\nThe unexplored connections between absorption properties and the\nlocations probed in galaxies, their morphologies, redshifts, and\nenvironments will ultimately be used to develop a global picture of\nkinematic and chemical evolution.\n\nAs a first step, we have observed 24 QSOs with the HIRES spectrograph\n(Vogt et~al.~1994). \nWe have resolved absorption profiles of the Mg~{\\sc ii}\n$\\lambda\\lambda2796,2803$ resonant doublet in $\\sim 50$ intervening\ngalaxies.\nMany of these galaxies have been ground--based imaged in the\nIR\/optical and spectroscopically confirmed to have the redshift seen\nin absorption (Steidel 1995; Steidel, Dickinson, \\& Persson 1996).\n\n\n\n\\section*{Gaseous Fragments and Kinematic Evolution}\n\nIn this contribution, we present partial results and a brief discussion\nof work to be published elsewhere (Churchill \\& Vogt 1996).\nIn Fig.~1, we show four HIRES\/Keck Mg~{\\sc ii} absorption profiles\nas seen in the spectra of background QSOs.\nThese absorption lines arise in low ionization gas, which also gives\nrise to Mg~{\\sc i} and Fe~{\\sc ii} transitions.\nGenerally, these profiles appear to exhibit ``characteristics''\nrelated to the location and structure probed by the QSO line of sight.\nIn particular, we note the complex high velocity spread in the\n$z=0.51$ galaxy toward Q1254 and the $z=0.92$ galaxy toward Q1206.\nOne could interpret the optically--thick components of these profiles\nas arising from the disks of these galaxies.\nThe HVC--like optically--thin components are more difficult to\nunderstand.\nHowever, they are highly suggestive of a picture in which material in\ngalaxy halos is comprised of kinematically and physically distinct\nclumps, consistent with the Searle--Zinn (1978) picture of halo formation.\n\n\nThere is a similarity between the features of the $z=1.17$ profile\ntoward Q1421 and that of the optically--thick components of Q1206.\nThese two QALs may arise in similar structures, or parts of\nthe galaxies.\nThe $z=1.55$ system toward Q1213 may be a merging or double galaxy,\nthe strength variations in the ``double'' profile being due to\nthe different line--of--sight orientations, morphologies, and\/or\nmasses of the galaxies.\nHST imaging would be decisive in testing these conjectures.\nSuch inferences can be drawn from QAL studies of local galaxies.\nBowen, Blades, \\& Pettini (1995) have observed a ``double'' profile\nsimilar to that of Q1213 that samples a line of sight passing\nthrough both M81 and the Milky Way and spans 400~km~s$^{-1}$.\nChurchill, Vogt, \\& Steidel (1995) have observed a possible double\ngalaxy at $z=0.74$ that exhibits a richly structured ``double''\nprofile spanning 300~km~s$^{-1}$.\nIn a ground--based image, there are two galaxies of nearly equal\nmagnitude (redshift?), each with impact parameter $\\sim 20$~kpc.\n\nIn Fig.~2, we present the probability, $P(\\Delta v)$, of observing any\ntwo clouds with line--of--sight velocity difference $\\Delta v$.\nThe cloud--cloud velocity dispersion within halos exhibits strong\nredshift evolution, becoming tighter as redshift decreases such that\nby a look--back time of $\\sim$ 8--10$h^{-1}$~Gyr\n($h=H_0\/100$~km~s$^{-1}$ Mpc$^{-1}$) Mg~{\\sc ii} absorbing clouds have\na mean velocity dispersion $\\sim 60$~km~s$^{-1}$.\nSuch {\\it pronounced}\\\/ evolution may be due to biasing in\nour sample selection, since we targeted an absorber population known\nto exhibit evolution (Steidel \\& Sargent 1992).\nSince there is no apparent evolution in the sizes of halos (Steidel \\&\nSargent 1992), or the number of clouds (this study), the implications\nare that an observable shift may be occurring in the mechanisms giving\nrise to significant amounts of high velocity material as early as\n$\\sim 8$ billion years ago.\nPerhaps we are seeing the epoch at which the frequency of dwarf\nsatellite galaxy accretion onto their primary slows considerably. \n\n\\acknowledgments\nWe would like to thank the Organizing Committee for hosting an\nenjoyable meeting and a small travel grant for CWC.\nThanks to Jane Charlton, Ken Lanzetta, and Chuck Steidel for \ninsightful on--going discussions.\nThis work has been supported in part by the Sigma Xi Grants--in--Aid of\nResearch program, the California Space Institute, and NASA\ngrant NAGW--3571.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\par\nThe $125~ {\\rm GeV}$ Higgs boson was found by the ATLAS and CMS collaborations in 2012, which pastes the last brick of the Standard Model (SM). While physicists were celebrating the success of the SM, several problems emerged in front of us. Definitely we all know by bottom of hearts that the SM by no means is the final theory of the Nature but only an effective one at the concerned energy scale. Its loopholes, such as the naturalness of Higgs boson and the vacuum stability, all compose serious challenge to our knowledge. Moreover, some experimental phenomena show slight deviations from the SM predictions which reveal traces of new physics beyond SM (BSM). Indeed, the target of high energy society is searching for BSM and it especially is the task of LHC which historically succeeded in the SM Higgs boson discovery. Among all possible signals of BSM, the most favorable and significant signal of BSM is the existence of a new Higgs-like boson(s) which is predicted by many new models about BSM.\n\n\\par\nRecently, a preliminary investigation based on the report by S. Durgut, which is available at the American Physical Society (APS) site \\cite{Durgut}, shows a structure around $18.4~ {\\rm GeV}$ in four-lepton final state by using the four-lepton events collected during the LHC Run I stage. Below, we just refer the report as D-report. After carefully analysis, they conclude that an enhancement exists at around $18.4~ {\\rm GeV}$ in the invariant mass distribution of $\\Upsilon l^+l^- ~ (l = e,\\, \\mu)$ \\cite{Durgut,Yi:2018fxo,phdpaper}.\n\n\\par\nThe enhancement is conjectured to occur through the process $pp \\rightarrow \\Upsilon l^+ l^- \\rightarrow \\mu^+\\mu^-l^+l^-$. Namely if $\\Upsilon l^+l^-$ comes from a unique enhancement, it would be a neutral boson $\\phi$, which is mainly produced via gluon-gluon fusion, and then sequently decays as $\\Upsilon\\Upsilon^* \\rightarrow \\mu^+\\mu^- l^+l^-$, where $\\Upsilon^*$ might be off-mass-shell due to the energy constraint. The mass of the new enhancement, if it indeed exists, is a few hundreds of MeV lower than the total mass of a $\\Upsilon$ pair. By analyzing the datasets collected by the CMS detector at a center-of-mass energy of $7~ {\\rm TeV}$ and $8~ {\\rm TeV}$ with an integrated luminosity of $25.6~ {\\rm fb}^{-1}$ \\cite{phdpaper}, and taking the kinematic requirements of $p_{T,l} > 2.0~ {\\rm GeV}$ and $\\left| \\eta_{l} \\right| < 2.4$ on the final-state leptons \\cite{Durgut,phdpaper}, the experimenters observed a peaking structure. By their rigorous analysis, the peak is located at $18.4 \\pm 0.1(stat.) \\pm 0.2(syst.)~ {\\rm GeV}$ for the $\\Upsilon \\mu^+\\mu^-$ and $\\Upsilon e^+e^-$ channels combined; event numbers are $44 \\pm 13$ and $35 \\pm 13$ for the $\\Upsilon \\mu^+\\mu^-$ and $\\Upsilon e^+e^-$ channels, respectively, and its significance is $3.6$ standard deviation after taking into account the look-elsewhere-effect \\cite{Durgut,phdpaper}. Moreover, the ANDY collaboration also claims that a significant peak at $m = 18.12 \\pm 0.15(stat.) \\pm 0.6(syst.)~ {\\rm GeV}$ in the dijet mass distribution is observed in Cu+Au collision at $\\sqrt{s_{NN}} = 200~{\\rm GeV}$ at the Relativistic Heavy Ion Collider \\cite{Bland:2019aha}. This result is in good agreement with the four-lepton signal observed at the $7~ {\\rm TeV}$ and $8~ {\\rm TeV}$ LHC by the CMS collaboration \\cite{Durgut,phdpaper}.\n\n\\par\nBecause this enhancement is close to the sum of the masses of four bottom quarks, thus some authors consider it to be a $bb\\bar{b}\\bar{b}$ tetraquark state with a mass in the range of $18.4~ {\\rm GeV} < m_{X_{bb\\bar{b}\\bar{b}}} < 20.3~ {\\rm GeV}$ \\cite{Berezhnoy:2011xn,Wu:2016vtq,Chen:2016jxd,Bai:2016int,Anwar:2017toa,Richard:2017vry,Wang:2017jtz}. The authors of Ref.\\cite{Karliner:2016zzc} show that $\\sigma(pp \\rightarrow X_{bb\\bar{b}\\bar{b}}[0^{++}] \\rightarrow 4l)\\leqslant 4~ {\\rm fb}$ at $\\sqrt{s} = 13~ {\\rm TeV}$ and $\\leqslant 2~ {\\rm fb}$ at $\\sqrt{s} = 8~ {\\rm TeV}$. In Ref.\\cite{Esposito:2018cwh} the authors emphasize that their work was motivated by the peak at $18.4~ {\\rm GeV}$ based on the tetraquark hypothesis, but the numerical result shows that the partial width for $X_{bb\\bar{b}\\bar{b}} \\rightarrow \\Upsilon\\mu^+\\mu^-$ is too small to tolerate the data currently observed at the LHC. In Ref.\\cite{Becchi:2020mjz} the authors calculate the decay width for $X_{bb\\bar b\\bar b} \\rightarrow \\Upsilon l^+ l^-$ and they prefer $X_{bb\\bar{b}\\bar{b}}$ to be a $2^{++}$ tetraquark state rather than a $0^{++}$ bound state. Furthermore, the authors of Ref.\\cite{Hughes:2017xie} believe that a $bb\\bar{b}\\bar{b}$ tetraquark should lie above the lowest noninteracting bottomonium-pair threshold.\n\n\\par\nBecause the mass is just a bit below the sum of two $\\Upsilon$ bosons, being driven by the expectation of searching for a BSM Higgs-like boson, it is tempted to conjecture the newly observed enhancement to be a $0^{++}$ fundamental boson. In this work, our purpose is to check if the idea could be tolerated by the experimental observation. To serve this goal, we calculate the production rate of $\\Upsilon l^+l^-$ at the LHC by assuming the peak observed in experiment to be real, and then estimate the full contribution from the $18.4~ {\\rm GeV}$ structureless BSM boson $\\phi(18.4)$ as well as the corresponding SM background. In this paper, the $0^{++}$ enhancement $\\phi(18.4)$ is assumed as a BSM Higgs-like boson with mass around $18.4~ {\\rm GeV}$. It should be noted that if $\\phi(18.4)$ were indeed a Higgs-like boson, and its width is large enough, a threshold effect would induce an asymmetric peak in the invariant mass spectrum of $\\Upsilon l^+l^-$ which could be observed in experiments. Thus we estimate the possibility by numerically calculating the production of $\\Upsilon l^+l^-$. We eventually find that the production rate induced by the Higgs-like boson $\\phi(18.4)$ is too small to be observed in the LHC with presently available experimental condition. It means that if we deliberately postulate a large width for $\\phi(18.4)$, the contribution of the supposed BSM model may generate an experimentally observed peak around $18.4~ {\\rm GeV}$, however the data says no.\n\n\\par\nDiscovering new physics beyond SM should begin with looking for a new extra Higgs-like boson(s), this strategy is commonly accepted by both experimentalists and theorists in the high energy physics society. So far by now, many BSM models predict various kinds of new Higgs-like bosons (for example, neutral, charged, $\\mathcal{CP}$-odd or $\\mathcal{CP}$-even etc., even doubly-charged bosons). Unfortunately, none of them was found in present experiments so far. When looking back, we find that almost all the particles predicted by those BSM models are much heavier than the SM scale, namely it varies from few hundreds of GeV to few hundreds of TeV. Such BSM particles cannot be produced in present experimental facilities. On other aspect, there does not exist a principle forbidding the existence of lighter BSM particles. For example, the two-Higgs-doublet-model may predict a new Higgs-like boson with a mass of $28~ {\\rm GeV}$ \\cite{Cici:2019zir}. By the general method adopted for searching TeV-scale particles at LHC, alternatively, we, in this work, explore a new Higgs-like boson at low energy regions. As a common sense the strategy can be traced back from our experience gained at lepton colliders, such as BES, Belle, etc. For example, in the scattering process $e^+e^- \\rightarrow J\/\\psi \\rightarrow \\text{{\\it final products}}$, the resonance ($J\/\\psi$) overwhelmingly dominates the portal, while the direct production just provides a continuous background. For the same cause, a direct production of four leptons from the gluon-gluon fusion at the proton-proton collider, i.e., $gg \\rightarrow \\Upsilon \\Upsilon^* \\rightarrow \\mu^+\\mu^- l^+l^-$ \\cite{Li:2009ug,Qiao:2009kg} where $\\Upsilon^*$ might be off-mass-shell, should just generate a background. If a medium Higgs-like boson $\\phi(18.4)$ indeed exists, it induces the portal of $\\Upsilon \\Upsilon^* \\rightarrow \\Upsilon l^+l^-$, a peak would appear in the invariant mass spectrum of $\\Upsilon l^+l^-$. With this assertion, we numerically calculate the contribution induced by the BSM Higgs-like boson $\\phi(18.4)$ to $pp \\rightarrow \\Upsilon \\Upsilon^* \\rightarrow \\Upsilon l^+l^-$ at the LHC. Comparing our numerical results with those in the D-report \\cite{Durgut}, we find that the assumption that the observed peak in the $\\Upsilon l^+l^-$ mass spectrum originates from a BSM Higgs-like boson decay should be ruled out.\n\n\n\\par\nThis work is organized as follows. After this introduction, in section II, we present our analytical calculation for $\\Upsilon l^+l^-$ production at the LHC in the framework of a BSM model, in which we assume that the interaction of the BSM Higgs-like boson with SM particles is in analogue to that of the SM Higgs boson. In section III, we numerically evaluate all corresponding quantities and illustrate the invariant mass distribution of the final state. The last section is devoted to our conclusion and a brief discussion.\n\n\n\\section{Analytical calculation for $pp \\rightarrow \\Upsilon l^+l^-$}\nAt the LHC, $\\Upsilon l^+l^-$ is mainly produced via gluon-gluon fusion \\cite{Georgi:1977gs,Anastasiou:2002yz}, i.e.,\n\\begin{eqnarray}\n\\sigma[pp \\rightarrow \\Upsilon l^+l^-] = \\int dx_1 dx_2\\, f(x_1, \\mu_F)\\, f(x_2, \\mu_F)\\, \\hat{\\sigma}[gg \\rightarrow \\Upsilon l^+l^-],\n\\label{eqn1}\n\\end{eqnarray}\nwhere $f(x, \\mu_F)$ is the gluon distribution function in proton, $\\mu_F$ is the factorization scale, while other production channels are neglected.\n\n\\par\nThe contribution of the BSM Higgs-like boson comes from the Breit-Wigner propagator $\\dfrac{1}{p^2-m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi}}$, where $p$ is the four-momentum flowing through the intermediate BSM Higgs-like boson. If we do not consider the interference with the SM background and neglect the $t$- and\n$u$-channel Feynman diagrams induced by the BSM Higgs-like boson, this contribution will be proportional to the square of the Breit-Wigner propagator $\\dfrac{1}{( s - m_{\\phi}^2 )^2 + m_{\\phi}^2 \\Gamma_{\\phi}^2}$, where $s = (k_1+k_2)^2$ and $k_i~ (i = 1, 2)$ are the four-momenta of the two initial-state gluons. When $s$ is close to $m_{\\phi}^2$, this factor turns into $\\dfrac{1}{m_{\\phi}^2 \\Gamma_{\\phi}^2}$ and a resonance would peak up from the\nbackground. However, if $s$ is far away from $m_{\\phi}^2$ (below or above), the contribution of $\\phi$ would be drowned into the background and no peak can be seen. In our case, $18.4~ {\\rm GeV}$ is slightly below the threshold of $2 m_{\\Upsilon}$. However, since its position is not too far from the threshold value and it possesses a relatively large width, the resonance effect still can manifest itself in the invariant mass spectrum of $\\Upsilon$ pair at the threshold. In one aspect, the mass of $\\phi$ cannot be larger than $2 m_{\\Upsilon}$, otherwise a peak at the $\\Upsilon$ pair invariant mass spectrum would be seen, but no such peak was experimentally observed.\n\n\\par\nTo evaluate the contribution of the supposed BSM Higgs-like boson $\\phi$ of $18.4~ {\\rm GeV}$ to the $\\Upsilon l^+l^-$ production at the LHC, we write up the complete expression where the Breit-Wigner propagator of $\\phi$ with a width observed in the concerned experiment would induce the peak in the $\\Upsilon l^+l^-$ invariant mass spectrum. Later our numerical results show that one only needs to account the contribution of the resonance above the threshold of $2 m_{\\Upsilon}$. Indeed, because $18.4~ {\\rm GeV}$ is smaller than 2$m_{\\Upsilon}$, $\\phi$ cannot be on its mass-shell for two on-shell $\\Upsilon$s, while the production rate for $pp \\rightarrow \\phi \\rightarrow \\Upsilon\\Upsilon^* \\rightarrow \\Upsilon l^+l^-$ is very tiny and can be neglected. Due to the extremely small decay width of $\\Upsilon$ ($\\Gamma_{\\Upsilon} \\sim 50~ {\\rm keV}$ and $\\Gamma_{\\Upsilon} \\ll \\Gamma_{\\phi}$), the parton-level cross section for the production of $\\Upsilon l^+l^-$ via gluon-gluon fusion can be written as\n\\begin{eqnarray}\n\\hat{\\sigma}[gg \\rightarrow \\Upsilon l^+l^-] \\simeq \\hat{\\sigma}[gg \\rightarrow \\Upsilon \\Upsilon]\n\\times\n2 \\, Br(\\Upsilon \\rightarrow l^+l^-).\n\\end{eqnarray}\nThe cross section $\\hat{\\sigma}[gg \\rightarrow \\Upsilon \\Upsilon]$ is given by\n\\begin{eqnarray}\n\\hat{\\sigma}[gg \\rightarrow \\Upsilon \\Upsilon] = \\int d \\Omega \\, \\left| \\mathcal{M}_{SM} + \\mathcal{M}_{\\phi} \\right|^2,\n\\label{sigmahat}\n\\end{eqnarray}\nwhere $\\mathcal{M}_{SM}$ and $\\mathcal{M}_{\\phi}$ represent the Feynman amplitudes in the SM and induced by the BSM Higgs-like boson $\\phi$, respectively. It is noted that the above formula is a general expression where we do not specially require the intermediate boson $\\phi$, if it indeed exists in the nature, to be real or virtual. The production of $\\Upsilon$ pair via gluon-gluon fusion at hadron colliders has been much investigated in the framework of the SM \\cite{Li:2009ug,Qiao:2009kg}. The $31$ Feynman diagrams for $gg \\rightarrow \\Upsilon \\Upsilon$ in the SM can be created with the help of {\\sc FeynArts} \\cite{Hahn:2000kx} package. We also calculate this process with the same input parameters as given in Ref.\\cite{Li:2009ug} for comparison, and find that our numerical result for the production cross section at the $14~ {\\rm TeV}$ LHC is in good agreement with the corresponding one of Ref.\\cite{Li:2009ug} within a tolerable calculation error. Then we step on to calculate the quantities concerning the new Higgs-like boson.\n\n\\par\nAs for the BSM contribution from the new Higgs-like boson $\\phi$, we consider only the $gg\\phi$ and $b\\bar{b}\\phi$ effective couplings for the Higgs-like boson. The Feynman diagrams induced by the BSM Higgs-like boson $\\phi$ can be classified into two categories. The diagrams in Fig.\\ref{fig1} are independent of the $b\\bar{b}\\phi$ coupling, while all the diagrams in Fig.\\ref{fig2} depend on both $gg\\phi$ and $b\\bar{b}\\phi$ couplings.\n\\begin{figure}[htbp]\n \\centering\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig1a.eps}}\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig1b.eps}} \\\\\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig1c.eps}}\n\\caption{Feynman diagrams for $gg \\rightarrow \\Upsilon\\Upsilon$ induced by the BSM Higgs-like boson $\\phi$ via the $gg\\phi$ effective coupling.}\n\\label{fig1}\n\\end{figure}\n\\begin{figure}[htbp]\n \\centering\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2a.eps}}\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2b.eps}} \\\\\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2c.eps}}\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2d.eps}}\n\\caption{Feynman diagrams for $gg \\rightarrow \\Upsilon\\Upsilon$ induced by the BSM Higgs-like boson $\\phi$ via both $gg\\phi$ and $b\\bar{b}\\phi$ effective couplings.}\n\\label{fig2}\n\\end{figure}\n\n\\par\nFollowing Refs.\\cite{Anastasiou:2015ema,Spira:2016ztx}, the effective coupling of the BSM Higgs-like boson to two gluons can be written as\n\\begin{eqnarray}\n\\label{VggH}\n\\mathcal{C}_{gg\\phi}^{\\mu \\nu}(k_1,\\, k_2)\n=\n-i\\, \\dfrac{g_{gg\\phi}(\\mu_R)}{m_{\\phi}} \\left[ 4k_1 \\cdot k_2 \\Big( g^{\\mu\\nu}-\\frac{k_1^\\nu k_2^\\mu}{k_1\\cdot k_2} \\Big) \\right],\n\\end{eqnarray}\nwhere $k_1,~ k_2$ and $\\mu,~ \\nu$ are the four-momenta and Lorentz indices of the two gluons, respectively, $g_{gg\\phi}(\\mu_R)$ is a dimensionless effective running coupling constant, and $\\mu_R$ is the renormalization scale. It is reasonable to assume that the evolution of the effective coupling constant $g_{gg\\phi}$ is the same as that of the QCD $\\alpha_s$, i.e., $\\dfrac{g_{gg\\phi}(\\mu_R)}{\\alpha_s(\\mu_R)}$ is independent of $\\mu_R$. Thus, we obtain the quark-level amplitude for the Feynman diagrams in Fig.\\ref{fig1} as\n\\begin{eqnarray}\n\\widetilde{\\mathcal{M}}_{\\phi}^{(g)}\n=\n-i \\frac{4 \\pi \\alpha_s(\\mu_R) \\lambda_{{\\rm color}}}{(p_1 + q_2)^2 (p_2 + q_1)^2}\\,\n\\epsilon_\\mu(k_1) \\epsilon_\\nu(k_2)\\,\n{\\rm Tr} \\Big[ v(p_2) \\bar{u}(p_1) \\gamma_{\\alpha} v(q_2) \\bar{u}(q_1) \\gamma_{\\beta} \\Big]\n\\Big(\n\\mathcal{S} + \\dfrac{\\mathcal{T} + \\mathcal{U}}{8}\n\\Big),\n\\end{eqnarray}\nwhere $\\lambda_{{\\rm color}} \\equiv \\delta^{ab} {\\rm Tr}(T^aT^b) = 4$, and $\\mathcal{S}$, $\\mathcal{T}$ and $\\mathcal{U}$ are given by\n\\begin{eqnarray}\n&&\n\\mathcal{S} =\n\\mathcal{C}^{\\mu\\nu}_{gg\\phi}(k_1,\\, k_2)\\,\n\\mathcal{C}^{\\alpha\\beta}_{gg\\phi}(p_1+q_2,\\, p_2+q_1)\n\\left\/\n\\Big[ (k_1 + k_2)^2 - m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi} \\Big]\n\\right.,\n \\\\\n&&\n\\mathcal{T} =\n\\mathcal{C}^{\\mu\\alpha}_{gg\\phi}(k_1,\\, p_1+q_2)\\,\n\\mathcal{C}^{\\nu\\beta}_{gg\\phi}(k_2,\\, p_2+q_1)\n\\left\/\n\\Big[ (p_1 + q_2 - k_1)^2 - m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi} \\Big]\n\\right.,\n \\\\\n&&\n\\mathcal{U} =\n\\mathcal{T}\\,\\Big|_{k_1 \\leftrightarrow k_2,~ \\mu \\leftrightarrow \\nu}\\,.\n\\end{eqnarray}\nThe effective coupling of the BSM Higgs-like boson to the bottom quarks is parameterized as\n\\begin{eqnarray}\n\\label{VbbH}\n\\mathcal{C}_{b\\bar{b}\\phi} = -i\\, g_{b\\bar{b}\\phi}(\\mu_R).\n\\end{eqnarray}\nWe assume that the evolution of the effective coupling constant $g_{b\\bar{b}\\phi}$ is the same as that of the bottom-quark $\\overline{{\\rm MS}}$ running mass $\\overline{m}_b(\\mu_R)$ \\cite{Bednyakov:2016onn,Tanabashi:2018oca}, i.e., $\\dfrac{g_{b\\bar{b}\\phi}(\\mu_R)}{\\overline{m}_b(\\mu_R)}$ is independent of $\\mu_R$. Then the quark-level amplitude for the Feynman diagrams in Fig.\\ref{fig2} can be expressed as\n\\begin{eqnarray}\n\\widetilde{\\mathcal{M}}_{\\phi}^{(b)}\n=\ni \\frac{4 \\pi \\alpha_s(\\mu_R) \\lambda_{{\\rm color}}}{(k_1 + k_2)^2 - m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi}}\\,\n\\epsilon_\\mu(k_1) \\epsilon_\\nu(k_2)\\,\n\\mathcal{C}^{\\mu\\nu}_{gg\\phi}(k_1,\\, k_2)\\,\n\\mathcal{C}_{b\\bar{b}\\phi}\n\\Big(\n\\mathcal{F}_a + \\mathcal{F}_b + \\mathcal{F}_c + \\mathcal{F}_d\n\\Big),\n\\end{eqnarray}\nwhere $\\mathcal{F}_a$, $\\mathcal{F}_b$, $\\mathcal{F}_c$ and $\\mathcal{F}_d$ are given by\n\\begin{eqnarray}\n&&\n\\mathcal{F}_a\n=\n\\dfrac{1}{(p_2 + q_1)^2}\\,\n{\\rm Tr}\n\\biggl[\nv(p_2) \\bar{u}(p_1) \\gamma^{\\alpha}\n\\dfrac{1}{(\\slashed{k}_1 + \\slashed{k}_2) - \\slashed{q}_2 - m_b}\nv(q_2) \\bar{u}(q_1) \\gamma_{\\alpha}\n\\biggr],\n\\\\\n&&\n\\mathcal{F}_b\n=\n\\dfrac{1}{(p_2 + q_1)^2}\\,\n{\\rm Tr}\n\\biggl[\nv(p_2) \\bar{u}(p_1)\n\\dfrac{1}{\\slashed{p}_1 - (\\slashed{k}_1 + \\slashed{k}_2) - m_b} \\gamma^{\\alpha}\nv(q_2) \\bar{u}(q_1) \\gamma_{\\alpha}\n\\biggr],\n\\\\\n&&\n\\mathcal{F}_c\n=\n\\mathcal{F}_b\\,\\Big|_{p_1 \\leftrightarrow q_1,~ p_2 \\leftrightarrow q_2}\\,,\n\\\\\n&&\n\\mathcal{F}_d\n=\n\\mathcal{F}_a\\,\\Big|_{p_1 \\leftrightarrow q_1,~ p_2 \\leftrightarrow q_2}\\,.\n\\end{eqnarray}\nWithin the framework of NRQCD \\cite{Hao:2006nf}, the hadron-level amplitudes $\\mathcal{M}_{\\phi}^{(g)}$ and $\\mathcal{M}_{\\phi}^{(b)}$ can be obtained from the quark-level amplitudes $\\widetilde{\\mathcal{M}}_{\\phi}^{(g)}$ and $\\widetilde{\\mathcal{M}}_{\\phi}^{(b)}$, respectively, by performing the following replacement:\n\\begin{eqnarray}\n&&\nv(p_2)\\bar{u}(p_1)\n~~\\longrightarrow~~\n\\dfrac{1}{2\\sqrt{2}} \\, \\rlap\/\\epsilon^*_{\\Upsilon} \\big( \\rlap\/p + m_{\\Upsilon} \\big)\n\\dfrac{1}{\\sqrt{m_b}}\n\\Psi_{\\Upsilon}(0) \\dfrac{1}{\\sqrt{N_c}}\n\\\\\n&&\nv(q_2)\\bar{u}(q_1)\n~~\\longrightarrow~~\n\\dfrac{1}{2\\sqrt{2}} \\, \\rlap\/\\epsilon^*_{\\Upsilon} \\big( \\rlap\/q + m_{\\Upsilon} \\big)\n\\dfrac{1}{\\sqrt{m_b}}\n\\Psi_{\\Upsilon}(0) \\dfrac{1}{\\sqrt{N_c}}\n\\\\\n&&\np_1 = p_2 = \\dfrac{p}{2}\n\\\\\n&&\nq_1 = q_2 = \\dfrac{q}{2}\n\\end{eqnarray}\n\n\\par\nThe analytic expression of the SM amplitude for $gg \\rightarrow \\Upsilon\\Upsilon$ (i.e., $\\mathcal{M}_{SM}$) can be obtained analogously, but is not presented here since it is too tedious. Through the standard manipulations, we obtain the cross section $\\hat{\\sigma}[gg \\rightarrow \\Upsilon\\Upsilon]$ (Eq.(\\ref{sigmahat})), and then a convolution with the gluon distribution function results in the cross section for $pp \\rightarrow \\Upsilon\\Upsilon$. In next section we will show our numerical results clearly.\n\n\n\\section{Numerical results}\n\\par\nS. Durgut reported a peak in the invariant mass distribution of $\\Upsilon l^+l^-$ at the energy of $18.4~ {\\rm GeV}$ \\cite{Durgut,phdpaper}. A naive conjecture suggests that the peak at $M_{\\Upsilon l^+l^-} \\sim 18.4~ {\\rm GeV}$ is induced by a BSM Higgs-like boson. Our goal is to check if this scenario works. In this work, the event samples are generated by using {\\sc FormCalc} \\cite{Hahn:2016ebn} package based on the Monte Carlo technique. In the numerical calculation, the mass of the BSM Higgs-like boson is set as $18.4~ {\\rm GeV}$, and thus we denote this Higgs-like boson as $\\phi(18.4)$. The factorization and renormalization scales are set to the transverse energy of the final-state $\\Upsilon$, i.e., $\\mu_F = \\mu_R = \\sqrt{m_{\\Upsilon}^2 + p_{T,\\Upsilon}^{\\,2}}$. The masses of $b$-quark and $\\Upsilon$ are taken as $m_b = 4.73~ {\\rm GeV}$ and $m_{\\Upsilon} = 9.46~ {\\rm GeV}$ \\cite{Tanabashi:2018oca}. Within the framework of NRQCD, the zero point wave function of $\\Upsilon$ and the branching ratios of $\\Upsilon$ to $\\mu^+\\mu^-$ and $e^+e^-$ are taken as $\\Psi_{\\Upsilon}^2(0) = 0.391~ {\\rm GeV}^3$ \\cite{Li:2009ug,Quigg:1977dd,Eichten:1994gt,Eichten:1995ch}, $Br(\\Upsilon \\rightarrow \\mu^+\\mu^-) = 2.48\\%$ and $Br(\\Upsilon \\rightarrow e^+e^-) = 2.38\\%$ \\cite{Tanabashi:2018oca}. The gluon distribution function and the strong coupling constant $\\alpha_s$ are adopted from {\\sc CT14LO} \\cite{Schmidt:2015zda}.\n\n\n\\par\nThe dependence of the cross section $\\sigma_{\\phi}[pp \\rightarrow \\Upsilon\\mu^+\\mu^-]$, defined by $\\big| \\mathcal{M}_{\\phi} = \\mathcal{M}_{\\phi}^{(g)} + \\mathcal{M}_{\\phi}^{(b)} \\big|^2$, on the effective coupling constants $g_{gg\\phi}$ and $g_{b\\bar{b}\\phi}$ at the $8~ {\\rm TeV}$ LHC is shown in Fig.\\ref{fig3}. Different colors of the points in Fig.\\ref{fig3} represent different values of $\\sigma_{\\phi}$. The parameter space region above the red line is excluded by the experimental constraint from the decay width of $\\phi(18.4)$, i.e.,\n\\begin{eqnarray}\n\\Gamma[\\phi(18.4) \\rightarrow gg]+ \\Gamma[\\phi(18.4) \\rightarrow b\\bar b] < \\Gamma[\\phi(18.4) \\rightarrow all] \\simeq 35~ {\\rm MeV}.\n\\end{eqnarray}\nIn the experimentally allowed region of the parameter space, the signal cross section $\\sigma_{\\phi}$ reaches its maximum at the parameter point ${\\rm A} = (0.0344,\\, 0.0829)$,\n\\begin{eqnarray}\n\\sigma_{\\phi}[ pp \\rightarrow \\Upsilon\\mu^+\\mu^- \\, @ ~ 8~ {\\rm TeV} ]\\Big|_{{\\rm A}}\n=\n0.853~ {\\rm fb}.\n\\end{eqnarray}\nIt is obvious that $\\sigma_{\\phi}[ pp \\rightarrow \\Upsilon\\mu^+\\mu^- \\, @ ~ 8~ {\\rm TeV} ] < 0.853~ {\\rm fb}$ in the whole experimentally allowed parameter space region. The purpose of this study is to investigate whether the existence of a BSM Higgs-like boson can fit the enhancement observed by the CMS collaboration. Therefore, we give preference to the parameter point A in the following discussion.\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{fig3.eps}\n\\caption{Dependence of $\\sigma_{\\phi}[pp \\rightarrow \\Upsilon \\mu^+\\mu^-]$ on the effective coupling constants $g_{gg\\phi}$ and $g_{b\\bar{b}\\phi}$ at the $8~ {\\rm TeV}$ LHC. The parameter space region above the red line is excluded by the experimental constraint of $\\Gamma[\\phi(18.4) \\rightarrow gg]+ \\Gamma[\\phi(18.4) \\rightarrow b\\bar b] < \\Gamma[\\phi(18.4) \\rightarrow all] \\simeq 35~ {\\rm MeV}$.}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\n\\par\nThe integrated cross sections and invariant mass spectra of $\\Upsilon \\mu^+\\mu^-$ for $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ at the $8$ and $13~ {\\rm TeV}$ LHC are provided in Tab.\\ref{tab1} and Fig.\\ref{fig4}, respectively. The contributions from $\\left| \\mathcal{M}_{SM} \\right|^2$, $\\left| \\mathcal{M}_{\\phi} \\right|^2 + 2 {\\rm Re}\\big( \\mathcal{M}_{SM}^{\\dag} \\mathcal{M}_{\\phi} \\big)$ and $\\left| \\mathcal{M}_{\\phi} \\right|^2$, which are regarded as the SM background and the new physics signals induced by the BSM Higgs-like boson with and without interference effect, are provided separately, and labeled with $B$, $S$ and $\\hat{S}$ respectively. Table \\ref{tab1} clearly shows that the interference between the BSM amplitude induced by $\\phi(18.4)$ and the SM amplitude for $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ is negative, and thus reduces the new physics signal induced by $\\phi(18.4)$ in $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ production. One can notice that the contribution of $\\phi(18.4)$ at colliding energy between two gluons being below $2 m_{\\Upsilon}$ is almost zero, but would jump up at $\\sqrt{s} = 2 m_{\\Upsilon}$. It is a standard threshold effect. One characteristic of the phenomenon is the observed ``peak\" is not in the symmetric Gaussian form. Anyhow, even though we suppose existence of a BSM Higgs-like boson which may decay into $\\Upsilon\\mu^+\\mu^-$, it is impossible to induce a peak at $18.4~ {\\rm GeV}$ at all. What's more, the extremely narrow peak at $M_{\\Upsilon \\mu^+\\mu^-} \\sim 18.4~ {\\rm GeV}$ in the invariant mass spectrum of $\\Upsilon \\mu^+\\mu^-$ in the D-report ($\\Gamma[\\phi(18.4) \\rightarrow all] < 35~ {\\rm MeV}$) \\cite{Durgut,phdpaper} gives a stringent constraint on the effective couplings of the Higgs-like boson $\\phi(18.4)$, especially the coupling to two gluons.\n\\begin{table}[h]\n\\begin{center}\n\\renewcommand\\arraystretch{1.8}\n\\begin{tabular}{cccc}\n\\hline\n\\hline\n~~$\\sqrt{s}$~ [TeV]~~ & ~~$\\sigma_{\\hat{S}}$~ [fb]~~ & ~~$\\sigma_{S}$~ [fb]~~ & ~~$\\sigma_{B}$~ [pb]~~ \\\\\n\\hline\n$8$ & $0.853$ & $-0.637$ & $1.041$ \\\\\n$13$ & $1.464$ & $-1.073$ & $1.811$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Integrated cross sections for $pp \\rightarrow \\Upsilon\\mu^+\\mu^-$ at the $8$ and $13~ {\\rm TeV}$ LHC. $B$ stands for the SM background, $S$ and $\\hat{S}$ represent the new physics signals induced by $\\phi(18.4)$ with and without interference effect, respectively. The decay width and the effective coupling constants of $\\phi(18.4)$ are taken as $\\Gamma_{\\phi} = 35~ {\\rm MeV}$, $g_{gg\\phi}(m_Z) = 0.0344$ and $g_{b\\bar{b}\\phi}(m_Z) = 0.0829$.}\n\\label{tab1}\n\\end{center}\n\\end{table}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{fig4a.eps}\n\\includegraphics[width=0.6\\textwidth]{fig4b.eps}\n\\caption{$\\Upsilon\\mu^+\\mu^-$ invariant mass distributions for $pp \\rightarrow \\Upsilon\\mu^+\\mu^-$ at the $8$ and $13~ {\\rm TeV}$ LHC. $B$ stands for the SM background and $\\hat{S}$ represents the new physics signal induced by $\\phi(18.4)$ without interference effect, respectively. The decay width and the effective coupling constants of $\\phi(18.4)$ are taken as $\\Gamma_{\\phi} = 35~ {\\rm MeV}$, $g_{gg\\phi}(m_Z) = 0.0344$ and $g_{b\\bar{b}\\phi}(m_Z) = 0.0829$.}\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\n\\par\nIn Tab.\\ref{tab2}, we present the production cross sections for both signal and SM background for $\\Upsilon e^+e^-$ final state. We can see clearly that the numerical results for the $pp \\rightarrow \\Upsilon e^+e^-$ channel are almost the same as those for $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ due to the lepton universality ($Br(\\Upsilon \\rightarrow e^+e^-) \\simeq Br(\\Upsilon \\rightarrow \\mu^+\\mu^-)$). Thus, we do not provide the invariant mass distribution of $\\Upsilon e^+e^-$ in this section.\n\\begin{table}[h]\n\\begin{center}\n\\renewcommand\\arraystretch{1.8}\n\\begin{tabular}{cccc}\n\\hline\n\\hline\n~~$\\sqrt{s}$~ [TeV]~~ & ~~$\\sigma_{\\hat{S}}$~ [fb]~~ & ~~$\\sigma_{S}$~ [fb]~~ & ~~$\\sigma_{B}$~ [pb]~~ \\\\\n\\hline\n$8$ & $0.819$ & $-0.611$ & $0.999$ \\\\\n$13$ & $1.405$ & $-1.030$ & $1.738$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{The same as Tab.\\ref{tab1} but for $\\Upsilon e^+e^-$ final state.}\n\\label{tab2}\n\\end{center}\n\\end{table}\n\n\n\\section{Discussions and conclusion}\n\\par\nBased on the data of the Run I of LHC at $7$ and $8~ {\\rm TeV}$, we investigate the origin of the peak at $18.4~ {\\rm GeV}$ in the invariant mass spectrum of $\\Upsilon l^+l^-$ newly reported in Refs.\\cite{Durgut,phdpaper}. We postulate it to be a $0^{++}$ BSM Higgs-like boson, and by the anzatz calculate the production rate of $\\Upsilon l^+l^-$ via gluon-gluon fusion at the LHC and discuss the effect of this BSM Higgs-like boson. In our calculations, we assume the effective couplings of the $0^{++}$ BSM Higgs-like boson to the gluons and bottom quarks have the same evolution behaviour as the corresponding ones of the SM Higgs boson.\n\n\\par\nFor the peak observed in the invariant mass spectrum of $\\Upsilon l^+l^-$ at $18.4~ {\\rm GeV}$ whose width was not accurately fixed yet, the situation might imply that the peak corresponds to a BSM Higgs-like boson which decays into $\\Upsilon\\Upsilon^*$ and later turns into $\\Upsilon l^+l^-$ and eventually goes to the four-lepton final state. The peak position is located at $18.4~ {\\rm GeV}$ which is lower than the threshold value of $2 m_{\\Upsilon}$, so that it impossibly directly decays into a real $\\Upsilon$ pair if we do not consider its width. If it possesses a relatively large width whose edge covers the region of $2 m_{\\Upsilon}$, it may result in an asymmetric peak at $M_{\\Upsilon l^+l^-} \\simeq 2 m_{\\Upsilon}$ in the invariant mass spectrum of $\\Upsilon l^+l^-$ via the threshold effect. We carefully analyze the possibility and our numerical results (Fig.\\ref{fig4}) assure that there cannot exist an even-not-very apparent asymmetric peak above $2 m_{\\Upsilon}$. Moreover, the contribution of the new Higgs-like boson to the portal $\\Upsilon l^+l^-$ would interfere with the SM contribution and accurate measurements may detect the variation. But all the numerical results do not manifest an appearance of a peak at $18.4~ {\\rm GeV}$.\n\n\\par\nFrom Tab.\\ref{tab1}, Tab.\\ref{tab2} and Fig.\\ref{fig4} we can notice that if the coupling constants are small and the width of the supposed Higgs-like boson is narrow ($\\Gamma[\\phi(18.4) \\rightarrow all] < 35~ {\\rm MeV}$), the cross section is $\\mathcal{O}(1~{\\rm fb})$, such a small cross section cannot be experimentally observed by the present facilities. Since the effect of the BSM Higgs-like boson $\\phi(18.4)$ cannot be detected in $\\Upsilon l^+l^-$ final state at the parameter point ${\\rm A} = (0.0344,\\, 0.0829)$, the whole parameter space region allowed by the constraint of $\\Gamma[\\phi(18.4) \\rightarrow gg]+ \\Gamma[\\phi(18.4) \\rightarrow b\\bar b] < \\Gamma[\\phi(18.4) \\rightarrow all] \\simeq 35~ {\\rm MeV}$ is entirely excluded by the peak observed in the $\\Upsilon l^+l^-$ invariant mass spectrum at $18.4~ {\\rm GeV}$. If this scenario is true, we would conclude that the peak at $18.4~{\\rm GeV}$ does not correspond to a $0^{++}$ BSM Higgs-like boson, but something else.\n\n\\par\nAll of our estimates are based on the experimental results reported in Refs.\\cite{Durgut,phdpaper}. Our numerical results decide that the peak may not corresponds to a $0^{++}$ BSM Higgs-like boson ($18.4~ {\\rm GeV}$). Definitely much more accurate measurements which will be carried out at future high energy facilities (including the updated LHC) will give more information about this peak.\n\n\\par\nBy our assumption, the observed peak is a BSM Higgs-like boson, if it is true, it would set a scale for the BSM and the significance is obvious. Indeed, for the peak appearing at the invariant mass spectrum of $\\Upsilon l^+l^-$, Refs.\\cite{Karliner:2016zzc,Esposito:2018cwh,Becchi:2020mjz} consider it to be a composite of $bb\\bar{b}\\bar{b}$, but all their study show that the decay width are too small to be currently observed at the LHC. The observation is important and following the data, the theoretical interpretation can be made. Since it implies new understanding on new physics beyond the SM and sets a new scale, obviously, the study along this line cannot be neglected. We hope the experimentalists of high energy physics to continue the investigation on the peak by more accurate measurement and analysis. The conclusion would greatly help theorists making a definite judgement to verify the validity of our ansatz or negate it.\n\n\\par\nNow let us make a brief summary and draw our conclusion (so far, but by no means for the future). In this work we are trying to investigate whether the enhancement observed at LHC is a structureless BSM boson. If it indeed is, it can contribute to the process of $pp \\rightarrow \\Upsilon l^+l^-$, but how it behaves, can it result in a peak at the invariant mass spectrum of $\\Upsilon l^+l^-$, in other words, does it induce the peak at $18.4~ {\\rm GeV}$ reported in Refs.\\cite{Durgut,phdpaper}? It demands a clear answer. Even though a BSM boson $\\phi$ exists and possesses a certain width, an inequality $m_{\\phi}+ \\Gamma_\\phi < 2 m_{\\Upsilon}$ holds. Our explicit computation indicates that $\\phi$ as an on-shell real particle may not directly contribute to $pp \\rightarrow \\phi \\rightarrow \\Upsilon\\Upsilon^* \\rightarrow \\Upsilon l^+l^-$. Thus even though a BSM Higgs-like boson $\\phi$ of $18.4~ {\\rm GeV}$ exists and may contribute to $pp \\rightarrow \\Upsilon l^+l^-$, the sizable rate only occurs above the threshold of 2$m_{\\Upsilon}$. But then $\\phi$ must be off-shell (or contributes via $t$- and $u$-channels), therefore our conclusion is that the experimentally observed peak located at $18.4~ {\\rm GeV}$ with a narrow width does not correspond to a BSM structureless Higgs-like boson. The peak of $18.4~ {\\rm GeV}$ must originate from other mechanism and its appearance cannot be a signature of existence of BSM as expected.\n\n\n\\vskip 10mm\n\\par\n\\noindent{\\large\\bf ACKNOWLEDGMENTS} \\\\\nThis work is supported in part by the National Natural Science Foundation of China (Grants No. 11675082, 11735010, 11775211, 11535002, 11805160, 11747040, 11375128, the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20170247), Special Grant of the Xuzhou University of Technology (No. XKY2016211, XKY2017215, XKY2018221), and the CAS Center for Excellence in Particle Physics (CCEPP).\n\n\n\\vskip 5mm\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nPG~1002+506 was discovered by the Palomar-Green UV-excess survey (Green,\nSchmidt, \\& Liebert 1986) and listed as a cataclysmic variable (CV). \nDuring a study of the CVs from this survey, Ringwald (1993) obtained\nultraviolet and red spectra, and tentatively reclassified it as a detached\nsubdwarf binary, noting H$\\alpha$ in strong emission, unresolved at\n10-\\AA\\ resolution. Several puzzling aspects were noted, however,\nincluding the near-constancy of the radial velocities throughout two\nnights, consistent with no change other than that attributable to\natmospheric dispersion in an unrotated slit. There was also no\nsignificant variation in the equivalent width of H$\\alpha,$ which one\nmight expect if this were a detached CV progenitor with the hot component\nirradiating the facing hemisphere of its companion. \n\nThat PG~1002+506 is not a CV was shown definitively by E. L. Robinson\n(1995, private communication): it does not flicker, or have the erratic\nvariability ubiquitous in CVs. This was found with high-speed\nsimultaneous {\\it UBVR\\\/} photometry taken in 1995 June with the Stiening\nphotometer on the McDonald Observatory 2.1-m telescope. In 25~min of\nphotometry with 1-s time resolution, all bands showed peak-to-peak\namplitudes of $<\\,2$\\%. \n\nThis and further spectra have forced another reclassification of this\nstar, as a high-latitude Be star. This is one of two known in the\nPalomar-Green catalog, the other being PG~0914+001 (Saffer et al.~1997). \nAn Oe star from this survey is also known, PG~2120+062 (Moehler, Heber, \\&\nDreizler 1994). \n\nFor reviews on Be stars, see Jaschek \\& Jaschek (1987) and Slettebak\n(1988). About one in five non-supergiant B stars shows emission, mainly\nin H$\\alpha$ but sometimes also in H$\\beta$ and higher Balmer lines. \nStruve (1931) attributed this to a disk, extruded by the star's rotation\nnear the breakup velocity, $ \\sqrt{ G M \/ R}.$ What excites the emission\nin Be stars is a long-standing mystery, however, as is their evolutionary\nstatus. Although Be stars often have an IR excess, PG~1002+506 is not an\nIRAS source (Neugebauer et al.~1988). \n\n\n\\section{Blue spectrum}\n\nA blue spectrum (Figure 1) was taken in service time with the Intermediate\nDispersion Spectrograph on the Isaac Newton Telescope on La Palma. This\n1800-s spectrum was taken in photometric conditions in $2''$ seeing,\nthrough a $1.73''$~slit, and has 1.5-\\AA\\ (FWHM) resolution. The slit was\naligned to the parallactic angle, to avoid atmospheric dispersion effects;\nthe spectrum was taken when PG~1002+506 was nearly overhead, at an airmass\nof 1.08. \n\nA spectral classification of B$5\\pm 1$ V was arrived at by comparing this\nspectrum to model atmospheres (Kurucz 1979) and published spectra (Jacoby,\nHunter, \\& Christian 1984; Jaschek \\& Jaschek 1987). That this is a\nmain-sequence star and not a subdwarf is shown by the presence of the H13\nand 14 lines. That it is not a giant or supergiant is shown by the widths\nof its Balmer lines, with FWZI of H$\\gamma$ of $ 31 \\pm 3$~\\AA. There is\nno spectroscopic evidence that this star is a binary. \n\n\n\n\\section{Radial Velocity}\n\nOn 1997 January 3 UT, two 10-min exposures were obtained with the Modular\nSpectrograph on the 2.4-m Hiltner Telescope at Michigan-Dartmouth-MIT\nObservatory, Kitt Peak, Arizona. The spectra covered from 4650 to 6727\n\\AA, and had 4-\\AA\\ (FWHM) resolution. The weather was poor, with $>\n1\\arcsec$ seeing and rising humidity that forced a shutdown just after\nthese spectra were taken. The spectrograph slit was set at the\nparallactic angle, even though PG~1002+506 was only one hour east of the\nmeridian. The 1\\arcsec\\ slit projected to 3 \\AA\\ on the detector. With\nthe mediocre seeing, we expect ``slit-painting'' velocity errors to be\nsmall, probably $< 5$ km~s$^{-1},$ based on experience with similar sharp\nlines in white dwarf\/red dwarf binaries (Thorstensen, Vennes, \\& Shambrook\n1994). The exposures were bracketed by HgNeXe exposures, for which the\nRMS residual was $< 0.05$~\\AA, and the maximum residuals for the weakest\nlines were $< 10$~km~s$^{-1}.$ Most lines had residuals around\n2~km~s$^{-1}.$\n\nH$\\alpha$ appears to be slightly resolved, and is in strong emission (see\nFigure 2), with an equivalent width of $17.8 \\pm 0.3 $~\\AA\\ and FWHM of\n$580 \\pm 30$ km~s$^{-1}.$ There is also emission in the core of H$\\beta.$\nBy convolving H$\\alpha$ with the derivative of a Gaussian with FWHM = 8\n\\AA\\ and taking the zero of the convolution as the velocity (Schneider \\&\nYoung 1980), we find heliocentric radial velocities of the spectra taken\nat HJD 2450451.90425 and 2450451.91140 of +29.3 and +28.9~km~s$^{-1},$\nrespectively. The velocities of the O~I $\\lambda$\\,6300 \\AA\\ night sky\nline were 1.6 and 0.7~km~s$^{-1},$ showing the accuracy of the wavelength\nscale. \n\nHowever, the emission lines in Be stars are well known to be variable in\nprofile over timescales of days or longer, and are therefore not reliable\nindicators of the systemic velocity. The spectra were therefore summed\ntogether and rectified, to remove continuum slope effects. The radial\nvelocity was then measured from the absorption wings of H$\\alpha$ by\nconvolving a positive and a negative Gaussian with the line profile and\ntaking the zero of this convolution as the velocity (Schneider \\& Young\n1980). In all cases the Gaussians had 4 channels FWHM. The separation\nbetween the Gaussians was varied, from 24 to 20 to 16 \\AA; the\ncorresponding heliocentric radial velocities are $-2.0,$ $-0.5,$ and\n$-4.1$~km~s$^{-1}.$ Finding the line's centroid by fitting and subtracting\na linear approximation of the continuum, numerical integration of the\nintensity, and taking the centroid (crudely, with the IRAF {\\it splot\\\/}\n`e' command) gave +0.4 km~s$^{-1}.$ We conclude that PG~1002+506 has a\nheliocentric radial velocity of $-2 \\pm 15$~km~s$^{-1}.$\n\n\n\\section{Model atmosphere analysis}\n\nWe have performed a model atmosphere analysis of the blue optical spectrum\nto estimate the atmospheric parameters $T_{\\rm eff}$ and log $g$, as well\nas the projected stellar rotation velocity $v \\sin i$. Our grid of\nsynthetic spectra was calculated with the radiative transfer code SYNSPEC\n(Hubeny, Lanz, \\& Jeffrey 1995), assuming the temperature and pressure\nstratifications of Kurucz (1991). The metal and helium abundances were\nheld fixed at the solar value. At the temperature and surface gravity of\nspectral type B5V, the assumption of LTE is well justified. The\ntemperature and gravity grid points were $T_{\\rm eff}$ = 13,000 --\n17,000~K in steps of 1,000~K, and log $g$ = 3.5 -- 5.0 in steps of 0.5\ndex. In addition each model was convolved with a rotational broadening\nfunction at projected rotation velocities $v \\sin i$ = 50 -- 350\nkm~s$^{-1}$ in steps of 50 km~s$^{-1}$ to produce a 3-dimensional fitting\ngrid. The stellar parameters were estimated by simultaneous variation\nusing a non-linear $\\chi^2$ minimization algorithm. Details of the\nsynthetic spectrum calculations and the fitting algorithm are given by\nSaffer et al.~(1994) and Saffer et al.~(1997). Due to the partial filling\nin of the lower Balmer lines by emission from the circumstellar material,\nwe have restricted the analysis to the portion of the spectrum blueward of\nH$\\beta.$\n\nThe best-fit stellar parameters are $T_{\\rm eff} = 14,900 \\pm 1200$~K,\n$\\log g = 4.20 \\pm 0.2$, and $v \\sin i = 340 \\pm 50$ km~s$^{-1}$ (see\nFigure 1). The quoted 1-$\\sigma$ errors are based on counting statistics\nand account for covariance for the fitting parameters; they also estimate\nsystematic errors. \n\n\n\\section{Evolutionary status}\n\nThe effective temperature, surface gravity, and very high rotational\nvelocity are fully consistent with a spectral classification of B5Ve. The\nbreakup velocity expected for this star is 540 km~s$^{-1}.$ The fit places\nthis star in the area of confusion in the $T_{\\rm eff}\/\\log g$ diagram\nwhere the Population I main- sequence intersects the Population II blue\nhorizontal branch (BHB) (Sch\\\"onberner 1993; Bertelli et al.~1994). For\nexample, PG~0832+676 at first appeared to be a young star far from the\nGalactic plane, but turned out to be a nearby blue evolved star, upon\nanalysis of high-resolution spectra (Hambly et al.~1996). However,\nidentification of PG~1002+506 as a BHB star is contradicted both by the\nemission reversals in the H$\\alpha$ and H$\\beta$ absorption lines, and by\nits high rotation velocity, since BHB stars are slow rotators (Peterson,\nRood, \\& Crocker 1995). \n\nAssuming PG~1002+506 to be of Population I origin, we used the derived\natmospheric parameters and the evolutionary tracks of Claret \\& Gimenez\n(1992) to estimate the stellar mass and evolutionary age (see Table 1). A\ndistance estimate was obtained from the absolute visual magnitude deduced\nfrom the stellar mass, atmospheric parameters, and bolometric corrections\nof Kurucz (1979). PG~1002+506 has $B = 15.36$ (Green et al.~1986).\nAssuming $B - V = -0.16$ for B5V stars (Allen 1973), and a reddening\n$E(B-V) < 0.01$, inferred from the map of Burstein \\& Heiles (1982), this\nwould imply a distance of 13.9 kpc, which for a Galactic latitude $b =\n51^{\\circ},$ corresponds to a z-distance of 10.8 kpc above the Galactic\nplane. Although large, this is not unheard of (Kilkenny 1992). For a\nGalactic longitude $l = 165^{\\circ},$ this would imply a galactocentric\nradius of 17.1 kpc, putting PG~1002+506 at the outskirts of the Galaxy. \n\n\n\\section{Kinematical analysis}\n\nAs the existence of young objects at large distances from the star forming\nregions of the Galactic disk is potentially interesting, we have performed\na kinematical analysis for PG~1002+506. Although no proper motion\ninformation is available, it is possible to use the observed radial\nvelocity of a star to constrain its evolutionary history. A detailed\ndescription of the method of analysis is given by Rolleston et al.~(1997). \n\nWe first consider a scenario whereby PG~1002+506 has a zero velocity\ncomponent parallel to the Galactic disk, and ejection has occurred\nperpendicular to the plane of the Galaxy. We have corrected the observed\nheliocentric velocity for the effects of differential rotation (Fich et\nal.~1989), assuming that the halo co-rotates with the disk, to determine\nthe stellar radial motion $(v_r)$ with respect to a standard of rest\ndefined by its local environment. Our initial assumption implies that the\nobserved radial velocity is a component of the stellar space motion\n$(v_z)$ perpendicular to the disk. We then attempt to show that PG\n1002+506 could have reached its present position in the Galactic halo\nwithin its evolutionary lifetime, while reproducing the observed radial\nvelocity, and calculating the required ejection velocity. These\ncalculations have adopted the gravitational potential function of House \\&\nKilkenny (1980). This analysis implicitly assumes that the star is ejected\nfrom the disk shortly after birth, consistent with cluster ejection\nsimulations. \n\nThe results of the kinematical analysis are given in Table 1. Given the\nlarge z-distance, it is not surprising to find the ``time of flight'' to\nbe larger than the evolutionary age. We have therefore considered the\neffects of errors in the derived atmospheric parameters and the radial\nvelocity measurement. By optimizing the values of $T_{\\rm eff}$ and $\\log\ng$ such that they are self-consistent within the errors, it is possible to\nincrease the evolutionary age, so that it is greater than the predicted\nflight time. For example, adopting values of $T_{\\rm eff} = 13,750$~K and\n$\\log g = 4.0$ would imply an age of 115 Myr for a mass of 4.0\n$M_{\\odot}.$ Allowing an error of 15 km~s$^{-1}$ in the observed\nheliocentric velocity also decreases the estimated flight time, but not\nsignificantly, to 84 Myr. \n\n\n\\section{Conclusions}\n\nPG~1002+506 appears to be a young, rapidly rotating B5Ve star at a\ndistance of 10.8 kpc from the Galactic plane, and at a galactocentric\nradius of 17.1 kpc. The kinematical analysis suggests that it could have\nattained its present Galactic position having been ejected from the disk\nshortly after its formation. Furthermore, the required ejection velocity\nof $\\approx 230$~km~s$^{-1}$ can also be produced by the known mechanisms\npredicted by Leonard (1993). A detailed atmospheric analysis with\nhigher-quality spectra should still be done, to determine abundances and\nconfirm that PG~1002+506 really is a distant main-sequence star, and not a\nnearby blue evolved star. If PG~1002+506 really is 10.8 kpc from the\nGalactic plane, interstellar absorption in this same spectrum would probe\na line through the Galactic halo otherwise difficult to acquire. \n\n\n\\acknowledgments\\noindent\nE.~Harlaftis took the blue spectrum with the Isaac Newton telescope, which\nis operated on La Palma by the Royal Greenwich Observatory at the Spanish\nObservatorio del Roque de los Muchachos of the Instituto de Astrofisica de\nCanarias. Michigan-Dartmouth-MIT Observatory is operated by a consortium\nof the University of Michigan, Dartmouth College, and the Massachusetts\nInstitute of Technology. Thanks also to Rob Robinson, Malcolm Coe,\nRichard Green, Uli Heber, Gerrie Peters, and Richard Wade, for helpful\ndiscussions. \n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIntegrable models may have very huge symmetries, that \nhelp us to study various behaviors of the systems.\nFor some well investigated models, we are able to \ncalculate correlation functions of observables \nby using representation theories of the symmetries, that\nhave been fruitfully studied within the language of \ninfinite dimensional Lie algebras and their suitable\ndeformation theories. We now have a lot of\nexamples of these symmetry algebras and their applications \nto many problems.\nWhat is the natural candidate for that symmetry which is \npresent in majority of the integrable models?\nIn other words, what is the ``universal'' symmetry among all these?\n\n\nThe $1+1$-dimensional conformal field theories (CFT's) \n\\cite{rBPZ} describe the \nuniversality classes of massless field theories in $1+1$-dimensions.\nIt is the guideline given in the celebrated paper\n\\cite{rBPZ} \nthat all the CFT's must be \nregarded as representations of the ``Virasoro algebra''\nregardless of the details of the models.\nNamely, the Virasoro algebra is the universal one in CFT.\nIndeed, through the Sugawara construction \\cite{rKZ,rGO},\nthe Virasoro algebra exists in any Kac-Moody algebra.\nThese models are successfully \napplied to the critical phenomena of \ntwo-dimensional classical statistical models, \nthe low temperature behavior of one-dimensional electron systems and so on.\nIn any sense, CFT is definitely quite well understood among \nother interacting field theories, \nbecause we have the infinite dimensional conformal symmetry.\n\nWe are gradually understanding the universal symmetry arising \nin off-critical models: massive integrable field\ntheories in 1+1 dimension (sine-Gordon model etc.) \\cite{rL}, \none-dimensional quantum spin chain systems ($XYZ$ model etc.) \\cite{rL},\ntwo-dimensional solvable lattice \nmodels (ABF models etc.) \\cite{rLP1,rLP2,rAJMP},\ndeformations of the KdV hierarchy and \ndiscretized soliton equations \\cite{rFR,rFr},\nCalogero-Sutherland(CS)-type quantum mechanical \nmodels \\cite{rAMOS,rAOS} and so on.\nIt had been recognized that nontrivial\nVirasoro-type symmetries exist in these off-critical theories;\nin \\cite{rLP1}, the existence of two-parameter\nVirasoro-type symmetry was conjectured, and \na one-parameter Virasoro-type Poisson structure was\nfound in \\cite{rFR}.\nIt was in the CS-type quantum \nmechanical model, that \nwe finally obtained the definition and an \nexplicit construction of the two-parameter\nVirasoro-type symmetry which we call \\v \\cite{rSKAO}.\nThis is already extended to the case of ${\\cal W}$-algebra \\cite{rFFr,rAKOS}.\nIt is really astonishing that all the Virasoro-type algebras \nrelated with the off-critical integrable models are \nobtained by taking suitable limits from \\v.\nHowever,\nwe have not fully understand \nthe meaning of ``universality'' played by this new Virasoro-type\nsymmetry \\v, so far. \nThe algebra \\v is one of the simplest example of \nelliptic algebras \\cite{rFIJKMY,rFFr}, since we have elliptic theta-functions\nin the operator product expansion (OPE) formulas.\nWe strongly hope that this Virasoro-type algebra \nwill be constructed in a canonical way from \nthe elliptic algebra ${\\cal A}_{q,p}$ \\cite{rFIJKMY}\\footnote{\nThe relations between the parameters in ${\\cal A}_{q,p}$ and \\v \nare $q_{\\!\\!~_{{\\cal V}ir}}=p_{\\!\\!\\!~_{{\\cal A}}}$ and\n$p_{\\!\\!~_{{\\cal V}ir}}={q_{\\!\\!\\!~_{{\\cal A}}}}^2$.}\nthrough a Sugawara-type construction and \nthat will give us a clue for the total understanding of \\v.\n\nIn this review, we will explain how this two parameter \nVirasoro-type algebra \\v\narose in the CS-type model, and\nanother aim is to accumulate\nas many problems and applications of \\v as possible.\nThis paper is organized as follows.\nIn Section 2, we study basic results of the Virasoro-type algebras \nstarting from the definition of the two-parameter Virasoro-type algebra \\v. \nIn Section 3, a Heisenberg realization of the Virasoro-type algebra and \nits applications to the CS-type models are presented.\nBased on this realization, we discuss\nrepresentation theories, futher applications and \nrelations with other several models, in Section 4.\nSection 5 is devoted to summary and comments.\n\n\\section{Quantum deformed Virasoro algebra \\v }\nIn this section we examine some of the fundamental properties of \nthe Virasoro-type algebra\n\\v \\cite{rSKAO} \nwhich can be directly derived from the defining relation. \nA Heisenberg realization and its application to \nvarious problems are studied in the next sections.\n\\subsection{definition of \\v}\nLet $p$ be a generic complex parameter with $|p| < 1$.\nLet us consider an associative algebra generated by\n$\\{T_n|n\\in \\bf{Z}\\}$ with the relation\n$$\nf(w\/z)T(z)T(w)-T(w)T(z)f(z\/w)\n= {\\rm const.}\\left[\n \\delta \\Bigl(\\frac{pw}{z}\\Bigr)-\n \\delta \\Bigl(\\frac{p^{-1}w}{z}\\Bigr)\\right],\n$$\nwhere $T(z)=\\sum_{n\\in\\bf{Z}}T_n z^{-n}$,\n$\\delta(x)=\\sum_{n \\in {\\bf Z}}x^n$ and \n$f(z)=\\pzs{l}f_l z^l$ is a structure function. We show that the \ncommutativity of the diagram (Yang-Baxter equation for $T(z)$)\n\\begin{eqnarray}\n\\begin{array}{ccc}\n T(x)T(y)T(z) &{\\displaystyle \\mathop{\\llra}^{f(z\/y)\\times} } &\nT(x)T(z)T(y) f(y\/z) \\\\\n& & + T(x)\\; \\delta{\\rm-function}\\\\\n{\\scriptstyle f(y\/x)\\times} \\Biggl\\downarrow & & \n{\\scriptstyle f(z\/x)\\times} \\Biggl\\downarrow \\\\\n& & \\\\\nT(y)T(x)T(z) f(x\/y) & & \nT(z)T(x)T(y) f(y\/z) f(x\/z)\\\\\n+ T(z)\\; \\delta{\\rm-function} & & + T(x)\\; \\delta{\\rm-function}\\\\\n& & + T(y)\\; \\delta{\\rm-function}\\\\\n{\\scriptstyle f(z\/x)\\times} \\Biggl\\downarrow & & \n{\\scriptstyle f(y\/x)\\times} \\Biggl\\downarrow \\\\\n& & \\\\\nT(y)T(z)T(x) f(x\/y) f(x\/z) &{\\displaystyle \\mathop{\\llra}^{f(z\/y)\\times} } & \nT(z)T(y)T(x) f(x\/y) f(x\/z)f(y\/z)\\\\\n+ T(z)\\; \\delta{\\rm-function} & & + T(z)\\; \\delta{\\rm-function}\\\\\n+ T(y)\\; \\delta{\\rm-function} & & + T(y)\\; \\delta{\\rm-function} \\\\\n& & + T(x)\\; \\delta{\\rm-function} \n\\end{array}\\label{YB-for-T}\n\\end{eqnarray}\ndetermines this \nstructure function $f(z)$ completely. \nHere the terms denoted by ``$\\delta$-function'' \nmean some combinations of \nthe $\\delta$-functions and the structure functions.\nThe commutativity of this diagram means\n\\begin{eqnarray}\n0&=&T(x) \\Biggl( \\delta(pz\/y) g(x\/z)-\\delta(py\/z) g(x\/y) \\Biggr) \\nonumber\\\\\n &+&T(y) \\Biggl( \\delta(px\/z) g(y\/x)-\\delta(pz\/x) g(y\/z) \\Biggr) \\label{asso}\\\\\n &+&T(z) \\Biggl( \\delta(py\/x) g(z\/y)-\\delta(px\/y) g(z\/x) \\Biggr), \\nonumber\n\\end{eqnarray}\nwhere\n$g(x) \\equiv f(x)f(x\/p) - f(1\/x) f(p\/x)$.\nNote that $g(x)=-g(p\/x)$.\nIt is an interesting exercise to see that the general solution to the \neq.\\ (\\ref{asso}) is\n$g(x)= c_1 \\Bigl(\\delta(x\/p)-\\delta(x) \\Bigr)$,\nwhere $c_1$ is a constant. \nHereafter, we just set $c_1=1$,\nsince there is no loss of generality.\nNote that the Yang-Baxter equation for $T(z)$ is ``not'' \ntrivially satisfied even if the current $T(z)$ has\nno spin degrees of freedom,\nbecause we have the $\\delta$-function term in the \nrelation (\\ref{e:a1.2}).\n{}From this we have\n\\begin{eqnarray*}\nf(x)f(x p)= \\alpha + \\sum_{n=1}^\\infty (1-p^{n})x^n ,\n\\end{eqnarray*}\nwhere $\\alpha$ is a constant.\nIt should be noted that this is the place where one more parameter comes in.\nIf we parameterize $\\alpha$ by introducing another parameter $q$ as\n$$\n\\alpha={1-p \\over (1-q)(1-t^{-1})}, \\qquad\\quad t=qp^{-1},\n$$\nwe have the difference equation \n$$\nf(x)f(x p)=\\alpha { (1-qx)(1-t^{-1} x) \\over (1-x)(1-p x)}.\n$$\nThis can be solved as\n\\begin{equation}\nf(x)=\\exp \\Biggl\\{ \\sum_{n=1}^\\infty \n{ (1-q^n)(1-t^{-n}) \\over 1+p^n} { x^n \\over n} \\Biggr\\}. \\label{structure}\n\\end{equation}\nWe arrive at the definition of the quantum deformed Virasoro algebra\n\\v \\cite{rSKAO}.\n\n\\proclaim Definition 1.\nLet $p$ and $q$ be complex parameters with\nthe conditions $|p|<1$ and $|q|<1$, and set $t=qp^{-1}$.\nThe associative algebra \\v is generated by the current\n$T(z)=\\sum_{n \\in {\\bf Z}} T_n z^{-n}$ satisfying the relation\n\\begin{equation}\nf(w\/z)T(z)T(w)-T(w)T(z)f(z\/w)\n= -\\frac{(1-q)(1-t^{-1})}{1-p} \\left[\n \\delta \\Bigl(\\frac{pw}{z}\\Bigr)-\n \\delta \\Bigl(\\frac{w}{pz}\\Bigr)\\right],\n\\label{e:a1.2}\n\\end{equation}\nwith the structure function (\\ref{structure}).\n\n\n\\noindent Note that the constant factor in the R.H.S. is so chosen \nthat our Heisenberg realization of this algebra becomes simple.\n\nWe now have the associative algebra \\v \nwhich exists in a nontrivial way,\nsince we have so chosen the structure function $f(z)$\nthat \nthe Yang-Baxter equation for the \\v current\n(\\ref{YB-for-T}) will not give us any more \nrelations than the quadratic relation (\\ref{e:a1.2}) for $T(z)$.\n\nThe defining relation (\\ref{e:a1.2}) can be written in terms of $T_n$ as \n\\begin{equation}\n[T_n \\, , \\, T_m]=-\\ps{l}f_l\\left(T_{n-l}T_{m+l}-T_{m-l}T_{n+l}\\right)\n-\\frac{(1-q)(1-t^{-1})}{1-p}(p^{n}-p^{-n})\\delta_{m+n,0},\n\\label{e:a1}\n\\end{equation}\nwhere $[A,B]=AB-BA$.\n\nThe relation \\eq{e:a1.2} is invariant\nunder the transformations\n\\begin{eqnarray}\n{\\rm (I)}&& \\qquad\\qquad T_n \\to -T_n, \n\\label{e:a1.3} \\\\\n{\\rm (II)}&& \\qquad\\qquad (q,t)\\to (q^{-1},t^{-1}),\n\\label{e:aa1} \\\\\n{\\rm (III)} &&\\qquad\\qquad q \\leftrightarrow t. \\label{e:aa2}\n\\end{eqnarray}\n\\par\n\nIn what follows, \nwe will frequently use the notation\n\\begin{equation}\nt=qp^{-1}=q^\\beta.\n\\end{equation}\nThis parameter $\\beta$ plays the role of the coupling constant of\nthe Calogero-Sutherland model. See Section \\ref{secCS}.\n\n\n\n\\subsection{special limits of \\v}\nHere we study some special limits of \\v, which \nexplains the connections among known examples of the \nVirasoro-type algebras.\n\n\n\\subsubsection{limit of $q\\rightarrow 1$: ordinary Virasoro algebra}\nLet us study the limit $q\\to 1$ ($\\beta$: fixed) by\nparameterizing $q=e^{h}$. \nSuppose that $T(z)$ has the following expansion in $h$\n\\begin{equation}\nT(z)=2+\\beta \\left( z^2 L(z)+\\frac{(1-\\beta)^2}{4\\beta} \\right)h^2\n + T^{(2)}(z)h^4 +\\cdots.\n\\label{pe:a7}\n\\end{equation}\nThis expansion is consistent with the invariance under transformation \n\\eq{e:aa1}.\nThe defining relation \\eq{e:a1.2} gives us the well known relations for\nthe ordinary Virasoro current\n$ L(z)=\\sum_{n \\in {\\bf Z}}L_{n}z^{-n-2}$, namely\n\\begin{equation}\n[L_n , L_m]=(n-m)L_{n+m}+\\frac{c}{12}(n^3-n)\\delta_{n+m,0},\n\\label{e:a8}\n\\end{equation}\nwhere the central charge $c$ is\n\\begin{equation}\nc=1-\\frac{6(1-\\beta)^2}{\\beta}.\n\\label{e:a9}\n\\end{equation}\nThis relation between the central charge of\nthe Virasoro algebra and the coupling constant\nof CS model is discussed in Section \\ref{secCS}.\n\n\\subsubsection{Frenkel-Reshetikhin's $q$-Virasoro algebra\n($\\beta\\rightarrow 0$)}\nLet us consider the limit $\\beta\\to 0$ ($q$: fixed).\nIn this limit, we obtain the classical $q$-Virasoro algebra\nfound by Frenkel and Reshetikhin \\cite{rFR}. Their algebra is \nthe first one among many $q$-deformed Virasoro algebras,\nwhich was obtained through the study of the quantum affine algebra\n$U_q(\\widehat{sl}_2)$ \\cite{rAOS2}.\nOther nice features of their algebra are\nthat it is no more a Lie algebra but a quadratic algebra \nwhich resembles the relation\n$\\{L(u),L(v)\\}=[r(u-v),L(u)L(v)]$ \nin the quantum inverse scattering method,\nand there exists a mysterious \nresemblance between their bosonic realization and the \nBaxter's\ndressed vacuum form in the Bethe ansatz method. \nSee Section \\ref{secSYMMETRIC}.\n\nIn this limit, $T_n$'s become commutative. \nHowever, we can define the Poisson bracket structure\ndefined by $\\{\\;,\\;\\}_{\\rm P.B.}=-\\lim_{\\beta\\rightarrow 0}\n[\\;,\\;]\/(\\beta \\ln q )$.\nThus we have\n\\begin{eqnarray}\n\\{T_n,T_m\\}_{\\rm P.B.}=\n\\sum_{l\\in {\\bf Z}}{1-q^l \\over 1+q^l} T_{n-l}T_{m+l}\n+ (q^n-q^{-n}) \\delta_{n+m,0}, \\label{qvirFR}\n\\end{eqnarray}\nwhich is the relations for the classical $q$-Virasoro algebra \\cite{rFR}.\n\nIn the paper \\cite{rFr}, deformations of the KdV hierarchy\nis studied. The $N$-th Korteweg-de Vries (KdV) hierarchy\nis a bihamiltonian integrable system with the \n$N$-th order differential operators.\nThey are deformed to the $q$-shift operators\n$$\n D^N- t_1(z) D^{N-1}+\\cdots +(-1)^{N-1} t_{N-2}(z) D\n+(-1)^N t_{N-1}(z),\n$$\nwhere $D\\cdot f(z)=f(qz)$.\nIt was shown there that the deformed bihamiltonian structure\nis given by the Poisson bracket for the \n$q$-${\\cal W}$ algebra of Frenkel and Reshetikhin \\cite{rFR}.\n\\proclaim Proposition 1. \\hspace{-2mm}\\cite{rFR}\nIn the case of $N=2$, the bihamiltonian structure is \ngiven by\n\\begin{eqnarray}\n\\{t_i(z),t_j(w)\\}_1&=&\\delta\\left( {wq\\over z}\\right)-\n\\delta\\left( {w\\over zq}\\right),\\\\\n\\{t_i(z),t_j(w)\\}_2&=&\\sum_{m \\in{\\bf Z}} \\left( {w\\over z}\\right)^m\n{1-q^m \\over 1+q^m} t(z)t(w)+\n\\delta\\left( {wq\\over z}\\right)-\n\\delta\\left( {w\\over zq}\\right).\n\\end{eqnarray}\n\n\\noindent\nThese Poisson brackets coincide with that of the \n$q$-Virasoro algebra of Frenkel and Reshetikhin (\\ref{qvirFR})\nwith two different choices of the structure function $f(z)$:\n{\\it i.e.}, $f(z)=1$ and $f(z)$ given by (\\ref{structure})\nwith $\\beta\\rightarrow 0$.\n\n\\subsubsection{Zamolodchikov-Faddeev algebra of sine-Gordon and XYZ models}\nThe $q$-Virasoro algebra can be interpreted as \nthe Zamolodchikov-Faddeev (ZF) algebra satisfied by a \nparticle excitation operator.\nThe structure function $f(x)$ determined by the associativity\nrelates with \nthe $S$ matrix characterized by the factorization property.\n\nFirst, we can rewrite the defining relation of the \n$q$-Virasoro algebra (\\ref{e:a1.2})\nas follows;\n\\begin{equation}\\label{eq:ZFXYZ}\nT(z_1) T(z_2) \n=\n S\\left( {z_1\\over z_2}\\right) T(z_2) T(z_1)\n+ C\\left( \\delta\\left( {z_1\\over pz_2}\\right) \n+ \\delta\\left( {pz_1\\over z_2}\\right) \\right), \n\\end{equation}\nwith\n\\begin{eqnarray}\nS(z) \n&\\!\\!\\!=\\!\\!\\!&\n{f(z) \\over f(z^{-1})}=\n {\\vartheta_1(zt^{-1};p) \\,\\vartheta_0(zt ;p) \\over \n \\vartheta_1(zt ;p) \\,\\vartheta_0(zt^{-1};p) },\\\\\nC \n&\\!\\!\\!=\\!\\!\\!&\n{(1-q)(1-t^{-1})\\over (1-p)f(p)}=\n {( q;p^2)_\\infty (t^{-1};p^2)_\\infty \\over \n (pq;p^2)_\\infty (pt^{-1};p^2)_\\infty},\n\\end{eqnarray}\nfor $|p|<1$.\nHere, \n$\\vartheta_1(z;p) = \n-ip^{1\\over 4} z^{1\\over 2}\n(p^2 z;p^2)_\\infty$$( z^{-1};p^2)_\\infty$$(p^2;p^2)_\\infty$ and\n$\\vartheta_0(z;p) = \n(p z;p^2)_\\infty \\times$ $\\times(pz^{-1};p^2)_\\infty$$(p^2;p^2)_\\infty$\nwith $(z;q)_\\infty \\equiv \\prod_{n\\geq 0}(1-zq^n)$.\n\nNext, let $p = e^{\\tau\\pi i}$\nand perform a modular transformation \n$\\tau\\rightarrow -1\/\\tau$ for theta functions\nand take a limit $-1\/\\tau \\rightarrow i\\infty$, {\\it i.e.}\n$p\\rightarrow 1$. Changing the parameterization as \n$z=p^{i{\\theta\\over\\pi}}$ and $t=p^{\\xi}$, we have \n\n\\proclaim Proposition 2. \n\\hspace{-2mm}\\cite{rL}\\footnote{The notations in \\cite{rL} are\n$x=p^{1\\over 2}$, $\\xi=\\beta\/(1-\\beta)$ and $\\epsilon=i\\tau$.}~\nIn the limit of $p\\rightarrow 1$, \nthe $q$-Virasoro generator satisfies \nthe following Zamolodchikov-Faddeev algebra\\footnote{\nThis ZF-equation should be understood in the sense of analytic continuation.}\n of the sine-Gordon model,\n\\begin{equation}\\label{eZFSG}\n{\\cal T}(\\theta_1) {\\cal T}(\\theta_2) = S\\left( {\\theta_1- \\theta_2}\\right) \n{\\cal T}(\\theta_2) {\\cal T}(\\theta_1), \\qquad\nS(\\theta) = {\\sinh\\theta + i\\sin\\pi\\xi \\over \\sinh\\theta - i\\sin\\pi\\xi},\n\\end{equation}\nwhere ${\\cal T}(\\theta)=\\lim_{-1\/\\tau \\rightarrow i\\infty}T(z)$.\n\n\\noindent\nNamely,\nthe $p=1$ $q$-Virasoro generator ${\\cal T}(z)$ creates \nthe basic particle (the first breather) with a rapidity $\\theta$\nof the sine-Gordon model, defined by the following Lagrangian density\n\\begin{equation}\\label{eLagrangianSG}\n{\\cal L}_{SG} = \n{1\\over2}(\\partial_\\mu\\phi)^2 + \n\\left( {m_0\\over b}\\right)^2 \\cos\\left( b\\phi\\right),\\qquad \nb^2=8\\pi\\beta=8\\pi{\\xi\\over 1+\\xi},\n\\end{equation}\nat the attractive range $0<\\xi<1$.\nThis identification is based on the \ncoincidence of the $S$-matrix for the \nbasic scalar particles in sine-Gordon model with\n$f(z)\/f(1\/z)$ in the limit of $p\\rightarrow 1$, and \nthe existence of\nsimple poles at the points $\\theta_1 = \\theta_2 \\pm i\\pi$\nin the delta function terms.\nFor the axiom of the ZF operator, see \\cite{rL2}.\nLukyanov also proposed in \\cite{rL} that,\nfor generic $p$,\neq.\\ \\eq{eq:ZFXYZ} can be interpreted as the ZF relation \nof the XYZ model,\n{\\it i.e.},\nthe $q$-Virasoro generator is \nthe basic scalar creation operator of this model.\n\n\\subsubsection{limit of $q\\rightarrow 0$}\nNext, study the $q\\rightarrow 0$ limit ($t$: fixed).\nThis limit is interesting from the point of view of the \nrepresentation theory of the Hall-Littlewood polynomials \\cite{rM}\nin terms of the Heisenberg algebra.\nIt is known by the work of Jing \\cite{rJ}\nthat the Hall-Littlewood polynomials are \nrealized by a multiple integral formula.\nThe integration kernel is \nsimply given by multiplying vertex \noperators on a vacuum state.\nAs for the number of the integration variables\nand the vertex operators, it is related with the \nshape of the Young diagram of each Hall-Littlewood polynomial.\nHis realization can be regarded as a deformation of \nthe determinant representation of the Schur polynomials.\nRecently, similar integral \nrepresentations are studied\nfor the Jack polynomials and the \nMacdonald polynomials \\cite{rSt,rM,rMY,rAMOS,rAOS}. \nHowever, the number of the integrals \nthere is much greater in general \nthan the case of the Schur or Hall-Littlewood\npolynomials. We will discuss this in \nSection \\ref{q0limit}\nby using the Heisenberg realization of \\v in the limit $q\\rightarrow0$.\n\nSo as to obtain well behaving generators at $q \\to 0$, \nlet us scale $T_n$ as \n\\begin{equation}\n{\\tilde T}_n = T_n p^{\\frac{|n|}{2}}.\n \\label{r1}\n\\end{equation}\nUsing this notation and \ntaking the limit ($q \\to 0$) of the relation (\\ref{e:a1.2}),\nwe have the commutation relation for \nthe deformed Virasoro algebra in this limit.\n\\proclaim Proposition 3. The commutation relations for \nthe deformed Virasoro generators\n${\\tilde T }_n$ are \n\\begin{eqnarray}\n\\left[{\\tilde T }_n,{\\tilde T }_m\\right] &=&\n-(1-t^{-1})\\sum_{\\ell =1}^{n-m}{\\tilde T }_{n-\\ell}{\\tilde T }_{m+\\ell}\n\\quad \\mbox{for}\n\\quad n > m > 0 \\quad \\mbox{or}\\quad 0>n>m, \n\\nonumber\\\\ \n\\left[{\\tilde T }_n,{\\tilde T }_0\\right] &=&\n -(1-t^{-1})\\sum_{\\ell =1}^{n}{\\tilde T }_{n-\\ell}{\\tilde T }_{\\ell}\n-(t-t^{-1})\\sum_{\\ell =1}^{\\infty} t^{-\\ell}\n{\\tilde T }_{-\\ell}{\\tilde T }_{n+\\ell} \\quad \\mbox{for}\\quad n > 0, \n\\nonumber\\\\ \n\\left[{\\tilde T }_0,{\\tilde T }_m\\right] &=&\n -(1-t^{-1})\\sum_{\\ell =1}^{-m}{\\tilde T }_{-\\ell}{\\tilde T }_{m+\\ell}\n-(t-t^{-1})\\sum_{\\ell =1}^{\\infty} t^{-\\ell}\n{\\tilde T }_{m-\\ell}{\\tilde T }_{\\ell} \\quad \\mbox{for}\\quad 0 > m, \n\\nonumber\\\\ \n\\left[{\\tilde T }_n,{\\tilde T }_m\\right] &\\!\\!=\\!\\!&\n-(1-t^{-1}){\\tilde T }_{m}{\\tilde T }_{n} \n-(t-t^{-1})\\sum_{\\ell =1}^{\\infty} t^{-\\ell}\n{\\tilde T }_{m-\\ell}{\\tilde T }_{n+\\ell} \n\\nonumber\\\\ \n& & +(1-t^{-1})\\mbox{\\rm sign}(n)\\delta_{n+m,0}\n\\quad \\mbox{for}\\quad n> 0> m, \n\\label{r2}\n\\end{eqnarray}\nwhere the function $\\mbox{\\rm sign}(n)$ is\n$1$, $0$ and $-1$ for $n>0$, $n=0$ and $n<0$, respectively.\n\n\n\\noindent\n\n\n\\subsection{highest weight modules of \\v}\nLet us define the Verma module of \\v.\nLet $\\kv{\\lambda}$ be the highest weight vector\nsuch that\n$T_0 \\kv{\\lambda}=\\lambda\\kv{\\lambda}$, $\\lambda\\in{\\bf C}$ and \n$T_n \\kv{\\lambda}=0$ for $n>0$.\nThe Verma module $M(\\lambda)$ is defined by\n$M(\\lambda)=\\mbox{\\v}\\kv{\\lambda}$.\nThe irreducible highest module $V(\\lambda)$ is obtained from\n$M(\\lambda)$ by removing all singular vectors and their descendants.\nRight modules are defined in a similar way from the\nlowest weight vector\n$\\bv{\\lambda}$\ns.t.\n$\\bv{\\lambda}T_0=\\lambda\\bv{\\lambda}$ and\n$\\bv{\\lambda}T_n=0$ for $n<0$.\nA unique invariant paring is defined by setting\n$\\bv{\\lambda}\\lambda \\rangle = 1$.\nThe Verma module $M(\\lambda)$ may have\nsingular vectors same as that of the ordinary Virasoro algebra.\nLet us introduce the (outer) grading operator $d$ which satisfies\n$[d , T_n]= n T_n$ and set $d\\ket{\\lambda}=0$. We call a vector\n$\\ket{v} \\in M(\\lambda)$ of level $n$ if $d\\ket{v}=-n\\ket{v}$.\n\nWhether there exist the singular vectors or not is\nchecked by calculating the Kac determinant.\nHere, we give some explicit forms of $f_{n}$\nwhich we will use for the calculations\n\\begin{eqnarray}\n& & f_{1}=\\frac{(1-q)(1-t^{-1})}{1+p},\\nonumber\n\\label{e:a11.2}\n\\\\\n& &\nf_{2}=\\frac{(1-q^{2})(1-t^{-2})}{2(1+p^{2})}+\n\\frac{(1-q)^2(1-t^{-1})^2}{2(1+p)^2}. \\nonumber\n\\label{e:a11.3}\n\\end{eqnarray}\n\\par\nAt level 1, the Kac determinant is the $1\\times 1$\nmatrix as follows\n\\begin{equation}\n\\bv{\\lambda}T_{1} T_{-1}\\kv{\\lambda}\n=\\frac{(1-q)(1-t)}{q + t}(\\lambda^2 - (p^{1\/2}+p^{-1\/2})^2 ).\n\\label{e:a15}\n\\end{equation}\nTherefore, there exist a singular vector at level 1 iff\n$\\lambda=\\pm \\left(p^{1\/2}+p^{-1\/2}\\right)$,\n since $q$ and $t$ are generic.\nThe signs $\\pm$ in the RHS are due to the symmetry \\eq{e:a1.3}.\n\\par\nAt level 2, the Kac determinant is\n\\begin{eqnarray}\n\\& \\hskip-10truem\n \\left|\n \\begin{array}{clcr}\n \\bv{\\lambda}T_{1}T_{1}T_{-1}T_{-1}\\kv{\\lambda} &\n \\bv{\\lambda}T_{1}T_{1}T_{-2}\\kv{\\lambda} \\\\\n \\bv{\\lambda}T_{2}T_{-1}T_{-1}\\kv{\\lambda} &\n \\bv{\\lambda}T_{2}T_{-2}\\kv{\\lambda} \\\\\n \\end{array}\n \\right|\n\n \\frac{(1-q^2)(1-q)^2 q^{-4}(1-t^2)(1-t)^2 t ^{-4}}{(q +t)^2 (q^2 +t^2)}\n\\nonumber \\\\\n\\&\\hskip35truemm\n\\times (\\lambda^2 qt-(q+t)^2 )( \\lambda^2 q^2t-(q^2+t)^2 )\n ( \\lambda^2 qt^2 - (q+t^2)^2 ).\n\\label{e:a18}\n\\end{eqnarray}\nThe vanishing conditions of the Kac determinant are\n\\begin{eqnarray}\n& \\mbox{(i)}&\\lambda =\\pm \\left(p^{1\/2} + p^{-1\/2}\\right),\n\\label{e:a19.1} \\\\\n& \\mbox{(ii)}&\\lambda =\\pm \\left(p^{1\/2}q^{1\/2} + p^{-1\/2}q^{-1\/2}\\right),\n\\label{e:a19.2} \\\\\n& \\mbox{(iii)}& \\lambda =\\pm \\left(p^{1\/2}t^{-1\/2} + p^{-1\/2}t^{1\/2}\\right).\n\\label{e:a19.3}\n\\end{eqnarray}\nIn the case (i), there is a singular vector at level 1.\nIn the cases (ii) and (iii), we have a singular vector at level 2.\nThe singular vector for the case (ii) is\n\\begin{equation}\n \\frac{qt^{-1\/2}(q+t)}{(1-q)^2(1+q)}T_{-1}T_{-1}\\kv{\\lambda}\n\\mp T_{-2}\\kv{\\lambda},\n\\label{e:a20.1}\n\\end{equation}\nand for (iii) is\n\\begin{equation}\n \\frac{q^{-1\/2}t(q+t)}{(1-t)^2(1+t)}T_{-1}T_{-1}\\kv{\\lambda}\n\\mp T_{-2}\\kv{\\lambda}.\n\\label{e:a21.1}\n\\end{equation}\n\n\nTo calculate the Kac determinant becomes difficult task\nwhen $N$ increases.\nWe have calculated up to level 4, and write down the\nconjectural form at level $N$.\n\n\\proclaim Conjecture 1. The Kac determinant at level-N is written as \n\\begin{equation}\\label{Kacconj}\n \\det{}_N\n =\n \\det\\Bigl(\\langle i\\ket{j}\\Bigr)_{1\\leq i,j\\leq p(N)}\n \\!\\!=\\!\\!\n \\prod_{\\scriptstyle r,s\\geq 1 \\atop \\scriptstyle rs\\leq N}\n \\Bigl(\\lambda^2-\\lambda_{r,s}^2\\Bigr)^{p(N-rs)}\n \\left(\\frac{(1-q^r)(1-t^r)}{q^r+t^r}\n \\right)^{p(N-rs)},\n\\end{equation}\nwhere \n$\n\\lambda_{r,s}=\nt^{r\/2}q^{-s\/2}+t^{-r\/2}q^{+s\/2}\n$\nand\nthe basis at level $N$ is defined\n$\\ket{1}=T_{-N}\\ket{\\lambda}$,\n$\\ket{2}=T_{-N+1}T_{-1}\\ket{\\lambda}$,$\\cdots,\n\\ket{p(N)}=T_{-1}^N\\ket{\\lambda}$, and $p(N)$ is the\nnumber of the partition of $N$.\n\n\\noindent\nWe remark that the $\\lambda$ dependence has essentially\nthe same structure as the case of the usual Virasoro algebra.\nTherefore, if $q$ and $t$ are generic, the character of the quantum Virasoro\nalgebra \\v,\nwhich counts the degeneracy at each level,\nexactly coincides with that of the usual Virasoro algebra.\nThe $\\lambda$-independent factor\nin the RHS will play an important role when we study the case\nthat $q$ is a root of unity.\n\n\\subsection{problem of obtaining a geometric interpretation of \\v}\nIt is remarkable that \\v arises in a variety of off-critical models\nin a universal way.\nAs for the geometric interpretation, however, \nwe have not\nhave a satisfactory answer yet.\nFor the ordinary Virasoro algebra with $c=0$, we have \nthe differential operator realization,\n$L_n=-z^{n+1} \\partial_z$,\nwhich explains that \nthe Virasoro algebra describes the Lie algebra structure \nof the tangent space of the conformal group.\nAs a natural deformation of this differential operator realization,\nis it possible to have a difference operator representation of\nthe deformed Virasoro algebra \n\\v for some parameters $q$ and $p$?\nIt may be possible to study a connection between \\v\nand the analysis over the local fields in the limit of \n$q\\rightarrow 0$ with fixed $t$. See Section \\ref{q0limit}.\n\n\n\\section{Free boson realization of \\v }\nIn this section, we present the Heisenberg realization of \\v\nand its applications to Calogero-Sutherland-type models.\n\n\n\\subsection{conformal field theory}\nOne of the simplest example of the conformal field theory is \nthe massless Klein-Gordon field in $1+1$ dimensions. \nWe briefly review how we can treat the \nVirasoro current in terms of the Klein-Gordon \nfield at the conformal point to \nprepare basic ideas and notations for the later discussions.\nAs for the detail, the readers are referred to the original or\nreview articles of the conformal field theory \\cite{rBPZ,rG}.\nThe \naction for the Klein-Gordon field $\\phi(x,\\tau)$ is \n$$\nS_{Eucl}=\\int d\\tau dx{1 \\over 2} \\left(\n (\\partial_\\tau \\phi)^2+ (\\partial_x \\phi)^2\n+ m^2 \\phi^2\\right).\n$$\nIf the system is massless $m=0$ then it acquires the infinite dimensional \nconformal symmetry. \nWe shall see how the generators of this conformal \ntransformation are realized in terms of the \nKlein-Gordon field $\\phi(x,\\tau)$.\nLooking at the \nequation of motion\n\\begin{eqnarray}\n\\partial_w \\partial_{\\bar w}\\phi(w,\\bar{w})=0,\\qquad\\qquad (w=x+i \\tau),\n\\end{eqnarray} \nwe have the decoupling of $\\phi$ into \nchiral and anti-chiral parts as\n$$\n\\phi(w,\\bar{w})=a(w)+\\bar{a}(\\bar{w}).\n$$\nTherefore,\nwe can study the chiral part and anti-chiral part separately.\nAfter the compactification of the \nspace into the segment \n$0\\leq x \\leq 2\\pi$ with the periodic boundary condition \nand introducing the conformal mapping $z=e^{iw}$, we arrive at\nthe expansion\n\\begin{eqnarray}\na(z)=Q+a_0\\ln z - \\sum_{n\\neq 0} {a_n \\over n} z^{-n}.\n\\end{eqnarray}\nThe Poisson brackets for these modes are\n\\begin{eqnarray}\n&& \\{ a_n,a_m\\}_{\\rm P.B.}=n \\delta_{n+m,0}\\qquad\n\\{ a_n,Q\\}_{\\rm P.B.}= \\delta_{n,0}.\n\\end{eqnarray}\nThe Virasoro current $L(z)$ is written as\n$$\nL(z)= \\sum_{n\\in{\\bf Z}} L_n z^{-n-2}=\n{1 \\over 2} \\left( \\partial a(z) \\right)^2.\n$$\nUsing the formula\n$$\n\\{ \\partial a(z),\\partial a(w)\\}_{\\rm P.B.}= {1 \\over z^2}\\delta'(w\/z),\n$$\nwe obtain\n\\begin{equation}\n\\{ L_n,L_m\\}_{\\rm P.B.}= (n-m) L_{n+m}.\n\\end{equation}\n\n\nTo quantize the system, we replace the Poisson brackets by\nthe commutators as\n\\begin{equation}\n[a_n,a_m]= n \\delta_{n+m}\\qquad [ a_n,Q]= \\delta_{n,0},\n\\end{equation}\nand define the Virasoro current with the\nnormal ordered product as\n$$\nL(z)= {1 \\over 2} :\\left( \\partial a(z) \\right)^2:.\n$$\nThe definition of this normal ordering is that \nwe shift every \nannihilation operators ({\\it i.e.}, $a_n$ with $n\\geq 0$)\nto the right of \ncreation operators ({\\it i.e.}, $a_n$ with $n<0$ and $Q$).\nFor example $:a_{-1}a_{2}:=a_{-1}a_{2}$,\n$:a_{3}a_{-2}:=a_{-2}a_{3}$ and so on.\nThis quantized Virasoro current obeys eq.\\ \\eq{e:a8} with $ c=1$.\nIn general the central charge $c$\ndepends on the model; $c=1\/2$ for real fermion, $c=3k\/(k+2)$ for \n$\\widehat{su}(2)_k$\nKac-Moody algebra, for example.\nOne more important construction of the \nVirasoro algebra which is relevant to our later discussion is\nthe Feigin-Fuchs construction\n$$\nL(z)= {1 \\over 2} :\\left( \\partial a(z) \\right)^2:+\n\\alpha_0 \\partial^2 a(z).\n$$\nFor this realization we have the central charge less than one\n$$\nc=1-12 \\alpha_0^2,\n$$\nif $\\alpha_0$ is real.\nFor the later discussion we will parameterize $\\alpha_0$ as\n$$\n\\alpha_0 =\\frac{1}{\\sqrt{2}}\\left({\\sqrt{\\beta}-\\sqrt{1\/\\beta}}\\right).\n$$\nWe will see that this parameter $\\beta $ has the meaning of the coupling\nconstant of the Calogero-Sutherland model. See Section \\ref{secCS}.\n\n\n\\subsection{singular vectors of the Virasoro algebra}\nWhat is very special in the representation space is\nthe singular vectors, which correspond to decoupled states from \nthe physical space. We review some of the \nexplicit formulas of the singular vectors.\n\nThe highest weight state $|h\\rangle$ is defined by\n$L_n |h\\rangle=0$ for $n>0$ and\n$L_0 |h\\rangle=h |h\\rangle$,\nand the Verma module $M(h)$ is spanned over the highest weight state\n$|h\\rangle$ \nas $M(h)=\\langle L_{-1},L_{-2},\\cdots\\rangle |h\\rangle$.\nThe singular vector $|\\chi\\rangle \\in M(h)$ at level $n$ is defined by\n$L_n |\\chi\\rangle = 0$ for $n>0$ and \n$L_0 |\\chi\\rangle = (h+n)|\\chi\\rangle$.\nBy a standard argument, it has null norm with any states\nin the Verma module;\n$\\langle * | \\chi \\rangle =0$.\nThe existence of such state depends crucially on the\nchoice of parameter $c$ and $h$.\nCelebrated Kac formula shows that if they are\nexplicitly parameterized as eq.\\ \\eq{e:a9} and \n\\begin{equation}\\label{eq:Kac}\nh_{rs}=\\frac{(\\beta r-s)^2-(\\beta-1)^2}{4\\beta},\n\\label{eq:KacFormula}\n\\end{equation}\nfor an arbitrary parameter $\\beta(\\neq0)\\in{\\bf C}$\nand integers $r$ and $s$ with $rs>0$, \nthere exists unique (up to normalizatin) null state of level $rs$.\nSome of the lower lying states can be explicitly\nobtained by solving the defining conditions.\nLet $|\\chi_{rs}\\rangle\\in M(h_{rs})$ be \nthe null state at level $rs$.\nFor example,\nwe obtain,\n\\begin{eqnarray}\\label{eq:Lower}\n|\\chi_{11}\\rangle&=& L_{-1}|h_{11}\\rangle,\\nonumber\\\\\n|\\chi_{12}\\rangle&=& (L_{-2}-{\\beta}L_{-1}^2)|h_{12}\\rangle,\\\\\n|\\chi_{22}\\rangle &=& \\left(L_{-4}+\n\\frac{2(\\beta^2-3\\beta+1)}{3(\\beta-1)^2}L_{-3}L_{-1}-\n\\frac{(\\beta+1)^2}{3\\beta}L_{-2}^2\\right.\\nonumber\\\\\n&&\\left.+\n\\frac{2(\\beta^2+1)}{3(\\beta-1)^2}L_{-2}L_{-1}^2-\n\\frac{\\beta}{3(\\beta-1)^2}L_{-1}^4\\right)\n|h_{22}\\rangle, \\nonumber \n\\end{eqnarray}\nand so on.\n\nSince we have the Feigin-Fuchs realization of $L(z)$,\nthe singular vectors are also written in\nthe bosonic creation oscillators $a_{-n}$ acting on the \nvacuum state $|\\alpha \\rangle$ \n({\\it i.e.}, $a_n|\\alpha\\rangle=0$ for $n> 0$ \nand $a_0|\\alpha\\rangle=\\alpha|\\alpha\\rangle$).\nIt is quite remarkable that all these singular vectors\ncan be regarded as the ``Jack symmetric polynomials''\nwhen we replace $a_{-n}$ by the power sum $\\sum_{i=1}^N x_i^n$.\nAs for the proof of this correspondence \nthe reader is referred to \\cite{rMY}.\n\nIn the Feigin-Fuchs construction, the Virasoro generators are \n\\begin{equation}\\label{eq:coulomb}\nL_n=\\frac{1}{2}\\sum_{m\\in {\\bf Z}} :a_{n+m} a_{-m}: -\\alpha_0\n (n+1) a_n.\n\\end{equation}\nThe highest weight state $|h_{rs}\\rangle$ is realized as \n$\\ket{\\alpha_{rs}}\\equiv e^{\\alpha_{rs}Q}|0\\rangle$,\nwith\n\\begin{equation}\n\\alpha_{rs}=\\frac{1}{\\sqrt 2}\\left((1+r)\\sqrt \\beta\n-(1+s)\\sqrt{1\/\\beta}\\right).\n\\label{eq:alphaRS}\n\\end{equation}\nIn terms of this free boson oscillators,\nthe null states (\\ref{eq:Lower}) are written as,\n\\begin{eqnarray}\n|\\chi_{11}\\rangle & \n& a_{-1} |\\alpha_{11}\\rangle, \\nonumber\\\\\n|\\chi_{12}\\rangle & \n& \\left(a_{-2}\n + \\sqrt{2\\beta}a_{-1}^2\\right)|\\alpha_{12}\\rangle, \\\\\n|\\chi_{22}\\rangle & \n& \\left( a_{-4}\n +\\frac{4\\sqrt{2\\beta}}{1-\\beta}\n a_{-3}a_{-1}\n -2\\frac{1+\\beta+\\beta^2}{\\sqrt{2\\beta}(1-\\beta)}a_{-2}^2\n -4 a_{-2}a_{-1}^2\n -\\frac{2\\sqrt{2\\beta}}{1-\\beta} a_{-1}^4\n \\right)|\\alpha_{22}\\rangle.\\nonumber\n\\end{eqnarray}\n\nTo translate these expressions into the Jack symmetric functions,\none can apply the rule,\n\\begin{equation}\na_{-n}\\rightarrow \\sqrt{\\frac{\\beta}{2}}\\sum_{i=1}^N x_i^n,\n\\qquad\n\\ket{\\alpha_{rs}}\\rightarrow 1.\n\\end{equation}\nUsing this rule, we have the correspondence \\cite{rMY}\n\\begin{equation}\n|\\chi_{rs}\\rangle \\sim J_{\\{ s^r\\}}(x;\\beta),\n\\end{equation}\nwhere the R.H.S. is the Jack symmetric polynomial for the \nrectangular diagram ${\\{ s^r\\}}$.\n\nIt was shown in \\cite{rAMOS} that the Jack polynomials for\narbitrary Young diagrams are realized as \nthe singular vector of the $W_N$ algebra.\n\n\n\n\\subsection{Calogero-Sutherland Hamiltonian, the Jack \npolynomials and the Virasoro algebra}\n\\label{secCS}\nIn this section, we explain the ``Calogero-Sutherland-Virasoro'' \ncorrespondence. As for the details, the readers are referred to \n\\cite{rAMOS} and references therein. \n\nThe Jack symmetric polynomials arise in \nthe Calogero-Sutherland (CS) model \\cite{rCS} as the wave functions of \nthe excited states of this model.\nAfter a suitable coordinate transformation, \nthe Hamiltonian and momentum of this system become\n\\begin{equation}\n {\\cal H}=\n \\sum_{i=1}^N D_i^2\n +\\beta\\sum_{i0$)\n\\begin{eqnarray}\na_n=\\sqrt{1 \\over 2\\beta}a^{\\rm old}_n ,\n\\qquad a_{-n}=\\sqrt{2 \\over \\beta}a^{\\rm old}_{-n}, \\qquad \na_0=\\sqrt{1 \\over 2\\beta}a^{\\rm old}_0,\\qquad\nQ= \\sqrt{2 \\over \\beta}Q^{\\rm old}.\n\\end{eqnarray}\nCorrespondingly we change the notation for \n$\\alpha_{rs}$ as\n$$\n\\alpha_{rs}=\\frac{1}{ 2}\\left((1+r)\\beta\n-(1+s)\\right),\n$$\nand write \n$|h_{rs}\\rangle= \\ket{\\alpha_{rs}}\\equiv e^{\\alpha_{rs}Q}|0\\rangle$\nas before.\n\nOne may derive bosonized Hamiltonian \nand momentum $\\widehat{\\cal H}$ and $\\widehat{\\cal P}$\nwhich satisfy\n$\n{\\cal O} \\pi_N \\langle 0|\\exp(\\beta \\sum_n{ a_n \\over n}p_n)\n=\\pi_N \\langle 0|\\exp(\\beta \\sum_n {a_n \\over n}p_n)\\widehat{\\cal O},\n\\label{e12}\n$\nwhere ${\\cal O}={\\cal H},{\\cal P}$, and \n$\\pi_N$ denotes the projection to the $N$-particle space.\nThey are given by,\n\\begin{equation}\n \\widehat{\\cal H}= \\beta \\sum_{n>0}\na_{-n} \\,L_n + (\\beta-1+\\beta N-2 a_0)\\, \\widehat{\\cal P},\n\\qquad\\quad\n\\widehat{\\cal P}=\\beta\\sum_{n=1}^\\infty a_{-n}a_n.\n\\end{equation}\nHere $L_n$'s are the annihilation operators of\nthe Feigin-Fuchs construction of the Virasoro algebra with\nthe center $c$ in \\eq{e:a9}.\nUsing these formulas, it is easily shown that\nthe singular vector $|\\chi_{rs}\\rangle$ of the Virasoro algebra is\nproportional to \nthe Jack polynomial for the Young diagram\n$\\lambda=\\{s^r\\}$ , since we have\n\\begin{equation}\n \\widehat{\\cal H}|\\chi_{rs}\\rangle =\n\\epsilon_{\\{s^r\\}}|\\chi_{rs}\\rangle.\n\\end{equation}\nwith $\\epsilon_{\\{s^r\\}}=(\\beta(N-r)+s)\\,rs$.\n\n\n\\subsection{screening currents}\nIn the Feigin-Fuchs construction we are able to have \ntwo weight-one primary fields $S_\\pm(z)$, which are called screening currents.\nThe condition of being weight-one primary gives us \nthe equation\n\\begin{eqnarray}\n[L_n,S_\\pm(z)]= \\partial_z \\left( z^{n+1} S_\\pm(z) \\right). \n\\end{eqnarray}\nWe have the solutions of this equation as follows\n\\begin{eqnarray}\n S_+(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{\\ps{n}\\beta \\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{-\\ps{n}2\\beta \\frac{a_{n}}{n}z^{-n}\n \\right\\} e^{\\beta Q}z^{2\\beta a_0}, \\label{Vscr1}\\\\\n S_-(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{-\\ps{n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{\\ps{n}2\\frac{a_{n}}{n}z^{-n}\\right\\}\n e^{- Q}z^{-2 a_0}. \\label{Vscr2}\n\\end{eqnarray}\n\nIn the Fock module with the highest weight state\n$\\ket{\\alpha_{r,s}}$, we have a singular vector $\\ket{\\chi_{r,s}}$\nat level $rs$. By using a screening current $S_+(z)$,\n$\\ket{\\chi_{r,s}}$ is given as follows \\cite{rKM,rTK}:\n\\begin{eqnarray}\n \\ket{\\chi_{r,s}}\n &\\!\\!=\\!\\!&\n \\oint\\prod_{j=1}^r\\frac{dz_j}{2\\pi i}\\cdot\n \\prod_{i=1}^r S_+(z_i)\n \\ket{\\alpha_{-r,s}} \n\\label{eq:VirSing}\\\\\n &\\!\\!=\\!\\!&\n \\oint\\prod_{j=1}^r\\frac{dz_j}{2\\pi iz_j}\\cdot\n \\prod_{i,j=1 \\atop i0&& \\\\\n&&\\\\\nq\\mbox{-deformation} \\Biggl\\downarrow &&\nq\\mbox{-deformation} \\Biggl\\downarrow \\\\\n && \\\\\n {\\rm Macdonald\\;difference\\;operator} &\n{\\displaystyle\\mathop{ \\llla}^{\\rm singular\\;vectors}}& \nT_n\\;{\\rm generates\\; the\\;} q\\mbox{-Virasoro algebra \\v } \\\\\n\\widehat{D}_{q,t} = \\ps{n} \\psi_{-n}T_{n}+\\cdots &&\\\\\n T_n P_{\\{s^r\\}}(x;q,t)=0\\quad {\\rm for}\\;n>0&& \\\\\n &&\n\\end{array}\n\\end{eqnarray*}\nHere, $\\psi_{-n}$ should be a suitable combination of \nthe creation operators $a_{-m}$'s with degree $n$.\n\nThe problem is ``to make this diagram commutative.''\nHowever, we have many unknown operators \n$\\psi_{-n}$ and $T_n$. \nTo have enough data to solve this problem, \nsome knowledge of the $q$-deformed screening operators\nmust be needed.\n\nBy studying the action of the bosonized Macdonald operator\n$\\widehat{D}_{q,t}$,\nwe are able to obtain bosonized realization for some of \nthe Macdonald polynomials.\nLet,\n\\begin{equation}\n \\exp\\left\\{ \\sum_{n=1}^\\infty \\frac{1-q^{\\gamma n}}{1-q^n}\n \\frac{a_{-n} }{n} z^n\\right\\}\n =\n \\sum_{n=0}^{\\infty} \\widehat Q_n^{(\\gamma)} z^n \\label{macQ}\n\\end{equation}\nthe states $\\widehat Q_n^{(\\gamma)}|0\\rangle$ with $\\gamma=\\beta$ or $-1$\nare the Macdonald polynomials $Q_\\lambda(x;q,t)$\ncorresponding to the Young diagram\nwith single row $(n)$ or single column $(1^n)$, respectively.\nAs for the difference between $P_\\lambda(x;q,t)$ and \n$Q_\\lambda(x;q,t)$, see \\cite{rM}.\nWe obtained other examples;\nfor the Young diagram with\ntwo rows $\\lambda=(\\lambda_1,\\lambda_2)$ or\ntwo columns ${}^t\\lambda=(\\lambda_1,\\lambda_2)$ we have\n\\begin{eqnarray}\n \\widehat Q_{(\\lambda_1,\\lambda_2)}^{({\\gamma})}|0\\rangle\n &\\!\\!=\\!\\!&\n \\sum_{\\ell=0}^{\\lambda_2}c^{({\\gamma})}(\\lambda_1-\\lambda_2,\\ell)\n \\widehat Q_{\\lambda_1+\\ell}^{({\\gamma})}\n \\widehat Q_{\\lambda_2-\\ell}^{({\\gamma})}\n |0\\rangle, \\nonumber\\\\\n c^{({\\gamma})}(\\lambda,\\ell)\n &\\!\\!=\\!\\!&\n \\frac{1-q^{\\frac{\\beta}{\\gamma}(\\lambda+2\\ell)}}\n {1-q^{\\frac{\\beta}{\\gamma}(\\lambda+\\ell)}}\n \\prod_{j=1}^{\\ell}\n \\frac{1-q^{\\frac{\\beta}{\\gamma}(\\lambda+j)}}\n {1-q^{\\frac{\\beta}{\\gamma}j}}\\cdot\n \\prod_{i=1}^{\\ell}\n \\frac{q^{\\gamma}-q^{\\frac{\\beta}{\\gamma}(i-1)}}\n {1-q^{\\gamma+\\frac{\\beta}{\\gamma}(\\lambda+i)}},\n\\end{eqnarray}\nwith $\\gamma=\\beta$ or $-1$, respectively.\n The Macdonald polynomials of single hook $(n,1^m)$ are,\n\\begin{equation}\n \\widehat Q_{(n,1^{m})}|0\\rangle =\n \\sum_{\\ell=0}^{m} \\frac{1-q^{n+\\ell}t^{m-\\ell}}{1-q} q^{m-\\ell}\\:\n \\widehat Q_{n+\\ell}^{(\\beta)}\\widehat Q_{m-\\ell}^{(-1)}|0\\rangle.\n\\end{equation}\nThese explicit formulas in terms of (\\ref{macQ})\nstrongly suggest that the screening currents\nfor \\v should be\n\\begin{eqnarray}\n S_+(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{\\ps{n}\\frac{1-t^n}{1-q^n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{{\\rm annihilation\\; part\\; for }S_+\n \\right\\} e^{\\beta Q}z^{2\\beta a_0},\\\\\n S_-(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{-\\ps{n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{{\\rm annihilation \\;part \\;for } S_- \\right\\}\n e^{- Q}z^{-2 a_0}.\n\\end{eqnarray}\nNote that this can be regarded as a good deformation of the\nscreening currents (\\ref{Vscr1}) and (\\ref{Vscr2}). \nIt is assumed that \nthe zero-mode parts are not deformed.\n\\\\\n\nIn \\cite{rSKAO}, it is shown that the problem of finding \nall these unknown parts for $\\widehat{D}_{q,t}$, $\\psi_n$ and \n$S^\\pm(z)$ is ``uniquely solved'' if we start from a nice \nansatz for the operators. The reason or mechanism\nbehind the process of solving the problem have not been \nwell investigated yet and seem somehow mysterious. \nWhat is clear so far is that the ``locality'' for \nthe operators seems quite important, {i.e.}, \nthe behaviors of the $\\delta$-functions\nin the OPE factors must be well controlled in some sence.\nThe mathematical structure of this ``locality'' should be \nunderstood in the future.\nTherefore,\nwe will not show the processes of solving this \nproblem here.\nLet us just summarize the results.\n\\proclaim Theorem 1. \\hspace{-2mm}\\cite{rSKAO} \nThe quantum deformed Virasoro current $T(z)$ is realized as\n\\begin{eqnarray}\\label{eq:qVirFFR1}\nT(z)&=& \\Lambda^+(z)+\\Lambda^-(z),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Lambda^+(z)\n &\\!\\!\\!\\!\\! = \\!\\!\\!\\!& p^{1\/2}\\exp\\left\\{-\\ps{n}\\frac{1-t^n}{1+p^n}\n\\frac{a_{-n}}{n}z^{n}t^{-n}p^{-n\/2}\\right\\}\n \\exp\\left\\{-\\ps{n}(1-t^n)\\frac{a_{n}}{n}z^{-n}p^{n\/2}\\right\\}\n q^{\\beta a_{0}}, \\nonumber \\\\\n\\Lambda^-(z)\n &=&\\!\\!\\! p^{-1\/2}\\exp\\left\\{\\ps{n}\\frac{1-t^n}{1+p^n}\n\\frac{a_{-n}}{n}z^{n}t^{-n}p^{n\/2}\\right\\}\n \\exp\\left\\{\\ps{n}(1-t^n)\\frac{a_{n}}{n}z^{-n}p^{-n\/2}\\right\\}\n q^{-\\beta a_{0}}.\n\\label{e:b1}\n\\end{eqnarray}\n\n\\noindent By studying the operator product expansions, it is \neasy to see that\nthis $T(z)$ satisfies the relation of \\v (\\ref{e:a1.2}).\nWe can observe that this formula has strong resemblance to the\ndressed vacuum form (DVF) in the algebraic Bethe ansatz.\nThis profound $q$-Virasoro-DVF correspondence \n($T(z)=\\Lambda^+(z)+\\Lambda^-(z)$) was\ndiscovered by Frenkel and Reshetikhin \\cite{rFR}.\n\n\\proclaim Theorem 2. \\hspace{-2mm}\\cite{rSKAO} \nThe Macdonald operator is written as\n\\begin{eqnarray}\n\\widehat{D}_{q,t}&=&\\frac{t^N}{t-1}\\left[\\oint\\frac{dz}{2\\pi\\i}\\frac{1}{z}\n\\psi(z)T(z) -p^{-1}q^{-2\\beta a_0}\\right] -\\frac{1}{t-1}\n\\nonumber \\\\\n&= &\\frac{t^N}{t-1}\\left[\\pzs{n}\\psi_{-n}T_n\n -p^{-1}q^{-2\\beta a_0}\\right]-\\frac{1}{t-1},\n\\label{e:b7.3}\n\\end{eqnarray}\nwhere the field $\\psi(z)$ is given by\n\\begin{equation}\n\\psi(z)=\\pzs{n}\\psi_{-n}z^n=p^{-1\/2}\\exp\\left\\{-\\ps{n}\\frac{1-t^n}{1+p^n}\n\\frac{a_{-n}}{n}z^np^{n\/2}t^{-n}\\right\\}\n q^{-\\beta a_0 }\n\\end{equation}\n\n\\proclaim Theorem 3. \\hspace{-2mm}\\cite{rSKAO} The screening currents for \\v\n\\begin{eqnarray}\n S_+(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{\\ps{n}\\frac{1-t^n}{1-q^n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{-\\ps{n}(1+p^n)\\frac{1-t^n}{1-q^n}\\frac{a_{n}}{n}z^{-n}\n \\right\\} e^{\\beta Q}z^{2\\beta a_0},\n \\label{e:c1.1}\\\\\n S_-(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{-\\ps{n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{\\ps{n}(1+p^n)\\frac{a_{n}}{n}z^{-n}p^{-n}\\right\\}\n e^{- Q}z^{-2 a_0},\n \\label{e:c1.2}\n\\end{eqnarray}\nsatisfies the \ncommutation relation:\n\\begin{eqnarray}\n \\Bigl[T_n,S_+(w)\\Bigr]\n &\\!\\!=\\!\\!&\n -(1-q)(1-t^{-1})\\frac{d_q}{d_q w}\n \\left((p^{-\\frac12}w)^{n+1}p^{\\frac12}:\\Lambda^-(p^{-\\frac12}w)S_+(w):\n \\right), \\label{e:c2.1}\\\\\n \\Bigl[T_n,S_-(w)\\Bigr]\n &\\!\\!=\\!\\!&\n -(1-q^{-1})(1-t)\\frac{d_t}{d_t w}\n \\left((p^{\\frac12}w)^{n+1}p^{-\\frac12}:\\Lambda^+(p^{\\frac12}w)S_-(w):\n \\right),\n \\label{e:c2.2}\n\\end{eqnarray}\nwhere\nthe difference operator with one parameter\nis defined by\n\\begin{equation}\n\\frac{d_\\xi}{d_\\xi z}g(z)=\\frac{g(z)-g(\\xi z)}{(1-\\xi)z}.\n\\label{e:c6}\n\\end{equation}\n\nSince the Kac determinant has the same structure as \nthe ordinary Virasoro algebra has, the structure of the\nsingular vectors are not changed in an essential manner.\nAs for the case of the Jack polynomials, \nwe can write down all the singular vectors at least\nformally in the following way\n\\begin{eqnarray}\\label{eq:qVirSing}\n \\ket{\\chi_{r,s}}\n &\\!\\!=\\!\\!&\n \\oint\\prod_{j=1}^r\\frac{dz_j}{2\\pi i}\\cdot\n \\prod_{i=1}^r S_+(z_i)\n \\ket{\\alpha_{-r,s}} ,\n\\end{eqnarray}\nhowever, we have to be very careful about the \nintegration cycle.\nRecently, very nice construction of \nthe integration cycles together with\nsome modification of the integration kernel\nis achieved in \\cite{rLP2,rJLMP}.\nA $q$-deformation of the Felder complex is \nconstructed in these works, and \nmathematical treatment becomes much ``easier'' than\nthe original case.\n\nFinally, we have as the Jack case,\n\n\\proclaim Theorem 4.\nThere exists a one to one correspondence between \nthe singular vectors $|\\chi_{r,s}\\rangle $ of the $q$-Virasoro algebra \\v and\nthe Macdonald functions $P_{\\{ s^r\\}}(x;q,t)$ \nwith the rectangular Young diagram ${\\{ s^r\\}}$ up to \nnormalization constants. \nIt is simply given by\n\\begin{eqnarray}\nP_{\\{ s^r\\}}(x;q,t) \\propto \n \\langle \\alpha_{r,s}| \n\\exp\\left\\{\\sum_{n=1}^\\infty \\frac{1-t^n}{1-q^n}\\frac{a_n}{n}\np_n\\right\\}|\\chi_{r,s}\\rangle,\n\\end{eqnarray}\nwhere $\\langle \\alpha_{r,s}| \\alpha_{r,s}\\rangle=1$.\n\nFor the general Young diagram, \nthe Macdonald polynomials correspond to\nthe singular vectors of $q$-${\\cal W}$ algebras \\cite{rAKOS}.\n\n\n\n\\subsection{\\v and the Hall-Littlewood polynomial}\n\\label{q0limit}\nIf we take the limit $q \\to 0$, the Macdonald \npolynomial $P_\\lambda(x;q,t)$ reduces to the \nHall-Littlewood polynomial $P_\\lambda(x;t)$ \\cite{rM}.\nLet us study the limit of \\v in $q \\to 0$,\nand the connection \nbetween \\v and the Hall-Littlewood polynomials $P_\\lambda(x;t)$.\nThe commutation relations are already given in (\\ref{r2}). \nThe Kac determinants at lower levels are calculated as\n\\begin{eqnarray}\n \\det{}_1 &\\!\\!=\\!\\!&\n \\langle \\lambda| {\\tilde T}_1 {\\tilde T}_{-1}|\\lambda\\rangle =1-t^{-1}, \\nonumber\\\\ \n \\det{}_2\n &\\!\\!=\\!\\!&\n \\left|\n \\begin{array}{cc}\n \\langle\\lambda| {\\tilde T}_{2}{\\tilde T}_{-2}|\\lambda\\rangle &\n \\langle\\lambda| {\\tilde T}_{2}{\\tilde T}_{-1}{\\tilde\nT}_{-1}|\\lambda\\rangle \\\\\n \\langle\\lambda| {\\tilde T}_{1}{\\tilde T}_{1}{\\tilde T}_{-2}|\\lambda\\rangle &\n \\langle\\lambda| {\\tilde T}_{1}{\\tilde T}_{1}{\\tilde T}_{-1}{\\tilde\nT}_{-1}|\\lambda\\rangle\n \\end{array}\n \\right| \\nonumber\\\\\n &\\!\\!=\\!\\!&\n (1-t^{-1})^2 (1-t^{-2}).\n \\label{r5}\n\\end{eqnarray}\nHere,\nwe observe that the Kac determinants do not depend on $\\lambda$.\nTherefore, if $t$ is generic, we have no singular vectors for any $\\lambda$.\nTo study the degeneration of the boson realization, \nwe have to restrict the zero-mode charge of the vacuum \n$|\\alpha_{r,s}\\rangle$ to $s= 0$, otherwise, we are not able to \nobtain nontrivial algebra.\nIt is easy to see that the operators\n$\\tilde{\\Lambda}^\\pm(z)\\equiv\n\\lim_{q\\rightarrow0}\\Lambda^\\pm(p^{\\pm1\/2}z)$ are well behaving ones.\nIntroducing the renormalized boson\n$b_n=- t^{n} a_n$, $b_{-n}= a_{-n}$ for $n>0$, \nwe have in $q\\rightarrow0$, \n\\begin{equation}\n[b_n,b_m]=n {1 \\over 1-t^{-|n|}}\\delta_{n+m,0},\n\\end{equation}\n\\begin{eqnarray}\n\\tilde{\\Lambda}^+(z)\n &=& t^{r\/2} \\exp\\left\\{\\ps{n} (1-t^{-n})\n\\frac{b_{-n}}{n}z^{n} \\right\\}\n \\exp\\left\\{-\\ps{n}(1-t^{-n})\\frac{b_{n}}{n}z^{-n}\\right\\}, \n\\label{eq:BosonHall} \\\\\n\\tilde{\\Lambda}^-(z)\n &=& t^{-r\/2} \\exp\\left\\{-\\ps{n}(1-t^{-n})\n\\frac{b_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{\\ps{n}(1-t^{-n})\\frac{b_{n}}{n}z^{-n}\\right\\},\\nonumber \n\\end{eqnarray}\non the Fock space spanned over $|\\alpha_{r,0}\\rangle$.\nNote that these are essentially the same as \nJing's operators $H(z)$ and $H^*(z)$ for the \nHall-Littlewood polynomial $P_\\lambda(x;t^{-1})$ \\cite{rJ}.\nUsing this notation, the rescaled \\v generator\n$ {\\tilde T }_n$ is expressed as\n\\begin{equation}\n{\\tilde T }_n =\\oint\\frac{dz}{2 \\pi i z}\n\\left(\\,\\, \\theta[\\,n\\leq0\\,] \\tilde{\\Lambda}^+(z) + \n \\theta[\\,n\\geq0\\,] \\tilde{\\Lambda}^-(z)\n\\,\\,\\right) z^n,\n\\end{equation}\nwhere $\\theta[P]=1$ or $0$ \nif the proposition $P$ is true or false, respectively. \nThis formula and the coincidence of our\n$\\tilde{\\Lambda}^+(z)$ with Jing's $H(z)$ means that \nin $q\\rightarrow0$ limit,\n``all the vectors in the Fock module'' are\nwritten in terms of the Hall-Littlewood polynomials.\n\nThe screening currents will disappear in this limit\nin the sense that $S_-(z)$ and\n$\\left[\\tilde{T}_n,S_+(w)\\right]$ become singular.\nThe disappearance of the singular vectors in the Fock module \nwhich is derived from the study of the Kac determinants\nis explained by this singular behavior of the \nscreening currents.\n\nIt seems interesting to study the relation between \\v and the\nHall algebra \\cite{rM} which is related with the analysis over \nthe local fields.\nIs it possible to have a geometric interpretation \nof \\v for $q=0$? \n\n\n\n\\section{Further aspects of \\v and relations with other models}\n\n\nHere we discuss {\\it i)}\nthe representation theories, {\\it ii)} application to the ABF model and \n{\\it iii)} a hidden elliptic algebra generated by screening current.\nWe will also study the limits of $\\beta=1$, $3\/2$ and $2$,\nand find some connections with the Kac-Moody algebras\nwhen $\\beta=1$, for $\\beta=3\/2$, the $q$-Virasoro generator\nis given by a BRST exact form and construct a topological model,\nand when $\\beta=2$, the $q$-Virasoro algebra relates\nwith $c=1$ ${\\cal W}_{1+\\infty}$ algebra. \n\n\n\\subsection{symmetric realization of \\v and vertex operators}\n\\label{secSYMMETRIC}\n\nWhen we introduced the Feigin-Fuchs realization,\nthe creation operators and the annihilation operators\nhad nice symmetry. However we destroyed this symmetry \nto make many formulas for the Jack and Macdonald \nsymmetric polynomials become simple.\nWe will study some applications of \\v to other \nsolvable systems. So, it helps us very much\nto have a ``symmetric expressions'' of \n$T(z)$ and $S_\\pm(z)$ \\cite{rFFr,rAKOS,rLP2,rKa,rAKMOS}.\n\nLet us introduce the fundamental Heisenberg algebra\n$h_n$ ($n\\in{\\bf Z}$), $Q_{h}$ having the commutation relations\n\\begin{eqnarray}\n [h_n,h_m]\n &\\!\\!=\\!\\!&\n \\frac{1}{n}\\frac{(q^{\\frac{n}{2}}-q^{-\\frac{n}{2}})\n (t^{\\frac{n}{2}}-t^{-\\frac{n}{2}})}{p^{\\frac{n}{2}}+p^{-\\frac{n}{2}}}\n \\delta_{n+m,0}\n ,\\qquad\n [h_n,Q_h]=\\frac12\\delta_{n,0}. \\label{boscom}\n\\end{eqnarray}\nThe correspondence to the bosonic oscillators \nin the last two subsections is $(n>0)$\n\\begin{equation}\nh_{ n}={t^n -1 \\over n} a_{ n},\\qquad\nh_{-n}= {1\\over n}{1-t^{-n}\\over 1+p^n} a_{-n},\\qquad\nh_0 = \\sqrt\\beta a_0,\\qquad\nQ_h ={\\sqrt\\beta\\over 2} Q.\n\\end{equation}\nBy these, \nthe Virasoro current $T(z)$ and the screening current $S_{\\pm}(z)$\n(with some modification) are written as\n\\begin{eqnarray}\n T(z) &\\!\\!\\!=\\!\\!\\!& \\Lambda^+(z)+\\Lambda^-(z),\n\\label{eq:qVirFFR2}\\\\\n \\Lambda^\\pm(z) &\\!\\!\\!=\\!\\!\\!&\n :\\exp\\left\\{\\pm\\sum_{n\\neq 0}h_np^{\\pm\\frac{n}{2}}z^{-n}\\right\\}:\n q^{\\pm\\sqrt{\\beta}h_0}p^{\\pm\\frac12}, \\\\\n S_{+}(z) &\\!\\!\\!=\\!\\!\\!& \n :\\exp\\left\\{- \\sum_{n\\neq 0}\n \\frac{p^{\\frac{n}{2}}+p^{-\\frac{n}{2}}}\n {q^{\\frac{n}{2}}-q^{-\\frac{n}{2}}}\n h_nz^{-n}\\right\\}:\n e^{2\\sqrt{\\beta}Q_h}z^{2\\sqrt{\\beta}h_0},\n\\label{eq:SC21}\\\\\n S_{-}(z) &\\!\\!\\!=\\!\\!\\!& \n :\\exp\\left\\{ \\sum_{n\\neq 0}\n \\frac{p^{\\frac{n}{2}}+p^{-\\frac{n}{2}}}\n {t^{\\frac{n}{2}}-t^{-\\frac{n}{2}}}\n h_nz^{-n}\\right\\}:\n e^{-{2\\over\\sqrt{\\beta}}Q_h}z^{-{2\\over\\sqrt{\\beta}}h_0}.\n\\label{eq:SC22}\n\\end{eqnarray}\nIf we introduce the isomorphisms $\\theta$ and $\\omega$ \nof the Heisenberg algebra\nrelated with (\\ref{e:aa1}), (\\ref{e:aa2}):\n\\begin{eqnarray}\n &&{\\rm (II')}\\;\\;\\;\\; \\theta:\n (q,t) \\mapsto (q^{-1},t^{-1}),\\quad \n h_n \\mapsto -h_n \\:(n\\neq 0),\n\\quad\n h_0 \\mapsto h_0, \\quad Q_h \\mapsto Q_h, \\cr\n && {\\rm (III')} \\;\\;\\;\\;\\omega: \\quad \n q \\leftrightarrow t \\quad, \\quad \n h_n \\mapsto -h_n,\n \\quad Q_h \\mapsto - Q_h , \\label{symm}\n\\end{eqnarray}\nthen\n$\\Lambda^-(z) = \\theta\\cdot \\Lambda^+(z) = \\omega\\cdot \\Lambda^+(z)$,\n$S_{-}(z) = \\omega\\cdot S_{+}(z)$ and\n$ \\theta\\cdot S_{\\pm}(z)= S_{\\pm}(z)$.\nHere $\\omega\\cdot \\beta$ should be understood as $1\/\\beta$.\nUnder the isomorphism $\\sigma$ such that:\n\\begin{equation}\n {\\rm (IV)}\\;\\;\\;\\; \\sigma:\n q \\leftrightarrow 1\/t, \\qquad \n \\sqrt\\beta \\leftrightarrow -\\sqrt{1\/\\beta},\n\\end{equation}\n$\\sigma\\cdot\\Lambda^\\pm(z) = \\Lambda^\\pm(z)$ and\n$\\sigma\\cdot S_\\pm(z) = S_\\mp(z)$.\n\n\nThe free boson realization for $T(z)$ is expressed as \nthe following deformed Miura transformation \\cite{rFR}\n\\begin{equation}\n:\\!\\left( p^D - \\Lambda^+(z) \\right) \\left( p^D - \\Lambda^-(z) \\right)\\!:\\,\n= p^{2D} - T(z) p^D + 1,\n\\label{eq:qMiura}\n\\end{equation}\nwhich has been generalized to define \nthe $q$-deformed $\\cal W$ algebra \\cite{rFFr,rAKOS}.\nBy using this transformation,\nFrenkel-Reshetikhin \\cite{rFR} proposed a generalization of\ntheir quasi-classical $q$-Virasoro algebra to $ABCD$-type cases.\nAn analogy to the Baxter's dressed vacuum form $Q$ defined by\n$:\\!\\left( p^D - \\Lambda^-(z) \\right) Q(z)\\!:\\, = 0$,\nso $\\Lambda^\\pm(z) = \\,:\\! Q(zp^{\\mp1})\\, Q(z)^{-1}\\!:$,\nseems to be of some interest.\n\n\nThe vertex operator defined by\n\\begin{equation}\nV_{2,1}(z) \\equiv \\,:\\! \\exp\\left\\{\n\\sum_{n\\neq0}{h_n\\over q^{n\\over2} - q^{-{n\\over2}} } z^{-n} \\right\\}\\!:\ne^{-\\sqrt\\beta Q_h} z^{-\\sqrt\\beta h_0},\n\\label{eq:Vertex21}\n\\end{equation}\nsatisfies\n$\\theta\\cdot V_{2,1}(z) = V_{2,1}(z)$ and\n\\begin{eqnarray}\ng\\left({w\\over z}p^{\\pm{1\\over2}}\\right) T(z) V_{2,1}(w) \n&\\!\\!\\!-\\!\\!\\!&\nV_{2,1}(w) T(z) g^{-1}\\left({z\\over w}p^{\\mp{1\\over2}}\\right)\\\\\n&\\!\\!\\!=\\!\\!\\!&\nt^{\\pm{1\\over4}}(t^{1\\over2} - t^{-{1\\over2}})\\,\n\\delta\\left(t^{\\pm{1\\over2}}{w\\over z}\\right)\n V_{2,1}(q^{\\pm{1\\over2}} w) p^{\\mp{1\\over2}},\n\\label{eq:T&V21}\n\\end{eqnarray}\nwith\n\\begin{equation}\ng(x) = \nt^{\\pm{1\\over4}}\n\\exp\\left\\{\\pm\\sum_{n>0}{1\\over n}\n{ t^{n\\over2} - t^{-{n\\over2}} \\over p^{n\\over2} - p^{-{n\\over2}} }x^n\n\\right\\}.\n\\end{equation}\nNote that\n$ V_{2,1}(q^{\\pm{1\\over2}} w) p^{\\mp{1\\over2}} =\n\\,:\\! \\Lambda^\\mp(t^{\\mp{1\\over2}}w) V_{2,1}(w)\\!:\\,$. \nIf we let\n$V_{1,2}(z) \\equiv \\sigma\\cdot V_{2,1}(z)$ and their fusion as\n\\begin{equation}\nV_{\\ell+1,k+1}(z) \\equiv \\,:\\!\n\\prod_{i=1}^\\ell V_{2,1}(q^{\\ell+1-2i\\over2\\ell}z)\n\\prod_{j=1}^k V_{1,2}(t^{k +1-2j\\over2k }z)\n\\!:,\n\\label{eq:VertexLK}\n\\end{equation}\nthen they also obey a similar commutation relation as \\eq{eq:T&V21}, \nwhich reduces to the usual defining relation for the Virasoro primary field\nof the conformal weight $h_{\\ell+1, k+1}$,\nin the limit $q\\rightarrow1$ \\cite{rAKMOS}.\nThe adjont action of the \\v generator $T(z)$ \non this fused vertex operator $V_{\\ell+1,k+1}$\nmay be closely connected with \na coproduct of the algebra \\v.\nSimilar but slightly different definition for fused vertex operators\n\\begin{equation}\n :\\!\n \\prod_{i=1}^{\\ell}V_{2,1}(q^{-\\frac{k}{2}}t^{\\frac{\\ell+1}{2}-i}z)\n \\prod_{j=1}^k V_{1,2}(t^{-\\frac{\\ell}{2}}q^{\\frac{k+1}{2}-j}z)\n \\!:, \\label{kade}\n\\end{equation}\nwas proposed in \\cite{rKa}. \nThe meaning of fused operators given by (\\ref{eq:VertexLK}) or\n(\\ref{kade}) has not been made clear yet. \n\n\n\nThe fundamental vertex operators $V_{2,1}(z)$ and $V_{1,2}(z)$,\nthat satisfy fermion like anti-commutation relation,\nare especially important.\nBecause\nthe $q$-Virasoro generator and screening currents are expressed by them\nas follows;\n\\begin{equation}\n \\Lambda^+(zp^{1\\over 2}) \n=\\,\\, : V_{2,1}^+(zq^{-{1\\over 2}}) V_{2,1}^{-}(zq^{1\\over 2}) : \np^{1\\over 2},\\qquad\n S^+(z) =\\,\\, : V_{2,1}^{-}(zp^{1\\over 2}) V_{2,1}^{-}(zp^{-{1\\over 2}}) :,\n\\label{eq:TSbyV21}\n\\end{equation}\nand the relations obtained by $\\omega$.\nHere\n$V^\\pm_{\\ell+1,k+1}(z)\\equiv V^{\\pm1}_{\\ell+1,k+1}(z)$.\nMoreover, \nthe boson and power-sum correspondence operator in eq.\\ \\eq{eq:BosonMacOp}\nis also realized as $:\\prod_{i=1}^N V_{2,1}(q^{1\\over2} x_i^{-1}):$.\nHence, they must play more important role in the $q$-Virasoro algebra.\n\n\n\n\\subsection{ABF model and \\v}\n\n\nIn the papers \\cite{rLP2}, the \nexplicit formula for the multipoint \ncorrelation functions is successfully obtained. \nWe review their method and the relation to \\v.\n\nThe $q$-Virasoro algebra can be applied to the off-critical phenomena,\nespecially to the ABF model in the regime \n\\uppercase\\expandafter{\\romannumeral3} \\cite{rABF}.\nLet $z\\equiv p^v$ and the vertex operators $\\Phi_\\pm(z)$ be\n\\begin{equation}\n\\Phi_+(z)\\equiv V_{2,1}(z),\\qquad\n\\Phi_-(z') \\equiv \n \\oint{dz\\over 2\\pi iz} V_{2,1}(z') S_+(z) z^\\beta f(v-v',\\pi),\n\\label{e:ABFVertex}\n\\end{equation}\nwhere\n$[v] \\equiv p^{{1\\over2} ((1-\\beta)v^2-v) }\n(z;q)_\\infty (qz^{-1};q)_\\infty (q;q)_\\infty $\nand \n\\begin{equation}\nf(v,w)\\equiv\n{[v+{1\\over2}-w]\\over[v-{1\\over2}]},\\qquad\n\\pi\\equiv\n-{2h_0 \\over \\sqrt\\beta-\\sqrt{1\/\\beta}}.\n\\end{equation}\nThe integration contour is a closed curve around the origin satisfying \n$p|z'|<|z|0}|\\RR,\\SSS\\rangle = 0,\\qquad\nh_0 |\\RR,\\SSS\\rangle = \n-{1\\over2}\\left( \\RR\\sqrt\\beta - \\SSS\\sqrt{1\/\\beta} \\right)|\\RR,\\SSS\\rangle.\n\\end{equation}\nSuppose \n$\\beta = \\QQ\/\\PP$ with coprime integers $\\PP>\\QQ\\in{\\bf N}$ and\nlet the screening charge \n$\\SC_+\\,:\\,\\cF{\\SSS}{\\RR}\\rightarrow\\cF{\\SSS}{\\RR-2}$ be\n\\begin{equation}\n\\SC_+= \\oint{dz\\over 2\\pi iz} S_+(z) z^\\beta f(v,\\pi),\n\\end{equation}\nand define the BRST charges $Q^+_j$ ($j\\in{\\bf Z}$) as\n\\begin{eqnarray}\nQ^+_{2j } =\n\\SC_+^\\RR \\,\\,\\,\\,&:&\n\\cF{\\SSS}{ \\RR-2j\\PP}\\,\\,\\,\\rightarrow\\,\\,\\cF{\\SSS}{-\\RR-2 j \\PP},\\cr\nQ^+_{2j+1} =\n\\SC_+^{\\PP-\\RR}\\!\\!\\!\\!&:&\n\\cF{\\SSS}{-\\RR-2j\\PP} \\rightarrow\\,\\,\\cF{\\SSS}{ \\RR-2(j+1)\\PP}.\n\\label{eq:BRST}\n\\end{eqnarray}\nWe also define the dual screening charge\n$\\SC_-\\,:\\,\\cF{\\SSS}{\\RR}\\rightarrow\\cF{\\SSS-2}{\\RR}$ and\nthe dual BRST charges $Q^-_j$\nby the replacement \n$\\sqrt\\beta\\leftrightarrow -\\sqrt{1\/\\beta}$, \n$q\\leftrightarrow 1\/t$ and\n$\\RR\\leftrightarrow \\SSS$.\n\n\n\\proclaim Proposition 4. \\hspace{-2mm}\\cite{rLP2,rJLMP}\nThe screening charges $\\SC_\\pm$ commute with each other and \nwith $q$-Virasoro generators\n\\begin{eqnarray}\n[\\SC_+,\\SC_-] \n&\\!\\!\\!=\\!\\!\\!&\n0,\\cr\n[T(z), \\SC_\\pm^{n_\\pm}]\n&\\!\\!\\!=\\!\\!\\!&\n0, \\quad \n{\\rm on}\\,\\,\\,\\, \\cF{\\RR_-}{\\RR_+},\\quad\n{\\rm with}\\,\\,\\,\\, n_\\pm\\equiv \\RR_\\pm\\,\\, {\\rm mod} \\,\\,P_\\pm,\n\\end{eqnarray}\nand are also nilpotent\n\\begin{equation}\nQ^\\pm_j Q^\\pm_{j-1} \n=\n\\SC_\\pm^{P_\\pm} = 0, \\quad P_\\pm>1.\n\\label{eq:nilpotent}\n\\end{equation}\n\nHence we can construct Felder type BRST complexes, \nfor example, by $\\SC_+$\n\\begin{equation}\n\\cdots\n\\BRS{\\SC_+^{ \\RR}}\\cF{\\SSS}{-\\RR+2\\PP}\n\\BRS{\\SC_+^{\\PP-\\RR}}\\cF{\\SSS}{ \\RR}\n\\BRS{\\SC_+^{ \\RR}}\\cF{\\SSS}{-\\RR}\n\\BRS{\\SC_+^{\\PP-\\RR}}\\cF{\\SSS}{ \\RR-2\\PP}\n\\BRS{\\SC_+^{ \\RR}}\n\\cdots.\n\\label{eq:Felder}\n\\end{equation}\n{}From the Kac determinant in (\\ref{Kacconj}),\nthe Fock module $\\cF{\\SSS}{\\RR}$ with $\\RR$, $\\SSS\\in{\\bf N}$ is reducible. \nTo obtain an irreducible one $\\cL{\\SSS}{\\RR}$, \nwe have to factor out the submodules by the Felder resolution.\nIn a special case, this irreducible module coincides with \nthe space of the ABF model.\n\n\nTo see this, we have to introduce a grading operator,\nwhich plays the role of the corner Hamiltonian in the ABF model,\n\\begin{equation}\nH_c = \\sum_{n>0} n^2 \n {p^{n\\over2}+p^{-{n\\over2}}\\over\n\\left( q^{n\\over2}-q^{-{n\\over2}}\\right) \n\\left( t^{n\\over2}-t^{-{n\\over2}}\\right) }\nh_{-n} h_n + h_0^2 -{1\\over 24}.\n\\label{eq:cornerHamil}\n\\end{equation}\nThis commutes with screening currents up to a total divergence\n\\begin{equation}\n[H_c,S_\\pm(z)] z^{\\beta^{\\pm1}} \n= {\\partial\\over\\partial z} \\left( S_\\pm(z) z^{\\beta^{\\pm1}}\\right),\n\\end{equation}\nand its eigenvalues $\\EPS{\\SSS}{\\RR}$ \non the Fock module $\\cF{\\SSS}{\\RR}$ are\n\\begin{equation}\n\\EPS{\\SSS}{\\RR}=\\DEL{\\SSS}{\\RR} - {c\\over 24} + n,\\qquad \nn\\in{\\bf Z}_{n\\geq0}\n\\end{equation}\nwith $\\DEL{\\SSS}{\\RR}$ in eq.\\ \\eq{eq:KacFormula}.\nWhen $\\QQ=\\PP-1$,\nthese values coincide with the eigenvalues of \nthe corner Hamiltonian of the ABF model\ncorresponding to the $1$-$d$ configurations given by the rule:\n{\\it i}) each height takes an integer value between \n$1$ and $\\PP-1$, {\\it ii}) the allowed values of difference\nin any neighboring heights are $\\pm1$, {\\it iii})\nthe height at the origin is $\\RR$,\n {\\it iv}) the asymptotic configuration is \n$\\cdots,\\SSS,\\SSS+1,\\SSS,\\SSS+1,\\cdots$.\nHowever, we should note that the \nmultiplicities of the bosonic Fock space and \nthat of ABF model are different.\n\nLukyanov and Pugai \\cite{rLP1,rLP2} showed that,\nafter the Felder-type BRST resolution by the dual screening current $S_-(z)$,\nthe multiplicities in the irreducible Fock module $\\cL{\\SSS}{\\RR}$ \ncoincide those of the ABF model.\nTherefore, ABF model is completely described by \nthe representation of the $q$-Virasoro algebra\nand the multi-point local height probabilities of ABF model \\cite{rFJMMN} are\nrealized as correlation functions of the vertex operators.\nFor example, the probability that \nthe heights at the same vertical column sites have the values \n$1\\leq\\RR_1,\\RR_2, \\cdots,\\RR_n\\leq\\PP-1$\nis proportional to\n\\begin{equation}\nTr_{\\cL{\\SSS}{\\RR_1}}\n\\left[ p^{2H_c}\n\\Phi_{-\\sigma_1 }(z_1 \/p)\\cdots\n\\Phi_{-\\sigma_{n-1}}(z_{n-1}\/p)\n\\Phi_{ \\sigma_{n-1}}(z_{n-1} )\\cdots\n\\Phi_{ \\sigma_1 }(z_1 )\n\\right],\n\\end{equation}\nwhere \n$\\sigma_s = \\RR_{s+1} - \\RR_s$.\n\n\n\n\n\\subsection{elliptic algebra generated by the screening currents and \n$k=1$ affine Lie algebra}\n\n\nThe properties of screening currents are quite important \nin the representation theory of the infinite-dimensional algebra;\nthey govern the irreducibility and the physical states\nas mentioned above subsections.\nMoreover they relate with hidden quantum symmetries.\n\n\n\\subsubsection{an elliptic algebra generated by $S^\\pm(z)$}\n\n\nHere, we show that the\nscreening currents generate an elliptic hidden symmetry,\nwhich reduces to the (quantum) affine Lie algebra with a special center\nwhen $c$ tends to $1$.\nLet us introduce a new current \n$\\Psi(z) \\equiv \\,:\\!S_+(q^{\\pm{1\\over2}}z)\\, S_-(t^{\\pm{1\\over2}}z)\\!:$, \n{\\it i.e.},\n\\begin{equation}\n\\Psi(z)\n= \\exp\\left\\{ \\sum_{n\\neq0} \n{ p^n-p^{-n} \\over (\\QINT qn)(\\QINT tn) } h_n z^{-n} \\right\\}\ne^{2\\alpha Q} z^{2\\alpha h_0},\n\\end{equation}\nwith $\\alpha = \\sqrt\\beta - 1\/\\sqrt\\beta$, \nthen we have\n\n\\proclaim Proposition 5.\nScreening Currents $S_\\pm(z)$ and $\\Psi(z)$ generate \nthe following elliptic two-parameter algebra;\n\\begin{eqnarray}\nf_{00}\\left({w\\over z}\\right) \\Psi(z) \\Psi(w) \n&\\!\\!\\!=\\!\\!\\!&\n\\Psi(w) \\Psi(z) f_{00}\\left({z\\over w}\\right),\n\\label{eq:ElliPP}\\\\\nf_{0\\pm}\\left({w\\over z}\\right) \\Psi(z) S_\\pm(w) \n&\\!\\!\\!=\\!\\!\\!&\nS_\\pm(w) \\Psi(z) f_{\\pm0}\\left({z\\over w}\\right),\n\\label{eq:ElliPS}\\\\\nf_{\\pm\\pm}\\left({w\\over z}\\right) S_\\pm(z) S_\\pm(w) \n&\\!\\!\\!=\\!\\!\\!&\nS_\\pm(w) S_\\pm(z) f_{\\pm\\pm}\\left({z\\over w}\\right),\n\\label{eq:ElliSS1}\n\\end{eqnarray}\n\\begin{equation}\n\\left[\\,S_+(z), S_-(w)\\,\\right]\n=\n{1\\over (p-1)w}\n\\left[\\, \n\\delta\\left(p^{ 1\\over2 }{w\\over z}\\right) \\Psi(t^{-{1\\over2}}w) -\n\\delta\\left(p^{-{1\\over2}}{w\\over z}\\right) \\Psi(q^{-{1\\over2}}w)\n\\,\\right],\n\\label{eq:ElliSS2}\n\\end{equation}\nwhere\n$f_{00}(x) =\nf_{++}(x) f_{+-}^2(xp^{1\\over2}) f_{--}(x)$ and \n$f_{0\\pm}(x) =\nf_{\\pm0}(x) =\nf_{+\\pm}(xq^{1\\over2}) f_{-\\pm}(xt^{1\\over2})$ \nwith\n\\begin{eqnarray}\nf_{+-}(x) \n=\nf_{-+}(x) \n&\\!\\!\\!=\\!\\!\\!&\n\\exp\\left\\{ -\\sum_{n>0}{1\\over n}(\\PINT pn)x^n \\right\\} x^{-1},\\\\\nf_{++}(x) \n&\\!\\!\\!=\\!\\!\\!&\n\\exp\n\\left\\{ -\\sum_{n>0}{1\\over n}{\\QINT tn \\over \\QINT qn} (\\PINT pn)x^n \\right\\} \nx^\\beta,\n\\end{eqnarray}\nand $f_{--}(x) = \\omega\\cdot f_{++}(x)$.\\\\\n\\indent\nThe relation between $\\Psi(z)$ and \nthe $q$-Virasoro generators $\\Lambda^\\pm(z)$ \nis simply written as \n\\begin{equation}\n[\\,\\Lambda^\\pm(z),\\Psi(w)\\,] =\n\\mp p^{\\mp{1\\over 2}} (\\QINT p1)\n\\delta\\left({w\\over z}\\right)\\,:\\!\\Lambda^\\pm(w)\\Psi(w)\\!:.\n\\end{equation}\n\n\n\nAs we shall see explicitly in the next subsection,\nin the limit of\n$q$ and $t$ tend to $0$ with $p$ and $t^{-{|n|\\over2}} h_n$ fixed,\nthe relations \\eq{eq:ElliPP}--\\eq{eq:ElliSS2}\nreduce to those of $k=1$ $U_q(\\widehat{sl}_2)$.\nTherefore, the algebra generated by \n$S^\\pm(z)$ and $\\Psi(z)$ can be regarded \nas an elliptic generalization of $U_q(\\widehat{sl}_2)$ with level-one.\nWe can regard $p$, $q$ and $t$ \nas three independent parameters. \nEven in this case, \nscreening currents \\eq{eq:SC21} and \\eq{eq:SC22} and new currents \n$\\Psi_\\pm(z) \\equiv \n\\,:\\!S_+(p^{\\pm{1\\over4}}z)\\, S_-(p^{\\mp{1\\over4}}z)\\!:$ \ngenerate an elliptic algebra. \nThese extended algebras may help us to investigate \nelliptic-type integrable models.\n\n\nIn the sense of analytic continuation,\nthese relations are also rewritten by \nusing elliptic theta functions\\cite{rFFr},\n\\begin{equation}\nS_\\pm(z) S_\\pm(w) \n=\nU_\\pm\\left({w\\over z}\\right) \nS_\\pm(w) S_\\pm(z)\n\\label{eq:ElliSS12}\n\\end{equation}\nwith\n\\begin{equation}%\nU_\\pm(x) =\n-x^{1-2\\beta}\n\\exp\\left\\{ \\sum_{n\\neq0}{1\\over n} \n{ q^{n\\over2} t^{-n} -q^{-{n\\over2}} t^n \\over q^{n\\over2}-q^{-{n\\over2}} }\nx^n \\right\\}\n=\n-x^{2(1-\\beta)} {\\vartheta_1(px;q)\\over\\vartheta_1(px^{-1};q)}, \n\\end{equation}\nand $U_-(x) = \\omega\\cdot U_+(x)$.\nNote that $U_\\pm(x)$ are quasi-periodic functions, namely for\n$U_+(x)$, we have\n\\begin{equation}\nU_+(qx) = U_+(x), \\qquad\nU_+(e^{2\\pi i}x) = e^{-4\\pi i\\beta} U_+(x).\n\\end{equation} \n\nIt should be noted that \nthe screening currents of $q$-${\\cal W}$ \nalgebra \\cite{rFFr,rAKOS} and\n$U_q(\\widehat{sl}_N)$ \\cite{rAOS} also obey similar elliptic relations.\n\n\n\n\\subsubsection{two $c=1$ limits}\n\n\nIt is possible to consider two different $c=1$ limits of \\v.\nThey are related with the (quantum) affine algebra of\n$A_1^{(1)}$-type.\n\\\\\n\n\\noindent(A) Let us consider the limit,\n$q \\to 0 $, $t \\to 0$, $p$ is fixed. \nIn this limit we must have $\\beta \\to 1$.\nthus, this is a ``$c=1$'' limit (see (\\ref{e:a9})).\nWe will see that the screening currents $S_\\pm(z)$ \nreduces to \nthe Frenkel-Jing realization of $U_{q}(\\widehat{ sl}_2)$ at \nlevel-one \\cite{rFJ}.\n\nIntroducing rescaled bosons as \n\\begin{eqnarray}\na_n =-h_n q^{|n|\/2}p^{-|n|\/4} [2n]\\qquad (n\\neq 0),\\qquad \na_0=-2 h_0,\\qquad\nQ = -2 Q_h,\n\\end{eqnarray}\nwe obtain\n\\begin{eqnarray}\n&&[a_n , a_m]= \\frac{[2n][n]}{n} \\delta_{n+m,0} , \\quad \n[a_n , Q]= 2\\delta_{n,0}, \\\\\n&&{S}_{-}(z) \\rightarrow\\exp\\left\\{\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_{-n} z^n p^{-\\frac{n}{4}}\\right\\}\n\\exp\\left\\{-\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_n z^{-n} p^{-\\frac{n}{4}}\\right\\}\ne^{Q}z^{a_0}, \n\\cr\n&&{S}_{+}(z)\\rightarrow \\exp\\left\\{-\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_{-n} z^n p^{\\frac{n}{4}}\\right\\}\n\\exp\\left\\{\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_n z^{-n} p^{\\frac{n}{4}}\\right\\}\ne^{-Q}z^{-a_0},\n\\label{eq:FJ}\n\\end{eqnarray}\nwhere $[n]=({p^{\\frac{n}{2}}-p^{-\\frac{n}{2}}})\/\n({p^{\\frac{1}{2}}-p^{-\\frac{1}{2}}})$. After \nreplacing $p^{\\frac{1}{2}}\\to q$,\nwe can identify screening currents ${S}_{\\pm}$ \nwith the Frenkel-Jing realization of the \nDrinfeld currents of $U_{q}(\\widehat{ sl}_2)$.\n\nIt is quite unfortunate that at this limit,\nthe \\v current becomes singular. Namely,\nit seems difficult to extract nontrivial\nobject from $T(z)$ at this limit. Thus it is still a challenging\nproblem to find a Sugawara construction for $U_{q}(\\widehat{ sl}_2)$.\nThe limit discussed in the next paragraph (B)\nmay be relevant to this problem.\n\\\\\n\n\n\\noindent(B)\nNext, we consider the limit of $\\beta \\to 1$ with $q$ fixed.\nThis is another ``$c=1$'' limit.\nIn this limit, the screening currents degenerate to \nthe Frenkel-Kac realization of the level-one $\\widehat{sl}_2$ \\cite{rFK}.\n\nIntroducing rescaled bosons as \n\\begin{eqnarray}\n a_n = \\frac{ 2 n}{q^{\\frac{n}{2}}-q^{-\\frac{n}{2}}} h_n \\quad\n (n \\neq 0), \\quad \n a_0 = 2h_0, \\quad\n Q = 2 Q_h,\n\\end{eqnarray}\nwe obtain\n\\begin{eqnarray}\n&&[a_n , a_m]= 2 n \\delta_{n+m,0} , \\quad \n[a_n , Q]= 2\\delta_{n,0}, \\\\\n&&{S}_{\\pm}(z) \\rightarrow\\exp\\left\\{ \\pm \\sum_{n\\neq 0}\n\\frac{a_{n}}{n} z^{-n} \\right\\}\ne^{\\pm Q}z^{\\pm a_0}.\n\\end{eqnarray}\n{}From these, it can be seen that the screening currents have\nreduced to the Frenkel-Kac realization of $\\widehat{sl}_2$.\n\nIn this limit, \\v survives and satisfies the relation\n\\begin{eqnarray}\n&&\\frac{\\sqrt{(1-q{w \\over z})(1-q^{-1}{w \\over z})}}{1-{w \\over z}}T(z)T(w)-\nT(w)T(z)\\frac{\\sqrt{(1-q{z \\over w})(1-q^{-1}{z \\over w})}}\n{1-{z \\over w}}\\\\\n&=&2(1-q)(1-q^{-1})\n{w \\over z} \\delta'\\left({w \\over z} \\right).\\nonumber\n\\end{eqnarray}\n\nIt is shown that all the vectors in the Fock space\nspanned over the vacuum $|0\\rangle$ are singular vectors of \nthis algebra\nby studying the Kac determinant at level-one.\nThis fact and the normalization of the \nbosonic oscillators in this limit mean that the \nthe Fock space is \nspanned by the Schur symmetric polynomials.\nThis degeneration is not accidental because the Macdonald \npolynomial $P_\\lambda(x;q,t)$ reduces to the \nSchur polynomial in the limit $t\\rightarrow q$.\n\nIn this ``$c=1$'' limit, \\v survives and acts on the Fock space\non which the bosonized currents of $U_q(\\widehat{sl}_2)$ can\nalso act. The relationship between these algebras\nhas not been made clear yet.\n\\\\\n\n\n\n\n\\subsection{$c=0$ topological $q$-superconformal model}\n\n\nThe $c=0$ Virasoro algebra\ndescribes a topological conformal model. \nWe shall call one of the screening currents of this model $G^+(z)$,\nand set $G^-(z)=:(G^+(z))^{-1}:$.\nThe screening charge $\\oint dz G^+(z)$ plays the role of the BRST\noperator.\nSince the model is topological, \nthe energy-momentum tensor $L_{c=0}(z)$ should be BRST \nexact. Actually, we have \n$\\left\\{ \\oint dz' G^+(z') , G^-(z) \\right\\} = 2 L_{c=0}(z)$.\n\nThere exists a similar structure in the $q$-Virasoro case\\footnote{\nThis was inspired by the discussion with T.~Kawai}.\nLet us consider the case when $c=0$, {\\it i.e.}, $\\beta=3\/2$ ($q=p^{-2}$).\nDenote one of the screening current $S_+(z)$ and \nthe normal ordering of its inverse\nas $G^+(z)$ and $G^-(z)$, respectively\n\\begin{equation}\nG^\\pm(z) =\n\\,:\\exp\\left\\{\n\\pm\\sum_{n\\neq0}{h_n \\over p^{n\\over 2}-p^{-{n\\over 2}}} z^{-n} \n \\right\\}:\\, \ne^{\\pm2\\sqrt\\beta Q}\nz^{\\pm2\\sqrt\\beta h_0}.\n\\end{equation}\n\\proclaim Proposition 6. The fields $T(z)$ and $G^\\pm(z)$ satisfy \nthe relations\n\\begin{eqnarray}\nf\\left( {w\\over z}\\right) T(z) T(w) - T(w) T(z) f\\left( {z\\over w}\\right) \n&\\!\\!\\! = \\!\\!\\!& \n-(p^{1\\over 2}+p^{-{1\\over 2}})(p^{3\\over 2}-p^{-{3\\over 2}}) \n\\left( \\delta\\left( {wp\\over z}\\right) \n- \\delta\\left( {w\\over zp}\\right) \\right), \\nonumber \\\\\nf\\left( {w\\over z}\\right) T(z) G^-(w) - G^-(w) T(z) f\\left( {z\\over w}\\right) \n&\\!\\!\\! = \\!\\!\\!& G^-(z) (p^{3\\over 2}-p^{-{3\\over 2}}) \n\\left( p^{2}\\delta\\left( {wp\\over z}\\right) \n-p^{-2}\\delta\\left( {w\\over zp}\\right)\\right), \\nonumber \\\\\n\\left\\{\\oint dz G^+(z), G^-(w) \\right\\} &\\!\\!\\! = \\!\\!\\!& \n{p^{-1}\\over w^2 (p-p^{-1})(p^{1\\over 2}-p^{-{1\\over 2}})}\n\\left( T(w) - (p^{1\\over 2}+p^{-{1\\over 2}}) \\right), \\nonumber\\\\\n\\left[ T(z), \\oint dw G^+(w) \\right] \n&\\!\\!\\! = \\!\\!\\!& 0 , \\qquad\\quad\n\\left\\{ G^\\pm(z),G^\\pm(w)\\right\\} = 0.\n\\label{eq:qTopological}\n\\end{eqnarray}\nwhere $f(x)$ is given by (\\ref{structure}) with $q=p^{-2}$.\n\n\\noindent\nNote that $G^-(z)$ is a primary field and\nits commutation relation with $T(z)$ is given by the same function $f(x)$\nin the defining relation of the $q$-Virasoro algebra.\n\nWe can regard these relations as a $c=0$ topological $q$-Virasoro algebra.\nThe screening charge $\\oint dz G_+(z)$ may play the role of BRST operator\nwhich reduces the bosonic Fock space to \nirreducible representation space of the $q$-Virasoro algebra.\nAt the value of the coupling constant $\\beta=3\/2$,\nthe central charge vanishes and \nthe entire Fock space contains only BRST trivial states,\nexcept for the vacuum state. \nThe $q$-Virasoro generator itself (up to a constant)\nis given by a BRST exact form.\nThus the $\\beta=3\/2$ $q$-Virasoro algebra is a topological field theory\nsame as $q=1$ case.\n\n\nThe relations between the currents $G^\\pm(z)$ and $\\Lambda^\\pm(z)$ are\nwritten as\n\\begin{eqnarray*}\nf\\left( {w\\over z}\\right) \\Lambda^\\pm(zp^{\\pm1}) G^+(w) \n- G^+(w) \\Lambda^\\pm(zp^{\\pm1}) f\\left( {z\\over w}\\right) \n&\\!\\!\\!=\\!\\!\\!& \\mp p^{\\mp1}(p^{3\\over 2}-p^{-{3\\over 2}})\n\\delta\\left( {w\\over z}p^{\\mp1}\\right) G^+(z) ,\\\\\nf\\left( {w\\over z}\\right) \\Lambda^\\pm(z) G^-(w) \n- G^-(w) \\Lambda^\\pm(z) f\\left( {z\\over w}\\right) \n&\\!\\!\\!=\\!\\!\\!& \\pm p^{\\pm2}(p^{3\\over 2}-p^{-{3\\over 2}})\n\\delta\\left( {w\\over z}p^{\\pm1}\\right) G^-(z) ,\n\\end{eqnarray*}\n\\vspace{-7mm}\n\\begin{eqnarray}\n\\left\\{G^+(z), G^-(w) \\right\\} \n&=&{p^{-3}\\over zw^2 (p-p^{-1})(p^{1\\over 2}-p^{-{1\\over 2}})} \\\\\n&\\times&\n\\left( \\Lambda^+\\left( {w}\\right)\\delta\\left( {w\\over zp}\\right) \n+ \\Lambda^-\\left( {w}\\right)\\delta\\left( {wp\\over z}\\right)\n-\\left( p^{1\\over 2}+p^{-{1\\over 2}}\\right)\n\\delta\\left( {w\\over z}\\right) \\right). \\nonumber\n\\end{eqnarray}\nTheir Fourier modes given by\n$G^\\pm(z) = \\sum_n G^\\pm_n z^{-n}$ and \n$\\Lambda^\\pm(z) = \\sum_n \\Lambda^\\pm_n z^{-n}$\nsatisfy \n$$\n\\left\\{G^+_{n+1}, G^-_{m+1} \\right\\} =\n{p^{-3}\\over (p-p^{-1})(p^{1\\over 2}-p^{-{1\\over 2}})}\n\\left( \\Lambda^+_{n+m-1} p^{-n} + \\Lambda^-_{n+m-1} p^n \n-\\left( p^{1\\over 2}+p^{-{1\\over 2}}\\right)\\delta_{n+m-1,0} \\right).\n$$\nNote that, \nthe relation between $G^+$ and $G^-$ can be \nexpressed in other ways.\nFor example, use the fact that \n$ G^+(z) G^-(w) {z\/w} + G^-(w) G^+(z) {w\/z} $ and\n$\\left\\{ G^+(z), G^-(w) \\right\\} z^{-r}{z^2 w^2\/(z+w)}$\nare also written by $\\Lambda^\\pm$ for any $r\\in{\\bf C}$.\n\nIn the $q=1$ case, $c=0$ topological algebra \ncan be constructed from \n$N=2$ superconformal algebra by the \noperation so-called ``twisting''\\cite{rEY}.\nWhat we have obtained here is a deformation of this twisted \nsuperconformal algebra. So far, \nwe have not been able to find a mechanism\nof ``untwisting'' in the deformed case.\nWe expect that our topological $q$-Virasoro algebra \nhelps us to find a \nsupersymmetric generalization of the $q$-Virasoro algebra \\v\nand a deformed twisting operation.\n\n\n\\subsection{$c=-2$ \\v and $c=1$ ${\\cal W}_{1+\\infty}$ algebra}\n\n\nSince $c=0$ Virasoro algebra is realized by the differential operator \n$L_n = - z^n D$ with $D = z \\partial_z$,\none can expect that $q$-Virasoro algebra \\v has a similar \nrepresentation by the difference \nor shift operator as\n$T_n \\sim z^n q^D$.\nHowever, this shift operator is nothing but the generating function of \nthe $c=0$ ${\\cal W}_{1+\\infty}$ generators.\nThus we expect some relations between the $q$-Virasoro and the ${\\cal W}_{1+\\infty}$ algebra.\nIndeed this is the case when $\\beta=2$, \nwe show a relation with $c=1$ ${\\cal W}_{1+\\infty}$ algebra.\n\nFirst, recall that the $q$-Virasoro generator is expressed by \nthe fundamental vertex operator and its dual $V_{2,1}^\\pm(z)$ \nas eq.\\ \\eq{eq:TSbyV21}.\nWhen $\\beta=2$, {\\it i.e.}, $c=-2$, ($q=1\/p$), \nthey reduce to the fermions such that\n\\begin{equation}\n\\{V_{2,1}^+ (z),V_{2,1}^- (w)\\}\n={1\\over z}\\delta\\left( {w\\over z}\\right),\\qquad\n\\{V_{2,1}^\\pm(z),V_{2,1}^\\pm(w)\\}=0.\n\\end{equation}\nOn the other hand, the generating function of $c=1$ ${\\cal W}_{1+\\infty}$ algebra\nis known to be also realized by a complex fermion.\nTherefore, we have found;\n\n\\proclaim Proposition 7.\nThe $\\beta=2$ ($c=-2$) $q$-Virasoro algebra $T(z) = \\Lambda^+(z) + \\Lambda^-(z)$ \ngenerates the $c=1$ ${\\cal W}_{1+\\infty}$ algebra\nand it is realized by fermions $V^\\pm_{2,1}(z)$ as follows;\n\\begin{equation}\n\\Lambda^\\pm(zq^{{1\\over 2}\\mp1}) =\n\\,:V_{2,1}^-(z) q^{\\mp D} V_{2,1}^+(z):\\, q^{\\mp{1\\over 2}}.\n\\label{eq:TbyFermion}\n\\end{equation}\n\n\nNext we show the relation between $q$-Virasoro and ${\\cal W}_{1+\\infty}$ algebras\nmore explicitly comparing their commutation relations.\nLet\n\\begin{equation}\nX^k(z) \\equiv\n\\,:\\exp\\left\\{\\sum_{n\\neq0}{1-q^{kn}\\over 1-q^n} h_n z^{-n} \\right\\}:\\, \nq^{k\\sqrt 2 h_0}\n\\equiv\n\\sum_{n\\in{\\bf Z}} X^k_n z^{-n}.\n\\end{equation}\nNote that the $q$-Virasoro generator is now $T(z)=X^1(z) + X^{-1}(z)$.\nThen\n\\begin{eqnarray}\\nonumber\n\\left[\\, X^k(z),X^\\ell(w) \\,\\right] \\&\n= {(q^k-1)(q^\\ell-1)\\over q^{k+\\ell}-1}\n\\left( X^{k+\\ell}(z) \\delta\\left( q^k{w\\over z}\\right) \n- X^{k+\\ell}(w) \\delta\\left( q^\\ell{z\\over w}\\right)\\right), \n\\cr\n\\left[\\, X^k(z),X^{-k}(w) \\,\\right] \\&\n= (q^{k\\over 2}-q^{-{k\\over 2}})^2 \\left( \n\\left( 1+\\sum_{n\\neq0}{1-q^{kn}\\over 1-q^n} \nn h_n z^{-n}\\right) \\delta\\left( q^k{w\\over z}\\right) \n-\\delta'\\left( q^k{w\\over z}\\right) \n\\right),\n\\end{eqnarray}\nfor $k,\\ell,n,m\\in{\\bf Z}$.\nTheir modes\n\\begin{equation}\nW^k_n = {X^k_n\\over q^k-1} - {1\\over 1-q^{-k}} \\delta_{n,0},\\qquad\nW^0_n = {n\\over q^n-1} h_n,\n\\end{equation}\nfor $k\\neq0$, satisfy\n\\begin{eqnarray}\n\\left[\\, W^k_n,W^\\ell_m \\,\\right] \\&\n= (q^{-km}-q^{-\\ell n}) W^{k+\\ell}_{n+m} + \n {q^{-km}-q^{-\\ell n}\\over 1-q^{-k-\\ell}} \\delta_{n+m,0},\n\\cr\n\\left[\\, W^k_n,W^{-k}_m \\,\\right] \\&\n= (q^{-km}-q^{kn}) W^0_{n+m} + n q^{kn}\\delta_{n+m,0}.\n\\label{eq:Winfty}\n\\end{eqnarray}\nThis is nothing but the algebra of \nthe generating functions for the $c=1$ ${\\cal W}_{1+\\infty}$ generators\n$W^k_n = W\\left( z^nq^{-kD}\\right)$\nin the notation of \\cite{rAFMO}.\n\nThe $c=0$ ${\\cal W}_{1+\\infty}$ algebra has a meaning of an area-preserving diffeomorphism\nand relates with classical membrane.\nWe expect this relation \nbetween the $q$-Virasoro algebra with the ${\\cal W}_{1+\\infty}$ algebra \nis a key for\na geometrical interpretation of \\v and\nthe quantization of the membrane.\n\n\n\n\n\\section{Summary and further issues}\n\n\nOur presentation has aimed to show that\nthe new Virasoro-type elliptic algebra \\v defined by eq.\\ \\eq{e:a1.2}\ncan be regarded as a universal symmetry of the massive integrable models.\n\n\nThe algebra \\v has two parameters $p$ and $q$ ($qp^{-1}=q^\\beta$)\nand it can be regarded as a generating function for \nseveral different Virasoro-type symmetry algebras\nappearing in solvable models.\nAt the limit of $q\\rightarrow 1$, the algebra \\v \nreduces to the ordinary Virasoro algebra \nwith the central charge $c$ related to $\\beta$ \\eq{e:a8}.\nWhen we consider the limit of $q\\rightarrow p$, \nit reduces to the \n$q$-Virasoro algebra of Frenkel-Reshetikhin \\eq{qvirFR}.\nWhen $q\\rightarrow 0$, \nwe obtain Jing's generating operators for\nthe Hall-Littlewood polynomials \\eq{eq:BosonHall}.\nThe topological algebra \\eq{eq:qTopological} and \nthe $c=1$ ${\\cal W}_{1+\\infty}$ algebra \\eq{eq:Winfty} are constructed from \nthe special cases $p=q^{-1\/2}$ and $p=1\/q$ respectively.\n\nOne of the peculiar features of the algebra \\v is its non-linearity.\nSince it is a quadratic algebra like the Yang-Baxter relation \nfor the transfer matrix,\nthe associativity is quite non-trivial \\eq{YB-for-T},\nand the Yang-Baxter equation \ndetermines the structure function uniquely, {\\it i.e.},\nfixes the algebra itself!\nMoreover it turns out to be the Zamolodchikov-Faddeev algebra\nof the particle-creation operators for the XYZ and \nthe sine-Gordon models \\eq{eq:ZFXYZ}.\n\nThe next essential nature is its infinite-dimensionality, \nwhich connects with the integrability of massive models.\nThe representation space of the algebra \\v possesses rich structure \nenough to describe the physical space of massive models.\nDespite its non-linearity, \nthe Kac determinant is very similar to that of the Virasoro case \n\\eq{Kacconj}. \n\n\nThe algebra \\v is realized by the free fields \n\\eq{eq:qVirFFR1} and \\eq{eq:qVirFFR2}\nin a quite simple way.\nIt shows us that the integrability of the model due to \\v symmetry\ncan be investigated in a natural manner in terms of the \nbosonic field.\nFurthermore this free field realization is described by \na deformed Miura transformation \\eq{eq:qMiura},\nand the deformed Miura transformation \nbrings about an interesting analogy with the dressed vacuum form.\nA generalization of this transformation gives\nthe $q$-deformed $\\cal W$ algebra \\cite{rFFr,rAKOS}.\n\nTo study infinite dimensional algebras, {\\it e.g.},\nnot only Virasoro and Kac-Moody algebras but also \\v,\none of the most essential oubjects in the representation theory is \nthe screening current. \nUsing the screening charge, one can\ndefine the physical states by the BRST method \\eq{eq:BRST},\nwrite down the null states \\eq{eq:qVirSing},\nand study the nontrivial monodromic property of the \nscreened vertex operators. \nThe null states \nrelate with the wave functions of the \nRuijsenaars model \\eq{eq:MacOp} which is \na relativistic generalization of the Calogero-Sutherland model. \nThe Hamiltonian of this model\nis realized by the positive modes of the \\v generators \\eq{e:b7.3}\nand causes the correspondence between the null states and excited states.\nThe monodromy matrix is connected to the ABF model.\nThe exchange relation of the vertex operators\npossesses a quantum group structure\ncharacterized by the solution of the face type \nYang-Baxter equation\\eq{eq:ExchangeABF}\nand leads to the identification of the deformed Virasoro vertex operators\nwith that of the ABF model.\nMoreover the screening currents show us an hidden symmetry;\nthat is an elliptic generalizaton of \n$k=1$ $U_q(\\widehat{sl}_2)$ algebra (Prop. 5).\n\\\\\n\n\nFinally, we mention some comments on further issues \ncoming from mathematical or physical points of view.\n\nMathematically, there are many things to be clarified.\nTo obtain a fusion of the ABF model or to find a suitable primary fields,\none needs a tensor product representation or \nsuitable adjoint action of the algebra \\v.\nIn other words, a co-product structure must be discovered. \nSince the algebra, however, is quadratic, it seems a highly non-trivial task.\nStudying the limit $q\\rightarrow 0$ may help us to reveal it.\n\nDispense with a help of free fields,\nthe correlation function must be determined\nby a difference equation coming from the Ward-Takahashi identity.\nTo find this equation, \nwe need to recognize a geometrical meaning of the algebra \\v;\nhow is a difference operator realized in \\v?\nRestricting the $q$-${\\cal W}_{1+\\infty}$ algebra \\cite{rKLR},\nwhich is defined as an algebra of higher pseud-difference operators,\nto the first order difference one,\nwe might obtain the algebra \\v and its difference operator realization.\nSeeking for a realization on the infinite $q$-wedge \\cite{rKMS}\nmight be able to connect both problems mentioned above; \na co-product and a geometrical interpretation.\n\nThe relation with the quantum affine Lie algebras also \nseems to be of some interest.\nFrenkel-Reshetikhin's $q$-Virasoro algebra was constructed \nby a $q$-Sugawara method \\cite{rFR} from \nthe $U_q(\\widehat{sl}_n)$ at the critical level.\nAre there any such constructions for the algebra \\v from \n$U_q(\\widehat{sl}_n)$ or, more hopefully, from \ntheir elliptic generalization ${\\cal A}_{q,p}$?\n\n\nPhysically, we anticipate many applications.\nThere are two approaches to investigate the 2-dimensional integrable models, \none is an Abelian method based on the algebraic Bethe-ansatz\nand the other is a non-Abelian one based on\nthe Virasoro algebra and its generalizations.\nThe latter approach describes the model more in detail,\nhowever its applicability is restricted \nonly to critical phenomena and some off-critical \ntrigonometric-type models.\nThe deformed Virasoro algebra \\v \nshould be a synthesis of the massive integrable models\nincluding elliptic-type models.\n\nIn the dual resonance model, which was a precursor of string theory,\nthe Veneziano amplitude has its generalizations\nto non-linearly rising Regge trajectories (see for example, \\cite{rFN}).\nHowever the absence of their operator representations\nhas disturbed their development, including to prove a no-ghost theorem. \nIn a special case, the amplitude reduces to $q$-beta function, \nwhich is very similar to a four-point function of our algebra \\v.\nWe hope that \\v gives \nan operator representation of a generalized Veneziano amplitude \nand opens further avenues to the exploration of new string theories.\n\nThe $c=0$ ${\\cal W}_{1+\\infty}$ algebra, an area-preserving diffeomorphism,\nis a symmetry of the classical membrane.\nThe relation between the $c=-2$ \\v algebra and $c=1$ ${\\cal W}_{1+\\infty}$ algebra \nmay be a key for\na geometrical interpretation of the algebra \\v and\nthe quantization of the membrane \nas the basic object of the $11(12)$-dimensional M(F) theory.\nOn the other hand, \nquantum membrane can be represented by a large $N$ matrix model \\cite{rDHN}, \nand the partition functions of more general conformal matrix models\nare described by eigenstates of the Calogero-Sutherland models \\cite{rAMOS}.\nDo the algebra \\v or eigenstates of the Ruijsenaars model \nrelate with a relativistic generalization of the quantum membrane? \n\nWhat is a field theoretical interpretation of the algebra \\v?\nThe low energy 4-dimensional $N=2$ super YM theories \nare described by some integrable models, \nperiodic Toda chain or elliptic CS model \\cite{rGKMMM}.\nIf we generalize them to 5D's one by Kaluza-Klein method\nwith an extra dimension compactified to a circle,\nthen they are described by the relativistic generalization of 4D's one\n\\cite{rN}.\nIt may suggest that the Kaluza-Klein with a radius $R$ can lead to \na relativistic generalization with the speed of light $1\/R$\nor a $q$-deformation with $q=e^R$ of a original theory.\nIs the algebra \\v understood as a Kaluza-Klein from CFT?\n\n\n\n\\vskip 5mm\n\n\\noindent{\\bf Acknowledgments:}\n\n\nWe would like to thank \nT.~Eguchi, D.~Fairlie, B.~Feigin, E.~Frenkel,\nJ.~Harvey, T.~Hayashi, T.~Inami, M.~Jimbo, \nS.~Kato, T.~Kawai, B.~Khesin, A.~Kuniba, \nE.~Martinec, Y.~Matsuo, T.~Miwa, K.~Nagatomo, N.~Nekrasov, \nP.~Wiegmann and Y.~Yamada\nfor discussions and encouragements.\nThis work is supported in part by Grant-in-Aid for Scientific\nResearch from Ministry of Science and Culture.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzoixp b/data_all_eng_slimpj/shuffled/split2/finalzzoixp new file mode 100644 index 0000000000000000000000000000000000000000..726aabb762128cdbbb641cb1518f63b99128a377 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzoixp @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nModels with multiple Higgs bosons provide one of the simplest\npossibilities for physics beyond the standard model. Indeed, two\nHiggs doublet models in particular have received a great deal of\nattention, and have arisen in a wide variety of contexts, including\nsupersymmetry or extra dimensions, as well as axion models. More\ngenerally, given our current lack of experimental data concerning the\nHiggs sector, it is natural to suppose that there may be more than a\nsingle Higgs boson waiting to be discovered at the weak scale.\n\nThe fundamental constraint on multi-Higgs doublet models comes from\nflavor physics. After diagonalizing the quark masses, the Yukawa\ncouplings of the neutral component of any extra Higgs boson will\ngenerically lead to tree-level flavor-changing neutral current (FCNC)\nprocesses, which are highly constrained by data. There are only a few\noptions generally considered for avoiding these difficulties. The\nfirst is simply to assume that the Yukawa couplings of any new Higgs boson\nto the standard model fermions are sufficiently small so as to be\nsafe. Generically, for Higgs boson masses of order the weak scale, this\nrequires Yukawa couplings of order $10^{-4}$ or less. The second\noption is to demand that only a single Higgs boson couples to standard\nmodel fermions of a given electric charge. This leads to two commonly\nconsidered scenarios, referred to as the Type I and II two Higgs\ndoublet models (2HDMs). In the Type I 2HDM, it is assumed that an\nadditional Higgs does not couple to any of the standard model\nfermions, while in the Type II model, a first Higgs couples only to\nthe up-type quarks, while a second couples only to the down-type\nquarks, as in supersymmetry. A third option often considered is that\nof ``minimal flavor violation\" \\cite{MFV}. In this scenario, it is assumed that\nthe full $U(3)^5$ flavor symmetry of the standard model is broken only\nby the Yukawa couplings of a single Higgs boson responsible for\ngenerating the fermion masses. The Yukawas are assumed to come from\nvacuum expectation values (vevs) of some set of fields transforming as\nbifundamentals under the flavor group. This results in all flavor\nviolation being of a size set by the ordinary standard model Cabbibo--Kobayashi--Maskawa (CKM)\nmatrix.\n\nOne feature which all of these scenarios have in common is that of\nvery small Yukawa couplings of the Higgs bosons to the first generation\nquarks and leptons. In the Type I and II 2HDMs this is required in\norder to avoid giving large masses to the first generation fermions,\nwhile in minimal flavor violation, this is required by virtue of the\nsmallness of the ordinary first-generation Yukawa couplings. The smallness of these Yukawa couplings can\nmake it difficult, for example, to explain a recent $Wjj$ anomaly at\nthe Tevatron \\cite{cdf} by using an extended Higgs sector. In this paper we will\nconsider a simple alternative scenario which is able to avoid this\nrequirement, address the $Wjj$ anomaly, and simultaneously suggest a mechanism for a\nstraightforward solution to the strong CP problem.\n\nOur setup assumes that a single Higgs boson $H$ dominates in providing the\nmasses for the standard model fermions. Beyond this, we will make\ntwo assumptions:\n\\begin{enumerate}\n\\item There is an $SU(3)_f$ flavor symmetry broken only by the $H$ Yukawa\n couplings. The 3 right-handed up quarks, 3 right-handed down quarks, and 3\nleft-handed doublet quarks are all taken to transform in triplet representations\nof the $SU(3)_f$ symmetry. In particular, a second Higgs doublet,\n $\\Phi=\\left(\\begin{array}{c} \\phi^+ \\\\ \\phi^0\\end{array}\\right)$ ,\n is assumed to have Yukawa couplings proportional to the identity\n matrix in a given canonical basis,\\footnote{A canonical basis is defined to be the basis of quark fields where the flavor symmetry $SU(3)_f$ is manifest.} preserving the $SU(3)_f$. Any further Higgs doublets beyond this should also have couplings of the same form.\\footnote{Large diagonal Yukawa couplings for $H$ may\nbe suppressed in an appropriate UV completion. See section 3 for an example.}\n\\item In the canonical basis, the $H$ Yukawa couplings are Hermitian matrices.\n\\end{enumerate}\n\nThe reason that such a scenario can be associated with a solution to the strong\nCP problem is straightforward; if CP is broken only\nspontaneously, then the strong CP parameter, $\\theta$, may be equal to zero\nin the original canonical basis. Diagonalizing the quark masses,\n$\\theta$ then {\\it remains} equal to zero due to the Hermiticity\nassumed for the Yukawa couplings. Spontaneous breaking of CP could take place through a variety\nof mechanisms already appearing in the literature, such as, for example, the Nelson\/Barr mechanism \\cite{NelsonCP, Barr}. We must simply require that our \"Hermitian Flavor Violation\" (HFV) structure emerges in the effective theory at low energies. This may be most easily accomplished if the $SU(3)_f$ flavor symmetry is respected by the sector of the UV theory responsible for CP violation. It may also be possible to have the same fields simultaneously break both the flavor $SU(3)_f$ and CP symmetries together; a candidate for this type of theory will be discussed in section 3.\n\nDue to the above assumed structure, the safety of the scenario from flavor-changing neutral currents\nis simple to understand at the qualitative level. After diagonalizing the quark masses,\nthe couplings of the neutral Higgs $\\phi^0$ remain unchanged, while\nthe charged Higgs $\\phi^+$ has flavor-changing interactions\nproportional to the corresponding CKM elements. In this way, the\nstructure of FCNC's is the same as in the standard model, with\nanalogous suppressions by the Glashow--Iliopoulos--Maiani (GIM) mechanism; we need only assume that\nthe $\\Phi$ Yukawa is somewhat smaller than the gauge\ncoupling of the weak interaction, depending on the $\\Phi$ mass. The only difference here is that, in the presence of the $H$ Yukawa couplings, there is no symmetry fixing the universality of the\n$\\phi^0$ interactions. In the standard model, the $Z^0$ couplings\nremain universal due to gauge invariance. This will lead to some\nsmall loop suppressed FCNCs, but not at a dangerous level. We will\ndemonstrate the safety of the flavor structure in more detail in\nsection 2, as well as discuss the limits on the $\\Phi$ Yukawa\ncouplings.\\footnote{An analogous 2HDM flavor scenario was discussed in reference \\cite{Argentina} but the motivation and underlying structure were different than what we consider here.} Section 3 will contain a proposal for a possible UV completion of our scenario, demonstrating a mechanism for realizing the required hierarchical, Hermitian structure of the $H$ Yukawa couplings, as well as addressing the strong CP problem.\n\nAs noted above, the key phenomenological difference between this model\nand more standard two Higgs doublet constructions is the presence of\nallowed large couplings of $\\Phi$ to the first generation fermions.\nAs a result, in section 4 we will discuss an explanation in this scenario for the excess in $Wjj$ events at CDF through resonant production of a heavy component of the new doublet.\\footnote{For other thoughts in this direction, see \\cite{Wang, Chen}} We will conclude in section 5.\n\n\\section{Flavor Constraints}\n\nIn the previous section, we presented a flavor structure which allows us to couple an additional Higgs boson to the standard model quarks, without having extremely suppressed couplings to the first generation. In this section we examine the major constraints on this model and place limits on the couplings of the additional Higgs boson. The Yukawa sector for this model is as follows:\n\n\\begin{equation}\n-{\\cal L}\\supset \\tilde{H} {\\bar Q_L}Y^U u_R + H {\\bar Q_L}Y^D d_R\n +\\tilde{\\Phi} {\\bar Q_L}G^U u_R + \\Phi {\\bar Q_L}G^D d_R,\n\\end{equation}\nwith $\\tilde{H} = i \\sigma_2 H^*$ and similarly for $\\Phi$.\nHere, the Yukawa couplings, $Y^U,Y^D,G^U,G^D$ are $3\\times 3$ matrices.\nWe assume that $Y^U$ and $Y^D$ are Hermitian matrices and $G^U, G^D$ are proportional to the identity matrix ${\\rm diag}(1,1,1)$ at some high-energy scale $\\Lambda_{UV}$, with constants of proportionality $g^U$ and $g^D$ respectively.\n\nOne of the most important constraints on the model comes from the up and down quark masses. Any term in the potential with an odd number of $\\Phi$ fields (and a corresponding odd number of $H$ fields) will lead to a $\\Phi$ vacuum expectation value and hence a contribution to the quark masses. While such terms may be taken to be absent at tree level in appropriate UV completions (see the next section for an example), they will still be generated at loop level due to the $\\Phi$ and $H$ Yukawa couplings. Indeed, the most important radiative corrections to the Higgs potential will come from top and bottom loops. These lead to contributions\n\\begin{equation}\n{\\cal L}_{mix}=-3\\left(\\frac{g^Dy_b+g^Uy_t}{8\\pi^2}\\right)\\Lambda_{UV}^2\\Phi^\\dagger H +h.c.\n\\end{equation}\nThe quark mass contributions induced by this operator are\n\\begin{eqnarray}\n\\Delta m_d =-3 g^D\\frac{g^Dm_b+g^Um_t}{8\\pi^2}\\frac{\\Lambda_{UV}^2}{m_\\phi^2}\\\\\n\\Delta m_u =-3 g^U\\frac{g^Dm_b+g^Um_t}{8\\pi^2}\\frac{\\Lambda_{UV}^2}{m_\\phi^2},\n\\end{eqnarray}\nwhere $m_\\phi$ is the mass of the CP-even scalar component of $\\Phi$. If we assume that these contributions are less than or equal to the values of the physical quark masses, then we may place the following approximate upper limits on $g^U$ and $g^D$:\n\\begin{eqnarray}\n g^U \\lesssim .007\\left(\\frac{m_\\phi}{300 {\\rm GeV}}\\right)\\left(\\frac{{\\rm TeV}}{\\Lambda_{UV}}\\right) \\;\\;\\;\\; \\\\\n g^D\\lesssim .06\\left(\\frac{m_\\phi}{300 {\\rm GeV}}\\right)\\left(\\frac{{\\rm TeV}}{\\Lambda_{UV}}\\right)\\;\\;\\;\\;\n\\end{eqnarray}\nwith the more severe constraint on $g^U$ due to $m_t\/m_b\\gg 1$. Although no tuning is needed when these constraints are satisfied, they can be relaxed if there is some degree of cancellation between the $\\Phi$ and $H$ contributions to the quark masses.\n\nWe next examine flavor-changing neutral current constraints. There are essentially two types of constraints we must consider. The first type come from loop corrections to FCNC processes which are present even in the strict limit that the $\\Phi$ couplings are proportional to the identity matrix. By construction, these types of corrections have the same structure as in the standard model, with $W^+$ or $H^+$ propagators replaced by $\\phi^+$, and generally obtain similar GIM style suppressions. In addition, we also have FCNC constraints coming from renormalization group (RG) running causing breaking of the perfect ${\\rm diag}(1, 1, 1)$ forms of the $\\Phi$ Yukawa couplings. We consider these in turn. Given the constraints already described above from the $\\Phi$ vev requirement, the most important FCNC process we must consider is $K^0-\\bar K^0$ mixing, and we will generally not discuss FCNC effects induced by the small coupling $g^U$.\n\n\nFor $K^0-\\bar K^0$ mixing, the experimental limits may be summarized as follows:\n In the below analysis, we will find three types of induced operators,\n\\begin{eqnarray}\n{\\cal O}_{LL}= \\bar s_R d_L \\bar s_R d_L \\;\\;\\;\\;\\;\\;\\;\\;\\;\\; & {\\cal O}_{RR}=\\bar s_L d_R \\bar s_L d_R \\;\\;\\;\\;\\;\\;\\;\\;\\;\\; & {\\cal O}_{LR}= \\bar s_R d_L \\bar s_L d_R,\n\\end{eqnarray}\nwhich all contribute to the $K^0-\\bar K^0$ mixing.\nThe current 95\\%\nconfidence level bounds on these operators, expressed in terms of the required mass scale suppressing the real ($\\Lambda_{\\rm Re}$) and imaginary ($\\Lambda_{\\rm Im}$) parts, are \\cite{constraints}\n\\begin{eqnarray}\n{\\cal O}_{LL},{\\cal O}_{RR}: \\;\\;\\;\\;\\;\\;\\;\\; &\\Lambda_{\\rm Re}\\geq 7.3\\times10^3 {\\rm TeV}, &\\Lambda_{\\rm Im}\\geq 10^5 {\\rm TeV}\\label{LLRRbound}\\\\\n{\\cal O}_{LR}: \\;\\;\\;\\;\\;\\;\\;\\; & \\Lambda_{\\rm Re}\\geq 1.7\\times10^4 {\\rm TeV}, &\\Lambda_{\\rm Im}\\geq 2.4 \\times 10^5 {\\rm TeV} \\label{LRbound}\n\\end{eqnarray}\n\nWe now consider the contribution to $K^0-\\bar K^0$ mixing coming from replacing one or both of the $W^\\pm$ lines with $\\phi^\\pm$ in the SM box diagrams. The expressions coming from these diagrams can be found in the Appendix. The box diagrams contribute dominantly to $K^0-\\bar K^0$ mixing through an induced operator of type ${\\cal O}_{LR}$. In Fig. (\\ref{KKMix}), we show the sizes of the mass scales in the real and imaginary parts of this operator in comparison with the experimental constraints, for a coupling $g^D$ of 0.06. \nExamining Fig. (\\ref{KKMix}), we see that the constraint coming from $\\Lambda_{\\rm Im}$ is somewhat similar in severity to that coming from the size of the induced down quark mass.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.47\\columnwidth]{ReKbarMix2.pdf}\\includegraphics[width=0.47\\columnwidth]{ImKbarMix2.pdf}\n\\caption{We plot the mass scales appearing in the real (left) and imaginary (right) parts of the effective operator of type ${\\cal O}_{LR}$ resulting from the $\\phi^+$ box diagrams for a Yukawa coupling $g^D$ of 0.06. The red lines show the constraints. More generally, the values of $\\Lambda_{\\rm Re}$ and $\\Lambda_{\\rm Im}$ scale inversely with the coupling $g^D$. \\label{KKMix} }\n\\end{figure}\n\n\nAs an aside, let us make a quick comment about the constraint coming from $b\\to s\\gamma$ decays. This constraint has been analyzed in \\cite{Mahmoudi:2009zx} for a two Higgs doublet model where the additional Higgs boson couples to the SM fermions diagonally. This analysis can be applied to our scenario. Taking $g^U$ small, we fall safely in the allowed parameter space for $g^D\\lesssim 0.1$.\\footnote{In \\cite{Mahmoudi:2009zx} they state that for $\\lambda_{tt}=0$ ($G^U_{33}$ in our notation, with $\\lambda_{bb} = G^D_{33}$),\n$b\\to s\\gamma$ decays are always safe. This is an artifact of an approximation they make which allows them to neglect the $\\lambda_{bb}^2$ contribution, which is not applicable to our case. However, the $b\\to s\\gamma$ contribution from the charged Higgs is invariant under an exchange $\\lambda_{tt} \\leftrightarrow \\lambda_{bb}$ (with the dominant diagram undergoing a parity transformation). In this way we may extract the limit for our case.}\n\nWe next consider the contribution to $K^0-\\bar K^0$ mixing from RG running breaking the universality of the $\\Phi$ couplings. The dominant effect comes from the wave-function renormalization of the $Q_L$ fields, due to the large top Yukawa coupling to the $H$ doublet.\nCalculating the wave function renormalization of $Q_L$, we find\n\\begin{equation}\n\\beta_{G^D}\\supset \\frac{G^D}{32\\pi^2}\\left(Y^UY^U\\right),\n\\end{equation}\nwhere we have neglected terms that are universal (since we only care about the breaking of universality here) and also smaller terms which are proportional to $Y^D$. We take the $\\Phi$ Yukawa couplings to be universal at a UV scale $\\Lambda_{UV}$, and then RG run down to the weak scale. Without loss of generality, we are free to diagonalize the up quark mass matrix $Y^U$ at the UV scale, before performing the running. Since we already know that we must take $\\Lambda_{UV}$ close to the TeV scale, we analyze the RG corrections using the leading-log approximation. The $G^D$ coupling at the EW scale is then\n\n\\begin{equation}\nG^D(M_{EW})=G^D(\\Lambda_{UV})+\\frac{G^D}{32\\pi^2}\\left(Y^UY^U\\right)\\ln\\left(\\frac{\\Lambda_{UV}}{M_{EW}}\\right).\n\\end{equation}\nAfter running the couplings to the weak scale, we then diagonalize $Y^D$ by redefining $d_{L_i}$ and $d_{R_j}$. Although the corrections to $G^D$ are diagonal in the basis we did the RG running, they are not universal. This non-universality gives family mixing when we rotate to the down quark mass eigenstates\n\n\\begin{equation}\nV^\\dagger G^D(M_{EW})V=G^D(\\Lambda_{UV})+\\frac{G^D}{32\\pi^2}V^{\\dagger}\\left(Y^UY^U\\right)V\\ln\\left(\\frac{\\Lambda_{UV}}{M_{EW}}\\right).\n\\end{equation}\nwhere $V$ is the CKM matrix. We will now show that these radiative corrections are small enough to be less constraining than earlier bounds presented in this section. We neglect all of the present contributions to $K^0-\\bar K^0$ mixing except those due to the top Yukawa coupling. The leading order contribution to the Lagrangian density resulting from scalar exchange is then\n\\begin{eqnarray}\n\\left(\\frac{g^Dy_t^2V^*_{ts}V_{td}}{64\\pi^2} \\ln\\left(\\frac{\\Lambda_{UV}}{M_{EW}}\\right)\\right)^2\\left[\\left(\\frac{1}{m_\\phi^2}-\\frac{1}{m_A^2} \\right)({\\cal O}_{LL}+{\\cal O}_{RR}) + 2\\left(\\frac{1}{m_\\phi^2}+\\frac{1}{m_A^2} \\right){\\cal O}_{LR} \\right],\n\\end{eqnarray}\nwhere $A$ is the CP-odd scalar, with $m_A$ being its mass. The coefficient of the ${\\cal O}_{LL}+{\\cal O}_{RR}$ operator then becomes\n\\begin{eqnarray}\n\\nonumber \\left(\\frac{g^D}{.1}\\right)^2\\left( \\frac{(150 {\\rm GeV})^2}{m_\\phi^2}-\\frac{(150 {\\rm GeV})^2}{m_A^2}\\right) &&\\!\\!\\!\\!\\!\\!\\!\\!\\! \\left(\\left(\\frac{\\ln (\\frac{\\Lambda_{UV}}{M_{EW}})}{3\\times 10^6{\\rm TeV}}\\right)^2-i\\left(\\frac{\\ln(\\frac{\\Lambda_{UV}}{M_{EW}})}{3\\times 10^6{\\rm TeV}}\\right)^2\\right),\n\\end{eqnarray}\nwhile that of the ${\\cal O}_{LR}$ operator is\n\\begin{eqnarray}\n\\nonumber \\left(\\frac{g^D}{.1}\\right)^2\\left( \\frac{(150 {\\rm GeV})^2}{m_\\phi^2}+\\frac{(150 {\\rm GeV})^2}{m_A^2}\\right) &&\\!\\!\\!\\!\\!\\!\\!\\!\\! \\left(\\left(\\frac{\\ln (\\frac{\\Lambda_{UV}}{M_{EW}})}{2\\times 10^6{\\rm TeV}}\\right)^2+i\\left(\\frac{\\ln(\\frac{\\Lambda_{UV}}{M_{EW}})}{2\\times 10^6{\\rm TeV}}\\right)^2\\right).\n\\end{eqnarray}\nThe strongest bounds on the above operators come from their imaginary parts, but by inspection of equations \\ref{LLRRbound} and \\ref{LRbound}, it is clear that they are quite a bit less constraining than the bounds coming from the scalar box diagram, or from the size of the induced down quark mass.\n\n\\section{A Possible UV Completion}\n\nHere we present a UV completion which can realize our scenario in the IR. To get a Hermitian Flavor Violation model, the couplings of the SM Higgs doublet $H$ and the extra doublet $\\Phi$ to the SM quarks must be very different; $H$ obviously needs to have a very hierarchical Yukawa\nmatrix, while $\\Phi$ must have an identity-like matrix. In\naddition, the $H$ Yukawa matrices are both required to be Hermitian.\n\nThe option we will consider here is to introduce a $Z_2$ symmetry to distinguish the two Higgs bosons, in addition to the $SU(3)_f$ flavor symmetry, with $Q_L$, $u_R$, and $d_R$ all taken as triplets under $SU(3)_f$ and even under $Z_2$. We\nassume $\\Phi$ is even while $H$ is odd under $Z_2$. In the symmetric\nlimit, $\\Phi$ has Yukawa matrices with both the up- and down-sectors\nproportional to the identity matrix, while $H$ does not have any\nYukawa couplings. The Yukawa matrices $Y^U$ and $Y^D$ need to be\ngenerated by exchange of heavy particles picking up\nsymmetry-breaking spurions. We assume that the $SU(3)_f$ symmetry is\nbroken by vevs of triplet spurions $v_1$, $v_2$, and $v_3$, while the\n$Z_2$ symmetry by a spurion $\\sigma$. CP is spontaneously broken in the\ntriplet VEVs, with the vev of $\\sigma$ assumed real. In order to communicate these symmetry breakings to the standard model sector, we introduce a set of Dirac fermions ${\\cal U}$,\n${\\cal U}'_i$, ${\\cal U}''_i$, ${\\cal D}$, ${\\cal D}'_i$, ${\\cal D}''_i$, where ${\\cal U}$ and ${\\cal D}$ are flavor triplets, and where fields of type $i$, including $v_i$, are charged under separate $U(1)_i$ abelian symmetries.\\footnote{We assume for simplicity that the flavor symmetries are gauged so that we don't have to worry about any light Goldstone modes, or possible Planck suppressed breaking effects.} The ${\\cal U}''_i$ and ${\\cal D}''_i$ fields are assumed even under the $Z_2$ symmetry, with all other new heavy quarks being odd. We will label the left and right handed components of these Dirac fermions with $L$ and $R$ subscripts, as usual. The full set of charges of the new fields are shown in Table 1\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{|c||c|c|c|c|}\n \\hline\n Field & $SU(3)_f$ & $U(1)^3$ & ${\\mathbb Z}_2$ & $SU(2)_W \\times\n U(1)_Y $ \\\\ \\hline\n ${\\cal U}$ & 3 & 0 & $-$ & $1_{2\/3}$ \\\\\n ${\\cal U}'$ & 1 & 3 & $-$ & $1_{2\/3}$ \\\\\n ${\\cal U}''$ & 1 & 3 & $+$ & $1_{2\/3}$ \\\\\n ${\\cal D}$ & 3 & 0 & $-$ & $1_{-1\/3}$ \\\\\n ${\\cal D}'$ & 1 & 3 & $-$ & $1_{-1\/3}$ \\\\\n ${\\cal D}''$ & 1 & 3 & $+$ & $1_{-1\/3}$ \\\\ \\hline\n $H$ & 1 & 0 & $-$ & $2_{1\/2}$ \\\\\n $\\Phi$ & 1 & 0 & $+$ & $2_{1\/2}$ \\\\\n $\\sigma$ & 1 & 0 & $-$ & $1_0$ \\\\\n $v$ & 3 & 3 & $+$ & $1_0$ \\\\ \\hline\n \\end{tabular}\n\\label{tab:charges}\n\\caption{Charges of fields in the example UV completion. Here a $U(1)^3$ charge of ``3\" means that there are three such fields with separate $U(1)$ charges of the form $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. }\n\\end{table}\n\nThe following set of interactions are permitted by the symmetries, and will be used to generate the appropriate $H$ Yukawa structure (showing only the up sector for brevity):\\footnote{Allowed heavy quark interactions with flipped chiralities may also be included and do not pose any difficulty.}\n\\begin{equation}\n {\\cal O}_1 = \\tilde{H} \\bar{Q}_L {\\cal U}_R, \\quad\n {\\cal O}_2 = \\bar{{\\cal U}}_L v_i {\\cal U}'_{R i}, \\quad\n\n {\\cal O}_3 = \\sigma \\bar{\\cal U}'_{L i} {\\cal U}''_{R i}, \\quad\n {\\cal O}_4 = \\bar{\\cal U}''_{L i} v^\\dagger_i u_R.\n\\end{equation}\nThe resulting $H$ Yukawa couplings then have the form\n\\begin{equation}\n Y_u = \\sum_i \\frac{\\langle v_i\\rangle \\langle \\sigma\\rangle \\langle\n v_i\\rangle^\\dagger}{M_{{\\cal U}} M_{{\\cal\n U}'_i} M_{{\\cal U}''_i}}\\ , \\qquad\n Y_d = \\sum_i \\frac{\\langle v_i\\rangle \\langle \\sigma\\rangle \\langle\n v_i\\rangle^\\dagger}{M_{{\\cal D}} M_{{\\cal\n D}'_i} M_{{\\cal D}''_i}}\\ .\n\\end{equation}\nNote that there is no contribution that mixes up different $i$'s thanks to the $U(1)_i$ flavor symmetries. This is crucial to ensure the Hermiticity of the Yukawa matrices.\nWithout loss of generality, we may make $SU(3)_f$ flavor rotations to put the $v$ vevs in form $v_1=(a,b,c)$, $v_2 =(d, e, 0)$, and $v_3 =(f, 0, 0)$. Assuming an inverse hierarchy among the heavy fermion masses, we may then\nobtain both Hermitian and hierarchical Yukawa matrices. In this construction we may take the $\\sigma$, $Z_2$ breaking vev to be of order TeV, along with one or more of the heavy quark masses, providing the effective $\\Lambda_{UV}$ cutoff on the dangerous loop diagrams discussed in section 2. The schematic form\nof the Yukawa matrices likely from this construction is\n\\begin{equation}\n Y^U \\approx \\left(\n \\begin{array}{ccc}\n \\lambda^8 & \\lambda^8 & \\lambda^8 \\\\\n \\lambda^8 & \\lambda^4 & \\lambda^4 \\\\\n \\lambda^8 & \\lambda^4 & 1\n \\end{array} \\right), \\qquad\n Y^D \\approx y_b \\left(\n \\begin{array}{ccc}\n \\lambda^3 & \\lambda^3 & \\lambda^3 \\\\\n \\lambda^3 & \\lambda^2 & \\lambda^2 \\\\\n \\lambda^3 & \\lambda^2 & 1\n \\end{array} \\right),\n\\end{equation}\nwith $y_b$ being roughly the ratio of the bottom and top masses.\n\nThere are a few operators which are allowed by all of the symmetries of Table 1, but which are nevertheless dangerous to our construction. These are\n\\begin{equation}\n \\bar{Q}_L v^i i{\\not\\!\\! D} v_i^\\dagger Q_L, \\qquad\n \\sigma H^\\dagger \\Phi, \\qquad\n \\sigma \\tilde{H} \\bar{Q}_L u_R, \\qquad\n \\sigma \\tilde{H} \\bar{Q}_L v_i v_i^\\dagger v_j v_j^\\dagger u_R.\n\\end{equation}\nThe first one leads to a non-universal Yukawa coupling to $\\Phi$ and\nhence flavor-changing neutral currents; the second one induces a\nvacuum expectation value for $\\Phi$, too-large fermion masses and\nhence fine-tuning; the third leads directly to too-large fermion masses; the last one destroys the Hermiticity of the\nYukawa matrix if $i \\neq j$. Taking sufficiently suppressed coefficients for these operators is technically natural, so long as the $v_i$ vevs and masses are taken to be at least a few orders of magnitude larger than the TeV scale, with corresponding small coefficients for the ${\\cal O}_2$ and ${\\cal O}_4$ operators. In general, the masses of the various fields, and coefficients of operators in this construction are somewhat flexible, and we will not discuss them in further detail.\n\nIt is a straightforward exercise to check that this model leads to no strong CP parameter at tree level by considering the phase of the determinant of the full quark mass matrix. At loop level, a highly suppressed contribution might arise after taking into account radiatively induced breaking of Hermiticity\/universality, as well as the small induced $\\Phi$ vev. A full calculation of such loop corrections in this particular UV completion is however beyond the scope of the present work.\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\\section{$Wjj$ events at the Tevatron}\n\nThe CDF collaboration recently reported on the production of $Wjj$\nwith an integrated luminosity of $4.3 {\\rm ~fb}^{-1}$ \\cite{cdf}.\nInvestigating the invariant mass distribution of the jet pair, they\nfound an excess of 253 events ($156\\pm 42$ electrons, $97\\pm 38$ muons)\nin the $120-160$ GeV range, which is well fit by a Gaussian peak\ncentered at $144\\pm 5$ GeV. Additionally, it has been reported that analysis of an additional $3{\\rm~fb}^{-1}$ sample collected by CDF shows this same feature,\ngiving a total significance of $4.1\\sigma$ for the combined $7.3{\\rm~fb}^{-1}$ data set \\cite{cdfnote}.\n\nHermitian Flavor violation provides a perfect setting for explaining the $Wjj$ anomaly\nwith a new $SU(2)$ doublet scalar.\\footnote{Several explanations for the $Wjj$ anomaly have been presented in the literature, including other $SU(2)$ doublet scalars \\cite{Wang, Babu, Dutta:2011kg}, a $Z$-prime \\cite{Hooper, Cheung}, new colored states \\cite{XPWang,Dobrescu,Carpenter:2011yj}, in supersymmetry \\cite{Kilic, Sato}, technicolor \\cite{Lane}, or string theory \\cite{stringy}, and within the context of the Standard model \\cite{He, Sullivan:2011hu,Plehn:2011nx}.} Indeed, what seems to be required is a large coupling of the scalar to the first generation quarks. However, as noted in the introduction, such couplings usually go hand in hand with\nexcessive flavor changing neutral currents. Hermitian flavor violation addresses this problem.\n\nProducing $Wjj$ events at the Tevatron through a new Higgs doublet can proceed via two primary mechanisms. The first\nis to simply have $t$-channel production of a $W$-boson along with the new scalar, followed by decay of the scalar to two jets. This scenario, however, typically requires large couplings which run into difficulty with the constraints discussed in section 2 as well as collider constraints, and we will not discuss it further here. The second option is to split the masses of the charged and neutral Higgs components, and consider resonant production of the heavier state. This state will then primarily decay into a $W$-boson plus the lighter state, with the lighter state then decaying to jets, as in Fig. (2) (in this case there will also be a small additional $t$-channel contribution).\nThe CDF collaboration's background-subtracted invariant mass distribution of the $l\\nu jj$ system,\\footnote{The neutrino momentum may be reconstructed up to a two-fold ambiguity using the on-shell mass condition for the $W$.} $M_{l\\nu jj}$, shows a peak in the $250\\sim 300{\\rm~GeV}$ range \\cite{cdfnote}, which one would expect from a heavy resonance at around this mass range.\nIn general, we may consider some flexibility in the permitted scalar spectrum, pertaining to\nthe choice of which state is producing the final state jets, as well as whether or not the scalar and pseudo-scalar components are split from one another. Hereafter we refer to the field(s) with mass $\\sim 150{\\rm~GeV}$ contributing to the excess as $\\Phi_l$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Wjj_resonant}\n\\caption{Production of $W$ boson and two jets via resonant production. \\label{fig:diagrams}}\n\\end{figure}\n\nSince the $\\Phi$ coupling to the $u_R$ quarks is relatively constrained by the requirement of not over-contributing to the up quark mass, as discussed in section 2, we will focus on the case where $\\Phi$ couples dominantly to the $d_R$ quarks.\\footnote{Though we focus on the case where $g^D$ is the dominant coupling, the results for coupling size in table 2 are essentially unchanged for the case of dominant $g^U$, with the exception of case 5, where the necessary initial partons required to produce a neutral resonance differ. The $\\sigma(Z\\Phi_l)$, $\\sigma(\\gamma\\Phi_l)$, and $\\sigma(jj)$ entries in table 2, however, differ for the dominant $g^U$ case (due to different initial partons).} Since the proton contains twice as many up quarks as down quarks, the best case for our model is to take $\\phi^+$ to be the heavier, resonantly produced state. At least one of the neutral components must then have a mass of $\\sim 150$GeV in order to explain the CDF excess.\nTo get an idea of the size of the coupling needed to account for the $Wjj$ excess, we generate $p\\bar{p}\\to\\Phi_l W^\\pm\\to l\\nu jj$ events with Madgraph\/MadEvent \\cite{Alwall:2007st}, which are then showered with Pythia \\cite{Sjostrand:2006za}, with detector simulation by PGS \\cite{pgs} using CDF parameters. We implement the cuts described in \\cite{cdf}, and require a total of $\\sim 250$ events to pass with a luminosity of $4.3{\\rm~fb}^{-1}$. We find good agreement between a $WW+WZ$ background created in this manner and the distribution in \\cite{cdf}, suggesting that this provides a reasonable estimate. The results for several scenarios are shown in table 2.\nFor a variety of mass spectra, we see that the size of the required $\\Phi$ Yukawa coupling is around $g^D \\sim .06$.\\footnote{This finding is consistent with the results of \\cite{Wang}, who considered a phenomenologically similar model.} As shown in section 2, a coupling of this size can evade all flavor constraints, as well as the constraint from the induced down quark mass, although the masses of $m_\\phi$ and $m_\\phi^+$ are preferred to be somewhat heavy, $ \\mathop{}_{\\textstyle \\sim}^{\\textstyle >} 300 {\\rm GeV}$. These constraints might be relaxed somewhat after taking into account QCD uncertainties, or if we didn't require producing the central value of the CDF excess. In particular, if we were to require producing only one standard deviation below the central value of the excess, then cases which previously required a coupling of .06 would instead require couplings of about .05.\\footnote{The constraints from flavor could also be weakened if one were to adopt case ``5u\" from Table 2. In that case, $\\Phi$ couples dominantly to the up sector, and fine tuning is required in order to keep the up quark mass small. Such a fine tuning might be considered acceptable depending on one's perspective on the origin of the fermion mass hierarchy. }\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c||c|c||c|c|c|c|}\n\\hline\n{} & $m_{\\phi}$, $m_{A}$, $m_{\\phi^\\pm}$ (GeV) & $g^D$ &$\\sigma(W^\\pm \\Phi_l)$ & $\\sigma(Z \\Phi_l)$ & $\\sigma(\\gamma \\Phi_l)$ & UA2 $\\sigma(jj)$\\\\ \\hline\n1 & 150 , 150 , 250 & 0.075 &4.1 pb & .032 pb & .008 pb& 2.0 pb\\\\ \\hline\n2 & 150 , 150 , 300 & 0.06 &1.7 pb & .020 pb & .005 pb& 1.3 pb\\\\ \\hline\n3 & 300 , 150 , 300 & 0.06 &1.6 pb& .430 pb & .003 pb& 0.6 pb\\\\ \\hline\n4 & 230 , 150 , 300 & 0.06 &1.7 pb& .016 pb & .003 pb& 0.6 pb\\\\ \\hline\n5d & 300 , 300 , 150 & 0.08 &1.6 pb& .028 pb & .016 pb& 5.2 pb\\\\ \\hline\n5u & 300 , 300 , 150 & (0.04) &1.5 pb& .008 pb & .004 pb& 1.3 pb\\\\ \\hline\n\\end{tabular}\n\\label{tab:couplings}\n\\caption{Size of Yukawa couplings which explain the CDF $Wjj$ excess. The cross sections $\\sigma(W^\\pm \\Phi_l)$, $\\sigma(Z \\Phi_l)$, and $\\sigma(\\gamma \\Phi_l)$ are calculated at Tevatron energy, with no cuts apart from requiring $p_T>30$ for the photon. $\\Phi_l$ refers to all fields with masses of $150{\\rm~GeV}$. $\\sigma(jj)$ refers to the dijet cross section for the process $p\\bar{p}\\to\\Phi_l\\to jj$ at $\\sqrt{s}=630{\\rm~GeV}$ and should be compared with the limit of ${\\mathcal O}(100{\\rm~GeV})$ \\cite{Alitti:1993pn}. The parentheses for model 5u indicate the value for the coupling $g^U$ rather than $g^D$.}\n\\end{center}\n\\end{table}\n\nWhile table 2 presents only a few benchmark points, the behavior suggests that various mass spectra could in principle be able to explain the $Wjj$ excess. There is some small variation in the required coupling with changes to the mass of the heavy resonance, as seen by comparison of scenarios 1 and 2. This reflects both a larger branching ratio $BR(\\phi^\\pm\\to W^\\pm\\phi^0)$ as well as a greater acceptance of events for the heavier resonance. \n\nIf the scalar and pseudo-scalar masses are split, then the required coupling may change slightly, but not significantly, compared with the degenerate case. As an example, consider taking the CP-even scalar component heavy, so that it is no longer within kinematic reach of the $\\phi^+$ decays. In that case, the size of the required $g^D$ coupling will remain essentially unchanged, at $\\sim .06$, as seen by comparison of scenarios 2, 3, and 4 in table 2. This follows because the width of the $\\phi^+$ resonance, $\\Gamma$, is cut in half.\\footnote{For these scenarios, $\\phi^\\pm$ decays dominantly to $W^\\pm\\phi^0$, with $BR(\\phi^\\pm\\to W^\\pm\\phi^0)\\approx 96\\%$ for scenario 2.} Indeed, in the tree level production diagram, we obtain an increased resonant enhancement from a $1\/\\Gamma^2$ in the propagator, yielding a factor of 4. There are half as many final states for the $\\phi^+$ decay, yielding a suppression by a factor of 2. Finally, due to the smaller width, there is half as much phase space volume for the initial quarks which can successfully hit the resonance. Taking into account the fact that the kinematics of the produced $Wjj$ events are unchanged from the degenerate case, and multiplying these factors together, we see that the overall event rate is essentially unchanged.\n\nAside from FCNC considerations, there are also direct collider constraints on two Higgs doublet models.\nOne might expect evidence of our additional Higgs sector in $\\gamma jj$ and $Zjj$ events. However, note that with resonant production, such events are quite suppressed, as shown in table 2, since the $\\gamma$ and $Z$ cannot be produced in $\\phi^+$ decays. Scenario 3 has the largest $Zjj$ cross section because the CP-even scalar is heavy enough for the resonant process $d\\bar{d}\\to \\phi\\to A+Z$. In contrast, the mass of $\\phi$ in scenario 4 lies below the threshold for decay to $A+Z$, so it does not receive such an enhancement.\n\n\nAdditionally, a new scalar with\na coupling to first generation quarks could be produced as an $s$-channel\nresonance and appear in dijet searches. Because of the large QCD\nbackgrounds, Tevatron dijet bounds are only significant for resonances\nheavier than those we consider here \\cite{Aaltonen:2008dn}. However,\nthe lower energy $p\\bar{p}$ collisions ($\\sqrt{s}=630 {\\rm ~GeV}$)\nobserved by the UA2 collaboration provide an opportunity for\nconstraining ${\\cal O}(100{\\rm ~GeV})$ dijet resonances. A search for\n$W_R'$ resonances using a $10.9 {\\rm~pb}^{-1}$ data sample places\nconstraints of ${\\mathcal O}(100\\rm~pb)$ for $\\sigma\\times BR(W'\\to jj)$ at\nthe 90\\% confidence level for a mass of $\\sim 150{\\rm\n ~GeV}$ \\cite{Alitti:1993pn}. Although we are considering a scalar\nresonance, they provide a guideline for our extended Higgs\nsector. Our scenarios are very safe from this bound, as shown in table 2.\n\n\\section{Discussions and Conclusions}\n\nIn this paper, we presented a novel type of two-doublet Higgs model\nthat allows for new $O(.1)$ Yukawa couplings to the light generations while\nnaturally suppressing FCNCs via a GIM-like mechanism. We also\ndiscussed phenomenological consequences at colliders. Thanks\nto the allowed large couplings of the up- and down-quarks to the extra\ndoublet, the production of the doublet states can be significant. In\nparticular, the bump in the $W+jj$ mass distribution reported by the\nCDF collaboration may be explained straightforwardly in this setup, while remaining consistent with phenomenological constraints.\nIn addition, Hermiticity of the Yukawa couplings in our scenario suggests\na possible solution to the strong CP problem through spontaneous CP breaking.\n\n\n\\section*{Acknowledgements}\n\nH.M. was supported in part by the U.S. DOE under Contract\nDE-AC03-76SF00098, in part by the NSF under grant PHY-04-57315, and in\npart by the Grant in-Aid for scientific research (C) 23540289 from\nJapan Society for Promotion of Science (JSPS). The work of T.T.Y was supported by JSPS Grant-in-Aid for Scientific Research (A) (22244021). This work was also\nsupported by the World Premier International Center Initiative (WPI\nProgram), MEXT, Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nA possible candidate for an extra hot thermal relic component is the\naxion particle produced thermally in the early universe. Axions\ntherefore can contribute to the hot dark matter component together\nwith the standard relic neutrino background. \nAxions may be produced in the early universe via thermal or non\nthermal processes, and arise as the solution to solve the strong CP\nproblem~\\cite{PecceiQuinn,Weinberg:1977ma,Wilczek:1977pj}. Axions are the Pseudo-Nambu-Goldstone\nbosons of a new global $U(1)_{PQ}$ (Peccei-Quinn) symmetry that is \nspontaneously broken at an energy scale $f_a$.\n The axion mass is given by\n\\begin{equation}\nm_a = \\frac{f_\\pi m_\\pi}{ f_a } \\frac{\\sqrt{R}}{1 + R}=\n0.6\\ {\\rm eV}\\ \\frac{10^7\\, {\\rm GeV}}{f_a}~,\n\\end{equation}\nwhere $f_a$ is the axion coupling constant, $R=0.553 \\pm 0.043 $ is the up-to-down quark masses\nratio and $f_\\pi = 93$ MeV is the pion decay constant. Non-thermal axions, as those produced by the misalignment mechanism, \nwhile being a negligible hot dark matter candidate, may constitute a fraction or the total cold dark matter component of the universe.\nWe do not explore such a possibility here. \nThermal axions will affect the cosmological observables in a very\nsimilar way to that induced by the presence of neutrino masses and\/or\nextra sterile neutrino species. \nMassive thermal axions as hot relics affect large scale structure,\nsince they will only cluster at scales larger than\ntheir free-streaming scale when they become non-relativistic,\nsuppressing therefore structure formation at small scales (large\nwavenumbers $k$). \nConcerning Cosmic Microwave Background (CMB) physics, \nan axion mass will also lead to a signature in the CMB photon\ntemperature anisotropies via the early integrated Sachs-Wolfe effect.\nIn addition, extra light species as thermal axions\nwill contribute to the dark radiation content of the universe, or, in\nother words, will lead to an increase of the effective number of relativistic degrees of freedom $N_{\\textrm{eff}}$, defined via\n\\begin{equation}\n \\rho_{rad} = \\left[1 + \\frac{7}{8} \\left(\\frac{4}{11}\\right)^{4\/3}N_{\\textrm{eff}}\\right]\\rho_{\\gamma} \\, ,\n\\end{equation}\nwhere $\\rho_{\\gamma}$ refers to the present photon energy density. \nIn the standard cosmological model in which a thermal axion content is\nabsent, the three active neutrino contribution leads to the canonical\nvalue of $N_{\\textrm{eff}}=3.046$ \\cite{Mangano:2005cc}. The extra contribution\nto $N_{\\textrm{eff}}$ arising from thermal axions can modify both the\nCMB anisotropies (via Silk damping) and\nthe light element primordial abundances predicted by Big Bang\nNucleosynthesis. \nThe former cosmological signatures of thermal axions have been extensively\nexploited in the literature to derive bounds on the thermal axion\nmass, see Refs.~\\cite{Melchiorri:2007cd,Hannestad:2007dd,Hannestad:2008js,Hannestad:2010yi,Archidiacono:2013cha,Giusarma:2014zza}.\n\nHowever, all the cosmological axion mass limits to date have assumed the usual simple power-law\ndescription for the primordial perturbations. The aim of this paper\nis to constrain the mass of the thermal axion using a non-parametric\ndescription of the Primordial Power Spectrum (PPS hereinafter) of the\nscalar perturbations, as introduced in Ref.~\\cite{Gariazzo:2014dla}.\nWhile in the simplest models of\ninflation~\\cite{\nGuth:1980zm,Linde:1981mu,Starobinsky:1982ee,Hawking:1982cz,Albrecht:1982wi,Mukhanov:1990me,Mukhanov:1981xt,Lucchin:1984yf, Lyth:1998xn,Bassett:2005xm,Baumann:2008bn}\nthe PPS has a scale-free power-law form, the PPS could be more complicated, presenting various features or a scale dependence.\nSeveral methods have been proposed in the literature to reconstruct\nthe shape of the PPS (see the recent work of Ref.~\\cite{Ade:2015lrj}). It has been\nshown~\\cite{Hunt:2013bha,Hazra:2014jwa} that there are small hints for deviations from the power-law form,\neven when using different methods and different data sets.\n\n\nThe energy scales at which the PPS was produced during inflation\ncan not be directly tested. We can only infer the PPS by measuring the\ncurrent matter power spectrum in the galaxy distribution and the power\nspectrum of the CMB fluctuations. The latter one, measured with exquisite \nprecision by the Planck experiment \\cite{Planck:2015xua,Ade:2013zuv,Ade:2013ktc}, \nis the convolution of the PPS with the transfer function. \nTherefore, in order to reconstruct the PPS, the assumption of an underlying cosmological\nmodel is a mandatory first step in order to compute the transfer function.\n\nHere we rather exploit a non-standard PPS approach, which can allow for a good fit to\nexperimental data even in models that deviates from the standard\ncosmological picture. In particular, we consider a thermal axion\nscenario, allowing the PPS to assume a more general shape than the\nusual power-law description. This will allow us to test the robustness of the cosmological thermal axion mass bounds (see Ref.~\\cite{Giusarma:2014zza} for a recent standard thermal axion analysis), as first performed in Ref.~\\cite{dePutter:2014hza} for the neutrino mass case. \n\nThe structure of the paper is as follows. Section \\ref{sec:method}\ndescribes the PPS modeling used in this study, as well as the\ndescription of the thermal axion model explored here and the\ncosmological data sets exploited to constrain such a model. In\nSec. \\ref{sec:results} we present and discuss the results arising from\nour bayesian analysis, performed through the Monte Carlo Markov Chains (MCMC) package \\texttt{CosmoMC} \\cite{Lewis:2002ah}, \nwhile the calculation of the theoretical observables is done through the Boltzman equations solver \n\\texttt{CAMB} (Code for Anisotropies in the Microwave Background) \\cite{Lewis:1999bs}.\nWe draw our conclusions in Sec.~\\ref{sec:concl}.\n\n\n\n\\section{Method}\n\\label{sec:method}\nIn this section we focus on the tools used in the numerical analyses performed here.\nSubsection~\\ref{sub:pps} describes the alternative model for the PPS\nof scalar perturbations used for the analyses here (see also Ref.~\\cite{Gariazzo:2014dla}),\nwhile in Subsection~\\ref{sub:model} we introduce the cosmological\nmodel and the thermal axion treatment followed in this study. Finally,\nwe shall present in Subsection~\\ref{sub:data}\n the cosmological data sets used in the MCMC analyses.\n\n\\subsection{Primordial Power Spectrum Model}\n\\label{sub:pps}\nThe primordial fluctuations in scalar and tensor modes are generated during the inflationary phase in the early universe.\nThe simplest models of inflation predict a power-law form for the PPS of scalar and tensor perturbations \n(see e.g. ~\\cite{Guth:1980zm,Linde:1981mu,Starobinsky:1982ee,Hawking:1982cz,Albrecht:1982wi,Mukhanov:1990me,Mukhanov:1981xt,Lucchin:1984yf,Lyth:1998xn,Bassett:2005xm,Baumann:2008bn}\nand references therein),\nbut in principle inflation can be generated by more complicated mechanisms, thus giving a different shape for the PPS \n(see Refs.~\\cite{Martin:2014vha,Kitazawa:2014dya} and references therein). In order to study how the cosmological constraints on the parameters change in more general inflationary scenarios, we assume a non-parametric form for the PPS.\n\nAmong the large number of possibilities, we decided to describe the\nPPS of scalar perturbations using a function to interpolate the PPS\nvalues in a series of nodes at fixed position. The interpolating\nfunction we used is named ``piecewise cubic Hermite interpolating\npolynomial'' (\\texttt{PCHIP}\\xspace) \\cite{Fritsch:1980} and it is a modified spline\nfunction, defined to preserve the original monotonicity of the point series that is interpolated.\nWe use a modified version of the original \\texttt{PCHIP}\\xspace algorithm \\cite{Fritsch:1984}, detailed in Appendix~A of Ref.~\\cite{Gariazzo:2014dla}.\n\nTo describe the scalar PPS with the \\texttt{PCHIP}\\xspace model, we only need to give the values of the PPS in a discrete number of nodes and to interpolate among them.\nWe use 12 nodes\nwhich span a wide range of $k$-values:\n\\begin{align}\nk_1 &= 5\\e{-6} \\, \\text{Mpc}^{-1} , \\nonumber\\\\\nk_2 &= 10^{-3} \\, \\text{Mpc}^{-1} , \\nonumber\\\\\nk_j &= k_2 (k_{11}\/k_2)^{(j-2)\/9} \\quad \\text{for} \\quad j\\in[3,10] , \\nonumber\\\\\nk_{11} &= 0.35 \\, \\text{Mpc}^{-1} , \\nonumber\\\\\nk_{12} &= 10\\, \\text{Mpc}^{-1} .\n\\label{eq:nodesspacing}\n\\end{align}\nWe choose equally spaced nodes in the logarithmic scale in the range $(k_2, k_{11})$, that is well constrained from the data \\cite{dePutter:2014hza},\nwhile the first and the last nodes are useful to allow for a non-constant behaviour of the PPS outside the well-constrained range.\n\nThe \\texttt{PCHIP}\\xspace PPS is described by\n\\begin{equation}\nP_{s}(k)=P_0 \\times \\texttt{PCHIP}\\xspace(k; P_{s,1}, \\ldots, P_{s,12})\n,\n\\label{eq:pchip}\n\\end{equation}\nwhere $P_{s,j}$ is the value of the PPS at the node $k_j$ divided by $P_0=2.36\\e{-9}$ \\cite{Larson:2010gs}.\n\\subsection{Cosmological and Axion Model}\n\\label{sub:model}\n\nThe baseline scenario we consider here is the $\\Lambda$CDM model, extended with hot thermal relics (the axions), together\nwith the PPS approach outlined in the previous section. For the numerical analyses \nwe use the following set of parameters, for which we assume flat priors in the intervals listed in Tab.~\\ref{tab:priors}:\n\\begin{equation}\\label{parameterPPS}\n\\{\\omega_b,\\omega_c, \\Theta_s, \\tau, m_a, \\sum m_\\nu, P_{s,1}, \\ldots, P_{s,12}\\}~,\n\\end{equation}\nwhere $\\omega_b\\equiv\\Omega_bh^{2}$ and $\\omega_c\\equiv\\Omega_ch^{2}$ \nare, respectively, the physical baryon and cold dark matter energy densities,\n$\\Theta_{s}$ is the ratio between the sound horizon and the angular\ndiameter distance at decoupling, $\\tau$ is the reionization optical depth, $m_a$ and $\\sum m_\\nu$ are the axion and the sum of three active neutrino masses (both in eV) and $P_{s,1}, \\ldots,\nP_{s,12}$ are the parameters of the \\texttt{PCHIP}\\xspace PPS. We shall also consider a scenario in which massive neutrinos are also present, to explore the expected degeneracy between the sum of the neutrino masses and the thermal axion mass~\\cite{Giusarma:2014zza}. \n\nIn order to compare the results obtained with the \\texttt{PCHIP}\\xspace PPS to the results obtained with the usual power-law PPS model, we describe the latter case with the following set of parameters:\n\\begin{equation}\\label{parameterPL}\n\\{\\omega_b,\\omega_c, \\Theta_s, \\tau, m_a, n_s, \\log[10^{10}A_{s}]\\}~,\n\\end{equation}\nwhere $n_s$ is the scalar spectral index, $A_{s}$ the amplitude of the primordial spectrum\nand the other parameters are the same ones described above. The case of several hot thermal relics for the standard scenario will not be carried out here, as it has been done in the past by several authors (see e.g.~\\cite{Giusarma:2014zza}).\nThe flat priors we use are listed in Tab.~\\ref{tab:priors}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{c|c}\nParameter & Prior\\\\\n\\hline\n$\\Omega_{\\rm b} h^2$ & $[0.005,0.1]$\\\\\n$\\Omega_{\\rm cdm} h^2$ & $[0.001,0.99]$\\\\\n$\\Theta_{\\rm s}$ & $[0.5,10]$\\\\\n$\\tau$ & $[0.01,0.8]$\\\\\n$m_a$ (eV) & $[0.1,3]$\\\\\n$\\sum m_\\nu$ (eV) & $[0.06,3]$\\\\\n$P_{s,1}, \\ldots, P_{s,12}$ & $[0.01,10]$\\\\\n$n_s$ & $[0.9, 1.1]$\\\\\n$\\log[10^{10}A_{s}]$ & $[2.7,4]$\\\\\n\\end{tabular}\n\\end{center}\n\\caption{\nPriors for the parameters used in the MCMC analyses.\n}\n\\label{tab:priors}\n\\end{table}\n\n\n\\begin{figure*}[!t]\n\\includegraphics[width=15cm]{ma_tot_loglog.pdf}\n\\caption{The left upper panel shows the temperature of decoupling as a function of the axion mass (solid curve), as well as the Big Bang Nucleosynthesis temperature, $T_{\\textrm{BBN}}\\simeq 1$~MeV (dashed curve). The right upper panel shows the axion contribution to the extra dark radiation content of the universe, while the bottom right plot depicts the free-streaming scale of an axion (solid curve) or a neutrino (dashed curve) versus the axion\/neutrino mass, in eV. The left bottom panel shows the current axion mass-energy density as a function of the axion mass.}\n\\label{fig:maref}\n\\end{figure*}\n\n\nConcerning the contribution of the axion mass-energy density to\nthe universe's expansion rate, we briefly summarize our treatment in the following.\nAxions decoupled in the early universe at a temperature $T_D$ given by\nthe usual freeze out condition for a thermal relic:\n\\begin{eqnarray}\n\\Gamma (T_D) = H (T_D)~,\n\\label{eq:decouplinga}\n\\end{eqnarray} \nwhere the thermally averaged interaction rate $\\Gamma$ refers to the $\\pi + \\pi \\rightarrow \\pi\n+a$ process:\n\\begin{eqnarray}\n\\Gamma = \\frac{3}{1024\\pi^5}\\frac{1}{f_a^2f_{\\pi}^2}C_{a\\pi}^2 I_a~,\n\\end{eqnarray}\nwith $C_{a\\pi} = \\frac{1-R}{3(1+R)}$ representing the axion-pion coupling constant and the integral $I_a$ reads as follows\n\\begin{eqnarray}\nI_a &=&n_a^{-1}T^8\\int dx_1dx_2\\frac{x_1^2x_2^2}{y_1y_2}\nf(y_1)f(y_2) \\nonumber \\\\\n&\\times&\\int^{1}_{-1}\nd\\omega\\frac{(s-m_{\\pi}^2)^3(5s-2m_{\\pi}^2)}{s^2T^4}~,\n\\end{eqnarray}\nin which $n_a=(\\zeta_{3}\/\\pi^2) T^3$ refers to the number density for axions in thermal equilibrium. The function $f(y)=1\/(e^y-1)$ is the pion thermal distribution and there are three different kinematical variables ($x_i=|\\vec{p}_i|\/T$, $y_i=E_i\/T$ ($i=1,2$) and $s=2(m_{\\pi}^2+T^2(y_1y_2-x_1x_2\\omega))$). \nThe freeze out equation above, Eq.~(\\ref{eq:decouplinga}), can be numerically solved, obtaining the axion decoupling temperature\n$T_D$ as a function of the axion mass $m_a$. Figure~\\ref{fig:maref} shows, in the left upper panel, the axion decoupling temperature as a function of the axion mass, in eV units. Notice that, the higher the axion mass, the lower the temperature of decoupling is. From the axion decoupling temperature it is possible to infer the present axion number density,\nrelated to the current photon density $n_\\gamma$ by \n\\begin{eqnarray}\nn_a=\\frac{g_{\\star S}(T_0)}{g_{\\star S}(T_D)} \\times \\frac{n_\\gamma}{2}~, \n\\label{eq:numberdens}\n\\end{eqnarray} \nwhere $g_{\\star S}$ represents the number of \\emph{entropic} degrees of\nfreedom, with $g_{\\star S}(T_0) = 3.91$. The contribution of the relic axion to the total mass-energy density of the universe will be given by the product of the axion mass times the axion number density. The quantity $\\Omega_a h^2$ at the present epoch is depicted in the bottom left panel of Fig.~\\ref{fig:maref}. Notice that, currently, a $1$~eV axion will give rise to $\\Omega_a h^2\\simeq 0.005$, while a neutrino of the same mass will contribute to the total mass-energy density of the universe with $\\Omega_\\nu h^2\\simeq 0.01$. Notice however that $\\Omega_a h^2$ represents the contribution from relic, thermal axion states. Non-thermal processes, as the misalignment production, could also produce a non-thermal axion population which we do not consider here, see the work of \\cite{DiValentino:2014zna} for the most recent cosmological constraints on such scenario. As previously stated, the\npresence of a thermal axion will also imply an extra radiation component at the BBN period:\n\\begin{equation}\n\\Delta N_{\\textrm{eff}} =\\frac{ 4}{7}\\left(\\frac{3}{2}\\frac{n_a}{n_\\nu}\\right)^{4\/3}~,\n\\end{equation}\nwhere $n_a$ is given by Eq.~(\\ref{eq:numberdens}) and $n_\\nu$ refers\nto the present neutrino plus antineutrino number density per flavour. The top right panel of Fig.~\\ref{fig:maref} shows the axion contribution to the radiation component of the universe as a function of the axion mass. Notice that the extra dark radiation arising from a $1$~eV axion is still compatible (at $95\\%$~CL) with the most recent measurements of $N_{\\textrm{eff}}$ from the Planck mission~\\cite{Planck:2015xua}. The last crucial cosmological axion quantity is the axion free streaming scale, i.e. the wavenumber $k_{\\rm {fs}}$ below which axion density perturbations will contribute to clustering once the axion is a non-relativistic particle. This scale is illustrated in Fig.~\\ref{fig:maref}, in the bottom right panel, together with that corresponding to a neutrino of the same mass. Notice that they cover the same scales for our choice of priors for $m_a$ and $\\sum m_\\nu$ and therefore one can expect a large correlation between these two quantities in measurements of galaxy clustering. We will explore this degeneracy in the following sections. We summarize the axion parameters in Tab.~\\ref{tab:axionparams}, where we specify the values of the decoupling temperature, $\\Delta N_{\\textrm{eff}}$, $\\Omega_a h^2$ and $k_{\\rm {fs}}$ for the range of axion masses considered here, $(0.1, 3)$~eV.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nAxion parameter & & \\\\\n\\hline\\hline\n$m_a$ (eV) & $0.1$ &$3$ \\\\\n$T_D$ (MeV) & $245.6$ &$43.2$ \\\\\n$\\Omega_a h^2$ & $0.0003$ &$0.016$ \\\\\n$\\Delta N_{\\textrm{eff}}$ & $0.18$ &$0.43$ \\\\ \n$k_{\\rm {fs}}$ ($h$\/Mpc) &$0.06$ &$1.46$\\\\\n\\hline\n\\end{tabular}\n\n\\end{center}\n\\caption{Values for the axion parameters, $T_D$, $\\Delta N_{\\textrm{eff}}$, $\\Omega_a h^2$ and $k_{\\rm {fs}}$ for the lower and upper prior choice of $m_a$ explored here. }\n\\label{tab:axionparams}\n\\end{table}\n\n\\subsection{Cosmological measurements}\n\\label{sub:data}\n\nOur baseline data set consists of CMB measurements. These include the\ntemperature data from the Planck satellite, see\nRefs.~\\cite{Ade:2013ktc,Planck:2013kta}, together with the\nWMAP 9-year polarization measurements, following\n\\cite{Bennett:2012fp}. We also consider high multipole data from the\nSouth Pole Telescope (SPT) \\cite{Reichardt:2011yv} as well as from the Atacama Cosmology Telescope (ACT) \\cite{Das:2013zf}. \nThe combination of all the above CMB data is referred to as the \\emph{CMB} data set.\n\n\nGalaxy clusters represent an independent tool to probe the cosmological\nparameters. Cluster surveys usually report their measurements by means\n of the so-called cluster normalization condition, $\\sigma_8\n \\Omega^\\gamma_m$, where $\\gamma \\sim \n 0.4$~\\cite{Allen:2011zs,Weinberg:2012es,Rozo:2013hha}.\nWe shall use here the cluster normalization condition as measured by\nthe Planck Sunyaev-Zeldovich (PSZ) 2013 catalogue~\\cite{Ade:2013lmv},\nreferring to it as the \\emph{PSZ} data set. The PSZ measurements of\nthe cluster mass function provide the constraint $\\sigma_8\n(\\Omega_m\/0.27)^{0.3}=0.764\\pm 0.025$. \nAs there exists a strong degeneracy between the value of the\n$\\sigma_8$ parameter and the cluster mass bias, it is possible to fix the value of the\nbias parameter accordingly to the results arising from numerical\nsimulations. In this last case, the error on the cluster\nnormalization condition from the PSZ catalogue is considerably reduced:\n$\\sigma_8 (\\Omega_m\/0.27)^{0.3}=0.78\\pm 0.01$. In our analyses, we shall consider the two PSZ measurements of the cluster normalization condition, to illustrate the impact of the cluster mass bias in the thermal axion mass bounds, as recently explored in Ref.~\\cite{Ade:2015fva} for the neutrino mass case. Figure \\ref{fig:sigma8} illustrates the prediction for the cluster normalisation condition, $\\sigma_8\n(\\Omega_m\/0.27)^{0.3}$, as a function of the thermal axion mass. We also show the current PSZ measurements with their associated $95\\%$~CL uncertainties, including those in which the cluster mass bias parameter is fixed. Notice that the normalisation condition decreases as the axion mass increases, due to the decrease induced in the $\\sigma_8$ parameter in the presence of axion masses: the larger the axion mass is, the larger the reduction in the matter power spectra will be. \n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=12.cm]{s8om03_ma.pdf}\n\\end{center}\n \\caption{Cluster normalisation condition, $\\sigma_8\n(\\Omega_m\/0.27)^{0.3}$, as a function of the thermal axion mass. We also show the current PSZ measurements with their associated $95\\%$~CL uncertainties.}\n\\label{fig:sigma8}\n\\end{figure*}\n\n\n\nTomographic weak lensing surveys are sensitive to the overall\namplitude of the matter power spectrum by measuring the correlations in the observed shape of distant \ngalaxies induced by the intervening large scale structure. \nThe matter power spectrum amplitude depends on both the $\\sigma_8$\nclustering parameter and the matter density $\\Omega_m$. Consequently, tomographic lensing surveys, via\nmeasurements of the galaxy power shear spectra, provide additional\nand independent constraints in the ($\\sigma_8$, $\\Omega_m$) plane. \nThe Canada-France-Hawaii Telescope Lensing Survey, CFHTLenS, with six tomographic redshift bins (from $z=0.28$ to\n$z=1.12$), provides a constraint on the relationship between $\\sigma_8$\nand $\\Omega_m$ of $\\sigma_8 (\\Omega_m\/0.27)^{0.46}=0.774\\pm\n0.040$~\\cite{Heymans:2013fya}. We shall refer to this data set as \\emph{CFHT}.\n\nWe also address here the impact of a gaussian prior on the Hubble constant\n$H_0=70.6\\pm3.3$ km\/s\/Mpc from an independent reanalysis of Cepheid\ndata~\\cite{Efstathiou:2013via}, referring to this prior as the \n\\emph{HST} data set. \n\nWe have also included measurements of the large scale structure of the\nuniverse in their geometrical form, i.e., in the form of Baryon Acoustic\nOscillations (BAO). Although previous studies in the literature have shown that, \nfor constraining hot thermal relics, the shape information contained in the galaxy power spectrum is more\npowerful when dealing simultaneously with extra relativistic species\nand hot thermal relic masses~\\cite{Hamann:2010pw,Giusarma:2012ph}, we exploit here the BAO signature, as the\ncontribution from the thermal axions to the relativistic number of\nspecies is not very large (see Tab.~\\ref{tab:axionparams}), and current measurements from galaxy surveys are\nmostly reported in the geometrical (BAO) form.\n\nThe BAO wiggles, imprinted in the power spectrum of the galaxy\ndistribution, result from the competition in the coupled photon-baryon\nfluid between radiation pressure and gravity. The BAO measurements\nthat have been considered in our numerical analyses include the\nresults from the WiggleZ~\\cite{Blake:2011en}, the\n6dF~\\cite{Beutler:2011hx} and the SDSS II surveys~\\cite{Percival:2009xn,Padmanabhan:2012hf}, at redshifts of $z=0.44,\n0.6, 0.73$, $z=0.106$ and $z=0.35$, respectively. We also include in our analyses as well the Data Release 11\n(DR11) BAO signal of the BOSS experiment~\\cite{Dawson:2012va}, which provides the most precise distant\nconstraints~\\cite{Anderson:2013zyy} measuring both the Hubble parameter and the angular diameter distance at an effective redshift of $0.57$. \nFigure \\ref{fig:dvboss} illustrates the spherically averaged BAO distance, $D_V(z) \\propto D^2_A(z)\/H(z)$, as a function of the axion mass, at a redshift of $z=0.57$, as well as the measurement from the BOSS experiment with $95\\%$~CL error bars~\\cite{Anderson:2013zyy}. Notice that, from background measurements only, there exists a strong degeneracy between the cold dark matter mass-energy density and the axion one. The solid black line in Fig.~ \\ref{fig:dvboss} shows the spherically averaged BAO distance if all the cosmological parameters are fixed, including $\\omega_c$. The spherically averaged BAO distance deviates strongly from the $\\Lambda$CDM prediction. However, if $\\omega_c$ is varied while $m_a$ is changed (in order to keep the total matter mass-energy density constant, see the dotted blue line in Fig.~\\ref{fig:dvboss}), the spherically averaged BAO distance approaches to its expected value in a $\\Lambda$CDM cosmology.\n\n\\begin{figure*}[!t]\n\\includegraphics[width=10.cm]{dv_ma.pdf}\n \\caption{The solid black line depicts the spherically averaged BAO distance $D_V(z)$, as a function of the axion mass, at a redshift of $z=0.57$, after keeping fixed all the remaining cosmological parameters, the cold dark matter included. The dashed blue line depicts the equivalent but keeping fixed the total matter mass-energy density (and consequently changing the cold dark matter $\\omega_c$). The bands show the measurement from the BOSS experiment (DR11) with its associated $95\\%$~CL error.}\n\\label{fig:dvboss}\n\\end{figure*}\n\n\n\\subsection{Compatibility of data}\nIt has been pointed out (see Sec. 5.5 of Ref.~\\cite{Ade:2013zuv} and also Refs.~\\cite{Giusarma:2014zza,Leistedt:2014sia}) that the value of $\\sigma_8$ reported by cluster measurements and the value estimated from Planck CMB measurements show a tension at the $\\sim 2\\sigma$ level. These discrepancies may arise due to the lack of a full understanding of the cluster mass calibrations. Although some studies in the literature, including the present one, show that in extended cosmological models with non-zero neutrino masses the discrepancies previously mentioned could be alleviated, the results from Ref.~\\cite{Leistedt:2014sia} show, using also Bayesian evidence, that a canonical $\\Lambda$CDM scenario with no massive neutrinos is preferred over its neutrino extensions by several combinations of cosmological datasets. Therefore, the results presented here and obtained when considering cluster data depend strongly on the reliability of low-redshift cluster data. If future data confirm current low-redshift cluster measurements, one could further test some of the possible beyond the $\\Lambda$CDM models using particle physics experiments. For instance, the existence of a full thermal sterile neutrino could be tested with neutrino oscillation experiments, and the active neutrino mass could also be tested by tritium experiments or, if the neutrino is a Majorana particle, by neutrinoless double beta decay searches.\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{table*}\n\\begin{center}\\footnotesize\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\n & CMB & CMB+HST & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO\\\\\n & & & & +HST & HST+CFHT & +HST+PSZ (fixed bias) & +HST+PSZ\\\\ \n\\hline\n\\hspace{1mm}\\\\\n\n$\\Omega_{\\textrm{c}}h^2$ & $0.127\\,^{+0.007}_{-0.007}$ & $0.122\\,^{+0.006}_{-0.006}$ & $0.122\\,^{+0.003}_{-0.003}$ & $0.121\\,^{+0.003}_{-0.003}$ & $0.120\\,^{+0.003}_{-0.003}$ & $0.118\\,^{+0.002}_{-0.002}$ & $0.119\\,^{+0.003}_{-0.004}$ \\\\\n\\hspace{1mm}\\\\\n\n$m_a$ [eV] & {\\rm {Unconstrained}} & $<1.31$ & $<0.89$ & $<0.91$ & $<1.29$ & $1.00\\,^{+0.50}_{-0.48}$ & $0.93\\,^{+0.70}_{-0.71}$ \\\\\n\\hspace{1mm}\\\\\n\n$\\sigma_8$ & $0.788\\,^{+0.079}_{-0.086}$ & $0.821\\,^{+0.052}_{-0.074}$ & $0.827\\,^{+0.044}_{-0.057}$ & $0.825\\,^{+0.045}_{-0.059}$ & $0.793\\,^{+0.049}_{-0.058}$ & $0.760\\,^{+0.023}_{-0.022}$ & $0.767\\,^{+0.046}_{-0.044}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\Omega_{\\textrm{m}}$ & $0.369\\,^{+0.070}_{-0.065}$ & $0.314\\,^{+0.045}_{-0.039}$ & $0.308\\,^{+0.016}_{-0.015}$ & $0.304\\,^{+0.016}_{-0.014}$ & $0.302\\,^{+0.016}_{-0.015}$ & $0.304\\,^{+0.016}_{-0.015}$ & $0.304\\,^{+0.016}_{-0.016}$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,1}$ & $<8.13$ & $<8.17$ & $<7.91$ & $<8.06$ & $<7.85$ & $<8.09$ & $<8.11$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,2}$ & $1.09\\,^{+0.42}_{-0.35}$ & $1.01\\,^{+0.43}_{-0.35}$ & $1.01\\,^{+0.40}_{-0.32}$ & $0.99\\,^{+0.42}_{-0.33}$ & $1.02\\,^{+0.43}_{-0.34}$ & $1.01\\,^{+0.42}_{-0.33}$ & $1.05\\,^{+0.43}_{-0.38}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,3}$ & $0.68\\,^{+0.39}_{-0.36}$ & $0.71\\,^{+0.39}_{-0.39}$ & $0.71\\,^{+0.39}_{-0.37}$ & $0.72\\,^{+0.39}_{-0.38}$ & $0.69\\,^{+0.39}_{-0.37}$ & $0.70\\,^{+0.40}_{-0.38}$ & $0.69\\,^{+0.40}_{-0.39}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,4}$ & $1.14\\,^{+0.24}_{-0.22}$ & $1.15\\,^{+0.24}_{-0.22}$ & $1.15\\,^{+0.23}_{-0.21}$ & $1.15\\,^{+0.23}_{-0.20}$ & $1.15\\,^{+0.23}_{-0.21}$ & $1.15\\,^{+0.23}_{-0.21}$ & $1.15\\,^{+0.22}_{-0.21}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,5}$ & $1.02\\,^{+0.11}_{-0.10}$ & $1.01\\,^{+0.11}_{-0.11}$ & $1.00\\,^{+0.11}_{-0.10}$ & $1.00\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.11}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,6}$ & $1.03\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.08}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.08}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,7}$ & $0.99\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.08}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.08}_{-0.07}$ & $0.96\\,^{+0.07}_{-0.06}$ & $0.95\\,^{+0.07}_{-0.06}$ & $0.96\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,8}$ & $0.94\\,^{+0.06}_{-0.06}$ & $0.95\\,^{+0.08}_{-0.07}$ & $0.95\\,^{+0.07}_{-0.06}$ & $0.95\\,^{+0.08}_{-0.07}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,9}$ & $0.92\\,^{+0.06}_{-0.05}$ & $0.94\\,^{+0.08}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.08}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,10}$ & $0.90\\,^{+0.06}_{-0.06}$ & $0.91\\,^{+0.08}_{-0.07}$ & $0.91\\,^{+0.07}_{-0.06}$ & $0.91\\,^{+0.08}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,11}$ & $1.25\\,^{+0.30}_{-0.28}$ & $1.24\\,^{+0.32}_{-0.31}$ & $1.23\\,^{+0.31}_{-0.31}$ & $1.24\\,^{+0.31}_{-0.31}$ & $1.22\\,^{+0.30}_{-0.31}$ & $1.22\\,^{+0.32}_{-0.28}$ & $1.23\\,^{+0.31}_{-0.30}$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,12}$ & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{$95\\%$~CL constraints on the physical cold dark matter density $\\Omega_{\\textrm{c}}h^2$, \nthe axion mass $m_a$ (in eV), the clustering parameter $\\sigma_8$, the relative matter energy density $\\Omega_{\\textrm{m}}$ and the $P_{s,j}$ parameters for the PPS nodes from the different combinations of data sets explored here in the $\\Lambda$CDM+$m_a$ model, considering the \\texttt{PCHIP}\\xspace PPS modeling.}\n\\label{tab:lcdm+ma+pchip}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table*}\n\\begin{center}\\footnotesize\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\n & CMB & CMB+HST & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO\\\\\n & & & & +HST & HST+CFHT & +HST+PSZ (fixed bias) & +HST+PSZ\\\\ \n\\hline\n\\hspace{1mm}\\\\\n\n$\\Omega_{\\textrm{c}}h^2$ & $0.124\\,^{+0.006}_{-0.005}$ & $0.124\\,^{+0.005}_{-0.005}$ & $0.122\\,^{+0.004}_{-0.004}$ & $0.121\\,^{+0.004}_{-0.004}$ & $0.120\\,^{+0.003}_{-0.003}$ & $0.119\\,^{+0.003}_{-0.003}$ & $0.120\\,^{+0.003}_{-0.003}$ \\\\\n\\hspace{1mm}\\\\ \n \n$m_a$ [eV] & $<1.83$ & $<1.56$ & $<0.84$ & $<0.83$ & $<1.16$ & $0.80\\,^{+0.53}_{-0.50}$ & $<1.26$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\sigma_8$ & $0.785\\,^{+0.064}_{-0.083}$ & $0.791\\,^{+0.057}_{-0.076}$ & $0.803\\,^{+0.041}_{-0.048}$ & $0.803\\,^{+0.041}_{-0.048}$ & $0.783\\,^{+0.047}_{-0.054}$ & $0.758\\,^{+0.028}_{-0.029}$ & $0.767\\,^{+0.045}_{-0.045}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\Omega_{\\textrm{m}}$ & $0.337\\,^{+0.048}_{-0.044}$ & $0.328\\,^{+0.041}_{-0.039}$ & $0.310\\,^{+0.025}_{-0.023}$ & $0.308\\,^{+0.024}_{-0.023}$ & $0.305\\,^{+0.025}_{-0.024}$ & $0.307\\,^{+0.027}_{-0.026}$ & $0.306\\,^{+0.027}_{-0.025}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\log[10^{10} A_s]$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.10\\,^{+0.05}_{-0.05}$ & $3.09\\,^{+0.05}_{-0.05}$ & $3.09\\,^{+0.05}_{-0.05}$ \\\\\n\\hspace{1mm}\\\\ \n \n$n_s$ & $0.961\\,^{+0.014}_{-0.015}$ & $0.963\\,^{+0.013}_{-0.014}$ & $0.968\\,^{+0.011}_{-0.011}$ & $0.969\\,^{+0.011}_{-0.011}$ & $0.971\\,^{+0.011}_{-0.011}$ & $0.973\\,^{+0.011}_{-0.011}$ & $0.972\\,^{+0.011}_{-0.011}$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{$95\\%$~CL constraints on \n$\\Omega_{\\textrm{c}}h^2$, \nthe axion mass $m_a$ (in eV), \n$\\sigma_8$, $\\Omega_{\\textrm{m}}$, ${\\rm{log}}(10^{10} A_s)$ and $n_s$\nfrom the different combinations of data sets explored here in the $\\Lambda$CDM+$m_a$ model, assuming the standard power-law PPS.}\n\\label{tab:lcdm+ma+pl}\n\\end{center}\n\\end{table*}\n\n\nTable \\ref{tab:lcdm+ma+pchip} depicts our results in the first scenario explored\nhere, in which the axion mass is a free parameter and the PPS is\ndescribed by the approach specified in Sec.~\\ref{sub:pps}. Concerning CMB measurements only, the\nbounds on the thermal axion masses are largely relaxed in the case in which the PPS\nis not described by a simple power-law, as can be noticed after comparing the results depicted in Tab.~\\ref{tab:lcdm+ma+pchip} with those shown in Tab.~\\ref{tab:lcdm+ma+pl}. This can be understood in terms of Fig.~\\ref{fig:plottt}, which\nillustrates the degeneracies in the temperature anisotropies between the\nthermal axion mass and the \\texttt{PCHIP}\\xspace PPS.\nFigure~\\ref{fig:plottt} shows the temperature anisotropies for a $\\Lambda$CDM model and a power-law PPS (solid red line), for a $2$~eV thermal axion\nmass and a power-law PPS (dashed blue line) and for a $\\Lambda$CDM model\nbut the PPS described by the \\texttt{PCHIP}\\xspace model explored here (dotted black line), with values\nfor the $P_{s,j}$ chosen to match the non-zero thermal axion mass curve, accordingly to their current allowed regions \n(see Tab.~\\ref{tab:lcdm+ma+pchip}). More concretely, we have used the following values for the PPS parameters: \n$P_{s,1}=1.15$, $P_{s,2}=1.073$, $P_{s,3}=1.058$, $P_{s,4}=1.03$, $P_{s,5}=0.99$, $P_{s,6}=0.97$,\n$P_{s,7}=0.966$, $P_{s,8}=0.932$, $P_{s,9}=0.91$, $P_{s,10}=0.86$, $P_{s,11}=0.84$ and $P_{s,12}=0.77$.\nNotice that the case of a $2$~eV thermal axion can be easily mimicked\nby a simple $\\Lambda$CDM model if the assumptions concerning the PPS \nshape are relaxed. We also add in this figure the measurements of the photon temperature anisotropies from the Planck 2013 data release~\\cite{Ade:2013zuv}. \n\n\n\n\\begin{figure*}[!t]\n\\includegraphics[width=11.1cm]{cl_compare-2.pdf}\n \\caption{Temperature anisotropies for the pure $\\Lambda$CDM model and a power-law PPS (solid red line), for a $2$~eV thermal axion\nmass and a power-law PPS (dashed blue line) and for the standard $\\Lambda$CDM model but the PPS described by the \\texttt{PCHIP}\\xspace model (dotted black line). The data points and the error bars in the left panel show the measurements of the photon temperature anisotropies arising from the Planck 2013 data release~\\cite{Ade:2013zuv}.}\n\\label{fig:plottt}\n\\end{figure*}\n\n\n\nThe addition to the CMB data of the HST prior on the Hubble constant\nprovides a $95\\%$~CL upper limit on the thermal axion mass of\n$1.31$~eV~\\footnote{There exists a very large degeneracy between $H_0$\n and the neutrino masses when restricting the numerical analyses to\n CMB measurements. The addition of the HST prior on the Hubble\n constant helps enormously in breaking this degeneracy,\n see~\\cite{Giusarma:2012ph}.}, while the further addition of the BAO\nmeasurements brings this constraint down to $0.91$~eV, as these last\ndata sets are directly sensitive to the free-streaming nature of the\nthermal axion. Notice that these two $95\\%$~CL upper bounds are very\nsimilar to the ones obtained when considering the standard power-law\npower spectrum, which are $1.56$~eV and $0.83$~eV for the CMB+HST and\nCMB+HST+BAO data combinations, respectively. \n\n\nInterestingly, when adding the CFHT bounds on the $\\sigma_8$-$\\Omega_m$ relationship, the bounds on the thermal axion mass become weaker. \nThe reason for that is due to the lower $\\sigma_8$ values preferred by weak lensing measurements, \nvalues that can be achieved by allowing for higher axion masses. \nThe larger the axion mass, the larger is the reduction of the matter power spectrum at small (i.e. cluster) scales, \nleading consequently to a smaller value of the clustering parameter $\\sigma_8$. \n\nIf we instead consider now the PSZ data set with fixed cluster mass\nbias, together with the CMB, BAO and HST measurements, a non-zero\nvalue of the thermal axion mass of $\\sim 1$~eV ($\\sim 0.80$~eV) is\nfavoured at $\\sim4\\sigma$ ($\\sim3\\sigma$) level, when considering the\n\\texttt{PCHIP}\\xspace (standard power-law) PPS approach~\\footnote{A similar effect when considering PSZ data for constraining either thermal axion or neutrino masses has also been found in Refs.~\\cite{Hamann:2013iba,Wyman:2013lza,Giusarma:2014zza,Dvorkin:2014lea,Archidiacono:2014apa}.}. However, these results must be regarded as an illustration of what could be achieved with future cluster mass calibrations, as the Planck collaboration has recently shown in their analyses of the 2015 Planck cluster catalogue~\\cite{Ade:2015fva}. When more realistic approaches for the cluster mass bias are used, the errors on the so-called cluster normalization condition are larger, and, consequently, the preference for a non-zero axion mass of $1$~eV is only mild in the \\texttt{PCHIP}\\xspace PPS case, while in the case of a standard power-law PPS such an evidence completely disappears.\n\n\nFigure~\\ref{fig:ma_pchip} (left panel) shows the $68\\%$ and $95\\%$~CL\nallowed regions in the ($m_a$, $\\Omega_c h^2$) plane for some of the\npossible data combinations explored in this study, and assuming the \\texttt{PCHIP}\\xspace PPS modeling. Notice that, when\nadding BAO measurements, lower values of the physical cold dark matter density are\npreferred. This is due to the fact that large scale structure allows\nfor lower axion masses than CMB data alone. \nThe lower is the thermal axion mass,\nthe lower is the amount of hot dark matter and consequently the lower should be\nthe cold dark matter component. \nThis effect is clear \nfrom the results shown in Tab.~\\ref{tab:lcdm+ma+pchip} and Tab.~\\ref{tab:lcdm+ma+pl}, where the values of the\nphysical cold dark matter density $\\Omega_c h^2$ and of the relative\ncurrent matter density $\\Omega_m$ arising from our numerical fits\nare shown, for the different data combinations considered here. \n\nThe right panel of Fig.~\\ref{fig:ma_pchip} shows the $68\\%$ and $95\\%$~CL\nallowed regions in the ($m_a$, $\\sigma_8$) plane in the \\texttt{PCHIP}\\xspace PPS scenario. The lower values of\nthe $\\sigma_8$ clustering parameter preferred by PSZ data (see the results shown in Tab.~\\ref{tab:lcdm+ma+pchip} and Tab.~\\ref{tab:lcdm+ma+pl}) are translated into a preference for non-zero thermal axion masses. Larger values of $m_a$ will enhance the matter power spectrum suppression at scales below the axion free-streaming scale, leading to smaller values of the $\\sigma_8$ clustering parameter, as preferred by PSZ measurements. The evidence for non-zero axion masses is more significant when fixing the cluster mass bias in the PSZ data analyses. \n\nFigure \\ref{fig:ma_pl} shows the equivalent to Fig.~\\ref{fig:ma_pchip} but for a standard power-law PPS. Notice that, except for the case in which CMB measurements are considered alone, the thermal axion mass constraints do not change significantly, if they are compared to the \\texttt{PCHIP}\\xspace PPS modeling. \nThis fact clearly states the robustness of the cosmological bounds on thermal axion masses and it is applicable to the remaining cosmological parameters, see Tabs.~\\ref{tab:lcdm+ma+pchip} and \\ref{tab:lcdm+ma+pl}. Note that, for the standard case of a power-law PPS, the preference for non-zero axion masses appears only when considering the (unrealistic) PSZ analysis with a fixed cluster mass bias. When more realistic PSZ measurements of the cluster normalization condition are exploited, there is no preference for a non-zero thermal axion mass. \n\n\n\\begin{figure*}[!t]\n\\begin{tabular}{c c}\n\\includegraphics[width=8.3cm]{lcdm_ma_pchip_omegac-ma.pdf}&\\includegraphics[width=8.3cm]{lcdm_ma_pchip_sigma8-ma.pdf}\\\\\n\\end{tabular}\n \\caption{The left panel depicts the $68\\%$ and $95\\%$~CL allowed\n regions in the ($m_a$, $\\Omega_c h^2$) plane for different possible\n data combinations, when a \\texttt{PCHIP}\\xspace{} PPS is assumed. The right panel shows the equivalent but in the\n ($m_a$, $\\sigma_8$) plane.}\n\\label{fig:ma_pchip}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\begin{tabular}{c c}\n\\includegraphics[width=8.3cm]{standard_ma_omegac-ma.pdf}&\\includegraphics[width=8.3cm]{standard_ma_sigma8-ma.pdf}\\\\\n\\end{tabular}\n \\caption{The left panel depicts the $68\\%$ and $95\\%$~CL allowed\n regions in the ($m_a$, $\\Omega_c h^2$) plane for different possible\n data combinations, when a power-law PPS is assumed. The right panel shows the equivalent but in the\n ($m_a$, $\\sigma_8$) plane.}\n\\label{fig:ma_pl}\n\\end{figure*}\n\n\\begin{figure}[!t]\n\\includegraphics[width=9cm]{ma_mnu.pdf}\n\\caption{$68\\%$ and $95\\%$~CL allowed\n regions in the ($\\sum m_\\nu$, $m_a$) plane, both in eV, for three different possible\n data combinations, when a \\texttt{PCHIP}\\xspace{} PPS is assumed.}\n\\label{fig:mamnu}\n\\end{figure}\nThe last scenario we explore here is a $\\Lambda$CDM+$m_a$+$\\sum m_\\nu$ universe, in which we consider two coexisting hot dark matter species: thermal axions and three active (massive) neutrinos.\nTable~\\ref{tab:lcdm+ma+mnu+pchip} illustrates the equivalent of Tab.~\\ref{tab:lcdm+ma+pchip} but including the active neutrino masses in the MCMC parameters. We do not perform here the analysis for the hot mixed dark matter model with the standard power-law matter power spectrum, as it was already presented previously in Ref.~\\cite{Giusarma:2014zza}. If we compare to the standard power-law case, we find that the bounds on the axion and neutrino masses presented here are very similar. Furthermore, no evidence for neutrino masses nor for a non-zero axion mass appears in this mixed hot dark matter scenario (except for the axion case and only if considering PSZ clusters with the bias fixed). The reason for that is due to the strong degeneracy between $m_a$ and $\\sum m_\\nu$, see Fig.~\\ref{fig:mamnu}, where one can notice that that these two parameters are negatively correlated: an increase in the axion mass will increase the amount of the hot dark matter component. In order to compensate the changes in both the CMB temperature anisotropies (via the early ISW effect) and in the power spectrum (via the suppression at small scales of galaxy clustering), the contribution to the hot dark matter from the neutrinos should be reduced. We have shown in Fig.~\\ref{fig:mamnu} three possible data combinations. Notice that for the case in which PSZ cluster measurements (with the bias fixed) are included the strong degeneracy between $m_a$ and $\\sum m_\\nu$ is partially broken, due to the smaller value of $\\sigma_8$ preferred by the former data set. However, these results strongly rely on the numerical results concerning the cluster mass bias and therefore the evidence for $m_a\\neq 0$ should be regarded as what could be obtained in if these measurements are further supported by independent data from future cluster surveys.\n\n\n\n\\begin{table*}\n\\begin{center}\\footnotesize\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\n & CMB & CMB+HST & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO & CMB+BAO \\\\\n & & & & +HST & +HST+CFHT & +HST+PSZ(fixed bias) & +HST+PSZ \\\\ \n\\hline\n\\hspace{1mm}\\\\\n\n$\\Omega_{\\textrm{c}}h^2$ & $0.130\\,^{+0.008}_{-0.007}$ & $0.125\\,^{+0.006}_{-0.007}$ & $0.121\\,^{+0.003}_{-0.003}$ & $0.121\\,^{+0.003}_{-0.003}$ & $0.119\\,^{+0.003}_{-0.003}$ & $0.118\\,^{+0.003}_{-0.003}$ & $0.118\\,^{+0.003}_{-0.003}$ \\\\\n\\hspace{1mm}\\\\ \n \n$m_a$ [eV] & $<2.48$ & $<1.64$ & $<0.81$ & $<0.86$ & $<1.23$ & $0.81\\,^{+0.59}_{-0.69}$ & $<1.46$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\sum m_\\nu$ [eV] & $<2.11$ & $<0.43$ & $<0.22$ & $<0.21$ & $<0.27$ & $<0.32$ & $<0.35$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\sigma_8$ & $0.700\\,^{+0.172}_{-0.202}$ & $0.803\\,^{+0.082}_{-0.091}$ & $0.833\\,^{+0.055}_{-0.058}$ & $0.834\\,^{+0.058}_{-0.064}$ & $0.787\\,^{+0.052}_{-0.055}$ & $0.766\\,^{+0.043}_{-0.044}$ & $0.757\\,^{+0.023}_{-0.022}$ \\\\\n\\hspace{1mm}\\\\ \n \n$\\Omega_{\\textrm{m}}$ & $0.486\\,^{+0.277}_{-0.193}$ & $0.356\\,^{+0.064}_{-0.062}$ & $0.309\\,^{+0.016}_{-0.015}$ & $0.308\\,^{+0.016}_{-0.015}$ & $0.306\\,^{+0.015}_{-0.015}$ & $0.308\\,^{+0.016}_{-0.016}$ & $0.308\\,^{+0.017}_{-0.016}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,1}$ & $<8.01$ & $<8.13$ & $<7.00$ & $<8.17$ & $<7.59$ & $<8.29$ & $<8.18$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,2}$ & $1.17\\,^{+0.42}_{-0.38}$ & $1.09\\,^{+0.41}_{-0.37}$ & $1.03\\,^{+0.40}_{-0.35}$ & $1.02\\,^{+0.39}_{-0.34}$ & $1.02\\,^{+0.40}_{-0.32}$ & $1.03\\,^{+0.36}_{-0.34}$ & $1.05\\,^{+0.40}_{-0.36}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,3}$ & $0.66\\,^{+0.37}_{-0.35}$ & $0.69\\,^{+0.38}_{-0.37}$ & $0.70\\,^{+0.38}_{-0.38}$ & $0.72\\,^{+0.38}_{-0.37}$ & $0.68\\,^{+0.37}_{-0.33}$ & $0.71\\,^{+0.40}_{-0.39}$ & $0.69\\,^{+0.39}_{-0.37}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,4}$ & $1.17\\,^{+0.23}_{-0.23}$ & $1.15\\,^{+0.23}_{-0.22}$ & $1.15\\,^{+0.22}_{-0.21}$ & $1.15\\,^{+0.21}_{-0.21}$ & $1.15\\,^{+0.20}_{-0.19}$ & $1.14\\,^{+0.21}_{-0.20}$ & $1.16\\,^{+0.22}_{-0.21}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,5}$ & $1.05\\,^{+0.15}_{-0.14}$ & $1.01\\,^{+0.11}_{-0.10}$ & $1.00\\,^{+0.11}_{-0.10}$ & $1.00\\,^{+0.11}_{-0.10}$ & $0.98\\,^{+0.11}_{-0.10}$ & $0.99\\,^{+0.11}_{-0.10}$ & $0.98\\,^{+0.11}_{-0.10}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,6}$ & $1.04\\,^{+0.09}_{-0.08}$ & $1.01\\,^{+0.08}_{-0.07}$ & $1.00\\,^{+0.07}_{-0.07}$ & $1.00\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,7}$ & $0.99\\,^{+0.06}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.06}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.98\\,^{+0.07}_{-0.07}$ & $0.95\\,^{+0.07}_{-0.06}$ & $0.95\\,^{+0.06}_{-0.06}$ & $0.95\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,8}$ & $0.93\\,^{+0.06}_{-0.05}$ & $0.94\\,^{+0.06}_{-0.06}$ & $0.95\\,^{+0.07}_{-0.07}$ & $0.95\\,^{+0.07}_{-0.07}$ & $0.93\\,^{+0.07}_{-0.05}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,9}$ & $0.91\\,^{+0.06}_{-0.05}$ & $0.93\\,^{+0.06}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.94\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ & $0.93\\,^{+0.06}_{-0.06}$ & $0.93\\,^{+0.07}_{-0.06}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,10}$ & $0.90\\,^{+0.06}_{-0.06}$ & $0.90\\,^{+0.07}_{-0.06}$ & $0.91\\,^{+0.07}_{-0.07}$ & $0.91\\,^{+0.08}_{-0.07}$ & $0.88\\,^{+0.07}_{-0.06}$ & $0.89\\,^{+0.07}_{-0.07}$ & $0.90\\,^{+0.07}_{-0.07}$ \\\\\n\\hspace{1mm}\\\\ \n \n$P_{s,11}$ & $2.18\\,^{+0.85}_{-0.77}$ & $2.07\\,^{+0.81}_{-0.80}$ & $2.12\\,^{+0.90}_{-0.86}$ & $2.15\\,^{+0.95}_{-0.94}$ & $1.64\\,^{+0.79}_{-0.75}$ & $1.83\\,^{+0.87}_{-0.86}$ & $1.84\\,^{+0.86}_{-0.87}$ \\\\\n\\hspace{1mm}\\\\ \n\n$P_{s,12}$ & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} & {\\rm {Unconstrained}} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\caption{$95\\%$~CL constraints on the physical cold dark matter density $\\Omega_{\\textrm{c}}h^2$, \nthe axion mass $m_a$, the sum of the active neutrino masses $\\sum m_\\nu$ (both in eV), the clustering parameter $\\sigma_8$, the relative matter energy density $\\Omega_{\\textrm{m}}$ and the $P_{s,j}$ parameters for the PPS nodes from the different combinations of data sets explored here in the $\\Lambda$CDM+$m_a$+$\\sum m_\\nu$ model, considering the \\texttt{PCHIP}\\xspace PPS modeling.}\n\\label{tab:lcdm+ma+mnu+pchip}\n\\end{center}\n\\end{table*}\n\n\nBesides the results concerning the thermal axion mass and the standard $\\Lambda$CDM parameters,\nwe also obtain constraints on the form of the PPS when modeled accordingly to the \\texttt{PCHIP}\\xspace scenario.\nThe 95\\% CL limits for the $P_{s,j}$ parameters are shown in Tab.~\\ref{tab:lcdm+ma+pchip}, \nwhile an example of the reconstructed PPS is given in Fig.~\\ref{fig:outputPPS}, \nwhere we show the 68\\%, 95\\% and 99\\% CL allowed regions arising from a fit to CMB data of the \\texttt{PCHIP}\\xspace PPS scale dependence, \n in the context of a $\\Lambda$CDM+$m_a$ model.\nWe do not show the corresponding figures obtained from all the other data combinations \nsince they are equivalent to Fig.~\\ref{fig:outputPPS}, as one can infer from the very small \ndifferences in the $95\\%$~CL allowed ranges for the $P_{s,j}$ parameters arising from different data sets, see Tab.~\\ref{tab:lcdm+ma+pchip}.\nNote that both $P_{s,1}$ and $P_{s,12}$ are poorly constrained at this confidence level:\nthe reason for that is the absence of measurements at their corresponding wavenumbers.\nAll the remaining $P_{s,j}$, with $j=2,\\ldots,11$ are\nwell-constrained. In particular, in the range between $k_5$\nand $k_{10}$ (see Eq.~(\\ref{eq:nodesspacing})), the $P_{s,j}$ are determined with few percent accuracy. Indeed, in the range covered between these nodes, the PPS does not present features and can be perfectly described by a power-law parametrization.\nAmong the interesting features outside the former range, we can notice in Fig.~\\ref{fig:outputPPS}\na significant dip at wavenumbers around $k=0.002\\, \\text{Mpc}^{-1}$, that comes from the dip at $\\ell=20-30$ in the CMB temperature power spectrum and a small bump around $k=0.0035\\, \\text{Mpc}^{-1}$, corresponding to the increase at $\\ell\\simeq40$. These features have been obtained in previous works \\cite{Hunt:2013bha,Hazra:2014jwa,Gariazzo:2014dla} using different methods and data sets. In addition, we obtain an increase of power at $k\\simeq0.2\\, \\text{Mpc}^{-1}$, necessary to compensate the effects of the thermal axion mass in both the temperature anisotropies and the large scale structure of the universe.\n\n\n\\begin{figure*}[!t]\n\\includegraphics[width=12cm]{lcdm_ma_pchip_PPSbands.pdf}\n\\caption{$68\\%$, $95\\%$ and $99\\%$~CL allowed regions for the \\texttt{PCHIP}\\xspace PPS scale dependence in the $\\Lambda$CDM+$m_a$ model, using CMB data only.\nThe bands are obtained with a marginalization of the posterior distribution for each different value of the wavenumber $k$ in a fine grid.\nThe black line represents the peak of the posterior distribution at each value of $k$.}\n\\label{fig:outputPPS}\n\\end{figure*}\n\n\\section{Conclusions}\n\\label{sec:concl}\nAxions provide the most elegant scenario to solve the strong CP problem, and may be produced in the early universe via both thermal and non-thermal processes. While non thermal axions are highly promising cold dark matter candidates, their thermal companions will contribute to the hot dark matter component of the universe, together with the (light) three active neutrinos of the standard model of elementary particles. Therefore, the cosmological consequences of light massive thermal axions are very much alike those associated with neutrinos, as axions also have a free-streaming nature, suppressing structure formation at small scales. Furthermore, these light thermal axions will also contribute to the dark radiation background, leading to deviations of the relativistic degrees of freedom $N_{\\textrm{eff}}$ from its canonically expected value of $N_{\\textrm{eff}}=3.046$. Based on these signatures, several studies have been carried out in the literature deriving bounds on the thermal axion mass~\\cite{Melchiorri:2007cd,Hannestad:2007dd,Hannestad:2008js,Hannestad:2010yi,Archidiacono:2013cha,Giusarma:2014zza}. \n\nNevertheless, these previous constraints assumed that the underlying\nprimordial perturbation power spectrum follows the usual power-law\ndescription governed, in its most economical form, by an amplitude\nand a scalar spectral index. Here we have relaxed such an assumption, \nin order to test the robustness of the cosmological axion mass\nbounds. Using an alternative, non-parametric description of the\nprimordial power spectrum of the scalar perturbations,\nnamed \\texttt{PCHIP}\\xspace and introduced in Ref.~\\cite{Gariazzo:2014dla}, we have shown that, in practice, when combining CMB measurements with low redshift cosmological probes, the axion mass constraints are only mildly sensitive to the primordial\npower spectrum choice and therefore are not strongly dependent on the\nparticular details of the underlying inflationary model. \nThese results agree with the findings of Ref.~\\cite{dePutter:2014hza} for the neutrino mass case. The tightest\n bound we find in the \\texttt{PCHIP}\\xspace primordial power spectrum approach is obtained when considering BAO measurements together with CMB data, with $m_a<\n 0.89$~eV at $95\\%$~CL. In the standard power-law primordial power\n spectrum modeling, the tightest bound is $m_a<\n 0.83$~eV at $95\\%$~CL, obtained when combining BAO, CMB and HST measurements. Notice that these\n bounds are very similar, confirming the robustness of the cosmological\n axion mass measurements versus the primordial power spectrum\n modeling. \n\nInterestingly, both weak lensing measurements and cluster number\ncounts weaken the thermal axion mass bounds. The reason for that is\ndue to the lower $\\sigma_8$ values preferred by \nthese measurements, which could be generated by a larger axion\nmass. More concretely, Planck cluster measurements provide a\nmeasurement of the so-called cluster normalization condition,\nwhich establishes a relationship between the clustering parameter\n$\\sigma_8$ and the current matter mass-energy density $\\Omega_m$. \nHowever, the errors on this relationship depend crucially on the\nknowledge of the cluster mass bias. A conservative approach for the\ncluster mass calibration results in a mild (zero) evidence for a\nnon-zero axion mass of $1$~eV in the \\texttt{PCHIP}\\xspace (power-law) PPS case.\nWe also illustrate a case in which the cluster mass bias is fixed, to forecast the expected results from future cosmological\nmeasurements. In this case, a non-zero\nvalue of the thermal axion mass of $\\sim 1$~eV ($\\sim 0.80$~eV) is\nfavoured at $\\sim4\\sigma$ ($\\sim3\\sigma$) level, when considering the\n\\texttt{PCHIP}\\xspace (power-law) PPS approach. \nWhen considering additional hot relics in our analyses, as the sum of the three active neutrino masses, the evidence for \na $\\sim 1$~eV thermal axion mass disappears almost completely. Furthermore, these values of axion masses correspond to an axion coupling constant \n$f_a= 6\\times 10^6$~GeV, which seems to be in tension with the limits extracted from the neutrino signal duration from SN 1987A~\\cite{Raffelt:2006cw,Raffelt:1990yz} (albeit these limits depend strongly on the precise axion emission rate and still remain rough estimates). Precise cluster mass calibration measurements are therefore mandatory to assess whether there exists a cosmological indication for non-zero axion masses, as the cluster mass bias is highly correlated with the clustering parameter $\\sigma_8$, which, in turn, is highly affected by the free-streaming nature of a hot dark matter component, as thermal axions.\n\n\\section{Acknowledgments}\nOM is supported by PROMETEO II\/2014\/050, by the Spanish Grant FPA2011--29678 of the MINECO and by PITN-GA-2011-289442-INVISIBLES. This work has been done within the Labex ILP (reference ANR-10-LABX-63) part of the Idex SUPER, and received financial state aid managed by the Agence Nationale de la Recherche, as part of the programme Investissements d'avenir under the reference ANR-11-IDEX-0004-02. EDV acknowledges the support of the European Research Council via the Grant number 267117 (DARK, P.I. Joseph Silk).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSince the seminal paper \\cite{GPV}, the theory of hydrodynamic limit of interacting particle systems has evolved into a powerful tool in the study of non-equilibrium properties of statistical systems of many components (see the book \\cite{KL} for a comprehensive exposition). Recently, and due to the infuence of physical and mathematical works about random walks in random environment, an increasing attention has been posed into particle systems evolving in random environments. Despite the early works \\cite{Fri}, \\cite{Qua}, \\cite{Kou}, we mention \\cite{FM}, \\cite{Qua} \\cite{Nag}, \\cite{JL}, \\cite{Fag}, \\cite{GJ}, \\cite{FJL}, \\cite{FL}, \\cite{Fag2}, \\cite{GJ2}. In \\cite{GJ}, \\cite{JL} the {\\em corrected empirical density} was introduced, which is nothing but a microscopic version of the compensated compactness lemma of Tartar \\cite{Tar}. Roughly speaking, when the inhomogeneous environment (random or not) has a divergence form and has a $\\Gamma$-limit, space homogenization of the environment and time homogenization of the interaction decouples, and the standard tools from the theory of hydrodynamic limit can be used to obtain the asymptotic behavior of the density of particles in a family of models, including the exclusion process and the zero-range process. \n\nIn this review, we give an unified approach to this problem, recovering previous results in \\cite{Nag}, \\cite{JL}, \\cite{Fag}, \\cite{FJL}, \\cite{Fag2}, in a simple way. In order to concentrate our efforts in the influence of the inhomogeneous environment on the asymptotics of the density of particles, we consider the simplest model of interacting particle systems, which is the symmetric exclusion process $\\eta_t^n$ in an unoriented graph. In this process, particles perform symmetric random walks on a graph $\\{X_n\\}_n$ with some rates $\\omega^n=\\{\\omega_{x,y}^n; x,y \\in X_n\\}$, conditioned to have at most one particle per site. We think of $\\{X_n\\}_n$ as a sequence of graphs embedding in some metric space $X$, and we are interested in the evolution of the measure $\\pi_t^n(dx)$ in $X$, obtained by giving a mass $a_n^{-1}$ to each particle. \n\nThis article is organized as follows. In Section \\ref{s1} we give precise definitions of the exclusion process, the inhomogeneous environment and we state our main result. We also define what we mean by an approximation $\\{X_n\\}_n$ of $X$ and by $\\Gamma$-convegence of the environment. In Section \\ref{s2} we introduce the corrected empirical density and we prove our main theorem. In Section \\ref{s3} we introduce the concept of energy solutions of the hydrodynamic equation, we prove uniqueness of such solutions and we obtain a substantial improvement of the main Theorem. The material of this Section is new and it gives a better understanding of the relation between $\\Gamma$-convergence of the environment and hydrodynamic limit of the particle system. In Section \\ref{s4} we discuss how to reobtain previous results in the literature relying in our main Theorem.\n\n\n\n\\section{Definitions and results}\n\\label{s1}\nIn this section we define the exclusion process in inhomogeneous environment and we recall some notions of $\\Gamma$-convergence that will be necessary in order to obtain the hydrodynamic limit of this process.\n\n\\subsection{Partitions of the unity and approximating sequences}\n\\label{s1.1}\nIn this section we fix some notation and we define some objects which will be useful in the sequel. \nLet $(X,\\mc B)$ be a Polish space. We assume that $X$ is $\\sigma$-compact.\nWe say that a sequence of functions $\\{\\mc U_i; i \\in I\\}$ is a {\\em partition of the unity} if:\n\\begin{description}\n \\item[i)] for any $i \\in I$, $\\mc U_i:X \\to [0,1]$ is a continuous function, \n \\item[ii)] for any $x \\in X$, $\\sum_{i \\in I} \\mc U_i(x)=1$,\n \\item[iii)] for any $x \\in X$, the set $\\{i \\in I; \\mc U_i(x)>0\\}$ is finite.\n\\end{description}\n\nWe say that the partition of the unity $\\{\\mc U_i;i \\in I\\}$ is {\\em regular} if $\\text{supp } \\mc U_i$ is compact for any $i \\in I$, and additionally $\\mc U_i(X) = [0,1]$. \nWe denote by $\\mc M_+(X)$ the set of Radon, positive measures in $X$. The symbol $\\{x_n\\}_n$ will denote a sequence of elements $x_n$ in some space, indexed by the set $\\bb N$ of positive integers. \n\n\nLet $\\{\\mc U_i\\}_i$ be a regular partition of the unity. We say that a sequence $\\{x_i ; i \\in I\\}$ in $X$ is a {\\em representative} of $\\{\\mc U_i\\}_i$ if $\\mc U_i(x_i)=1$ for any $i \\in I$. Notice that we have $x_i \\neq x_j$ for $i \\neq j$. \n\nLet $\\{\\mc U_i^n; i \\in I_n\\}_n$ be a sequence of regular partitions of the unity. We say that a measure $\\mu \\in \\mc M_+(X)$ is the scaling limit of the sequence $\\{\\mc U_i^n\\}_n$ if there exists a sequence $\\{a_n\\}_n$ of positive numbers such that for any sequence $\\{x_i^n;i \\in I_n\\}$ of representatives of $\\{\\mc U_i^n\\}_n$ we have\n\\[\n\\lim_{n \\to \\infty} \\frac{1}{a_n} \\sum_{i \\in I_n} \\delta_{x_i^n} = \\mu\n\\]\nwith respect to the vague topology, where $\\delta_x$ is the Dirac mass at $x \\in X$. We call $\\{a_n\\}_n$ the {\\em scaling} sequence. \n\nFrom now on, we fix a sequence $\\{\\mc U_i^n\\}_n$ of regular partitions of the unity with scaling limit $\\mu$, scaling sequence $\\{a_n\\}_n$ and we assume that $\\mu(A)>0$ for any non-empty, open set $A \\subseteq X$. \nFix a sequence $\\{x_i^n;i \\in I_n\\}$ of representatives of $\\{\\mc U_i^n\\}_n$. Define $X_n = \\{x_i^n; i \\in I_n\\}$. \nSince $\\{\\mc U_i^n\\}$ is a partition of the unity, the induced topology in $X_n$ coincides with the discrete topology. For $x=x_i^n$, we will denote $\\mc U_x^n= \\mc U_i^n$. Define\n\\[\n\\mu_n(dx) = \\frac{1}{a_n} \\sum_{x \\in X_n} \\delta_x(dx).\n\\]\n\nBy definition, $\\mu_n \\to \\mu$ in the vague topology. We denote by $\\mc L^2(\\mu_n)$ the Hilbert space of functions $f: X_n \\to \\bb R$ such that $\\sum_{x \\in X_n} f(x)^2 <+\\infty$, equipped with the inner product\n\\[\n\\< f, g\\>_n = \\frac{1}{a_n} \\sum_{x \\in X_n} f(x)g(x).\n\\]\n\nWe define $\\mc L^2(\\mu)$, $\\mc L^1(X_n)$ and $\\mc L^1(\\mu)$ in the analogous way and we denote $\\ = \\int fg d\\mu$.\nWe denote by $\\mc C_c(X)$ the set of continuous functions $f: X \\to \\bb R$ with compact support. In the same spirit, we denote by $\\mc C_c(X_n)$ the set of functions $f: X_n \\to \\bb R$ with finite support. We define the projection $S_n: \\mc C_c(X) \\to \\mc C_c(X_n)$ by taking\n\\[\n\\big( S_n G\\big)(x) = a_n \\int G \\mc U_x^n d\\mu.\n\\]\n\nThis operator, under suitable conditions, can be extended to a bounded operator from $\\mc L^2(X)$ to $\\mc L^2(X_n)$. Notice that $\\int S_n G d\\mu_n = \\int G d\\mu$. Therefore $S_n$ is continuous from $\\mc L^1(\\mu)$ to $\\mc L^1(X_n)$.\n\n\\subsection{$\\Gamma$-convergence}\n\n\nDefine $\\bar{ \\bb R} = [-\\infty,+\\infty]$. Let $(Y,\\mc F)$ be a topological space, and let $F_n,F: Y \\to \\bar{ \\bb R}$. We say that $F_n$ is $\\Gamma$-convergent to $F$ if:\n\\begin{description}\n \\item[i)] For any sequence $\\{y_n\\}_n$ in $Y$ converging to $y \\in Y$,\n \\[\n F(y) \\leq \\liminf_{n \\to \\infty} F_n(y_n).\n \\]\n \\item[ii)] For any $y \\in Y$ there exists a sequence $\\{y_n\\}_n$ converging to $y$ such that\n \\[\n \\limsup_{n \\to \\infty} F_n(y_n) \\leq F(y).\n \\]\n\\end{description}\n\nAn important property of $\\Gamma$-convergence is that it implies {\\em convergence of minimizers} in the following sense:\n\n\\begin{proposition}\n\\label{p1}\nLet $F_n, F: Y \\to \\bar{\\bb R}$ be such that $F_n$ is $\\Gamma$-convergent to $F$. Assume that there exists a relatively compact set $K \\subseteq Y$ such that for any $n$,\n\\[\n\\inf_{y \\in Y} F_n(y) =\\inf_{y \\in K} F_n(y).\n\\]\n\nThen,\n\\[\n\\lim_{n \\to \\infty} \\inf_{y \\in K} F_n(y) = \\min_{y \\in Y} F(y).\n\\]\nMoreover, if $\\{y_n\\}_n$ is a sequence in $K$ such that $\\lim_n (F_n(y_n)-\\inf_K F_n)=0$, then any limit point $y$ of $\\{y_n\\}_n$ satisfies $F(y) = \\min_Y F$.\n\\end{proposition}\n\nA useful property that follows easily from the definition is the stability of $\\Gamma$-convergence under continuous perturbations:\n\n\\begin{proposition}\n\\label{p2}\nLet $F_n,F: Y \\to \\bar{\\bb R}$ be such that $F_n$ is $\\Gamma$-convergent to $F$. Let $G_n: Y \\to \\bb R$ be such that $G_n$ converges uniformly to a continuous limit $G$. Then, $F_n+G_n$ is $\\Gamma$-convergent to $F+G$.\n\\end{proposition}\n\n\\subsection{The exclusion process in inhomogeneous environment}\n\nIn this section we define the exclusion process in inhomogeneous environment as a system of particles evolving in the set $X_n$. \nLet $\\omega^n = \\{\\omega_{x,y}^n;x,y \\in X_n\\}$ be a sequence of non-negative numbers such that $\\omega^n_{x,x}=0$ and $\\omega^n_{x,y}=\\omega^n_{y,x}$ for any $x,y \\in X_n$. We call $\\omega^n$ the {\\em environment}. We define the exclusion process $\\eta_t^n$ with environment $\\omega^n$ as a continuous-time Markov chain of state space $\\Omega_n = \\{0,1\\}^{X_n}$ and generated by the operator\n\\[\nL_n f(\\eta) = \\sum_{x,y \\in X_n} \\omega_{x,y}^n \\big[f(\\eta^{x,y}) - f(\\eta)\\big],\n\\]\nwhere $\\eta$ is a generic element of $\\Omega_n$, $f: \\Omega_n \\to \\bb R$ is a function which depends on $\\eta(x)$ for a finite number of elements $x \\in X_n$ (that is, $f$ is a {\\em local function}) and $\\eta^{x,y} \\in \\Omega_n$ is defined by\n\\[\n\\eta^{x,y}(z) = \n\\begin{cases}\n\\eta(y), & \\text{if } z=x\\\\\n\\eta(x), & \\text{if } z=y\\\\\n\\eta(z), & \\text{if } z \\neq x,y.\\\\\n\\end{cases}\n\\]\n\nIn order to have a well-defined Markovian evolution for any initial distribution $\\eta_0^n$, we assume that $\\sup_x \\sum_{y \\in X_n} \\omega^n_{x,y} <+\\infty$.\nWe interpret $X_n$ as a set of sites and $\\eta_t^n(x)$ as the number of particles at site $x \\in X_n$ at time $t$. Since $\\eta_t^n(x) \\in \\{0,1\\}$, there is at most one particle per site at any given time: this is the so-called {\\em exclusion rule}. Notice that the dynamics is conservative in the sense that no particles are annihilated or destroyed.\n\nOur interest is to study the collective behavior of particles for the sequence of processes $\\{\\eta_\\cdot^n\\}_n$. In order to do this, we introduce the {\\em empirical density of particles} as the measure-valued process $\\pi_t^n$ defined by\n\\[\n\\pi_t^n(G) = \\frac{1}{a_n} \\sum_{x \\in X_n} \\eta_t^n(x) S_n G(x)\n\\]\nfor any $G \\in \\mc C_c(X)$. Using Riesz's theorem, it is not difficult to check that $\\pi_t^n$ is effectively a positive Radon measure in $X$. \nObserve that when $\\eta_0^n(x)=1$ for any $x \\in X_n$, then $\\eta_t^n(x)=1$ for any $x \\in X_n$ and any $t \\geq 0$. In this situation, the empirical process $\\pi_t^n$ is identically equal to the measure $\\mu$. Notice that the random variable $\\pi_t^n$ defined in this way corresponds to a process defined in the space $\\mc D([0,\\infty), \\mc M_+(X))$ of c\\`adl\\`ag paths with values in $\\mc M_+(X)$. For functions $G:X_n \\to \\bb R$, we define $\\pi_t^n(G) = a_n^{-1} \\sum_x \\eta_t^n(x) G(x)$.\n\n\\subsection{$\\Gamma$-convergence of the environment}\n\nIn this section we will make a set of assumptions on the environment $\\{\\omega^n\\}_n$ which will allows us to obtain an asymptotic result for the sequence $\\{\\pi_\\cdot^n\\}_n$. We start with two assumptions about the sequence of partitions of the unity $\\{\\mc U_x^n\\}_n$. Our first assumption corresponds to a sort of ellipticity condition on the partitions of the unity $\\{\\mc U_x^n\\}_n$: \n\n\\begin{description}\n\\item[\\bf (H1)] There exists $\\Theta <+\\infty$ such that \n\\[\n\\sup_{x \\in X_n} a_n \\int \\mc U_x^n d\\mu \\leq\\Theta \\text{ for any $n>0$.}\n\\]\n\n\\end{description}\n\nUnder this condition, the projection $S_n$ satisfies $||S_n G||_\\infty \\leq \\theta ||G||_\\infty$, and by interpolation $S_n$ can be extended to a continuous operator from $\\mc L^2(\\mu)$ to $\\mc L^2(X_n)$. Our second condition states that $S_n$ is close to an isometry when $n \\to \\infty$:\n\\begin{description}\n \\item[\\bf{(H2)}] For any $F \\in \\mc L^2(\\mu)$, we have\n \\[\n \\lim_{n \\to \\infty} \\_n = \\.\n \\] \n\\end{description}\n\nNow we are ready to discuss on which sense we will say that the environment $\\omega^n$ converges. \nFor a given function $F: X_n \\to \\bb R$ of finite support, we define $\\mc L_n F$ by\n\\[\n\\mc L_n F(x) = \\sum_{y \\in X_n} \\omega_{x,y}^n \\big(F(y)-F(x)\\big).\n\\]\n\nIt turns out that $\\mc L_n$ can be extended to a non-positive operator in $\\mc L^2(X_n)$. In fact, for any function $F$ of finite support, the {\\em Dirichlet form}\n\\[\n\\_n = \\frac{1}{2 a_n} \\sum_{x, y \\in X_n} \\omega_{x,y}^n \\big(F(y)-F(x)\\big)^2\n\\]\nis clearly non-negative. For a function $G \\in \\mc L^2(\\mu)$, define $\\mc E_n(G) = \\$. Notice that $\\mc E_n: \\mc L^2(\\mu) \\to \\bar{\\bb R}$ is a quadratic form. Now we are ready to state our first hypothesis about the environment:\n\\begin{description}\n \\item[{\\bf (H3)}] There exists a non-negative, symmetric operator $\\mc L: D(\\mc L) \\subseteq \\mc L^2(\\mu) \\to \\mc L^2(\\mu)$ such that $\\mc E_n$ is $\\Gamma$-convergent to $\\mc E$, where $\\mc E(G) = -\\int G \\mc L G d\\mu$. \n\\end{description}\n\nOur second hypothesis about the environment $\\omega^n$ concerns to its $\\Gamma$-limit $\\mc L$:\n\\begin{description}\n \\item[{\\bf (H4)}] There exists a dense set $\\mc K \\subseteq \\mc C_c(X)$ such that $\\mc K$ is a kernel for the operator $\\mc L$, and for any $G \\in \\mc K$, $\\mc LG$ is continuous and $\\int |\\mc L G| d\\mu <+\\infty$.\n\\end{description}\n\n\\subsection{Hydrodynamic limit of $\\eta_t^n$}\n\nIn this section we explain what we understand as the hydrodynamic limit of $\\eta_t^n$. We say that a sequence $\\{\\nu_n\\}_n$ of distributions in $\\Omega_n$ is {\\em associated } to a function $u:X \\to \\bb R$ if for any function $G \\in \\mc C_c(X)$ and any $\\epsilon >0$ we have\n\\[\n\\lim_{n \\to \\infty} \\nu_n \\Big\\{ \\Big|\\frac{1}{a_n} \\sum_{x \\in X_n} \\eta(x) G(x) - \\int G(x)u(x)\\mu(dx)\\Big|>\\epsilon\\Big\\}=0. \n\\]\n\nNotice that we necessarily have $0 \\leq u(x) \\leq 1$ for any $x \\in X$, since $\\eta(x) \\in \\{0,1\\}$.\nFix an initial profile $u_0:X \\to [0,1]$ and take a sequence of distributions $\\{\\nu_n\\}$ associated to $u_0$. Let $\\eta_t^n$ be the exclusion process with initial distribution $\\nu_n$. We denote by $\\bb P_n$ the law of $\\eta_t^n$ in $\\mc D([0,\\infty),\\Omega_n)$ and by $\\bb E_n$ the expectation with respect to $\\bb P_n$. The fact that $\\{\\nu_n\\}_n$ is associated to $u_0$ can be interpreted as a law of large numbers for the empirical measure $\\pi_0^n$: $\\pi_0^n(dx)$ converges in probability to the deterministic measure $u_0(x) \\mu(dx)$. We say that the hydrodynamic limit of $\\eta_t^n$ is given by the equation $\\partial_t u = \\mc L u$ if for any $t>0$, the empirical measure $\\pi_t^n(dx)$ converges in probability to the measure $u(t,x) \\mu(dx)$, where $u(t,x)$ is the solution of the equation $\\partial_t u = \\mc L u$ with initial condition $u_0$. Before stating our main result in a more precise way, we need some definitions. \n\nFor $F, G \\in D(\\mc L)$, define the bilinear form $\\mc E(F,G) = - \\int F \\mc L G d\\mu$. Notice that $\\mc E(F,G)$ is still well defined if only $G \\in D(\\mc L)$. \nWe say that a function $u:[0,T] \\times X \\to [0,1]$ is a weak solution of (\\ref{echid}) with initial condition $u_0$ if $\\int_0^T \\int u_t^2 d\\mu dt <+\\infty$ and for any differentiable path $G: [0,T] \\to \\mc K$ such that $G_T \\equiv 0$ we have\n\\[\n\\ +\\int_0^T \\Big\\{ \\<\\partial_t G_t,u_t\\> -\\mc E(G_t,u_t)\\Big\\}dt=0.\n\\]\n\n\n\\begin{theorem}\n\\label{t1}\nLet $\\{\\nu_n\\}_n$ be associated to $u_0$ and consider the exclusion process $\\eta_t^n$ with initial distribution $\\nu_n$. Assume that $\\int \\pi_0^n(dx)$ is uniformly finite:\n\\begin{description}\n \\item[{\\bf (H5)}] \n \\[\n \\lim_{M \\to \\infty} \\sup_n \\nu_n\\Big\\{\\frac{1}{a_n} \\sum_{x \\in X_n} \\eta(x) >M \\Big\\} =0.\n \\]\n\\end{description}\n\nThen, the sequence of processes $\\{\\pi_\\cdot^n(dx)\\}_n$ is tight and the limit points are concentrated on measures of the form $u(t,x)\\mu(dx)$, where $u(t,x)$ is a weak solution of the {\\em hydrodynamic equation}\n\\begin{equation}\n\\label{echid}\n\\left\\{\n\\begin{array}{rcl}\n\\partial_t u & = & \\mc L u, \\\\\nu(0,\\cdot) & = & u_0(\\cdot).\\\\\n\\end{array}\n\\right.\n\\end{equation}\n\nIf such solution is unique, the process $\\pi_\\cdot^n(dx)$ converges in probability with respect to the Skorohod topology of $\\mc D([0,\\infty),\\mc M_+(X))$ to the deterministic trajectory $u(t,x)\\mu(dx)$.\n\\end{theorem}\n\n\nUsually in the literature, hydrodynamic limits are obtained in finite volume, since the pass from finite to infinite volume is non-trivial. Assumption {\\bf (H5)} is in this spirit: it is automatically satisfied when the cardinality of $X_n$ is of the order of $a_n$ (on which case $\\mu(X)<+\\infty$), and it is very restrictive when $X_n$ is infinite. For simplicity, we restrict ourselves to the case on which {\\bf (H5)} is satisfied. \n\n\n\\section{Hydrodynamic limit of $\\eta_t^n$: proofs}\n\\label{s2}\nIn this section we obtain the hydrodynamic limit of the process $\\eta_t^n$. The strategy of proof of this result is the usual one for convergence of stochastic processes. First we prove tightness of the sequence of processes $\\{\\pi_\\cdot^n\\}_n$. Then we prove that any limit point of this sequence is concentrated on solutions of the hydrodynamic equation. Finally, a uniqueness result for such solutions allows us to conclude the proof. However, the strategy outlined above will not be carried out for $\\{\\pi_\\cdot^n\\}_n$ directly, but for another process $\\hat \\pi_\\cdot^n$, which we call the {\\em corrected} empirical process.\n\n\\subsection{The corrected empirical measure}\n\\label{s2.1}\n\nIn this section we define the so-called corrected empirical measure, relying on the $\\Gamma$-convergence of the environment. First we need to extract some information about convergence of the operators $\\mc L_n$ to $\\mc L$ from the $\\Gamma$-convergence of the associated Dirichlet forms.\n\nTake a general Hilbert space $\\mc H$ and let $\\mc A$ be a non-negative, symmetric operator defined in $\\mc H$. By Lax-Milgram theorem, we know that for any $\\lambda>0$ and any $g \\in \\mc H$, the equation $(\\lambda+\\mc A)f =g$ has a unique solution in $\\mc H$. Moreover, the solution $f$ is the minimizer of the functional $f \\mapsto \\+\\lambda||f||^2-2\\$. Fix $\\lambda>0$. For a given function $G \\in \\mc L^2(\\mu)$, define the functionals\n\\[\n\\mc E_n^G(F) = \\mc E_n(F) +\\lambda\\_n -2 \\_n,\n\\]\n\\[\n\\mc E^G(F) = \\mc E(F) +\\lambda\\-2 \\.\n\\]\n\nBy Proposition \\ref{p2}, $\\mc E_n^G$ is $\\Gamma$-convergent to $\\mc E^G$. In particular, a sequence of minimizers $F_n$ of $\\mc E_n^G$ converge to the minimizer $F$ of $\\mc E^G$. Notice that $F_n$ is not uniquely defined in general, although $S_n F_n$ it is. By the discussion above, $(\\lambda-\\mc L_n) S_n F_n = S_n G$ and $(\\lambda -\\mc L) F =G$. Since the operator norm of $S_n$ is bounded by $\\Theta$, we conclude that the $\\mc L^2(X_n)$-norm of $S_n F_n-S_n F$ converges to 0 as $n \\to \\infty$.\nBy {\\bf (H2)}, we conclude that $\\mc E_n(F_n)$ converges to $\\mc E(F)$. \n\n\n\nNow we are ready to define the corrected empirical measure $\\hat \\pi_t^n$. Take a function $G \\in \\mc K$ and define $H=(\\lambda-\\mc L) G$. Define $G_n$ as a minimizer of $\\mc E_n^H$. Notice that in this way $S_n G_n$ is uniquely defined. Then we define \n\\[\n\\hat \\pi_t^n(G) = \\frac{1}{a_n} \\sum_{x \\in X_n} \\eta_t^n(x) S_n G_n(x).\n\\]\n\nIn order to prove that $\\hat \\pi_t^n(G)$ is well defined, we need to prove that $\\sum_x S_n G_n(x)$ is finite. Remember that $(\\lambda - \\mc L_n) S_n G_n = S_n H$. Consider the continuous-time random walk with jump rates $\\omega_{x,y}^n$. Remember that the condition $\\sup_x \\sum_y \\omega_{x,y}^n$ ensures that this random walk is well defined. Let $p_t^n(x,y)$ be its transition probability function. An explicit formula for $S_n G_n$ in terms of $p_t^n(x,y)$ is\n\\[\nS_n G_n (x) = \\int_0^\\infty e^{-\\lambda t} \\sum_{y \\in X_n} p_t^n(x,y) S_n H(y) dt.\n\\]\n\nSince $\\sum_x p_t(x,y)=1$ for any $y \\in X_t$, we conclude that\n\\[\n\\frac{1}{a_n} \\sum_{x \\in X_n} S_n G_n(x) = \\frac{1}{\\lambda} \\int H d\\mu\n\\]\nand in particular $S_n G_n$ is summable. We conclude that $\\hat \\pi_t^n(G)$ is well defined. Notice that it is not clear at all if $\\hat \\pi_t^n$ is well defined as a measure in $X$. \n\n\\subsection{Tightness of $\\{\\pi_\\cdot^n\\}_n$ and proof of Theorem \\ref{t1}}\n\nIn this section we prove tightness of $\\{\\pi_\\cdot^n\\}_n$ and we prove Theorem \\ref{t1}. As we will see, we rely on the corrected empirical measure, which turns out to be the right object to be studied. \nBy {\\bf (H5)}, we have\n\\[\n\\lim_{n \\to \\infty} \\bb P_n \\Big( \\sup_{0 \\leq t <+\\infty} \\big|\\pi_t^n(G) - \\hat \\pi_t^n(G)\\big|>\\epsilon\\Big) =0.\n\\]\n\nNotice that {\\bf (H5)} can be substituted by the following condition, which can be sometimes proved directly.\n\\begin{description}\n \\item[{\\bf (H5')}] For any $G \\in \\mc K$,\n \\[\n \\lim_{n \\to \\infty} \\frac{1}{a_n} \\sum_{x \\in X_n} \\big| S_n G_n(x) - S_n G(x)\\big| =0.\n \\]\n\\end{description}\n\nIn particular, $\\{\\pi_\\cdot^n(G)\\}_n$ is tight if and only if $\\{\\hat \\pi_\\cdot^n(G)\\}_n$ is tight. The usual way of proving tightness of $\\{\\hat \\pi_\\cdot^n(G)\\}_n$ is to use a proper martingale decomposition. A simple computation based on Dynkin's formula shows that\n\\begin{equation}\n\\label{ec1}\n\\mc M_t^n(G) = \\hat \\pi_t^n(G) - \\hat \\pi_0^n(G) - \\int_0^t \\pi_s^n(\\mc L_n S_n G_n) ds\n\\end{equation}\nis a martingale. The quadratic variation of$\\mc M_t^n(G)$ is given by\n\\[\n\\<\\mc M_t^n(G)\\> = \\int_0^t \\frac{1}{a_n^2} \\sum_{x,y \\in X_n} \\big(\\eta_s^n(y)-\\eta_s^n(x)\\big)^2 \\omega^n_{x,y} \\big( S_n G_n(y) -S_n G_n(x)\\big)^2 ds.\n\\]\nIn particular, $\\<\\mc M_t^n(G)\\> \\leq t a_n^{-1} \\mc E_n(G_n)$. At this point, the convenience of introducing the corrected empirical process becomes evident. By definition, $\\mc L_n S_n G_n = S_n \\mc L G +\\lambda(S_n G_n - S_n G)$. Since $H= (\\lambda-\\mc L)G$, the function $G$ is the minimizer of $\\mc E^H$. Therefore, $G_n$ converges to $G$ in $\\mc L^2(X)$. By {\\bf (H2)}, the $\\mc L^2(X_n)$-norm of $S_n G_n -S_n G$ goes to 0 and $\\mc E_n(G_n)$ converges to $\\mc E(G)$. \n\nWe conclude that $\\mc M_t^n(G)$ converges to 0 as $n \\to \\infty$, and in particular the sequence $\\{\\mc M_\\cdot^n(G)\\}_n$ is tight. In the other hand, the integral term in (\\ref{ec1}) is equal to\n$\\int_0^t \\pi_s^n(\\mc L G)ds$.\n\nNotice that $\\pi_s^n(\\mc L G) \\leq \\int |\\mc L G| d \\mu$ for any $t \\geq 0$, from where we conclude that the integral term is of bounded variation, uniformly in $n$. Tightness follows at once. Since $\\{\\hat \\pi_0^n(G)\\}_n$ is tight by comparison with $\\{\\pi_0^n(G)\\}_n$, we conclude that $\\{\\hat \\pi_\\cdot^n(G)\\}_n$ is tight, which proves the first part of Theorem \\ref{t1}. As a by-product, we have obtained tightness for $\\{\\pi_\\cdot^n\\}_n$ as well, and the convergence result\n\\[\n\\lim_{n \\to \\infty}\\Big\\{ \\pi_t^n(G) -\\pi_0^n(G) - \\int_0^t \\pi_s^n(\\mc L G) ds\\Big\\} =0\n\\]\nfor any $G \\in \\mc K$. Notice that we have exchanged $\\hat \\pi_t^n(G)$ by $\\pi_t^n(G)$. Let $\\pi_\\cdot$ be a limit point of $\\{\\pi_\\cdot^n\\}_n$. Then, $\\pi_\\cdot$ satisfies the identity\n\\[\n\\pi_t(G) -\\pi_0(G) - \\int_0^t \\pi_s(\\mc L G) ds =0\n\\] \nfor any function $G \\in \\mc K$. By hypothesis, $\\pi_0(dx) = u_0(x)\\mu(dx)$. Repeating the arguments for a function $G_t(x) = G_0(x) + t G_1(x)$ with $G_0,G_1 \\in \\mc K$, we can prove that\n\\[\n\\pi_t(G_t) - \\pi_0(G_0) -\\int_0^t \\pi_s((\\partial_t+\\mc L)G_s)ds=0\n\\] \nfor any piecewise-linear trajectory $G_\\cdot:[0,T] \\to \\mc K$. The same identity holds by approximation for any smooth path $G_\\cdot: [0,T] \\to \\mc C_c(X)$, which proves that the process $\\pi_\\cdot$ is concentrated on weak solutions of the hydrodynamic equation. When such solutions are unique, the process $\\pi$ is just a $\\delta$-distribution concentrated on the path $u(t,x)\\mu(dx)$. Since compactness plus uniqueness of limit points imply convergence, Theorem \\ref{t1} is proved. \n\n\\section{Energy solutions and energy estimate}\n\\label{s3}\nIn this section we define what we mean by {\\em energy solutions} of Equation (\\ref{echid}), we prove that any limit point of the empirical measure $\\{\\pi_\\cdot^n\\}$ is concentrated on energy solutions of (\\ref{echid}) and we give a simple criterion for uniqueness of such solutions. \n\n\\subsection{Energy solutions} \n\nLet $\\mc E : H \\to \\bar{\\bb R}$ be a quadratic form defined over a Hilbert space $H$ of inner product $\\<\\cdot,\\cdot\\>$. We say that $\\mc E$ is {\\em closable} if for any sequence $\\{f_n\\}_n$ converging in $H$ to some limit $f$ such that $\\mc E(f_n-f_m)$ goes to $0$ as $n, m \\to \\infty$, we have $f=0$. \nLet $\\mc E: H \\to \\bar{\\bb R}$ be closable. We define $\\mc H_1=\\mc H_1(\\mc E)$ as the closure of the set $\\{f\\in H; \\mc E(f)<+\\infty\\}$ under the norm $||f||_1 = (\\mc E(f)+ \\)^{1\/2}$. \n\nWe say that a dense set $K \\subseteq H$ is a {\\em kernel} of $\\mc E$ if $\\mc H_1$ is equal to the closure of $K$ under the norm $||\\cdot||_1$. We say that a symmetric operator $\\mc L: D(\\mc L) \\subseteq H \\to H$ generates $\\mc E$ if $\\mc E(f)=\\$ for $f \\in D(\\mc L)$ and $D(\\mc L)$ is a kernel of $\\mc E$. \n\nFix $T >0$. For a function $u: [0,T] \\to H$ we define the norm\n\\[\n||u||_{1,T} = \\Big( \\int_0^T ||u_t||_1^2 dt \\Big)^{1\/2}\n\\]\nand we define $\\mc H_{1,T}$ as the Hilbert space generated by this norm. \nGiven a closable form $\\mc E$ generated by the operator $\\mc L$, we say that a trajectory $u: [0,T] \\to H$ is an {\\em energy solution} of (\\ref{echid}) if $u \\in \\mc H_{1,T}$ and for any differentiable trajectory $G: [0,T] \\to \\mc H_1$ with $G(T)=0$ we have\n\\[\n\\ + \\int_0^T \\Big\\{ \\<\\partial_t G_t, u_t\\> - \\mc E(G_t,u_t)\\Big\\} dt =0.\n\\]\n\nIn other words, an energy solution of (\\ref{echid}) is basically a weak solution belonging to $\\mc H_{1,T}$. In fact, by taking suitable approximations of $G$, it is enough to prove this identity for trajectories $G$ such that $G_t \\in K$ for any $t \\in [0,T]$, where $K$ is any kernel of $\\mc E$ contained in $D(\\mc L)$. Notice that the norm in $\\mc H_{1,T}$ is stronger than the norm $\\int_0^T u_t^2 dt$, and therefore a weak solution is effectively weaker than an energy solution of (\\ref{echid}).\n\n\\subsection{The energy estimate}\n\nIn this section we prove that the limit points of the empirical measure are concentrated on energy solutions of (\\ref{echid}). For simplicity, we work on finite volume. From now on we assume that $X$ is compact. Therefore, there exists a constant $\\kappa$ such that the cadinality of $X_n$ is bounded by $\\kappa a_n$. We have the following estimate.\n\n\\begin{theorem}\n\\label{t2}\nFix $T >0$. Let $\\{H^i: X_n \\times X_n \\times [0,T] \\to \\bb R; i=1,\\dots,l\\}$ be a finite sequence of functions. There exists a constant $C=C(T)$ such that\n\\begin{multline}\n\\label{en.est}\n\\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\Big\\{ \\frac{2}{a_n} \\sum_{x,y \\in X_n} \\omega^n_{x,y} H_{x,y}^i(t) \\big( \\eta_t^n(y) -\\eta_t^n(x)\\big) \\\\\n\t-\\frac{1}{a_n} \\sum_{x,y \\in X_n} \\omega_{x,y}^n (H_{x,y}^i)^2 \\eta_t^n(x)\\Big\\} dt \\Big]\n\\leq C + \\frac{\\log l}{a_n}.\n\\end{multline}\n\\end{theorem}\n\n\\begin{proof}\n\nBefore starting the proof of this theorem, we need some definitions. Fix $\\rho >0$. Denote by $\\nu^\\rho$ the product measure in $\\Omega_n$ defined by \n\\[\n\\nu^\\rho\\big(\\eta(x_1)=1,\\dots,\\eta(x_k)=1\\big) = \\rho^k.\n\\]\n\nIt is not difficult to check that the measure $\\nu^\\rho$ is left invariant under the evolution of $\\eta_t$. For two given probability measures $P_1$, $P_2$, we define the entropy $H(P_1|P_2)$ of $P_1$ with respect to $P_2$ as\n\\[\nH(P_1|P_2)= \n\\begin{cases}\n+ \\infty, & \\text{ if $P_1$ is not absolutely continuous with respect to $P_2$}\\\\\n\\int \\log \\frac{dP_1}{dP_2} d P_1 & \\text{ otherwise. }\\\\\n\\end{cases}\n\\]\n\nFor $\\eta \\in \\Omega_n$, denote by $\\delta_\\eta$ the Dirac measure at $\\eta$. It is not difficult to see that $H(\\delta_\\eta|\\nu^\\rho) \\leq C(\\rho) a_n$ for any $\\eta \\in \\Omega_n$, where $C(\\rho)$ is a constant that can be chosen independently from $n$. Let us denote by $\\bb P^\\rho$ the distribution in $D([0,T],\\Omega_n)$ of the process $\\eta_t^n$ with initial distribution $\\nu^\\rho$. By the convexity of the entropy, $H(\\bb P_n|\\bb P^\\rho) \\leq C(\\rho,T)a_n$ for a constant $C(\\rho,T)$ not depending on $n$. The following arguments are standard and can be found in full rigor in \\cite{KL}. Let us denote by $F^i(s)$ the function (depending on $H^i(s)$ and $\\eta_s^n$) under the time integral in \\eqref{en.est}. By the entropy estimate,\n\\begin{align*}\n\\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T F^i(t)dt\\Big]\n\t\\leq \\frac{H(\\bb P_n|\\bb P^\\rho)}{a_n} +\n\t\\frac{1}{a_n} \\log \\bb E^\\rho \\Big[ \\exp\\big\\{ \\sup_{i=1,\\dots,l} a_n \\int_0^T F^i(t)dt\\big\\}\\Big].\n\\end{align*}\n\nIn order to take the supremum out of the expectation, we use the inequalities $\\exp\\{\\sup_i b_i\\} \\leq \\sum_i\\exp\\{b_i\\}$ and $\\log\\{\\sum_i b_i\\} \\leq \\log l + \\sup_i \\log b_i$, valid for any real numbers $\\{b_i, i=1,\\dots,l\\}$. In this way we obtain the bound\n\\begin{equation}\n \\label{ec2}\n\\begin{split}\n\\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T F^i(t)dt\\Big]\n\t&\\leq C(\\rho,T) +\\frac{\\log l}{a_n} \\\\\n\t&\\quad+\\sup_{i=1,\\dots,l} \\frac{1}{a_n} \\log \\bb E^\\rho \\Big[ \\exp\\big\\{ a_n\\int_0^T F^i(t)dt\\big\\}\\Big].\n\\end{split}\n\\end{equation}\n\n\nTherefore, it is left to prove that the last supremum is not positive. It is enough to prove that the expectation $ \\bb E^\\rho \\big[ \\exp\\big\\{ \\int_0^T F^i(t)dt\\big\\}\\big]$ is less or equal than $1$ for any function $F^i$. From now on we drop the index $i$. By Feynman-Kac's formula plus the variational formula for the largest eigenvalue of the operator $F(t)+L_n$, we have\n\\[\n\\frac{1}{a_n} \\log \\bb E^\\rho \\Big[ \\exp\\big\\{ a_n\\int_0^T F(t)dt\\big\\}\\Big]\n\t\\leq \\int_0^T \\sup_f \\big\\{ \\_\\rho -\\_\\rho\\},\n\\]\nwhere we have denoted by $\\<\\cdot,\\cdot\\>_\\rho$ the inner product in $\\mc L^2(\\nu_\\rho)$ and the supremum is over functions $f \\in \\mc L^2(\\nu_\\rho)$. A simple computation using the invariance of $\\nu_\\rho$ shows that\n\\[\n\\_\\rho = \\sum_{x,y \\in X_n} \\omega_{x,y}^n \\int \\big[f(\\eta^{x,y})-f(\\eta)\\big]^2 \\nu_\\rho(d\\eta).\n\\]\nRecall the expression for $F(t)$ in terms of $H$. We will estimate each term of the form $2 a_n^{-1} \\_\\rho$ separatedly:\n\\begin{align*}\n\\frac{2}{a_n} \\_\\rho\n\t&= \\frac{2}{a_n} H_{x,y} \\<\\eta(x), f(\\eta^{x,y})^2-f(\\eta)^2\\>_\\rho \\\\\n\t& \\leq \\frac{2}{a_n} \\Big\\{ \\frac{(H_{x,y})^2\\beta_{x,y}^n}{2} \\<\\eta(x), (f(\\eta^{x,y})+f(\\eta))^2\\>_\\rho \\\\\n\t&\\quad+ \\frac{1}{2 \\beta_{x,y}^n} \\<\\eta(x),(f(\\eta^{x,y})-f(\\eta))^2\\>_\\rho \\Big\\}.\n\\end{align*}\n\nChoosing $\\beta_{x,y}^n = 1\/\\omega_{x,y}^n$ and putting this estimate back into (\\ref{ec2}), we obtain the desired estimate.\n\\qed\n\\end{proof}\n\nTake $G^i \\in \\mc K$ and take $H_{x,y}^i = S_n G_n^i(y) - S_n G_n^i(x)$, with $G_n^i$ defined as in Section \\ref{s2.1}. Recall the identity $\\mc L_n S_n G_n^i= S_n \\mc L G^i + \\lambda(S_n G_n^i -S_n G^i)$. The energy estimate (\\ref{en.est}) gives\n\\[\n \\bb E_n \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\big(2 \\hat \\pi_t^n(\\mc L G^i) - \\mc E_n(G_n^i) \\big)dt \\Big] \\leq C(\\rho,T) + C_1(l,n),\n\\]\nwhere $C_1(l,n)$ is a constant that goes to 0 when $l$ is fixed and $n \\to \\infty$. Take a limit point of the sequence $\\{\\pi_\\cdot^n\\}_n$. We have already seen that $\\hat \\pi_t^n(\\mc L G^i)$ converges to $\\pi_t(\\mc L G)$. Therefore, the process $\\pi_\\cdot$ satisfies\n\\[\n E \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\big(2 \\pi_s(\\mc L G^i) - \\mc E(G^i)\\big)dt\\Big] \\leq C(\\rho,T).\n\\]\n\nSimilar arguments prove that for piecewise linear trajectories $\\{G^i_t; i=1,\\dots,l\\}$ in $\\mc K$, we have\n\\[\n E \\Big[ \\sup_{i=1,\\dots,l} \\int_0^T \\big(2 \\pi_s(\\mc L G^i(t)) - \\mc E(G^i(t))\\big)dt\\Big] \\leq C(\\rho,T).\n\\]\n\nSince $l$ is arbitrary and piecewise linear trajectories with values in $\\mc K$ are dense in $\\mc H_{1,T}$, we conclude that $E[||\\pi_\\cdot||_{1,T}^2] <+\\infty$, from where we conclude that $||\\pi_\\cdot||_{1,T}$ is finite $a.s.$\nWe establish this result as a theorem.\n\n\\begin{theorem}\n \\label{t3}\n Let $\\eta_t^n$ an exclusion process as in Theorem \\ref{t1}.\n If one of the following conditions is satisfied,\n \\begin{enumerate}\n \\item[i)]\n $X$ is compact,\n \\item[ii)]\n Assumption {\\bf (H5')} holds and the entropy density is finite:\n \\[\n \\sup_{n} \\frac{H(\\bb P_n| \\bb P^\\rho)}{a_n} <+\\infty,\n \\]\n\\end{enumerate}\nthen any limit point of the sequence $\\{\\pi^n_\\cdot(dx)\\}_n$ is concentrated on energy solutions of the hydrodynamic equation (\\ref{echid}). In particular, since such energy solutions are unique, the sequence $\\{\\pi_\\cdot^n(dx)\\}_n$ is convergent.\n\\end{theorem}\n\n\\subsection{Uniqueness of energy solutions}\n\nIn this section we prove uniqueness of energy solutions for (\\ref{echid}). Since the equation is linear, it is enough to prove uniqueness for the case $u_0 \\equiv 0$. Let $u_t$ be a solution of (\\ref{echid}) with $u_0 \\equiv 0$. Then,\n\\[\n \\int_0^T \\big\\{ \\<\\partial_t G_t , u_t\\> - \\mc E(G_t,u_t)\\big\\}dt =0\n\\]\nfor any differentiable trajectory in $\\mc H_{1,T}$ with $G_T=0$. Take $G_t = - \\int_t^T u_s ds$. Then $\\partial_t G_t = u_t$ and the first term above is equal to $\\int_0^T\\dt$. An approximation procedure and Fubini's theorem shows that the second term above is equal to\n\\[\n \\frac{1}{2} \\mc E\\Big(\\int_0^T u_t dt\\Big).\n\\]\n\nBoth terms are non-negative, so we conclude that $\\int_0^T \\dt =0$ and $u_t \\equiv 0$.\n\n\\section{Applications}\n\\label{s4}\nIn this section we give some examples of systems on which Theorems \\ref{t1} and \\ref{t3} apply. In the literature, the sequence $\\omega^n$ is often referred as the set of {\\em conductances} of the model. Unless stated explicitely, in these examples, $X$ will be equal to $\\bb R^d$ or the torus $\\bb T^d = \\bb R^d \/ \\bb Z^d$. The set $X_n$ will be equal to $n^{-1} \\bb Z^d$ and we construct the partitions $\\{\\mc U_x^n\\}$ in the canonical way, taking $\\mc U_x^n$ as a continuous, piecewise linear function with $\\mc U_x^n(x)=1$ and $\\mc U_x^n(y)=0$ for $y\\in X_n$, $y \\neq x$.\n\n\\subsection{Homogenization of ergodic, elliptic environments}\n\nLet $(\\Omega,\\mc F,P)$ be a probability space. Let $\\{\\tau_x; x \\in \\bb Z^d\\}$ be a family of $\\mc F$-mesurable maps $\\tau_x:\\Omega \\to \\Omega$ such that\n\\begin{description}\n\\item[i)] $P(\\tau_x^{-1} A) = P(A)$ for any $A \\in \\mc F$, $x \\in \\bb Z^d$.\n\\item[ii)] $\\tau_x \\tau_{x'} = \\tau_{x+x'}$ for any $x,x' \\in \\bb Z^d$.\n\\item[iii)] If $\\tau_x A =A$ for any $x \\in \\bb Z^d$, then $P(A)=0$ or $1$.\n\\end{description}\n\nIn this case, we say that the family $\\{\\tau_x\\}_{x \\in \\bb Z^d}$ is ergodic and invariant under $P$. Let $a=(a_1,\\dots,a_d): \\Omega \\to \\bb R^d$ be an $\\mc F$-measurable function. Assume that there exists $\\epsilon_0>0$ such that\n\\[\n\\epsilon_0 \\leq a_i(\\omega) \\leq \\epsilon_0^{-1} \\text{ for all } \\omega \\in \\Omega \\text{ and } i=1,\\dots,d.\n\\] \n\nWe say in this situation that the environment satisfies the {\\em ellipticity condition}. Fix $\\omega \\in \\Omega$. Define $\\omega^n$ by $\\omega^n_{x,x+e_i\/n}=\\omega^n_{x+e_i\/n,x}= n^2 a_i(\\tau_nx \\omega)$, $\\omega^n_{x,y}=0$ if $|y-x| \\neq 1\/n$. Here $\\{e_i\\}_i$ is the canonical basis of $\\bb Z^d$. In this case, $a_n = n^d$ and $\\mu$ is the Lebesgue measure in $\\bb R^d$.\nIn \\cite{PV}, it is proved that there is a positive definite matrix $A$ such that the quadratic form $\\mc E_n$ associated to $\\omega^n$ is $\\Gamma$-convergent to $\\mc E(f)=\\int \\nabla f \\cdot A \\nabla f dx$, $P-a.s.$ In particular, Theorem \\ref{t1} applies with $\\mc L f= \\text{div}(A \\nabla f)$. This result was first obtained in \\cite{GJ}. \n\n\\subsection{The percolation cluster}\n\nLet $e=\\{e^i_x; x \\in \\bb Z^d, i=1,\\cdots,d\\}$ be a sequence of i.i.d. random variables, with $P(e_x^i=1) = 1- P(e_x^i=0) =p$ for some $p=(0,1)$. Define for $x,y \\in X_n$, $\\omega^n_{x,x+e_i\/n}=\\omega^n_{x+e_i\/n,x}= n^2 e_{nx}^i$, $\\omega^n_{x,y}=0$ if $|y-x| \\neq 1\/n$. Fix a realization of $e$. We say that two points $x, y \\in X_n$ are connected if there is a finite sequence $\\{x_0=x,\\dots,x_l=y\\} \\subseteq X_n$ such that $|x_{i-1}-x_i|=1\/n$ and $\\omega_{x_{i-1},i}^n =1$ for any $i$. Denote by $\\mc C_0$ the set of points connected to the origin. It is well known that there exists $p_c \\in (0,1)$ such that $\\theta(p)= P(\\mc C_0 \\text{ is infinite })$ is 0 for $pp_c$. Fix $p>p_c$. Define $a_n = n^d$ and $\\mu_0(dx) = \\theta(p) dx$. In \\cite{F2}, it is proved that there exists a constant $D$ such that, $P-a.s$ in the set $\\{\\mc C_0 \\text{ is infinite }\\}$, the quadratic form $\\mc E_n$ associated to the environment $\\omega^n$ restricted to $\\mc C_0$ is $\\Gamma$-convergent to $\\mc E(f) = \\theta(p) D \\int (\\nabla f)^2 dx$. Theorem \\ref{t1} applies with $\\mc L = D \\Delta$, assuming that the initial measures $\\nu_n$ put mass zero in configurations with particles outside $\\mc C_0$.\nThis result was first obtained in \\cite{Fag}, relying on a duality representation of the simple exclusion process.\n\n\\subsection{One-dimensional, inhomogeneous environments} \n\nIn dimension $d=1$, the $\\Gamma$-convergence of $\\mc E_n$ can be studied explicitely. For nearest-neighbors environments ($\\omega^n_{x,y} =0$ if $|x-y|=1$), $\\Gamma$-convergence of $\\mc E_n$ is equivalent to convergence in distribution of the measures\n\\[\nW_n(dx) = \\frac{1}{n} \\sum_{x \\in \\bb Z} (\\omega_{x,x+1}^n)^{-1} \\delta_{x\/n}(dx).\n\\] \n\nLet $W(dx)$ be the limit. We assume that $W(dx)$ gives positive mass to any open set. For simplicity, suppose that $W(\\{0\\})=0$. Otherwise, we simply change the origin to another point with mass zero. For two functions $f,g: \\bb R \\to \\bb R$ we say that $g = df\/dW$ if\n\\[\nf(x) = f(0) + \\int_0^x g(y) W(dy).\n\\]\nThen $\\mc E_n$ is $\\Gamma$-convergent to the quadratic form defined by $\\mc E(f) = \\int (df\/dW)^2 dW$. In this case, $\\mc L = d\/dx d\/dW$. A technical difficulty appears if $W(dx)$ has atoms. In that case, there is no kernel $\\mc K$ for $\\mc L$ contained in $\\mc C_c(\\bb R)$. To overcome this point, we define for $x \\leq y$, $d_W(x,y)=d_W(y,x)= W((x,y])$. The function $d_W$ is a metric in $\\bb R$, and in general $\\bb R$ is {\\em not} complete under this metric: an increasing sequence $x_n$ converging to $x$ is always a Cauchy sequence with respect to $d_W$, but $d_W(x_n,x)\\geq W(\\{x\\})$, which is non-zero if $x$ is an atom of $W$. Define $\\bb R_W = \\bb R \\cup \\{x-; W(\\{x\\})>0\\}$. It is easy to see that $\\bb R_W$ is a complete, separable space under the natural extension of $d_W$, and that continuous functions in $\\bb R_W$ are in bijection with c\\`adl\\`ag functions in $\\bb R$ with discontinuity points contained on the set of atoms of $W(dx)$. It is not difficult to see that the set of $W$-differentiable functions in $\\mc C_c(\\bb R_W)$ is a kernel for $\\mc L$ and that Theorems \\ref{t1} and \\ref{t3} apply to this setting. In \\cite{FJL}, the remarkable case on which $W(dx)$ is a {\\em random}, self-similar measure (an $\\alpha$-stable subordinator) was studied in great detail. \n\n\\subsection{Finitely ramified fractals} \n\nLet us consider the following sequence of graphs in $\\bb R^2$. Define $a_0=(0,0)$, $a_1=(1\/2,\\sqrt 3\/2),$ and $a_2= (1,0)$ and define $\\varphi_i: \\bb R^2 \\to \\bb R^2$ by taking $\\varphi_i(x) = (x+a_i)\/2$. Define $X_0= \\{a_0,a_,a_2\\}$ and $X_{n+1}= \\cup_i \\varphi_i(X_n)$ for $n \\geq 0$. For $x,y \\in X_0$ we define $\\omega^0_{x,y}=1$, we put $\\omega_{x,y}^0=0$ if $\\{x,y\\} \\subsetneq X$ and inductively we define\n\\[\n\\omega_{x,y}^{n+1} = 5 \\sum_i \\omega^n_{\\varphi^{-1}_i(x), \\varphi^{-1}(y)}.\n\\]\n\nThe set $X_n$ is a discrete approximation of the Sierpinski gasket $X$ defined as the unique compact, non-empty set $X$ such that $X = \\cup_i \\varphi_i(X)$. Here we are just saying that $\\omega_{x,y}^n = 5^n$ if $x,y$ are neighbors in the canonical sense. In this case $a_n=3^n$ and $\\mu$ is the Hausdorff measure in $X$. It has been proved \\cite{Kig} that the quadratic forms $\\mc E_n$ converge to a certain Dirichlet form $\\mc E$ which is used to define an abstract Laplacian in $X$. In particular, Theorems \\ref{t1} and \\ref{t3} apply to this model. This result was obtained in \\cite{Jar3} in the context of a zero-range process. The same result can be proved for general {\\em finitely ramified fractals}, in the framework of \\cite{Kig}.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nStars with initial masses in the range 0.8--8~M$_{\\odot}$ \nend their life with a phase of catastrophic mass-loss.\nDuring the asymptotic giant branch (AGB) phase, they develop \na superwind leading to mass-loss rates up to \n10$^{-4}$M$_{\\odot}$\\,yr$^{-1}$. This superwind enriches the \nISM with newly synthesized elements.\n\nThe mass-loss mechanism of AGB stars is likely due to \npulsations from the star and radiation pressure on dust \ngrains. Shocks due to pulsation extend the atmosphere, so \nthat the material ejected by the star becomes dense and \ncold enough for dust to form. Due to its opacity, dust \nabsorbs the radiation from the star and is driven away by \nradiation pressure, carrying the gas along through friction.\nTheoretical models (Winters et al.\\ 2000), show that the \nmass loss evolves from a pulsation driven regime \ncharacterized by a low mass-loss rate and a slow expansion \nvelocity to a dust-driven regime with a high mass-loss rate \nand a high expansion velocity ($>$5km.s$^{-1}$).\n \nStudying the effect of metallicity on the mass-loss \nprocess is important to understand the formation of dust \naround AGB stars in the early Universe. At low metallicity, \nless seeds are present for dust formation, so one might \nexpect dust formation to be less efficient and thus the \nmass-loss rates to be lower.\n\nTheoretical work by Bowen \\& Willson (1991) predicts that for \nmetallicities below [Fe\/H]$=-1$ dust-driven winds fail, and \nthe wind must stay pulsation-driven. However, observational \nevidence for any metallicity dependence is still very limited \n(Zijlstra 2004). More recent observational (Groenewegen et \nal.\\ 2007) and theoretical works (Wachter et al.\\ 2008; \nMattsson et al. 2008) indicate that the mass-loss rates from \ncarbon stars in metal-poor environments are similar to our \nGalaxy.\n\nTo obtain the first observational evidence on mass-loss rates \nat low metallicity, we have carried out several surveys with \nthe {\\it Spitzer Space Telescope} of stars in nearby dwarf \ngalaxies. These show significant mass-loss rates down to \nZ=1\/25 Z${_\\odot}$ (Lagadec et al.\\ 2007b; Matsuura et al.\\ \n2007; Sloan et al.\\ 2009), but only for carbon-rich stars. \nThe current evidence indicates that oxygen-rich stars have \nlower mass-loss rates at lower metallicities. For carbon \nstars, no evidence for a dependency of mass-loss rate on\nmetallicity has yet been uncovered. Consequently, (Lagadec \n\\& Zijlstra 2008) have proposed that the carbon-rich dust \nplays an important role in triggering the AGB superwind.\n\nThe main uncertainty arises from the unknown expansion \nvelocity. This parameter is needed to convert the density\ndistribution to a mass-loss rate. The expansion velocity is \nalso in itself a powerful tool. Hydrodynamical simulations \n(Winters et al.\\ 2000), have shown that radiation-driven \nwinds have expansion velocity in excess of 5\\,km\\,s$^{-1}$, \nwhile pulsation-driven winds are slower.\n\nThere is some evidence that expansion velocities are lower \nat low metallicity, from measurement of OH masers (Marshall \net al.\\ 2004). However, OH masers are found only in \noxygen-rich stars, which appear to have suppressed mass-loss\nat low metallicities. We lack equivalent measurements for\nmetal-poor carbon-rich stars, which do reach substantial \nmass-loss rates. For these, the only available velocity \ntracer is CO. Currently, extra-galactic stars are too \ndistant for CO measurements. However, there are a number of \ncarbon stars in the Galactic Halo, which are believed to \nhave similarly low metallicity. One CO measurement exists\nfor a Galactic Halo star: its expansion velocity has been \nestimated to be $\\sim$3.2 km\\,s$^{-1}$ through the \n$^{12}$CO $J=2 \\rightarrow 1$ transition (Groenewegen et \nal.\\ 1997). This is much lower than typical expansion \nvelocities, which are in the range 10-40 km\\,s$^{-1}$ for stars with similar colours,\n and thus optical depths (Fig.\\ref{histo_vexp}).\n\nA large number of metal-poor carbon stars have recently \nbeen discovered in the Galactic Halo (Totten \\& Irwin 1998; \nMauron et al.\\ 2004, 2005, 2007). These stars may have been \nstripped from the Sagittarius Dwarf Spheroidal galaxy (Sgr\ndSph), which has a metallicity of [Fe\/H]$\\sim-1.1$ (Van de \nBergh 2000). These stars are the closest metal-poor carbon \nstars known, and they are bright enough to be detected in CO \nusing ground-based millimeter telescopes.\n\nWe have carried out observations of six Halo carbon stars in \nthe CO J $= 3 \\rightarrow 2$ transition. Here, we report the \nresults of these observations.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{halo_plot.ps}\n\\caption{\\label{histo_vexp} Distribution of the expansion \nvelocity for the observed Halo carbon stars compared \nwith stars from the disc with similar J$-$K colors.}\n\\end{center}\n\\end{figure}\n\n \n\n\n\n\n\\section{Sample selection}\n\nMauron et al.\\ (2004, 2005, 2007) and Mauron (2008) have \ndiscovered $\\sim$ 100 carbon stars in the Galactic Halo, \nadding to the sample of $\\sim$ 50 described by Totten \\& \nIrwin (1998). All of these stars are spectroscopically \nconfirmed carbon stars. Only the brightest can be detected \nin the sub-millimeter range. We selected the six stars with \nthe highest IRAS 12$\\mu$m flux observable with the James \nClerk Maxwell Telescope (JCMT, Mauna Kea, Hawaii). The \nemission from an AGB star at 12$\\mu$m is due to thermal \nemission from the dust in the envelope. Thus one expects \nthe stars with the largest 12$\\mu$m flux to be the brightest \nin CO. All the observed stars have 3$$\\\\\n &&&days&kpc&kpc &kpc &kpc\\\\\n\\hline\n\n\nIRAS 04188+0122 & 192.1775&-31.9867&359 &7.3 &6.0 &6.4& 6.5$\\pm$0.6\\\\\nIRAS 08427+0338 & 223.4859&+26.8173&288 &6.6 &5.3 &5.1& 5.5$\\pm$0.8\\\\\nIRAS 11308-1020 & 273.6969&+47.7772&- &2.8 &2.1 &- & 2.5$\\pm$0.5\\\\\nIRAS 16339-0317 & 012.7346&+27.7944&- &5.6 &4.2 &- & 4.9$\\pm$1.0\\\\\nIRAS 12560+1656 & 312.2528&+79.4127&- &13.3&10.7&- & 12.0$\\pm$1.9\\\\\nIRAS 18120+4530 & 073.0530&+25.3482&408 &7.6 &5.7 &6.7& 6.7$\\pm$0.9\\\\ \n\n\n\n\\hline \\\\\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig_all_mc.ps}\n\\caption{\\label{fig_dist} Absolute K magnitude (M$_K$) of carbon stars as a fonction of their J-K infrared colours. \nDiamonds and squares represent LMC and SMC carbon stars respectively. The vertical dashed line represent the limit \nunder which relation Eq.\\ref{dist_lmc} is no longer valid.}\n\\end{center}\n\\end{figure}\n\n\nMost of the methods to determine mass-loss rates rely on an\naccurate distance to the observed stars. To determine \nthese distances, we applied three methods. Two are based on \nnear-infrared colours; the third is the period-luminosity \nrelationship.\n\n\nThe first method uses the infrared colours of the observed stars, \nusing the relation between M$_K$ and J$-$K determined by\nSloan et al. (2008). They found from a sample of carbon\nstars in the Small Magellanic Cloud (SMC) that:\n \n\\begin{equation} \n\\label{dist_smc}\nM_K = -9.18 + 0.395(J-K) \n\\end{equation} \n\nMauron et al.\\ (2008) compared the near-infrared 2MASS \nphotometry of stars in the Halo and the Large Magellanic\nCloud (LMC) and found a different relationship from the one \nabove between J$-$K and M$_K$. Their relation gives fainter \nM$_K$ values at a given J$-$K colour than that of Sloan et \nal., with the difference increasing with redder colours. We \nhave recalibrated the relation for the LMC, using the samples \nof stars confirmed to be carbon-rich with the Infrared \nSpectrograph on {\\it Spitzer} by Zijlstra et al.\\ (2006), \nBuchanan et al.\\ (2006), Leisenring et al.\\ (2008), and Sloan \net al.\\ (2008). We find that:\n\n\\begin{equation} \n\\label{dist_lmc}\nM_K = -9.94 + 0.754(J-K),\n\\end{equation} \n\n\\noindent for J$-$K colours greater than 2.0. \n This \ncalibration of the J$-$K relation is our second method.\nFig.\\, \\ref{fig_dist} compares the SMC and LMC calibrations of the relation\nbetween M$_K$ and J$-$K.\n\n\n\n\nThe scatter in the LMC sample is 0.81 magnitudes about the\nfitted line, compared to 0.51 magnitudes for the SMC sample.\nThe two fitted lines yield nearly identical distances at\nJ$-$K = 2, but as the colour grows redder, the samples\ndiverge from each other. At J$-$K = 5, the difference in\n$M_K$ is a full magnitude. For the colours in our sample,\nthe two methods yield results differening by 0.41 to 0.64\nmagnitudes, comparable to the spread in the SMC sample.\nThe different slopes in the two samples may result from\ntheir different metallicities, but that is only speculation\non our part.\n\n\nThe third method utilises the period-luminosity relation for \ncarbon Miras described by Feast et al.\\ (2006): \n \n\\begin{equation} \nM_{\\rm bol}=-2.54 \\log P+2.06, \n\\end{equation} \nWith an uncertainty of 0.24 magnitudes. \nWe derived the bolometric magnitudes using the equation for\nbolometric correction derived by Whitelock et al.\\ (2006), \nafter converting all of the photometry to the SAAO system as \ndescribed by Lagadec et al.\\ (2008):\n \n\\begin{eqnarray} \n\\nonumber {\\rm BC_K} & = & +\\, 0.972 + 2.9292\\times(J-K) \n -1.1144\\times(J-K)^2 \\\\ \n & & +0.1595\\times(J-K)^3 -9.5689\\,10^3(J-K)^4 \n\\end{eqnarray} \n\\noindent \n\nTable \\ref{gal} presents the obtained distances, D$_1$, D$_2$ \nand D$_3$ respectively. For those stars without periods, the \nfinal estimated distance is the average of D$_1$ and D$_2$. \nFor those stars with periods, the D$_1$ and D$_2$ values\nbracket D$_3$ in two of the three cases. Consequently, we \nfirst averaged D$_1$ and D$_2$, then averaged the result \nwith D$_3$ to arrive at our final estimate of the distance.\nThe uncertainties in the final estimated distances are the the standard deviation of the individual\ndistances. \n\n\\subsection{Galactic location of the stars}\n\\label{location}\n\nKnowing the distance to the stars and their $V_{\\rm lsr}$ \nallows us to study their location in the Galaxy. All six\nstars examined here have been classified previously as \nmembers of the Halo, based on their distances from the \nGalactic plane. Our CO observations give us velocity \ninformation for these stars, which we can compare to Galactic \nrotation models. Table \\ref{gal} lists the Galactic \ncoordinates l and b.\n\nFig.\\ref{aitoff} shows the location of our six carbon stars\non an Aitoff projection. The dashed line schematically \nrepresents the Sgr dSph orbit (Ibata et al. 2001). One of \nour stars, IRAS\\,12560+1656, lies very close to this orbit. Its \n$V_{\\rm lsr}$ is in the range of the observed $V_{\\rm lsr}$ \nfor stars in the Sgr dSph stream. IRAS\\,12560+1656 very likely belongs \nto this stream. Our observations are thus certainly the \nfirst CO observations of an extragalactic AGB star. Two \nother stars, IRAS\\,16339-0317 and IRAS\\,18120+4530, have large negative \n$V_{\\rm lsr}$, fully consistent with membership in the \nHalo. Finally, IRAS\\,04188+0122, IRAS\\,08427+0338, and IRAS\\,11308-1020 have a distance, \nlocation and $V_{\\rm lsr}$ consistent with membership in the \nthick disc. These three last stars thus have a metallicity \nbetween that of the thin Galactic disc and the Galactic \nHalo, the average metallicity of the thin disc being $\\sim$-0.17 while the one of the thick disc is \n$\\sim$-0.48 (Soubiran et al., 2003).\nOur sample thus contains three metal-poor AGB stars and three \nAGB stars with intermediate metallicity.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=13cm,angle=-90]{zz-myplot-last1.ps}\n\\caption{\\label{aitoff} The location of the six carbon\nstars on an Aitoff projection of the sky. Open circles \nrepresent Halo carbon stars from Mauron et al.\\ (2004, \n2005, 2007) and Totten \\& Irwin (1998) with J$-$K $>$1.2 \nand K$>$ 6 to eliminate carbon stars in the disc. Filled \nsymbols represent the six stars in our sample.}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Low expansion velocities in the Halo}\n\n\\label{lowvexp}\nThe present observations allow us to directly measure the \nexpansion velocity for a sample of carbon stars in the Halo.\nThe stars we observed are quite red (3$<$J$-$K$<$4), and have \nsubstantial circumstellar dusty envelopes responsible \nfor the observed reddening. Our measured expansion \nvelocities are in the range 3--16.5 km\\,s$^{-1}$. The \nexpansion velocity of carbon stars increases during the \nevolution on the AGB (Scho\\\"ier, 2007), i.e.\\ when the dusty \nenvelope becomes optically thicker. To compare the expansion \nvelocities we measured in Halo carbon stars with carbon stars\nin the disc, we took a sample of carbon stars in the disc \nwith colours similar to our sample. We selected all of the\ncarbon stars with 3$<$J$-$K$<$4 in the extensive catalogue of \nCO observations of evolved stars by Loup et al.\\ (1993). \nFig.\\ref{histo_vexp} compares the distribution of expansion \nvelocities in the two samples. The Halo carbon stars clearly\nhave a lower mean expansion velocity.\n\nThe three stars in the Halo and the Sgr dSph stream have \n$V_{\\rm exp}$ in the range 3--8.5 km\\,s$^{-1}$, while the \nthree stars associated with the thick disc have velocities \nranging from 11.5 to 16.5 km\\,s$^{-1}$. The latter range is\nat the low end of expansion velocities for AGB stars with\nsimilar near-infrared colours in the thin disc.\n\n\\subsection{Origin of the low expansion velocities}\n\nSection \\ref{location} and \\ref{lowvexp} have shown that the \nthree stars we observed in the Halo and Sgr dSph stream have \nlow expansion velocities. The stars we observed in the thick \ndisc have expansion velocities intermediate between those \ntypically observed in the Halo and the thin disc. This \ndifference could arise from differences in metallicity, with\nthe more metal-poor carbon stars having the slower winds.\n\nMattsson et al.\\ (2008) and Wachter et al.\\ (2008) have \nrecently conducted theoretical investigations of the winds \nfrom metal-poor carbon stars. Both studies show that \nmetal-poor carbon stars can develop high mass loss rates, \nleading to the formation of a large dusty envelope, in \nagreement with spectroscopic observations from {\\it Spitzer} \nof AGB stars in metal-poor galaxies (Zijlstra et al.\\ 2006; \nSloan et al.\\ 2006; Groenewegen et al.\\ 2007; Lagadec et \nal.\\ 2007; Matsuura et al.\\ 2007; Leisenring et al.\\ 2008;\nLagadec et al.\\ 2009; Sloan et al.\\ 2009). \n\nWachter et al.\\ (2008) predict that the outflow velocities \nfrom carbon stars should be lower in metal-poor environments,\nbecause of the lower gas-to-dust mass ratio and because the \nformation of less dust leads to less efficient acceleration \nof the wind outside of the sonic region. This interpretation\nis consistent with our interpretation that the low expansion \nvelocities we have observed in the Halo are due to their low \nmetallicities.\n\nHydrodynamical models (Winters et al.\\ 2000, Wachter et al.\\ \n2008) distinguish two types of models. Model A applies to\ncases where the radiation pressure on dust is efficient. \nMass-loss rates can exceed 10$^{-7}$M$_{\\odot}$\\,yr$^{-1}$, \nand expansion velocities can climb above $\\sim$5 km\\,$s^{-1}$.\nModel B applies to cases where pulsations drive the mass loss.\nIn these cases, the mass-loss rates and expansion velocities\nare smaller. The mass loss occurs in a two-step process, with\nstars first losing mass due to pulsations, followed by \nacceleration due to radiation pressure on the dust grains. The \nhigh mass-loss rate and low expansion velocity of TI~32 does not \nfit either of these models, possibly because metal-poor carbon\nstars can develop strong mass-loss from pulsation alone. \n\n\n\\section{Conclusions and perspectives}\n\nWe have detected the CO J\\,$=$\\,3$\\rightarrow$\\,2 in six carbon stars \nselected as Halo stars. Only one carbon star had been \ndetected in CO previously. Comparison of the infrared \nobservations and radiative transfer models indicates that \nthese stars are losing mass and producing dust. Their \nmass-loss rates are larger than their nuclear burning rates, \nso their final evolution will be driven by this mass-loss \nphenomenon.\n\nWe show that three of the observed stars are certainly \nmembers of the thick disc, while one is in the Sgr dSph stream \nand two are in the Halo. The CO observation of the Sgr dSph \nstream star are thus the first identified millimetre \nobservations of an extragalactic AGB star. The expansion \nvelocity we determined from our CO observations are lower \nthan those of carbon stars in the thin disc with similar \nnear-infrared colours. The observed carbon stars with the \nlowest expansion velocities are Halo or Sgr dSph stream carbon \nstars. There is a strong indication that the expansion winds \nare lower in metal-poor environments, which agrees with recent \ntheoretical models (Wachter et al.\\ 2008). \n\nSo far, the effect of metallicity on the mass-loss from \ncarbon-rich AGB stars has studied primarily from infrared \nobservations of AGB stars in Local Group galaxies, mostly \nwith the {\\it Spitzer Space Telescope}. Infrared \nobservations measure the infrared excess, which can \nbe converted to a mass-loss rates assuming an expansion \nvelocity for the circumstellar material. So far all of the \nmass-loss rates have been estimated using the assumptions \nthat the expansion velocity is independent of the \nmetallicity. The results presented here show that this \nassumption needs to be reconsidered.\n\nWe have recently obtained spectra of some of the present\nsample with the Infrared Spectrograph on {\\it Spitzer}.\nCombining our CO observations and these spectra together \nand comparing them to {\\it Spitzer} spectra from carbon \nstars in other galaxies in the Local Group will allow us \nto quantitatively study the mass-loss from carbon-rich AGB \nstars in the Local Group and its dependence on metallicity. \nThis will be the subject of a forthcoming paper.\n\n\n\n\n\\section*{Acknowledgments}\nEL wishes to thank the JAC staff for their great help \ncarrying out this program, and Rodrigo Ibata for useful \ndiscussions about membership of stars to the Sgr stream. \nwe thank the referee C. Loup for her useful comments that helped improving the quality of the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn~\\cite{Polyak} Polyak suggested a quantization $l_q(L)\\in\n\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ of the Bennequin \ninvariant of a\ngeneric cooriented oriented wave front $L\\subset{\\mathbb R}^2$. \nIn this paper we construct an \ninvariant $S(L)$ which is, in a sense, \na generalization of $l_q(L)$ to the case of\na wave front on an arbitrary surface $F$. \n\nIn the same paper~\\cite{Polyak} Polyak \nintroduced Arnold's~\\cite{Arnold} $J^+$-type\ninvariant of a front $L$ on an oriented surface $F$. It takes values in\n$H_1(ST^*F,\\frac{1}{2}{\\mathbb Z})$. \nWe show that $S(L)\\in \\frac{1}{2}{\\mathbb Z}[H_1(ST^*F)]$ is a refinement of this\ninvariant in the sense that it is taken to \nPolyak's invariant under the natural mapping \n$\\frac{1}{2}{\\mathbb Z}[H_1(ST^*F)]\\rightarrow H_1(ST^*F,\\frac{1}{2}{\\mathbb Z})$. \n\nFurther we generalize $S(L)$ to the case where $L$ is a wave front\non an orbifold.\n\nInvariant $S(L)$ is constructed in two steps. The first one consists in \nlifting of $L$ to the Legendrian knot\n$\\lambda$ in the $S^1$-fibration\n$\\pi:ST^*F\\rightarrow F$. The second step can be applied to any knot in an\n$S^1$-fibration, and \nit involves the structure of the fibration in a\ncrucial way. This step allows us to define the\n$S_K$ invariant of a knot $K$ in the total space $N$ of an \n$S^1$-fibration. Since ordinary knots are considered up to a rougher\nequivalence relation (ordinary isotopy versus Legendrian isotopy), in order\nfor $S_K$ to be well defined it has to take values in a quotient of \n${\\mathbb Z}[H_1(N)]$. This invariant is\ngeneralized to the case of a knot in a Seifert fibration, and this allows\nus to\ndefine $S(L)$ for wave fronts on orbifolds.\n\nAll these invariants are Vassiliev invariants of order one in an\nappropriate sense. \n\nFor each of these invariants we introduce its version with values in \nthe group of formal linear combinations of the \nfree homotopy classes of oriented curves in the\ntotal space of the corresponding fibration.\n\nThe first invariants of this kind were constructed by\nFiedler~\\cite{Fiedler} in the case\nof a knot $K$ in a ${\\mathbb R}^1$-fibration over a surface and by Aicardi in the\ncase of\na generic oriented cooriented wave front $L\\subset {\\mathbb R}^2$. \nThe connection between\nthese invariants and $S_K$ is discussed in~\\cite{Tchernov}. \n\n\n\nThe space $ST^*F$ is naturally fibered over a surface $F$ with a fiber\n$S^1$. In~\\cite{Turaev} Turaev introduced a shadow description \nof a knot $K$ in\nan oriented three dimensional manifold $N$ fibered over an oriented surface \nwith a fiber \n$S^1$. A shadow presentation of a knot $K$ is a generic projection of $K$,\ntogether with an assignment of numbers to regions. It describes a knot type\nmodulo a natural action of $H_1(F)$.\nIt appeared to be a very useful\ntool. Many invariants of knots in $S^1$-fibrations, in particular quantum\nstate sums, can be expressed\nas state sums for their shadows. In this paper\nwe construct shadows of Legendrian liftings of wave fronts. \nThis allows one to use any\ninvariant already known for shadows in the case of wave fronts. \n\nHowever, in this paper shadows are used mainly for the purpose of depicting\nknots in $S^1$-fibrations.\n\n\n\n\n\\section{Shadows}\\label{sh-def}\n\\subsection{Preliminary constructions}\\label{prelim}\n\nWe say that a one-dimensional submanifold $L$ of a total space $N^3$ of \na fibration \n$\\pi:N^3\\rightarrow F^2$ is {\\em generic\nwith respect to $\\pi$\\\/ } if $\\pi\\big|_L$ is a generic immersion.\nRecall that an immersion of 1-manifold into a surface is said to\nbe {\\em generic\\\/} if it has neither self-intersection points with\nmultiplicity greater than $2$ nor self-tangency points, and at\neach double point its branches are transversal to each other. An\nimmersion of (a circle) $S^1$ to a surface is called a {\\em curve\\\/}.\n\n\n\nLet $\\pi$ be an oriented $S^1$-fibration of $N$ over an oriented\nclosed surface $F$.\n\n$N$ admits a fixed point free involution which preserves fibers.\nLet $\\tilde N$ be the quotient of $N$ by\nthis involution, and let $p:N\\rightarrow\\tilde{N}$ be the corresponding double\ncovering. Each fiber of $p$ (a pair of antipodal points) is\ncontained in a fiber of $\\pi$. Therefore, $\\pi$ factorizes through\n$p$ and we have a fibration $\\tilde{\\pi}:\\tilde{N}\\rightarrow F$.\nFibers of $\\tilde\\pi$ are projective lines. They are\nhomeomorphic to circles.\n\n\nAn isotopy of a link $L\\subset N$ is said to be vertical with respect to\n$\\pi$ if each point of $L$ moves along a fiber of $\\pi$. It is clear\nthat if two links are vertically isotopic, then their projections\ncoincide. Using vertical isotopy we can modify each generic\nlink $L$ in such a way that any two points of $L$ belonging to the\nsame fiber lie in the same orbit of the involution. \nDenote the obtained generic link by $L'$.\n\n\nLet $\\tilde{L}=p(L')$. It is obtained from $L'$ by gluing together\npoints lying over the same point of $F$. Hence $\\tilde{\\pi}$ maps\n$\\tilde{L}$ bijectively to $\\pi(L)=\\pi(L')$. Let $r:\\pi (L)\\rightarrow\n\\tilde{L}$ be an inverse bijection. It is a section of\n$\\tilde{\\pi}$ over $\\pi (L)$.\n\nFor a generic non-empty collection of curves on a surface by a {\\em region\\\/}\nwe mean the closure of a connected component of the complement of\nthis collection. Let $X$ be a region for $\\pi (L)$ on $F$. Then \n$\\tilde{\\pi}\\big|_X$ is a trivial fibration. Hence we can identify it with\nthe projection $S^1 \\times X\\rightarrow X$. Let $\\phi$ be a\ncomposition of the section $r\\big|_{\\partial X}$ with the projection to $S^1$.\nIt maps $\\partial X$ to $S^1$.\nDenote by $\\alpha_X$ the degree of $\\phi$. (This is actually an\nobstruction to an extension of $r\\big|_{\\partial X}$ to $X$.) One can\nsee that $\\alpha_X$ does not depend on the choice of the trivialization\nof $\\tilde{\\pi}$ and on the choice of $L'$.\n\n\\subsection{Basic definitions and properties}\n\\begin{defin}\\label{def-shadowlink}\nThe number $\\frac{1}{2}\\alpha _X$ corresponding to a region $X$ is\ncalled the {\\em gleam\\\/} of $X$ and is denoted by $\\protect\\operatorname{gl}(X)$.\nA {\\em shadow\\ $s(L)$ of a generic link $L\\subset N$\\\/} is a (generic)\ncollection of curves $\\pi (L)\\subset F$ with the gleams assigned to each\nregion $X$. The sum of\ngleams over all regions is said to be the {\\em total gleam\\\/} of the\nshadow. \n\\end{defin}\n\n\\begin{emf}\\label{prop1-gleam}\nOne can check that for any region $X$ the integer $\\alpha _X$ is congruent\nmodulo 2 to the number of corners of $X$. Therefore, $\\protect\\operatorname{gl}(X)$ is an integer if\nthe region $X$ has even number of corners and half-integer otherwise.\n\\end{emf}\n\n\\begin{emf}\\label{prop2-gleam}\nThe total gleam of the shadow is equal to the Euler number of $\\pi$.\n\\end{emf}\n\n\\begin{defin}\\label{def-shadow}\nA {\\em shadow \\\/} on $F$ is a generic collection of curves together\nwith the numbers $\\protect\\operatorname{gl}(X)$ assigned to each region $X$. These numbers\ncan be either integers or half-integers, and they should satisfy the\nconditions of~\\ref{prop1-gleam}~and~\\ref{prop2-gleam}.\n\\end{defin}\n\nThere are three local moves $S_1,S_2$, and $S_3$ of shadows shown in\nFigure~\\ref{shad3.fig}. They are similar to the well-known\nRiedemeister moves of planar knot diagrams.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{shad3.eps}\n \\end{center}\n\\caption{}\\label{shad3.fig}\n\\end{figure}\n\n\\begin{defin} Two shadows are said to be {\\em shadow equivalent\\\/} if\nthey can be transformed to each other by a finite sequence of moves\n$S_1,S_2,S_3$, and their inverses.\n\\end{defin}\n\n\\begin{emf}\n There are two more important shadow moves $\\bar S_1$ and $\\bar S_3$ \nshown in Figure~\\ref{shad5.fig}. They are similar to the previous versions of \nthe first and the third Riedemeister\nmoves and can be expressed \nin terms of $S_1,S_2$, $S_3$, and their inverses.\n\\end{emf}\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{shad5.eps}\n \\end{center}\n\\caption{}\\label{shad5.fig}\n\\end{figure}\n\n\n\\begin{emf}\\label{actionconstruct}\nIn~\\cite{Turaev} the action of \n$H_1(F)$ on the set of all isotopy\ntypes of links in $N$ is constructed as follows. \nLet $L$ be a generic link in $N$ and $\\beta$ an\noriented (possibly self-intersecting) curve on $F$ presenting\na homology class $[\\beta ]\\in H_1(F)$. Deforming $\\beta $ we can\nassume that $\\beta $ intersects $\\pi(K)$ transversally at a finite\nnumber of points different from the self-intersection points of $\\pi(K)$.\nDenote by $\\alpha =[a,b]$ a small segment of $L$ such that\n$\\pi(\\alpha )$ contains exactly one intersection point $c$ of $\\pi(L)$\nand $\\beta $. Assume that $\\pi(a)$ lies to the left, and\n$\\pi(b)$ to the right of $\\beta $. Replace $\\alpha $ by the arc\n$\\alpha '$ shown in Figure~\\ref{shad13.fig}.\n\\begin{figure}[htbp]\n\\begin{center}\n\\epsfxsize 5cm\n \\leavevmode\\epsffile{shad13.eps}\n \\end{center}\n\\caption{}\\label{shad13.fig}\n\\end{figure}\n We will call this\ntransformation of $L$ a {\\em fiber fusion\\\/} over the point $c$.\nAfter we apply fiber fusion to $L$ over all points of $\\pi(L)\\cap\\beta$\nwe get a new generic link $L'$ with $\\pi(L)=\\pi(L')$. One can notice that\nthe shadows of $K$ and $K'$ coincide. Indeed, each time $\\beta $\nenters a region $X$ of $s(L)$, it must leave it. \nHence the contributions of the newly inserted arcs to\nthe gleam of $X$ cancel out. Thus links belonging to one\n$H_1(F)$-orbit always produce the same shadow-link on $F$.\n\n\\end{emf}\n\n\\begin{thm}[Turaev \\cite{Turaev}]\\label{action}\nLet $N$ be an oriented closed manifold, $F$ an oriented surface, and \n$\\pi:N\\rightarrow F$ an\n$S^1$-fibration with the Euler number $\\chi (\\pi)$. The\nmapping that associates to each link $L\\subset N$ its shadow equivalence\nclass on $F$ establishes a bijective correspondence between the\nset of isotopy types of links in $N$ modulo the action of\n$H_1(F)$ and the set of all shadow equivalence classes on $F$ \nwith the total gleam $\\chi(\\pi)$. \n\\end{thm}\n\n\n\\begin{emf}\\label{homolfusion}\nIt is easy to see that all links whose projections represent \n$0\\in H_1(F)$ and whose shadows coincide are homologous \nto each other. To prove this,\none looks at the description of a fiber fusion and notices that to each\nfiber fusion where we add a positive fiber corresponds another where we\nadd a negative one. Thus the numbers of positively and negatively \noriented fibers we add are equal, and they cancel\nout.\n\\end{emf}\n\n\\begin{emf}\\label{shadowgeneral}\nAs it was remarked in~\\cite{Turaev} it is easy to transfer the construction \nof shadows and Theorem~\\ref{action} to the \ncase where $F$ is a non-closed oriented surface and $N$ is an oriented\nmanifold. In order\nto define the gleams of the regions that have a non-compact closure or\ncontain components of $\\partial F$, we have to choose a section of the fibration \nover all boundary components and ends of $F$. In the case of non-closed $F$\nthe total gleam of the shadow is equal to the obstruction \nto the extension of the section to the entire surface.\n\\end{emf}\n\n\n\\section{Invariants of knots in $S^1$-fibrations.}\n\\subsection{Main constructions}\n\nIn this section we deal with knots in an $S^1$-fibration\n$\\pi$ of an oriented three-dimensional manifold $N$ over an oriented \nsurface $F$. We do not assume $F$ and $N$ to be\nclosed. As it was said in~\\ref{shadowgeneral}, all theorems \nfrom the previous section are applicable in this case.\n\n\n\n\n\n\n\n\\begin{defin}[of $S_K$]\\label{SK} Orientations of $N$ and $F$ determine an \norientation of a fiber of the fibration. Denote by $f\\in H_1(N)$ \nthe homology class of a positively oriented fiber.\n\n\nLet $K\\subset N$ be an oriented knot which is generic with respect to $\\pi$.\nLet $v$ be a double point of $\\pi (K)$. The fiber $\\pi^{-1}(v)$ divides $K$ \ninto two arcs that inherit the orientation from $K$. Complete each arc\nof $K$ to an oriented knot by adding the arc of $\\pi^{-1}(v)$ \nsuch that the orientations of these\ntwo arcs define an orientation of their union. The orientations of $F$ and\n$\\pi(K)$ allow one to identify a small neighborhood of $v$ in $F$ with a\nmodel picture shown in Figure~\\ref{shad1.fig}a. Denote the knots\nobtained by the operation above by $\\mu^+_v$ and $\\mu^-_v$ as shown in\nFigure~\\ref{shad1.fig}. We will often call this construction a {\\em\nsplitting\\\/} of $K$ (with respect to the orientation of $K$).\n\nThis splitting can be described in terms of shadows as follows.\nNote that $\\mu ^+_v$ and $\\mu ^+_v$ are not in general position. We slightly \ndeform them in a neighborhood of $\\pi^{-1}(v)$, so that\n$\\pi(\\mu ^+_v)$ and $\\pi(\\mu ^+_v)$ \ndo not have double points in the neighborhood of $v$.\nLet $P$ be a neighborhood of $v$ in $F$\nhomeomorphic to a closed disk. \nFix a section over $\\partial P$ such that the intersection\npoints of $K\\cap\\pi^{-1}(\\partial P)$ belong to the section. Inside $P$ we can\nconstruct Turaev's shadow (see \\ref{shadowgeneral}). \nThe action of $H_1(\\protect\\operatorname{Int} P)=e$ \non the set of the isotopy types of links is trivial. \nThus the part of $K$ can be reconstructed in the unique way \n(up to an isotopy fixed on $\\partial P$) from the shadow over $P$\n(see~\\ref{shadowgeneral}). \nThe shadows for $\\mu^+_v$ and $\\mu^-_v$ are \nshown in Figures~\\ref{shad1.fig}a~and~\\ref{shad1.fig}b respectively. \n\n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 10 cm\n\\leavevmode\\epsffile{shad1.eps}\n \\end{center}\n\\caption{}\\label{shad1.fig}\n\\end{figure}\n\n\nRegions for the shadows $s(\\mu ^+_v)$ and \n$s(\\mu ^-_v)$ are, in fact, unions of regions for $s(K)$. One should think of \ngleams as of measure, so that the gleam of a region is the sum \nof all numbers inside.\n\n\nLet $H$ be the quotient of the group ring ${\\mathbb Z} [H_1(N)]$ (viewed as a\n${\\mathbb Z}$-module) \nby the submodule generated by $\\bigl\\{ [K]-f, [K]f-e \\bigr\\} $. \nHere by $[K]\\in H_1(N)$ we denote the homology class represented by the image\nof $K$.\n\nFinally define $S_K\\in H$ by the following\nformula, where the summation is taken over all double points $v$\nof $\\pi (K)$:\n\n\\begin{equation}\\label{eqSK}\nS_K=\\sum_v \\bigl([\\mu ^+_v]-[\\mu ^-_v]\\bigr).\n\\end{equation}\n\\end{defin}\n\n\\begin{emf}\\label{sum-property} Since $\\mu^+_v\\cup\\mu^-_v=K\\cup\\pi^{-1}(v)$\nwe have\n\\begin{equation}\n[\\mu ^+_v][\\mu ^-_v]= [K]f.\n\\end{equation}\n\\end{emf}\n\n\n\n\\begin{thm}\\label{correct2} $S_K$ is an isotopy invariant of the knot $K$.\n\\end{thm}\n\n\nFor the proof of Theorem~\\ref{correct2} see Subsection~\\ref{pfcorrect2}.\n\n\n\n\\begin{emf}\\label{stupid}\nIt follows from~\\ref{sum-property} \nthat $S_K$ can also be described as an element of ${\\mathbb Z}[H_1(N)]$ equal to the\nsum of $\\bigl([\\mu^+_v]-[\\mu^-_v]\\bigr)$ over all double points for \nwhich the sets $\\{ [\\mu ^+_v],[\\mu ^-_v] \\}$ and $\\{ e,f \\}$ are disjoint. \nNote\nthat in this case we do not need to factorize ${\\mathbb Z}[H_1(N)]$ to make $S_K$\nwell defined.\n\\end{emf}\n\n\n\\begin{emf}\\label{homotopvers}\nOne can obtain an invariant similar to $S_K$ with\nvalues in the free ${\\mathbb Z}$-module generated by the set of all free \nhomotopy classes of oriented curves in $N$. To do this one substitutes\nthe homology classes of $\\mu ^+_v$ and $\\mu ^-_v$ in~\\eqref{eqSK} \nwith their free homotopy\nclasses and takes the summation over the set of all double points $v$ of \n$\\pi (K)$ such that neither one of the knots $\\mu ^+_v$ and $\\mu ^-_v$ \nis homotopic to\na trivial loop and neither one of them is homotopic to a positively\noriented fiber\n(see~\\ref{stupid}).\n\nTo prove that this is indeed an invariant of $K$ one can easily modify the\nproof of Theorem~\\ref{correct2}. \n\\end{emf}\n\n\\subsection{$S_K$ is a Vassiliev invariant of order one}\n\n\\begin{emf} Let $\\pi:N\\rightarrow F$ be an $S^1$-fibration over a\nsurface. Let $K\\subset N$ be a knot generic with respect to $\\pi$ and $v$ a\ndouble point of $\\pi(K)$. A modification of pushing of one branch of $K$ \nthrough the other along the fiber $\\pi ^{-1}(v)$ \nis called a {\\em modification of $K$ along the fiber\\\/} $\\pi^{-1}(v)$.\n\\end{emf}\n\n\\begin{emf}\\label{helpdelta}\nIf a fiber fusion increases by one the gleam $\\gamma$ \nin Figure~\\ref{shad1.fig}b, then $[\\mu ^+_v]$ is multiplied by $f$. \nIf a fiber fusion increases by one the gleam $\\alpha$ \nin Figure~\\ref{shad1.fig}c, then $[\\mu ^-_v]$ is multiplied by $f^{-1}$. \nThese facts are easy to verify.\n\\end{emf} \n\n\n\\begin{emf}\\label{vassiliev2} \nLet us find out how $S_K$ changes under the modification along a fiber over a\ndouble point $v$. Consider a singular knot $K'$\n(whose only singularity is a point $v$ of transverse\nself-intersection). Let $\\xi_1$ and $\\xi_2$ be the homology classes of the\ntwo loops of $K'$ adjacent to $v$. The two\nresolutions of this double point correspond to adding $\\pm\\frac{1}{2}$ to\nthe gleams of the regions adjacent to $v$ in two ways shown in\nFigures~\\ref{shad14.fig}b and~\\ref{shad14.fig}c. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 12cm\n \\leavevmode\\epsffile{shad14.eps}\n \\end{center}\n\\caption{}\\label{shad14.fig}\n\\end{figure} \n\nUsing~\\ref{helpdelta} one verifies that under the corresponding modification \n$S_K$ changes by \n\\begin{equation}\\label{type2}\n(f-e)(\\xi_1+\\xi_2).\n\\end{equation}\n\n\nThis means that the first derivative of $S_K$ depends\nonly on the homology classes of the two loops adjacent to the singular\npoint. Hence the second derivative of $S_K$ is $0$. Thus it is a Vassiliev \ninvariant of order one in the usual sense.\n\n\n\nFor similar reasons\nthe version of $S_K$ with values in the free ${\\mathbb Z}$-module generated \nby all free homotopy classes of oriented curves in $N$ is also a Vassiliev\ninvariant of order one.\n\\end{emf}\n\n\n\\begin{thm}\\label{realization2}$ $\n\\begin{description}\n\\item[\\textrm{I}] If $K$ and $K'$ are two knots representing the same free\nhomotopy class, then $S_{K}$ and $S_{K'}$ are congruent modulo \nthe submodule generated by elements of form\n\\begin{equation}\\label{type}\n(f-e)(j+[K]j^{-1})\n\\end{equation}\nfor $j\\in H_1(N)$.\n\n\\item[\\textrm{II}] If $K$ is a knot, and $S\\in H$ is congruent to $S_{K}$ modulo the \nsubmodule generated by elements of \nform~\\eqref{type} (for $j\\in H_1(N)$), \nthen there exists a knot $K'$ such that:\n\n\\begin{description}\n\\item[a] $K$ and $K'$ represent the same free homotopy class;\n\n\n\\item[b] $S_{K'}=S$.\n\\end{description}\n\\end{description}\n\\end{thm}\n\nFor the proof of Theorem~\\ref{realization2} see\nsubsection~\\ref{pfrealization2}. \n\n\n\\subsection{Example.}\nIf $N$ is a solid torus $T$ fibered over a disk, \nthen we can calculate the value\nof $S_K$ directly from the shadow of $K$.\n\n\n\\begin{defin} Let $C$ be an oriented closed curve in ${\\mathbb R}^2$ and $X$ a\nregion for $C$. Take a point $x\\in\\protect\\operatorname{Int} X$ and connect it to a point near\ninfinity by a generic oriented path $D$. \nDefine the sign of an intersection point of $C$ and $D$ as \nshown in Figure~\\ref{shad12.fig}. Let $\\protect\\operatorname{ind}_C X$ be the sum over\nall intersection points of $C$ and $D$ of the signs of these points.\n\\end{defin}\n\n\\begin{figure}[htb]\n\\begin{center}\n \\epsfxsize\\hsize\\advance\\epsfxsize -0.5cm\n \\leavevmode\\epsffile{shad12.eps}\n \\end{center}\n\\caption{}\\label{shad12.fig}\n\\end{figure}\n\nIt is easy to see that $\\protect\\operatorname{ind}_C(X)$ is independent on the choices \nof $x$ and $D$.\n\n\\begin{defin} Let $K\\subset T$ be an oriented knot which is\ngeneric with respect to $\\pi$, and let $s(K)$ be its\nshadow. Define $\\sigma(s(K))\\in {\\mathbb Z}$ as the following sum \nover all regions $X$ for $\\pi (K)$: \n\\begin{equation}\n\\sigma(s(K))=\\sum_X\\protect\\operatorname{ind}_{\\pi(K)}(X)\\protect\\operatorname{gl}(X).\n\\end{equation}\n\\end{defin}\n\n\nDenote by $h\\in {\\mathbb Z}$ the image of $[K]$ under the natural identification of \n$H_1(T)$ with ${\\mathbb Z}$. \n\n\\begin{lem}\\label{homtor}\n$\\sigma(s(K))=h.$ \n\\end{lem}\n\n\\begin{emf}Put \n\n\\begin{equation}\nS'_K=\\sum t^{\\sigma(s(\\mu ^+_v))}-t^{\\sigma(s(\\mu ^-_v))},\n\\end{equation}\nwhere the sum is taken over all double points $v$ of $\\pi (K)$\nsuch that $\\{ 0,1\\}$ \nand $\\{ \\sigma(s(\\mu ^+_v)),\\sigma(s(\\mu ^-_v))\\}$ are disjoint\n(see~\\ref{stupid}).\n \nLemma~\\ref{homtor} implies that $S'_K$ is the image of $S_K$ under the\nnatural identification of ${\\mathbb Z}[H_1(T)]$ with the ring of\nfinite Laurent polynomials (see~\\ref{stupid}).\n\n\\end{emf}\n\nOne can show~\\cite{Tchernov} that $S'_K$ and Aicardi's partial\nlinking polynomial of $K$ (which was introduced in~\\cite{Aicardi})\ncan be explicitly expressed in terms of each other. \n\n\n\n\\subsection{Further generalizations of the $S_K$ invariant}\nOne can show that an invariant similar to $S_K$ can be introduced in the\ncase where $N$ is oriented and $F$ is non-orientable.\n\n\\begin{defin}[of $\\tilde S_K$]\\label{tildeSK}\nLet $N$ be oriented and $F$ non-orientable.\nLet $K\\subset N$ be an oriented knot generic with respect to $\\pi$, and \nlet $v$ be a double point of $\\pi(K)$. Fix an orientation of a small\nneighborhood of $v$ in $F$. Since $N$ is oriented\nthis induces an orientation\nof the fiber $\\pi^{-1}(v)$. Similarly to the definition of $S_K$\n(see~\\ref{SK}), we split our knot with respect to the \norientation and obtain two\nknots $\\mu_1^+(v)$ and $\\mu_1^-(v)$. Then we take the other orientation \nof the neighborhood of $v$ in $F$, and in the same way we obtain another pair\nof knots $\\mu_2^+(v)$ and $\\mu_2^-(v)$. The element \n$\\bigl( [\\mu_1^+(v)]-[\\mu_1^-(v)]+[\\mu_2^+(v)]-[\\mu_2^-(v)]\\bigr)\\in\n{\\mathbb Z}[H_1(N)]$ does not depend on which orientation of the neighborhood of $v$\nwe choose first. \n\nSimilarly to the definition of $S_K$, we can describe all\nthis in terms of shadows as it is shown in Figure~\\ref{shadun2.fig}.\nThese shadows are constructed with respect to the same orientation of the\nneighborhood of $v$.\n\n\n\n\nLet $f$ be the homology class of a fiber of $\\pi$ oriented in some way. \nAs one can easily prove $f^2=e$, so it\ndoes not matter which orientation we choose to define $f$.\nLet $\\tilde H$ be the quotient of ${\\mathbb Z} [H_1(N)]$ (viewed as a ${\\mathbb Z}$-module) by the \n${\\mathbb Z}$-submodule\ngenerated by $\\Bigl \\{ [K]-f+e-[K]f=(e-f)([K]+e) \\Bigr\\}$.\nFinally define $\\tilde S_K\\in \\tilde H$ by the\nfollowing formula, where the summation is taken \nover all double points\n$v$ of $\\pi (K)$: \n\\begin{equation}\\label{eqtildeSK}\n\\tilde \nS_K=\\sum_v \\Bigl([\\mu _1^+(v)]-[\\mu _1^-(v)]+[\\mu _2^+(v)]-[\\mu\n_2^-(v)]\\Bigr).\n\\end{equation}\n\\end{defin}\n\n\\begin{figure}[htb]\n \\begin{center}\n\n \\epsfxsize 12cm\n \\leavevmode\\epsffile{shadun2.eps}\n \\end{center}\n\\caption{}\\label{shadun2.fig}\n\\end{figure}\n\n\\begin{thm}\\label{correct3} $\\tilde \nS_K$ is an isotopy invariant of the knot $K$.\n\\end{thm}\n\nThe proof is essentially the same as the proof of Theorem~\\ref{correct2}.\n \n\n\n\\begin{emf} One can easily prove that $\\tilde S_K$ \ninvariant satisfies\nrelations similar to~\\eqref{type2}. In particular, $\\tilde S_K$ is also a\nVassiliev invariant of order one. \n\nOne can introduce a version of this invariant with values in the free\n${\\mathbb Z}$-module generated by all free homotopy classes of oriented curves in \n$N$. \nTo do this, we substitute the homology classes of $\\mu _1^+(v),$\n$\\mu _1^-(v),$ $\\mu _2^+(v)$, and $\\mu_2^-(v)$ with the corresponding \nfree homotopy classes. The summation should be taken over the set of all\ndouble points of $\\pi(K)$ for which neither one of $\\mu _1^+(v),\n\\mu _1^-(v),\\mu _2^+(v)$, and $\\mu_2^-(v)$ is homotopic to a trivial loop and\nneither one of them is homotopic to a fiber of $\\pi$. To prove that this is\nindeed an invariant of $K$, one easily modifies the proof of\nTheorem~\\ref{correct2}. \n\\end{emf} \n\n\n\n\n\n\n\n\\section{Invariants of knots in Seifert fibered spaces}\n\nLet $(\\mu, \\nu)$ be a pair of relatively prime integers. Let \n$$D^2=\\Bigl\\{ (r,\\theta); 0\\leq r\\leq 1, 0\\leq \\theta \\leq 2\\pi\\Bigr\\}\\subset\n{\\mathbb R} ^2$$\nbe the unit disk defined in polar coordinates. A fibered solid torus of \ntype $(\\mu, \\nu )$ is the quotient space of the cylinder $D^2\\times I$ via\nthe identification $\\bigl (\\bigl (r,\\theta \\bigr ),1\\bigr )=\n\\bigl (\\bigl (r,\\theta+\\frac{2\\pi \\nu}{\\mu}\\bigr\n),0\\bigr)$. The fibers are the images of the curves ${x}\\times I$. The\ninteger $\\mu$ is called the index or the multiplicity. For $|\\mu| >1$ the\nfibered solid torus is said to be {\\em exceptionally fibered,\\\/} and the\nfiber that is the image of $0\\times I$ is called the {\\em exceptional fiber\\\/}. \nOtherwise the\nfibered solid torus is said to be {\\em regularly fibered,\\\/} \nand each fiber is a {\\em regular fiber\\\/}.\n\n\\begin{defin}\\label{Seifert}\nAn orientable three manifold $S$ is said to be a \n{\\em Seifert fibered manifold\\\/}\nif it is a union of pairwise disjoint closed curves, called fibers, such\nthat each one has a closed neighborhood which is a union of fibers and\nis homeomorphic to a fibered solid torus by a fiber preserving\nhomeomorphism.\n\\end{defin}\n\nA fiber $h$ is called {\\em exceptional\\\/} if $h$ has a neighborhood \nhomeomorphic to\nan exceptionally fibered solid torus (by a fiber preserving homeomorphism), \nand $h$ corresponds via the\nhomeomorphism to the exceptional fiber of the solid torus. If $\\partial S\\neq\n\\emptyset$,\nthen $\\partial S$ should be a union of regular fibers. \n\nThe quotient space obtained from a Seifert fibered manifold $S$ by\nidentifying each fiber to a point is a 2-manifold. It is called the orbit\nspace and the images of the exceptional fibers are called\n{\\em the cone points.\\\/}\n\n\n\\begin{emf}\\label{numbers}\nFor an exceptional fiber $a$ of\nan oriented Seifert fibered manifold there is a unique pair of relatively\nprime integers $(\\mu_a, \\nu_a)$ such that $\\mu_a>0$, $|\\nu_a|<\\mu_a$, and a\nneighborhood of $a$ \nis homeomorphic (by a fiber preserving homeomorphism) to a\nfibered solid torus of type $(\\mu_a, \\nu_a)$. We call the pair $(\\mu_a,\n\\nu_a)$ {\\em the type of the exceptional fiber\\\/} $a$. We also call this\npair the type of the corresponding cone point. \n\\end{emf}\n\n\nWe can define an invariant of an oriented knot\nin a Seifert fibered manifold that is similar to the $S_K$ invariant.\n\nClearly any $S^{1}$-fibration can be viewed as a Seifert fibration without\ncone points. This justifies the notation in the definition below.\n\n\\begin{defin}[of $S_K$]\\label{SeifertSK}\nLet $N$ be an oriented Seifert fibered manifold with an oriented orbit space \n$F$. Let $\\pi :N\\rightarrow F$ be the corresponding fibration and $K\\subset\nN$ an oriented knot in general position with respect to $\\pi$. Assume\nalso that $K$ does not intersect the exceptional fibers. For each double\npoint $v$ of $\\pi (K)$ we split $K$ into $\\mu ^+_v$ and $\\mu ^-_v$ (see\n~\\ref{SK}). Let $A$ be the set of all exceptional\nfibers. Since $N$ and $F$ are oriented, we have an induced orientation of\neach exceptional fiber $a\\in A$. For $a\\in A$ set $f_a$ to be the homology \nclass of the fiber with this orientation. For $a\\in A$ of type $(\\mu_a,\n\\nu_a)$ (see~\\ref{numbers})\nset $N_1(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } mod\n\\text{ }2\\pi\\in (0,\\pi]\\bigr\\}$, \n$N_2(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } mod\n\\text{ }2\\pi\\in (0,\\pi)\\bigr\\}$. Define $R^1_a, R^2_a\\in {\\mathbb Z}[H_1(N)]$ by the\nfollowing formulas:\n\n\\begin{equation}\nR^1_a=\\sum_{k\\in N_1(a)}\\bigl([K]f_a^{\\mu_a-k}-f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl(f_a^{\\mu_a-k}-[K]f_a^k\\bigr),\n\\end{equation}\n\\begin{equation}\nR^2_a=\\sum_{k\\in N_1(a)}\\bigl(f_a^{\\mu_a-k}-[K]f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl([K]f_a^{\\mu_a-k}-f_a^k\\bigr).\n\\end{equation}\n\nLet $H$ be the quotient of ${\\mathbb Z} [(H_1(N)]$ (viewed as a ${\\mathbb Z}$-module) \nby the free ${\\mathbb Z}$-submodule \ngenerated by \n$\\Bigl \\{[K]f-e, [K]-f, \\bigl \\{R^1_a, R^2_a\\bigr\\} _{a\\in A}\\Bigr \\}$.\nFinally, define $S_K\\in H$ by the following formula, where the \nsummation is taken over all double points $v$ of\n$\\pi (K)$:\n\n\\begin{equation}\\label{SSK}\nS_K=\\sum_v\\bigl( [\\mu ^+_v]-[\\mu ^-_v]\\bigr).\n\\end{equation}\n \n\\end{defin}\n\n\\begin{thm}\\label{correctSSK}\n$S_K$ is an isotopy invariant of the knot $K$.\n\\end{thm}\n\nFor the proof of Theorem~\\ref{correctSSK} see Subsection~\\ref{correctSSK}.\n\nWe introduce a similar invariant in the case \nwhere $N$ is oriented and $F$ is non-orientable.\n\n\\begin{defin}[of $\\tilde S_K$]\\label{tilde1SK}\n\nLet $N$ be an oriented Seifert fibered manifold with a non-orientable orbit\nspace $F$. Let $\\pi:N\\rightarrow F$ be the corresponding fibration and\n$K\\subset N$ an oriented knot in general position with respect to $\\pi$.\nAssume also that $K$ does not intersect the exceptional fibers. For each\ndouble point $v$ of $\\pi(K)$ we split $K$ into \n$\\mu_1^+(v),$ $\\mu_1^-(v),$ $\\mu_2^+(v)$, and $\\mu_2^-(v)$ \nas in~\\ref{tildeSK}. The element \n$\\bigl([\\mu_1^+(v)]-[\\mu_1^-(v)]+[\\mu_2^+(v)]-[\\mu_2^-(v)]\\bigr)\\in\n{\\mathbb Z}[H_1(N)]$ \nis well\ndefined. \n\nDenote by $f$ the homology class of a regular fiber oriented in some way. \nNote that $f^2=e$, so the orientation we use to define $f$ does not matter. \nFor a cone point $a$ denote by $f_a$ \nthe homology class of the fiber $\\pi^{-1}(a)$ oriented in some\nway.\n \nFor $a\\in A$ of type $(\\mu_a,\\nu_a)$ set \n$N_1(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } \nmod\n\\text{ }2\\pi\\in (0,\\pi]\\bigr\\}$, \n$N_2(a)=\\bigl\\{ k\\in \\{1,\\dots,\\mu_a\\}| \\frac{2\\pi k\\nu_a}{\\mu_a}\\text{ } mod\n\\text{ }2\\pi\\in (0,\\pi)\\bigr\\}$. \n\nDefine $R_a\\in {\\mathbb Z}[H_1(N)]$ by the\nfollowing formula:\n\n\\begin{multline}\nR_a=\n\\sum_{k\\in N_1(a)}\\Bigl([K]f_a^{\\mu_a-k}-f_a^k+\nf_a^{k-\\mu_a}-[K]f_a^{-k}\\Bigr)\\\\\n-\\sum_{k\\in N_2(a)}\\Bigl(f_a^{\\mu_a-k}-[K]f_a^k+\n[K]f_a^{k-\\mu_a}-f_a^{-k}\\Bigr)\n\\end{multline}\n\n\n\n\n\n\n\nLet $\\tilde H$ be the quotient of\n${\\mathbb Z}[H_1(N)]$ (viewed as a ${\\mathbb Z}$-module) by the free\n${\\mathbb Z}$-submodule generated by $\\Bigl\\{(e-f)([K]+e),\\{R_a\\}_{a\\in A}\\Bigr\\}$.\n\nOne can prove that under the change of the orientation of $\\pi^{-1}(a)$ \n(used to define $f_a$) $R_a$ goes to \n$-R_a$. Thus $\\tilde H$ is well defined. \nTo show this, one verifies that if $\\mu_a$ is odd, then $N_1(a)=N_2(a)$. Under\nthis change each term from the first sum (used to define $R_a$) goes to minus\nthe corresponding term from the second sum and vice versa. (Note that\n$f^2=e$.) If $\\mu_a=2l$ is even, then $N_1(a)\\setminus\\{l\\}=N_2(a)$. \nUnder this change each term with $k\\in N_1(a)\\setminus \\{l\\}$ goes to minus the\ncorresponding term with $k\\in N_2(a)$ and vice versa. \nThe term in the first sum\nthat corresponds to $k=l$ goes to minus itself.\n\nFinally define $\\tilde S_K\\in \\tilde H$ as \nthe sum over all double points $v$ of $\\pi(K)$:\n\\begin{equation}\\tilde S_K=\n\\sum_v\\Bigl([\\mu^+_1(v)]-[\\mu^-_1(v)]+\n[\\mu^+_2(v)]-[\\mu^-_2(v)]\n\\Bigr).\n\\end{equation} \n\\end{defin}\n\n\n\\begin{thm}\\label{correcttildeSSK}\n$\\tilde S_K$ is an isotopy invariant of $K$.\n\\end{thm}\n\n\nThe proof is a straightforward generalization of the proof of\nTheorem~\\ref{correctSSK}. \n\n\n\n\n\n\n\\begin{emf} \nOne can easily verify that $S_K$ and $\\tilde S_K$ \nsatisfy relations similar to~\\eqref{type2}. \nHence both of them are Vassiliev invariants of order one\n(see~\\ref{vassiliev2}). \n\n\\end{emf}\n \n\\section{Wave fronts on surfaces}\n\\subsection{Definitions}\nLet $F$ be a two-dimensional manifold. A {\\em contact element\\\/}\nat a point in $F$ is a one-dimensional vector subspace of the tangent plane. \nThis subspace divides the tangent plane into two half-planes. A choice of\none of them is called a {\\em coorientation\\\/} of a contact element. \nThe space of all cooriented contact\nelements of $F$ is a spherical cotangent bundle $ST^*F$. \nWe will also denote it by $N$. It is an $S^1$-fibration over $F$. The\nnatural contact structure on $ST^*F$ is a\ndistribution of hyperplanes given by the condition that a velocity vector of\nan incidence point of a contact element belongs to the element. \nA {\\em Legendrian\\\/} curve $\\lambda$ in $N$ is an immersion of $S^1$ into\n$N$ such that for each $p\\in S^1$ \nthe velocity vector of $\\lambda$ at $\\lambda(p)$ \nlies in the contact plane. The naturally cooriented \nprojection $L\\subset F$ of a Legendrian curve $\\lambda\\subset N$ \nis called {\\em the wave front\\\/} of\n$\\lambda$. \nA cooriented wave front may be uniquely lifted to a Legendrian curve\n$\\lambda \\subset\nN$ by taking a coorienting normal direction as a contact element at each\npoint of the front.\nA wave front is said to be generic if it is an immersion\neverywhere except a finite number of points, where it has cusp\nsingularities,\nand all multiple points are double points with transversal self-intersection.\nA cusp is the projection of a point where the corresponding Legendrian\ncurve is tangent to the fiber of the bundle.\n\n\n\n\\subsection{Shadows of wave fronts}\n\n\n\\begin{emf}\\label{orientST*F}\nFor any surface $F$ the space $ST^*F$ is canonically oriented. The\norientation is constructed as follows. \nFor a point $x\\in F$ fix an\norientation of $T_xF$. It induces an orientation of the fiber over $x$. \nThese two orientations determine an orientation of three dimensional planes\ntangent to the points of the fiber over $x$. A straightforward \nverification shows \nthat this orientation is independent on the orientation of $T_xF$ we choose.\nHence the orientation of $ST^*F$ is well defined. \n\n\nThus for oriented $F$ the shadow of a generic knot in $ST^*F$ is well defined \n(see~\\ref{prelim} and~\\ref{shadowgeneral}). Theorem~\\ref{front-shadow}\ndescribes the shadow of a Legendrian lifting of a generic cooriented wave front\n$L\\subset F$. \n\\end{emf}\n\n\n\\begin{defin} Let $X$ be a connected component of $F\\setminus L$. We denote \nby $\\chi \\protect\\operatorname{Int} (X)$ the Euler characteristic of $\\protect\\operatorname{Int} (X)$, \nby $C^i_X$ the number of cusps in the boundary of the region $X$ pointing \ninside\n$X$ (as in Figure~\\ref{pic2.fig}a), by $C^o_X$ the number of cusps \nin the boundary of $X$\npointing outside (as in Figure~\\ref{pic2.fig}b),\nand by $V_X$ the number of corners of $X$ where locally the picture\nlooks in one the two ways shown in Figure~\\ref{pic2.fig}c. It\ncan happen that a cusp point is pointing both inside and outside of $X$. In\nthis case it contributes both in $C^i_X$ and in $C^o_X$. If the corner of the\ntype shown in Figure~\\ref{pic2.fig}c enters twice in $\\partial X$, \nthen it should be counted twice. \n\\end{defin}\n\n\\begin{figure}[htb]\n\\begin{center}\n \\epsfxsize\\hsize\\advance\\epsfxsize -0.5cm\n \\leavevmode\\epsffile{pic2.eps}\n \\end{center}\n\\caption{}\\label{pic2.fig}\n\\end{figure}\n\n\n\n\n\n\n\\begin{thm}\\label{front-shadow}\nLet $F$ be an oriented surface and\n$L$ a generic cooriented wave front on $F$ corresponding to a \nLegendrian curve $\\lambda$. \nThere exists a small deformation of $\\lambda$ in the class of all smooth \n(not \nonly Legendrian) curves such that the resulting curve is generic with respect\nto the projection, and the shadow of this curve can be constructed in the \nfollowing way. We \nreplace a small neighborhood of each cusp of $L$ with a smooth simple arc. \nThe gleam of an arbitrary region $X$ that has a compact closure and does not\ncontain boundary components of $F$ is calculated by the following formula:\n\\begin{equation}\n\\protect\\operatorname{gl}_X=\\chi \\protect\\operatorname{Int} (X)+\\frac {1}{2}(C^i_X-C^o_X-V_X).\n\\end{equation}\n\\end{thm}\n\nFor the proof of Theorem~\\ref{front-shadow} see \nSubsection~\\ref{pffront-shadow}.\n\n\\begin{rem}\nThe surface $F$ in the statement of Theorem~\\ref{front-shadow} is not\nassumed to be compact.\n\nNote that as we mentioned in~\\ref{shadowgeneral}, the gleam of a region $X$ \nthat does not have compact closure or contains boundary components is\nnot well defined unless we fix a section over all ends of $X$ and components of\n$\\partial F$ in $X$.\n\nThis theorem first appeared in~\\cite{Tchernov}. A similar result was\nindependently obtained by Polyak~\\cite{MPolyak}. \n\\end{rem}\n\n\\begin{emf} A self-tangency point $p$ of a wave front is said to be a\npoint of a {\\em dangerous self-tangency\\\/} if the coorienting normals of the two \nbranches coincide at $p$ (see Figure~\\ref{pic8.fig}). Dangerous\nself-tangency points correspond to self-intersection of the Legendrian curve. \nHence a generic deformation of the front $L$ \nnot involving {\\em dangerous\\\/} self-tangencies\ncorresponds to an isotopy of the Legendrian knot $\\lambda$.\n\n\\begin{figure}[htb]\n\\begin{center}\n \\epsfysize 1.1cm\n \\leavevmode\\epsffile{pic8.eps}\n \\end{center}\n\\caption{}\\label{pic8.fig}\n\\end{figure}\n\nAny generic deformation of a wave front $L$ corresponding to an isotopy\nin the class of the Legendrian knots \ncan be split into a sequence of modifications shown in \nFigure~\\ref{pic6.fig}. The construction of Theorem~\\ref{front-shadow}\ntransforms these generic modifications of wave fronts to shadow moves:\nIa and Ib in Figure~\\ref{pic6.fig} are transformed to the \n$\\bar S_1$ move for shadow diagrams, IIa, IIb, II'a, II'b, II'c, and\nII'd are transformed to the $S_2$ move, \nfinally IIIa and IIIb are transformed to $S_3$ and $\\bar S_3$ respectively.\n\\end{emf}\n\n\\begin{figure} [htbp]\n\\begin{center}\n \\epsfxsize 12.5cm\n \\leavevmode\\epsffile{pic6.eps}\n \\end{center}\n\\caption{}\\label{pic6.fig}\n\\end{figure}\n\n\n\n\\begin{emf} Thus for the Legendrian lifting of a wave front \nwe are able to calculate all invariants that we can calculate for shadows. \nThis includes the analogue of the linking number for the\nfronts on ${\\mathbb R}^2$ (see~\\cite{Turaev}), the second order Vassiliev invariant\n(see~\\cite{Shumakovitch}), \nand quantum state sums (see~\\cite{Turaev}).\n\\end{emf}\n\n\\subsection{Invariants of wave fronts on surfaces.}\nIn particular, the $S_K$ invariant gives rise to an invariant of a generic\nwave front. This invariant appears to be related to the formula for the\nBennequin invariant of a wave front introduced by Polyak in~\\cite{Polyak}.\n\nLet us recall the corresponding results and definitions of~\\cite{Polyak}. \n\nLet $L$ be a generic cooriented oriented wave front on an oriented surface\n$F$.\nA branch of a wave front is said to be positive (resp.\nnegative) if the frame of coorienting and orienting vectors defines\npositive (resp. negative) orientation of the surface $F$. \nDefine the {\\em sign\\\/} $\\epsilon (v)$ of a double point $v$ of $L$ to be\n$+1$ if the signs of both branches of the front intersecting at $v$\ncoincide and\n$-1$ otherwise. Similarly \nwe assign a positive (resp. negative) sign to a cusp point\nif the coorienting vector turns in a positive (resp. negative)\ndirection while traversing a small neighborhood of the cusp point along the\norientation. We denote half of the number of \npositive and negative cusp points by \n$C^+$ and $C^-$ respectively.\n\n\nLet $v$ be a double point of $L$. The orientations of $F$ and $L$ allow one\nto distinguish the two wave fronts $L^+_v$ and $L^-_v$ obtained by splitting \nof $L$ in $v$ with respect to orientation and\ncoorientation (see Figures~\\ref{pic9.fig}a.1 and~\\ref{pic9.fig}b.1).\n(Locally one of the two fronts lies to the left and another to the right of\n$v$.) \n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 10cm \n \\leavevmode\\epsffile{pic9.eps}\n \\end{center}\n\\caption{}\\label{pic9.fig}\n\\end{figure}\n\nFor a Legendrian curve $\\lambda$ in $ST^*{\\mathbb R} ^2$ denote by $l(\\lambda)$ its\nBennequin invariant described in the works of \nTabachnikov~\\cite{Tabachnikov} and Arnold~\\cite{Arnold} with the sign\nconvention of~\\cite{Arnold} and~\\cite{Polyak}. \n\n\\begin{thm}[Polyak~\\cite{Polyak}]\\label{BenR2}\nLet $L$ be a generic oriented cooriented wave front on ${\\mathbb R}^2$\nand $\\lambda$ the corresponding Legendrian curve. Denote by $\\protect\\operatorname{ind}(L)$ \nthe degree of the mapping taking a point $p\\in S^1$ \nto the point of $S^1$ corresponding to the direction of \nthe coorienting normal of $L$ at $L(p)$. Define $S$ as the \nfollowing sum over all double points of $L:$\n\\begin{equation}\nS=\\sum_v (\\protect\\operatorname{ind}(L^+_v)-\\protect\\operatorname{ind}(L^-_v)-\\epsilon (v)).\n\\end{equation}\nThen \n\\begin{equation}\nl(\\lambda )=S+(1-\\protect\\operatorname{ind}(L))C^++(\\protect\\operatorname{ind}(L)+1)C^- +\\protect\\operatorname{ind}(L)^2.\n\\end{equation}\n\\end{thm}\n\n\nIn~\\cite{Polyak} it is shown that the Bennequin invariant of a wave front on\nthe ${\\mathbb R}^2$ plane admits quantization. Consider a formal quantum parameter\n$q$. Recall that for any $n\\in {\\mathbb Z}$ the corresponding quantum number\n$[n]_q\\in {\\mathbb Z}[q,q^{-1}]$ is a finite Laurent polynomial in $q$ defined by \n\\begin{equation}\n[n]_q=\\frac{q^n-q^{-n}}{q-q^{-1}}.\n\\end{equation}\nSubstituting quantum integers instead of integers in~\\ref{BenR2}\nwe get the following theorem.\n\n\\begin{thm}[Polyak~\\cite{Polyak}]\\label{QBen}\nLet $L$ be a generic cooriented oriented wave \nfront on ${\\mathbb R}^2$ and $\\lambda$ the corresponding\nLegendrian curve. Define $S_q$ by the following\nformula, where the sum is taken over the set of all \ndouble points of $L:$\n\\begin{equation}\\label{Sq}\nS_q=\\sum_v[ind(L^+_v)-ind(L^-_v)-\\epsilon (v)]_q.\n\\end{equation}\nPut\n\\begin{equation}\nl_q(L)=S_q+[1-ind(L)]_qC^++[ind(L)+1]_qC^-+[ind(L)]_qind(L).\n\\end{equation} \nThen $l_q(\\lambda)=l_q(L)\\in\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ is invariant \nunder isotopy in the class of the Legendrian knots. \n\\end{thm}\n \n\nThe $l_q(\\lambda)$ invariant can be expressed~\\cite{Aicpriv} \nin terms of the partial linking\npolynomial of a generic cooriented oriented wave front\nintroduced by Aicardi~\\cite{Aicardi}.\n\nThe reason why this invariant takes values in $\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ and\nnot in ${\\mathbb Z}[q,q^{-1}]$ is that the number of positive (or negative) cusps can\nbe odd. This makes $C^+$ ($C^-$) a half-integer.\n\nLet $\\lambda ^{\\epsilon}_v$ with $\\epsilon =\\pm$ be the Legendrian lifting \nof the front $L^{\\epsilon}_v$. Let $f\\in \nH_1(ST^*F)$ be the homology class of a positively\noriented fiber.\n\n\n\\begin{thm}[Polyak~\\cite{Polyak}]\\label{Ben} Let $L$ be a generic oriented \ncooriented wave front on an oriented surface $F$. Let $\\lambda$ be the corresponding \nLegendrian curve. Define $l_F(\\lambda)\\in H_1(ST^*F,\\frac{1}{2}{\\mathbb Z})$ \nby the following formula: \n\\begin{equation}\nl_{F}(\\lambda)=\\Bigl(\\prod_v[\\lambda ^+_v][\\lambda ^-_v]^{-1}\nf^{-\\epsilon (v)}\\Bigr)(f[\\lambda]^{-1})^{C^+}([\\lambda]f)^{C^-}\n\\end{equation}\n(We use the multiplicative notation for the group operation in\n$H_1(ST^*F)$.)\n\n \nThen $l_{F}(\\lambda)$ is invariant under isotopy in the class of the\nLegendrian knots.\n\\end{thm}\n\nThe proof is straightforward. One verifies that $l_{F}(\\lambda)$ is \ninvariant under all oriented versions of non-dangerous self-tangency, \ntriple point, cusp crossing, and cusp birth moves of the wave front.\n\nIn~\\cite{Polyak} this invariant is denoted by $I_\\Sigma^+(\\lambda)$ and, in\na sense, it\nappears to be a natural generalization of Arnold's $J^+$\ninvariant~\\cite{Arnold} \nto the case of an\noriented cooriented wave front on an oriented surface.\n\n\nNote that in the situation of Theorem~\\ref{BenR2} the indices of all\nthe fronts involved are the images of the homology classes of their\nLegendrian liftings under the natural identification of $H_1(ST^*{\\mathbb R}\n^2)$ with ${\\mathbb Z}$. If one replaces everywhere in~\\ref{BenR2} indices with \nthe corresponding homology classes and puts $f$ instead of $1$, then the\nonly difference between the two formulas is the term $\\protect\\operatorname{ind} ^2(L)$. (One has\nto remember that we use the multiplicative notation for the operation \nin $H_1(ST^*F)$.)\n \n\\begin{emf} The splitting of a Legendrian knot $K$ into \n$\\mu^+_v$ and $\\mu^-_v$\n(see~\\ref{SK}) can be done up to an isotopy in the\nclass of the Legendrian knots. Although this can be done in many ways, there \nexists the simplest way. The projections $\\tilde L^+_v$ and $\\tilde L^-_v$\nof the Legendrian curves created by the splitting are shown in \nFigure~\\ref{pic5.fig}. (This fact follows from Theorem~\\ref{front-shadow}.) \n\nLet $\\tilde\\lambda ^{\\epsilon}_v$ with $\\epsilon =\\pm$ be the Legendrian lifting \nof the front $\\tilde L^{\\epsilon}_v$.\n\\end{emf}\n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 12 cm\n \\leavevmode\\epsffile{pic5.eps}\n \\end{center}\n\\caption{}\\label{pic5.fig}\n\\end{figure}\n\n\n\n\\begin{thm}\\label{splitwave} Let $L$ be a generic oriented cooriented wave\nfront on an oriented surface $F$. Let $\\lambda$ be the corresponding\nLegendrian curve. \nDefine $S(\\lambda)\\in \\frac{1}{2}{\\mathbb Z}\\bigl[ H_1(ST^*F)\\bigr]$ by the following \nformula: \n\\begin{equation}\\label{SFL}\nS(\\lambda)=\\sum_v\\Bigl([\\tilde \\lambda ^+_v]-[\\tilde \\lambda\n^-_v]\\Bigr)\n+(f-[\\lambda])C^+ +([\\lambda]f-e)C^-.\n\\end{equation} \nThen $S(\\lambda)$ is invariant under isotopy in the class of the Legendrian\nknots. \n\\end{thm}\n\nThe proof is straightforward. One verifies that $S(\\lambda)$ is indeed\ninvariant under all oriented versions of non-dangerous self-tangency, \ntriple point, cusp crossing, and cusp birth moves of the wave front.\n\n\\begin{emf}\\label{homvers}\nBy taking the free homotopy classes of $\\tilde \\lambda^+_v$ and $\\tilde\n\\lambda^-_v$ instead of the homology classes one obtains a different \nversion of the $S(\\lambda)$ invariant. It takes values in the group of\nformal half-integer linear combinations of \nthe free homotopy classes of oriented \ncurves in $ST^*F$.\nIn this case the terms $[\\lambda]$ and $f$ in~\\eqref{SFL} should be\nsubstituted with the free homotopy classes of $\\lambda$ and of a \npositively oriented fiber respectively. The terms $[\\lambda]f$ and $e$\nin~\\eqref{SFL} should be substituted with the free homotopy classes of \n$\\lambda$ with a positive fiber added to it and the class of a \ncontractible curve\nrespectively. Note that $f$ lies in the center of $\\pi_1(ST^*F)$, so that the\nclass of $\\lambda$ with a fiber added to it is well defined.\n\nA straightforward verification shows that this version of $S(\\lambda)$ is also\ninvariant under isotopy in the class of the Legendrian knots.\n\\end{emf}\n\n \n\n\\begin{thm}\\label{splitting} Let $L$ be a generic oriented cooriented wave\nfront on an oriented surface $F$. Let $\\lambda$ be the corresponding \nLegendrian curve. \nLet $S(\\lambda)$ and $l_F(\\lambda)$ be the invariants\nintroduced in~\\ref{splitwave} and~\\ref{Ben} respectively. Let\n\\begin{equation}\n\\protect\\operatorname{pr}:\\frac{1}{2}{\\mathbb Z}\\bigl [H_1(ST^*F)\\bigr]\\rightarrow \nH_1(ST^* F,\\frac{1}{2}{\\mathbb Z})\n\\end{equation}\nbe the mapping defined as\nfollows: for any $n_i\\in \\frac{1}{2}{\\mathbb Z}$ and $g_i\\in H_1(ST^*F),$\n\\begin{equation}\n\\sum n_i g_i\\mapsto \\prod g_i^{n_i}.\n\\end{equation}\nThen $\\protect\\operatorname{pr}(S(\\lambda))=l_F(\\lambda)$.\n\\end{thm}\n \nThe proof is straightforward: one must verify that \n\\begin{equation}\\label{svyaz}\n[\\lambda ^+_v][\\lambda ^-_v]^{-1}f^{-\\epsilon (v)}=\n[\\tilde \\lambda ^+_v][\\tilde \\lambda ^-_v]^{-1}\n\\text{ in } H_1(ST^*F). \n\\end{equation}\n(Recall that we use a multiplicative notation for the group operation\nin $H_1(ST^*F)$.)\n\nThis means that $S_F(\\lambda)$ is a refinement of Polyak's invariant \n$l_F(\\lambda)$.\n \n\n\\begin{emf}\\label{quant} \nOne can verify that there is a unique linear combination \n$\\sum_{m\\in {\\mathbb Z}}n_m[m]_q=l_q(\\lambda)$ with $n_m$ being non-negative\nhalf-integers such that $n_0=0$, and if $n_m>0$, then $n_{-m}=0$. \nTo prove this one must verify that\n$\\{\\frac{1}{2}[n]_q|0< n\\}$ is a basis for the ${\\mathbb Z}$-submodule of \n$\\frac{1}{2}{\\mathbb Z}[q,q^{-1}]$ \ngenerated by the quantum numbers and use the identity $n[m]_q=-n[-m]_q$. \n\\end{emf} \n\n \nThe following theorem shows that if $L\\subset {\\mathbb R}^2$, then $S(\\lambda)$ and\nPolyak's quantization $l_q(\\lambda)$ (see~\\ref{QBen}) of the Bennequin \ninvariant can be explicitly expressed in terms of each other. \n\n\\begin{thm}\\label{equivalence}\nLet $f\\in H_1(ST^*{\\mathbb R}^2)$ be the class of a positively oriented fiber. Let \n$L$ be a generic oriented cooriented wave front on ${\\mathbb R}^2$, \n$\\lambda$ the corresponding Legendrian\ncurve, and $f^h$ the homology class realized by it. Let\n$l_q(\\lambda)-[h]_qh=\\sum_{m\\in Z} n_m[m]_q$ be written in the form described \nin~\\ref{quant} and $S(\\lambda)=\\sum_{l\\in Z}k_l f^l$. \nThen\n\\begin{equation}\\label{rel1}\nl_q(\\lambda)=[h]_qh+\\sum_{k_l>0} k_l[2l-h-1]_q,\n\\end{equation}\nand\n\\begin{equation}\\label{rel2}\nS(\\lambda)=\\sum_{n_m > 0}\nn_m(f^{\\frac{h+1+m}{2}}-f^{\\frac{h+1-m}{2}}).\n\\end{equation}\n\\end{thm}\nFor the proof of Theorem~\\ref{equivalence} see Subsection~\\ref{pfequivalence}.\n\nOne can show that for $n_m>0$ both $\\frac{h+1+m}{2}$ and $\\frac{h+1-m}{2}$\nare integers, so that the sum~\\eqref{rel2} takes values \nin $\\frac{1}{2}{\\mathbb Z}[H_1(ST^*{\\mathbb R}^2)]$.\n\nNote that the $l_q(\\lambda)$ invariant was defined only for fronts on the\nplane ${\\mathbb R}^2$. Thus $S(\\lambda)$ is, in a sense, a generalization of\nPolyak's\n$l_q(\\lambda)$ to the case of wave fronts on an arbitrary oriented surface $F$.\n\n\\begin{emf}\nThe splitting of the Legendrian knot $K$ into $\\mu_1^+(v)$, $\\mu_1^-(v)$, \n$\\mu_2^+(v)$, and $\\mu_2^-(v)$ (which was used to define $\\tilde S(K)$, \nsee~\\ref{tildeSK}) can be done up to an isotopy in the class of the\nLegendrian knots. Although this can be done in many ways, there is the\nsimplest one. The projections $\\tilde L_1^+(v)$, $\\tilde L_1^-(v)$, \n$\\tilde L_2^+(v)$, and $\\tilde L_2^-(v)$ are shown in\nFigure~\\ref{front1.fig}. (This fact follows from \nTheorem~\\ref{front-shadow}.) \n\n\nThis allows us to introduce an invariant similar to $S(\\lambda)$ \nfor generic oriented cooriented \nwave fronts on a non-orientable surface $F$ in the\nfollowing way. \n\n\nLet $L$ be a generic wave front on a\nnon-orientable surface $F$. Let $v$ be a double point of $L$. Fix some \norientation of a small neighborhood of $v$ in $F$. The orientations of the\nneighborhood and $L$ allow one to distinguish the wave fronts $L^+_1$,\n$L^-_1$, $L^+_2$, and $L^-_2$ obtained by the two splittings of $L$ with\nrespect to the orientation and coorientation (see Figure~\\ref{front1.fig}).\nLocally the fronts with the upper indices plus and minus are located\nrespectively to the right and to the left of $v$.\nTo each double\npoint $v$ of $L$ we associate an element\n$\\Bigl([\\tilde\\lambda^+_1(v)]-[\\tilde\\lambda^-_1(v)]+[\\tilde\\lambda^+_2(v)]-\n[\\tilde\\lambda^-_2(v)]\\Bigr)\\in {\\mathbb Z}[H_1(ST^*(F)]$. Here we denote by lambdas\nthe Legendrian curves corresponding to the wave fronts appearing under\nthe splitting. Clearly \nthis element does not depend on the orientation of the\nneighborhood of $v$ we have chosen.\n\n\\begin{figure}[htb]\n\\begin{center}\n\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{front1.eps}\n \\end{center}\n\\caption{}\\label{front1.fig}\n\\end{figure}\n\n\nFor a wave front $L$ let $C$ be half of the number of cusps of $L$. Denote\nby $f$ the homology class of the fiber of $ST^*F$ oriented in some way. \nNote that $f^2=e$, so it does not matter which orientation of the fiber we\nuse to define $f$.\n\\end{emf}\n\n\\begin{thm}\\label{unorientsplit}\nLet $L$ be a generic cooriented oriented wave front on a non-orientable surface $F$\nand $\\lambda$ the corresponding Legendrian curve.\nDefine $\\tilde S(\\lambda)\\in\\frac{1}{2}{\\mathbb Z}\\bigl[ H_1(ST^*(F))\\bigr]$ by the following\nformula, where the summation is taken over the set of all double points of\n$L:$\n\\begin{equation}\\label{tildeSdef}\n\\tilde S(\\lambda)=\\sum_v\\Bigl([\\tilde \\lambda ^+_1(v)]-\n[\\tilde \\lambda ^-_1(v)]+[\\tilde \\lambda ^+_2(v)]-[\\tilde \\lambda\n^-_2(v)]\\Bigr)+C\\bigl([\\lambda]f-e+f-[\\lambda]\\bigr).\n\\end{equation}\nThen $\\tilde S(\\lambda)$ is invariant under isotopy in the class of \nthe Legendrian knots.\n\\end{thm}\n\nThe proof is straightforward. One verifies that $\\tilde S(\\lambda)$ is\nindeed invariant under all oriented versions of non-dangerous\nself-tangency, triple point passing, cusp crossing, and cusp birth moves of\nthe wave front.\n\nThe reason we have $\\tilde S(\\lambda)\\in \\frac{1}{2}{\\mathbb Z}[H_1(ST^*F)]$ is that\nif $L$ is an orientation reversing curve, then \nthe number of cusps of $L$ is odd. In this case $C$ is a\nhalf-integer.\n\n\n\\section{Wave fronts on orbifolds}\n\\subsection{Definitions}\n\\begin{defin}\\label{orbifold}\n\nAn {\\em orbifold\\\/} is a surface $F$ with the additional structure\nconsisting of: \n\n1) set $A\\subset F$;\n\n2) smooth structure on $F\\setminus A$;\n\n3) set of homeomorphisms $\\phi_a$ of neighborhoods $U_a$ of $a$ in $F$ \nonto ${\\mathbb R}^2\/G_a$ such that $\\phi_a(a)=0$ and\n$\\phi_a\\Big|_{U_a\\setminus a}$ is a diffeomorphism.\nHere $G_a=\\bigl\\{e^{\\frac{2\\pi\nk}{\\mu_a}}\\big| k\\in\\{1,\\dots,\\mu_a\\}\\bigr\\}$ is a group\nacting on ${\\mathbb R}^2={\\mathbb C}$ by multiplication. ($\\mu _a\\in {\\mathbb Z}$ is assumed to be\npositive.) \n\\end{defin}\n\nThe points $a\\in A$ are called {\\em cone points\\\/}.\n\nThe action of $G$ on ${\\mathbb R}^2$ induces the action of $G$ on\n$ST^*{\\mathbb R}^2$. This makes $ST^*{\\mathbb R}^2\/G$ a Seifert fibration over\n${\\mathbb R}^2\/G$. Gluing together the pieces over neighborhoods of $F$ we obtain a\nSeifert fibration $\\pi:N\\rightarrow F$. The fiber over a cone point $a$ is an\nexceptional fiber of type $(\\mu_a,-1)$ (see~\\ref{numbers}). \n \nThe natural contact structure on $ST^*{\\mathbb R}^2$ is invariant under \nthe induced action of $G$. Since $G$ acts freely on $ST^*{\\mathbb R}^2$, this implies\nthat $N$ has an induced contact structure. As before, the naturally\ncooriented projection\n$L\\subset F$ of a generic Legendrian curve $\\lambda$ is called \n{\\em the front of\\\/} $\\lambda$.\n \n\\subsection{Invariants for fronts on orbifolds}\nFor oriented $F$ we construct an\ninvariant similar to $S(\\lambda)$. It corresponds to the $S_K$\ninvariant of a knot in a Seifert fibered space. \nFor non-orientable surface $F$\nwe construct an analogue of $\\tilde S(\\lambda)$. It corresponds to\nthe $\\tilde S_K$ invariant of a knot in a Seifert fibered space.\n\nNote that any surface $F$ can be viewed as an orbifold without cone\npoints. This justifies the notation below.\n\n \nLet $F$ be an oriented surface. The orientation of $F$ induces an \norientation of all fibers. Denote by $f$ \nthe homology class of a positively oriented fiber. For a cone point $a$\ndenote by $f_a$ the homology class of a positively oriented fiber\n$\\pi^{-1}(a)$. For a generic oriented cooriented wave front $L\\subset F$\ndenote by $C^+$ (resp. $C^-$) half of the number \nof positive (resp. negative) cusps of $L$.\nNote that for a double point $v$ of a generic front $L$\nthe splitting into $\\tilde L^+_v$ and $\\tilde L^-_v$ is well defined. \nThe corresponding Legendrian\ncurves $\\tilde \\lambda^+_v$ and $\\tilde \\lambda^-_v$ in \n$N$ are also well defined.\n\nFor $a\\in A$ of type $(\\mu_a,-1)$ \nput $N_1(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{-2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi]\\bigr\\}$, \n$N_2(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi)\\bigr\\}$. \nDefine $R^1_a, R^2_a\\in\n{\\mathbb Z}[H_1(N)]$ by the following formulas: \n\n\\begin{equation}\nR^1_a=\\sum_{k\\in N_1(a)}\\bigl([\\lambda]f_a^{\\mu_a-k}-f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl(f_a^{\\mu_a-k}-[\\lambda]f_a^k\\bigr),\n\\end{equation}\n\\begin{equation}\nR^2_a=\\sum_{k\\in N_1(a)}\\bigl(f_a^{\\mu_a-k}-[\\lambda]f_a^k\\bigr)-\n\\sum_{k\\in N_2(a)}\\bigl([\\lambda]f_a^{\\mu_a-k}-f_a^k\\bigr).\n\\end{equation}\n\n\n\nSet $J$ to be the quotient of\n$\\frac{1}{2}{\\mathbb Z}[H_1(N)]$ by the free Abelian \nsubgroup generated by \n$\\Bigl\\{\\{\\frac{1}{2}R_1(a),\\frac{1}{2}R_2(a)\\}_{a\\in A}\\Bigr\\}$. \n\n \n\\begin{thm}\\label{orbifold1}\nLet $L$ be a generic cooriented oriented wave front on $F$ and\n$\\lambda$ the corresponding Legendrian curve.\n\nThen $S(\\lambda)\\in J$ defined by the sum over \nall double points of $L$,\n\\begin{equation}\nS(\\lambda)=\n\\sum\\Bigl([\\tilde \\lambda^+(v)]-[\\tilde \\lambda^-(v)]\\Bigr)+(f-[\\lambda])C^++\n([\\lambda]f-e)C^-,\n\\end{equation}\nis invariant under isotopy in the class of the Legendrian knots.\n\\end{thm} \n\nFor the proof of Theorem~\\ref{orbifold1} see Subsection~\\ref{pforbifold1}.\n\n\nLet $F$ be a non-orientable surface. \nDenote by $f$ the homology class of a regular fiber oriented in some way. \nNote that $f^2=e$, so the orientation we use to define $f$ does not matter. \nFor a cone point $a$ denote by $f_a$ \nthe homology class of the fiber $\\pi^{-1}(a)$ oriented in some\nway. For a generic oriented cooriented wave front $L\\subset F$\ndenote by $C$ half of the number of cusps of $L$.\nNote that for a double point $v$ of a generic front $L$ \nthe element \n$\\bigl([\\tilde\\lambda^+_1(v)]-[\\tilde\\lambda^-_1(v)]+\n[\\tilde\\lambda^+_2(v)]-[\\tilde\\lambda^-_2(v)]\\bigr)\\in {\\mathbb Z}[H_1(N)]$ \nused to introduce\n$\\tilde S(\\lambda)$ is well defined.\n \nFor $a\\in A$ of type $(\\mu_a,-1)$ \nput $N_1(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big |\\frac{-2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi]\\bigr\\}$, and \n$N_2(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big |\\frac{-2k\\pi}{\\mu_a}\n\\protect\\operatorname{mod} 2\\pi\n\\in(0,\\pi)\\bigr\\}$. \nDefine $R_a\\in\n{\\mathbb Z}[H_1(N)]$ by the following formula:\n\\begin{multline}\nR_a=\n\\sum_{k\\in N_1(a)}\\Bigl([\\lambda]f_a^{\\mu_a-k}-f_a^k+\nf_a^{k-\\mu_a}-[\\lambda]f_a^{-k}\\Bigr)\\\\\n-\\sum_{k\\in N_2(a)}\\Bigl(f_a^{\\mu_a-k}-[\\lambda]f_a^k+\n[\\lambda]f_a^{k-\\mu_a}-f_a^{-k}\\Bigr).\n\\end{multline}\n\n\n\n\n\n\n\nPut $\\tilde J$ to be the quotient of\n$\\frac{1}{2}{\\mathbb Z}[H_1(N)]$ by a free Abelian subgroup generated by \n$\\Bigl\\{\\{\\frac{1}{2}R_a\\}_{a\\in A}\\Bigr\\}$.\n\nSimilarly to~\\ref{tilde1SK}, one \ncan prove that under the change of the orientation of $\\pi^{-1}(a)$ \n(used to define $f_a$) $R_a$ goes to \n$-R_a$. Thus $\\tilde J$ is well defined. \n\n\n\\begin{thm}\\label{orbifold2}\nLet $L$ be a generic cooriented oriented wave front on $F$ and\n$\\lambda$ the corresponding Legendrian curve.\n\n\nThen $\\tilde S(\\lambda)\\in \\tilde J$ defined by \nthe summation over all double points of $L$,\n\\begin{equation}\n\\tilde S(\\lambda)=\n\\sum\\Bigl([\\tilde \\lambda^+_1(v)]-[\\tilde \\lambda^-_1(v)]+\n[\\tilde \\lambda^+_2(v)]-[\\lambda^-_2(v)]\n\\Bigr)+\\bigl(([\\tilde \\lambda]f-e+f-[\\lambda]\\bigr)C,\n\\end{equation}\nis invariant under isotopy in the class of the Legendrian knots.\n\\end{thm} \n\nThe proof is a straightforward generalization of the proof of\nTheorem~\\ref{orbifold1}. \n\n\n\n\n\n\n\n\n\\section{Proofs}\n\n\\subsection{Proof of Theorem~\\ref{correct2}.}\\label{pfcorrect2}\nTo prove the theorem it suffices to show that $S_K$ is invariant\nunder the elementary isotopies. They project to: \na death of a double point, cancellation of two double points, and passing\nthrough a triple point. \n\nTo prove the invariance, we fix a homeomorphic to a closed disk \npart $P$ of $F$ containing the projection of one of the elementary\nisotopies.\nFix a section over the boundary of $P$ such that the \npoints of $K\\cap\\pi^{-1}(\\partial P)$ belong to the section. Inside $P$ we can\nconstruct the Turaev shadow (see \\ref{shadowgeneral}). \nThe action of $H_1(\\protect\\operatorname{Int} P)=e$ \non the set of isotopy types of links is trivial (see~\\ref{action}). \nThus the part of $K$ can be reconstructed in the unique way \nfrom the shadow over $P$. \nIn particular, one can compare the homology classes\nof the curves created by splitting at a double point inside $P$. \nHence to prove the theorem, it suffices to verify the invariance under the\noriented versions of the moves $S_1,S_2$, and $S_3$.\n\nThere are two versions of the oriented move $S_1$ shown in\nFigures~\\ref{shad11.fig}a and~\\ref{shad11.fig}b. \n\nFor Figure~\\ref{shad11.fig}a the term $[\\mu^+_v]$ appears to be equal to\n$f$. From~\\ref{sum-property} we know that \n$[\\mu^+_v][\\mu^-_v]=[K]f$, so that $[\\mu^-_v]=[K]$. Hence\n$[\\mu^+_v]-[\\mu^-_v]=f-[K]$ and is equal to zero in $H$. In the same way we\nverify that $[\\mu^+_v]-[\\mu^-_v]$ (for $v$ shown in\nFigure~\\ref{shad11.fig}b) is equal to $[K]f-e$. It is also zero in $H$. \nThe summands corresponding to other double points do not change under this move, \nsince it does\nnot change the homology classes of the knots created by the splittings.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 6cm\n \\leavevmode\\epsffile{shad11.eps}\n \\end{center}\n\\caption{}\\label{shad11.fig}\n\\end{figure}\n\n There are three oriented versions of the $S_2$ move. We show\nthat $S_K$ does not change under one of them. The proof for\nthe other two is the same or easier. We choose the version \ncorresponding to the upper part of Figure~\\ref{shad2.fig}. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 8cm\n \\leavevmode\\epsffile{shad2.eps}\n \\end{center}\n\\caption{}\\label{shad2.fig}\n\\end{figure}\nThe summands corresponding to the double points not in this figure are \npreserved under the move, since it does not change the homology classes \nof the corresponding knots. So it suffices to show that the terms \nproduced by \nthe double points $u$ and $v$ in this figure cancel out. \nNote that the shadow $\\mu ^-_v$ is transformed to $\\mu^+_u$\nby the $\\bar S_1$ move. It is known that \n$\\bar S_1$ can be expressed in terms of $S_1,S_2$, and \n$S_3$, thus it also does not change the homology classes of the \nknots created by the splittings. Hence $[\\mu^+_u]$ and $[\\mu^-_v]$ cancel out. \nIn the same way one \nproves that $[\\mu ^-_u]$ and $[\\mu ^+_v]$ also cancel out. \n\nThere are two oriented versions of the $S_3$ move: $S'_3$ and \n$S''_3$, shown in Figures~\\ref{shad7.fig}a and~\\ref{shad7.fig}b respectively.\n\\begin{figure}[htb]\n \\begin{center}\n \\epsfxsize 7cm\n \\leavevmode\\epsffile{shad7.eps}\n \\end{center}\n\\caption{}\\label{shad7.fig}\n\\end{figure}\nThe $S''_3$ move can be expressed in terms of $S'_3$ and \noriented versions of $S_2$ and $S_2^{-1}$. To prove this we use \nFigure~\\ref{shad6.fig}. There are two ways to get from \nFigure~\\ref{shad6.fig}a to Figure~\\ref{shad6.fig}b. One is to apply \n$S''_3$. Another way is to apply three times the oriented version of \n$S_2$ to obtain Figure~\\ref{shad6.fig}c, \nthen apply $S'_3$ to get Figure~\\ref{shad6.fig}d, \nand finally use three times the oriented \nversion of $S_2^{-1}$ to end up at Figure~\\ref{shad6.fig}b.\n\\begin{figure}[htb]\n \\begin{center}\n \\epsfxsize 9cm\n \\leavevmode\\epsffile{shad6.eps}\n \\end{center}\n\\caption{}\\label{shad6.fig}\n\\end{figure}\n\nThus it suffices to verify the invariance under $S'_3$. The terms\ncorresponding to the double points not in Figures~\\ref{shad8.fig}a \nand~\\ref{shad8.fig}b are preserved for the same reasons as above. \nThe terms coming from double points $u$ in \nFigure~\\ref{shad8.fig}a and $u$ in Figure~\\ref{shad8.fig}.b are the \nsame. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{shad8.eps}\n \\end{center}\n\\caption{}\\label{shad8.fig}\n\\end{figure}\nThis holds also for the $v$- and $w$-pairs of double points in these two\nfigures. We prove this statement only for the $u$-pair of double points. For\n$v$- and $w$-pairs the proof is the same or simpler. There is only\none possibility: either the dashed line belongs to both $\\pi(\\mu^+_u)$\nin Figures~\\ref{shad8.fig}a.1 and~\\ref{shad8.fig}b.1 respectively or to\nboth $\\pi(\\mu ^-_u)$ in Figures~\\ref{shad8.fig}a.2 \nand~\\ref{shad8.fig}b.2 respectively. \nWe choose the one to which it does not belong. \nSumming up gleams on each of the two sides of it we immediately see\nthat the corresponding shadows are the same on both pictures. \nThus the homology classes of the corresponding knots are equal. \nBut $[\\mu ^+_u][\\mu ^-_u]=[K]f$ (see~\\ref{sum-property}),\nthus the homology classes of the knots represented by the \nother shadows are also equal. \nThis completes the proof of Theorem~\\ref{correct2}.\\qed\n\n\\subsection{Proof of Theorem~\\ref{realization2}.}\\label{pfrealization2}\nI: $K'$ can be obtained from $K$ by a sequence of isotopies and\nmodifications along fibers. Isotopies do not change $S$. The modifications \nchange $S$ by\nelements of type~\\eqref{type2}. To complete the proof we use the\nidentity $\\xi_1 \\xi_2=[K]$.\n\nII: We prove that for any $i\\in H_1(N)$ there exist\ntwo knots $K_1$ and $K_2$ such that they represent the same free homotopy\nclass as $K$, \n\\begin{equation}\nS_{K_1}=S_K+(f-e)([K]i^{-1}+i),\\text{ and}\n\\end{equation}\n\\begin{equation} \nS_{K_2}=S_K-(f-e)([K]i^{-1}+i).\n\\end{equation}\nClearly this would imply the second statement of the theorem.\n\nTake $i\\in H_1(N)$. Let $K_i$ be an oriented knot in $N$ such that $[K_i]=i$. \nThe space $N$ is oriented, hence the tubular neighborhood $T_{K_i}$ of $K_i$ \nis homeomorphic to an oriented solid torus $T$. Deform $K_i$, so that\n$K_i\\cap T_{K_i}$ is a small arc (see Figure~\\ref{aicardi9.fig}). \nPull one part of the arc along $K_i$ in $T_{K_i}$ under the other part\nof the arc (see Figure~\\ref{aicardi1.fig}). This isotopy creates \ntwo new double points \n$u$ and $v$ of $\\pi(K)$. (Since $T_{K_i}$ may be knotted, it might happen that \nthere are other new double points, but we do not need them for\nour construction.) Making a fiber modification along the part of \n$\\pi^{-1}(u)$ that lies in $T$ one obtains $K_2$. Making a fiber\nmodification along the part of $\\pi^{-1}(v)$ that lies in $T$ one obtains\n$K_1$. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 7cm\n \\leavevmode\\epsffile{aicardi9.eps}\n \\end{center}\n\\caption{}\\label{aicardi9.fig}\n\\end{figure}\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 7cm\n \\leavevmode\\epsffile{aicardi1.eps}\n \\end{center}\n\\caption{}\\label{aicardi1.fig}\n\\end{figure}\nThis completes the proof of Theorem~\\ref{realization2}.\n\\qed\n\n\n\\subsection{Proof of Theorem~\\ref{homtor}.}\\label{pfhomtor}\nIt is easy to verify that any two shadows with the same projection can be \ntransformed to each other by a sequence of fiber fusions. \nOne can easily create a trivial knot with an ascending diagram \nsuch that its projection is any desired curve. \nThis implies that any two shadows \non ${\\mathbb R}^2$ can be transformed to each other by a sequence of fiber fusions, \nmovements $S_1,S_2,S_3$, and their inverses. \nA straightforward verification shows that $\\sigma(s(K))$ does \nnot change under the moves $S_1,S_2,S_3$, and their inverses. \nUnder fiber fusions the homology class of the knot and the element \n$\\sigma$ change in the same way. \nTo prove this, we use Figure~\\ref{shad16.fig}, where Figure~\\ref{shad16.fig}a shows the shadow before the application \nof the fiber fusion (that adds $1$ \nto the homology class of the knot) and \nFigure~\\ref{shad16.fig}b after. In this diagrams \nthe indices of the regions are denoted by Latin letters. Now one easily \nverifies that $\\sigma$ also increases by one. Finally, for \nthe trivial knot with a trivial shadow diagram its homology class \nand $\\sigma (s(K))$ are both equal to $0$.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 9cm\n \\leavevmode\\epsffile{shad16.eps}\n \\end{center}\n\\caption{}\\label{shad16.fig}\n\\end{figure}\nThis completes the proof of Theorem~\\ref{homtor}.\n\\qed\n\n\\subsection{Proof of Theorem~\\ref{correctSSK}.}\\label{pfcorrectSSK}\nIt suffices to show that $S_K$ does\nnot change under the elementary isotopies of the knot. \nThree of them correspond in the projection to: a \nbirth of a small loop, passing through a point of self-tangency, and passing\nthrough a triple point. The fourth one is \npassing through an exceptional fiber.\n\nFrom the proof of Theorem~\\ref{correct2} one gets that $S_K$\nis invariant under the first three of the elementary isotopies described above. \nThus it suffices to prove invariance under passing through an exceptional \nfiber $a$.\n\nLet $a$ be a singular fiber of type $(\\mu_a, \\nu_a)$ (see~\\ref{numbers}). \nLet $T_a$ be a neighborhood of $a$\nwhich is fiber-wise isomorphic to the standardly fibered solid torus with\nan exceptional fiber of type $(\\mu_a, \\nu_a)$. \n\nWe can assume that the move proceeds as follows. At the start \n$K$ and $T_a$\nintersect along a curve lying in the meridional disk $D$ of $T_a$. The part of\n$K$ close to $a$ in $D$ is an arc $C$ of a circle of a very large radius. This\narc is symmetric with respect to the $y$ axis passing through $a$ in $D$.\nDuring the move this arc slides along the $y$ axis through the fiber $a$\n(see Figure~\\ref{seif2.fig}). \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 11cm\n \\leavevmode\\epsffile{seif2.eps}\n \\end{center}\n\\caption{}\\label{seif2.fig}\n\\end{figure}\n\n\nClearly two points $u$ and $v$ of $C$ after this move are in the same \nfiber if and only if they are symmetric with respect to the $y$ axis, and the \nangle formed by $v,a,u$ in $D$ is less or equal to $\\pi$ and is equal \nto $\\frac{2l\\pi}{\\mu_a}$ for some $l\\in \\{1,\\dots,\\mu_a\\}$ \n(see Figure~\\ref{seif2.fig}).\nThey are in the same fiber \nbefore the move if and only if the angle formed by $u,a,v$ in $D$ \nis less than $\\pi$ and is equal to $\\frac{2l\\pi}{\\mu_a}$ for \nsome $l\\in \\{1,\\dots,\\mu_a\\}$ (see Figure~\\ref{seif2.fig}).\n \nConsider a double point $v$ of $\\pi\\big|_D(K)$ that appears \nafter the move and\ncorresponds to the angle $\\frac{2l\\pi}{\\mu_a}$. There is a unique\n$k\\in N_1(a)$ such that \n$\\frac{2\\pi\\nu_a k}{\\mu_a}\\protect\\operatorname{mod} 2\\pi=\\frac{2l\\pi}{\\mu}$. Note that to make\nthe splitting of $[K]$ into $[\\mu^+_v]$ and $[\\mu^-_v]$ \nwell defined, we do not need the two points of $K$ projecting to $v$ \nto be antipodal in $\\pi^{-1}(v)$. This allows one to compare \nthese homology classes with $f_a$. For the orientation of $C$ shown in \nFigure~\\ref{seif2.fig} one verifies that connecting $v$ to $u$ along the orientation of\nthe fiber we are adding $k$ fibers $f_a$. (Note that the\nfactorization we used to define the exceptionally fibered torus was\n$\\bigl(\\bigl(r,\\theta\\bigr),1\\bigr)=\n\\bigl(\\bigl(r,\\theta+\\frac{2\\pi\\nu}{\\mu} \\bigr),0\\bigr)$.) \nThus $[\\mu^-_v]=f_a^k$ (see\nFigure~\\ref{seif1.fig}). From~\\ref{sum-property} we know that\n$[\\mu^+_v][\\mu^-_v]=[K]f$. Hence $[\\mu^+_v]=[K]f_a^{\\mu_a-k}$. \n\n\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{seif1.eps}\n \\end{center}\n\\caption{}\\label{seif1.fig}\n\\end{figure} \n\nAs above, to each double point $v$ of $\\pi\\big|_D(K)$ before this move\nthere corresponds $k\\in N_2(a)$. For this double point $[\\mu^+_v]=f_a^{\\mu_a-k}$ and\n$[\\mu^-_v]=[K]f_a^k$. \n\nSumming up over the corresponding values of $k$ we see that \n$S_K$ changes by $R^1_a$ under this move. Recall that $R^1_a=0\\in H$. Thus \n$S_K$ is invariant under the move.\n\nFor the other choice of the orientation of $C$ the value of $S_K$ \nchanges by $R^2_a=0\\in H$.\n\nThus $S_K$ is invariant under all elementary isotopies, and this proves\nTheorem~\\ref{correctSSK}.\n\\qed \n\n\n\n\n\n\n\n\n\n \n\n\n\\subsection{Proof of Theorem~\\ref{front-shadow}.}\\label{pffront-shadow}\nDeform $L$ in the neighborhoods of all double points of $L$ \n(see Figure~\\ref{pic1.fig}), \nso that the two points of the Legendrian knot corresponding \nto the double point of $L$ are antipodal in the fiber. \n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{pic1.eps}\n \\end{center}\n\\caption{}\\label{pic1.fig}\n\\end{figure}\nAfter we make the quotient of the fibration by the ${\\mathbb Z}_2$-action, \nthe projection of the \ndeformed $\\lambda$ is not a cooriented front anymore but a front \nequipped with a normal field of lines. \n(This corresponds to the factorization $S^1\\rightarrow{\\mathbb R} P^1$.) \nUsing Figure~\\ref{pic3.fig} one calculates \nthe contributions of different cusps and double points to the total \nrotation number of the line field under traversing the boundary in the \ncounter clockwise direction.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 12cm\n \\leavevmode\\epsffile{pic3.eps}\n \\end{center}\n\\caption{}\\label{pic3.fig}\n\\end{figure} \n\n\nThese contributions are as follows:\n\\begin{equation}\n\\begin{cases}\n 1 & \\text{for every cusp point pointing inside $X$};\\\\\n -1 & \\text{for every cusp point pointing outside $X$};\\\\\n -1 & \\text{for every double point of the type shown in\nFigure~\\ref{pic2.fig}c};\\\\\n 0 & \\text{for the other types of double points.}\n\\end{cases}\n\\end{equation}\n\nTo get the contributions to gleams, we divide these numbers by $2$ \n(as we do in the construction of shadows, see~\\ref{prelim}).\n\nIf the region does not have cusps and double points in its boundary, \nthen the obstruction to an extension of the section over $\\partial X$ \nto $X$ is equal to $\\chi(\\protect\\operatorname{Int} X)$.\n\n\nThis completes the proof of Theorem~\\ref{front-shadow}.\n\\qed\n\n\\subsection{Proof of Theorem~\\ref{equivalence}.}\\label{pfequivalence}\nA straightforward verification shows that \n\\begin{equation}\\label{useful1}\n\\protect\\operatorname{ind} \\tilde L_u^+-\\protect\\operatorname{ind} \\tilde L_u^-=\n\\protect\\operatorname{ind} L^+_u-\\protect\\operatorname{ind} L^-_u-\\epsilon(u),\n\\end{equation} \n\\begin{equation}\\label{useful2}\n\\protect\\operatorname{ind} \\tilde L^+_u+ \\protect\\operatorname{ind} \\tilde L^-_u=\\protect\\operatorname{ind} L+1,\n\\end{equation}\nand\n\\begin{equation}\\label{useful3}\n\\protect\\operatorname{ind} L^+_u + \\protect\\operatorname{ind} L^-_u = \\protect\\operatorname{ind} L\n\\end{equation}\nfor any double point $u$ of $L$.\n\nLet us prove~\\eqref{rel1}. We write down the formal sums used to\ndefine $S(\\lambda)$ and $l_q(\\lambda)$ and start to reduce them in a\nparallel way as described below. \n\nWe say that a double point is essential if $[\\tilde\\lambda^+_u]\\neq \n[\\tilde\\lambda^-_u]$. \n\nFor non-essential $u$ we see that the term \n$\\bigl([\\tilde\\lambda^+_u]-[\\tilde\\lambda^-_u]\\bigr)$ in\n$S(\\lambda)$ is zero. Using~\\eqref{useful1} we get that the term\n$[\\protect\\operatorname{ind} L^+_v- \\protect\\operatorname{ind} L^-_v-\\epsilon (v)]_q$ in \n$l_q(\\lambda)$ is also zero. \n\nThe index of a wave front coincides with the homology class of its lifting\nunder the natural identification of $H_1(ST^*F)$ with ${\\mathbb Z}$. This fact\nand~\\eqref{useful2} imply that if we have \n$[\\tilde \\lambda^+_u]=[\\tilde \\lambda^-_v]$ for two double points $u$ and $v$, \nthen $[\\tilde \\lambda^-_u]=[\\tilde \\lambda^+_v]$. \nHence $\\bigl([\\tilde \\lambda^+_u]-\n[\\tilde \\lambda^-_u]\\bigr)=-\\bigl([\\tilde \\lambda^+_v]-\n[\\tilde \\lambda^-_v]\\bigr)$,\nand these two terms cancel out. Identity~\\eqref{useful1} implies that the \nterms $[\\protect\\operatorname{ind} L^+_u-\\protect\\operatorname{ind} L^-_u-\\epsilon(u)]_q$ and \n$[\\protect\\operatorname{ind} L^+_v-\\protect\\operatorname{ind} L^-_v-\\epsilon(v)]_q$ also cancel out. \n\nFor similar reasons, if for a double point $u$ the term \n$\\bigl ([\\tilde \n\\lambda^+_u]-[\\tilde \\lambda^-_u]\\bigr)$ is equal to \n$\\bigl([\\lambda]-f\\bigr)$, so that we can simplify $S(\\lambda)$ by crossing\nout the term and decreasing the coefficient $C^+$ by one. \nThen \n$[\\protect\\operatorname{ind} L^+_u-\\protect\\operatorname{ind} L^-_u-\\epsilon(u)]_q=[\\protect\\operatorname{ind} L-1]_q$, and we can simplify \n$l_q(\\lambda)$ by crossing out the term and decreasing \nthe coefficient $C^+$ by one.\n\n\nSimilarly if the input of double point $u$ into $S(\\lambda)$ is \n$\\bigl(e-[\\lambda]f\\bigr)$, then we reduce the two sums in the parallel way\nby crossing out the corresponding terms and decreasing by one the coefficients\n$C^-$. \n\nWe make the cancellations described above in both $S(\\lambda)$ and\n$(l_q(\\lambda)-[h]_qh)$ in a parallel way until we can \nnot reduce $S(\\lambda)$ any more. \nIn this reduced form the terms of the form $k_lf^l$ with\n$k_l>0$ correspond to \nthe terms of type $[\\tilde \\lambda^+_u]$ for some double points $u$ of $L$.\n(The case where a term of this type correspond to cusps is \ntreated separately below.) \nIdentities~\\eqref{useful1} \nand~\\eqref{useful2} imply that the contribution of the corresponding \ndouble points into $l_q(\\lambda)$ is $k_l[2l-h-1]_q$. \n\nIn the case where $k_lf^l$ term comes from the cusps and not from the double\npoints of $L$, one can easily verify that the corresponding input of cusps \ninto $(l_q(\\lambda)-[h]_qh)$ can still be written as $k_l[2l-h-1]q$.\n\nThus $l_q(\\lambda)=[h]_qh+\\sum_{k_l>0} k_l[2l-h-1]_q$, \nand we have proved~\\eqref{rel1}.\n\nLet us prove~\\eqref{rel2}. \nAs above we reduce $S(\\lambda)$ and\n$(l_q(\\lambda)-[h]_qh)$ in a parallel way. Note that the\ncoefficient at\neach $[m]_q$ was positive from the very beginning by the definition\nof $l_q(\\lambda)$, and it stays positive under the cancellations described \nabove. \nAfter this reduction each term $n_m[m]_q$ \nis a contribution of $n_m$ double points. (The case where it is a\ncontribution of \ncusps is treated separately as in the proof of~\\eqref{rel2}.) \nLet $u$ be one of these double points.\nThen from~\\eqref{useful1} and~\\eqref{useful2} we get\nthe following system of two equations in variables $\\protect\\operatorname{ind} \\tilde L^+_u$ and \n$\\protect\\operatorname{ind} \\tilde L^-_u$:\n\\begin{equation}\n\\begin{cases}\n\\protect\\operatorname{ind} \\tilde L^+_u-\\protect\\operatorname{ind} \\tilde L^-_u=m &\\\\\n\\protect\\operatorname{ind} \\tilde L^+_u+\\protect\\operatorname{ind} \\tilde L^-_u=\\protect\\operatorname{ind} L+1.&\n\\end{cases}\n\\end{equation}\n\nSolving the system we get that $[\\tilde \\lambda\n^+_u]=f^{\\frac{m+h+1}{2}}$ and $[\\tilde \\lambda\n^-_u]=f^{\\frac{h+1-m}{2}}$.\n\nThis proves identity~\\eqref{rel2} and Theorem~\\ref{equivalence}.\n\\qed \n \n\n\n\\subsection{Proof of Theorem~\\ref{orbifold1}.}\\label{pforbifold1}\nThere are five elementary isotopies of a generic front $L$ on an orbifold\n$F$. Four of them are: the birth of two cusps, passing through a\nnon-dangerous self-tangency point, passing through a triple point,\nand passing of a branch through a cusp point. For all possible oriented\nversions of these moves a straightforward calculation shows that \n$S(\\lambda)\\in \\frac{1}{2}{\\mathbb Z}[H_1(N)]$ is preserved.\n\nThe fifth move is more complicated. It corresponds to a generic passing of \na wave front lifted to ${\\mathbb R}^2$ \nthrough the preimage of a cone point $a$. \nWe can assume that this\nmove is a symmetrization by $G_a$ of the following move. \nThe lifted front in the neighborhood of $a$ is an arc $C$ of a circle\nof large radius with center at the $y$ axis, and during this move \nthis arc slides through $a$ along $y$ (see Figure~\\ref{orbi2.fig}).\n\nClearly after this move points $u$ and $v$ on the arc $C$ turn out to be \nin the same fiber if and only if they are symmetric with respect \nto the $y$ axis, and the angle formed by $v,a,u$ is less or equal to $\\pi$ \nand is equal to $\\frac{2k\\pi}{\\mu_a}$ for some $k\\in \\{1,\\dots,\\mu_a\\}$ \n(see Figure~\\ref{orbi2.fig}). \nWe denote the set of such numbers $k$ by \n$\\bar N_1(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{2k\\pi}{\\mu_a}\n\\in(0,\\pi]\\bigr\\}$. \n\nTwo points $u$ and $v$ on the arc $C$ are in the same fiber \nbefore the move if and only if they are symmetric with respect \nto the $y$ axis, and the angle formed by $u,a,v$ is less than\n$\\pi$ and is equal $\\frac{2k\\pi}{\\mu_a}$ for \nsome $k\\in \\{1,\\dots,\\mu_a\\}$ (see Figure~\\ref{orbi2.fig}).\nWe denote the set of such numbers $k$ by \n$\\bar N_2(a)=\\bigl\\{k\\in\\{1,\\dots,\\mu_a\\}\\big|\\frac{2k\\pi}{\\mu_a}\n\\in(0,\\pi)\\bigr\\}$.\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 11cm\n \\leavevmode\\epsffile{orbi2.eps}\n \\end{center}\n\\caption{}\\label{orbi2.fig}\n\\end{figure} \n\n\nThe projection of this move for the orientation of $L'$ drawn in\nFigure~\\ref{orbi2.fig} is shown in Figure~\\ref{orbi1.fig}. \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\epsfxsize 10cm\n \\leavevmode\\epsffile{orbi1.eps}\n \\end{center}\n\\caption{}\\label{orbi1.fig}\n\\end{figure} \n\n\nSplit the wave front in\nFigure~\\ref{orbi1.fig} at the double point $v$ (appearing after the move) \nthat corresponds to some\n$k\\in \\bar N_1(a)$. \nThen $\\tilde\\lambda^-_v$ is a front with two positive cusps that\nrotates $k$ times around $a$ in the clockwise direction. Hence\n$[\\tilde\\lambda^-_v]=ff_a^{-k}=f_a^{\\mu_a-k}$. We know that \n$[\\tilde\\lambda^+_v][\\tilde\\lambda^-_v]=[\\lambda]f$ and that\n$f_a^{\\mu_a}=f$. \nThus\n$[\\tilde\\lambda^+_v]=[\\tilde\\lambda]f_a^k$.\n \nIn the same way we verify that if we split the front \nat the double point $v$ \n(existing before the move) that corresponds to some $k\\in \\bar N_2(a)$,\nthen $[\\tilde \\lambda^+_v]=f_a^k$ and $[\\tilde \\lambda^-_v]=[K]f_a^{|\\mu_a|-k}$. \n\nNow making sums over all corresponding numbers $k\\in \\{1,\\dots,\\mu_a\\}$\nwe get\nthat under this move $S(\\lambda)$ changes by \n\\begin{equation}\n\\bar R^1_a=\\sum_{k\\in\\bar N_1(a)}\\bigl([\\lambda]f_a^k-f_a^{\\mu_a-k}\\bigr)-\n\\sum_{k\\in\\bar N_2(a)}\\bigl(f_a^k-[\\lambda]f_a^{\\mu_a-k}\\bigr).\n\\end{equation}\nA straightforward verification shows that $R^1_a=\\bar R^1_a$. (Note that the \nsets $N_1(a)$ and $N_2(a)$ are different from $\\bar N_1(a)$ and $\\bar N_2(a)$.)\n\nRecall that $R^1_a=0\\in J$. Thus $S(\\lambda)$ is invariant under the move. \n\nFor the other choice of the orientation of $C$ the value of \n$S(\\lambda)$ changes by $R^2_a=0\\in J$. \n\nHence $S(\\lambda)$ is invariant under all elementary\nisotopies, and we have proved Theorem~\\ref{orbifold1}.\n\\qed\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpqzo b/data_all_eng_slimpj/shuffled/split2/finalzzpqzo new file mode 100644 index 0000000000000000000000000000000000000000..79db49ec1e15ce690399a35c42d927133d4cf3c6 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzpqzo @@ -0,0 +1,5 @@ +{"text":"\\section{Results}\n\\label{sec:results}\n\nOdors are mixtures of odorant molecules that are ligands of olfactory receptors.\nAny odor can be described by a vector~$\\boldsymbol c = (c_1, c_2, \\ldots, c_{{N_{\\rm l}}})$ that specifies the concentrations of all ${N_{\\rm l}}$ possible ligands. During a single sniff, the ligands in the odor~$\\boldsymbol c$ come in contact with the~${N_{\\rm r}}$ different odor receptors.\nIn the simplest case, the sensitivity of receptor~$n$ to ligand~$i$ can be described by a single number~$S_{ni}$ and the total excitation~$e_n$ of receptor~$n$ is given by~\\cite{McGann2005, Lin2006}\n\\begin{align}\n\te_n &= \\sum_i S_{ni} c_i\n\t\\;.\n\t\\label{eqn:excitation}\n\\end{align}\n\nTypical receptors have a non-linear dose-response curve~\\cite{Reisert2009} and the output~$a_n$ is thus a non-linear function of~$e_n$.\nMoreover, receptors are subject to noise~\\cite{Lowe1995}, \\mbox{e.\\hspace{0.125em}g.}\\@\\xspace, from stochastic binding, which limits the number of distinguishable outputs.\nTo capture both effects, we consider receptors with only two output states, which corresponds to large noise~\\cite{Koulakov2007}.\nIn this case, the activity~$a_n$ of receptor~$n$ is given by\n\\begin{align}\n\ta_n &= \\begin{cases}\n\t\t0 & e_n < 1 \\\\\n\t\t1 & e_n \\ge 1\t\n\t\\end{cases}\n\t\\;,\n\t\\label{eqn:receptor_activity}\n\\end{align}\n\\mbox{i.\\hspace{0.125em}e.}\\@\\xspace, the receptor is active if its excitation~$e_n$ exceeds a threshold.\n\\Eqsref{eqn:excitation}--\\eqref{eqn:receptor_activity} describe the mapping of the odor~$\\boldsymbol c$ to the activity pattern~$\\boldsymbol a=(a_1, a_2, \\ldots, a_{{N_{\\rm r}}})$, where the receptor array is characterized by the sensitivity matrix~$S_{ni}$, see \\figref{fig:schematic}C.\nThis activity pattern is then analyzed by the brain to infer the odor~$\\boldsymbol c$.\nSuch a distributed representation of odors in activity patterns has been compared to compressed sensing \\cite{Stevens2015};\nhere we focus on how this representation can be tuned to match the structure of natural odors. \n\nWe assume that the structure of natural odors in a given environment can be captured by a probability distribution $P_{\\rm env}(\\boldsymbol c)$ from which odors are drawn.\n$P_{\\rm env}(\\boldsymbol c)$ can encode, for example, the fact that some ligands are more common than others or that some ligands are strongly correlated or anti-correlated in their occurrence. \nSince natural odor statistics are hard to measure~\\cite{Wright2005}, we work with a broad class of distributions~$P_{\\rm env}(\\boldsymbol c)$ characterized by a few parameters.\nWe define $p_i$ to be the probability with which ligand~$i$ occurs in a random odor.\nThe correlations between the occurrence of ligands are captured by a covariance matrix~$p_{ij}$.\nWe expect $p_i$ to be small since any given natural odor typically contain tens to hundreds of ligands~\\cite{Knudsen1993, Lin2006}, which is a small subset of all ${N_{\\rm l}} \\gtrsim 2100$ ligands~\\cite{Wright2005}.\nWhen a ligand~$i$ is present, we assume its concentration $c_i$ has mean~$\\mu_i$ and standard deviation~$\\sigma_i$.\nThus, the full natural odor statistics $P_{\\rm env}(\\boldsymbol c)$ are parameterized by $p_i$, $\\mu_i$, and $\\sigma_i$ for all ligands~$i$ and a covariance matrix~$p_{ij}$ in our model.\n\n\\subsection{Optimal receptor arrays}\nAn optimal receptor array must tailor receptor sensitivities $S_{ni}$ so that the odors-to-activity mapping \ngiven by \\Eqsref{eqn:excitation}--\\eqref{eqn:receptor_activity}\ndedicates more activity patterns to more frequent or more important odors as specified by $P_{\\rm env}(\\boldsymbol c)$. In information-theoretic terms, the array must maximize the mutual information $I(\\boldsymbol c, \\boldsymbol a)$~\\cite{Atick1992}. \nIn our model, the mapping from~$\\boldsymbol c$ to~$\\boldsymbol a$ is deterministic and $I$ can be written as the entropy of the output distribution~$P(\\boldsymbol a)$, \n\\begin{align}\n\tI &= -\\sum_{\\boldsymbol a} P(\\boldsymbol a) \\log_2 P(\\boldsymbol a)\n\t\\;,\n\t\\label{eqn:MI_def}\n\\end{align}\nwhere the sum is over all possible activity patterns~$\\boldsymbol a$.\nNote that $P(\\boldsymbol a) = \\int \\text{d} \\boldsymbol c \\, P(\\boldsymbol a | \\boldsymbol c) P_{\\rm env} (\\boldsymbol c)$, where $P(\\boldsymbol a | \\boldsymbol c)$ describes the mapping from $\\boldsymbol c$ to $\\boldsymbol a$.\nConsequently, $I$ depends on $S_{ni}$ and the odor environment $P_{\\rm env}(\\boldsymbol c)$.\nIn fact, $I$ is maximized by sensitivities~$S_{ni}$ that are tailored to $P_{\\rm env}(\\boldsymbol c)$ such that all activity patterns~$\\boldsymbol{a}$ are equally likely \\cite{Atick1992,Laughlin1981}.\n\nThe mutual information~$I$ can be approximated~\\cite{Sessak2009} in terms of the mean activities $\\mean{a_n}$ and the covariance between receptors, $\\cov(a_n, a_m)= \\mean{a_n a_m} - \\mean{a_n} \\mean{a_m}$, encoded by $P(\\boldsymbol a)$,\n\\begin{align}\n\tI &\\approx \n\t- \\!\\sum_n \\bigl[\\mean{a_n}\\log_2 \\mean{a_n} + (1 - \\mean{a_n})\\log_2(1 - \\mean{a_n})\\bigr]\n\t\\notag \\\\ & \\quad\n\t- \\frac{8}{\\ln 2} \\sum_{n 1$, where $e_{\\rm b}$ and $e_{\\rm d}$ are the excitations caused by the~$s_{\\rm b}$ shared and the $s - s_{\\rm b}$ different ligands, respectively.\nApproximating the probability distribution of the excitations as a log-normal distribution, we can calculate the expected distance~$h$, see SI.\n\\figref{fig:mixture_discrimination}B shows that this approximation (solid lines) agrees well with numerical calculations (symbols).\nThe figure also shows that mixtures can only be distinguished well if the concentration of the constituents is in the right range.\nThis is because receptors are barely excited for too small concentrations while they are saturated for large concentrations. \nThe distance~$h$ also strongly depends on the number~$s_{\\rm b}$ of shared ligands between the two mixtures, which has also been shown experimentally~\\cite{Bushdid2014}.\nThe distance vanishes for $s_{\\rm b} = s$, but \\figref{fig:mixture_discrimination}B shows that a single different ligand can be sufficient to distinguish mixtures in the right concentration range (green line).\nThis range increases with the width~$\\lambda$ of the sensitivity distribution, similar to the range over which concentrations can be measured, see \\Eqref{eqn:concentration_range}.\n\n\\begin{figure}[t]\n\t\\centerline{\n\t\t\\includegraphics[width=\\figwidth]{\"Figures\/Fig5\"}\n\t}\n\t\\caption{%\n\t\tThe discriminability of mixtures strongly depends on the concentrations at which odors are presented. \n\t\t(A) Maximal mixture size~$s_{\\rm max}$ (from \\Eqref{eqn:mixture_size_max}) as a function of the ligand concentration~$c$ for different widths~$\\lambda$ of the sensitivity distribution at ${N_{\\rm r}}\/\\eta=300$.\n\t\tDotted lines indicate where $c$ is below the detection threshold for single ligands.\n\t\t(B)~Mean difference~$h$ in the activation pattern of two mixtures of size~$s=10$ as a function of $c$ for different numbers $s_{\\rm b}$ of shared ligands and widths~$\\lambda$.\n\t\tAnalytical results (lines) are compared to numerical simulations (symbols).\n\t\t\\label{fig:mixture_discrimination}\n\t}\n\\end{figure}%\n\n\\subsection{Experimentally measured receptor arrays}\n\nThe response of receptors to individual ligands has been measured experimentally for flies~\\cite{Muench2015} and humans~\\cite{Mainland2015}.\nWe use these published data to estimate the statistics of realistic sensitivity matrices as described in the SI.\n\\figref{fig:sensitivities} shows the histograms of the logarithms of the sensitivities for flies and humans.\nBoth histograms are close to a normal distribution, with similar standard deviations~$\\lambda_{\\rm exp} \\approx 1.1$, which implies log-normally distributed sensitivities.\nUsing a simple binding model between receptors and ligands, $\\lambda_{\\rm exp}$ can also be interpreted as the standard deviation of the interaction energies, see SI.\nConsequently, these interaction energies exhibit a similar variation on the order of one $k_{\\rm B} T$ for both organisms, which could be caused by the biophysical similarity of the receptors.\n\n\\begin{figure}[t]\n\t\\centerline{\n\t\t\\includegraphics[width=\\figwidth]{\"Figures\/Fig6\"}\n\t}\n\t\\caption{%\n\t\tSensitivities of olfactory receptors appear to be log-normally distributed for (A) flies~\\cite{Muench2015} and (B) humans~\\cite{Mainland2015}.\n\t\tThe histograms of the logarithms of $n$ entries of the sensitivity matrix (orange) are compared to a normal distribution (blue) with the same mean and standard deviation~$\\lambda_{\\rm exp}$.\n\t\t\\label{fig:sensitivities}\n\t}\n\\end{figure}%\n\nWe next use the measured log-normal distribution for the sensitivities to compare the concentration resolution~$R$ predicted by \\Eqref{eqn:concentration_resolution} to measured 'just noticeable relative differences'~$R^{-1}$~\\cite{Koulakov2007}.\nFor humans (${N_{\\rm r}} = 300$), the measured values are as low as \\unit[4]{\\%}~\\cite{Cain1977}, which implies $\\eta\\lambda \\approx 4.8$.\nUsing $\\lambda \\approx 1.1$, this suggest that about 4 receptors have to be activated until a change in concentration can be registered.\nAdditionally, our theory predicts that humans can sense concentrations over about $2.6$ orders of magnitude, which follows from \\Eqref{eqn:concentration_range} for $\\lambda = 1.1$, $\\eta = 1$, and ${N_{\\rm r}} = 300$.\nHowever, we are not aware of any measurements of the concentration range for humans.\n\nOur theory also predicts the maximal number of ligands that can be distinguished as a function of the concentration~$c$ of the individual ligands.\nFor $\\lambda \\approx 1.1$, we expect that the maximal number~$s_{\\rm max}$ of ligands in a mixture is around 20 if individual ligands can be detected, see \\figref{fig:mixture_discrimination}A.\nExperimental studies report similar numbers, \\mbox{e.\\hspace{0.125em}g.}\\@\\xspace, $s_{\\rm max} \\approx 15$~\\cite{Jinks1999} and $s_{\\rm max} < 30$~\\cite{Weiss2012}.\nHowever, \\figref{fig:mixture_discrimination}A shows that $s_{\\rm max}$ strongly depends on the concentration of the individual ligands and thus on experimental details.\nSimilarly, how well mixtures can be discriminated also depends strongly on the ligand concentration.\n\\figref{fig:mixture_discrimination}B shows that the concentration range over which mixtures can be distinguished is less than an order of magnitude for $\\lambda \\approx 1.1$.\n\n\\section{Discussion}\n\nWe studied how arrays of olfactory receptors can be used to measure odor mixtures, focusing on the combinatorial code of olfaction, \\mbox{i.\\hspace{0.125em}e.}\\@\\xspace, how the combined response of multiple receptors can encode the composition (quality) and the concentration (quantity) of odors.\nSuch arrays are optimal if each receptor responds to about half of the encountered odors and the receptors have distinct ligand binding profiles to minimize correlations.\n\nOur simple model of binary receptors can in principle distinguish a huge number of odors, since there are $\\sim10^{90}$ different output combinations for ${N_{\\rm r}}=300$.\nHowever, it is not clear whether all outputs are achievable and how they are used to distinguish odors.\nWe showed that the mean receptor sensitivity must be tailored to the mean concentration to best use the large output space.\nAnother important parameter of receptor arrays is the fraction of receptors that is activated by a single ligand, which is equivalent to the sparsity~$\\xi$ in the simple case of binary sensitivities.\nIf $\\xi$ is small, combining different ligands typically leads to unique output patterns that allow to identify the mixtures, but the concentration of isolated ligands cannot be measured reliably, since only few receptors are involved.\nConversely, if $\\xi$ is large, mixtures of multiple ligands will excite almost all receptors, such that neither the odor quality nor the odor quantity can be measured reliably.\nHowever, here, the concentration of an isolated ligand can be measured precisely.\nWe discussed this property in detail for sensitivities that are log-normally distributed, where the width~$\\lambda$ controls whether mixtures can be distinguished well or concentrations can be measured reliably.\nInterestingly, experiments find that individual ligands at moderate concentration only excite few glomeruli~\\cite{Saito2009}, but natural odors at native concentrations can excite many~\\cite{Vincis2012}.\nThis could imply that the sensitivities are indeed adapted such that each receptor is excited about half the times for natural odors.\n\nOur model implies that having more receptor types can improve all properties of the receptor array.\nIn particular, both the concentration resolution $R$ and the typical distance $h$ between mixtures are proportional to ${N_{\\rm r}}$, a prediction that can be tested experimentally.\nFor instance, mice, with ${N_{\\rm r}} \\approx 1000$ receptor types, are very good at identifying a single odor in a mixture~\\cite{Rokni2014}, but flies, with ${N_{\\rm r}} = 52$~\\cite{Muench2015}, should perform much worse.\nHowever, quantitative comparisons might be difficult since the discrimination performance strongly depends on the normalized concentration~$c\\bar S$ at which odors are presented.\nIn fact, we predict that mixtures can hardly be distinguished if the concentration of the individual ligands is changed by an order of magnitude, see \\figref{fig:mixture_discrimination}B.\n\nOur results also apply to artificial chemical sensor arrays known as 'artificial noses'~\\cite{Albert2000, Stitzel2011}.\nHaving more sensors improves the general performance of the array, but it is also important to tune the sensitivity of individual sensors.\nHere, sensors should be as diverse as possible while still responding to about half the incoming mixtures.\nUnfortunately, building such chemical sensors is difficult and their binding properties are hard to control~\\cite{Stitzel2011}.\nIf the sensitivity matrix of the sensor array is known, our theory can be used to estimate the information~$I_n$ that receptor $n$ contributes as\n$\nI_n \\approx \n\t\tH_{\\rm b}(\\mean{a_n})\n\t\t- \\frac{4}{\\ln 2} \\sum_{m \\neq n} \\! \\cov(a_n, a_m)^2\n$\nwhere $H_{\\rm b}(p) = - p\\log_2 p - (1 - p)\\log_2(1 - p)$, such that $I = \\sum_n I_n$, see \\Eqref{eqn:MI_est}.\nThis can then be used for identifying poor receptors that contribute only little information to the overall results.\n\nOur focus on the combinatorial code of the olfactory system certainly neglects intricate details of the system.\nFor instance, we consider sensitivity matrices with independent entries, but biophysical constraints will cause chemically similar ligands to excite similar receptors~\\cite{Malnic1999, Hallem2006}.\nThis is important because it makes it difficult to distinguish similar ligands~\\cite{Perez2015} and it might thus be worthwhile to dedicate more receptors to such a part of chemical space.\nAdditionally, receptors or glomeruli might interact with each other, \\mbox{e.\\hspace{0.125em}g.}\\@\\xspace, causing inhibition reducing the signal upon binding a ligand~\\cite{Ukhanov2010}.\nWe can in principle discuss inhibition in our model by allowing for negative sensitivities, but more complicated features cannot be captures by the linear relationship in \\Eqref{eqn:excitation}.\nOne important non-linearity is the dose-response curve of individual receptor neurons~\\cite{Reisert2009}, which we approximate by a step function, see \\Eqref{eqn:receptor_activity}.\nThis simplification reduces the information capacity of a single glomerulus to $\\unit[1]{bit}$, while it is likely higher in reality. \nHowever, we expect that allowing for multiple output levels would only increase the concentration resolution and not change the discriminability of mixtures very much~\\cite{Koulakov2007}.\nIt would be interesting to see how such an extended model can measure heterogenous mixtures with ligands at different concentrations.\n\n\\begin{acknowledgments}\nWe thank Michael Tikhonov and Carl Goodrich for helpful discussions and a critical reading of the manuscript.\nThis research was funded by\nthe National Science Foundation through DMR-1435964, \nthe Harvard Materials Research Science and Engineering Center DMR-1420570,\nthe Division of Mathematical Sciences DMS-1411694, and\nthe German Science Foundation through ZW 222\/1-1.\nMPB is an investigator of the Simons Foundation. \n\n\\end{acknowledgments}\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nFullerenes, hollow clusters made up of carbon atoms bonded by sp$^2$ orbitals, have interesting conducting properties. The origin of such a behavior \nis their delocalized $\\pi$ frontier molecular orbitals, \nwhat gives rise to a high conductance when an electric field is applied on the molecule, e.g. by an external potential bias. It has been discussed that free-electron and tight-binding (TB) models can capture the main features of the electronic transport through nearly spherical fullerenes \\cite{Manousakis91, Mizorogi03}. The high conductivity of C$_{60}$ has lead to speculate on the possibility of considering it as a conducting spherical shell \\cite{Amusia06}.\nParticularly, several authors have studied the molecular junctions of the C$_{60}$ fullerene under different types of connections, such as, \na substrate and a STM tip \\cite{Paulsson99, Neel08}, one-dimensional leads \\cite{Saffarzadeh08}, carbon nanotubes \\cite{Shokri11}, gold clusters \\cite{Bilan12} or break junctions \\cite{Lortscher13}. \nFurthermore, the stability and strong hybridization of C$_{60}$ with metallic surfaces make it also a feasible anchoring group with high conductance \\cite{Bilan12}. \nIn the search of similar suitable molecular junctions, other larger icosahedral fullerenes C$_n$ from the same family $n=60 k^2$ ($k$ integer) have also been shown to be stable \\cite{Yu09, Dunlap06}, while the conductance of others, \nsuch as C$_{20}$ or its complexes have also been explored \\cite{Otani04, An09, Ji12}. On the other hand, by doping with boron and nitrogen, fullerene-based molecular junctions were found to have negative resistance \\cite{Yaghobi11}. \n\nA number of methods have been applied to the study of energetics and stability of buckyonions, namely, icosahedral fullerenes encapsulated by larger ones \\cite{Maiti93, Guerin97, Heggie97, Heggie98, Dodziuk00, Glukhova05, Baowan07, Enyashin07, Pudlak09, Xu08, Charkin13}.\nCarbon nano-onions are interesting structures between fullerenes and multi-wall carbon nanotubes, having high thermal stability and chemical reactivity compared to CNTs. They also have the characteristic high contact area and affinity for noble metals, what make them interesting as anchoring groups for molecular electronics. \nFor instance, it has also been shown that onion-like nanoparticles can be used as electrochemical capacitors, also called supercapacitors, with high discharge rates of up to three order of magnitud higher than conventional supercapacitors \\cite{Pech10}.\nThe static polarizability, closely related to the response of the electronic charges to applied static fields, have been studied for onions formed by members of the icosahedral ${60 k^2}$ family, using both phenomenological models and first-principle methods \\cite{Iglesias-Groth03, Zope08, Gueorguiev04}, showing their capability to partially screen static external electric fields. The conductance of onion-like structures functionalized with sulfide-terminated chains has been measured between a gold substrate and a gold STM tip \\cite{Sek13}.\n\nThe present work is aimed to study the electronic transmission of single-wall and multi-wall fullerenes weakly attached to metallic leads. \nIn the next Section we discuss the TB \nmodel and the influence of the curvature and finite size of the layers on it, \nas well as the Green function-based method for the calculation of the transmission.\nIn Section III, we present our results for the dependence of the transmission function on the electron energy $T(E)$. \nWe study how $T(E)$ is affected by the relative angular orientation of the shells, the number of intershell connections included in the TB \nmodel and the number of shells of the onions. Finally in Section IV, we summarize our conclusions on the systems studied.\n\\section{Model and calculation method}\n\\subsection{Single-wall fullerenes: curvature and finite size \\label{one shell model}}\nThe Hamiltonian of a $n$-atom fullerene $C_n$ is described in the TB\napproximation with one $\\pi$ orbital per site\n\\begin{equation}\nH_n = t_n \\sum_{\\langle ij\\rangle} c_i^\\dagger c_j + {\\rm H.c.},\n\\end{equation}\nwhere the summation runs on nearest neighbor atom pairs $\\langle ij\\rangle$, and the operator $c_i^\\dagger$ ($c_j$) creates (annihilates) an electron in the $\\pi$ orbital centered at the atom $i$ ($j$). The constant on-site energy has been taken as zero.\n\nA single parameter $t_n$ is used for the hopping integral between nearest neighbor atoms for a given fullerene.\nAlthough no bond dimerization effect is included, it has been shown that its sole effect is to slightly break some degeneracies in the spectrum with no major qualitative effect \\cite{Manousakis91}. \nIn order to take into account the effect of the curvature of the shell for the various fullerenes, we consider the\nhopping parameter $t_n$ to be a function of the mean radius $R_n$ of the (nearly) spherical shell of $n$ C atoms and the mean inter-atomic distance $d_n$ \\cite{Dresselhaus02}\n\\begin{equation}\nt_n=t \\left[1-\\frac{1}{2}\\left(\\frac{d_n}{R_n}\\right)^2\\right].\n\\end{equation}\nThe value $t =-2.73$ eV is a suitable hopping for graphene and is chosen to correctly reproduce the HOMO-LUMO gap for C$_{60}$, $E_g=-1.90$ eV, as obtained from DFT calculations \\cite{Saito91}. Although the TB \nHamiltonian only depends on the topology of the molecule (i.e., on the atoms bonded), the molecular geometry affects the hopping integral. \n\\begin{comment}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\hline \\hline \nSystem & $d_n$ (\\AA) & $R_n$ (\\AA) & $t_n$ (eV) \\\\ \\hline\nC$_{60}$ & 1.43 & 3.54 & 2.51 \\\\\nC$_{240}$ & 1.42 & 7.05 & 2.68 \\\\\nC$_{540}$ & 1.42 & 10.58 & 2.71 \\\\ \n \\hline\nC$_{20}$ & 1.42 & 1.99 & 2.04 \\\\\nC$_{180}$ & 1.42 & 6.10 & 2.66 \\\\\nC$_{500}$ & 1.42 & 10.18 & 2.70 \\\\ \n\\hline \\hline\n\\end{tabular}\n\\caption{\\label{d R t} Mean interatomic C-C distances, mean radii and hopping integral for the single-wall icosahedral fullerenes of the families $60k^2$ and $20k^2$ ($k$ integer).}\n\\end{center}\n\\end{table}\n\\end{comment}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccc|ccccc}\n\\hline \\hline \nC$_n$ & $d_n$ (\\AA) & $R_n$ (\\AA) & $t_n$ (eV) &&& C$_n$ & $d_n$ (\\AA) & $R_n$ (\\AA) & $t_n$ (eV) \\\\ \\hline\nC$_{60}$ & 1.43 & 3.54 & 2.51 &&& C$_{20}$ & 1.42 & 1.99 & 2.04 \\\\\nC$_{240}$ & 1.42 & 7.05 & 2.68 &&& C$_{180}$ & 1.42 & 6.10 & 2.66 \\\\\nC$_{540}$ & 1.42 & 10.58 & 2.71 &&& C$_{500}$ & 1.42 & 10.18 & 2.70 \\\\\n\\hline \\hline\n\\end{tabular}\n\\caption{\\label{d R t} Mean interatomic C-C distances, mean radii and hopping integral for the single-wall icosahedral fullerenes of the families $60k^2$ and $20k^2$ ($k$ integer).}\n\\end{center}\n\\end{table}\nTable \\ref{d R t} shows that the mean radius of the shell is the main geometrical variable, with the mean inter-atomic distance being approximately constant for the various fullerenes C$_n$ of the families $n=60k^2$ and $n=20k^2$ ($k$ integer). The inter-shell separations are approximately constant for successive fullerenes of each family ($\\sim3.5$\\AA, \\ close to the inter-layer separation in bulk graphite, for the former, and $\\sim 2$\\AA \\ for the latter).\n\\subsection{Intershell interactions in two-wall fullerenes \\label{two-shells}}\nWe shall study single-wall fullerenes and bilayer and trilayer onions composed by two and three concentric shells of icosahedral symmetry ($I_h$), C$_{n_1}$@C$_{n_2}\\ldots$. \nIn onions, it has been shown that the resulting structures for each shell are much the same those ones of the isolated fullerenes. Therefore, we assume the geometry for each shell in the onion to be the same as that of isolated C$_n$, taken from Yoshida data base. The most stable mutual orientation is the one that preserves the $I_h$ symmetry of the composite system \\cite{Heggie97, Heggie98}. \nFurthermore, in the onion-like family C$_{60k^2}$, it is reasonable to assume the strength of their intershell interactions as similar to those between the layers of graphite, due to the similarity in their interlayer separation.\nFigure \\ref{bilayer}a shows a scheme of a part of two adjacent parallel layers in graphite separated a distance of 3.35 \\AA. Each layer is constituted by two triangular lattices whose sites are denoted as A and B. The subscript 1 or 2 indicates which layer each site belongs. In graphite, the distances and number of nearest neighbors are exactly determined by the relative position between the plane infinite layers of identical geometry. \nIn two-wall buckyonions, nevertheless, the inner and outer fullerenes have finite sizes with different number of atoms, different geometrical structures due of their different number of hexagons (e.g., in the family 60$n^2$, every member have 12 pentagons) and eventually different orientations from each other (even though both have $I_h$ symmetry), as shown in Figures \\ref{bilayer}b and \\ref{bilayer}c. \nTherefore, although it is usually thought of intershell interactions in buckyonions as locally similar to interlayer interactions in bilayer graphene, a more thorough consideration of the intershell hopping is needed due to the differences mentioned above.\n\\begin{figure*}\n\\includegraphics[angle=-90,scale=0.5]{figure1.eps}\n\\caption{\\label{bilayer} (a) Scheme of a part of a two adjacent layers of graphite showing the interlayer connections between the lattices A and B from the upper and lower layers, (b)-(c) Schemes of two portions of the three layers of the onions considered in this work and their connections: (b) with the centers of pentagons aligned, and (c) with the centers of hexagons aligned. These and others configurations occurs in all the multiwall fullerenes considered here, making variable the number of nearest neighbors of the atoms of the shells. (d) Average number of connections $f$ in two-wall buckyonions per atom belonging to the inner sheel as a function of a maximum range for including neighbors $R_{\\rm cutoff}$, Eq. (\\ref{def f_links}), for C$_{60}$@C$_{240}$ with $I_h$ symmetry, C$_{240}$@C$_{540}$ with $I_h$ symmetry, C$_{60}$@C$_{240}$ without symmetry (symmetry $C_1$) and for two layers of graphite, for comparison purpose. In graphite only $f=0.5$ and $f=5$ are possible for the range of $R_{\\rm cutoff}$ shown, and the step-like variation of $f$ is due to the constrain imposed by the geometrical alignment between the two infinite layers. The three shadowed regions highlight ranges of $R_{\\rm cutoff}$ where $f$ has also a step-like behavior for the two-wall onions. (d) Scheme of the single-wall C$_{60}$, C$_{240}$ and C$_{540}$ fullerenes.}\n\\end{figure*}\nIn graphite the hopping parameter $\\gamma_0=t$ describes the covalent bonding arising from $sp^2$ hybridization in a layer.\nInter-layer Van der Waals interactions are described by the parameters $\\gamma_1 = t_\\perp\\approx 0.4$ eV (hopping energy between atoms A1 and A2), $\\gamma_3 \\approx 0.3$ eV (between B1 and B2 atoms) and $\\gamma_4 \\approx 0.04$ eV (hopping energy for A1-B2 and A2-B1 pairs) \\cite{CastroNeto09, Castro10}. \nIn bilayer onions, no such two lattices exists due to the finite size and the curvature of the layers. Nevertheless, similar Van de Waals interactions between nearest neighbor (NN) and next-nearest neighbors (NNN) atoms exist. Taking into account that for graphite $\\gamma_1\\approx \\gamma_3$ and $\\gamma_4\\approx 0$, we take a single value $t_\\perp=0.35$ eV for the hopping between pairs of the NN and NNN between layers \\cite{Pudlak09}. \nThe intershell interaction is included through the hopping integral $t_\\perp$ between pairs of NN and NNN atoms as follows: \nwe define a cutoff radius $R_{\\rm cut off}$ such that every pair of atoms, at ${\\bf r}_i$ and ${\\bf r}_j$, belonging to adjacent shells and separated a distance shorter than $R_{\\rm cut off}$ is assigned a hopping $t_\\perp$, i.e.,\n\\begin{equation}\nt_{ij} = \\left\\{ \n\\begin{array}{cc}\nt_\\perp, & |{\\bf r}_i-{\\bf r}_j|\\le R_{\\rm cut off} \\\\\n0 , & {\\rm otherwise}.\n\\end{array}\n\\right.\n\\label{def t_ij}\n\\end{equation}\nIt should be noted that, due to the faceting of the icosahedral symmetry, the number of intershell connections is not isotropic. \nAs an illustration, Figures \\ref{bilayer}b and \\ref{bilayer}c show two portions of a trilayer onion within a solid angle around the directions joining the centers of the pentagons and hexagons, respectively. The lines joining atoms in adjacent shells represent the intershell connections $t_{ij}=t_\\perp$ for a given cutoff radius. Due to the non sphericity of the shells, the outermost pentagon is not connected but the outermost hexagon it is.\nWe characterize the number of pairs of atoms connected at a given $R_{\\rm cutoff}$, by the mean number of neighbors (in the outer shell) `felt' by atoms in the inner shell, \n\\begin{equation}\nf(R_{\\rm cutoff})=N_{\\rm connect}\/N_{\\rm inner},\n\\label{def f_links}\n\\end{equation}\nwhere $N_{\\rm connect}$ is the number of intershell connections for a given $R_{\\rm cutoff}$, and $N_{\\rm inner}$ is the number of atoms in the inner shell.\nFigure \\ref{bilayer}d shows the variation of $f$ with $R_{\\rm cut off}$ for C$_{60}$@C$_{240}$ with $I_h$ symmetry, C$_{240}$@C$_{540}$ with $I_h$ symmetry, C$_{60}$@C$_{240}$ without symmetry (symmetry $C_1$) and for two adjacent layers of graphite. In the latter, there are wide ranges of $R_{\\rm cut off}$ where the number of connections keeps constant. Thus, due to the parallel orientation of the infinite layers, there are only some discrete values at which the number of connections increases due to the inclusion of neighbors farther to a given atom. For the onions with $I_h$ symmetry, \nthere are common ranges of $R_{\\rm cutoff}$ that show step-like behavior for both onions (see regions shadowed in Figure \\ref{bilayer}d). In the next section we present results of calculations with a number of intershell connections (characterized by $f$) chosen in those regions. Such a step-like dependence is in contrast to the approximately linear dependence for C$_{60}$@C$_{240}$ with symmetry $C_1$, when the symmetry axes of the $I_h$ fullerenes are misaligned. \nThe faceting of the fullerenes induced by the $I_h$ symmetry is particularly noticeable for the largest layers, as can be seen en Figure \\ref{bilayer}e for C$_{60}$, C$_{240}$ and C$_{540}$. Therefore, the fraction of carbon atoms having a NN or NNN within the range $|{\\bf r}_i-{\\bf r}_j|\\le R_{\\rm cut off}$ is smaller for the larger fullerenes. \nHence, the TB Hamiltonian for the onions becomes that of the isolated layers with an inter-layer interaction term\n\\begin{equation}\nH_{\\rm onion} = \\sum_n^{N} H_n + t_\\perp \\sum_{\\langle ij\\rangle} (c_i^\\dagger c_j + c_j^\\dagger c_i ),\n\\label{H_onion}\n\\end{equation}\nwhere site $i$ belongs to a inner shell, and site $j$ is its NN or NNN on the adjacent outer shell.\n\\subsection{Electronic transport}\nWhen a molecule is attached between two metallic leads and subject to a potential bias, the charge current flowing through it can be calculated with the Landauer equation \\cite{Landauer86}\n\\begin{equation}\nI=\\frac{2e}{h}\\int dE \\ T(E) \\left[f_L(E)-f_R(E) \\right],\n\\end{equation}\nwhere $f_L$ and $f_R$ are the Fermi distributions at the left (L) and right (R) leads. At low temperatures, the transmission function represents the dimensionless conductance (in units of the quantum $e^2\/2h$) and is calculated as\n\\begin{equation}\nT(E) = 4{\\rm Tr}({\\bf \\Gamma}^L {\\bf G}^r(E) {\\bf \\Gamma}^R {\\bf G}^a(E)),\n\\end{equation}\nwhere ${\\bf G}^a$ and ${\\bf G}^r$ are the matrix representation of the advanced and retarded Green functions ${\\bf G}^{r,a}=(E {\\bf 1}-{\\bf H}\\pm i0)^{-1}$, and ${\\bf \\Gamma}^L$ and ${\\bf \\Gamma}^R$ are the spectral densities of the leads \\cite{Cuevas10}.\n\nIn the wide band approximation, the Green function of the connected system can be obtained by using Dyson equation, as\n\\begin{equation}\nG_{1n}^r = \\frac{g_{1n}}{1-\\Gamma^2(g_{11}g_{nn}-|g_{1n}|^2)-i\\Gamma(g_{11}+g_{nn})}.\n\\label{connected G1n}\n\\end{equation}\nwhere $g_{ij}$ is the retarded Green function of the isolated system, and $\\Sigma_L=\\Sigma_R=i\\Gamma$ are the self-energies of the leads, considered to be energy independent. Throughout this work, we connect the fullerenes to the leads, using $\\Gamma=0.05$ eV, through two carbon atoms located diametrally opposite to each other, one atom in the vertex of a pentagon and its corresponding one obtained by applying the operation of inversion with respect to the center of symmetry.\n\\section{Results and discussion}\n\\subsection{Effect of the relative orientation between adjacent shells}\nThe approximately spherical form of the fullerenes, particularly the smaller ones, allows to treat onions as a family of concentric spherical shells for the calculation of some properties, such as the determination of radii of equilibrium, intershell distances, static polarizability or photoionization cross section \\cite{Xu96, Ruiz04, Kidun06, Dolmatov08}. For other applications, however, a more accurate description of their geometry is relevant. \nIn particular, the trend in larger fullerenes to approach faceted icosahedral forms makes very relevant the relative orientation of adjacent layers, even for concentric shells having individually $I_h$ symmetry, as already shown in Figure \\ref{bilayer}d for the average number of connections per atom $f$. \nThe influence of the relative orientation on the quantum transmission is shown in Figure \\ref{orientation} for two bilayer onions: C$_{60}$@C$_{240}$, and C$_{240}$@C$_{540}$. Two orientations were considered: one where the onion has overall $I_h$ symmetry, that is, with the symmetry axes of the individual shell aligned; and one where both shells are concentric but the individual symmetry axes are rotated an arbitrary relative angle from each other ($C_1$ symmetry).\n\\begin{figure*}\n\\includegraphics[scale=0.9]{figure2.eps} \n\\vspace{-0.0cm}\n\\caption{\\label{orientation} Transmission function for (a) C$_{60}$@C$_{240}$, and (b) C$_{240}$@C$_{540}$ buckyonions for the concentric shells rotated an arbitrary angle with respect to each other ($C_1$ symmetry), and with their symmetry axes aligned ($I_h$ symmetry).}\n\\end{figure*}\nThe differences in $T(E)$ between the icosahedral ($I_h$) and the non symmetrical ($C_1$) onion are visible in Figure \\ref{orientation}. In the onion with misalignment between the shells, some degeneracies in the energy spectrum are broken, as reflected in the occurrence of multiple resonant states with many peaks and antiresonances close to each other leading to a trend to the formation of narrow bands for the larger onion C$_{240}$@C$_{540}$. The transmission function of the non symmetrical onions have rapid variations from perfect to vanishing transmission with slight variations of the Fermi energy $E$, while $T(E)$ for the $I_h$ onions is a well behaved smooth function with a few peaks of perfect transmission in the range around the HOMO-LUMO gap. The gap itself increases when the shells become disoriented from each other what also corresponds to a less stable configuration, as previously reported \\cite{Heggie97}.\n\\subsection{Dependence on the number of intershell hopping connections \\label{connections}}\nAs mentioned in Section \\ref{two-shells}, the relative orientation of hexagons belonging to adjacent shells, the choice of the cutoff radius for defining NN and NNN sites of a given atom and the faceting of the larger fullerenes preclude a unique definition of the number of intershell connections to be included in the Hamiltonian (\\ref{H_onion}). Both the interlayer distance in graphite and intershell distance in the $60 k^2$ family are close to 3.5 \\AA. Thus, we took two cutoff radii close to this value and a larger one (see shadowed regions of Figure \\ref{bilayer}d), to include a bigger number of NN and NNN pairs in the intershell interaction Hamiltonian.\nIn the following we show the results of calculated $T(E)$ for icosahedral onions ($I_h$@$I_h$) and the aforementioned three choices of $R_{{\\rm cutoff}}$. In those three regions, $f=1$, 1 and 4 for C$_{60}$@C$_{240}$, and $f=0.5$, 1 and 2 for C$_{240}$@C$_{540}$. \nIn Figure \\ref{R_cutoff}, the transmission $T(E)$ is depicted for the three values of $f$, as indicated in the legends. It should be noticed that for the smaller onion C$_{60}$@C$_{240}$, the three $f$ values give almost the same curve (Fig. \\ref{R_cutoff}a). \nTherefore, this increase of connectivity does not affect the transmission throughout the composite system. Therefore, the main paths of transmission between shells are along the pairs formed by the atoms of the inner shell with its closest neighbor in the outer one.\n\\begin{figure*}\n\\includegraphics[scale=0.90]{figure3.eps} \n\\vspace{-0.1cm}\n\\caption{\\label{R_cutoff} Transmission function $T(E)$ of the bilayer and trilayer onions (a) C$_{60}$@C$_{240}$, (b) C$_{240}$@C$_{540}$ and (c) C$_{60}$@C$_{240}$@C$_{540}$ for three different cutoff radii chosen in the shadowed regions of Figure \\ref{bilayer}. The resulting mean number of intershell connections $f$, Eq. (\\ref{def f_links}), are indicated in each curve.}\n\\end{figure*}\nFigure \\ref{R_cutoff}b shows that the dependence of the transmission on the number of intershell connections is more important for the larger onion C$_{240}$@$C_{540}$. \nFor larger fullerenes the deviation from the spherical shape becomes more noticeable as the number of atoms increases. Thus the C$_{240}$ shell is less spherical than C$_{60}$, and C$_{540}$ is clearly faceted. This departure from sphericity favours the hopping from the inner to the outer shell along certain directions, namely, those in which the icosahedral faces of both shells are closer to each other. Therefore, the larger the number of intershell links the better the quantum transmission. Interestingly, the increasing in the number of connections mainly affect the states above the highest occupied-lowest unoccupied (HOMO-LUMO) gap, noticeably the LUMO and LUMO+1 ones.\n\nFinally, Figure \\ref{R_cutoff}c shows $T(E)$ for the trilayered onion C$_{60}$@C$_{240}$@$C_{540}$ which is notoriously similar to the one of the bilayer onion of Fig \\ref{R_cutoff}b, thus showing that the two external shells are the most relevant for the transmission, with only small corrections coming from the innermost C$_{60}$ shell. This effect is discussed in greater detail in Section \\ref{shells}. \nThe bilayer onion have four peaks above the gap in range shown, namely, those corresponding to LUMO, LUMO+1, LUMO+2 and LUMO+3. The two central peaks (LUMO+1, LUMO+2) are almost degenerates in C$_{240}$@$C_{540}$ but become better resolved in the trilayered onion.\n\\subsection{Influence of the number of onion shells \\label{shells}}\nWe shall show here that the outermost shell greatly determines the most relevant features of the transmission spectra of bilayered and trilayered onions. The importance of the influence of the inner shells on $T(E)$ decreases inwards.\nIn Figure \\ref{R_cutoff}c we observed that the most noticeable effects when adding C$_{60}$ to the two external shells are the occurrence of an antiresonance between the LUMO and LUMO+1 peaks, and the widening of the narrow Fano-like profile after the LUMO+2 peak (at $E \\sim0.2$ eV.). Such a type of effects are analogous to the modifications to the transmission function through a chain introduced by adding a lateral site or chain to the system.\n\\begin{figure*}\n\\includegraphics[scale=0.90]{figure4.eps} \n\\vspace{-0.0cm}\n\\caption{\\label{shell number} Transmission function for (a)C$_{60}$@C$_{240}$ and (b) C$_{240}$@C$_{540}$ compared to those for the single-wall fullerenes.}\n\\end{figure*}\nIn the following we fix the number of connections in trilayer onions by chosing $f=4$ ($f=2$) between the two outermost (innermost) shells.\nFigure \\ref{shell number} shows the transmission for the bilayer onions C$_{60}$@C$_{240}$ and C$_{240}$@C$_{540}$ as compared to those for the corresponding single-wall fullerenes. It can be seen that for energies below the gap, $T(E)$ for the onions is very similar to the one of the outer fullerene, i.e., the one for C$_{240}$ in Figure \\ref{shell number}(a), and for C$_{540}$ in Figure \\ref{shell number}(b). \nThe peaks above the gap preserves similar features both in C$_{60}$@C$_{240}$ and C$_{240}$, although with a relative energy shift of the peaks.\n\\begin{figure*}\n\\includegraphics[scale=0.90]{figure5.eps}\n\\vspace{-0.0cm}\n\\caption{\\label{family 20} transmission function for (a)C$_{20}$@C$_{180}$ and C$_{180}$@C$_{500}$, and (b) C$_{20}$@C$_{180}$@C$_{500}$ compared to those for the single-wall fullerenes.}\n\\end{figure*}\nIn Figure \\ref{shell number}b similar considerations can be made about the comparison the transmission through C$_{240}$@C$_{540}$ and C$_{540}$; the latter provides most of the features observed in the former, particularly for energies below the gap. The intershell connections in C$_{240}$@C$_{540}$ eventually contributes to the occurrence of antiresonances, such as that at $\\sim-1.4$ eV, not present in $T(E)$ for C$_{540}$.\nIn other cases, it softens the vanishing of transmission, such as in the antiresonance of C$_{540}$ at $\\sim-1.1$ eV which becomes in a finite transmission for C$_{240}$@C$_{540}$. \nRoughly speaking, introducing C$_{60}$ as a third innermost shell does not greatly modify the transmission of the bilayered onion.\nThe influence of the external shell on the transmission can be interestingly shown in the onions formed from fullerenes belonging to the family $20 k^2$, which also have icosahedral symmetry. Thus, C$_n$ fullerenes with $n=20$, 80, 180, 320, $500\\ldots$ are predicted to be stable, with a radii increasing by approximately 2 \\AA \\ from each member from the family to the next one. The radii of equilibrium for the family $60 k^2$ were shown to be accurately determined by a continuous spherically symmetric Lennard-Jones model \\cite{Baowan07}. Application of the same model for the multiwall onions of the $20 k^2$ family (with $k$ odd integer), results an intershell distance of about 4 \\AA, not far from the intershell distance for the $60 k^2$ family or the interlayer distance in graphite. Therefore, we show calculations for the single-wall, two-wall and three-wall fullerenes obtained from C$_{20}$, C$_{180}$ and C$_{500}$.\nOur TB\ncalculations show that C$_{20}$ and C$_{500}$ are gapless, while C$_{180}$ presents a gap of $\\approx 1.5$ eV. In Figures \\ref{family 20}a and \\ref {family 20}b, C$_{180}$ is the outer and inner shell respectively. As a consequence, in Figure \\ref{family 20}a, C$_{20}$@C$_{180}$ shows a region of vanishing transmission within the gap, while it keeps finite for C$_{180}$@C$_{500}$, except for the already discussed narrow antiresonances arising from the intershell connections. Hence, the transmission function is strongly sensitive to the electronic structure of the external shell. As a final example of this property, Figure \\ref{family 20}b, shows the transmission for the trilayered C$_{20}$@C$_{180}$@ C$_{500}$ as compared to those of the single-wall fullerenes. It is seen that the most of the features of the onion are reproduced by the transmission through C$_{500}$, with some Fano-like antiresonances originated in the transmission spectrum of the inner C$_{180}$ and, to less extent, in the innermost C$_{20}$.\nIt can bee seen that the three-wall onion (Figure Figure \\ref{family 20}b) is well described by the two-wall one (Figure \\ref{family 20}a). Interestingly, the Fano-like resonance of C$_{20}$@C$_{180}$@ C$_{500}$ at $E\\approx 0.1$ eV is not present in C$_{180}$@ C$_{500}$ but it is in C$_{20}$@C$_{180}$; therefore, such a peak of conductance is an effect from the coupling of the two inner sheells.\n\\section{Conclusions}\nIn this work we have studied theoretically the quantum transmission through single-wall fullerenes and bilayered and trilayered onions of icosahedral symmetry, when attached to metallic leads, by using a TB\nHamiltonian and Green functions methods. Although the Van der Waals interactions between onion shells are supposed to be similar to those between graphite layers, the finite size of the fullerenes, their curvature and relative orientations need some analysis for including the intershell hopping parameter. We include in the model the effect of finite size and curvature through a parametrization of the hopping integral as a function of the number of atoms of the shell. The number of connections from a given atom to others belonging to adjacent shells was studied by introducing a cutoff radius for the interaction. \nWe found that misalignment of the symmetry axes produces breaking of the level degeneracies of the individual shells, giving rise some narrow quasi-continuum bands instead of the localized discrete peaks of the individual fullerenes. Most of the features of the transmission through the onions are already visible in the transmission function of the single-wall fullerene forming the outer shell. The main modifications between them are antiresonances arising from the coupling between the outer layer with the next innermost one. For three-wall onions, the transmission becomes barely sensitive to the most internal shell. Interestingly, when the fullerene of the external shell is gapless, the transmission of the onion does not vanish along finite ranges of energy. This property could be useful for designing multilayered fullerenes with tailored conductance by properly growing the outermost layers.\n\\section*{Acknowledgments}\nWe acknowledge financial support for this project from CONICET (PIP 11220090100654\/2010) and SGCyT(UNNE) through grant PI F007\/11.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nTwisted 2D van der Waals materials have emerged as an elegant platform to engineer and study correlated quantum phases with unprecedented experimental control \\cite{Andrei2020, Novoselov2016, Cao2018, Balents2020, Wu2018, Regan2020}. At certain \\textit{magic} angles, the electronic structure is deformed into exceedingly narrow bands in a moir\\'e Brillouin zone. The small bandwidth yields pronounced correlation effects and generally makes multiple neighboring quantum phases experimentally accessible in a single fabricated device through application of electrostatic gating \\cite{Cao2018}. \n\nRecently, a related paradigm of twisted bilayer structures was introduced that does not rely on engineering of correlated flat bands, but can produce interesting new phases by combining known non-trivial properties of constituent monolayers \\cite{Can2021,volkov2021}. It was shown that two cuprate monolayers, stacked at a twist angle $\\theta$, can give rise to a spontaneously time-reversal symmetry broken state by virtue of simple electron tunneling between the layers. Most notably, in a finite range of angles around the critical twist $\\theta_c=45^\\circ$, the ground state of an otherwise nodal $d$-wave superconductor becomes fully gapped and acquires a finite Chern number. At exactly $45^\\circ$ the gapped phase persists up to the native critical temperature of the cuprate monolayer, thus furnishing the first known example of a high-$T_c$ topological superconductor. \n\nPioneering experimental work on very thin twisted Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$ (BSCCO) flakes succeeded in fabricating bilayers at various twist angles \\cite{Zhao2021}. Measurements of the interlayer Josephson current, Fraunhofer interference patterns and half-integer Shapiro steps in samples close to the $45^\\circ$ are suggestive of a $\\mathcal{T}$-broken phase \\cite{Tummuru2022}. Strong twist angle dependence of the critical current has been reported elsewhere \\cite{Lee2021}. \n\nIt was later noted by Song, Zhang and Vishwanath \\cite{Song2021} that twist angle and momentum dependence of the interlayer tunneling matrix element $g_{\\bm{k}}$, arising from the symmetry of the copper active orbitals, can play an important role in the emergence of the $\\mathcal{T}$-broken phase. As originally argued by Xiang and Wheatley \\cite{Xiang_1996}, the matrix element has the form\n\\begin{equation}\\label{gk}\n g_{\\bm{k}}=g_0 \\cos{2\\theta}+g_1\\mu_{\\bm{k}}(\\theta\/2)\\mu_{\\bm{k}}(-\\theta\/2),\n\\end{equation}\nwhich we generalized here to a twisted bilayer geometry following \\cite{Song2021}. The $g_0$ term represents the direct tunneling between copper $d_{x^2-y^2}$ orbitals while the $g_1$ term describes the `oxygen-assisted' tunneling process with form factor $\\mu_{\\bm{k}}(\\theta)=[\\cos{(R_\\theta k_x)}-\\cos{(R_\\theta k_y)}]\/2$ and $R_\\theta$ the rotation matrix. Crucially, the part of $g_{\\bm{k}}$ that remains non-vanishing at $\\theta=45^\\circ$ contains the form factor $\\mu_{\\bm{k}}$ that suppresses tunneling at the nodes of the $d$-wave order parameter. While spontaneous $\\mathcal{T}$-breaking is still found to occur in this case, the ground state remains gapless and hence does not support the gapped topological phase with non-zero Chern number predicted in Ref.\\,\\cite{Can2021}.\n\nIn the present work we consider the twisted bilayer problem within a family of incoherent tunneling models \\cite{Graf1993,Radtke1995,Radtke1996,Radtke_1997,Turlakov2001} in which the transfer of electrons between two adjacent CuO$_2$ layers is mediated by impurities that are inherently present in the otherwise inert `spacer' layers. Such incoherent tunneling models have been shown to yield better agreement with experimentally measured $c$-axis transport properties of nominally clean single-crystal cuprates than models where momentum is strictly conserved \\cite{Ginsberg1994, Sheehy_2004}. Because random impurities break all spatial symmetries of the system, the form of the interlayer coupling is required to respect the crystal symmetry constraints only on average. One may thus expect that incoherent tunneling models will evade the difficulties noted above and produce a fully gapped topological phase near $\\theta=45^\\circ$. \n\nBased on a perturbative diagrammatic treatment within a simplified continuum model we show that the incoherent tunneling model indeed delivers the same phenomenology as the coherent model of Ref.\\,\\cite{Can2021} while respecting with the point group symmetries of the physical system. Importantly, we show that for sufficiently slowly varying disorder the ground state near $\\theta=45^\\circ$ is gapped in the $\\mathcal{T}$-breaking region and topologically non-trivial. These results are then confirmed in a more realistic setting through a full numerical diagonalization of a lattice model with parameters chosen to reproduce the actual cuprate band structure in the vicinity of the Fermi level. The effect of interface inhomogeneity on Josephson effects in twisted bilayers was recently considered in Ref.\\,\\cite{volkov2021} where it was found that sufficiently strong disorder can leave the system in a topologically trivial state around 45$^\\circ$. This is consistent with our deductions.\n\nThe paper is structured as follows. After summarizing the origin of the $\\mathcal{T}$-broken phase in the language of group theory (Sec.~\\ref{sec:group theory}), in Sec.~\\ref{sec:model} we introduce a model of incoherent interlayer tunneling within a continuum framework and show that a fully-gapped $\\mathcal{T}$-broken phase emerges in the vicinity of $\\theta=45^\\circ$. The phase is topological as evidenced by chiral edge modes traversing the gap that appear in the disorder-averaged boundary spectral function. In Sec.~\\ref{sec:latticemodel}, we supplement our continuum results with a lattice model that simultaneously captures the characteristic cuprate Fermi surface geometry, the moir\\'e effects and disorder in interlayer coupling. Concluding remarks appear in Sec.~\\ref{sec:conclusions}.\n\n\n\n\\section{Group theoretical discussion of $\\mathcal{T}$-breaking in twisted cuprates}\n\\label{sec:group theory}\n\nThe phenomenology of $\\mathcal{T}$-breaking in twisted cuprates can be captured by a two-component Landau-Ginzburg theory with complex order parameters $\\Psi_1$ and $\\Psi_2$ with a relative phase $\\varphi$. The $\\varphi$-dependent part of free energy is of the general form \n\\begin{align}\n \\mathcal{F}(\\varphi) = -B \\, |\\Psi_1\\Psi_2| \\cos \\varphi + C \\, |\\Psi_1 \\Psi_2|^2 \\cos 2\\varphi.\n \\label{eq:LG}\n\\end{align}\nTime reversal symmetry will be spontaneously broken whenever the free energy develops two minima that are related by $\\mathcal{T}: \\varphi \\rightarrow -\\varphi$. The Josephson coupling term, proportional to $\\cos \\varphi$, has only a single minimum at $\\varphi=0$ or $\\pi$, depending on the sign of $B$.\nPresence the fourth order term proportional to $C\\cos 2\\varphi$ is therefore necessary to break $\\mathcal{T}$. Additionally, one must have $C>0$, since otherwise the two minima of $\\cos 2\\varphi$ occur at $\\varphi=0,\\pi$ which map to themselves under $\\mathcal{T}$. In Ref.\\,\\cite{Can2021} it was argued that $C$ is indeed positive based on microscopic mean-field calculations. We confirm that $C$ remains positive in the case of incoherent interlayer coupling in Sec.~\\ref{sec:freeenergy}.\n\nGiven $C>0$, the fourth order term is minimized at $\\varphi=\\pm\\pi\/2$. Then, $\\mathcal{T}$-breaking occurs as a consequence of the competition among the two terms in Eq.~\\eqref{eq:LG}. Specifically, $\\mathcal{T}$ will be broken when \n\\begin{align}\n 4 C \\left| \\Psi_1 \\Psi_2 \\right| > \\left| B\\right| \\,.\n \\label{eq:trsbreaking}\n\\end{align}\nA special situation clearly arises if symmetry requires $B$ to vanish; then Eq.~\\eqref{eq:trsbreaking} is guaranteed to be satisfied for any $C>0$. \n\nNext we describe a set of symmetry requirements under which the coefficient $B$ vanishes and the system is forced into the $\\mathcal{T}$-broken phase. The order parameters $\\Psi_1,\\Psi_2$ transform according to irreducible representations (irreps) of the point group of the crystal. Two cases must be distinguished: (a) $\\Psi_1$ and $\\Psi_2$ transform under two \\textit{different} 1D irreps or (b) $(\\Psi_1,\\Psi_2)$ transform under a 2D irrep \\cite{Annett1990,Poniatowski2022}. The latter case is considered a generic pathway to $\\mathcal{T}$-breaking that occurs immediately upon entering the SC phase. The former case generically yields two successive phase transitions with distinct critical temperatures, $T_{c}$ and $T_c'$, with $\\mathcal{T}$-breaking setting at the lower one $T_c'$. Note that $T_c'$ can be zero or negative, in which case the $\\mathcal{T}$-broken phase is physically not accessible \\cite{Annett1990, Kaba.2019}.\n\nThe point symmetries of an untwisted cuprate bilayer form the point group $D_{4h}$. Here, the $d_{x^2-y^2}$ and $d_{xy}$ order parameters transform according to the 1D irreps $B_{1g}$ and $B_{2g}$, respectively. At arbitrary twist angle, inversion and mirror symmetries are broken and the point group reduces to $D_4$ with $d$-wave irreps $B_1$ and $B_2$. Thus, given the $d$-wave nature of the order parameter in cuprates, only pathway (b) to $\\mathcal{T}$-breaking is possible and no definite symmetry-based arguments can be made.\n\nPrecisely at ${45^{\\circ}}$, however, the symmetry group is enlarged to the non-crystallographic point group $D_{4d}$ which contains an additional $8$-fold improper rotation $S_8$ of the quasicrystalline lattice. Most notably, among the irreps of $D_{4d}$ \\textit{only} the 2D $E_2$-irrep supports $d$-wave order. Thus, the Josephson coupling term $ -B \\, |\\Psi_1\\Psi_2| \\cos \\varphi$ must necessarily be absent at $45^\\circ$. This is so because it descends from the $-B(\\Psi_1\\Psi_2^\\ast +{\\rm c.c.})$ term in the free energy which is not invariant under $S_8: (\\Psi_1,\\Psi_2)\\rightarrow (\\Psi_2,-\\Psi_1)$. Thus, $\\mathcal{T}$-breaking can be viewed as a fundamental consequence of the point group at $\\theta= 45^\\circ$. We summarize our key arguments as follows: Two-component order parameters that transform under a 2D irrep naturally break $\\mathcal{T}$. At $45^\\circ$ twist angle, because the point group of the bilayer is $D_{4d}$, any $d$-wave order parameter must necessarily transform under a 2D irrep. Therefore, the superconducting state breaks $\\mathcal{T}$ right below $T_c$. \n\nThe phase diagram of twisted bilayer cuprates derived in Ref.~\\cite{Can2021} can then be understood from continuity arguments. It is expected that the $\\mathcal{T}$-breaking phase will not be limited to the exact $45^\\circ$-twist, but will extend to a range of twist angles in its vicinity. Since at twists slightly away from $45^\\circ$ the order parameters transform under two 1D-irreps, two distinct critical temperatures are permitted and $\\mathcal{T}$-breaking will no longer coincide with the critical temperature $T_c$ of the spontaneous $U(1)$-symmetry breaking. This naturally leads to the wedge-shaped $\\mathcal{T}$-broken domain in the phase diagram explicitly computed in Ref.~\\cite{Tummuru2022}. Our symmetry arguments will be manifest in a microscopic description of the bilayer system. \n\n\n\n\n\\section{Incoherent tunneling}\n\\label{sec:model}\n\n\\subsection{Background and model definition}\n\nExperimental measurements of the $c$-axis transport in bulk crystals of BSCCO and other cuprates, summarized for example in Ref.\\,\\cite{Ginsberg1994}, have been interpreted as evidence of interlayer tunneling dominated by disorder-mediated, incoherent processes. The $c$-axis superfluid stiffness, accessible through the measurement of the $c$-axis London penetration depth \\cite{Hosseini1998, Panagopoulos2003}, provides a particularly clear evidence. Experimentally, the temperature dependence of the $c$-axis superfluid stiffness in clean single crystals was observed to follow an approximate power-law behavior $\\rho_c = a-b T^\\alpha$ with $\\alpha\\simeq2$ at low temperatures, whereas the in-plane stiffness showed a $T$-linear dependence \\cite{Hardy1993}. The latter is the canonical behavior expected of a clean $d$-wave superconductor, reflecting the presence of low-energy excitations with a Dirac spectrum \\cite{Hirschfeld1993}. Models with coherent tunneling between CuO$_2$ predict the same linear $T$-dependence for the $c$-axis stiffness \\cite{Klemm1995}, in clear disagreement with experimental data. If the interlayer tunneling were dominated by the oxygen-assisted processes (the $g_1$ term in Eq.\\ \\eqref{gk}) then theory predicts $\\rho_c = a-b T^5$ \\cite{Xiang_1996}, again at variance with experiment. \n \nAs demonstrated in Refs.\\ \\cite{Radtke1995,Radtke1996,Radtke_1997,Sheehy_2004} a description that captures the correct $\\sim T^2$ scaling along the $c$-axis (while preserving the $T$-linear behavior in the $ab$ plane) can be given using the incoherent $c$-axis tunneling approach. In the following we shall review the relevant model and then apply it to the problem of a twisted bilayer.\n\nA minimal model of the uncoupled bilayer system consists of the second-quantized Hamiltonian\n\\begin{align}\n\t\\mathcal{H}_0 = \\sum_{\\mathbf{k}l}\\Psi_{\\mathbf{k}l}^\\dagger\n\tH_{\\mathbf{k}l}\n\t\\Psi_{\\mathbf{k}l}\n\\end{align}\nwhere $l=1,2$ denotes the layer index, Nambu-Gorkov spinors $\\Psi_{\\mathbf{k}l}=(c_{\\mathbf{k}l\\uparrow },\\, c_{-\\mathbf{k}l\\downarrow })^T$. In the BCS approximation, we have\n\\begin{align}\n\tH_{\\mathbf{k}l}\n\t=\n\t\\xi_{\\mathbf{k}} \\sigma_z \n\t+ \\Delta_{\\mathbf{k}l}' \\sigma_x\n\t- \\Delta_{\\mathbf{k}l}'' \\sigma_y,\n\\end{align}\nwith Pauli matrices $\\sigma_j$ acting in the Nambu space and $\\Delta_{\\mathbf{k}l}'$, $\\Delta_{\\mathbf{k}l}''$ denoting real and imaginary parts of the superconducting gap function. We adopt units such that $\\hbar=e=k_B=m_e=a_0=1$, where mass is measured in units of electron mass $m_e$ and length scales in units of lattice constant $a_0$. To make the model tractable we assume a simple parabolic band dispersion given by $\\xi_{\\mathbf{k}}=\\mathbf{k}^2\/2m-\\mu$ in each layer. (In Sec.\\ \\ref{sec:latticemodel} we consider a more realistic band structure and show that it leads to similar results.) The two superconducting $d$-wave order parameters are\n\\begin{align}\n\t\\Delta_{\\mathbf{k}1} &= \\Delta e^{i\\varphi\/2} \\cos(2\\alpha_\\mathbf{k} -\n\t\\theta) \n\t\\nonumber\n\t\\\\\n\t\\Delta_{\\mathbf{k}2} &= \\Delta e^{-i\\varphi\/2} \\cos(2\\alpha_\\mathbf{k} +\n\t\\theta)\n\t\\label{eq:op}\n\t\\,,\n\\end{align}\nwhere $\\varphi$ is the phase difference and $\\alpha_\\mathbf{k}$ denotes the polar angle of $\\mathbf{k}$. \n\nThe layers are coupled by the term\n\\begin{align}\\label{tun}\n\t\\mathcal{H}' = \\sum_{\\mathbf{kq}} \\gamma_{\\mathbf{q}} c_{\\mathbf{k},1}^\\dagger\n\tc_{\\mathbf{k-q},2} + \\textrm{h.c.}\n\\end{align}\nand $\\mathcal{H}=\\mathcal{H}_0 + \\mathcal{H}'$ constitutes the full model. The lack of momentum conservation in Eq.\\ \\eqref{tun} is the defining feature of the incoherent tunneling models and originates, physically, from the disorder present in the spacer layers separating the copper-oxygen planes. The disorder is captured via a set of Gaussian-distributed random variables $\\gamma_\\mathbf{q}$ of average $\\overline{\\gamma_\\mathbf{q}}=0$ and variance given by\n\\begin{align}\n\t\\overline{\\gamma_\\mathbf{q}^* \\gamma_\\mathbf{q+p}} &=\n\t\\frac{1}{N}\\frac{4\\pi g^2}{3\\Lambda^2}\n\t\\delta_{\\mathbf{p},0}e^{-\\mathbf{q}^2\/\\Lambda^2} \\,.\n \\label{eq:incotunnterm}\n\\end{align}\nThe scale $\\Lambda$ defines the characteristic momentum change that an electron undergoes when tunneling between the two layers. The factor $1\/3$ is chosen to reproduce the phase diagram of the coherent model of Ref.~\\cite{Can2021} in the limit $\\Lambda\\rightarrow 0$ for the same value of $g$. For simplicity we have neglected any $\\theta$-dependence of the interlayer coupling although we expect the randomness to be stronger in twisted samples due to the increase in interface roughness, added strain, and moir\\'e lattice modulations. \n\nThe above form of incoherent interlayer tunneling is consistent with all lattice symmetries because $\\gamma_\\mathbf{q}$ vanishes on average. This constitutes the key difference to a coherent coupling of the form $\\sum_{\\mathbf{k}}(g\\, c_{\\mathbf{k},1}^\\dagger c_{\\mathbf{k},2} + \\textrm{h.c.})$. As was pointed out in Ref.~\\cite{Song2021}, at $45^\\circ$ twist the two participating Cu $d$-orbitals transform under a 2D representation of $D_{4d}$ and a coherent tunneling term is therefore not invariant under $S_8: (c_{\\mathbf{k},1},c_{\\mathbf{k},2})\\rightarrow (c_{\\mathbf{k},2}, -c_{\\mathbf{k},1})$. It thus vanishes by virtue of the same argument as the Josephson coupling in Eq.~\\eqref{eq:trsbreaking}. This is indeed owed to the coincidence of the atomic Cu orbitals transforming under the same representation as the superconducting order parameters.\n\nIt is instructive to consider the limit $\\Lambda \\rightarrow 0$ of the incoherent tunneling in Eq.~\\ref{eq:incotunnterm}. Here, one has $\\overline{\\gamma_\\mathbf{q}^* \\gamma_\\mathbf{q+p}} = g^2\\delta_{\\mathbf{q},0}\\delta_{\\mathbf{p},0}\/3$ and momentum is conserved in the interlayer tunneling process. Yet, $\\Lambda \\rightarrow 0$ is not the clean limit in the sense that, in real space, it corresponds to the case where macroscopic regions are correlated with the same random value of interlayer tunneling $g\/\\sqrt{3}$. From the viewpoint of disorder-induced incoherence, the random values of $g$ should only be correlated in the vicinity of an impurity which sets the appropriate scale for $1\/\\Lambda$. In the discussion below, the $\\Lambda\\rightarrow 0$ thus serves as an abstract but convenient reference point that connects the present model to the calculations in the original work \\cite{Can2021}. We will refer to it as the \\textit{coherent} limit.\n\n\n\n\\subsection{Free energy and phase diagram}\n\\label{sec:freeenergy}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{diagram-figure}\n\t\\caption{Diagrammatic expansion of the interlayer current (a) at order $g^2$ and (b-c) at order $g^4$. Full lines correspond to electronic propagators $G$ and dashed lines correspond to impurity vertices paired by disorder average. The open circle denotes the current vertex $j_\\mathbf{q}$ defined in the main text and impurity vertices $\\gamma_\\mathbf{q}$ are given by black dots.}\n\t\\label{fig:diagrams}\n\\end{figure}\n\nThe physics of $\\mathcal{T}$-breaking is captured by the $\\varphi$-dependence of the free energy. To determine the free energy we begin by implementing the global gauge transformation \n$\n\\left( c_{\\mathbf{k}1},\\,c_{\\mathbf{k}2} \\right)\n\\rightarrow\n\\left( c_{\\mathbf{k}1}e^{i\\varphi\/4},\\,c_{\\mathbf{k}2}e^{-i\\varphi\/4} \\right)\n$\nwhich moves the superconducting phase difference from the order parameters in Eq.~\\eqref{eq:op} to $\\mathcal{H}'$ according to\n\\begin{align}\n\t\\mathcal{H}' \\rightarrow \n\t\\mathcal{H}' =\n\t\\sum_{\\mathbf{kq}} \\gamma_{\\mathbf{q}} e^{i\\varphi\/2}c_{\\mathbf{k},1}^\\dagger\n\tc_{\\mathbf{k-q},2} + \\textrm{h.c.} \\,.\n\\end{align}\nIn this gauge, the disorder-averaged interlayer current is given by\n\\begin{align}\n\tJ &= \\sum_{\\mathbf{kq}} i e^{i\\varphi\/2}\\overline{\\gamma_\\mathbf{q}\n\t\\langle c_{\\mathbf{k},1}^\\dagger c_{\\mathbf{k-q},2} \\rangle} + \\textrm{h.c.} \n\t\\nonumber\n\t\\\\\n\t&= \\text{Tr}\n\t\\left[ \\overline{\n\t\tj_\\mathbf{q} G(\\mathbf{k},\\mathbf{k-q},\\omega_n)}\n\t\\right]\\,,\n\t\\label{eq:current}\n\\end{align}\nwhere $G(\\mathbf{k},\\mathbf{k'},\\tau)=\\langle T_\\tau c_{\\mathbf{k}}(\\tau) c_{\\mathbf{k'}}^\\dagger(0) \\rangle$ is the full imaginary time ordered Green's function of the disordered system and the current vertex is\n\\begin{align}\n\tj_\\mathbf{q} =i\\gamma_\\mathbf{q} \n\t\\begin{pmatrix}\n\t\t0 & e^{i\\sigma_z\\varphi \/2}\\\\\n\t\t-e^{-i\\sigma_z\\varphi \/2} & 0\n\t\\end{pmatrix} \\,.\n\\end{align}\nNote that the trace is to be performed over all momenta $\\mathbf{k},\\mathbf{q}$ and Matsubara frequencies $\\omega_n=(2n+1)\\pi\/\\beta$, in addition to interlayer and Nambu indices.\n\nFrom the Josephson relation $J(\\varphi)=2\\partial \\mathcal{F}(\\varphi)\/\\partial \\varphi$ one then obtains the functional dependence of the free energy on the interlayer phase difference $\\varphi$ by simple integration. We expand Eq.~\\eqref{eq:current} up to fourth order in $g$ while treating $\\mathcal{H}'$ as a perturbation. Three different terms arise, which are diagrammatically represented in Fig.~\\ref{fig:diagrams}. Panel (a) corresponds to the term \n\\begin{align}\n J_c^{(2)} = \\text{Tr}\\left[\\overline{j_{\\mathbf{q}} G_0(\\mathbf{k-q},\\omega_n)\n \tu_{-\\mathbf{q}} G_0(\\mathbf{k},\\omega_n)}\n \\right] \\,\n \\label{eq:j1}\n\\end{align}\nwhich is quadratic in $g$. Here, $G_0(\\mathbf{k},\\omega_n)=\\left( i\\omega_n - H_{\\mathbf{k}} \\right)^{-1}$ is the unperturbed, translationally invariant Green's function with $H_{\\mathbf{k}}={\\rm diag}(H_{\\mathbf{k}1},H_{\\mathbf{k}2})$ and\n\\begin{align}\n\tu_\\mathbf{q} =\\gamma_\\mathbf{q} \n\t\\begin{pmatrix}\n\t\t0 & \\sigma_ze^{i\\sigma_z\\varphi \/2}\\\\\n\t\t\\sigma_z e^{-i\\sigma_z\\varphi \/2} & 0\n\t\\end{pmatrix} \\,.\n\\end{align}\nis the impurity vertex. The disorder average acts on $\\gamma_\\mathbf{q}$ factors and is performed according to Eq.\\ \\eqref{eq:incotunnterm}.\nThe diagrams in Fig.~\\ref{fig:diagrams}(b-c) represent terms of order $g^{4}$:\n\\begin{equation}\n\\begin{aligned}\n\tJ_c^{(4)} &= \n\t2\\, \\text{Tr} \\left[ j_{\\mathbf{q}} G_0(\\mathbf{k},\\omega_n)\n\t\tu_{\\mathbf{q}'} G_0(\\mathbf{k-q'},\\omega_n)\n\t\tu_{-\\mathbf{q}'}\n\t\t\\right.\n\t\t\\nonumber\n\t\t\\\\\n\t\t& \\quad\\quad\n\t\t\\times\n\t\t\\left.\n\t\tG_0(\\mathbf{k},\\omega_n)\n\t\tu_{-\\mathbf{q}}G_0(\\mathbf{k+q},\\omega_n)\n\t\\right] \\\\\n\t&\\quad +\n\t\\text{Tr}\\left[ \n\t j_{\\mathbf{q}} \n\t G_0(\\mathbf{k},\\omega_n)\n\t\tu_{\\mathbf{q}'} \n\t\tG_0(\\mathbf{k-q'},\\omega_n)\n\t\tu_{-\\mathbf{q}}\n\t\t\\right.\n\t\t\\nonumber\n\t\t\\\\\n\t\t& \\quad\\quad\n\t\t\\times\n\t\t\\left.\n\t\tG_0(\\mathbf{k-q'+q},\\omega_n)\n\t\tu_{-\\mathbf{q}'}G_0(\\mathbf{k+q},\\omega_n)\n\t\\right]\n\\end{aligned}\n\\end{equation}\nwhere impurity averaging is assumed but not explicitly shown for clarity of notation. Evaluating the traces, we obtain the current of the form \n\\begin{align}\n\tJ = J_c^{(2)} + J_c^{(4)} = J_{c1}(\\theta) \\sin \\varphi - J_{c2}(\\theta) \\sin 2\\varphi \\,,\n\t\\label{eq:jc}\n\\end{align}\nwith coefficients, to lowest order of $g$,\n\\begin{align}\n J_{c1}&=\n\t4\n\t\\sum_{n\\mathbf{k}} j_{n\\mathbf{k}}\n\n\n \n \n \n \n \n \n \n \\label{eq:jc1}\n \\\\\n\tJ_{c2}&=\n\t8\n\t\\sum_{n\\mathbf{k}}\n\tj_{n\\mathbf{k}}^2\n\t+\n\t4\\sum_{n\\mathbf{kqq'}} \n\t\\left|\\gamma_\\mathbf{q} \\right|^2\n \\left|\\gamma_\\mathbf{q'} \\right|^2\n \\label{eq:jc2}\n \\\\\n &\n \\times\n f_{n\\mathbf{k},1}f_{n\\mathbf{k+q+q'},1}\n f_{n\\mathbf{k+q},2}f_{n\\mathbf{k+q'},2}\n\t\\nonumber\n\\end{align}\nHere, we have defined\n\\begin{align}\n\\label{eq:convolution}\n j_{n\\mathbf{k}} &= \\left(f_{n\\mathbf{k},1} * \\left|\\gamma_{n\\mathbf{k}} \\right|^2 \\right)\n f_{n\\mathbf{k},2}\n \\\\\n f_{n\\mathbf{k},l} &= \\frac{\\Delta_{\\mathbf{k},i}}{\\omega_n^2+E_{\\mathbf{k},l}^2}\n \\label{eqn:f-function}\n\\end{align}\nand $(*)$ denotes a convolution integral, $a_{\\mathbf{k}}*b_{\\mathbf{k}} = \\sum_{\\mathbf{q}}a_{\\mathbf{q}}b_{\\mathbf{k-q}}$. The quasiparticle dispersion of the unperturbed bands is given by $E_{kl} = \\sqrt{\\xi_{\\mathbf{k}}^2+\\Delta_{\\mathbf{k}l}^2}$. \n\n\nFrom Eq.~\\eqref{eq:jc} one obtains the free energy\n\\begin{align}\n 2\\mathcal{F}=-J_{c1}(\\theta) \\cos \\varphi + \\frac{J_{c2}(\\theta)}{2} \\cos 2\\varphi +\\textrm{const}\\,.\n\\end{align}\nThe $\\mathcal{T}$-breaking phase transition occurs as a consequence of competition between $\\cos \\varphi$ and $\\cos 2 \\varphi$ terms. Clearly, $J_{c2}>0$ and the ground state acquires a finite phase difference for \n\\begin{align}\n 2J_{c2}>|J_{c1}|,\n \\label{eq:crit}\n\\end{align}\nwhere it spontaneously breaks $\\mathcal{T}$. From our discussion in Sec.~\\ref{sec:group theory} it follows that that $J_{c1}$ must vanish at twist of $\\theta=\\pi\/4$. Explicitly, one can see this result as follows. The functions $E_{\\mathbf{k},l}, \\, \\left|\\gamma_\\mathbf{k} \\right|^2$ transform under the $A_{1g}$ irrep of $D_{4h}$ whereas the $f_{\\mathbf{k},1},f_{\\mathbf{k},2}$ transform under $B_{1g}$ and $B_{2g}$, respectively. We note that convolution with the $A_{1g}$-symmetric impurity distribution $\\left|\\gamma_\\mathbf{k} \\right|^2$ does not change the symmetry of the convolution integral. Hence, $j_{\\mathbf{k}}$ transforms under $B_{1g}\\otimes B_{2g}=A_{2g}$ and all terms in Eq.~\\eqref{eq:jc1} average to zero at ${45^{\\circ}}$-twist. However, $j_{\\mathbf{k}}^2$ is $A_{1g}$-symmetric and $J_{c2}$ will consequently be finite and positive. Thus it is clear that condition \\eqref{eq:crit} is generally satisfied at $\\theta=45^\\circ$ and $\\mathcal{T}$ will always be broken as soon as the system enters the SC state below $T_c$.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{phase-diagram.pdf}\n\t\\caption{Phase diagram of incoherently coupled twisted bilayer cuprates. For a given $\\Lambda$, the inside of the cone-shaped region breaks $\\mathcal{T}$. Black-dashed lines mark the phase boundary in the clean limit, previously introduced in \\cite{Can2021}. For increasing degree of momentum non-conservation $\\Lambda$, the $\\mathcal{T}$-breaking phase boundaries shrink towards $45^\\circ$.}\n\t\\label{fig:phasediagram}\n\\end{figure}\n\nWe conclude that impurity-mediated tunneling must not qualitatively change the $\\mathcal{T}$-breaking phase diagram relative to the model of Ref.\\ \\cite{Can2021}. The incoherent tunneling, however, shifts the phase boundaries. As shown in Appendix \\ref{apdx:lambda-scaling}, $J_{c1}\\sim 1\/\\Lambda$ and $J_{c2}\\sim 1\/\\Lambda^2$. Since $J_{c1}$ is only weakly dependent on $\\theta$, and $J_{c1}$ vanishes linearly around $45^\\circ$ twist, it follows from Eq.~\\eqref{eq:crit} that the width of the $\\mathcal{T}$-breaking phase space is proportional to $J_{c2}(\\theta=0)\/J_{c1}(\\theta=0) \\sim 1\/\\Lambda$. In the perfectly incoherent limit, $\\Lambda\\rightarrow\\infty$, the free energy becomes independent of $\\varphi$ and the $\\mathcal{T}$-breaking phase disappears.\n\nTo quantitatively ascertain the effect of incoherent tunneling on the phase diagram, we numerically evaluate the coefficients $J_{ci}$. In principle, all Matsubara sums can be evaluated analytically, at the cost of removing the simple convolution structure in Eq.~(\\ref{eq:convolution}). This leaves three remaining momentum integrals to be numerically evaluated at complexity $\\mathcal{O}(N^3)$ where $N$ is the number of $\\mathbf{k}$-points of the 2D mesh used to perform the integrals. A more efficient approach is to exploit the convolution structure of Eq.~(\\ref{eq:convolution}) using the fast Fourier transform (fft) algorithm and numerically evaluate $M$ Matsubara frequencies, affording evaluation of diagrams Fig.~\\ref{fig:diagrams}(a-b) at order $\\mathcal{O}(MN\\log N)$. The crossed diagram Fig.\\ \\ref{fig:diagrams}(c) does not possess a convolution structure. As we show in Appendix \\ref{apdx:complexity}, it can be evaluated at a cost of $\\mathcal{O}(MN^2)$.\n\nThe resulting phase diagram is shown in Fig.~\\ref{fig:phasediagram} for coupling strength $g=\\SI{10.5}{\\milli\\electronvolt}$ and several values of $\\Lambda$. We see that the $\\mathcal{T}$-breaking phase space is largest in the `clean' limit $\\Lambda\\rightarrow 0$ where it extends between $(45\\pm 6)^\\circ$ at $T=0$. Increasing $\\Lambda$ gradually reduces the extent of the $\\mathcal{T}$-broken phase which eventually vanishes in the perfectly incoherent limit when $\\Lambda \\sim k_F$, i.e. when impurity correlations are on the scale of the lattice constant. Physically, this occurs because at this level of incoherence the Cooper pair essentially looses all memory of its momentum structure in the process of tunneling between layers.\n\n\n\n\\subsection{Spectral gap and topological superconductivity}\n\nHaving discerned the fate of the $\\mathcal{T}$-breaking phase in the presence of impurity-mediated tunneling we proceed to examine the topological properties of the resulting ground state. In the clean limit, $\\mathcal{T}$-breaking establishes a topological phase with Chern number $\\mathcal{C}=4$ \\cite{Can2021}. Since the disordered model is connected to the clean case by taking the limit $\\Lambda \\rightarrow 0$, it is reasonable to expect the same $\\mathcal{C}=4$ phase as long as the quasiparticle gap does not close.\n\nHere, we show that these expectations are indeed met. To this end, we evaluate the Green's function\n\\begin{align}\n G(\\mathbf{k},\\omega_n) = [G_0-\\Sigma(\\mathbf{k},\\omega_n)]^{-1}\n \\label{eq:gfd}\n\\end{align}\nin the Born approximation where\n\\begin{align}\n \\Sigma_{\\tau\\tau'} &=\n \\sum_{\\mathbf{q}} u_{\\mathbf{q}} \\, G_0(\\mathbf{k-q},\\omega_n)\\, u_{\\mathbf{-q}}\n \\\\\n &=-\\delta_{\\tau,\\tau'}\\,f_{\\mathbf{k},\\bar{\\tau}} (i\\omega_n + \\xi_{\\mathbf{k}} \\sigma_z + e^{i\\tau\\sigma_z \\varphi}\\Delta_{\\mathbf{k},\\bar{\\tau}} \\sigma_x ) \n * \\left|\\gamma_{\\mathbf{k}}\\right|^2\n \\nonumber \\,.\n \\end{align}\n with layer-indices $\\tau=\\pm 1$. Here, we regularized the continuum model on a square lattice using \n \\begin{align}\n \\xi_\\mathbf{k}&=-2t (\\cos k_x +\\cos k_y) - \\mu\n \\nonumber\n \\\\\n \\Delta_{\\mathbf{k},\\tau}&= \\Delta [(\\cos k_x - \\cos k_y) \\cos \\theta \n \\label{eq:alignedgap}\n \\\\\n & \\qquad\\quad\n + \\,\\tau \\sin k_x \\sin k_y \\, e^{-i\\varphi}\\sin \\theta ]\n \\nonumber\n \\end{align}\n with parameters chosen to match the continuum model.\n \nFollowing the method introduced in Ref.~\\cite{Pinon.2020}, we compute a spatially resolved Green function\n\\begin{align}\n G_B(x, k_y,\\omega_n) &= G(x, k_y) - G(x, k_y) \n T(k_y)\n G(-x, k_y) \n \\nonumber\n \\\\\n T(k_y) &= \\left[ \n \\frac{1}{\\sqrt{N}} \\sum_{k_x} G(k_x, k_y)\n \\right]^{-1} \n \\label{eq:transfm}\n\\end{align}\nin the presence of a strong repulsive potential at $x=0$ which simulates an edge and thus allows us to inspect the edge modes of the disordered system. We outline the method and give a derivation of Eq.~\\eqref{eq:transfm} in Appendix~\\ref{apdx:surfacegf}. \n\nIn Fig.~\\ref{fig:edgemode} we plot the analytically continued boundary spectral function\n\\begin{align}\n A_B(x,k_y,\\omega) = -\\frac{1}{\\pi} \\text{Im} \\left[G_B(x,k_y,-i\\omega+\\eta)\\right]\n\\end{align}\nat the edge $(x=1)$ as well as the bulk spectral function. We clearly observe two chiral edge modes traversing the bulk gap thus confirming the non-trivial topology of the system. The edge modes display a degeneracy in the layer degree of freedom, suggesting that the model is in a topological phase with Chern number $\\mathcal{C}=4$. The bulk gap is reduced but remains finite as $\\Lambda$ increases.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{spectrum.pdf}\n\t\\caption{Bulk (a-b) and boundary (c-d) spectrum for incoherently coupled cuprate bilayers with $\\Lambda=0$ (left) and $\\Lambda\/k_F=0.08$ (right) at $45^\\circ$ twist angle. The spectrum shows chiral edge modes traversing the bulk gap which is reduced but finite for increased $\\Lambda$. Edge modes, which are in fact degenerate, indicate a Chern number $\\mathcal{C}=4$. }\n\t\\label{fig:edgemode}\n\\end{figure}\n\n\n\n\\section{Lattice model}\n\\label{sec:latticemodel}\n\n\\begin{figure*}[t]\n \\includegraphics[width=16cm]{latt_summ.pdf}\n \\caption{(a) Illustration of the geometry of the bilayer lattice model at an incommensurate angle of $\\sim 43^\\circ$. (b) Disorder averaged free energy of the bilayer at zero temperature as a function of the phase difference. The minima $\\varphi_{\\rm min}$ are situated away zero at small disorder strengths. (c) Dependence of the order parameter amplitude and phase as a function of temperature for $\\tilde{\\Lambda}=0.2$. One could view it as a vertical cut at a specific twist in the phase diagram of Fig.~\\ref{fig:phasediagram}, with the onset of a non-zero phase marking the phase boundary.}\n \\label{fig:latt_F}\n\\end{figure*}\n\nSo far we have looked at the role of disorder in a continuum formulation of a twisted bilayer. The two-site unit cell of the regularized model allowed for analytical expressions for the layer Green's function to which we have systematically added incoherent interlayer tunneling and calculated the free energy up to fourth order in $g$. Another approach to tackle the problem and corroborate the results in a more general setting is to perform a BCS mean field theory on two twisted square lattices that represent the two CuO planes. In this case one can incorporate more realistic band structures but it is also harder to obtain the Green's functions analytically using Feynman diagrams. The reason is twofold: For one, at an arbitrary twist the lattice model is not commensurate. Secondly, at commensurate twist angles close to $45^\\circ$, the moir\\'{e} unit cell contains many sites. To get around this, we perform a brute force disorder average wherein several disorder realizations with the same microscopic parameters are taken into account. While it limits us to real space, such a treatment is exact because all orders in perturbation theory are implicitly accounted for.\n\nFor each layer, we consider a square lattice Hubbard model with nearest neighbor density-density interactions such that a mean-field decoupling produces a $d$-wave order parameter. Including the interlayer tunneling processes with amplitudes $g_{ij}$, the bilayer is described by\n\\begin{eqnarray}\n &{\\cal H} = & -t \\sum_{\\langle ij \\rangle \\sigma l} c^\\dag_{i \\sigma l} c_{j \\sigma l} - t'\\sum_{\\langle \\langle ij \\rangle \\rangle \\sigma l} c^\\dag_{i \\sigma l} c_{j \\sigma l} \n - \\mu \\sum_{i \\sigma l} n_{i \\sigma l} \\nonumber \\\\\n &+&\\sum_{\\langle ij \\rangle l}\\left(\\Delta_{ij, l}c^\\dag_{i\\uparrow\n l}c^\\dag_{j\\downarrow l}+{\\rm h.c.}\\right) \n - \\sum_{i j \\sigma} g_{ij} c^\\dag_{i \\sigma 1} c_{j \\sigma 2},\n \\label{eq:hm_latt}\n\\end{eqnarray}\nwhere $l$ is a layer index, $t$ ($t'$) is the (next-)nearest-neighbor hopping amplitude, $\\mu$ is the chemical potential that controls on-site particle density $n_{i \\sigma l}$ and $\\Delta_{ij,l}$ denotes the complex order parameter on the bond connecting sites $i$ and $j$ on layer $l$. Considering a fully coherent interlayer tunneling, Ref.~\\cite{Can2021} employs a circularly symmetric, exponentially decaying form $g_{ij} = e^{-(r_{ij}-c)\/\\rho}$ which connects sites $i$ and $j$ separated by $\\bm{r}_{ij}$. Therein $c$ in the interlayer separation and $\\rho$ is defined by the radial extent of the participating orbitals. The twist angle $\\theta$ between the layers determines connectivity and the strength of the interlayer tunnelings. The free energy of this model shows a double-well structure for twist angles around $45^\\circ$ \\cite{Can2021}. \n\nTo incorporate incoherent processes, we introduce a random tunneling factor that vanishes on average, but encodes the correlation between different processes depending on spatial separation. That is, \n\\begin{equation}\n g_{ij} = g_{\\bm{R}} ~ e^{-(r_{ij}-c)\/\\rho} \n \\label{eq:g_inco}\n\\end{equation}\nwhere $\\bm{R} = (\\bm{r}_{i} + \\bm{r}_{j})\/2$ denotes the center of mass location of the hopping and \n\\begin{align}\n \\overline{g_{\\bm{R}}} &= 0, \\nonumber \\\\\n \\overline{g_{\\bm{R}}g_{\\bm{R}'}} &= g^2 \\exp\\left[-\\frac{\\tilde{\\Lambda}^2}{4} (\\bm{R}-\\bm{R}')^2 \\right].\n\\end{align}\nAnalogous to the parameter $\\Lambda$ in the continuum model, $\\tilde{\\Lambda}$ sets the length scale for the correlation between different tunneling amplitudes and is indicative of disorder strength. We distinguish the two simply because of the slightly differing definitions. To simulate the Fermi surface of optimally doped BSCCO with hole pocket around $(\\pi,\\pi)$, we set $t=153$meV, $t' = -0.45t$ and $\\mu = -1.35t$ \\cite{Bille2001}. Further, we choose $c=2.2$ and $\\rho=0.4$ (in units of the lattice constant) to set interlayer distances. The $d$-wave order parameters in cuprates originates in the CuO planes and the interlayer coupling is a minor perturbation that does not influence the order parameter magnitude. In other words, temperature dependence of the gap in each layer is independent of twist and coupling strength strength $g$, which we peg at 20meV. Therefore, we use a $\\Delta$ calculated self-consistently in a monolayer, which has a maximum of $\\sim 40$meV at 0K in accordance with experimental findings in cuprates \\cite{Dama2003, Fischer2007}.\n\nTo look for ${\\cal T}$-breaking we examine the free energy of the system, which can be calculated from the eigenvalues $E_i$ of the BdG Hamiltonian \\eqref{eq:hm_latt}:\n\\begin{equation}\n {\\cal F}_{\\rm BdG}= \\sum_i E_i - 2 k_B T \\sum_{i}\\ln\\left[2\\cosh{(E_{i}\/2 k_B T)}\\right].\n\\end{equation}\nIn particular, for a given twist $\\theta$ and disorder parameter $\\tilde\\Lambda$, we draw from the distribution \\eqref{eq:g_inco} and average the free energy over 50 independent realizations. We choose a square bilayer sample as shown in Fig.~\\ref{fig:latt_F}(a), but the results are independent of the shape. Further, the exact number of sites in the system depends on the cut and the twist angle, but the free energy does not show an appreciable change beyond $\\sim 900$ sites per layer. In agreement with the continuum model, Fig.~\\ref{fig:latt_F}(b) shows that the presence of ${\\cal T}$-breaking free energy minima is controlled by $\\tilde\\Lambda$. Namely, small values of $\\tilde\\Lambda$ support the ${\\cal T}$-broken ground state while larger values do not. \n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nTwisted bilayers of high-$T_c$ cuprates hold the potential for realizing topological superconductivity, wherein a topological gap is spontaneously induced. As per the symmetry informed momentum space form factors, which determine the electron hopping between interlayer Cu atoms, the tunneling amplitude vanishes along the nodal directions and a spectral gap may not appear. In this work we highlight that an important aspect to consider in such an analysis is the disorder mediated tunneling. Not only does disorder appear naturally due to oxygen doping and interfacial defects, but incorporating momentum non-conservation has been shown to better represent experimental data in clean single crystals.\n\nUsing perturbative diagrammatic calculations and disorder averaging on the lattice, we find that an experimentally motivated incoherent tunneling model that respects all point group symmetries of the physical system gives rise to a qualitatively similar phase diagram as obtained in Ref.\\,\\cite{Can2021}. Specifically we find a substantial range of twist angles around $45^\\circ$ and temperatures where spontaneous ${\\cal T}$-breaking occurs and produces a fully gapped topological phase with non-zero Chern number. The angular extent of the ${\\cal T}$-broken phase depends on the disorder length scale $\\Lambda^{-1}$ where the coherent limit $\\Lambda\\to 0$ recovers the phase diagram of Ref.\\ \\cite{Can2021} and increasing $\\Lambda$ corresponds to a shrinking extent of the topological phase. Only when the incoherence length scale is comparable to the Fermi momentum, the twist angle for spontaneous ${\\cal T}$ breaking is reduced to exactly ${45^{\\circ}}$. \n\nFrom an experimental point of view, the inhomogeneity due to oxygen doping the BiO planes of untwisted BSCCO was found to be correlated over $\\approx 14 \\si{\\angstrom}$ \\cite{Pan2001}. Since the CuO plane lattice constant is $\\approx 5 \\si{\\angstrom}$, that amounts to a correlations over 3 unit cells, i.e., $\\tilde{\\Lambda} \\approx 0.3$. In a twisted geometry, one may expect the characteristic length scale to decrease and, hence, the estimate for $\\tilde{\\Lambda}$ could shift up. That said, the role of complex atomic arrangements, moir\\'e length scales and strong correlations are difficult to incorporate into such a heuristic reasoning. One would probably have to await data from complementary experimental probes, such as transport and optical response, to discern the nature of the superconducting state around ${45^{\\circ}}$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{jc.pdf}\n\t\\caption{Critical current $J_c$ of the twisted bilayer as a function as interlayer coherence scale $\\Lambda$. Incoherence significantly reduces $J_c$. The color scale denotes temperature $T$ in panel (a) and twist angle $\\theta$ in (b).}\n\t\\label{fig:jc}\n\\end{figure}\n\nIt was noted in Ref.\\,\\cite{Song2021} that the measured critical current density $J_c$ in both twisted and untwisted Bi2212 is about factor of 500 smaller than the theory prediction based on the slave-boson mean field theory of a $t$-$J$ model used in that study. We checked that a similar discrepancy occurs in the calculation using BCS mean field theory of Ref.\\,\\cite{Can2021}. As indicated in Fig.\\,\\ref{fig:jc} the discrepancy is somewhat reduced in the incoherent tunneling model (by about one order of magnitude at large $\\Lambda$) but nevertheless significant disagreement with experiment persists. As noted in Ref.\\,\\cite{Sheehy_2004} this is a known problem that affects superconductors in the cuprate family and becomes increasingly severe in the underdoped part of their phase diagram. A phenomenological fix can be implemented \\cite{Sheehy_2004} by restricting the momentum sums in the expression for $J_c$ to patches of linear size $\\sim x$ (the hole doping) around the nodal points of the $d$-wave order parameter. This modification leaves the temperature dependence of $\\rho_{ab}(T)$ and $\\rho_{c}(T)$ unchanged, but reduces their $T=0$ magnitude to experimentally observed values. It similarly fixes the problem with $J_c$. As with many aspects of cuprates a truly microscopic understanding of this phenomenon remains a challenge to the theory community. With regards to twisted cuprate bilayers it would be interesting to explore the effect of the phenomenological fix outlined above on the phase diagram but we leave this to future work. \n\n\n\\section*{Acknowledgments}\n\nWe are grateful to S. Egan, O. Can, X. Cui, \\'E. Lantagne-Hurtubise, X.-Y. Song, A. Vishwanath, P. Volkov and S.Y.F. Zhao for helpful discussions and correspondence. This research was supported in part by NSERC and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. We thank the Max Planck-UBC-UTokyo Center for Quantum Materials for fruitful collaborations and financial support. R.H. acknowledges the Joint-PhD program of the University of British Columbia and the University of Stuttgart.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\nAt the LHC, top quark pairs ($t\\bar{t}$) will be produced mainly via gluon fusion \\mbox{($\\sim$ 90\\%)}. A recent prediction for the cross section at $\\sqrt{s}=14$ TeV is a next-to-leading order (NLO) calculation with soft-gluon next-to-leading-log (NLL) resummation \\cite{xsecNLONLL}: $\\sigma_{t\\bar{t}}^{NLO+NLL}=908^{+82}_{-85}$ (factorisation and renormalisation scales) $^{+30}_{-29}$ (parton distribution function (PDF)) pb. As $V_{tb} \\approx 1$, top quarks nearly always decay into a $W$-boson and a $b$-quark. A $W$-boson decays roughly 2\/3$^{rd}$ of the cases into two quarks and 1\/3$^{rd}$ into a lepton and a neutrino. Therefore, $t\\bar{t}$ decay can be divided into three channels: 4\/9$^{th}$ fully hadronic, 4\/9$^{th}$ semileptonic and 1\/9$^{th}$ dileptonic. The fully hadronic channel suffers from large QCD multi-jet background. The presence of missing transverse energy (from undetectable neutrino's) and at least one lepton in the two other channels creates the possibility to select $t\\bar{t}$ events while reducing background considerably. These two channels are used in ATLAS for $t\\bar{t}$ cross section measurements with the first 100pb$^{-1}$ integrated luminosity of data \\citep{ATLAS-CSC} and will be discussed in the following sections.\n\n\n\\section{SINGLE LEPTON CHANNEL}\nTwo complementary measurements in the single lepton channel are investigated: a counting method and a likelihood fit. Events are selected that passed the electron (muon) trigger {\\texttt{e22i}} ({\\texttt{mu20}}) and contain missing transverse energy $\\not\\negthickspace{E}_{T} > 20$ GeV, an isolated electron (muon) with transverse momentum $p_{T} > 20$ GeV\/c and pseudo rapidity $\\lvert \\eta \\rvert < 2.5$, three jets with $p_{T} > 40$ GeV\/c and $\\lvert \\eta \\rvert < 2.5$ and an additional fourth jet with $p_{T} > 20$ GeV\/c and $\\lvert \\eta \\rvert < 2.5$. The hadronic top is reconstructed by taking the invariant mass $M_{jjj}$ of the three jet combination with the highest $p_{T}$. To reduce background from jet combinatorics and other processes, at least one di-jet combination is required to be compatible with the $W$-boson mass $\\lvert M_{jj}-M_{W} \\rvert < 10$ GeV\/c$^{2}$. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.3\\textwidth]{TopMassPlot_muon_Figure2_100pb.eps}\n\\hspace{.1\\textwidth}\n\\includegraphics[width=.3\\textwidth]{CSC_MassPlot_Mwcut_muon_Fit_100pb.eps}\n\\caption{(a) Expected invariant three jet mass distribution of correctly reconstructed $t\\bar{t}$ events (white), incorrectly reconstructed events due to jet combinatorics (dark shaded) and background processes (light shaded). The histogram is normalised to 100pb$^{-1}$ using the total available statistics (1 fb$^{-1}$). (b) Fit of a Gaussian and Chebychev polynomial to a Monte Carlo generated pseudo-experiment corresponding to 100pb$^{-1}$ of data.} \n\\label{fig:top_mass_plot}\n\\end{figure*}\n\n\\begin{samepage}\nIn the counting method the cross section is determined by subtracting the number of expected background events from the number of observed events in the $M_{jjj}$ distribution divided by the selection efficiency and integrated luminosity. The main background comes from $W$+ jets and single top. When tagging of $b$-quark jets is possible, the purity can be increased by a factor four, while the efficiency is only reduced by a factor two. The largest systematic uncertainty ($\\sim$10\\%) in this analysis is the background normalisation.\n\\end{samepage}\n\nIn the likelihood fit, only the correctly reconstructed $t\\bar{t}$ events that end up in the peak of the distribution are used for the cross section measurement. The peak is fitted with a Gaussian, while the combined background shape from jet combinatorics and other processes such as $W$+ jets is described by a Chebychev polynomial. The cross section is calculated from the number of $t\\bar{t}$ events in the peak divided by the overall efficiency for a $t\\bar{t}$ event to end up in the peak and the integrated luminosity. The fit is sensitive to the shapes of the distribution and therefore this is the main systematic uncertainty ($\\sim$10\\%). The expected uncertainties on the cross section for 100 pb$^{-1}$ of data are:\n\\begin{center}\n\\begin{tabular}{l r r r r}\n Likelihood method & 7 (stat) & $\\pm 15$ (syst) & $\\pm 3$ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n Counting method & 3 (stat) & $\\pm 16$ (syst) & $\\pm 3$ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n\\end{tabular}\n\\end{center}\n\n\n\\section{DIFFERENTIAL CROSS SECTION}\nThe measurement of the differential cross section as function of the invariant mass $M_{t\\bar{t}}$ of the $t\\bar{t}$ system provides an important check of the Standard Model. Deviations from the $t\\bar{t}$ continuum indicate the presence of new physics, for example new heavy resonances decaying into a $t\\bar{t}$ pair. Semileptonic events are selected using the same criteria as mentioned in the previous section, i.e. without $b$-tagging. The $t\\bar{t}$ pair is reconstructed by adding up the vectors of the four highest $p_{T}$ jets, the lepton and $\\not\\negthickspace{E}_{T}$. By using a $W$- and top mass constraint in a least square fit, assigning jets to the (anti-)top, a better result can be obtained than with the default event reconstruction only combining the four jets with the lepton and $\\not\\negthickspace{E}_{T}$ using a $W$-mass constraint on the leptonic side. The expected mass resolution ranges from 5\\% to 9\\% between 200 and 850 GeV\/c$^{2}$. \n\nAlso, the double differential cross section for $t\\bar{t}$ is sensitive to possible new physics. In this measurement $b$-tagging is used in addition to the default single lepton $t\\bar{t}$ selection criteria to improve purity up to 45\\%. The hadronic top is reconstructed by finding the highest $p_{T}$ combination of a $b$-tagged jet with a di-jet combination $\\lvert M_{jj} - M_{W}\\rvert < 20$ GeV\/c$^{2}$ close to it. The region which is covered by this method is 50 GeV\/c $< p_{T} <$ 280 GeV\/c and rapidity $\\lvert y \\rvert < 2$. The statistical error on the distribution will be around 30\\% for 100 pb$^{-1}$ (10\\% for 1 fb$^{-1}$). The main systematic uncertainty comes from the jet energy scale and initial\/final state radiation ($\\sim$15\\%).\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.3\\textwidth]{ditopmasscomp_linscale_simple.eps} \n\\hspace{.1\\textwidth}\n\\includegraphics[width=.3\\textwidth]{AllHadTopPtYReco_1fb.eps}\n\\caption{(a) Normalised di-top mass distribution for the default (dotted line) and the improved (dashed line) reconstruction and (b) reconstructed $p_{T}$ and $y$ distribution of hadronically decaying top quarks, normalised to 1 fb$^{-1}$.} \n\\label{fig:differential_xsec}\n\\end{figure*}\n\n\n\\section{DILEPTON CHANNEL}\nFor the cross section determination in the dilepton channel three measurements are considered. The three measurements start with the preselection of events with two high-$p_{T}$ opposite signed leptons ($e^{+}e^{-}$, $\\mu^{+}\\mu^{-}$ and $e^{\\pm}\\mu^{\\mp}$) and then have different additional requirements. These events are efficiently selected by a combination of single and dilepton triggers. The uncertainty is expected to be small due to the trigger {\\texttt{OR}} condition between the channels and the high statistics available for efficiency measurements such as the 'tag-and-probe' method using events with $Z$-bosons.\n\nIn the cut and count method the cross section is determined by comparing the number of observed events with the number of expected background events (from Monte Carlo). The optimum selection criteria are determined to be: two isolated opposite signed leptons with $p_{T}>20$ GeV\/c, a veto on events with $M_{\\ell^{+}\\ell^{-}}$ around $Z$ peak (85-95 GeV\/c$^{2}$), two jets of at least 20 GeV\/c and $\\not\\negthickspace{E}_{T}>30$ GeV. With a S\/B of 4.3 the main background processes remaining are $Z\\rightarrow \\tau^{+} \\tau^{-}$ and semi-leptonic $t\\bar{t}$ where one of the jets fakes a lepton. The jet energy scale introduces the largest systematic uncertainty.\n\nThe inclusive template method is based on the observation that the three dominant sources of isolated leptons which can be selected in the $e\\mu$ channel are $t\\bar{t}$, $W^{+}W^{-}$ and $Z\\rightarrow\\tau^{+}\\tau^{-}$. These three processes can be separated by looking at the 2-D plane spanned by $\\not\\negthickspace{E}_{T}$ and $N_{jet}$. After scanning different configurations of background templates, the template with the highest probability is selected. The fit has free parameters including the $t\\bar{t}$, $W^{+}W^{-}$ and $Z\\rightarrow\\tau^{+}\\tau^{-}$ cross section. Contamination from processes with a fake lepton is reduced by using tight isolation criteria for the electron and a veto on events with $\\not\\negthickspace{E}_{T}$ aligned along the reconstructed muon. The acceptance of the two leptons and the shapes of the 2-D templates determine the systematic uncertainties.\n\nFor the third method a log-likelihood function is used to extract the parameters $N_{sig}$ and $N_{bkg}$ given the fixed total number of events $N_{tot}$. The signal and background input functions are determined by fitting Chebychev polynomials to Monte Carlo generated distributions. The sum of the semi-leptonic $t\\bar{t}$, $W^{+}W^{-}$ and $Z\\rightarrow\\tau^{+}\\tau^{-}$ is considered as a single background distribution. The two variables for the distributions are $\\Delta\\varphi$ between: (i) the highest $p_{T}$ lepton and $\\not\\negthickspace{E}_{T}$ and (ii) the highest $p_{T}$ jet and $\\not\\negthickspace{E}_{T}$. Like for the cut and count method, the jet energy scale is the dominant systematic uncertainty. Expected uncertainties on the cross section for 100 pb$^{-1}$ of data are:\n\\begin{center}\n\\begin{tabular}{l r r r r}\n Cut and Count method & 4 (stat) & $^{+5}_{-2}$ (syst) & $\\pm 2 $ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n Template method & 4 (stat) & $\\pm 4$ (syst) & $\\pm 2 $ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n Likelihood method & 5 (stat) & $^{+8}_{-5}$ (syst) & $\\pm 2 $ (PDF) $\\pm 5$ (luminostiy) $\\%$ \\\\\n\\end{tabular} \n\\end{center}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.3\\textwidth]{zpeak_resized.eps} \n\\includegraphics[width=.3\\textwidth]{tt_nj_met.eps}\n\\includegraphics[width=.3\\textwidth]{L0_MET_edited_resized.eps}\n\\caption{(a) Invariant di-lepton mass distribution in the cut \\& count method, (b) $t\\bar{t}$ template of the $N_{jet}$ vs $\\not\\negthickspace{E}_{T}$ distribution for the inclusive template fit and (c) a likelihood fit to the signal (dotted blue line) and background (dotted red line).} \n\\label{fig:dilepton_channel}\n\\end{figure*}\n\n\n\\begin{acknowledgments}\nThe author has been supported by the Netherlands Organisation of Scientific Research (NWO) under VIDI research grant 680.47.218.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{introduction}\nKirchberg proved in 1994 that tensorial absorption of $\\mathcal{O}_\\infty$ and pure infiniteness were equivalent for simple separable nuclear C$^*$-algebras. This theorem can be reinterpreted as the equivalence of $\\mathcal{Z}$-stability and strict comparison for simple separable nuclear traceless C$^*$-algebras (see \\cite{Ro}), an equivalence that can be considered in stably finite algebras, too. This and the results of \\cite{TW} led to the following conjecture:\n\n\\begin{conjs}[T-Winter, 2008]\\label{conjecture}\nLet $A$ be a simple unital nuclear separable C$^*$-algebra. The following are equivalent:\n\\begin{enumerate}\n\\item[(i)] $A$ has finite nuclear dimension;\n\\item[(ii)] $A$ is $\\mathcal{Z}$-stable;\n\\item[(iii)] $A$ has strict comparison of positive elements.\n\\end{enumerate}\n\\end{conjs}\n\n\\noindent\nIt is expected that these conjecturally equivalent conditions will characterize those algebras which are determined up to isomorphism by their Elliott invariants. In the absence of even the weakest condition, (iii), one cannot classify AH algebras using only the Elliott invariant (\\cite{T3}). While it is possible that this can be corrected with an enlarged invariant, the jury is still out. Combining the main result of \\cite{W} with that of \\cite{R} yields (i) $\\Rightarrow$ (ii), while R\\o rdam proves (ii) $\\Rightarrow$ (iii) in \\cite{Ro}. This note yields the following result.\n\\begin{thms}\\label{main}\nConjecture \\ref{conjecture} holds for AH algebras.\n\\end{thms}\n\\noindent\nWe proceed by proving (ii) implies (iii) for AH algebras (Corollary \\ref{maincor}) and appealing to \\cite[Corollary 6.7]{W}. It should be noted that our contribution is quite modest: we simply verify the hypotheses of the main result of \\cite{W}. \n\n\n\\section{Strict comparison and almost divisibility}\n\nLet $A$ be a unital C$^*$-algebra, and $\\mathrm{T}(A)$ its simplex of tracial states. Let $\\mathrm{Aff}(\\mathrm{T}(A))$ denote the continuous $\\mathbb{R}$-valued affine functions on $\\mathrm{T}(A)$, and let $\\mathrm{lsc}(\\mathrm{T}(A))$ denote the set of bounded lower semicontinuous strictly positive affine functions on $\\mathrm{T}(A)$. Set $\\mathrm{M}_{\\infty}(A) = \\cup_n \\mathrm{M}_n(A)$. For positive \n$a,b \\in \\mathrm{M}_{\\infty}(A)$, we write $a \\precsim b$ if there is a sequence $(v_n)$ in $\\mathrm{M}_{\\infty}(A)$ such that $v_nbv_n^* \\to a$ in norm (for the norm, view $\\mathrm{M}_\\infty(A)$ sitting naturally inside $A \\otimes \\mathcal{K}$). We write $a \\sim b$ if $a \\precsim b$ and $b \\precsim a$. Set $W(A) = \\{a \\in \\mathrm{M}_\\infty(A) \\ | \\ a \\geq 0 \\}\/\\sim$, and let $\\langle a \\rangle$ denote the equivalence class of $a$. $W(A)$ can be made into an ordered Abelian monoid by setting\n\\[\n\\langle a \\rangle + \\langle b \\rangle = \\langle a \\oplus b \\rangle \\ \\ \\mathrm{and} \\ \\ \\langle a \\rangle \\leq \\langle b \\rangle \\Leftrightarrow a \\precsim b.\n\\]\n$W(A)$ is the original {\\it Cuntz semigroup} of $A$. \n\nWe say that $W(A)$ is {\\it almost divisible} if for any $x \\in W(A)$ and $n \\in \\mathbb{N}$, there exists $y \\in W(A)$ such that \n\\[\nny \\leq x \\leq (n+1)y.\n\\]\nIf $\\tau \\in \\mathrm{T}(A)$, we define $d_\\tau: W(A) \\to \\mathbb{R}^+$ by\n\\[\nd_\\tau(\\langle a \\rangle) = \\lim_{n \\to \\infty} \\tau(a^{1\/n}).\n\\]\nThis map is known to be well-defined, additive, and order preserving, and for a fixed positive $a \\in \\mathrm{M}_{\\infty}(A)$, $A$ simple, the map $\\tau \\mapsto d_\\tau(a)$ belongs to $\\mathrm{lsc}(\\mathrm{T}(A))$. If $a \\precsim b$ whenever $d_\\tau(a) < d_\\tau(b)$ for every $\\tau \\in \\mathrm{T}(A)$, then we say that $A$ has {\\it strict comparison}. \n\nIt is implicit in Conjecture \\ref{conjecture} that a unital simple separable nuclear C$^*$-algebra with strict comparison of positive elements should have almost divisible Cuntz semigroup, but no general method has yet been found to establish this fact. Positive results have been limited to particular classes of C$^*$-algebras. Here we handle the case of AH algebras.\n\n\n\n\\begin{props}\\label{dense}\nLet $A$ be a unital, simple, stably finite C$^*$-algebra with strict comparison of positive elements. Suppose that for any $f \\in \\mathrm{Aff}(\\mathrm{T}(A))$ and $\\epsilon>0$ there is positive $a \\in \\mathrm{M}_\\infty(A)$ such that\n\\[\n|f(\\tau) - d_\\tau(a)| < \\epsilon, \\ \\forall \\tau \\in \\mathrm{T}(A).\n\\]\nIt follows that $W(A)$ is almost divisible.\n\\end{props}\n\n\\begin{proof}\nLet $g \\in \\mathrm{lsc}(\\mathrm{T}(A))$ be given. Then there is a strictly increasing sequence $(f_i)$ of strictly positive functions in $\\mathrm{Aff}(\\mathrm{T}(A))$ with the property that $\\sup_i f_i(\\tau) = g(\\tau)$. The function $f_i - f_{i-1}$ is strictly positive and continuous, and so achieves a minimum value $\\epsilon_i >0$ on the compact set $\\mathrm{T}(A)$. Passing to a subsequence, we may assume that $\\epsilon_i < \\epsilon_{i-1}$. By hypothesis, we can find, for each $i$, a positive $a_i \\in \\mathrm{M}_\\infty(A)$ such that\n\\[\n|f_i(\\tau)-d_\\tau(a_i)|<\\epsilon_{i+1}\/3.\n\\]\nIt follows that $(\\tau \\mapsto d_\\tau(a_i))_{i \\in \\mathbb{N}}$ is a strictly increasing sequence in $\\mathrm{lsc}(\\mathrm{T}(A))$ with supremum\n$g$. By strict comparison, we have $a_i \\precsim a_{i+1}$, i.e., $(\\langle a_i \\rangle)_{i \\in \\mathbb{N}}$ is an increasing sequence in $W(A)$. \\cite[Theorem 1]{cei} then guarantees the existence of a supremum $y$ for this sequence in $W(A \\otimes \\mathcal{K}) \\supseteq W(A)$. The map $d_\\tau$ is supremum preserving for each $\\tau$, and we conclude that $d_\\tau(y) = g(\\tau), \\ \\forall \\tau \\in \\mathrm{T}(A)$. \n\nNow let $x \\in W(A)$ and $n \\in \\mathbb{N}$ be given, and set $h(\\tau) = d_\\tau(x)$ for each $\\tau \\in \\mathrm{T}(A)$. It is straightforward to find $g \\in \\mathrm{lsc}(\\mathrm{T}(A))$ with the property that \n\\[\nng < h < (n+1)g.\n\\]\nWe may moreover find $x \\in W(A \\otimes \\mathcal{K})$ such that $d_\\tau(x) = g(\\tau)$, as in the first part of the proof. By strict comparison, any representative $a$ for $x$ (that is, $\\langle a \\rangle =x$ in $W(A \\otimes \\mathcal{K})$) will satisfy\n\\[\nn\\langle a \\rangle \\precsim x \\precsim (n+1)\\langle a \\rangle,\n\\]\nso it remains only to prove that $a$ can be chosen to lie in $\\mathrm{M}_\\infty(A)$, rather than $A \\otimes \\mathcal{K}$. Let $\\mathbf{1_k}$ denote the unit of $\\mathrm{M}_k(A)$. Since $g$ is bounded, we have $g(\\tau) < k = d_\\tau(\\mathbf{1_k})$ for some $k \\in \\mathbb{N}$ and all $\\tau$. By strict comparison, then, $\\langle a \\rangle$ is dominated by a Cuntz class in $W(A)$. It then follows from \\cite[Theorem 4.4.1]{5guys} that there is positive $b \\in \\mathrm{M}_\\infty(A)$ such that $\\langle b \\rangle = \\langle a \\rangle$, completing the proof.\n\\end{proof}\n\n\\begin{cors}\\label{maincor}\nLet $A$ be a unital simple AH algebra with strict comparison. It follows that $A$ is $\\mathcal{Z}$-stable.\n\\end{cors}\n\n\\begin{proof}\n$A$ is stably finite, and satisifes the hypothesis of Proposition \\ref{dense} concerning the existence of suitable $a$ for each $f$ and $\\epsilon$ by \\cite[Theorem 5.3]{bpt}. It therefore has strict comparison and almost divisible Cuntz semigroup. These hypotheses, together with the fact that $A$ is simple, nuclear, separable, and has locally finite nuclear dimension allow us to appeal to \\cite[Theorem 6.1]{W} and conclude that $A$ is $\\mathcal{Z}$-stable.\n\\end{proof}\n\nWe must concede that Proposition \\ref{dense} closes a gap in the proof of \\cite[Theorem 1.2]{T2}. There, we proved that the hypotheses of Propostion \\ref{dense} were satisfied for a simple unital ASH algebra with slow dimension growth but neglected to explain how this guarantees almost divisibility for $W(A)$ as opposed to $W(A \\otimes \\mathcal{K})$. While this could have been done in several ways, our appeal here to the recent article \\cite{5guys} was the most efficient one. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzprlq b/data_all_eng_slimpj/shuffled/split2/finalzzprlq new file mode 100644 index 0000000000000000000000000000000000000000..f1431d55af9c6c75ae1b60875e14c8cd88b3d40a --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzprlq @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe Euler characteristic, defined as the alternating sum of the Betti numbers, is a key invariant of topological spaces of finite type (such as cell complexes built out of a finite number of cells). One can define an invariant $\\widetilde \\chi(G)$ for a group $G$ by substituting group cohomology for singular cohomology, but unless $G$ has a finite-type $K(G,1)$-space this invariant lacks many desirable features of topological Euler characteristics. This is unfortunate because many of the most interesting groups have torsion, and groups with torsion never have finite-type $K(G,1)$-spaces. A solution was proposed by C.T.C.\\ Wall, who observed that if $G$ has a torsion-free subgroup $H$ of finite index that {\\it does} have a finite-type classifying space then the rational number $\\chi(H)\/[G:H]$ is an invariant of $G$ \\cite{Wall61}. In particular, this number does not depend on the choice of $H$. Wall called it the {\\it rational Euler characteristic} of $G$, denoted $\\chi(G)$. This rational Euler characteristic is better behaved than $\\widetilde \\chi(G)$; for example if $1\\to A\\to B\\to C\\to 1$ is a short exact sequence of groups then $\\chi(B)=\\chi(A)\\chi(C)$, assuming $\\chi(A), \\chi(B) $ and $\\chi(C)$ are all defined. \n\nIt turns out that rational Euler characteristics of arithmetic groups can often be expressed in terms of zeta functions; this ultimately depends on the Gauss-Bonnet theorem (see \\cite{Serre71, Serre79} for details and a guide to the literature). Remarkably, Harer and Zagier showed that the rational Euler characteristics of mapping class groups are also given by zeta functions, e.g.\\ the rational Euler characteristic of the mapping class group of a once-punctured surface of genus $g$ is equal to $\\zeta(1-2g)$ \\cite{HZ86}. This was later reproved by Penner \\cite{penner1986moduli} and by Kontsevich \\cite{Kon92}, each using asymptotic methods related to perturbation expansions from quantum field theory. \n\n We are concerned here with the rational Euler characteristic of the group $\\Out(F_n)$ of outer automorphisms of a finitely-generated free group. This group shares many features with both mapping class groups and arithmetic groups, though it belongs to neither class. In 1987 Smillie and Vogtmann found a generating function for $\\chi(\\Out(F_n))$ and computed its value for $n\\leq 11$ \\cite{SV87a}. From the results of these computations, they conjectured that $\\chi(\\Out(F_n))$ is always negative and the absolute value grows faster than exponentially. In 1989 Zagier simplified the generating function and computed $\\chi(\\Out(F_n))$ for $n\\leq 100$; this added strong evidence for these conjectures without providing a proof \\cite{Zagier}. The only general statements previously known about the value of $\\chi(\\Out(F_n))$ are that it is non-zero for even values of $n$, and certain information was established about the primes dividing the denominator \\cite{SV87a, SV87b}. \n\n\n\nIn this article we show that $\\chi(\\Out(F_n))$ is negative for all $n$ and prove that its asymptotic growth rate is controlled by the classical gamma and log functions:\n\n\\begin{thmx}\\label{thm:SVconj}%\n The rational Euler characteristic of $\\Out(F_n)$ is strictly negative, $\\chi\\left( \\Out(F_n) \\right) < 0$, for all $n \\geq 2$ and its magnitude grows more than exponentially,\n \\begin{align*}\n \\chi\\left(\\Out(F_n) \\right) &\\sim - \\frac{1}{\\sqrt{2 \\pi}} \\frac{\\Gamma(n-\\frac32)}{\\log^2 n} \\text{ as } n \\to \\infty.\n \\end{align*}\n\\end{thmx}\n\n\n \n\n\n\n\nThe proof of Theorem~\\ref{thm:SVconj} is based on the following theorem, in which we produce an asymptotic expansion, with respect to the asymptotic scale $\\{(-1)^k\\G(n+\\frac{1}{2}-k)\\}_{k\\geq0}$ in the limit $n\\to \\infty$, whose coefficients are closely related to $\\chi(\\Out(F_n))$.\n\n\n\\begin{thmx}\\label{thm:asymptotic_expansion}\n\\begin{align*}\n \\sqrt{2 \\pi}e^{-n} n^n &\\sim \\sum_{ k\\geq 0 } \\Ch_k (-1)^k \\Gamma( n + \\frac12 - k ) \\text{ as } n\\rightarrow \\infty,\n\\end{align*}\nwhere $\\Ch_k$ is the coefficient of $z^k$ in the formal power series $\\exp\\left( \\sum_{n\\geq 1} \\chi\\left( \\Out (F_{n+1}) \\right) z^n \\right)$.\n\\end{thmx}\n\n\nWe then relate this to a certain expansion of the Lambert $W$-function. The Lambert $W$-function is a well-studied function with a long history \\cite{corless1996lambertw}. Eventually, we are able to use results of Volkmer \\cite{volkmer2008factorial} about the coefficients of this second expansion to prove Theorem~\\ref{thm:SVconj}. \nIn Proposition~\\ref{prop:efficient_chn} we also exploit the connection with the Lambert $W$-function to give an efficient recursive algorithm for computing $\\chi(\\Out(F_n))$. \n\n\nIn Section~\\ref{sec:core} we show that there is a close relationship between $\\chi(\\Out(F_n))$ and the classical zeta function by considering the Connes-Kreimer Hopf algebra ${\\mathbf{H}}$ of 1-particle-irreducible graphs, i.e.\\ graphs with no separating edges. Briefly, the formula in \\cite{SV87a} for $\\chi(\\Out(F_n))$ can be regarded as the integral of a certain character $\\tau$ of ${\\mathbf{H}}$ on the space spanned by admissible connected graphs with fundamental group $F_n$, with respect to the ``measure'' $\\mu(\\G)=1\/|\\Aut(\\G)|$. Proposition~\\ref{prop:bernoulli_graphs_sum} shows that integrating $\\tau^{-1}$ (the inverse of $\\tau$ in the group of characters) with respect to the same measure produces $\\zeta(-n)\/n=\\frac{B_{n+1}}{n(n+1)},$ where $B_n$ is the $n$-th Bernoulli number. \n\n\n\nThe asymptotic expansion in Theorem~\\ref{thm:asymptotic_expansion} is strikingly reminiscent of the well-known Stirling asymptotic expansion of the gamma function in the asymptotic scale $\\{\\sqrt{2\\pi} e^{-n} n^{n-\\frac12-k} \\}_{k\\geq 0}$,\n\\begin{align*}\n \\Gamma(n) &\\sim \n \\sum_{k\\geq 0} \\widehat b_k \\sqrt{2\\pi} e^{-n} n^{n-\\frac12-k} \\text{ as } n \\to \\infty.\n\\end{align*}\nThe coefficients of this asymptotic expansion are related to the Bernoulli numbers as well: $\\widehat b_k$ is the coefficient of $z^k$ in the formal power series $\\exp\\left( \\sum_{n \\geq 1} \\frac{B_{n+1}}{n(n+1)} z^n \\right)$. We will explore this analogy in more detail in Section~\\ref{sec:asymptotic_expansions}.\nGiven this intriguing parallel between the numbers $\\chi(\\Out(F_n))$ and the Bernoulli numbers, which are very prominent objects in number theory, it would be interesting to look for a number theoretic meaning for the numbers $\\chi(\\Out(F_n))$ as well. The algorithm \ngiven in Proposition~\\ref{prop:efficient_chn} may be helpful for investigations in this direction. \n\nAs was pointed out in \\cite{SV87a}, non-vanishing of $\\chi(\\Out(F_n))$ implies that the kernel of the natural map from $\\Out(F_n)$ to $GL(n,\\mathbb{Z})$ does not have finitely-generated homology. Magnus proved in 1935 that this kernel is finitely generated and asked whether it is finitely presented, which would imply that the second homology is finitely generated \\cite{Magnus}. Bestvina, Bux and Margalit showed in 2007 that the homology in dimension $2n-4$ is not finitely generated \\cite{BBM}. However Magnus' question is still unanswered for $n>3$. \n\n\nTheorem~\\ref{thm:SVconj} implies that for large $n,$ torsion-free subgroups of finite index in $\\Out(F_n)$ have a huge amount of homology in odd dimensions. We would like to say the same is true for the whole group $\\Out(F_n)$. One way to prove this is to compare the asymptotic growth rate of $\\chi(\\Out(F_n))$ with that of the ``naive'' Euler characteristic $\\widetilde\\chi(\\Out(F_n))$. Brown \\cite{Brown82} showed that the difference between $\\widetilde \\chi$ and $\\chi$ can be expressed in terms of rational Euler characteristics of centralizers of finite-order elements of $\\Out(F_n)$. Harer and Zagier used this to compare the rational and naive Euler characteristics for surface mapping class groups, using the fact that centralizers of finite-order elements are basically mapping class groups of surfaces of lower complexity. Centralizers in $\\Out(F_n)$ are more difficult to understand, but preliminary results obtained by combining the methods of this paper with results of Krsti\\'{c} and Vogtmann \\cite{KrVo93} indicate that the ratio $\\widetilde\\chi(\\Out(F_n))\/ \\chi(\\Out(F_n))$ tends to a positive constant. Note that \nGalatius proved that the stable rational homology of $\\Out(F_n)$ vanishes \\cite{Galatius}, so this would show that there is a huge amount of {\\em unstable} homology in odd dimensions. This is completely mysterious, as only one odd-dimensional class has ever been found, by a very large computer calculation in rank $7$ \\cite{Bartholdi}, and this calculation gives no insight into where all of these odd-dimensional classes might come from. \n\n\n \n\n\nFinally we recall that, by the work of Kontsevich, the cohomology of $\\Out(F_n)$ with coefficients in a field of characteristic zero can be identified with the homology of the rank $n$ subcomplex of his Lie graph complex \\cite{kontsevich1993formal}. \nOur results therefore apply to the Euler characteristic of this graph complex as well. Kontsevich himself wrote down asymptotic formulas for the Euler characteristics of three of his graph complexes in \\cite{kontsevich1993formal}; see Chapter 5 of \\cite{gerlits2002} for a detailed derivation of these formulas. The connection with graph complexes is explained a little further in Section~\\ref{sec:kontsevich}. \nMore discussion of the relation of the current paper with ideas from topological quantum field theory---with further relations to Kontsevich's work---can be found in Section~\\ref{sec:tqft}. \n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe thank Dirk Kreimer for support during this project. \nMB would like to thank Karen Yeats and Sam Yusim for discussions on the subject.\nMB was supported by the NWO Vidi grant 680-47-551.\nKV would like to thank Peter Smillie for discussions. KV was partially supported by the Royal Society Wolfson Research Merit Award WM130098 and by a Humboldt Research Award.\n\n\n\\newcommand\\vertex[1]{\\fill #1 circle (.15)}\n\\newcommand{{\\rm A}}{{\\rm A}}\n\\section{Graphs and rational Euler characteristics}\n\nIn this section we give variations on the results of \\cite{SV87a}. The idea is to use the action of $\\Out(F_n)$ and closely related groups on contractible spaces of finite graphs\nto deduce information about the homology of the groups, including the rational Euler characteristic. \n\n\\subsection{Combinatorial graphs}\n\nWe begin with a combinatorial definition of a graph and related terms.\n\n\\begin{definition} A {\\em graph} $\\G$ consists of a finite set $H(\\G)$ called {\\em half-edges} together with\n\\begin{itemize}\n\\item a partition of $H(\\G)$ into parts called {\\em vertices} and\n\\item an involution $\\iota_\\G\\colon H(\\G)\\to H(\\G)$.\n\\end{itemize}\nThe {\\em valence} $|v|$ of a vertex $v$ is the number of half-edges in $v$. A {\\em leaf} of $\\G$ is a fixed point of the involution $\\iota_\\G$ and an {\\em edge} is an orbit of size $2$. \nA {\\em graph isomorphism} $\\G\\to\\G'$ is a bijection $H(\\G)\\to H(\\G')$ that preserves the vertex partitions and commutes with the involutions. \n\\end{definition} \n\n\\begin{notation} Let $\\G$ be a graph.\n\\begin{itemize}\n\\item $\\Aut(\\G)$ is the group of isomorphisms $\\G\\overset{\\cong}{\\rightarrow} \\G$. \n\\item $L(\\G)$ is the set of leaves of $\\G$, and $s(\\G)=|L(\\G)|$.\n\\item $E(\\G)$ is the set of edges of $\\G$ and $e(\\G)=|E(\\G)|$.\n\\item $V(\\G)$ is the set of vertices of $\\G$ and $v(\\G)=|V(\\G)|$.\n\\end{itemize}\n\\end{notation} \n\nThe graph with one vertex, $n$ edges, $s$ leaves and $2n+s$ half-edges is called a {\\em thorned rose}, and will be denoted $R_{n,s}$. If $n=0$ we will also call $R_{0,s}$ a {\\em star graph} (see Figure~\\ref{fig:graphs}).\n\nWith the exception of Section~\\ref{sec:core}, we will only consider {\\em admissible} graphs, where a graph is admissible if all vertices have valence at least $3$. \n\n\\begin{figure}\n\\begin{center}\n \\begin{tikzpicture} [thick, scale=.25] \\fill (0,0) circle (.2); \\draw (0,0) to [out=135, in=180] (0,4.1); \\draw (0,0) to [out=45, in=0] (0,4.1); \\draw (0,0) to [out=45, in=110] (4,1); \\draw [thick] (0,0) to [out=-30, in=-70] (4,1); \\draw (0,0) to [out=135, in=70] (-4,1); \\draw (0,0) to [out=210, in=-110] (-4,1); \\foreach \\x in {-120, -100, -80, -60} { \\draw (0,0) to (\\x:3); } \\begin{scope} [xshift = 15cm]; \\fill (0,0) circle (.2); \\foreach \\x in {0,72,144,216,288} { \\draw (0,0) to (\\x:3); } \\end{scope} \\end{tikzpicture}\n\\end{center}\n\\caption{Thorned rose $R_{3,4}$ and star graph $R_{0,5}$}\\label{fig:graphs}\n\\end{figure}\n \n\\begin{definition} Let $\\G$ be a graph. A {\\em subgraph} of $\\G$ is a graph $\\gamma$ with $H(\\gamma)=H(\\G)$, $V(\\gamma)=V(\\G)$, and $E(\\gamma)\\subseteq E(\\G)$.\n\\end{definition}\nA graph $\\G$ always has itself as a subgraph. There is a unique ``trivial'' subgraph $\\gamma_0$ with involution the identity, so $H(\\gamma_0)=H(\\G)$,$V(\\gamma_0)=V(\\G)$, $E(\\gamma_0)=\\emptyset$ and $L(\\gamma_0)=H(\\G).$\n\n\\subsection{Topological graphs} Every combinatorial graph $\\G$ has a {\\em topological realization} as a 1-dim\\-en\\-sional $CW$-complex. For each element of $V(\\G)$ we have a $0$-cell called a {\\em vertex} and for each element of $E(\\G)$ we have a $1$-cell called an {\\em edge}. For each element of $L(\\G)$ we have both a $0$-cell, called a {\\em leaf vertex} and a $1$-cell connecting the leaf vertex to a (non-leaf) vertex. \nBy our definition each connected component of the topological realization must have at least one vertex. \nNote that graphs may have multiple edges and they may have loops at a vertex. \nThe thorned rose $R_{n,s}$ defined in the last section has $(s+1)$ $0$-cells and $(n+s)$ 1-cells. \n\nThe valence of a point $x$ is the minimal number of components of a deleted neighborhood of $x$. \nIn an admissible graph the vertices are at least trivalent and the leaf vertices are univalent, so there are no bivalent $0$-cells. \n\nA graph isomorphism is a cellular homeomorphism, up to isotopy. Since admissible graphs have no bivalent $0$-cells, any homeomorphism is a cellular homeomorphism. \n\nNotice that, by our definition, the topological realization of a subgraph of $\\G$ is not a subcomplex of the topological realization of $\\G$. Rather, it can be described as a graph obtained from $\\G$ by cutting some of its edges, thus forming pairs of leaves. To make the result a $CW$-complex we have to add $0$-cells (leaf vertices) to the ends of the cut edges. A subgraph can also be visualized as the closure of a sufficiently small neighborhood of a subcomplex. \n \n In the remainder of this section we will work with the topological realization of a graph instead of using the combinatorial definition, so that we may freely use topological concepts such as connectivity, fundamental group and homotopy equivalence. \n\n\nFor $s\\geq 0$ let \n\\begin{itemize}\n\\item $\\GG$ denote the set of isomorphism classes of finite admissible graphs,\n\\item $\\GG_s\\subset \\GG$ be the subset consisting of admissible graphs with exactly $s$ leaves, \n\\item $\\GG^c \\subset \\GG$ and $\\GG_s^c \\subset \\GG_s$ be the respective subsets of connected graphs. \n\\end{itemize} \n \n\\subsection{Groups}\n\\label{sec:groups}\n\nFor any $n$ and $s$ we define ${\\rm A}_{n,s}$ to be the group of homotopy classes of homotopy equivalences of the thorned rose $R_{n,s}$ that fix the leaf vertices $\\{b_1,\\ldots,b_s\\},$ i.e.\\ ${\\rm A}_{n,s}=\\pi_0(HE(\\G, b_1,\\ldots,b_s))$. The groups ${\\rm A}_{n,s}$ generalize $\\Out(F_n) \\cong {\\rm A}_{n,0}$ and $\\Aut(F_n) \\cong {\\rm A}_{n,1}$. If $n=0$ then $R_{0,s}$ is a graph with no loops and at least $3$ leaves as we are insisting on at least one vertex, which is at least trivalent. So, ${\\rm A}_{0,s}$ is only defined for $s\\geq 3$, where it is the trivial group. If $n=1$ then $R_{1,s}$ is a loop with $s\\geq 1$ leaves, and there is a short exact sequence $$1\\to \\mathbb{Z}^{s-1}\\to {\\rm A}_{1,s}\\to \\mathbb{Z}\/2\\mathbb{Z}\\to 1.$$ \n If $n\\geq 2$ and $s\\geq 0$ there is a short exact sequence $$1\\to F_{n}^{s}\\to {\\rm A}_{n,s}\\to \\Out(F_{n})\\to 1.$$\n See \\cite{CHKV16} for background on the groups ${\\rm A}_{n,s}$.\nThese groups appear, for example, in the context of homology stability theorems \\cite{Hat95}, the bordification of Outer space and virtual duality \\cite{BF00, BSV18}, and assembly maps for homology \\cite{CHKV16}. \n \n\\subsection{Complexes of graphs and the rational Euler characteristic}\nIf a group $G$ is virtually torsion-free and acts properly and cocompactly on a contractible $CW$-complex $X$, then the rational Euler characteristic $\\chi(G)$ can be calculated using this action, by the formula $$\\chi(G)=\\sum_{\\sigma\\in \\mathcal C} \\frac{(-1)^{\\dim \\sigma}}{|\\text{Stab}(\\sigma)|},$$\nwhere $\\mathcal C$ is a set of representatives for the cells of $X$ mod $G$ (see, e.g.\\ \\cite{Bro82}, Proposition~(7.3)). \n \n \n For any $s\\geq 0$ the group ${\\rm A}_{n,s}$ is virtually torsion-free and acts properly and cocompactly on a contractible cube complex $K_{n,s}$. To describe $K_{n,s}$ it is convenient to label the leaves of a graph, so that two graphs $\\G$ and $\\G'$ are isomorphic if there is a graph isomorphism $\\G\\to\\G'$ that preserves leaf-labels; an isomorphism class is then called a {\\em leaf-labeled graph}. We use the notation $\\lG$, $\\lG_s$, $\\lG^c$ and $\\lG^c_s$ instead of $\\GG$, $\\GG_s$, $\\GG^c$ and $\\GG^c_s$ to denote the respective set of leaf-labeled graphs and $\\PAut(\\G)$ to denote the set of automorphisms of a graph that fix the leaves.\n \n \\begin{figure}\n \\begin{center}\n \\begin{tikzpicture} [scale=.55] \\draw (6.25,-1.75) to (6.25,3.5) to (11.75,3.5) to (11.75,-1.75) to (6.25,-1.75); \\begin{scope}[xshift=7.75cm, yshift=.25cm, scale=.4] \\coordinate (a) at (0,-.1); \\coordinate (b) at (3,4.5); \\coordinate (c) at (6,0); \\coordinate (one) at (-1.5,-.6); \\coordinate (two) at (7.5,-.6); \\coordinate (three) at (7.5,.6); \\draw (a) to (b) to (c) to (a); \\draw (a) to [out=90, in= 180 ] (b); \\draw[thick, red] (a) to (b) to (c); \\draw (a) to (one); \\node [left] (x) at (one) {$1$}; \\draw (c) to (two); \\node [below right] (x) at (two) {$3$}; \\draw (c) to (three); \\node [above right] (x) at (three) {$2$}; \\node (x) at (-.1,3.1) {$a$}; \\node (y) at (2.8,-.8) {$b$}; \\draw (0,2.4) to (.5,2.7) to (.75,2.3); \\draw (3,-.25) to (3.5,0) to (3,.25); \\vertex{(a)};\\vertex{(b)};\\vertex{(c)}; \\end{scope} \\begin{scope}[xshift=12.25cm, yshift=3.75cm,scale=.25] \\coordinate (a) at (0,0); \\coordinate (b) at (3,4.5); \\coordinate (c) at (6,0); \\draw (a) to (b) to (c) to (a); \\draw (a) to [out=90, in= 180 ] (b); \\draw (a) to (-1.5,-.6); \\draw (c) to (7.5,-.6);\\draw (c) to (7.5,.6); \\vertex{(a)};\\vertex{(b)};\\vertex{(c)}; \\end{scope} \\begin{scope}[xshift=12.5cm,yshift=.5cm, scale=.25] \\coordinate (a) at (0,0); \\coordinate (b) at (3,4.5); \\coordinate (c) at (6,0); \\draw (a) to (b) to (c) to (a); \\draw (a) to [out=90, in= 180 ] (b); \\draw[thick, red] (b) to (a); \\vertex{(a)};\\vertex{(b)};\\vertex{(c)}; \\draw (a) to (-1.5,-.6); \\draw (c) to (7.5,-.6); \\draw (c) to (7.5,.6); \\end{scope} \\begin{scope}[xshift=8cm, yshift=4cm,scale=.25] \\coordinate (a) at (0,0); \\coordinate (b) at (3,4.5); \\coordinate (c) at (6,0); \\draw (a) to (b) to (c) to (a); \\draw (a) to [out=90, in= 180 ] (b); \\draw[thick, red] (b) to (c); \\vertex{(a)};\\vertex{(b)};\\vertex{(c)}; \\draw (a) to (-1.5,-.6); \\draw (c) to (7.5,-.6); \\draw (c) to (7.5,.6); \\end{scope} \\begin{scope}[xshift=12cm, yshift=-2.75cm, scale=.25] \\coordinate (c) at (6,0); \\coordinate (n) at (1.5,2.25); \\coordinate(r) at (0,3); \\vertex{(c)};\\vertex{(n)}; \\draw (n) to[out=90, in=70] (r); \\draw (n) to [out=210, in=-110] (r); \\draw (c) to [out=190, in= -90] (n);\\draw (c) to [out=120, in= 20 ](n); \\draw (n) to (.5,.75);\\draw (c) to (7.5,-.6); \\draw (c) to (7.5,.6); \\end{scope} \\begin{scope}[xshift=8cm, yshift=-3cm, scale=.25] \\coordinate (c) at (6,0); \\coordinate (n) at (1.5,2.25); \\coordinate(r) at (0,3); \\draw (n) to[out=90, in=70] (r); \\draw (n) to [out=210, in=-110] (r); \\draw (c) to [out=190, in= -90] (n); \\draw [thick, red] (c) to [out=120, in= 20 ](n); \\vertex{(c)};\\vertex{(n)}; \\draw (n) to (.5,.75); \\draw (c) to (7.5,-.6); \\draw (c) to (7.5,.6); \\end{scope} \\begin{scope}[xshift=4.25cm, yshift=.5cm, scale=.25] \\coordinate (a) at (0,0);\\coordinate (m) at (4.5,2.25); \\vertex{(a)};\\vertex{(m)}; \\draw (a) to (m); \\draw[thick, red] (a) to (m); \\draw (a) to [out=80, in= 150] (m); \\draw (a) to [out=-15, in= -105 ](m); \\draw (a) to (-1.5,-.6); \\draw (m) to (6,3.5); \\draw (m) to (6.2,2); \\end{scope} \\begin{scope}[xshift=4.25cm, yshift=3.75cm, scale=.25] \\coordinate (a) at (0,0); \\coordinate (b) at (3,4.5); \\coordinate (c) at (6,0); \\coordinate (m) at (4.5,2.25); \\coordinate (n) at (1.5,2.25); \\vertex{(a)};\\vertex{(m)}; \\draw (a) to (m); \\draw (a) to [out=80, in= 150] (m); \\draw (a) to [out=-15, in= -105 ](m); \\draw (a) to (-1.5,-.6); \\draw (m) to (6,3.5); \\draw (m) to (6.2,2); \\end{scope} \\begin{scope}[xshift=5cm, yshift=-2.5cm, scale=.25] \\vertex{(0,0)}; \\draw (0,0) to [out=45, in=110] (4,1); \\draw (0,0) to [out=-30, in=-70] (4,1); \\draw (0,0) to [out=135, in=70] (-4,1); \\draw (0,0) to [out=210, in=-110] (-4,1); \\draw (0,0) to (0,-2); \\draw (0,0) to (-.5,2); \\draw (0,0) to (.5,2); \\end{scope} \\end{tikzpicture}\n \\caption{A $2$-dimensional cube $(\\G,\\varphi)$ in $K_{2,3}$. Here $\\varphi$ is the red subgraph, $\\pi_1(\\G)=F\\langle a, b\\rangle$ and the leaves are labeled $1,2,3$. The marking is indicated by arrows and labels on the edges in the complement of a maximal tree.}\\label{fig:cube}\n \\end{center}\n \\end{figure}\n\nThe cube complex $K_{n,s}$ has one cube for each equivalence class of triples $(\\G, \\varphi, g)$, where \n\\begin{itemize}\n \\item $\\G\\in \\lG^c_s$ is connected with $s$ labeled leaves and with $\\pi_1(\\G) \\cong F_n$,\n \\item $\\varphi$ is a \\textit{subforest} of $\\G,$ i.e.\\ a subgraph with no cycles, \n \\item $g\\colon R_{n,s}\\to \\G$ is a leaf-label-preserving homotopy equivalence, called a \\textit{marking} and\n \\item $(\\G,\\varphi,g)\\sim (\\G',\\varphi',g')$ if there is a leaf-label-preserving graph isomorphism $h\\colon\\G\\to\\G'$ sending $\\varphi$ to $\\varphi'$ such that $h\\circ g$ is homotopic to $g'$ through leaf-label-preserving homotopies.\n\\end{itemize}\nAn example of a cube in $K_{2,3}$ is depicted in Figure~\\ref{fig:cube}.\nContractibility of $K_{n,s}$ was proved for $s=0, n\\geq 2$ by Culler and Vogtmann \\cite{CV86} and in general by Hatcher \\cite{Hat95} (see also \\cite{HV96}). (For $n\\geq 2$, $K_{n,s}$ was originally described as a simplicial complex, but its simplices naturally group themselves into cubes, as was done, e.g.\\ in \\cite{HV98}.) \n\n Smillie and Vogtmann \\cite{SV87a} considered only the case $s=0,$ but their arguments apply verbatim for graphs with leaves. We define a function\n \\[\n\\tau(\\G)=\\sum_{\\varphi\\subset \\G} (-1)^{e(\\varphi)},\n\\] \nwhere the sum is over all subforests $\\varphi\\subset \\G,$ including the trivial subgraph, and $e(\\varphi)$ is the number of edges in $\\varphi$. \nFor instance, \n$\\tau(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=.7 of vm] (v0); \\coordinate [right=.7 of vm] (v1); \\draw (v0) circle(.7); \\draw (v1) circle(.7); \\filldraw (vm) circle (1pt); \\end{tikzpicture}%\n}\n) = 1$, \nas it has only the trivial subforest and\n$\\tau(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\draw (vm) to (-.3,-1.25);\\draw (vm) to (.3,-1.25); \\coordinate [left=.7 of vm] (v0); \\coordinate [right=.7 of vm] (v1); \\draw (v0) circle(.7); \\draw (v1) circle(.7); \\filldraw (vm) circle (1pt); \\end{tikzpicture}%\n}\n) = 1$ for the same reason (recall that a leaf is not an edge).\nWe have $\\tau(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (v0); \\coordinate [right=1.5 of v0] (v1); \\coordinate [left=.7 of v0] (i0); \\coordinate [right=.7 of v1] (o0); \\draw (v0) -- (v1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\draw (i0) circle(.7); \\draw (o0) circle(.7); \\end{tikzpicture}%\n}\n) = 0$, as it has two forests whose respective contributions to the sum cancel (in fact $\\tau$ always vanishes on graphs with a separating edge as ensured by Lemma~(2.3) of \\cite{SV87a}) and \n$\\tau(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=1 of vm] (v0); \\coordinate [right=1 of vm] (v1); \\draw (v0) -- (v1); \\draw (vm) circle(1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\end{tikzpicture}%\n}\n)=-2\n$, as the graph $\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=1 of vm] (v0); \\coordinate [right=1 of vm] (v1); \\draw (v0) -- (v1); \\draw (vm) circle(1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\end{tikzpicture}%\n}$ has four forests with contributions $-1,-1,-1$ and $1$.\n\nThe following theorem relates the rational Euler characteristics of the groups ${\\rm A}_{n,s}$ to the function $\\tau$. \n \\begin{theorem}\\label{thm:SV}%\n \\[\n \\chi({\\rm A}_{n,s})=\\sum \\frac{\\tau(\\G)}{|\\PAut(\\G)|},\n\\]\nwhere the sum is over all connected leaf-labeled graphs $\\Gamma$ with $s$ leaves and fundamental group $F_n$. \\end{theorem} \n \\begin{proof} \n The group ${\\rm A}_{n,s}$ acts properly and cocompactly on $K_{n,s}$. It acts transitively on markings, assuming the graphs are leaf-labeled, so there is one orbit of cubes for each isomorphism class of pairs $(\\G,\\varphi)\\in\\lG^c_s$ with fundamental group $F_n$. The dimension of this cube is $e(\\varphi)$, and the stabilizer of $(\\G,\\varphi, g)$ is isomorphic to the group $\\PAut(\\G,\\varphi)$ of automorphisms of $\\G$ that fix the leaves and send $\\varphi$ to itself. Therefore we have\n $$\\chi({\\rm A}_{n,s})=\\sum_{\\sigma\\in \\mathcal C} \\frac{(-1)^{\\dim \\sigma}}{|\\text{Stab}(\\sigma)|}=\\sum_{(\\G,\\varphi)} \\frac{(-1)^{e(\\varphi)}}{|\\PAut(\\G,\\varphi)|},$$\n where the sum is over all isomorphism classes of pairs $(\\G,\\varphi)$ of leaf-labeled graphs $\\G\\in \\lG_s^c$ and forests $\\varphi\\subset \\G$. The full automorphism group $\\PAut(\\G)$ acts on the set of forests in $\\G$, and an orbit is an isomorphism class of pairs $(\\G,\\varphi)$, so the orbit-stabilizer theorem now gives\n $$\\sum_{(\\G,\\varphi)} \\frac{(-1)^{e(\\varphi)}}{|\\PAut(\\G,\\varphi)|} = \\sum_{\\G\\in\\lG^c_s} \\sum_{ \\varphi\\subset \\G} \\frac{(-1)^{e(\\varphi)}}{|\\PAut(\\G)|}=\\sum_{\\G\\in\\lG^c_s} \\frac{\\tau(\\G)}{|\\PAut(\\G)|}$$\n as desired.\n \n \n Note that for $n\\geq 2$ and $s=0$ we have $\\lG_0^c=\\GG_0^c,$ ${\\rm A}_{n,0} \\cong \\Out(F_n)$ and $\\PAut(\\G) \\cong \\Aut(\\G)$, so this is Proposition~(1.12) of \\cite{SV87a}. \n \\end{proof}\n \n\n\\begin{example} \nUsing this theorem we can immediately verify that\n\\begin{gather*}\n\\chi({\\rm A}_{2,0}) = \\chi(\\Out(F_2))= \\\\\n\\frac{\\tau(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=.7 of vm] (v0); \\coordinate [right=.7 of vm] (v1); \\draw (v0) circle(.7); \\draw (v1) circle(.7); \\filldraw (vm) circle (1pt); \\end{tikzpicture}%\n}\n)}{|\\PAut(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=.7 of vm] (v0); \\coordinate [right=.7 of vm] (v1); \\draw (v0) circle(.7); \\draw (v1) circle(.7); \\filldraw (vm) circle (1pt); \\end{tikzpicture}%\n}\n)|}\n+\n\\frac{\\tau(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (v0); \\coordinate [right=1.5 of v0] (v1); \\coordinate [left=.7 of v0] (i0); \\coordinate [right=.7 of v1] (o0); \\draw (v0) -- (v1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\draw (i0) circle(.7); \\draw (o0) circle(.7); \\end{tikzpicture}%\n}\n)}{|\\PAut(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (v0); \\coordinate [right=1.5 of v0] (v1); \\coordinate [left=.7 of v0] (i0); \\coordinate [right=.7 of v1] (o0); \\draw (v0) -- (v1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\draw (i0) circle(.7); \\draw (o0) circle(.7); \\end{tikzpicture}%\n}\n)|}\n+\n\\frac{\\tau(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=1 of vm] (v0); \\coordinate [right=1 of vm] (v1); \\draw (v0) -- (v1); \\draw (vm) circle(1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\end{tikzpicture}%\n}\n)}{|\\PAut(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=1 of vm] (v0); \\coordinate [right=1 of vm] (v1); \\draw (v0) -- (v1); \\draw (vm) circle(1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\end{tikzpicture}%\n}\n)|}\n=\n\\frac{1}{8} + \\frac{0}{8} + \\frac{-2}{12} = \n-\\frac{1}{24}.\n\\end{gather*}\n\n\\end{example}\n \n \n \\begin{corollary}\\label{cor:SV}\n $$ \\frac{\\chi({\\rm A}_{n,s})}{s!}= \\sum_{\\substack{\\G\\in\\GG^c_{s}\\\\ \\pi_1(\\G) \\cong F_n}} \\frac{\\tau(\\G)}{|\\Aut(\\G)|}.$$\n \\end{corollary}\n\n %\n %\n \\begin{proof} \n $\\Aut(\\G)$ acts on the set of leaf-labelings of $\\G \\in \\lG_s^c$. The orbits are the leaf-labeled graphs, and the stabilizer of a labeling is $\\PAut(\\G)$, giving $|\\Aut(\\G)|=|\\text{Orbit}(\\G)| |\\PAut(\\G)|. $ There are $s!$ labelings of $\\G.$ Each orbit has the same size, so the size of each orbit is $s!\/\\ell(\\G)$, where $\\ell(\\G)$ is the number of leaf-labeled graphs with underlying graph $\\G$. Therefore,\n \\begin{align*} \n |\\Aut(\\G)|&=\\frac{s!}{\\ell(\\G)} |\\PAut(\\G)|. \\qedhere\n \\end{align*}\n \\end{proof}\n\n\n\\section{Formal power series} \nFor the rest of this article it will be convenient to use $|\\G|=e(\\G)-v(\\G)$ instead of the rank of $\\pi_1(\\G)$ to filter the set of graphs. For connected graphs $\\G$ this is only a minor change of notation as $\\text{rank}(\\pi_1(\\G)) = e(\\G)-v(\\G) +1 =|\\G|+1$. %\n \nConsistent with this shift, we define $\\ch_n = \\chi(\\Out(F_{n+1}))$ and consider the formal power series\n\\begin{align*}\nT(z)=\\sum_{n=1}^\\infty \\chi(\\Out(F_{n+1})) z^n=\\sum_{n=1}^\\infty \\ch_n z^n.\n\\end{align*}\nBy Theorem~\\ref{thm:SV} with $s=0$ we have \n\\begin{align}\n\\label{eqn:def_Tz_cntd_graph_sum}\nT(z)=\\sum_{n=1}^\\infty \\left(\\sum_{\\substack{\\G\\in\\GG^c_0\\\\ \\pi_1(\\G) \\cong F_{n+1}}}\\frac{\\tau(\\G)}{|\\Aut(\\G)|}\\right) z^n = \\sum_{\\G\\in\\GG^c_0}\\frac{\\tau(\\G)}{|\\Aut(\\G)|}z^{|\\G|}.\n\\end{align}\nFor general $A_{n,s}$ we define a bivariate generating function for the Euler characteristic by\n\\begin{align}\n\\label{eqn:def_Tzx_Ans}\nT(z,x)= \\sum_{s\\geq 3} \\ch({\\rm A}_{0,s})z^{-1}\\frac{x^s}{s!} + \\sum_{s\\geq 1} \\ch({\\rm A}_{1,s}) \\frac{x^s}{s!} + \\sum_{n\\geq 1}\\sum_{s\\geq 0} \\ch({\\rm A}_{n+1,s})z^n\\frac{x^s}{s!}.\n\\end{align}\nRecall that the groups ${\\rm A}_{n,s}$ are only defined for $s \\geq 3$ if $n=0$, for $s \\geq 1$ if $n=1$ and for $s\\geq 0$ if $n \\geq 2$. \nObviously, $T(z,0) = T(z)$ and by Corollary~\\ref{cor:SV}\n\\begin{align}\n\\label{eqn:def_Tzx_cntd_graph_sum}\nT(z,x)=\\sum_{\\G\\in\\GG^c}\\frac{\\tau(\\G)}{|\\Aut(\\G)|}z^{|\\G|}x^{s(\\G)},\n\\end{align}\nwhere $s(\\G)$ is the number of leaves in $\\G$. \n\nThe relationships between the groups ${\\rm A}_{n,s}$, which were described in Section~\\ref{sec:groups}, imply the following functional relation between $T(z)$ and the bivariate generating function $T(z,x)$.\n \n\\begin{proposition} \n \\label{prop:Tzx_leaves_identity}\n $$T(z,x) = \\frac{e^x-\\frac{x^2}{2}-x-1}{z} + \\frac{x}{2} + T(ze^{-x}).$$\n\\end{proposition}\n\n\\begin{proof} \nThe groups ${\\rm A}_{0,s}$ are trivial, so we have $\\ch({\\rm A}_{0,s})=1$ for all $s \\geq 3$. \n For the groups ${\\rm A}_{1,s}$ we have the short exact sequence $$1\\to \\mathbb{Z}^{s-1}\\to {\\rm A}_{1,s}\\to \\mathbb{Z}\/2\\mathbb{Z}\\to 1.$$ \nThus $\\chi({\\rm A}_{1,s})=0$ if $s\\geq 2$ and $\\chi({\\rm A}_{1,1})=\\chi(\\mathbb{Z}\/2\\mathbb{Z})=\\frac{1}{2}$.\nFor the groups ${\\rm A}_{n+1,s}$ with $n\\geq 1$ the short exact sequence $$1\\to F_{n+1}^{s}\\to {\\rm A}_{n+1,s}\\to \\Out(F_{n+1})\\to 1,$$\ngives $\\chi({\\rm A}_{n+1,s})=\\chi(\\Out(F_{n+1}))(-n)^{s}=\\ch_n (-n)^{s}$. \n\nPutting these together into eq.~\\eqref{eqn:def_Tzx_Ans} gives \n\\begin{align*}\n T(z,x)&=\\sum_{s\\geq 3} \\frac{x^s}{s!}z^{-1} + \\frac{x}{2} +\\sum_{n\\geq 1}\\sum_{s\\geq 0} \\ch_n\\frac{ (-n)^s}{s!} z^nx^s \\\\\n & =\\sum_{s\\geq 3} \\frac{x^s}{s!}z^{-1} + \\frac{x}{2} +\\sum_{n\\geq 1}\\ch_n \\sum_{s\\geq 0} \\frac{(-n)^s}{s!} x^s z^n \\\\\n & =\\sum_{s\\geq 3} \\frac{x^s}{s!}z^{-1} + \\frac{x}{2} +\\sum_{n\\geq 1}\\ch_n e^{-nx} z^n\\\\\n &=\\sum_{s\\geq 3} \\frac{x^s}{s!}z^{-1} + \\frac{x}{2} + T(ze^{-x}). \\qedhere\n\\end{align*}\n\\end{proof}\n\n\n\n \n\n\\section{Algebraic graph combinatorics}\n\\label{sec:graphical_enumeration}\n\nAlthough Theorem~\\ref{thm:SV} gives an explicit expression for the coefficients of $T(z)$ and $T(z,x)$, we will use an implicit equation, which the generating function $T(z,x)$ must satisfy, to prove Theorem \\ref{thm:asymptotic_expansion}. This implicit equation together with the identity from Proposition~\\ref{prop:Tzx_leaves_identity} determines the coefficients $\\chi(\\Out(F_n))$ completely.\n\nTo formulate this implicit equation, it is convenient to use the \\textit{coefficient extraction operator} notation: For an arbitrary formal power series $f(x)$ the notation $[x^n] f(x)$ means `the $n$-th coefficient in $x$ of $f(x)$.'\n\n\\begin{proposition}\n\\label{prop:Tzx_graph_counting_identity}\n\\begin{align}\n\\label{eqn:Tzx_implicit_equation}\n 1 &= \\sum_{\\ell \\geq 0} (-z)^\\ell (2\\ell-1)!! [x^{2\\ell}] \\exp\\left( T(z,x) \\right),\n\\end{align}\nwhere $(2\\ell-1)!!=(2\\ell)! \/ ( \\ell! 2^\\ell )$ is the double factorial.\n\\end{proposition}\nIn the remainder of this section we will first give a combinatorial reformulation of this identity and then prove it. \n\n\n\\subsection{The exponential formula}\nThe exponential of the generating function $T(z,x)$ in \\eqref{eqn:Tzx_implicit_equation} has a straightforward combinatorial interpretation. While $T(z,x)$ can be expressed as a sum over connected graphs as we did in \\eqref{eqn:def_Tzx_cntd_graph_sum}, the generating function $\\exp(T(z,x))$ can be expressed as a sum over all graphs.\nThe reason for this is that the function $\\tau$ is multiplicative on disjoint unions of graphs, so we can apply the \\textit{exponential formula} given, for example, in \\cite[5.1]{stanley1997enumerative2}.\n\nBriefly, the argument behind the exponential formula is that if $\\phi$ is a function on graphs that is multiplicative on disjoint unions, i.e.\\ $\\phi(\\G_1 \\sqcup \\G_2) = \\phi(\\G_1) \\phi(\\G_2)$, then \n\\begin{align*}\n\\sum_{\\substack{\\G \\in \\GG \\\\ |C(\\G)| = n}} \\frac{\\phi(\\G)}{|\\Aut\\G|} = \\frac{1}{n!} \\sum_{\\gamma_1, \\ldots, \\gamma_n \\in \\GG^c} \\prod_{i=1}^n \\frac{\\phi(\\gamma_i)}{|\\Aut \\gamma_i|},\n\\end{align*}\nwhere we sum over all graphs with $n$ connected components on the left hand side and over all $n$ tuples of connected graphs on the right hand side. The factor $1\/n!$ accounts for the number of ways to order the connected components of the graphs. If we sum this equation over all $n \\geq 0$ and use $e^x = \\sum_{n \\geq 0} x^n\/n!$, we get\n\\begin{lemma}[Exponential formula]\n\\label{lmm:exponential_formula}\nLet $\\phi$ be a function from graphs to a power series ring that is multiplicative on disjoint unions (i.e.\\ $\\phi(\\G_1 \\sqcup \\G_2) = \\phi(\\G_1) \\phi(\\G_2)$). If the coefficient in $\\phi(\\G)$ of a given monomial is non-zero for only finitely many graphs $\\G$, then \n\\begin{align*}\n\\sum_{\\G \\in \\GG} \\frac{\\phi(\\G)}{|\\Aut\\G|}\n= \\exp\\left( \n\\sum_{\\G \\in \\GG^c} \\frac{\\phi(\\G)}{|\\Aut\\G|}\n\\right).\n\\end{align*}\n\\end{lemma}\nThe finiteness condition on the function $\\phi$ is necessary to ensure that $\\sum_{\\G \\in \\GG} \\frac{\\phi(\\G)}{|\\Aut\\G|}$ exists in the respective power series space.\n \n\\begin{corollary}\n\\label{crll:disconnected_exp}\n\\begin{align*}\n\\exp({T(z)}) &= \\sum_{\\G\\in\\GG_0}\\frac{\\tau(\\G)}{|\\Aut(\\G)|}z^{|\\G|}\\\\\n\\exp({T(z,x)}) &= \\sum_{\\G\\in\\GG}\\frac{\\tau(\\G)}{|\\Aut(\\G)|}z^{|\\G|}x^{s(\\G)}.\n\\end{align*}\n\\end{corollary}\n\n\n\\begin{proof}\nLet $\\phi_1$ be the function defined by $\\phi_1(\\G) = \\tau(\\G) z^{|\\G|}$ for $\\G \\in \\GG_0$ and $\\phi_1(\\G)=0$ for $\\G \\in \\GG_s$ with $s \\geq 1$. This function is multiplicative on disjoint unions of graphs, because $\\tau$ is (Lemma~(2.2) of \\cite{SV87a}) and both $|\\G|$ and $s(\\G)$ are additive graph invariants. The first statement follows by applying Lemma~\\ref{lmm:exponential_formula} to $\\phi_1$ and using eq.\\ \\eqref{eqn:def_Tz_cntd_graph_sum}. For the second statement apply Lemma~\\ref{lmm:exponential_formula} to the function $\\phi_2(\\G) = \\tau(\\G) z^{|\\G|} x^{s(\\G)}$ for all $\\G \\in \\GG$, note that there are only a finite number of admissible graphs with fixed $|\\G|$ and $s(\\G)$ and apply eq.\\ \\eqref{eqn:def_Tzx_cntd_graph_sum}. \n\\end{proof}\nNote that the power series $T(z)$ and $\\exp(T(z))$ carry the same information. \nRecall that $\\ch_n$ is the coefficient of $z^n$ in $T(z)$, and denote the coefficient of $z^n$ in $\\exp(T(z))$ by $\\Ch_n$. \nThe coefficients $\\ch_n$ and $\\Ch_n$ are related by\n\\begin{align}\n\\label{eqn:exp_pwrsrs_relation}\n \\ch_n&=\\Ch_n-\\frac{1}{n}\\sum_{k=1}^{n-1}k \\ch_k \\Ch_{n-k} \\text{ for } n \\geq 1.\n\\end{align}\nThis recursive relation can be derived by taking the formal derivative of $\\exp(T(z))$ which results in the (formal) differential equation $\\frac{d}{dz}\\exp({T(z)}) = e^{T(z)} \\frac{d}{dz}T(z)$. Note that it is also important that $\\ch_0 = 0$ for $\\exp(T(z))$ to make sense as a power series with $\\mathbb{Q}$ as coefficient ring. \n\nWe can immediately use the relationship between the coefficients $\\Ch_n$ and $\\ch_n$ to prove the following statement which will be helpful later while proving that the rational Euler characteristic of $\\Out(F_n)$ is always negative.\n\\begin{lemma}\n\\label{lmm:exp_negative}\nIf $\\Ch_n < 0$ for all $n \\geq 1$, then $\\ch_n < 0$ for all $n \\geq 1$.\n\\end{lemma}\n\\begin{proof}\nThis follows by induction on $n$ on eq.\\ \\eqref{eqn:exp_pwrsrs_relation}. \n\\end{proof}\nBecause $\\ch_n = \\chi(\\Out(F_{n+1}))$, proving $\\Ch_n < 0$ for all $n\\geq 1$ is therefore sufficient to show that $\\chi(\\Out(F_n)) < 0$ for all $n\\geq 2$.\n\n\\subsection{Convolution identities}\n\nBy Corollary~\\ref{crll:disconnected_exp}, the statement of Proposition~\\ref{prop:Tzx_graph_counting_identity} is equivalent to the identity\n\\begin{align}\n\\label{eqn:graph_sum_tau_identity}\n1=\\sum_{\\ell\\geq 0} (-z)^\\ell(2\\ell-1)!! \\sum_{\\G\\in\\GG_{2\\ell}} \\frac{\\tau(\\G)}{|\\Aut(\\G)|} z^{|\\G|}.\n\\end{align}\n\nIf $\\gamma$ is a subgraph of $\\G$, we denote by $\\G\/\\gamma$ the graph that one obtains from $\\G$ by collapsing each edge that is in $\\gamma$ to a point.\nWe will use the following \\textit{convolution identity} for $\\tau$ to prove eq.\\ \\eqref{eqn:graph_sum_tau_identity} and therefore also Proposition~\\ref{prop:Tzx_graph_counting_identity}. \n\\begin{proposition}\n\\label{prop:tau_identity}\nIf $\\G$ is a graph with at least one cycle, then \n\\begin{align*}\n \\sum_{\\gamma \\subset \\G} \\tau(\\gamma) (-1)^{e(\\G\/\\gamma)} = 0.\n\\end{align*}\nwhere the sum is over all subgraphs of $\\G$, including the trivial subgraph and $\\G$ itself. \n\\end{proposition}\n\n This statement can be seen as an identity in the incidence algebra of the subgraph poset of a graph. We will discuss a related viewpoint in Section~\\ref{sec:core}, where we will interpret it as an inverse relation in the group of characters of the Hopf algebra of core graphs.\n\n\\begin{proof}\nRecall that $\\tau(\\G) = \\sum_{\\varphi \\subset \\G} (-1)^{e(\\varphi)}$ where the sum is over all subforests of $\\G$. Therefore,\n\\begin{gather*}\n \\sum_{\\gamma \\subset \\G} \\tau(\\gamma) (-1)^{e(\\G\/\\gamma)}\n =\n (-1)^{e(\\G)} \\sum_{\\gamma \\subset \\G} \\tau(\\gamma) (-1)^{e(\\gamma)}\n =\n \\\\\n (-1)^{e(\\G)} \\sum_{\\gamma \\subset \\G} \\sum_{\\substack{ \\varphi \\subset \\gamma\\\\ \\text{ forest } \\varphi}} (-1)^{e(\\varphi)}(-1)^{e(\\gamma)}\n =\n (-1)^{e(\\G)} \\sum_{\\substack{ \\varphi \\subset \\G\\\\ \\text{ forest } \\varphi}} (-1)^{e(\\varphi)} \\sum_{\\substack{\\gamma \\subset \\G\\\\ \\gamma \\supset \\varphi}} (-1)^{e(\\gamma)}.\n\\end{gather*}\nThe set of subgraphs of $\\G$ containing $\\varphi$ is in bijection with the set of subsets of $E(\\G) \\setminus E(\\varphi)$. Because $\\G$ has at least one cycle, $E(\\G) \\setminus E(\\varphi)$ is not empty and the alternating sum \n\\begin{align*}\n\\sum_{\\substack{\\gamma \\subset \\G\\\\ \\gamma \\supset \\varphi}} (-1)^{e(\\gamma)} \n&= \n(-1)^{e(\\varphi)}\\sum_{E' \\subset E(\\G) \\setminus E(\\varphi)} (-1)^{|E'|} = 0. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary} \n\\label{crll:tau_identity_summed}\n\\begin{align*}\n1= \\sum_{\\G\\in \\GG_0} \\sum_{\\gamma\\subset\\G} \\frac{\\tau(\\gamma)(-1)^{e(\\G\/\\gamma)}}{|\\Aut(\\G)|}z^{|\\G|}.\n\\end{align*}\n\\end{corollary}\n\\begin{proof} Since all non-trivial graphs in $\\GG_0$ have cycles, Proposition~\\ref{prop:tau_identity} implies that the only non-zero contribution to the sum comes from the empty graph.\n\\end{proof}\n\nTo eventually obtain the statement of Proposition~\\ref{prop:Tzx_graph_counting_identity}, we transform this identity using the following proposition, which is an elementary application of labeled counting.\n\n\n\n\n\\begin{proposition}\n \\label{prop:convoluted_graph_sum}\nLet $\\phi$ be a function from graphs to a formal power series ring such that for each monomial $m$ and each integer $\\ell\\geq 0$, the coefficient of $m$ in $\\phi(\\G)$ is non-zero for only finitely many graphs $\\G\\in \\GG_{2\\ell}.$ \nThen\n $$\\sum_{\\G\\in \\GG_0} \\sum_{\\gamma\\subset\\G} \\frac{\\phi(\\gamma)w^{e(\\G\/\\gamma)}}{|\\Aut(\\G)|}=\\sum_{\\ell\\geq 0} w^\\ell (2\\ell-1)!! \\sum_{\\gamma \\in \\GG_{2\\ell}} \\frac{\\phi(\\gamma)}{|\\Aut \\gamma|},$$\n where $w$ is a formal variable. \n\\end{proposition}\n\n\\begin{proof} \n To prove the proposition we will use (totally) labeled graphs. \n Here a {\\em labeling} of $\\G$ with $e(\\G)$ edges, $v(\\G)$ vertices and $s(\\G)$ leaves consists of \n\\begin{itemize}\n\\item ordering the edges, i.e.\\ labeling them $1,\\ldots,e(\\G)$,\n\\item orienting each edge, \n\\item ordering the vertices, i.e.\\ labeling them $1,\\ldots,v(\\G)$,\n\\item ordering the leaves, i.e.\\ labeling them $1,\\ldots,s(\\G)$. \n\\end{itemize}\nThe set of labeled graphs with $s$ leaves will be denoted $\\LG_s$. \n\n\nThe advantage of using labeled graphs instead of unlabeled graphs is that a sum of terms $1\/|\\Aut(\\G)|$ over unlabeled graphs on $v$ vertices and $e$ edges becomes a sum of $1\/(v!e!2^e)$ over labeled graphs using the orbit-stabilizer theorem. The group $\\Aut(\\G)$ acts on the set of labelings of $\\G$, an orbit is a labeled graph and all stabilizers are trivial. This simplifies expressions that involve these automorphism groups. In particular, abbreviating $v=v(\\G), e=e(\\G)$ and $d=e(\\gamma)$ we have\n \\begin{align*}\n \\sum_{\\G\\in \\GG_0} \\sum_{\\gamma\\subset\\G} \\frac{\\phi(\\gamma)w^{e(\\G\/\\gamma)}}{|\\Aut(\\G)|}\n=\\sum_{\\G\\in \\LG_0} \\sum_{\\gamma\\subset\\G}\\frac{w^{e-d}}{e!v!2^{e}}\\phi(\\gamma).\n\\end{align*}\nA subgraph $\\gamma$ inherits a labeling from $\\G$: the vertices are the same, so they have the same labels. The ordering and orientation on the edges of $\\G$ induces an ordering and orientation on the edges of $\\gamma$, giving a labeling on these. The edges not in $\\gamma$ also have an induced ordering, and we use that to order the leaves of $\\gamma$ by the following rule: If there are $\\ell$ edges in $E(\\G)\\setminus E(\\gamma)$, label the leaf corresponding to the initial half of the $i$-th edge by $i$, and the leaf corresponding to the terminal half by $i+\\ell$. \n\nWe now change the order of summation. Remembering that $\\gamma$ has an even number of leaves, we get\n\\begin{align*}\n\\sum_{\\G\\in \\LG_0} \\sum_{\\gamma\\subset\\G}\\frac{w^{e-d}}{e!v!2^{e}}\\phi(\\gamma)&=\n \\sum_{\\ell\\geq 0}\\sum_{\\gamma\\in\\LG_{2\\ell}} \\sum_{\\G\\in \\LG_0, \\G\\supset \\gamma}\\frac{w^{e-d}}{e!v!2^{e}}\\phi(\\gamma)\\\\\n&=\\sum_{\\ell\\geq 0}\\sum_{\\gamma\\in\\LG_{2\\ell}} \\sum_{\\G\\in \\LG_0, \\G\\supset \\gamma}\\frac{w^{\\ell}}{(\\ell+d)!v!2^{\\ell+d}}\\phi(\\gamma).\n \\intertext{\nWe next note that a labeling on $\\gamma$ specifies an isomorphism type of $\\Gamma$ containing $\\gamma$, using the rule that the $i$-th leaf should be connected to the $(i+\\ell)$-th leaf. This also orders the edges in $E(\\Gamma)\\setminus E(\\gamma)$ and orients them from $i$ to $i+\\ell$. The edges of $\\gamma\\subset \\G$ are ordered and oriented as subsets of $E(\\gamma)$. There are $\\binom{d+\\ell}{\\ell}$ ways of shuffling the two orderings to get a total ordering on $E(\\G)$ that induces the given orderings on $E(\\gamma)$ and $E(\\G)\\setminus E(\\gamma)$. Thus the last expression becomes \n}\n &=\\sum_{\\ell\\geq 0}\\sum_{\\gamma\\in\\LG_{2\\ell}} \\binom{d+\\ell}{\\ell} \\frac{w^{\\ell}}{(d+\\ell)!v!2^{d+\\ell}}\\phi(\\gamma) \\\\\n &=\\sum_{\\ell\\geq 0} \\frac{w^\\ell}{2^\\ell}\\sum_{\\gamma\\in\\LG_{2\\ell}} \\frac{(d+\\ell)!}{\\ell!d!}\\frac{1}{(d+\\ell)!v!2^{d}}\\phi(\\gamma) \\\\\n &=\\sum_{\\ell\\geq 0} \\frac{1}{\\ell!2^\\ell} w^\\ell \\sum_{\\gamma\\in\\LG_{2\\ell}} \\frac{\\phi(\\gamma)}{d!v!2^{d}}.\n \\intertext{\nWe now translate back to unlabeled graphs to get \n}\n& =\\sum_{\\ell \\geq 0} \\frac{1}{\\ell!2^\\ell} w^\\ell \\sum_{\\gamma\\in\\GG_{2\\ell}} \\frac{\\phi(\\gamma)(2\\ell)!}{|\\Aut(\\gamma)|}\\\\\n& =\\sum_{\\ell \\geq 0} \\frac{(2\\ell)!}{\\ell!2^\\ell} w^\\ell \\sum_{\\gamma\\in\\GG_{2\\ell}} \\frac{\\phi(\\gamma)}{|\\Aut(\\gamma)|}\\\\\n& =\\sum_{\\ell \\geq 0}(2\\ell-1)!! w^\\ell \\sum_{\\gamma\\in\\GG_{2\\ell}} \\frac{\\phi(\\gamma)}{|\\Aut(\\gamma)|}.\n\\qedhere\n\\end{align*}\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:Tzx_graph_counting_identity}]\n Use Proposition~\\ref{prop:convoluted_graph_sum} with $\\phi(\\Gamma) = \\tau(\\Gamma) z^{|\\Gamma|}$ and $w = -z$, together with the observation that $(-1)^{e(\\G\/\\gamma)}z^{|\\G|}=z^{|\\gamma|}(-z)^{e(\\G)-e(\\gamma)}$. We get\n$$\n \\sum_{\\G \\in \\GG_0} \\sum_{\\gamma \\subset \\G} \\frac{ \\tau(\\gamma)(-1)^{e({\\G\/\\gamma})}}{|\\Aut \\G|}z^{|\\G|} \n =\\sum_{\\ell\\geq 0} (-z)^\\ell(2\\ell-1)!! \\left(\\sum_{\\gamma\\in\\GG_{2\\ell}} \\frac{\\tau(\\gamma)}{|\\Aut(\\gamma)|} z^{|\\gamma|}\\right).\n$$\nApply Corollary~\\ref{crll:tau_identity_summed} to obtain eq.\\ \\eqref{eqn:graph_sum_tau_identity} and Corollary~\\ref{crll:disconnected_exp} after that.\n\\end{proof}\n\\section{The Hopf algebra of core graphs}\\label{sec:core}\n\nA graph with no separating edges is called a {\\it core graph}, {\\it bridgeless} or {\\it 1-particle irreducible graph}. Let ${\\mathbf{H}}$ denote the $\\mathbb{Q}$-vector space generated by all finite core graphs. In contrast to the rest of the article, we will include graphs with bivalent edges as generators of ${\\mathbf{H}}$. \nThe vector space ${\\mathbf{H}}$ can be made into an algebra whose multiplication is induced by disjoint union of generators; here we identify all graphs with no edges with the neutral element $\\mathbb{I}$ for this multiplication. (Thus a topological graph representing the neutral element is a (possibly empty) disjoint union of isolated vertices and star graphs.)\n\nThe algebra ${\\mathbf{H}}$ can also be equipped with a coproduct $\\Delta \\colon {\\mathbf{H}} \\to {\\mathbf{H}} \\otimes {\\mathbf{H}}$, defined by \n\\begin{align}\n \\label{eqn:coproduct}\n\\Delta(\\G)=\\sum_{\\gamma\\subset\\G} \\gamma\\otimes \\G\/\\gamma,\n\\end{align}\nwhere the sum is over all core subgraphs of $\\G$.\n\n\\begin{example}\nThe graph \n$\n{\n\\def 2ex {1.3ex}\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v0) to[bend left=20] (v2); \\draw (v0) to[bend right=20] (v3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) to[bend right=60] (v3); \\draw (v2) to[bend left=60] (v3); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n}\n$ has seven mutually non-isomorphic core subgraphs---including the trivial subgraph graph (identified with $\\mathbb{I}$) and the graph $\n{\n\\def 2ex {1.3ex}\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v0) to[bend left=20] (v2); \\draw (v0) to[bend right=20] (v3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) to[bend right=60] (v3); \\draw (v2) to[bend left=60] (v3); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n}\n$ itself.\nThe coproduct is given by \n{\n\\def 2ex {2ex}\n\\begin{gather*}\n\\Delta \n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v0) to[bend left=20] (v2); \\draw (v0) to[bend right=20] (v3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) to[bend right=60] (v3); \\draw (v2) to[bend left=60] (v3); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n=\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v0) to[bend left=20] (v2); \\draw (v0) to[bend right=20] (v3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) to[bend right=60] (v3); \\draw (v2) to[bend left=60] (v3); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n\\otimes\n\\mathbb{I}\n+\n2~\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\coordinate[above =.5 of v2] (e2); \\coordinate[below =.5 of v3] (e3); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v2) -- (e2); \\draw (v3) -- (e3); \\draw (v0) to[bend left=20] (v2); \\draw (v0) to[bend right=20] (v3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) -- (v3); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n\\otimes \n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v) ; \\def .4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture {.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\coordinate (i2) at ([shift=(255:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\coordinate (i3) at ([shift=(285:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\coordinate (i4) at ([shift=(315:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture}\n+\n2~\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\coordinate[above left=.5 of v2] (e2); \\coordinate[below left=.5 of v3] (e3); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v2) -- (e2); \\draw (v3) -- (e3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) to[bend right=60] (v3); \\draw (v2) to[bend left=60] (v3); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n\\otimes \n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.4 of v0] (m); \\coordinate[right=.4 of m] (v1); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture}\n+\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\coordinate[above left=.5 of v2] (e2); \\coordinate[below left=.5 of v3] (e3); \\coordinate[above right=.5 of v2] (h2); \\coordinate[below right=.5 of v3] (h3); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v2) -- (h2); \\draw (v3) -- (h3); \\draw (v2) -- (e2); \\draw (v3) -- (e3); \\draw (v0) to[bend left=20] (v2); \\draw (v0) to[bend right=20] (v3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n\\otimes \n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v) ; \\def .8}; \\def \\rud {.4}; \\coordinate (s1) at ([shift=(30:\\rad)]v); \\coordinate (s2) at ([shift=(150:\\rad)]v); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\draw (v) to[out=90,in=120] (s1) to[out=-60,in=0-30] (v); \\draw (v) to[out=210,in=240] (s2) to[out=60,in=90] (v); \\filldraw (v) circle (1pt); \\end{scope} \\end{tikzpicture {.8}; \\def .4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture {.4}; \\coordinate (s1) at ([shift=(30:.8}; \\def \\rud {.4}; \\coordinate (s1) at ([shift=(30:\\rad)]v); \\coordinate (s2) at ([shift=(150:\\rad)]v); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\draw (v) to[out=90,in=120] (s1) to[out=-60,in=0-30] (v); \\draw (v) to[out=210,in=240] (s2) to[out=60,in=90] (v); \\filldraw (v) circle (1pt); \\end{scope} \\end{tikzpicture)]v); \\coordinate (s2) at ([shift=(150:.8}; \\def \\rud {.4}; \\coordinate (s1) at ([shift=(30:\\rad)]v); \\coordinate (s2) at ([shift=(150:\\rad)]v); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\draw (v) to[out=90,in=120] (s1) to[out=-60,in=0-30] (v); \\draw (v) to[out=210,in=240] (s2) to[out=60,in=90] (v); \\filldraw (v) circle (1pt); \\end{scope} \\end{tikzpicture)]v); \\coordinate (i1) at ([shift=(225:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\coordinate (i2) at ([shift=(255:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\coordinate (i3) at ([shift=(285:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\coordinate (i4) at ([shift=(315:.4}; \\coordinate [above=.4 of v] (m); \\coordinate (i1) at ([shift=(225:\\rud)]v); \\coordinate (i2) at ([shift=(255:\\rud)]v); \\coordinate (i3) at ([shift=(285:\\rud)]v); \\coordinate (i4) at ([shift=(315:\\rud)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\filldraw (v) circle (1pt); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture)]v); \\draw (v) -- (i1); \\draw (v) -- (i2); \\draw (v) -- (i3); \\draw (v) -- (i4); \\draw (v) to[out=90,in=120] (s1) to[out=-60,in=0-30] (v); \\draw (v) to[out=210,in=240] (s2) to[out=60,in=90] (v); \\filldraw (v) circle (1pt); \\end{scope} \\end{tikzpicture}\n\\\\\n+\n4~\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\coordinate[above left=.5 of v2] (e2); \\coordinate[below left=.5 of v3] (e3); \\coordinate[above right=.5 of v2] (h2); \\coordinate[below right=.5 of v3] (h3); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v2) -- (e2); \\draw (v3) -- (e3); \\draw (v2) -- (h2); \\draw (v3) -- (h3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) -- (v3); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n\\otimes \n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.4 of v0] (m); \\coordinate[right=.4 of m] (v1); \\coordinate[right=.3 of v1] (v2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below =.5 of v1] (o1); \\coordinate[above =.5 of v1] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\draw (v2) circle (.3); \\draw (m) circle (.4); \\end{scope} \\end{tikzpicture}\n+\n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\coordinate[above left=.5 of v2] (e2); \\coordinate[below left=.5 of v3] (e3); \\coordinate[above right=.5 of v2] (h2); \\coordinate[below right=.5 of v3] (h3); \\draw (v2) -- (h2); \\draw (v3) -- (h3); \\draw (v2) -- (e2); \\draw (v3) -- (e3); \\draw (v2) to[bend right=60] (v3); \\draw (v2) to[bend left=60] (v3); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}\n\\otimes \n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.4 of v0] (m1); \\coordinate[right=.4 of m1] (v1); \\coordinate[right=.4 of v1] (m2); \\coordinate[right=.4 of m2] (v2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v2] (o1); \\coordinate[above right=.5 of v2] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v2) -- (o1); \\draw (v2) -- (o2); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\draw (m1) circle (.4); \\draw (m2) circle (.4); \\end{scope} \\end{tikzpicture}\n+\n\\mathbb{I} \n\\otimes \n\\begin{tikzpicture}[x=2ex,y=2ex,baseline={([yshift=-.7ex]current bounding box.center)}] \\begin{scope}[node distance=1] \\coordinate (v0); \\coordinate[right=.5 of v0] (v4); \\coordinate[above right= of v4] (v2); \\coordinate[below right= of v4] (v3); \\coordinate[below right= of v2] (v5); \\coordinate[right=.5 of v5] (v1); \\coordinate[above right= of v2] (o1); \\coordinate[below right= of v2] (o2); \\coordinate[below left=.5 of v0] (i1); \\coordinate[above left=.5 of v0] (i2); \\coordinate[below right=.5 of v1] (o1); \\coordinate[above right=.5 of v1] (o2); \\draw (v0) -- (i1); \\draw (v0) -- (i2); \\draw (v1) -- (o1); \\draw (v1) -- (o2); \\draw (v0) to[bend left=20] (v2); \\draw (v0) to[bend right=20] (v3); \\draw (v1) to[bend left=20] (v3); \\draw (v1) to[bend right=20] (v2); \\draw (v2) to[bend right=60] (v3); \\draw (v2) to[bend left=60] (v3); \\filldraw (v0) circle(1pt); \\filldraw (v1) circle(1pt); \\filldraw (v2) circle(1pt); \\filldraw (v3) circle(1pt); \\end{scope} \\end{tikzpicture}.\n\\end{gather*}%\n\nNote that the complete contraction $\\G\/\\G$ has no edges, so is identified with $\\mathbb{I}$. Also note that we omitted isolated vertices of the subgraphs on the left hand side of the tensor product since isolated vertices are also identified with $\\mathbb{I}$.\n\\end{example}\n\n\n\nThis coproduct is coassociative, making ${\\mathbf{H}}$ into a bialgebra. In fact Kreimer showed that ${\\mathbf{H}}$ has a Hopf algebra structure \\cite{kreimer2009core}. The unit $u \\colon \\mathbb{Q} \\to {\\mathbf{H}}$ sends $u \\colon q \\mapsto q \\mathbb{I}$. The co-unit $\\epsilon \\colon {\\mathbf{H}}\\to \\mathbb{Q}$ sends $\\mathbb{I}$ to $1\\in \\mathbb{Q}$ and all other graphs to $0$.\nThe antipode $S\\colon {\\mathbf{H}}\\to {\\mathbf{H}}$ can be defined inductively by $S(\\mathbb{I})=\\mathbb{I}$ and, \n\\begin{align*} \n S(\\G)=-\\sum_{\\gamma\\subsetneq \\G} S(\\gamma)\\G\/\\gamma \\text{ for } \\Gamma \\neq \\mathbb{I},\n\\end{align*}\nwhere the sum is over all core subgraphs of $\\G$ which are not equal to $\\G$. This recursion terminates since the graphs $\\gamma$ in the sum have fewer independent cycles (i.e. smaller first Betti number) than $\\G$. The result is a polynomial in core graphs. \nWe refer the reader to \\cite{sweedler1969hopf} for a general account of Hopf algebras or \\cite[Ch.\\ 3]{borinsky2018graphs} for more information about this specific Hopf algebra.\n \n A {\\em character} on ${\\mathbf{H}}$ is a linear map $\\phi$ which satisfies $\\phi(\\G_1 \\G_2) = \\phi(\\G_1) \\phi(\\G_2).$ The convolution $\\phi \\star \\psi$ of two characters \nis defined by $$(\\phi\\star \\psi)(\\Gamma)=\\sum_{\\gamma \\subset \\G} \\phi(\\gamma) \\psi(\\Gamma\/\\gamma),$$ where we again sum over all core subgraphs of $\\G$. Because ${\\mathbf{H}}$ is a Hopf algebra, the set of all characters from ${\\mathbf{H}}$ to any commutative algebra forms a group under the convolution product. This follows from the antipode being the inverse to the identity map, $\\id \\colon {\\mathbf{H}} \\to {\\mathbf{H}}$, in the sense that $\\id \\star S = S \\star \\id= u \\circ \\epsilon$. The map $u \\circ \\epsilon \\colon {\\mathbf{H}} \\to {\\mathbf{H}}$ is the identity element of the $\\star$-group of characters ${\\mathbf{H}} \\to {\\mathbf{H}}$. It satisfies $u \\circ \\epsilon(\\mathbb{I}) = \\mathbb{I}$ and $u \\circ \\epsilon(\\G) = 0$ if $\\G \\neq \\mathbb{I}$. If $\\phi$ is a character ${\\mathbf{H}} \\to \\mathcal{A}$ which maps to a unital commutative algebra $\\mathcal{A}$, then $\\phi^{\\star -1} := \\phi \\circ S$ is the inverse of $\\phi$ under the star product in the sense that $$ \\phi^{\\star -1} \\star \\phi = \\phi \\star \\phi^{\\star -1} = u_{\\mathcal{A}} \\circ \\epsilon,$$ where $u_{\\mathcal{A}}$ is the unit of $\\mathcal{A}$.\n\nBecause $\\tau$ is multiplicative on disjoint unions of graphs, it induces a character ${\\mathbf{H}} \\to \\mathbb{Q}$. We can define the even simpler character $\\sigma(\\Gamma) = (-1)^{e(\\Gamma)}$ and formulate Proposition~\\ref{prop:tau_identity} in the Hopf algebra language:\n\\begin{proposition}\n$\\tau \\star \\sigma = \\sigma \\star \\tau = u_\\mathbb{Q} \\circ \\epsilon$. \n\\end{proposition}\n\n\\begin{proof}\nBy Proposition~\\ref{prop:tau_identity} and the definition of the $\\star$ product $\\tau \\star \\sigma = u_\\mathbb{Q} \\circ \\epsilon$. Because the characters form a group, we also have $\\sigma \\star \\tau = u_\\mathbb{Q} \\circ \\epsilon$.\n\\end{proof}\n\n\nAlthough the Hopf algebra ${\\mathbf{H}}$ and its coproduct are defined only on core graphs, we can also consider the maps $\\tau$ and $\\sigma$ on the space of all graphs.\n The linear space ${\\mathbf{G}}$ which is generated by all graphs can be made into a (left) ${\\mathbf{H}}$-comodule by defining a \\textit{coaction}, $\\rho \\colon {\\mathbf{G}} \\to {\\mathbf{H}} \\otimes {\\mathbf{G}}$, using the formula \\eqref{eqn:coproduct} with $\\rho$ in place of $\\Delta$. The left side of the tensor product in \\eqref{eqn:coproduct} will always be a core graph and can naturally be associated with an element in ${\\mathbf{H}}$. The star product applied on characters of ${\\mathbf{G}}$ in the same way as on characters of ${\\mathbf{H}}$ becomes an \\textit{action} this way. See \\cite[Ch.\\ 3]{borinsky2018graphs} for details. \n\nApplying $\\sigma$ to the weighted sum of all connected graphs with no leaves gives an especially interesting result: \n\\begin{proposition}\n\\label{prop:bernoulli_graphs_sum}\n\\begin{align*}\n \\sum_{\\substack{\\Gamma\\in \\GG^c_0\\\\|\\G|=n}}\\frac{\\sigma(\\G)}{|\\Aut(\\G)|}=\\frac{\\zeta(-n)}{n}=-\\frac{B_{n+1}}{n(n+1)} \\text{ for all } n \\geq 1.\n\\end{align*}\n\\end{proposition}\nThis statement is not new. It follows as a special case from `Penner's model' \\cite{penner1986moduli} (see also \\cite[Appendix\\ D]{Kon92}). \nThe sum could be thought of as the integral of $\\sigma$ over the space of connected graphs with measure $\\mu(\\G)=1\/|\\Aut(\\G)|$, whereas integrating its convolutive inverse $\\tau(\\G)$ over the same space with the same measure gives $\\ch_n$ by Corollary~\\ref{cor:SV}. \n\nIn Section~\\ref{sec:laplace} we will give a proof of Proposition~\\ref{prop:bernoulli_graphs_sum} as a special case of Corollary~\\ref{crll:graph_laplace}. Here, we can immediately verify it for $n=1$:\n\\begin{align*}\n\\frac{\\sigma(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=.7 of vm] (v0); \\coordinate [right=.7 of vm] (v1); \\draw (v0) circle(.7); \\draw (v1) circle(.7); \\filldraw (vm) circle (1pt); \\end{tikzpicture}%\n}\n)}{|\\Aut(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=.7 of vm] (v0); \\coordinate [right=.7 of vm] (v1); \\draw (v0) circle(.7); \\draw (v1) circle(.7); \\filldraw (vm) circle (1pt); \\end{tikzpicture}%\n}\n)|}\n+\n\\frac{\\sigma(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (v0); \\coordinate [right=1.5 of v0] (v1); \\coordinate [left=.7 of v0] (i0); \\coordinate [right=.7 of v1] (o0); \\draw (v0) -- (v1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\draw (i0) circle(.7); \\draw (o0) circle(.7); \\end{tikzpicture}%\n}\n)}{|\\Aut(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (v0); \\coordinate [right=1.5 of v0] (v1); \\coordinate [left=.7 of v0] (i0); \\coordinate [right=.7 of v1] (o0); \\draw (v0) -- (v1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\draw (i0) circle(.7); \\draw (o0) circle(.7); \\end{tikzpicture}%\n}\n)|}\n+\n\\frac{\\sigma(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=1 of vm] (v0); \\coordinate [right=1 of vm] (v1); \\draw (v0) -- (v1); \\draw (vm) circle(1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\end{tikzpicture}%\n}\n)}{|\\Aut(\n{\n\\begin{tikzpicture}[x=1ex,y=1ex,baseline={([yshift=-.6ex]current bounding box.center)}] \\coordinate (vm); \\coordinate [left=1 of vm] (v0); \\coordinate [right=1 of vm] (v1); \\draw (v0) -- (v1); \\draw (vm) circle(1); \\filldraw (v0) circle (1pt); \\filldraw (v1) circle (1pt); \\end{tikzpicture}%\n}\n)|}\n&=\n\\frac{1}{8} + \\frac{-1}{8} + \\frac{-1}{12} = \n-\\frac{1}{12} = -\\frac{B_2}{2}.\n\\end{align*}\n\n\nThe Bernoulli numbers are classical objects with a long history, and it is well-known that $B_{2n+1}$ vanishes for $n \\geq 1$ and that the sign of $B_{2n}$ is $(-1)^{n+1}$ for $n\\geq 1$. To analyse similar properties of the numbers $\\ch_n$, we will make heavy use of \\textit{asymptotic expansions}. We will go into the details after a short digression about the relation of our methods with perturbative methods used in quantum field theory. \n\n\n\\section{Renormalized topological quantum field theory}\\label{sec:tqft}\n\nOur approach to analyzing the numbers $\\ch_n$ is in line with an established technique for analyzing topological objects by using \\textit{perturbative quantum field theory} or equivalently \\textit{Feynman diagram techniques} \\cite{bessis1980quantum}. The term \\textit{topological quantum field theory} is used for a quantum field theory whose observables are topological invariants \\cite{witten1988topological}. See also \\cite{kontsevich1993formal,kontsevich1994feynman} for further aspects of this theory and \\cite{conant2003theorem} for a more detailed account focused on group cohomology. \n\nOne prominent application of topological quantum field theory is intersection theory in the moduli space of complex curves, as developed by Witten \\cite{witten1990two} and Kontsevich \\cite{Kon92}. Penner \\cite{penner1986moduli} had already applied perturbative quantum field theory techniques to reprove the result of Harer and Zagier %\non the rational Euler characteristic of the mapping class group. In the course of his study of intersection theory Kontsevich gave a simplified version of Penner's proof \\cite[Appendix\\ D]{Kon92}. This simplified proof involves a formula similar to the one in Proposition~\\ref{prop:bernoulli_graphs_sum}. \n\nWe can endow our approach to studying the numbers $\\ch_n$ with a quantum field theoretical interpretation, in a spirit similar to the work of Penner and Kontsevich. Here is a brief, heuristic indication of how this goes.\n\nWe start with the statement of Proposition~\\ref{prop:Tzx_graph_counting_identity} and immediately apply Proposition~\\ref{prop:Tzx_leaves_identity} to obtain the equation\n\\begin{align*}\n 1 &= \\sum_{\\ell \\geq 0} (-z)^\\ell (2\\ell-1)!! [x^{2\\ell}] \\exp\\left( \\frac{e^x-\\frac{x^2}{2}-x-1}{z} + \\frac{x}{2} + T(ze^{-x}) \\right).\n\\end{align*}\nNow flip the sign of $z$ to get\n\\begin{align}\n\\label{eqn:Tzx_graph_counting_identity_qft_expl}\n 1 &= \\sum_{\\ell \\geq 0} z^\\ell (2\\ell-1)!! [x^{2\\ell}] \\exp\\left( -\\frac{e^x-\\frac{x^2}{2}-x-1}{z} + \\frac{x}{2} + T(-ze^{-x}) \\right).\n\\end{align}\nFor the remainder of this section we regard $z$ not as a formal variable, but rather as a positive real number. We then recall the Gaussian integrals \n\\begin{align*}\n \\frac{1}{\\sqrt{2\\pi z}} \\int_\\mathbb{R} x^{2 \\ell} e^{-\\frac{x^2}{2 z}} dx &= z^\\ell (2\\ell-1)!! \\text{ for all } \\ell \\geq 0 \\\\\n \\frac{1}{\\sqrt{2\\pi z}} \\int_\\mathbb{R} x^{2 \\ell+1} e^{-\\frac{x^2}{2 z}} dx &=0 \\text{ for all } \\ell \\geq 0.\n\\end{align*}\nSubstituting these into eq.\\ \\eqref{eqn:Tzx_graph_counting_identity_qft_expl} gives\n\\begin{align*}\n 1 &= \\sum_{\\ell \\geq 0} \\frac{1}{\\sqrt{2\\pi z}} \\int_\\mathbb{R} x^{2 \\ell} e^{-\\frac{x^2}{2 z}} dx [x^{2\\ell}] \\exp\\left( -\\frac{e^x-\\frac{x^2}{2}-x-1}{z} + \\frac{x}{2} + T(-ze^{-x}) \\right).\n\\end{align*}\nThis integral is not convergent since we are no longer regarding $z$ as a formal variable, but we will disregard this issue for this heuristic argument. In the same laissez-faire spirit, we ignore convergence issues and interchange summation with integration to obtain\n\\begin{align}\n \\label{eqn:renormalization_condition}\n 1 &= \\frac{1}{\\sqrt{2\\pi z}} \\int_\\mathbb{R} \\exp\\left( -\\frac{e^x-x-1}{z} + \\frac{x}{2} + T(-ze^{-x}) \\right) dx.\n\\end{align}\nThis integral is again not well-defined, as the series $T(z)$ does not converge to a function of $z$ in any finite domain: it is only a formal power series with a vanishing radius of convergence. However, we can interpret the right hand side of this equation as a `path-integral' of a \\textit{zero-dimensional quantum field theory} with the \\textit{action} $-(e^x-x-1)$, where the parameter $z$ takes the role of \\textit{Planck's constant} $\\hbar$. The additional terms in the exponent $\\frac{x}{2} + T(-ze^{-x})$ can be interpreted as \\textit{counterterms} or \\textit{renormalization constants} which \\textit{renormalize} the quantum field theory in a generalized sense. In fact, equation~\\eqref{eqn:renormalization_condition} can be interpreted as a \\textit{renormalization condition} of a quantum field theory. \n\nIn Kontsevich's proof of the Harer-Zagier formula, a topological quantum field theory was constructed whose perturbative expansion encoded the geometric invariants of interest. As we have seen above, our method can also be interpreted as an application of quantum field theory to the analysis of the invariants $\\ch_n$. However, instead of using the coefficients of the perturbative expansion directly, we use the coefficients of the renormalization constants to express the quantities which are of interest. We might therefore say that we are using a \\textit{renormalized topological quantum field theory} to encode $\\ch_n$. \n\nThis is consistent with the interpretation of $\\tau$ as a character on the core Hopf algebra. Connes and Kreimer \\cite{connes2000renormalization} showed that the renormalization procedure in quantum field theory can be seen as the solution of a Riemann-Hilbert problem using a Birkhoff decomposition. The Birkhoff decomposition can be formulated elegantly as an inversion in the group of characters of a certain Hopf algebra. In our topological case, which is much simpler than the full physical picture, this interpretation boils down to the brief exposition in Section~\\ref{sec:core}. Consult \\cite{borinsky2017renormalized} for a general treatment of renormalized zero-dimensional quantum field theory in a Hopf algebraic framework. \n\nAfter these expository remarks we now return to our rigorous treatment of the Euler characteristic of $\\Out(F_n)$. \n\n\n\n\\newcommand{\\llrrparen}[1]{%\n \\left(\\mkern-3mu\\left(#1\\right)\\mkern-3mu\\right)}\n\\section{Asymptotic expansions}\n\\label{sec:asymptotic_expansions}\n\n\n\nAn often useful approach to studying a generating function such as $T(z)= \\sum_{n\\geq 1} \\ch_n z^n$ is to interpret it as an analytic function in $z$ and then use analytic techniques to study the nature of its coefficients \\cite{flajolet2009analytic}. However, in our case this standard approach is doomed to fail, at least if it is applied naively, as the coefficients of $T(z)$ turn out to grow factorially so the power series $T(z)$ has a vanishing radius of convergence.\n\nWe will circumvent this problem by using an \\textit{asymptotic expansion} of a certain function to describe the coefficients of $T(z)$. In contrast to Taylor expansions of analytic functions, asymptotic expansions are not necessarily convergent in any non-vanishing domain of $\\mathbb{C}$. \n\n\n\\subsection{Asymptotic notation}\n\nIn this section we fix the notation we use for asymptotic expansions and prove a basic property that we will use repeatedly. \nWe begin by recalling the {\\em big $\\mathcal{O}$} and {\\em small $o$} notation.\nLet $f, g$ and $h$ be functions defined on a domain $D$ and let $L$ be a limit point of $D$. \nThe notation $f(x)=g(x) + \\mathcal{O}(h(x))$ means $f-g\\in \\mathcal{O}(h)$, where $\\mathcal{O}(h)$ is the set of all functions $u$ defined on $D$ such that \n\\begin{align*}\n \\limsup_{x\\to L} \\left| \\frac{u(x)}{h(x)} \\right| < \\infty.\n\\end{align*}\nSimilarly, $f(x)=g(x)+o(h(x))$ means $f-g\\in o(h),$ where $o(h)$ consists of all functions $u$ that satisfy $\\lim_{x\\to L} \\frac{u(x)}{h(x)} = 0$. %\n\nAn {\\it asymptotic scale} on $D$ with respect to a limit $L$ is a sequence of functions $\\{\\varphi_k\\}_{k\\geq 0}$ with the property $\\varphi_{k+1} \\in o(\\varphi_k)$ for $k \\geq 0$. A common example, for functions with domain $\\mathbb{R}$ and limit $L=\\infty$, is $\\varphi_k(x)=x^{-k}$.\n\n\\begin{definition}\n \\label{def:asymptotic_expansion}\n An {\\it asymptotic expansion} of a function $f$ defined on $D$ with respect to the limit $L$ and the asymptotic scale $\\{\\varphi_k\\}_{k\\geq0}$ is a sequence of coefficients $c_k$ such that\n\\begin{align*}\n f(x) = \\sum_{k=0}^{R-1} c_k\\varphi_k(x) + \\mathcal{O}(\\varphi_R(x)) \\text{ for all } R \\geq 0,\n\\end{align*}\nwhere the $\\mathcal{O}$ refers to the limit $x \\rightarrow L$. %\nWe will write this infinite set of $\\mathcal{O}$ relations as, \n\\begin{align*}\n f(x)\\sim \\sum_{k\\geq 0} c_k\\varphi_k(x) \\text{ as } x \\rightarrow L.\n\\end{align*}\n\\end{definition}\nAsymptotic expansions are widely used in mathematical analysis, the physical sciences and engineering to obtain very accurate approximations to functions. \nA detailed introduction to asymptotic expansions can be found in de Bruijn's book \\cite{de1981asymptotic}. A key feature of asymptotic expansions is that, for a given function $f$, limit $L$ and asymptotic scale $\\{\\varphi_k\\}_{k \\geq 0}$, the coefficients $c_k$ are unique if they exist. We will make use of this property in the proof of Theorem~\\ref{thm:asymptotic_expansion}.\n\nThe coefficients of an asymptotic expansion depend on the choice of the asymptotic scale. However, under certain conditions we can translate between asymptotic expansions in different asymptotic scales:\n\\begin{lemma}\n \\label{lmm:scale_change}\n Suppose $\\Phi=\\{\\varphi_k\\}_{k\\geq0}$ and $\\Psi=\\{\\psi_m\\}_{m\\geq0}$ are two asymptotic scales on a domain $D$ with respect to the same limit $L$, and suppose $f$ has an asymptotic expansion in $\\Phi$\n\\begin{equation}\n \\label{eqn:scale_change_original}\n f(x) \\sim \\sum_{k \\geq 0} c_k \\varphi_k(x) \\text{ as } x\\rightarrow L.\n\\end{equation}\nIf each $\\psi_m$ also has an asymptotic expansion in $\\Phi$\n\\begin{align}\n \\label{eqn:scale_change_scale_relation}\n\\psi_m(x)\\sim \\sum_{k\\geq m} c_{m,k} \\varphi_k(x) \\text{ as } x\\rightarrow L\n\\end{align}\nwith $c_{m,m}\\neq 0$, then $f$ has an asymptotic expansion in $\\Psi$\n\\begin{align*}\n f(x) \\sim \\sum_{m \\geq 0} c'_m \\psi_m(x) \\text{ as } x\\rightarrow L,\n\\end{align*}\n where the coefficients $c_m'$ are implicitly determined by the infinite triangular equation system $c_k = \\sum_{m= 0}^{k} c_m' c_{m,k}$ for all $k\\geq 0$.\n\\end{lemma}\n \n\n\\begin{proof}\nBy the definition of an asymptotic expansion we have\n \\begin{align*}\n \\psi_m- \\sum^{R-1}_{k=m} c_{m,k} \\varphi_k\\in \\mathcal{O}(\\varphi_R) \\text{ for all } R \\geq m \\geq 0.\n \\end{align*}\nWe can multiply a function in $\\mathcal{O}(h)$ by a constant or add a finite number of functions in $\\mathcal{O}(h)$ without changing the $\\mathcal{O}$ class. Thus multiplying by $c_m'$ and then adding from $m=0$ to $R-1$ gives\n \\begin{align*}\n \\sum_{m=0}^{R-1} c'_m \\psi_m - \\sum_{m=0}^{R-1}\\sum^{R-1}_{k=m} c'_mc_{m,k} \\varphi_k \\in \\mathcal{O}(\\varphi_R) \\text{ for all } R \\geq 0.\n \\end{align*}\n Changing the order of summation and using the definition of the constants $c'_m$ gives \n \\begin{align*}\n \\sum_{m=0}^{R-1} c'_m \\psi_m- \\sum_{k=0}^{R-1}\\sum_{m=0}^{k} c'_mc_{m,k} \\varphi_k = \\sum_{m=0}^{R-1} c'_m \\psi_m - \\sum_{k=0}^{R-1}c_k \\varphi_k \\in \\mathcal{O}(\\varphi_R) \\text{ for all } R \\geq 0.\n \\end{align*}\nBy eq.\\ \\eqref{eqn:scale_change_original} we have $f- \\sum_{k=0}^{R-1} c_k\\varphi_k \\in \\mathcal{O}(\\varphi_R)$, so combining this with the above gives $$f- \\sum_{m=0}^{R-1} c_m'\\psi_m \\in \\mathcal{O}(\\varphi_R) \\text{ for all } R\\geq 0.$$\nIt remains only to check that $\\mathcal{O}(\\varphi_R)=\\mathcal{O}(\\psi_R)$. This follows from eq.\\ \\eqref{eqn:scale_change_scale_relation}, which implies $\\psi_R=c_{R,R}\\varphi_R + \\mathcal{O}(\\varphi_{R+1}),$ together with the assumption that $c_{R,R}\\neq 0$ and the fact that $\\varphi_{R+1}\\in o(\\varphi_R)$. \n \\end{proof}\n\n\n\n In this paper the domain of our functions will mostly be the natural numbers, i.e.\\ our functions are sequences $f\\colon\\mathbb{N} \\rightarrow \\mathbb{R}$, and the limit will almost always be $\\infty$, but the asymptotic scale will vary. \n\n\\subsection{Stirling's approximation}\nArguably, one of the most studied asymptotic expansions is \\textit{Stirling's approximation}. This is an asymptotic expansion of the gamma function \n\\begin{align}\n \\label{eqn:stirling_approximation}\n \\Gamma(n) \\sim \\sum_{k \\geq 0} \\widehat b_k \\sqrt{2\\pi} e^{-n}n^{n-\\frac12-k} \\text{ as } n \\rightarrow \\infty,\n\\end{align}\nwhere $\\widehat b_k$ is the coefficient of $z^k$ in $\\exp\\left( \\sum_{k=1}^\\infty \\frac{B_{k+1}}{k(k+1)} z^k \\right)$. See for instance \\cite[Sec.\\ 3.10]{de1981asymptotic} for a proof.\nStirling's approximation is used extensively as a tool for approximating the value of $\\Gamma(n)$ for large $n$. %\nWe, however, will view eq.\\ \\eqref{eqn:stirling_approximation} as an asymptotic expansion of $\\Gamma(n)$ in the asymptotic scale $\\{\\sqrt{2 \\pi} e^{-n} n^{n-\\frac12-k}\\}_{k\\geq0}$ and use it merely as a tool to encode and manipulate the coefficients $\\widehat b_k$. \n\nRecall that the gamma function satisfies $\\Gamma(z+1) = z\\Gamma(z);$ this ensures that the sequence of functions $\\{\\Gamma( n -k + \\frac12 )\\}_{k\\geq0}$ forms an asymptotic scale in the limit $n \\rightarrow \\infty$. The statement of Theorem~\\ref{thm:asymptotic_expansion} gives an asymptotic expansion of $f(n)= \\sqrt{2 \\pi} e^{-n} n^{n}$ in this scale, whose coefficients coincide with those of the formal power series $\\exp(T(z))$; we can think of this as a kind of ``inverted'' Stirling's approximation. \n \nAlthough there is a large and growing literature on Stirling's approximation (see \\cite{borwein2018gamma} for a recent survey), such an asymptotic expansion of $\\sqrt{2 \\pi} e^{-n} n^{n}$ does not seem to have been studied previously. \nThis type of `inverted Stirling's approximation' might also be relevant for other applications: many problems dictate or suggest an inherent asymptotic scale. For instance, it might be natural to work in the asymptotic scale $\\{(2(n-k)-1)!!\\}_{k\\geq 0}$, where $(2(n-k)-1)!! = 2^{n-k} \\Gamma(n-k+\\frac12)\/\\sqrt{\\pi}$, for counting problems whose solution involves double factorials. Moreover, power series with coefficients which have an asymptotic expansion in the scale $\\{\\Gamma(n-k+\\beta)\\}_{k\\geq 0}$ with $\\beta \\in \\mathbb{R}$ have a rich algebraic structure; for instance they are closed under multiplication and functional composition \\cite{borinsky2018generating}.\n\nTo establish the asymptotic expansion in Theorem~\\ref{thm:asymptotic_expansion} we will start with a trivial asymptotic expansion for the constant function $1$ in the scale $\\{n^{-k}\\}_{k\\geq0}$, then use Lemma~\\ref{lmm:scale_change} to change to the scale $\\{\\psi_m\\}_{m\\geq0}$, where \n\\begin{align} \\label{eqn:psi_defn}\n\\psi_m(n) = \\frac{\\Gamma( n -m +\\frac12 )}{\\sqrt{2 \\pi} e^{-n} n^{n}}.\n\\end{align}\nIn order to apply the lemma, we need to find asymptotic expansions for the functions $\n\\psi_m(n)$. We do this using the following variant of Stirling's approximation. \n\\begin{proposition}\n \\label{prop:graph_stirling}\n Let $\\Psi=\\{\\psi_m\\}_{m\\geq0}$ be the asymptotic scale with domain $\\mathbb{N}$ and limit $\\infty$ defined in eq.\\ \\eqref{eqn:psi_defn}.\n Then each $\\psi_m$ has an asymptotic expansion in the asymptotic scale $\\{n^{-k}\\}_{k\\geq0}$ given by\n\\begin{align*}\n %\n\\psi_m(n)\n\\sim \\sum_{k \\geq m} c_{m,k} n^{-k}\n\\text{ as } n \\rightarrow \\infty,\n\\end{align*}\nwhere $c_{m,k}$ is the coefficient of $z^k$ in the formal power series \n \\begin{align}\n \\label{eqn:graph_stirling_asymp_psi}\nz^m\n\\sum_{\\ell \\geq 0}\nz^{\\ell} (2\\ell-1)!!\n[x^{2\\ell}]\ne^{-\\frac{1}{z}\\left(e^x - \\frac{x^2}{2} - x - 1\\right) + x \\left( \\frac12 - m \\right)}.\n\\end{align}\n\\end{proposition}\n\nWe will prove this proposition using \\textit{Laplace's method}, which serves as a connection between graphical enumeration and asymptotic expansions. We will introduce this method in the next section and therefore postpone the proof of Proposition~\\ref{prop:graph_stirling} until then. \n\nAssuming Proposition~\\ref{prop:graph_stirling} we are now ready to prove Theorem~\\ref{thm:asymptotic_expansion}.\n\n\\begin{repthmx}{thm:asymptotic_expansion}\n The function $\\sqrt{2 \\pi}e^{-n} n^n$ has the following asymp\\-totic expansion in the asymptotic scale $\\{(-1)^k \\Gamma( n + \\frac12 - k ) \\}_{k\\geq0}$,\n\\begin{align*}\n \\sqrt{2 \\pi}e^{-n} n^n &\\sim \\sum_{ k\\geq 0 } \\Ch_k (-1)^k \\Gamma\\left( n + \\frac12 - k \\right) \\text{ as } n\\rightarrow \\infty,\n\\end{align*}\nwhere $\\Ch_k$ is the coefficient of $z^k$ in the formal power series $\\exp\\left( \\sum_{n\\geq 1} \\chi( \\Out (F_{n+1}) ) z^n \\right)$.\n\\end{repthmx}\n\n\\begin{proof}\n The constant function $f(n)\\equiv 1$ has a trivial asymp\\-totic expansion in the asymptotic scale $\\{n^{-k}\\}_{k\\geq0}$, namely\n \\begin{align*}\n 1 &\\sim \\sum_{k \\geq 0} c_k n^{-k} \\text{ as } n \\rightarrow \\infty,\n \\end{align*}\n with coefficients $c_0 = 1$ and $c_k=0$ for all $k\\geq 1$. Using Lemma~\\ref{lmm:scale_change} and Proposition~\\ref{prop:graph_stirling} we can change the asymptotic scale from $\\{n^{-k}\\}_{k\\geq0}$ to the scale $\\Psi$ as defined in Proposition~\\ref{prop:graph_stirling}, giving\n \\begin{align*}\n %\n 1 &\\sim \\sum_{m \\geq 0} c_m' \\psi_m(n) \\text{ as } n \\rightarrow \\infty,\n \\end{align*}\nwhere the coefficients $c_m'$ are uniquely determined by the triangular equation system \n\\begin{align}\n\\label{eqn:triangular_system} \nc_k = \\sum_{m = 0}^{k} c_m' c_{m,k} \\text{ for all } k \\geq 0\n\\end{align}\nand the coefficients $c_{m,k}$ are those defined in the statement of Proposition~\\ref{prop:graph_stirling}. Namely, \n$c_{m,k}$ is the coefficient of $z^k$ in the formal power series given in eq.\\ \\eqref{eqn:graph_stirling_asymp_psi}. It follows from this power series representation that $c_{m,m} \\neq 0$ for all $m \\geq 0$, which justifies our application of Lemma~\\ref{lmm:scale_change} and guarantees that the linear equation system~\\eqref{eqn:triangular_system} can be uniquely solved for the coefficients $c_m'$. \n\nBy definition of $\\psi_m$ in eq.\\ \\eqref{eqn:psi_defn}, this asymptotic expansion becomes\n \\begin{align*}\n 1 &\\sim \\sum_{m \\geq 0} c'_m\\frac{\\Gamma( n -m +\\frac12 )}{\\sqrt{2 \\pi} e^{-n} n^{n}} \\text{ as } n \\rightarrow \\infty.\n \\end{align*}\n Multiplying both sides by $\\sqrt{2 \\pi} e^{-n} n^{n}$ gives \n \\begin{align*}\n \\sqrt{2 \\pi} e^{-n} n^{n} &\\sim \\sum_{m \\geq 0} c'_m\\Gamma( n -m +\\frac12 ) \\text{ as } n \\rightarrow \\infty.\n \\end{align*}\n It remains to show that $c_m'=(-1)^m\\Ch_m$. \nFrom Proposition~\\ref{prop:Tzx_graph_counting_identity} \nand Proposition~\\ref{prop:Tzx_leaves_identity} we have\n \\begin{align*}\n1 &= \\sum_{\\ell \\geq 0} (-z)^\\ell (2\\ell-1)!! [x^{2\\ell}] \\exp\\left( T(z,x) \\right)\\\\\n&= \n\\sum_{\\ell \\geq 0} (-z)^\\ell (2\\ell-1)!! [x^{2\\ell}] \\exp\\left( \\frac{e^x-\\frac{x^2}{2}-x-1}{z} + \\frac{x}{2} + T(ze^{-x}) \\right)\\\\\n &=\\sum_{\\ell \\geq 0} z^{\\ell} (2\\ell-1)!! [x^{2\\ell}] e^{-\\frac{1}{z} \\left( e^x-\\frac{x^2}{2} - x - 1 \\right) + \\frac12 x} \\exp\\left( T(-ze^{-x}) \\right).\n \\end{align*}\n Expanding the second exponential in $z$ gives,\n \\begin{align*}\n 1 &= \\sum_{\\ell \\geq 0} z^{\\ell} (2\\ell-1)!! [x^{2\\ell}] e^{-\\frac{1}{z} \\left( e^x-\\frac{x^2}{2} - x - 1 \\right) + \\frac12 x } \\sum_{m \\geq 0} z^{m} e^{-mx}(-1)^m\\Ch_m \\\\\n &= \\sum_{m \\geq 0}(-1)^m\\Ch_m z^{m} \\sum_{\\ell \\geq 0} z^{\\ell} (2\\ell-1)!! [x^{2\\ell}] e^{-\\frac{1}{z} \\left( e^x-\\frac{x^2}{2} - x - 1 \\right) + \\left(\\frac12-m\\right) x},\n \\end{align*}\nwhere $\\Ch_k$ is the coefficient of $z^k$ in the formal power series $\\exp\\left( \\sum_{n\\geq 1} \\ch_n z^n \\right)$.\nBy eq.\\ \\eqref{eqn:graph_stirling_asymp_psi} this is\n$$1 = \\sum_{m \\geq 0} (-1)^m \\Ch_m \\sum_{k\\geq m} c_{m,k} z^k=\\sum_{k\\geq 0}\\sum_{m\\leq k} (-1)^m \\Ch_m c_{m,k} z^k.$$\n Because $[z^k] 1 = c_k$, we can also write this as $c_k = \\sum_{m \\leq k} (-1)^m \\Ch_m c_{m,k}$ for all $k\\geq 0$. Therefore, we constructed a solution of the triangular equation system in \\eqref{eqn:triangular_system}. %\nBecause the coefficients $c_m'$ are unique, it follows that $c_m' = (-1)^m \\Ch_m$ as claimed.\n\\end{proof}\n\n\\begin{remark} The coefficients $c_{m,k}$ of the asymptotic expansion of the functions $\\psi_{m}$ given in Proposition~\\ref{prop:graph_stirling} (eq.\\ \\eqref{eqn:graph_stirling_asymp_psi}) can also be written in terms of Bernoulli numbers if we use the conventional expression of Stirling's approximation given in eq.\\ \\eqref{eqn:stirling_approximation}. Slightly abusing the $\\sim$ notation me may write the asymptotic expansion for $\\psi_m(n)$ as\n\\begin{align*}\n\\frac{\\Gamma( n -m +\\frac12 )}{\\sqrt{2 \\pi} e^{-n} n^{n}} &\\sim \n\\frac{\\sqrt{2\\pi} \\left(n-m+\\frac12\\right)^{n-m} e^{-n+m-\\frac12}\\exp\\left( \\sum_{k\\geq1} \\frac{B_{k+1}}{k(k+1)} \\left(n-m+\\frac12\\right)^{-k} \\right)}{\\sqrt{2 \\pi} e^{-n} n^{n}}\n\\\\\n&=\nn^{-m}\\left(\\frac{n-m+\\frac12}{n}\\right)^{n-m} e^{m - \\frac12} \\exp\\left(\\sum_{k\\geq1} \\frac{B_{k+1}}{k(k+1)} \\left(n-m+\\frac12\\right)^{-k} \\right).\n\\end{align*}\nWriting $z=\\frac{1}{n}$ this becomes\n$$z^m\\left(\\left({1-z\\left(m-\\frac12\\right)}\\right)^{\\frac{1}{z}-m}e^{m - \\frac12} \\exp\\left(\\sum_{k\\geq1} \\frac{B_{k+1}}{k(k+1)} \\left(\\frac{1}{z}-m+\\frac12\\right)^{-k} \\right)\\right).\n$$\nSince the coefficients of the asymptotic expansion for $\\psi_m(n)$ are given by the above power series as well as by the power series in eq.\\ \\eqref{eqn:graph_stirling_asymp_psi}, the series are equal, giving the following identity for Bernoulli numbers, for all $m\\geq 0$. \n\\begin{align*}\n&\\sum_{\\ell \\geq 0}\nz^{\\ell} (2\\ell-1)!!\n[x^{2\\ell}]\n\\exp\\left(-\\frac{1}{z}\\left(e^x - \\frac{x^2}{2} - x - 1\\right) + x \\left( \\frac12 - m \\right) \\right) \\\\\n&=\\exp\\left( \n \\left(m-\\frac{1}{z}\\right) \\log \\frac{1}{1-z\\left(m-\\frac12\\right)} + m - \\frac12 +\\sum_{k\\geq1} \\frac{B_{k+1}}{k(k+1)} z^k \\left(1-z\\left(m-\\frac12\\right)\\right)^{-k} \\right) \n \\\\\n&=\\exp\\left( \n\\sum_{k\\geq1} \\frac{z^k}{k(k+1)} \\left( \\left(m-\\frac12 k \\right) \\left(m-\\frac12\\right)^k + \\frac{B_{k+1}}{\\left(1-z\\left(m-\\frac12\\right)\\right)^{k}} \\right) \\right)\n\\end{align*}\n\nThis identity actually holds for all $m\\in \\mathbb{R}$. However, it is unclear how to prove such an identity without asymptotic techniques. The special case $m=\\frac12$ lies at the heart of the proof of Proposition~\\ref{prop:bernoulli_graphs_sum}. De Bruijn also discusses this case using Laplace's method and writes that the identity is `by no means easy to verify directly' \\cite[Sec.\\ 4.5]{de1981asymptotic}.\n\\end{remark}\n \\subsection{Laplace's method: A bridge between graphical enumeration and asymptotics}\n\\label{sec:laplace}\n A common source of asymptotic expansions is \\textit{Laplace's method}. Laplace's method is, as one might guess from the name, quite an old technique. It is usually used to extract asymptotic information from a complicated integral without evaluating it in full generality. We will use Laplace's method in the opposite way, as we are going analyze the properties of a complicated number sequence by associating it with a relatively simple integral. This way, the method will serve as a bridge between graphical enumeration as described in Section~\\ref{sec:graphical_enumeration} and the analytic world of integrals and their asymptotic expansions.\n \n\n\\begin{lemma}[Laplace's method]\n \\label{lmm:laplace_method}\nLet $f$ and $g$ be real-valued functions on a domain $D \\subset \\mathbb{R}$ with $0$ in its interior. Suppose both $f$ and $g$ are analytic in a neighborhood of $0$, that $g(0)=g'(0)=0$, $g''(0)=-1,$ and $0$ is the unique global supremum of $g$. Finally, assume that the integral\n \\begin{align*}\n \\int_D | f(x) | e^{n g(x)} dx\n \\end{align*}\n exists for sufficiently large $n$.\n Then the sequence $I(n)$ given by the integral formula\n\\begin{align}\n \\label{eqn:integral_I}\n I(n) = \\sqrt{\\frac{n}{2 \\pi}} \\int_D f(x) e^{n g(x)} dx\n\\end{align}\nadmits an asymptotic expansion with asymptotic scale $\\{n^{-k}\\}_{k\\geq0}$, \n\\begin{align}\n \\label{eqn:laplace_expansion}\n I(n) \\sim \\sum_{k \\geq 0} c_k n^{-k} \\text{ as } n \\rightarrow \\infty,\n\\end{align}\nwhere $c_k$ is the coefficient of $z^k$ in the formal power series,\n\\begin{align}\n\\label{eqn:laplace_coeffs}\n \\sum_{\\ell \\geq 0} z^\\ell (2\\ell-1)!! [x^{2\\ell}] f(x) e^{\\frac{1}{z} \\left( g(x) + \\frac{x^2}{2} \\right) }.\n\\end{align}\n\\end{lemma}\nA quite similar statement is given in \\cite[Thm. B7]{flajolet2009analytic}. Unfortunately, only a partial proof is given there. For the convenience of the reader we provide a proof in the appendix. The argument revolves around approximating the integral in eq.\\ \\eqref{eqn:integral_I} with a Gaussian integral. It closely follows the arguments in \\cite[Sec.\\ 4.4]{de1981asymptotic} and \\cite[Thm. B7]{flajolet2009analytic}.\n\n\n\nWe wrote the coefficients of the asymptotic expansion in eq.\\ \\eqref{eqn:laplace_coeffs} suggestively to illustrate the close relationship of asymptotic expansions which come from Laplace's method \nand generating functions of graphs such as the one in Proposition~\\ref{prop:convoluted_graph_sum}. We will use this relationship in the following Corollary, which we will need to give the relation between graphs and the zeta function stated in Proposition~\\ref{prop:bernoulli_graphs_sum}. \n\\begin{corollary}\n \\label{crll:graph_laplace}\n Let $f$ be the constant function $f(x)\\equiv 1$, and assume $g$ is analytic near $0$ with Taylor series $$g(x)=-\\frac{x^2}{2} + \\sum_{s\\geq 3} x^s \\frac{b_s}{s!}.$$\n Then for all $k\\geq 0$ the coefficients $c_k$ of the asymptotic expansion in eq.\\ \\eqref{eqn:laplace_coeffs} can be written as a weighted sum over graphs,\n\\begin{align}\n c_k = \\sum_{ \\substack{ \\G \\in \\GG_0\\\\ |\\G| = k} }\\frac{ \\prod_{v \\in V(\\Gamma)} b_{|v|} }{|\\Aut \\G|},\n\\end{align}\nwhere $|v|$ is the \\textit{valence} of the vertex $v$. \\end{corollary}\n\\begin{proof}\n Let $\\phi: \\GG \\rightarrow \\mathbb{R}\\llrrparen{z}$ be the function from the set of graphs to the space of Laurent series in $z$ defined by setting $\\phi(\\G) = 0$ if $\\G$ contains an edge and $\\phi(\\G) = \\prod_{v \\in V(\\G)} \\left(z^{-1}b_{|v|}\\right)$ if $\\G$ has no edges. There are only finitely many graphs with $2\\ell$ leaves which have no edges, and the function $\\phi$ is multiplicative on the disjoint union of graphs, so we may apply Proposition~\\ref{prop:convoluted_graph_sum} and Lemma~\\ref{lmm:exponential_formula} to get\n\\begin{align*}\n \\sum_{\\G \\in \\GG_0}\\frac{w^{e(\\G)} \\prod_{v \\in V(\\G)} (z^{-1}b_{|v|})}{|\\Aut \\G|} = \\sum_{\\ell \\geq 0} w^\\ell (2\\ell-1)!! [x^{2\\ell}] \\exp\\left( \\sum_{\\gamma \\in \\GG^c} x^{s(\\gamma)} \\frac{\\phi(\\gamma)}{|\\Aut \\gamma|} \\right),\n\\end{align*}\nwhere we used the fact that a graph has only one subgraph with no edges. The only graphs without edges which are also connected are the star graphs $R_{0,s}$. This together with the fact that $R_{0,s}$ has the symmetric group $\\Sigma_s$ as automorphism group gives \n\\begin{align*}\n\\sum_{\\gamma \\in \\GG^c} x^{s(\\G)} \\frac{\\phi(\\gamma)}{|\\Aut \\gamma|} = \\sum_{s\\geq 3} x^s\\frac{\\phi(R_{0,s})}{|\\Aut R_{0,s}|}\n= \\frac{1}{z} \n\\sum_{s\\geq 3} x^s \\frac{b_s}{s!}.\n\\end{align*}\n Setting $w = z$ results in,\n\\begin{align*}\n \\sum_{\\G \\in \\GG_0}\\frac{\\prod_{v \\in V(\\G)} b_{|v|}}{|\\Aut \\G|}z^{|\\G|} = \\sum_{\\ell \\geq 0} z^{\\ell} (2\\ell-1)!! [x^{2\\ell}] \\exp\\left( \\frac{1}{z}\n \\sum_{s\\geq 3} x^s \\frac{b_s}{s!} \\right). \n\\end{align*}\nThe right hand side is now exactly the power series given in eq.\\ \\eqref{eqn:laplace_coeffs} that determines $c_k$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:bernoulli_graphs_sum}]\nWe start with Euler's integral representation of the gamma function \n$$\\G(n)=\\int_{0}^\\infty u^ne^{-u}\\frac{du}{u}.$$ Substituting $u = n e^x$ gives\n\\begin{align*}\n \\Gamma( n ) &= \\int_{-\\infty}^\\infty n^n e^{nx} e^{-ne^x}dx\n = e^{-n} n^{n} \\int_{-\\infty}^\\infty e^{-n\\left(e^x-x - 1 \\right) } dx.\n\\end{align*}\nWe can now apply Lemma~\\ref{lmm:laplace_method} with $g(x) =-(e^x-x-1)$, $f(x)=1$ and $D=\\mathbb{R}$ to get\n an asymptotic expansion\n\\begin{align*}\n \\Gamma( n ) &= \\sqrt{2\\pi} e^{-n} n^{n-\\frac12} \\sum_{k \\geq 0} c_k n^{-k}.\n\\end{align*}\nBy Corollary~\\ref{crll:graph_laplace} and because $-(e^x-x-1) = -\\sum_{s\\geq3} \\frac{x^s}{s!}$ the coefficients satisfy,\n\\begin{align*}\n c_k = \\sum_{ \\substack{ \\G \\in \\GG_0\\\\ |\\G| = k} }\\frac{ (-1)^{v(\\Gamma)} }{|\\Aut \\G|} \\text{ for all } k \\geq 0.\n\\end{align*}\nStirling's approximation in eq.\\ \\eqref{eqn:stirling_approximation} gives another expression for the coefficients $c_k$. Because the different ways to express the asymptotic expansion of $\\Gamma(n)$ with the same scale and limit must coincide, we get\n\\begin{align*}\n\\sum_{ \\substack{ \\G \\in \\GG_0} }\\frac{ (-1)^{v(\\Gamma)} }{|\\Aut \\G|} z^{|\\Gamma|}\n=\n\\exp\\left( \\sum_{k=1}^\\infty \\frac{B_{k+1}}{k(k+1)} z^k \\right).\n\\end{align*}\nSince taking the formal logarithm restricts the sum on the left to connected graphs (Lemma~\\ref{lmm:exponential_formula}) we get\n\\begin{align*}\n\\sum_{ \\substack{ \\G \\in \\GG_0^c} }\\frac{ (-1)^{v(\\Gamma)} }{|\\Aut \\G|} z^{|\\Gamma|}\n=\n \\sum_{k=1}^\\infty \\frac{B_{k+1}}{k(k+1)} z^k.\n\\end{align*}\nNow notice that \n$\\sigma(\\Gamma) = (-1)^{e(\\Gamma)} = (-1)^{|\\Gamma|} (-1)^{v(\\Gamma)}$ and $B_{k+1} =0$ for all even $k > 0$, giving \n\\begin{align*} \n\\sum_{ \\substack{ \\G \\in \\GG_0^c\\\\ |\\G| = n} }\\frac{ \\sigma(\\Gamma) }{|\\Aut \\G|}&= (-1)^n\\frac{B_{n+1}}{n(n+1)}= - \\frac{B_{n+1}}{n(n+1)} =\\frac{\\zeta(-n)}{n}. \\qedhere\n\\end{align*}\n\\end{proof}\n\nWe now turn to the proof of Proposition~\\ref{prop:graph_stirling}, which follows along similar lines.\n\\begin{proof}[Proof of Proposition~\\ref{prop:graph_stirling}]\n Assume $n,m \\geq 0$ with $n \\geq \\max\\{1,m\\}$. Start with Euler's integral and substitute $u = n e^x$ to obtain\n\\begin{align*}\n \\Gamma\\left( n -m +\\frac12 \\right) &= \\int_0^\\infty u^{n-m+\\frac12} e^{-u} \\frac{du}{u} \n = e^{-n} n^{n-m+\\frac12} \\int_{-\\infty}^\\infty e^{-n\\left(e^x-x - 1 \\right) + x \\left( \\frac12 - m \\right) } dx.\n\\end{align*}\nTherefore,\n\\begin{align*}\n\\psi_m (n)= \n\\frac{\\Gamma( n -m +\\frac12 )}{\\sqrt{2 \\pi} e^{-n} n^{n} } = \nn^{-m}\\sqrt{\\frac{n}{2\\pi}} \n\\int_{-\\infty}^\\infty e^{-n\\left(e^x-x-1 \\right) + x \\left( \\frac12 - m \\right) } dx.\n\\end{align*}\nThe condition $n \\geq \\max\\{1,m\\}$ guarantees that the integral exists. \nThe functions $f(x) =e^{ x \\left( \\frac12 - m \\right)}$ and $g(x) = -(e^x-x - 1)$, defined on $D= \\mathbb{R}$, satisfy the conditions of Lemma~\\ref{lmm:laplace_method}, so we can apply Laplace's method to obtain\n \\begin{align*}\n n^m \\psi_m(n) \\sim \\sum_{k \\geq 0} c_{m,k}' n^{-k} \\text{ as } n \\rightarrow \\infty,\n\\end{align*}\nwhere $c'_{m,k}$ is the coefficient of $z^k$ in the power series\n\\begin{align*}\n\\sum_{\\ell \\geq 0} z^{\\ell} (2\\ell-1)!! [x^{2\\ell}] e^{- \\frac{1}{z} \\left(e^x- \\frac{x^2}{2} -x - 1 \\right) + x \\left( \\frac12 - m \\right)}.\n\\end{align*}\nFrom Definition \\ref{def:asymptotic_expansion} and the fact that $n^{-m}\\mathcal{O}(n^{-R+m}) = \\mathcal{O}(n^{-R})$, it follows that\n\\begin{align*}\n \\psi_m(n) \\sim \\sum_{k \\geq m} c_{m,k-m}' n^{-k} \\text{ as } n \\rightarrow \\infty.\n\\end{align*}\nSetting $c_{m,k} := c_{m,k-m}'$ gives eq.\\ \\eqref{eqn:graph_stirling_asymp_psi}.\n\\end{proof}\n\nWe have now completed all of the steps in the proof of Theorem~\\ref{thm:asymptotic_expansion}.\nBefore we continue with the proof of Theorem~\\ref{thm:SVconj}, we will briefly discuss the relationship of our considerations with Kontsevich's Lie graph complex.\n\\subsection{Lie graph complex}\n\\label{sec:kontsevich} Kontsevich's {\\em Lie graph complex} $\\mathfrak{L}_*$ computes the Chevalley-Eilenberg homology of a certain infinite-dimensional Lie algebra $\\ell_\\infty$ associated to the Lie operad. %\nIn \\cite{kontsevich1993formal} Kontsevich remarked that the orbifold Euler characteristic of the subcomplex $\\mathfrak{L}^{(n)}_*$ spanned by connected graphs with fundamental group $F_n$ can be encoded as coefficients of the asymptotic expansion of the integral\n\\begin{align}\n \\label{eqn:integral_kontsevich}\n \\sqrt{\\frac{n}{2 \\pi}} \\int_{D} \\exp\\left(-n \\sum_{s \\geq 2} \\frac{x^s}{s(s-1)}\\right) \\sim\n\\sum_{k \\geq 0} c_k n^{-k} \\text{ as } n \\to \\infty,\n\\end{align}\nwhere $D$ is a small domain that contains a neighborhood of $0$ and $c_k$ is the $z^k$ coefficient of the power series $\\exp( \\sum_{n \\geq 1} \\chi(\\mathfrak{L}_*^{(n+1)}) z^n )$. %\nObserving that $- \\sum_{s \\geq 2} \\frac{x^s}{s(s-1)} = -\\sum_{s \\geq 2} (s-2)! \\frac{x^s}{s!}$ and using Corollary~\\ref{crll:graph_laplace} together with the exponential formula (Lemma~\\ref{lmm:exponential_formula}), we may conclude that \n\\begin{align*}\n \\chi(\\mathfrak{L}_*^{(n)}) = \\sum_{\\substack{\\G \\in \\GG^c_0\\\\ \\pi_1(\\G) \\cong F_n}} \\frac{\\xi(\\G)}{|\\Aut \\G|},\n\\end{align*}\nwhere $\\xi$ is the function given by $$\\xi(\\G) = (-1)^{|V(\\G)|} \\prod_{v \\in V(\\G)} (|v|-2)!$$ This formula for $\\chi(\\mathfrak{L}_*^{(n)})$ also follows directly from counting graphs whose vertices are dressed with Lie operad elements. \nWe have $\\chi(\\mathfrak{L}_*^{(n)}) = \\ch_{n-1}$, because\n\\begin{align*}\n H_k(\\mathfrak{L}_*^{(n)}) \\cong H^{2n-2-k}(\\Out(F_n)).\n\\end{align*}\nThis was first observed by Kontsevich \\cite{kontsevich1993formal}; see \\cite{conant2003theorem} for a detailed proof. The statements in Theorems~\\ref{thm:SVconj} and \\ref{thm:asymptotic_expansion}, therefore apply verbatim to the orbifold Euler characteristic of $\\mathfrak{L}_*^{(n)}$. It is, however, unclear what role the map $\\xi$ and the Lie graph complex play in the interesting Hopf algebraic duality between $\\tau$ and $\\sigma$ explained in Section~\\ref{sec:core}. \n\nThe integral in eq.\\ \\eqref{eqn:integral_kontsevich} gives another representation of the coefficients $\\ch_n,$ but the descriptive power of this representation is limited: it seems that the integral does not evaluate to a `known' function, which could facilitate the extraction of information about the coefficients $\\ch_n$. Recall that the fact that two functions have the same asymptotic expansion does not imply their equality, so it does not follow from the considerations above and Theorem~\\ref{thm:asymptotic_expansion} that the left hand side of eq.\\ \\eqref{eqn:integral_kontsevich} is equal to $\\sqrt{2 \\pi}e^{-n} n^n$. %\n\n\n\n\n\\section{The Lambert \\texorpdfstring{$W$}{W}-function}\nIn this section we prove that the coefficients of the asymptotic expansion in Theorem~\\ref{thm:asymptotic_expansion} are all negative. The first statement of Theorem~\\ref{thm:SVconj}, that $\\chi(\\Out(F_n)) < 0$ for all $n \\geq 2$, follows then by Lemma~\\ref{lmm:exp_negative}.\n\n\\subsection{Singularity analysis}\n\nWe will accomplish this by using a second method to obtain the asymptotic expansion of the sequence $\\sqrt{2 \\pi} e^{-n} n^n$ with respect to the asymptotic scale $\\{(-1)^k\\Gamma(n-k+\\frac{1}{2})\\}_{k\\geq0}$. This second method is \\textit{singularity analysis}. By Theorem~\\ref{thm:asymptotic_expansion} and because of the uniqueness of asymptotic expansions, we therefore obtain another expression for the coefficients $\\hat\\chi_n$ of $\\exp(\\sum_{n\\geq 1} \\ch_n z^n)$. This expression will involve the Lambert $W$-function, which is defined as the solution of the functional equation $W(z) e^{W(z)}= z$ \\cite{corless1996lambertw}. Eventually, we will use a theorem of Volkmer \\cite{volkmer2008factorial} to show that the coefficients of the asymptotic expansion are negative. \n\n\\begin{figure}\n\\begin{tikzpicture} \\begin{axis}[ height=\\figureheight, tick align=outside, tick pos=left, width=\\figurewidth, x grid style={white!69.01960784313725!black}, xlabel={\\(\\displaystyle z\\)}, xmajorgrids, xmin=-1, xmax=1, xtick style={color=black}, y grid style={white!69.01960784313725!black}, ylabel={\\(\\displaystyle W(z)\\)}, ymajorgrids, ymin=-5, ymax=5, ytick style={color=black} ] \\addplot [semithick, black] table {%\n-0.367879441171442 -1\n-0.367322540037808 -0.945960576192766\n-0.365569562332727 -0.891921152385532\n-0.362489397863951 -0.837881728578298\n-0.357940064029995 -0.783842304771064\n-0.351767902646442 -0.72980288096383\n-0.343806721099256 -0.675763457156596\n-0.333876874118769 -0.621724033349363\n-0.321784282228105 -0.567684609542129\n-0.307319382664605 -0.513645185734895\n-0.290256008301518 -0.459605761927661\n-0.270350189808666 -0.405566338120427\n-0.247338875984067 -0.351526914313193\n-0.220938566862327 -0.297487490505959\n-0.19084385385887 -0.243448066698725\n-0.156725860840461 -0.189408642891491\n-0.118230579620611 -0.135369219084257\n-0.0749770929619213 -0.0813297952770232\n-0.0265556777246851 -0.0272903714697893\n0.0274742196695564 0.0267490523374447\n0.0875861437909174 0.0807884761446787\n0.154288935633532 0.134827899951913\n0.228129192355259 0.188867323759146\n0.309693891352915 0.24290674756638\n0.399613189339125 0.296946171373614\n0.498563407764693 0.350985595180848\n0.607270216650625 0.405025018988082\n0.72651202965913 0.459064442795316\n0.857123624045936 0.51310386660255\n1 0.567143290409784\n}; \\addplot [semithick, black, dashed] table {%\n-0.0336897349954273 -5\n-0.0376055021646619 -4.86206896551724\n-0.0419426179058552 -4.72413793103448\n-0.0467400631717225 -4.58620689655172\n-0.0520391328982212 -4.44827586206897\n-0.0578832678204224 -4.31034482758621\n-0.0643177859261937 -4.17241379310345\n-0.0713894875400297 -4.03448275862069\n-0.0791461025317493 -3.89655172413793\n-0.0876355415898728 -3.75862068965517\n-0.0969049056948192 -3.62068965517241\n-0.106999198646334 -3.48275862068966\n-0.117959676476869 -3.3448275862069\n-0.129821754505718 -3.20689655172414\n-0.142612377291115 -3.06896551724138\n-0.156346738389076 -2.93103448275862\n-0.171024215124495 -2.79310344827586\n-0.186623357930594 -2.6551724137931\n-0.203095743525044 -2.51724137931034\n-0.220358465453742 -2.37931034482759\n-0.238284993397005 -2.24137931034483\n-0.256694082986828 -2.10344827586207\n-0.275336359427955 -1.96551724137931\n-0.293878129430117 -1.82758620689655\n-0.31188189506695 -1.68965517241379\n-0.328782948102461 -1.55172413793103\n-0.343861311642715 -1.41379310344828\n-0.356208164845436 -1.27586206896552\n-0.364685732545771 -1.13793103448276\n-0.367879441171442 -1\n}; \\addplot [semithick, black, dotted] table {%\n-0.367879441171442 -5\n-0.367879441171442 5\n}; \\addplot [semithick, black, dotted] table {%\n0 -5\n0 5\n}; \\end{axis} \\end{tikzpicture}\n\\caption{Plot of the two real branches of the Lambert $W$-function. The solid line depicts the principal branch $W_0$, the dashed line the other real branch, $W_{-1}$. Both branches share a square root type singularity at $z = -1\/e$. The $W_{-1}$ additionally has a logarithmic singularity at $z=0$. The locations of the singularities are indicated with dotted lines.\n}\n\\label{fig:lambertW}\n\\end{figure}\n\n\\begin{proposition}\n \\label{prop:lambert_rep}\n The coefficient $\\Ch_k$ of $z^k$ in $\\exp\\left(\\sum_{n\\geq 1} \\ch_n z^n\\right)$ satisfies\n\\begin{align}\n \\Ch_k = -2 \\frac{\\Gamma(k +\\frac12 )}{\\sqrt{2\\pi}} v_{2k-1} \\text{ for all } k\\geq 0,\n\\end{align}\nwhere the $\\{v_k\\}_{k\\geq -1}$ are the coefficients of the following expansion involving the derivative of the principal branch of the Lambert $W$-function in the vicinity of its branch-point at $z=-\\frac{1}{e}$, \n\\begin{align}\n \\label{eqn:ck_def_lambert}\n z W_0'(z) &= \\sum_{k= -1}^{\\infty} (-1)^{k+1} v_{k} (1+ez)^{\\frac{k}{2}}.\n\\end{align}\n\\end{proposition}\nIn Figure~\\ref{fig:lambertW}, the principal branch $W_0$ of the Lambert $W$-function is depicted with a solid line. Note that the index $k$ in the summation starts with $-1$. We chose this notation to be consistent with Volkmer \\cite{volkmer2008factorial}, who proved a couple of interesting properties of the numbers $v_k$ motivated by a problem posed by Ramanujan. Most important for our considerations, he shows in \\cite[Thm.\\ 3]{volkmer2008factorial} that $v_k > 0$ for all $k \\geq 1$. He proves this by deriving the following integral representation for the coefficients $v_k$ \\cite[Thm.\\ 2]{volkmer2008factorial},\n\\begin{align*}\n v_k &= - \\frac{1}{2\\pi} \\int_0^\\infty (1+z)^{-\\frac{k}{2}-1} \\frac{\\Im W_{-1}(e^{-1} z)}{|1+W_{-1}(e^{-1}z)|^2} dz \\text{ for all }k \\geq 1,\n\\end{align*}\nwhere $\\Im$ denotes the imaginary part of a complex number and $W_{-1}$ is the branch of the Lambert $W$-function which is real and decreasing on the interval $(-\\frac{1}{e}, 0)$. This branch is drawn with a dashed line in Figure~\\ref{fig:lambertW}. The integrand is strictly negative since $\\Im W_{-1}(z) \\in (-2\\pi, -\\pi)$ for $z \\in (0,\\infty)$ \\cite{corless1996lambertw}.\n\n\n\\begin{corollary}\n \\label{crll:negative_T}\n For all $n\\geq 2$, $\\chi\\left( \\Out(F_{n}) \\right) < 0$.\n\\end{corollary}\n\\begin{proof}\n Apply Proposition~\\ref{prop:lambert_rep}, the fact that $\\Gamma(k+\\frac12) > 0$ and \\cite[Thm.\\ 3]{volkmer2008factorial} to get \n$\\Ch_n < 0$ for all $n \\geq 1$. The result now follows from Lemma~\\ref{lmm:exp_negative}.\n\\end{proof}\n\nAs already mentioned, we will use singularity analysis to prove Proposition~\\ref{prop:lambert_rep}. The basic observation behind singularity analysis is the following: the radius of convergence of the Taylor expansion of a function $f$ is equal to norm of the singularity of $f$ in $\\mathbb{C}$ which is closest to the origin. This singularity is called the \\textit{dominant singularity} of the function. The radius of convergence is also equal to the limit $\\limsup_{n\\rightarrow \\infty} |a_n|^{-\\frac{1}{n}}$ where $f(z) = \\sum_{n=0}^\\infty a_n z^n$. The radius of convergence therefore determines the exponential growth rate of the coefficients $a_n$. In many cases, the detailed nature of the function's dominant singularity determines the asymptotic behaviour of the coefficients completely. To illustrate these notions, we will start by proving one of the most basic statements from the framework of singularity analysis. For other required statements from this framework, we will refer to the literature. \nA very detailed and instructive introduction to singularity analysis can be found in Flajolet's and Sedgewick's book \\cite[Part 2]{flajolet2009analytic}. \n\\begin{lemma}\n \\label{lmm:dominant_singularity}\n If $g$ is a generating function with power series expansion $g(z)= \\sum_{n=0}^\\infty b_n z^n,$ which has radius of convergence $r$, then \n\\begin{align*}\n b_n \\in o\\left(C^{-n}\\right) \\text{ for all } 0< C < r.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nBy elementary calculus, $r^{-1} = \\limsup_{n \\rightarrow \\infty} |b_n|^{\\frac{1}{n}}.$ Therefore for every $\\delta > 0$ there exists an $n_0$ such that $|b_n|^{\\frac{1}{n}} < r^{-1}+\\delta$ for all $n \\geq n_0.$ It follows that $$|b_n| < (r^{-1} +\\delta)^n = \\left( \\frac{r}{1 + \\delta r} \\right)^{-n} \\text{ for all } n \\geq n_0.$$\nBecause we can choose any $\\delta > 0$, the statement follows. This argument also works if $r = \\infty$. \n\\end{proof}\n\nSuppose we can decompose a generating function $h(z)=\\sum_{n\\geq 0}d_nz^n$ as a sum $h(z)=f(z)+g(z)$ with $f(z)=\\sum_{n\\geq 0}a_nz^n$ and $g$ analytic in a disk around $0\\in \\mathbb{C}$ of radius larger than $1$.\nThen by Lemma~\\ref{lmm:dominant_singularity} there is a constant $C>1$ such that \n\\begin{align*}\n d_n = a_n + o(C^{-n}).\n\\end{align*}\nThis is especially useful if the coefficients $a_n$ have an asymptotic expansion,\n\\begin{align*}\n a_n \\sim \\sum_{k \\geq 0 } c_k \\varphi_k(n) \\text{ as } n \\to \\infty,\n\\end{align*}\nwith an asymptotic scale $\\{\\varphi_k\\}_{k \\geq 0}$ which satisfies $o(C^{-n}) \\subset \\mathcal{O}(\\varphi_k(n))$ for all $k \\geq 0$. In this common case, we may neglect terms contributed by $g$ to the generating function $h$\nand conclude that \n\\begin{align*}\n d_n \\sim \\sum_{k \\geq 0 } c_k \\varphi_k(n) \\text{ as } n \\to \\infty.\n\\end{align*}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture} [scale=1.5] \\fill [gray!20] ([shift=(20:1.3cm)] 0,0) arc (20:340:1.3); \\fill [white] (20:1.3) to (.5,0) to (340:1.3) to (20:1.3); \\draw [thick] ([shift=(0:.8)] 0,0) arc (0:20:.5); \\draw[->] (-2,0) to (2,0); \\draw[->] (0,-2) to (0,2); \\fill (0:.5) circle (.025); \\node [above] (one) at (0:.5) {$1$}; \\node [below] (Re) at (2,0) {$\\Re z$}; \\node [left] (Im) at (0,2) {$\\Im z$}; \\node (fi) at (1,.15) {$\\phi$}; \\node (Delta) at (-.4,.4) {$\\Delta$}; \\end{tikzpicture}\n\\caption{The region $\\Delta \\subset \\mathbb{C}$ in the statement of Lemma~\\ref{lmm:singularity_analysis}}\\label{fig:pacman}\n\\end{center}\n\\end{figure}\n\nTo prove Proposition~\\ref{prop:lambert_rep} we will need\n\\begin{lemma}[Basic singularity analysis {\\cite[Cor.\\ 3]{flajolet1990singularity}}]\n \\label{lmm:singularity_analysis}\n Let $f:\\mathbb{C} \\rightarrow \\mathbb{C}$ be analytic at $0$ with an isolated singularity at $1$, such that $f(z)$ can be analytically continued to an open domain of the form $\\Delta = \\{ z : |z| < R, z \\neq 1, | \\arg(z-1)| > \\phi \\} \\subset \\mathbb{C}$ with some $R > 1$ and $0 < \\phi < \\pi\/2$ (see Figure~\\ref{fig:pacman}). \n %\n If $f(z)$ has the following asymptotic behaviour in $\\Delta$ for $R \\geq 0$,\n \\begin{align}\n \\label{eqn:singular_limit_exp}\n f(z) &= \\sum_{k=0}^{R-1} c_k (1-z)^{\\alpha_k} + \\mathcal{O}\\left((1-z)^{A}\\right) \\text{ as } z \\rightarrow 1^{-},\n \\end{align}\n where $c_k\\in \\mathbb{R}$ and $\\alpha_0 \\leq \\alpha_1 \\leq \\ldots \\leq \\alpha_{R-1} < A \\in \\mathbb{R}$, then the coefficients $a_n = [z^n] f(z)$ have the asymptotic behaviour, \\begin{align}\n \\label{eqn:singular_limit_asymp}\n a_n = \\sum_{k = 0}^{R-1} c_k \\binom{ n - \\alpha_{k} -1}{n} + \\mathcal{O}(n^{-A-1}) \\text{ as } n \\rightarrow \\infty.\n \\end{align}\n\\end{lemma}\n Note that eq.\\ \\eqref{eqn:singular_limit_asymp} is not an asymptotic expansion in the sense of Definition \\ref{def:asymptotic_expansion}, because we did not specify an asymptotic scale. \n\n\\begin{proof}[Proof of Proposition~\\ref{prop:lambert_rep}]\nThe principal branch of the Lambert $W$-function has the series representation \\cite{corless1996lambertw},\n\\begin{align*}\n W_0(z) &= \\sum_{n\\geq1} (-1)^{n+1} \\frac{n^{n-1}}{n!} z^n.\n\\end{align*}\nBy acting with $z \\frac{d}{d z}$, we obtain the expansion\n\\begin{align}\n \\label{eqn:Wprime_expansion}\n z W_0'(z) &= \\sum_{n\\geq1} (-1)^{n+1} \\frac{n^{n}}{n!} z^n.\n\\end{align}\nThe function $W_0$ is analytic in the cut plane $\\mathbb{C} \\setminus [-1\/e,-\\infty)$ and has an expansion in the vicinity of the branch point at $z=-1\/e$,\n\\begin{align*}\n W_0(z) &= -1 + \\sqrt{2(1+ez)} - \\frac{2}{3} (1+ez) + \\frac{11}{72} \\left(\\sqrt{2(1+ez)}\\right)^{3} + \\ldots\n\\end{align*}\nwhich is convergent if $z \\in [-1\/e,0)$ \\cite[Sec.\\ 4]{corless1996lambertw} (see also Figure~\\ref{fig:lambertW}).\nTherefore, the function $z W_0'(z)$ has an expansion of the form\n\\begin{align*}\n z W_0'(z) &= \\sum_{k = -1}^\\infty (-1)^{k+1} v_{k} (1+ez)^{\\frac{k}{2}}.\n\\end{align*}\nUsing the basic version of singularity analysis from Lemma~\\ref{lmm:singularity_analysis}, we can obtain the asymptotic behaviour of the sequence $e^{-n} \\frac{n^{n}}{n!}$ from this: \nfirst we rescale the $z$-variable of $z W_0'(z)$ to obtain the expansion,\n\\begin{align*}\n -\\frac{z}{e} W_0'\\left(-\\frac{z}{e} \\right) &= \\sum_{k = -1}^{R-1} (-1)^{k+1} v_k (1-z)^{\\frac{k}{2}} + \\mathcal{O}\\left((1-z)^{\\frac{R}{2}}\\right) \\text{ as } z \\rightarrow 1^{-} \\text{ for all } R \\geq 0.\n\\end{align*}\nAs $z W_0'(z)$ is analytic in the cut plane $\\mathbb{C} \\setminus [-1\/e,-\\infty)$, the function $-\\frac{z}{e} W_0'\\left(-\\frac{z}{e} \\right)$ is analytic in another cut plane $\\mathbb{C} \\setminus [1,\\infty)$. \nAs $\\Delta \\subset \\mathbb{C} \\setminus [1,\\infty)$, we can apply Lemma~\\ref{lmm:singularity_analysis} and eq.\\ \\eqref{eqn:Wprime_expansion} to get\n\\begin{align*}\n [z^n]\\frac{-z}{e} W_0'\\left(-\\frac{z}{e} \\right) &= \n -e^{-n} \\frac{n^{n}}{n!} =\n \\sum_{k = -1}^{R-1} (-1)^{k+1} v_{k} \\binom{n-\\frac{k}{2} -1}{n} + \\mathcal{O}\\left(n^{-\\frac{R}{2}-1}\\right) \\text{ for all } R \\geq 0,\n\\end{align*}\nwhere we used $\\alpha_k = \\frac{k}{2}$ and $A = \\frac{R}{2}$.\nThe even contributions in the sum over $k$ vanish since the first argument of the binomial coefficient is an integer that is smaller than the second. Therefore, \n\\begin{align*}\n -e^{-n} \\frac{n^{n}}{n!} &= \\sum_{k = 0}^{R-1} v_{2k-1} \\binom{n-k -\\frac12}{n} + \\mathcal{O}\\left(n^{-R- \\frac12}\\right) \\text{ for all } R \\geq 0.\n\\end{align*}\nThe binomial coefficient can be expressed in terms of $\\Gamma$ functions $\\binom{n-k -\\frac12}{n} = \\frac{\\Gamma(n-k +\\frac12)}{n! \\Gamma(\\frac12 - k)}$. As a consequence of the reflection formula $\\Gamma(z)\\Gamma(1-z) =\\frac{\\pi}{\\sin(\\pi z)}$, we have $\\Gamma\\left(\\frac12 - k\\right) = \\frac{(-1)^k \\pi}{\\Gamma(k + \\frac12)}$. Hence,\n\\begin{align*}\n -e^{-n} \\frac{n^{n}}{n!} &= \\frac{1}{\\pi n!}\n \\sum_{k = 0}^{R-1}(-1)^k v_{2k-1} \\Gamma\\left( n - k + \\frac12 \\right) \\Gamma\\left(k+\\frac12\\right) + \\mathcal{O}\\left(n^{-\\frac{R}{2}-1}\\right) \\text{ for all } R \\geq 0.\n\\end{align*}\nThe statement follows from the uniqueness of asymptotic expansions and the property of the $\\Gamma$ function that $\\mathcal{O}\\left((n!) n^{-R-\\frac12}\\right) = \\mathcal{O}\\left(\\Gamma\\left(n-R+\\frac12\\right)\\right)$.\n\\end{proof}\n\n\n Proposition~\\ref{prop:lambert_rep} together with known techniques for evaluating the various expansion coefficients of the Lambert $W$-function provides an efficient way to calculate the numbers $\\ch_n$: \n\n\\begin{proposition}\n\\label{prop:efficient_chn}\nThe numbers $\\ch_n$ and $\\Ch_n$ can be calculated using the recursion equations,\n \\begin{gather*}\n \\ch_n=\\Ch_n-\\frac{1}{n}\\sum_{k=1}^{n-1}k \\ch_k \\Ch_{n-k} \\\\\n \\Ch_n = - (2n-1)!!\\left( \\frac12 (2n-1) \\mu_{2n-1} -(2n+1) \\mu_{2n+1} \\right)\n\\\\\n %\n \\mu_n = \\frac{n-1}{n+1}\\left( \\frac{\\mu_{n-2}}{2} + \\frac{\\alpha_{n-2}}{4} \\right) -\\frac{\\alpha_{n}}{2} - \\frac{\\mu_{n-1}}{n+1} \\\\\n \\alpha_n = \\sum_{k=2}^{n-1} \\mu_k \\mu_{n+1-k},\n \\end{gather*}\n for all $n \\geq 1$ with $\\alpha_0=2, \\alpha_1= -1, \\mu_{-1} = 0, \\mu_0 = -1, \\mu_1 = 1$ and $\\Ch_0 = 1$.\n\\end{proposition}\n\\begin{proof}\nThe coefficients $\\mu_n$ are the expansion coefficients of the Lambert-$W$ function near its branch point:\n$W_0(z) = \\sum_{n \\geq 0} \\mu_n \\left( 2 ( 1+ez) \\right)^{\\frac{n}{2}}.$ The recursion for $\\mu_n$ is given in \\cite[eqs.\\ (4.23) and (4.24)]{corless1996lambertw}; it follows from the differential equation which $W$ satisfies. \nWe can adapt \\cite[eq.\\ (2.11)]{volkmer2008factorial} to the notation of \\cite{corless1996lambertw} (compare \\cite[eq.\\ (2.1)]{volkmer2008factorial} with the definition of $\\mu_n$) to get an expression for $v_n$ in terms of $\\mu_n$:\n\\begin{align*}\n v_n = (-1)^{n+1} 2^{\\frac{n}{2}} \\left( \\frac12 n \\mu_n -(n+2) \\mu_{n+2} \\right)\n\\end{align*}\nThe equation for $\\Ch_n$ follows using Proposition~\\ref{prop:lambert_rep} and $(2n-1)!! = 2^{n+\\frac12}\\Gamma(n+\\frac12)$. Finally, we use eq.~\\eqref{eqn:exp_pwrsrs_relation} to translate from $\\Ch_n$ to $\\ch_n$.\n\\end{proof}\n\n Written in power series notation with $T(z) = \\sum_{n\\geq 1} \\ch_n z^n$ and $\\exp(T(z)) = \\sum_{n\\geq 0} \\Ch_n z^n$, the first few coefficients are\n \\begin{gather*}\nT(z)= - \\frac{1}{24} z - \\frac{1}{48} z^{2} - \\frac{161}{5760} z^{3} - \\frac{367}{5760} z^{4} - \\frac{120257}{580608} z^{5} + \\ldots \\\\\n\\exp(T(z))= 1 - \\frac{1}{24} z - \\frac{23}{1152} z^{2} - \\frac{11237}{414720} z^{3} - \\frac{2482411}{39813120} z^{4} - \\frac{272785979}{1337720832} z^{5} + \\ldots\n \\end{gather*}\n With this approach we calculated the value of $\\ch_n$ up to $n=1000$ with basic computer algebra tools. \n \nIn addition to being able compute the value of $\\ch_n$ for very large $n$, we can also determine the explicit asymptotic behavior of the coefficients for large $n$. We do that in the next section.\n\n\\subsection{The asymptotic growth of \\texorpdfstring{$\\chi(\\Out(F_n))$}{chi(Out(Fn))}}\n\n\\begin{proposition}\n \\label{prop:chi_outfn_asymptotics}\n The Euler characteristic of $\\Out(F_n)$ has the leading asymptotic behaviour,\n \\begin{align}\n \\chi(\\Out(F_n))=\n - \\frac{1}{\\sqrt{2\\pi}} \\frac{\\Gamma(n -\\frac32 )}{\\log^2 n} + \\mathcal{O}\\left( \\frac{\\log \\log n }{\\log^3 n}\\Gamma\\left(n -\\frac32 \\right) \\right) \\text{ as } n\\to \\infty.\n \\end{align}\n\\end{proposition}\nWe will prove Proposition~\\ref{prop:chi_outfn_asymptotics} by applying a stronger version of singularity analysis to determine the asymptotic behaviour of the coefficients $v_k$. Proposition~\\ref{prop:lambert_rep} and a classic theorem by Wright \\cite{wright1970asymptotic} will eventually enable us to deduce the asymptotic behaviour of the sequence $\\chi(\\Out(F_n))$.\n\\begin{lemma}\n \\label{lmm:asymp_vk}\n The coefficients $v_{k}$ have the leading asymptotic behaviour,\n \\begin{align}\n v_{k} &= -\\frac{1}{k(\\log k)^2} + \\mathcal{O}\\left( \\frac{\\log \\log k}{k(\\log k)^3}\\right) \\text{ as } k \\to \\infty.\n \\end{align}\n\\end{lemma}\n\\begin{proof}\n In addition to the expansion in eq.\\ \\eqref{eqn:ck_def_lambert}, the numbers $v_k$ are the coefficients of the following expansion of the other real branch of the Lambert $W$-function \\cite{volkmer2008factorial}, \n \\begin{align*}\n z W_{-1}'(z) &= -\\sum_{k = -1}^{\\infty}v_k (1+ez)^{\\frac{k}{2}} \\text{ for } z \\in \\left(-\\frac{1}{e},0\\right).\n \\end{align*}\nThe discrepancy between the two expansions is given by the two different choices for the branch of the square root. We first consider the odd coefficients $v_{2k-1}$. Setting $w = 1+ez$ we define\n \\begin{align*}\n g(w)= \\frac12 \\sqrt{w} \\left( z W_0'\\left(z\\right) - z W_{-1}'\\left(z\\right) \\right) = \\sum_{k = 0}^{\\infty}v_{2k-1} w^{k}.\n \\end{align*}\n The function $g(w)$ can be analytically continued to $w=0$. Moreover, $g(w)$ has no other singularities in a $\\Delta$-domain as defined in Lemma~\\ref{lmm:singularity_analysis}: the dominant singularity of $g(w)$ comes from the logarithmic singularity of $W_{-1}$ at $z=0$ (see Figure~\\ref{fig:lambertW}), so is located at $w=1$ after the variable change. The principal branch $W_0$ is analytic at $z=0$. Neither $W_0$ nor $W_{-1}$ has any other singularities in the relevant domain.\n\n Because the differential equation $W'(z) = \\frac{W(z)}{z(1+W(z))}$ is satisfied by every branch of the Lambert $W$-function, we have\n \\begin{align*}\n g(w) &= \\frac12 \\sqrt{w} \\left( \\frac{W_{0}(z)}{1+W_{0}(z)} - \\frac{W_{-1}(z)}{1+W_{-1}(z)} \\right) = \n - \\frac12 \\sqrt{w} \\frac{W_{-1}\\left(\\frac{w-1}{e}\\right)}{1+W_{-1}\\left(\\frac{w-1}{e}\\right)} + \\text{`analytic'} \\text{ as } w\\rightarrow 1^- \\\\\n &=\n \\frac12 \\frac{\\sqrt{w}}{1+W_{-1}\\left(\\frac{w-1}{e}\\right)} + \\text{`analytic'} \\text{ as } w\\rightarrow 1^{-},\n \\end{align*}\n where we are able to neglect contributions which are analytic at $w=1$ since, by Lemma~\\ref{lmm:dominant_singularity}, they will eventually contribute only exponentially suppressed terms asymptotically.\n The function $W_{-1}$ has the singular behaviour \\cite[Sec.\\ 4]{corless1996lambertw},\n \\begin{align*}\n W_{-1}(z) = \\log(-z) + \\mathcal{O}\\left(\\log(-\\log(-z))\\right) \\text{ as } z \\rightarrow 0^{-}.\n \\end{align*}\n Thus, we get the singular expansion for $g(w)$,\n \\begin{align*}\n g(w) &= \n \\frac12 \\frac{\\sqrt{ 1- (1-w)}}{1+\\log\\left(\\frac{1-w}{e} \\right)+ \\mathcal{O}\\left(\\log(-\\log(\\frac{1-w}{e}))\\right)}+ \\text{`analytic'} \\text{ as } w\\rightarrow 1^{-}\n\\\\\n &= \n- \\frac12 \\left(\\log\\frac{1}{1-w}\\right)^{-1} + \\mathcal{O}\\left( \\frac{\\log(-\\log\\left(1-w \\right))}{\\left(\\log\\left(1-w \\right)\\right)^2}\\right) + \\text{'analytic'} \\text{ as } w\\rightarrow 1^{-}.\n \\end{align*}\n With this knowledge we may use a more general statement from singularity analysis to extract the asymptotics of the coefficients of $g(w)$, for instance \\cite[Cor.\\ 6]{flajolet1990singularity}. More details are given in \\cite[Sec.\\ VI.2]{flajolet2009analytic}, where one can find the `asymptotic transfer law' $[w^k] \\left(\\log\\frac{1}{1-w} \\right)^{-1}= -\\frac{1}{k(\\log k)^2} + \\mathcal{O}\\left(\\frac{1}{k(\\log k)^3}\\right)$ for $k\\rightarrow \\infty$ in Table VI.5. Also `transferring' the $\\mathcal{O}$ term in the singular expansion of $g$ into its corresponding asymptotic term for the coefficients \\cite[Cor.\\ 6]{flajolet1990singularity} gives,\n \\begin{align*}\n [w^k]g(w)&= v_{2k-1} = \\frac12 \\frac{1}{k (\\log k)^2} + \\mathcal{O}\\left( \\frac{\\log\\log k}{k (\\log k)^3} \\right) \\text{ as } k \\rightarrow \\infty.\n \\end{align*}\n We note that the asymptotic behaviour of the even coefficients $v_{2k}$ follows analogously by starting with \n $g(w)= \\frac12 \\left(- z W_0'\\left(z\\right) - z W_{-1}'\\left(z\\right) \\right) = \\sum_{k = 0}^{\\infty}v_{2k} w^{k}$, although we will not need this for the present article.\n \\end{proof}\n\nThe only remaining task for proving Theorem~\\ref{thm:SVconj} is to transfer our knowledge of the asymptotic behaviour of $v_k$ to the coefficients $\\ch_n$. To deduce the asymptotic behaviour of these coefficients, we will use a classical theorem by Wright in the theory of graphical enumeration.\n\\begin{lemma}[Thm.\\ 2 of \\cite{wright1970asymptotic} with $R=1$] \n \\label{lmm:wright_connected_asymptotics}\nLet $f(x)= \\sum_{n \\geq 0} c_n x^n$ be a power series in $\\mathbb{R}[[x]]$, and let $\\exp(f(x)) = \\sum_{n \\geq 0} \\hat{c}_n x^n$. Suppose \n $c_0 = 0$, $\\hat{c}_0 = 1,$ and $\\hat{c}_{n-1} \\in o(\\hat{c}_{n})$ as $n\\to\\infty$ as well as \n \\begin{align}\n \\label{eqn:center_sum}\n \\sum_{k = 1}^{n-1} \\hat{c}_k \\hat{c}_{n-k} \\in \\mathcal{O}(\\hat{c}_{n-1}) \\text{ as } n\\to \\infty.\n \\end{align}\nThen \n $c_n = \\hat{c}_n + \\mathcal{O}(\\hat{c}_{n-1})$ as $n \\to \\infty$.\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:chi_outfn_asymptotics}]\nLet $T(z) = \\sum_{n\\geq 1} \\ch_n z^n$, and $\\exp(T(z)) = \\sum_{n\\geq 0} \\Ch_n z^n$.\n\n We have to verify that $\\Ch_n$ satisfies the conditions of Lemma~\\ref{lmm:wright_connected_asymptotics}. The only condition that is not immediate is eq.\\ \\eqref{eqn:center_sum}. By Proposition~\\ref{prop:lambert_rep} and Lemma~\\ref{lmm:asymp_vk} we have\n \\begin{align*}\n \\Ch_n &= - \\frac{1}{\\sqrt{2\\pi}} \\frac{\\Gamma(n +\\frac12 )}{n \\log^2 n} + \\mathcal{O}\\left( \\frac{\\log \\log n }{n \\log^3 n}\\Gamma(n +\\frac12 ) \\right)\\\\\n &= - \\frac{1}{\\sqrt{2\\pi}} \\frac{\\Gamma(n -\\frac12 )}{\\log^2 (n+1)} + \\mathcal{O}\\left( \\frac{\\log \\log n }{\\log^3 n}\\Gamma(n -\\frac12 ) \\right).\n \\end{align*}\n From this it follows that we can find a constant $C \\in \\mathbb{R}$ such that $|\\Ch_n| \\leq C \\frac{\\Gamma(n -\\frac12 )}{\\log^2 (n+1)}$ for all $n \\geq 1$.\n Recall that $\\Gamma(x)$ is \\textit{log-convex} on the interval $x\\in(0,\\infty)$, i.e.\\ $\\log(\\Gamma(x))$ is a convex function on this interval \\cite{artin2015gamma}. The function $-\\log(\\log(1+x))$ is convex on this interval as well, since its second derivative $\\frac{1+\\log(1+x)}{(1+x)^2 \\log^2(1+x)}$ is positive. If $f(x)$ is convex on the interval $[a,b]$, then $f(b+a-x)$ is also convex on $[a,b]$. If another function $g(x)$ is convex on this interval, then $f(x) + g(x)$ is too. Therefore, \n \\begin{align*}\n \\log( \\Gamma(n-x-\\frac12) ) + \\log( \\Gamma(x-\\frac12) ) - 2 \\log \\log(1+n-x) - 2 \\log \\log(1+x)\n \\end{align*}\n is convex for $x\\in(\\frac12 , n-\\frac12)$. Because $e^x$ is an increasing function,\n $\\frac{\\Gamma(n-x-\\frac12)\\Gamma(x-\\frac12)}{ \\log^2(1+n-x) \\log^2(1+x) }$\n is also convex on $x\\in(\\frac12, n-\\frac12)$. This also implies convexity on the smaller interval $[2,n-2]$. The usual inequality for convex functions now gives\n \\begin{align*}\n \\frac{\\Gamma(n-x-\\frac12)\\Gamma(x-\\frac12)}{ \\log^2(1+n-x) \\log^2(1+x) }\n \\leq \\frac{\\Gamma(n-2-\\frac12)\\Gamma(2-\\frac12)}{ \\log^2(1+n-2) \\log^2(1+2) } \\text{ for all } x \\in [2,n-2],\n \\end{align*}\n and we can estimate\n \\begin{gather*}\n \\left|\\sum_{k=1}^{n-1} \\Ch_{n-k} \\Ch_k \\right| \\leq 2 | \\Ch_{n-1} \\Ch_1 | +\\sum_{k=2}^{n-2} | \\Ch_{n-k} \\Ch_k| \n \\leq 2 | \\Ch_{n-1} \\Ch_1 | + C^2 \\sum_{k=2}^{n-2} \\frac{\\Gamma(n-k -\\frac12 )\\Gamma(k -\\frac12 )}{\\log^2 (1+n-k)\\log^2 (1+k)} \\\\\n \\leq 2 | \\Ch_{n-1} \\Ch_1 | + C^2 (n-3) \\frac{\\Gamma(n-2 -\\frac12 )\\Gamma(2 -\\frac12 )}{\\log^2 (1+n-2)\\log^2 (1+2)}.\n \\end{gather*}\n It follows that $\\sum_{k=1}^{n-1} \\Ch_{n-k} \\Ch_k \\in \\mathcal{O}(\\Ch_{n-1})$, so Lemma~\\ref{lmm:wright_connected_asymptotics} can be applied to give \n\\begin{align*}\n \\ch_n &= \\Ch_n + \\mathcal{O}(\\Ch_{n-1}) = - \\frac{1}{\\sqrt{2\\pi}} \\frac{\\Gamma(n -\\frac12 )}{\\log^2 n} + \\mathcal{O}\\left( \\frac{\\log \\log n }{\\log^3 n}\\Gamma\\left(n -\\frac12 \\right) \\right),\n\\end{align*}\nbecause $\\Ch_{n-1} \\in \\mathcal{O}\\left( \\frac{\\log \\log n }{\\log^3 n}\\Gamma\\left(n -\\frac12 \\right) \\right)$.\n\\end{proof}\n\nThe asymptotic behavior of $\\chi(\\Out(F_n))$ now follows by combining our results.\n\\begin{repthmx}{thm:SVconj}\n The rational Euler characteristic of $\\Out(F_n)$ is strictly negative, $\\chi\\left( \\Out(F_n) \\right) < 0$, for all $n \\geq 2$ and its magnitude grows more than exponentially,\n \\begin{align*}\n \\chi\\left(\\Out(F_{n}) \\right) &\\sim - \\frac{1}{\\sqrt{2 \\pi}} \\frac{\\Gamma(n-\\frac32)}{\\log^2(n)}.\n \\end{align*}\n\\end{repthmx}\n\n\\begin{proof}\n Apply Corollary~\\ref{crll:negative_T} and Proposition~\\ref{prop:chi_outfn_asymptotics}.\n\\end{proof}\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section*{Supplemental Material}\n\\begin{center}\n{\\Large \\bf Supplemental Material}\\\\ \n\\end{center}\n\\vspace*{12pt}\n\\begin{enumerate}[\\bf I.]\n\n\\item {\\bf Thermodynamic stability of stoichiometric vs non-storichiometric Mg$_M$O$_x$ clusters}\n\n\\item {\\bf Details on the implemented GA schemes}\n\n\\item {\\bf Performance of reaxFF}\n\n\\item {\\bf O$_2$-adsorption energy on MgO$_x$, with functionals corrected by the experimental value of O$_2$ binding energy}\n\n\\item {\\bf O$_2$-adsorption energy on Mg$_2$O$_x$ and Mg$_3$O$_x$ clusters}\n\n\\item {\\bf Mg$_2$O$_x$ phase diagrams with various functionals}\n\n\\item {\\bf Effect of translational, rotational, vibrational contributions to the free energy on Mg$_2$O$_x$ phase diagram}\n\n\\item {\\bf Effect of anharmonic contributions to configurational free energy}\n\n\\item {\\bf Examples of spin densities on non-stochiometric Mg$_M$O$_x$}\n\n \n\\end{enumerate}\n\n\n\n\\begin{figure*}[b!]\n{\\bf \\Large I. Thermodynamic stability of stoichiometric vs non-storichiometric Mg$_M$O$_x$ clusters \\\\}\n\\includegraphics[width=0.8\\columnwidth,clip]{fig-1-SI.png}\n\\caption[]{Free energy of formation of thermodynamically most stable non-stoichiometric (Mg$_M$O$_x$ with $M \\neq x$) relative to stoichiometric ($M = x$) clusters at several $(T,p_{\\textrm{O}_2})$ conditions. The geometries were optimized with PBE+vdW, and the electronic energy was calculated using PBE0+vdW. The label of the horizontal axis shoes, below the amount $M$ of Mg atoms, the amount $x$ of O atoms for the thermodynamically most stable non-stoichiometric cluster at the corresponding $M$, at $p_{\\textrm{O}_2} = 1$~atm and $T=300$ K. For the same thermodynamic condition, the third line reports the spin multiplicity $\\mathcal{M}$ of the lowest free-energy non-stoichiometric Mg$_M$O$_x$ (the stoichiometric clusters are all singlets, i.e., $\\mathcal{M}=1$).}\n\\label{SM:I}\n\\end{figure*} \n\n\\clearpage\n\\newpage\n\\begin{center}\n{\\bf \\Large II. Details on the implemented GA scheme \\\\} {\\bf The benchmark and full detail on validation is found in \\cite{long}\\\\} \n\\end{center}\n\nSchematically, our cGA algorithm proceeds as follows (all terms in italic will be explained afterwards):\n\\begin{enumerate}[(1)]\n \\item Selection of a composition of the clusters and formation of an initial pool of random structures, locally optimized by a classical force field (FF).\n \\item Evaluation of the {\\em fitness} function for all structures (using FF binding energy).\n \\item GA global optimization using the classical FF. This consists in the iteration of steps (i)--(v): \n \\begin{enumerate}[(i)]\n \\item {\\em Selection} of two structures (in GA jargon, {\\em parents}).\n \\item Assemblage of a trial structure ({\\em child}) through {\\em crossover} and {\\em mutation}.\n \\item Local optimization (force minimization) of child structure using the classical FF.\n \\item Evaluation of the {\\em fitness} function. Comparison of the optimized child with existent structures; reject if {\\em similar}, jump to (i). \n \\item Check whether convergence has been reached. If so, stop FF-GA and go the next step, DFT-GA.\n \\end{enumerate}\n \\item Formation of a new pool of structures using best fit structures from FF-GA, locally optimized at the DFT level (PBE+vdW, {\\em low-level settings}).\n \\item Calculation of fitness function for all structures (using energy at the PBE0+vdW level).\n \\item GA scheme using DFT. In practice iteration of steps (a)--(i):\n \\begin{enumerate}[(a)]\n \\item {\\em Selection} of two structures.\n \\item Assemblage of a child structure through {\\em crossover} and {\\em mutation}.\n \\item Local optimization of the child structure with PBE+vdW, {\\em low-level settings}.\n \\item Comparison of the optimized child with existent structures. {\\em Early rejection} if {\\em similar}; jump to (a). \n \\item Further local optimization of the child with PBE+vdW, {\\em high-level settings}.\n \\item Harmonic analysis of the optimized child; if unstable, perturb along the unstable mode and go back to (c). \n \\item Evaluation of {\\em fitness} function based on PBE0+vdW total energy. \n \\item Check whether convergence has been reached. If so, stop.\n \\end{enumerate}\n\\end{enumerate}\n\nIn the following, we analyze one by one the key words introduced in the detailed scheme above.\n\n \n\\subsubsection*{\\bf Fitness function}\n\\label{ff}\nEach cluster $i$ in the population is assigned a normalized fitness value, $\\rho_i$, based on its total energy (binding energy for the FF):\n\\begin{equation}\n\\label{eqn1}\n\\rho_i=\\frac{\\epsilon_i}{\\sum_i\\epsilon_i}\n\\end{equation}\nand $\\epsilon_i$ is the relative energy of the $i^{th}$ cluster as defined below:\n\\begin{equation}\n\\label{eqn2}\n\\epsilon_i=\\frac{E_\\textrm{max}-E_i}{E_\\textrm{max}-E_\\textrm{min}}\n\\end{equation}\nWhere $E_i$ is the total energy of the $i^{th}$ cluster of the population and $E_\\textrm{min}, E_\\textrm{max}$ correspond to the dynamically updated lowest and highest total energies in the population, respectively. \n\nWith this definition, low (more negative) energy clusters have high fitness and high (less negative) energy clusters have low fitness. \n\n\\subsubsection*{\\bf Selection rule}\n\\label{sr}\n We use a ``roulette-wheel'' selection criterion \\cite{roulette} with selection probability proportional to the value of the normalized fitness function. The idea is that the lower the total (or binding) energy (i.e., large negative value) of a certain configuration, the larger the probability to be chosen from the population. A cluster is picked at random and is selected for mating if its normalized fitness value ($\\rho_i$) is greater than $\\textrm{Rand}[0,1]$, a randomly generated number between 0 and 1 (i.e., if $\\rho_i > \\textrm{Rand}[0,1]$); where $\\rho_i$ is the normalized fitness function defined in section-\\ref{ff}. \n \nThe above `` best-fit'' selection scheme can take significantly long time to reach another basin in the PES. In such situations, adding a little diversity by selecting one ``bad'' (high-energy) structure in the population is found to help in moving out to the next basin. Therefore, we define a complementary fitness function $\\tilde{\\rho}_j = (1-\\rho_j)$ and we select one structure with high $\\rho_i$ and another with high $\\tilde{\\rho}_j$. This choice, when the the mixing ratio among different selection rules is optimized, greatly helps the convergence of the GA scheme and we also show how we optimized the mixing ratio among different selection rules.\n\n\\subsubsection*{\\bf Crossover}\n\\label{sec:cross}\nThe crossover operator takes care of combining the two parent clusters selected as explained above. It is implemented as a modified version of the cut-and-splice crossover operator of Deaven and Ho.\\cite{crossover} In our implementation of the cut-and-splice operation, first a random rotation is given (keeping the center of geometry of the cluster at the origin of the coordinate system) to both the parent clusters. Both clusters are then cut horizontally parallel to the $xy$-plane ($z=0$). Atoms with positive $z$-value are selected from one cluster and atoms with negative $z$-value are selected form the other cluster. These complementary fragments are spliced together. Importantly, this cut-and-splice operation does not ensure the preservation of the chosen cluster size (i.e., the total number of atoms) and the specific composition. We have adapted here three different kind of crossovers to maintain size and composition. \n\n(i) Crossover-1: Strictly speaking, this is a combined crossover and mutation (see below) step. After cut-and-splice we always maintain the same ordering of atoms that is given in the parent clusters. As an example, let us consider a small cluster like Mg$_2$O$_2$. In the parent cluster the ordering of atomic coordinate is given as Mg(1), O(2), Mg(3), O(4). When the cut-and-splice operation is applied, we get, for instance, a child with atomic coordinates from cluster one as Mg(1), Mg(3) (i.e., above the $xy$-plane) and that of from cluster two as Mg(3'), O(4') (i.e., below the $xy$-plane). Therefore, the entire atomic coordinates of the child are [Mg(1), Mg(3)], [Mg(3'), O(4')]. If this is the case, we replace the species of Mg(3) with O (without changing its coordinate) to impose the correct composition to the child. Thus the new ordering of atoms is: Mg(1), O(2), Mg(3), O(4) (i.e., the same composition as the parents). Therefore, it is possible that after the cut-and-splice operation a Mg atom of the parent cluster is replaced by an O atom in the child and vice versa.\n\nAlso the total spin of the clusters is left free to evolve together with the spatial coordinates of the atoms. In this way we sample on equal footing the configurational space of atomic coordinates and the spin.\nThe crossover of the spin coordinates is performed in the following way: when we create a new child by grabbing the atomic coordinates from the parents as explained above, we also make note of the atom-projected spin moments (via Hirshfeld partitioning of the electron density) for each atom. Such spin moments are given as initial moments of the individual atoms of the child. During the optimization process, these atom-projected moments are left free to change. \n\n(ii) Crossover-2: In this procedure, after cut-and-splice we check whether the stoichiometry of the parents is maintained in the child. If it is maintained, we accept the child, otherwise we reject it and we iterate until until the child has the required stoichiometry.\\cite{r15,r16} The advantage of this procedure is that it helps to maintain winning features of the parent molecule but most of the time it takes many iterations to obtain a valid child, even for a moderately sized cluster. In case one or more pairs of atom are too close, we adopt the same remedy as for crossover-1. The spin coordinates are taken care of the same as in the crossover-1 case.\n\n(iii) Crossover-3: After re-orientation of the selected parent clusters we take all the metal (Mg) atoms from one parent molecule and all the oxygen atoms from another parent molecule. This crossover helps introducing diversity in the genetic pool, but the rate of rejection during the assemblage of the child can be rather high due to the high likelihood that two atoms are too close.\n\n\\subsubsection*{\\bf Mutation}\n\\label{mut}\nAfter crossover, which generates a child, mutation is introduced.\nDifferent mutation operators can be defined. We have adopted a) a translation between the two halves of the clusters (this is performed if atoms coming from the two different parents find themselves too close upon splicing of the two halves) and b) exchange of the atom species without perturbing their coordinates \n\n\\subsubsection*{\\bf Similarity of structures}\n\\label{ss}\nIn order to decide whether a newly found structure was already seen previously during the GA scan, after the local optimization we a) compare the energy of the new structure with that of all the others seen before and b) use a criterion based on the distances between all the atoms' pairs. In practice, we construct a coarse grained radial distribution function (rdf) of the clusters, consisting of 14 bins conveniently spaced. Each bin contains the (normalized) number of atom-pairs whose distance is between the distances that define the boundaries of the bin. For each cluster we have then a 14-dimensional rdf-array \nand the euclidean distance (i.e., the square root of the sum of the squared difference between corresponding elements in the two arrays) between the arrays arranged for two clusters is evaluated.\nIf this distance (note that it is a pure number) is greater than a convenient threshold (we used 0.01), then the structures are considered as different.\n\n\\subsubsection*{\\bf Local optimization and early rejection scheme}\n\\label{cascade}\n\nAlthough the geometry and the energy of the structures is not fully converged with PBE+vdW @ {\\em low-level settings}, we have realized that there is a one-to-one mapping between the structures found at this level and those fully converged. In other words, if two structures are {\\em similar} according to PBE+vdW @ {\\em low-level settings}, they are also at the PBE0+vdW @ {\\em high-level settings} (see below). Furthermore, if a structure at the PBE0+vdW @ {\\em high-level settings} is within $\\sim 0.5$ eV from the running GM, with PBE+vdW @ {\\em low-level settings} the structure is not further than 0.2 eV from the same running GM (with energy evaluated with PBE+vdW @ {\\em low-level settings}). This implies that, with our conservative choice of not optimizing with the {\\em high-level settings} structures that with {\\em low-level settings} result positive to the {\\em similarity} test or are more than 1.5 eV higher in energy than the current GM, we are not risking to reject structures that would eventually result in the GM or close to it. \n\nIn the PBE+vdW, {\\em high-level settings} optimization, atomic forces were converged to less than $10^{-6}$ eV\/\\AA. The grid settings where set to ``tight'' and the basis set was tier-2. In cascade, for the structure optimized in this way (i.e., without further optimization), we evaluated the PBE0+vdW energy with the tier-2 basis set. This energy is later used for the calculation of fitness of that particular cluster. \nThe difference in binding energy between an isomer optimized with PBE0+vdW forces (tight \/ tier 1 \/ forces converged to 10$^{-5}$ eV\/\\AA) and the same optimized with PBE+vdW (tight \/ tier 2 \/ forces converged to 10$^{-5}$ eV\/\\AA), when the energy of both geometries is evaluated via PBE0+vdW (tight \/ tier 2), is small, i.e. at most 0.04 eV among all cases we checked. The computational cost of the PBE0+vdW further optimization would be thus not worthy (we estimated a gain of up to a factor 2 of overall computational time just by skipping the latter optimization)\n\n\\subsubsection*{\\bf Parallelization}\nThe operation of selecting from the genetic pool two structures for the mating and the subsequent local optimization of the child, is an operation that can be performed at any moment also when a local optimization of a child is already running.\nThe algorithm is thus suitable for a very efficient parallelization. \n\nOn top of FHI-aims parallelization (i.e. local optimization is run in parallel on an optimized number of CPUs) we add a second level of parallelization, i.e., we run at the same time several local optimizations, independently. The only communication among such replicas is the selection of the parents that is performed from a common genetic pool. The latter is also updated by each replica at the end of each local optimization.\nThe local optimizations run independently, i.e., each replica can start a new mating + local optimization cycle right after one is concluded; hence, there is no idling time between cycles.\nThus, we have $n$ local optimizations running in parallel, each requiring $p$ cores\n, that fill the $n\\times p$ cores required for the algorithm. The scaling behavior is about O($p^{1.5}$) with the number of cores for the local optimization part \\cite{aimsp} . The number $p$ is indeed tuned in order to be sure that the speed-up is still O($p^{1.5}$) for the specific system. The scaling with respect to the $n$ replicas is linear, because the replicas are for the most of the time independent and only at the beginning and at the end of each local optimization, information is shared among the replicas.The first level of parallelization is performed within the FHI-aims code, by means of the MPI environment. The second level is script based: The total $n\\times p$ number of cores is divided into $n$ groups, $n$ subdirectories are created and in each of them a cycle of local optimization job runs, each using $p$ cores.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\begin{figure}[t]\n\\includegraphics[width=0.45\\textwidth]{.\/images\/Intro_v3.pdf}\n\\vspace{-0.3cm}\n\\centering\n\\caption{(a) T-SNE \\cite{van2013barnes} visualization of\nthe BEV features in different UDA scenarios, in which features are obviously separated by domain. (b) The performance of our method compared with previous works \\cite{li2022bevdepth,wang2021exploring,li2022unsupervised}. Both methods are built on BevDepth with a ResNet-50 backbone and evaluated on the target domain.}\n\\label{fig:intro}\n\\vspace{-0.4cm}\n\\end{figure}\nThe camera-based 3D object detection has attracted increasing attention, especially in the field of autonomous driving \\cite{arnold2019survey, chen2017multi, chen2016monocular}. Nowadays, it has obtained obvious advancements driven by Bird-Eye-View (BEV) perception methods \\cite{philion2020lift, huang2021bevdet, li2022bevdepth,li2022unifying, li2022bevstereo} and large scale labeled autonomous driving datasets \\cite{geiger2012we, caesar2020nuscenes, sun2020scalability}. However, due to the vast variety of perception scenes \\cite{wang2021exploring, li2022unsupervised}, the camera-based methods can suffer significant performance degradation caused by domain shift or data distribution variation.\n\n\nRecently, though mono-view 3D detection methods \\cite{li2022towards, li2022unsupervised} carry out the UDA setting on camera parameters or annotation methods variation, domain adaptation problem on many real-world scenarios is still unexplored in both Mono-view \\cite{cai2020monocular, wang2021fcos3d, brazil2019m3d, ding2020learning, li2022diversity, zhang2021objects, simonelli2019disentangling} and Multi-view\\cite{philion2020lift, wang2022detr3d, liu2022petr, liu2022petrv2, chen2022polar, jiang2022polarformer, li2022bevformer, reading2021categorical, huang2021bevdet, li2022unifying, li2022bevdepth} settings. As shown in Fig.\\ref{fig:intro}, we discover the tremendous domain gap in the scene, weather, and day-night changing scenarios from source to target data in nuscenes \\cite{caesar2020nuscenes}, which leads to inferior performance of baseline \\cite{li2022bevdepth} (only 0.174, 0.159, and 0.05 NDS). Therefore, we attempt to transfer a multi-view 3D detector from a labeled source domain to an unlabeled target domain.\n\nIn UDA, the main challenge for Multi-view LSS-based BEV perception is the entangle of domain shift on multiple latent spaces: \n(1)\\textit{2D images latent space. } Since multi-view images contain abundant semantic information, it will result in a manifold domain shift when scenarios change.\n(2) \\textit{3D voxel latent space.}\nVoxel features that are constructed by domain-specific images feature and unreliable depth prediction will assemble more domain shift. \n(3) \\textit{BEV latent space.}\nDue to the shift in the above aspects, the constructed BEV feature results in an accumulation of domain shift and leading to alignment noises. As shown in Fig.\\ref{fig:intro} (a), we visualize the distribution of BEV features, which are obviously separated by domains on different cross domain scenarios. \n\n\n\n\n\n\n\n\n\n\nTo this end, we propose a novel Multi-level Multi-space Alignment Teacher-Student ($M^{2}ATS$) framework to disentangle domain shift problems in multiple latent spaces, which consists of a Depth-Aware Teacher (DAT) model and Multi-space Feature Aligned (MFA) Student model. \nDAT introduces composite depth-aware information to construct better voxel and BEV features in the target domain, which contains less source domain specific information. \nTo construct composite depth-aware information, DAT adaptively screened out reliably predicted depth by uncertainty estimation to compensate for lidar. It promotes representation consistency between DAT and student model through transferring domain-invariant multi-space knowledge and pseudo labels, thus addressing domain shift in pixel and instance level respectively. \nIn order to assist DAT in further bridging global level domain gaps on multiple latent spaces, we propose MFA in the student model. It aligns three task-relevant features of two domains, including multi-view images, 3D voxel, and BEV features. In the overall $M^{2}ATS$ framework, DAT and MFA compensate each other to address the domain shift of multiple latent spaces in multi-level.\n\nAs far as we know, we are the first to study the cross domain problem in Multi-View 3D object detection. We design three classical and one continual changing UDA scenarios, which are \\textbf{Scene} (from Boston to Singapore), \\textbf{Weather} (from sunny to rainy and foggy), \\textbf{Day-night}, and \\textbf{Changing Foggy degree} in \\cite{caesar2020nuscenes}. For continual changing scenarios, we construct cross domain experiments with the continuously increased density of Fog, which gradually enlarges the domain gap. Our proposed method achieves competitive performance in all scenarios (shown in Fig. \\ref{fig:intro} (b)). Compared with the previous state-of-the-art (SOTA) UDA method (i.e., STM3D\\cite{li2022unsupervised}), it improves the NDS by 2.5, 3.2, and 5.2\\% respectively in three classical scenarios.\n\nThe main contributions are summarized as follows:\n\n\\textbf{1)} We explore the unsupervised domain adaptation (UDA) problem for BEV perception of Multi-view 3D object detection. We propose a novel Multi-level Multi-space Alignment Teacher-Student ($M^{2}ATS$) framework to address multi-latent space domain shift.\n\n\\textbf{2)} In $M^{2}ATS$, we propose a Depth-Aware Teacher (DAT) to construct uncertainty-guided depth-aware information, and then transfer the domain-invariant pixel-level features and instance-level pseudo label to the student model. $M^{2}ATS$ contains a Multi-space Feature Aligned (MFA) student model which aligns multi-space features between two domains and compensates DAT to further alleviate the domain shift at global-level.\n\n\\textbf{3)} We conduct extensive experiments on the four challenging UDA scenarios, achieving SOTA performance compared with previous Mono-view 3D and 2D detection UDA methods. And we provide a Foggy-nuScene dataset.\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{.\/images\/framework_v8.pdf}\n\\centering\n\\vspace{-0.22cm}\n\\caption{The framework of Multi-level Multi-space Alignment Teacher-Student ($M^{2}ATS$), which is composed of the Depth-Aware Teacher (DAT) and Multi-space Feature Aligned (MFA) student model. In \\textbf{the bottom part}, the DAT model takes target domain input and adopts depth-aware information to construct domain-invariant features, which transfers pixel (multi-space features) along with instance-level (pseudo label) knowledge to the student model. In \\textbf{the upper part}, the MFA student model aligns multi-space features (\\textcolor{red}{red circle}) at the global level between two domains. $M^{2}ATS$ framework aims to comprehensively address the multi-latent space domain shift problem.}\n\\label{fig:method}\n\\vspace{-0.42cm}\n\\end{figure*}\n\\section{Related work}\n\\subsection{Camera-based 3D object detection}\nNowadays, 3D Object Detection plays an important role in autonomous driving and machine scene understanding. \nTwo paradigms are prominent in this aspect: Single-view \\cite{cai2020monocular, wang2021fcos3d, brazil2019m3d, ding2020learning, liu2021autoshape, manhardt2019roi, barabanau2019monocular, li2022diversity, zhang2021objects, simonelli2019disentangling} and Multi-view\\cite{philion2020lift, wang2022detr3d, liu2022petr, liu2022petrv2, chen2022polar, jiang2022polarformer, li2022bevformer, reading2021categorical, huang2021bevdet, li2022unifying, li2022bevdepth, huang2022bevdet4d, li2022bevstereo}. In Single-view detection, previous works can be categorized into several streams, i.e. leveraging CAD models \\cite{liu2021autoshape, manhardt2019roi, barabanau2019monocular}, setting prediction targets as key points \\cite{li2022diversity, zhang2021objects}, and disentangling transformation for 2D and 3D detection \\cite{simonelli2019disentangling}. Specifically, FCOS3D \\cite{wang2021fcos3d} can predict 2D and 3D attributes synchronously. M3D-RPN \\cite{brazil2019m3d} considers single-view 3D object detection task as a standalone 3D region proposal network. In order to establish a more reliable 3D structure, D4LCN \\cite{ding2020learning} alters 2D depth prediction with pseudo LiDAR representation. \\cite{cai2020monocular} calculates the depth of the objects by integrating the actual height of the objects. To better leverage the depth information in the detection process, MonoDTR\\cite{huang2022monodtr} proposes an end-to-end depth-aware transformer network. However, taking into account the precision and practicality of detection, more and more multi-view 3D object detectors are proposed. \n\n\nThe Multi-view paradigm can be categorized into two branches, namely transformer-based \\cite{carion2020end} and LSS-based \\cite{philion2020lift}. \nFirst of all, to extend DETR \\cite{carion2020end} into 3D detection, DETR3D \\cite{wang2022detr3d} first predicts 3D bounding boxes with a transformer network. Inspired by DETR3D, some works adopt object queries \\cite{liu2022petr, liu2022petrv2, chen2022polar, jiang2022polarformer} or BEV grid queries \\cite{li2022bevformer} to extract features from images and utilize attention method, resulting in better 2D-to-3D transformation. However, transformer-based methods don't project image features onto BEV representation. Following LSS \\cite{philion2020lift}, some methods \\cite{reading2021categorical, huang2021bevdet, li2022unifying} predict a distribution over lidar depth and generate a point cloud with multi-view image features for 3D detection. Specifically, BevDepth \\cite{li2022bevdepth} introduces depth supervision and speeds up the operation of voxel pooling. Bevdet4d \\cite{huang2022bevdet4d} and BevStereo \\cite{li2022bevstereo} thoroughly explore temporal information in the task and concatenate volumes from multiple time steps. In this paper, we adopt BevDepth \\cite{li2022bevdepth} as the baseline 3D object detector for its simple and powerful working flow, along with its great potential in cross domain feature extraction.\n\n\\subsection{UDA in 3D object detection}\nDomain Adaptive Faster R-CNN \\cite{chen2018domain} first probes the cross domain problem in object detection. Based on \\cite{ganin2015unsupervised}, most previous works \\cite{cai2019exploring, saito2019strong, wang2021exploring, xu2020exploring, xu2020cross, yu2022cross} follow the cross domain alignment strategy\nand explore the influence of domain shift in multi-level features. As for 3D object detection, \\cite{luo2021unsupervised, li2022unsupervised, zhang2021srdan} investigate Unsupervised Domain Adaptation (UDA) strategies for point cloud 3D detectors. In particular, \\cite{luo2021unsupervised, zhang2021srdan} adopt alignment methods to align the feature and instance level information between two domains. STM3D \\cite{li2022unsupervised} develop self-training strategies to realize UDA by consistent and high-quality pseudo\nlabels. Recently, some works \\cite{barrera2021cycle, acuna2021towards, ng2020bev, saleh2019domain} investigate the cross domain strategies in BEV perception, which aim to reduce the simulation-to-real domain shift. In terms of camera-based monocular 3d object detection, \\cite{li2022towards, li2022unsupervised} first attempt to disentangle the camera parameters and guarantee the geometry consistency in cross domain phase. In contrast, we dedicate ourselves to solving the domain gap in multi-view 3d object detection tasks, which infer 3D scenes from the BEV perspective. We propose a novel Multi-level Multi-space Alignment\nTeacher-Student framework to deal with the accumulation of domain shift on multi-latent spaces.\n\n\n\\section{Methods}\n\\subsection{Overall framework}\n\\label{sec:overall}\nIn this paper, we study the Unsupervised Domain Adaptation (UDA) problem in LSS-based Vision-Centric Bird-Eye-View (BEV) perception \\cite{philion2020lift,li2022bevdepth}, where the accumulation of multi-latent space domain shift existed. As shown in Fig. \\ref{fig:method}, \\textbf{Multi-level Multi-space Alignment Teacher-Student} ($M^{2}ATS$) framework contains a Depth-Aware Teacher (DAT) and Multi-space Feature Aligned (MFA) Student model. Along with transferring multi-latent space knowledge from DAT to the student model, the MFA student model simultaneously aligns the task-relevant features between two domains in the same latent spaces. We discuss each component and its interactions in the following.\n\n\\textbf{Depth-Aware Teacher model}. As shown in Fig. \\ref{fig:method}, DAT extracts features and generates pseudo label from the target domain data, and transfers the pixel and instance-level knowledge to the student model. Specifically, due to the unreliable depth prediction and sparse property of target lidar, it suffers domain shift and incompletion when constructing voxel and BEV features. \nTherefore, we introduce a depth-aware mechanism that completes the sparse lidar by the predicted depth and adopts uncertainty estimation to reserve reliable depth prediction. DAT then adopts depth-aware information as an intermediate to construct domain-invariant voxel and BEV features and generates reliable pseudo labels to fully exploit target domain data. The goal is to align the representation of DAT and the student model, thus alleviating domain shift accumulation.\n\n\\textbf{Multi-space Feature Aligned} student model.\nAs shown in Fig. \\ref{fig:method}, the MFA student model receives the transferred knowledge and deals with domain shift at the global level. It extracts features from both source and target domains while introducing an alignment mechanism to pull close the representation of two domains in multi-latent space. The alignments lie in the 2D image, voxel, and BEV features, which aim to obtain domain-invariant features and alleviate domain shift at global level. MFA and DAT compensate each other to address the domain shift in the same latent spaces.\n\n\\subsection{Preliminary}\n\\label{sec:setup}\n\n\n\nFor the UDA setting \\cite{zhao2020review}, we are provided by labeled source domain $D_{s} = \\{\\{I^i_{s}\\}^M_{j=1},L^i_{s},G^i_{s}\\}^{N_{s}}_{i=1}$ and unlabeled target domain $D_{t} = \\{\\{I^i_{t}\\}^M_{j=1},L^i_{s}\\}^{N_{t}}_{i=1}$ of N samples and M camera views, in which $I^i$, $L^i$, and $G^i$ denote images, lidar, and detection ground truth respectively. We adopt an encoder \\cite{he2016deep} to extract image features, and use lidar data $L^i$ to supervise its depth prediction. These joint projects to 3D voxel feature $VF^i$, which are then voxel pooled to BEV feature $BF^i$. Note that, we only utilize lidar supervision during training following previous camera-based works \\cite{li2022bevdepth}.\n\n\\subsection{Depth-aware teacher}\n\\label{sec:DAT}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.38\\textwidth]{.\/images\/unc_v3.pdf}\n\\centering\n\\vspace{-0.2cm}\n\\caption{The detailed process of constructing depth-aware information. The uncertainty map is estimated by MC Dropout \\cite{gal2016dropout}.}\n\\label{fig:unc}\n\\vspace{-0.38cm}\n\\end{figure}\n\nSince detection perception is generated from voxel and BEV features which are initially constructed by depth prediction, the performance depends heavily on the accuracy of the depth sub-network estimation \\cite{li2022bevdepth,li2022bevstereo}. However, cross domain transferring significantly aggravating depth prediction error \\cite{liu2022unsupervised}. Therefore, in Depth-Aware Teacher (DAT) model, our goal is to leverage depth-aware information to construct domain-invariant depth information along with corresponding voxel and BEV features.\n\nTo this end, we construct composite depth information in the teacher model by combining sparse lidar data with reliable and domain-invariant depth prediction. Note that, domain shift lies in the original depth prediction of the target domain, we thus adaptively screened out reliable prediction of the target domain by depth uncertainty estimation. In contrast to uncertainty guidance, a more simple selection mechanism, which selects based on confidence score, moves decision boundaries to low-density regions in sampling-based approaches. However, it may lead to inaccuracy due to the poor calibration of neural networks. Moreover, the domain gap issue will result in more severe miscalibration \\cite{guo2017calibration}. Therefore, we adopt Dropout methods \\cite{gal2016dropout,ghiasi2018dropblock} with uncertainty measurement instead of confidence predictions. Specifically, similar to \\cite{rizve2021defense}, instead of designing a particular uncertainty measure, we pioneering introduce uncertainty guidance to ignore depth prediction of the source domain specified. And we offer a new solution to address the domain shift in dense prediction task, which leverages an uncertainty map to select reliable and domain-invariant pixel prediction in the target domain. When utilizing reliable depth information to build the following features in the target, the source data trained model has a better feature representation and concentrates more on depth estimation tasks without domain shift influence. \nThe uncertainty map of depth prediction is: \n\n\\begin{equation}\n \n D^i_{map} = \\left\\{\n \\begin{array}{l}\n 1 , \\quad \\mathcal{U}(D^i)\\le \\mathcal{E}_{thresh}\\\\ \n 0 , \\quad \\mathcal{U}(D^i)\\ge \\mathcal{E}_{thresh}\n \\end{array}\n\\right.\n\\end{equation}\nwhere $\\mathcal{U}(D^i)$ is the uncertainty of $i^{th}$ pixel depth prediction, and $\\mathcal{E}_{thresh}$ is the selected threshold, which is decided on the variance of the uncertainty values. As shown in Fig.\\ref{fig:unc}, we utilize an uncertainty map to obtain reliable depth prediction and composite target sparse Lidar data, constructing depth-aware information.\n\nInspired by the prevalent soft teacher \\cite{sohn2020simple, xu2021end}, the rest of the teacher model is built with mean teacher mechanism \\cite{tarvainen2017mean}. The weights of teacher model $\\mathcal{T}_{DAT}$ at time step $t$ is the exponential moving average (EMA) of successive student model $\\mathcal{S}_{TGT.}$, shown below:\n\\begin{equation}\n\\label{eq:2}\n \\mathcal{T}_{DAT.}^{t} = \\alpha \\mathcal{T}_{DAT.}^{t-1} + (1-\\alpha) \\mathcal{S}_{TGT.} ^{t}\n\\end{equation}\n, where $\\alpha$ is a smoothing coefficient. With the help of depth-aware information and mean-teacher mechanism, the DAT model can continuously transfer multi-latent space (\\ie, images, voxel, BEV, output) knowledge to the student, including pixel-level features and instance-level pseudo labels. The student model thus can concentrate more on task-driven knowledge learning without domain shift.\n\n\\subsection{Multi-space feature aligned student}\n\\label{sec:MFA}\nIn student model learning, aside from the pixel and instance-level transferred knowledge from DAT, we further introduce Multi-space Feature Alignment (MFA) method to deal with the domain shift at the global level. In order to further alleviate the domain specific feature influence, we pull close the features representation of the source and target domain in multiple latent spaces, including 2D images, voxel, and BEV. Specifically, we utilize the domain adversarial training method \\cite{ganin2016domain} which inserts gradient reversal layers and reverses back-propagated gradients for domain discriminators to optimize the model and extract domain-invariant features. Different from the previous alignment method \\cite{wang2021exploring, yu2022cross}, LSS-based approaches hold multi-space features so that we align multi-space global information synchronously. The MFA loss $\\mathcal{L}_{MFA}$ is shown in Eq.\\ref{eq:MFA}, where ${F}_{s,l}$ and ${F}_{t,l}$ is the source and target domain features at $l$ latent space $(L\\in\\{images, voxel, BEV\\})$ and $D$ denotes the domain discriminator.\n\\begin{equation}\n\\label{eq:MFA}\n\\mathcal{L}_{MFA}({F}_{s},{F}_{t}) = \\sum_{l\\in L}(\\log{D}({F}_{s,l}) + \\log(1-{D}({F}_{t,l}))\n\\end{equation}\n\nIn the $M^{2}ATS$ framework, DAT and MFA compensate each other to deal with the multi-latent spaces domain shift problem in multi-level. \n\n\n\n\\subsection{Training objectives and inference}\n\\label{sec:loss}\nThe $M^{2}ATS$ framework contains a Depth-Aware Teacher (DAT) model and Multi-space Feature Aligned (MFA) Student model, the teacher model is updated by EMA operation. When adopting the DAT model to transfer multi-space features to the student, the knowledge transfer loss $\\mathcal{L}_{MKT}$ is shown as follows:\n\\begin{equation}\n\\mathcal{L}_{MKT} = \\sum_{l\\in L}\\frac{1}{W_{l}'\\times H_{l}'}\\sum_{i\\in \\mathcal{P}}||F_{Te,l}^{{i}}-F_{St,l}^{{i}}||^{2}\n\\label{eq:KT}\n\\end{equation}\nwhere $F_{Te,l}^{i}$ and $F_{St,l}^{i}$ stand for the $i^{th}$ pixel value from DAT model and student model at $l$ latent space, $L\\in\\{images, voxel, BEV\\}$ . $W_{l}^{'}$ and $H_{l}^{'}$ stand for width and height of the transferred features, $\\mathcal{P} =\\{1,2,..,W_{l}'\\times H_{l}'\\}$. We thus pull close the distance of the feature between model $\\mathcal{T}_{DAT}$ and $\\mathcal{S}_{TGT}$ with MSE loss. With the help of pixel-level domain-invariant knowledge transfer, we can reduce the domain shift in the three latent spaces. Meanwhile, the integrated domain adaptation loss $\\mathcal{L}_{DA}$ is shown in Eq.\\ref{eq:domain}.\n\\begin{equation}\n \\mathcal{L}_{DA} = \\lambda_1*\\mathcal{L}_{UNC} + \\lambda_2*\\mathcal{L}_{MKT} + \\lambda_3*\\mathcal{L}_{MFA}\n\\label{eq:domain}\n\\end{equation}\n, where $\\mathcal{L}_{UNC}$ is the detection loss \\cite{li2022bevdepth} penalized by target domain pseudo label, which is generated by teacher model.\nIn order to maintain the balance of loss penalties, $\\lambda_1$ is set to 1, $\\lambda_2$ and $\\lambda_3$ is set to 0.1.\nDuring inference, same with other camera-based methods \\cite{li2022bevdepth,li2022bevformer,li2022bevstereo}, we only adopt multi-view camera data.\n\n\n\n\n\\section{Evaluation}\nWe conduct extensive experiments to demonstrate the effectiveness of the Multi-level Multi-space Alignment Teacher-Student ($M^{2}ATS$) framework. \nIn Sec~\\ref{sec:4.1}, the details of the setup of UDA scenarios and implementation details are given. \nIn Sec~\\ref{sec:4.2}, we evaluate the cross domain performance of $M^{2}ATS$ in the three classical and one continual changing scenarios, including scene, weather, day-night, and continuously increased foggy degree Adaptation. \nThe comprehensive ablation studies are conducted in Sec~\\ref{ablation}, which investigate the impact of each component. \nFinally, we provide qualitative analysis to better present the effectiveness of our proposed framework in Sec~\\ref{sec:4.4}.\n\n\n\\begin{table*}[t]\n \\centering\n \\caption{Results of different methods for scene adaptation scenario on the validation set \\cite{caesar2020nuscenes}, from Boston to Singapore. DA means utilizing the domain adaption method, and R50 and R101 adopt Resnet 50 and 101 as the backbone.} \n \\vspace{-0.2cm}\n \n \\setlength{\\tabcolsep}{1.3mm}{\n \\begin{tabular}{c|c|c|c|cccccc}\n \\hline\n & Method & Backbone & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 & mATE \u2193 & mASE \u2193 & mAOE \u2193 & mAVE \u2193 & mAAE \u2193 \\\\\n \\hline\\hline\n & BEVDet\\cite{ huang2021bevdet} & R50 & 0.126 & 0.117 & 0.873 & 0.787 & 1.347& 1.302 & 0.666 \\\\\n Baseline & BEVDepth\\cite{li2022bevdepth} & R50 & 0.174 & 0.115 & 0.888 & 0.412 & 1.031 & 1.056 & 0.527 \\\\\n & BEVDepth\\cite{li2022bevdepth} & R101 & 0.187 & 0.115 & 0.874 & 0.391 & 0.944 & 1.021 & 0.501 \\\\\n \\hline\n \\multirow{2}{*}{DA} & SFA\\cite{wang2021exploring}(BEVDepth) & R50 & 0.181 & 0.124 & 0.856 & 0.411 & 1.023 & 1.075 & 0.540 \\\\\n \n & STM3D\\cite{li2022unsupervised}(BEVDepth) & R50 & 0.183 & 0.129 & 0.840 & 0.421 & 1.050 & 1.055 & 0.550 \\\\\n \n \\hline\n \\multirow{2}{*}{DA} & Ours(BEVDepth) & R50 & \\textbf{0.208} & \\textbf{0.148} & \\textbf{0.813} & \\textbf{0.402} & \\textbf{0.907} & \\textbf{1.134} & \\textbf{0.536} \\\\\n &Ours(BEVDepth) & R101 & \\textbf{0.211} & \\textbf{0.166} & \\textbf{0.758} & \\textbf{0.427} & \\textbf{1.127} & \\textbf{1.108} & \\textbf{0.535} \\\\\n\n \\hline\n \n \n \n \\end{tabular}%\n }\n \\label{tab:scene}%\n \\vspace{-0.1cm}\n\\end{table*}%\n\n\\begin{table*}[t]\n \\centering\n \\caption{Results of different methods for weather adaptation scenarios on the validation set \\cite{caesar2020nuscenes}, from Sunny to Rainy and Foggy-3.} \n \\vspace{-0.2cm}\n \n \\setlength{\\tabcolsep}{1.3mm}{\n \\begin{tabular}{c|c|c|ccc|ccc}\n \\hline\n & & & & Target Rainy & & & Target Foggy-3 & \\\\\n & Method & Backbone & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 & mATE \u2193 & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 & mATE \u2193 \\\\\n \\hline\\hline\n & BEVDet\\cite{ huang2021bevdet} & R50 & 0.232 & 0.207 & 0.818 & 0.135 & 0.072 & 0.867 \\\\\n Baseline & BEVDepth\\cite{li2022bevdepth} & R50 & 0.268 & 0.196 & 0.824 & 0.159 & 0.079 & 0.882 \\\\\n & BEVDepth\\cite{li2022bevdepth} & R101 & 0.272 & 0.212 & 0.842 & 0.202 & 0.122 & 0.804 \\\\\n \\hline\n \\multirow{2}{*}{DA} & SFA\\cite{wang2021exploring}(BEVDepth) & R50 & 0.281 & 0.200 & 0.840 & 0.228 & 0.133 & 0.840 \\\\\n \n & STM3D\\cite{li2022unsupervised}(BEVDepth) & R50 & 0.276 & 0.212 & 0.820 & 0.234 & 0.145 & 0.721 \\\\\n \n \\hline\n \\multirow{2}{*}{DA} & Ours(BEVDepth) & R50 & \\textbf{0.305} & \\textbf{0.243} & \\textbf{0.819} & \\textbf{0.266} & \\textbf{0.173} & \\textbf{0.805} \\\\\n & Ours(BEVDepth) & R101 & \\textbf{0.308} & \\textbf{0.247} & \\textbf{0.726} & \\textbf{0.271} & \\textbf{0.174} & \\textbf{0.793} \\\\\n \\hline\n \n \n \n \\end{tabular}%\n }\n \\label{tab:weather}%\n \\vspace{-0.20cm}\n\\end{table*}%\n\n\\begin{table*}[t]\n \\centering\n \\caption{Results of different methods for day-night adaptation scenario on the validation set \\cite{caesar2020nuscenes}. } \n \\vspace{-0.25cm}\n \n \\setlength{\\tabcolsep}{1.1mm}{\n \\begin{tabular}{c|c|c|c|cccccc}\n \\hline\n & Method & Backbone & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 & mATE \u2193 & mASE \u2193 & mAOE \u2193 & mAVE \u2193 & mAAE \u2193 \\\\\n \\hline\\hline\n & BEVDet\\cite{ huang2021bevdet} & R50 & 0.010 & 0.009 & 0.990 & 0.977 & 1.078 & 1.509&0.984 \\\\\n Baseline & BEVDepth\\cite{li2022bevdepth} & R50 & 0.050 & 0.012 & 0.042 & 0.646 & 1.129 & 1.705 & 0.915 \\\\\n & BEVDepth\\cite{li2022bevdepth} & R101 & 0.062 & 0.036 & 1.033 & 0.706 & 0.973 & 1.447 & 0.895 \\\\\n \\hline\n \\multirow{2}{*}{DA} & SFA\\cite{wang2021exploring}(BEVDepth) & R50 & 0.092 & 0.032 & 0.995 & 0.556 & 0.993 & 1.480 & 0.948 \\\\\n \n & STM3D\\cite{li2022unsupervised}(BEVDepth) & R50 & 0.070 & 0.035 & 0.979 & 0.549 & 1.063 & 1.587 & 0.937 \\\\\n \n \\hline\n \\multirow{2}{*}{DA} & Ours(BEVDepth) & R50 & \\textbf{0.132} & \\textbf{0.054} & \\textbf{0.711} & \\textbf{0.465} & \\textbf{1.072} & \\textbf{1.504} & \\textbf{0.772} \\\\\n & Ours(BEVDepth) & R101 & \\textbf{0.188} & \\textbf{0.127} & \\textbf{0.189} & \\textbf{0.484} & \\textbf{0.820} & \\textbf{1.784} & \\textbf{0.711} \\\\\n \\hline\n \n \n \n \\end{tabular}%\n }\n \\label{tab:day}%\n \\vspace{-0.1cm}\n\\end{table*}%\n\n\n\n\\subsection{Experimental setup}\n\\label{sec:4.1}\n\\subsubsection{Datasets and adaptation scenarios}\nWe evaluate our proposed framework on nuscenes \\cite{caesar2020nuscenes}, which is a large-scale autonomous-driving dataset. In order to pave the way for Unsupervised Domain Adaptation (UDA) in multi-view 3d object detection, we split the nuscenes into different paired source-target domain data. We introduce three classical and challenging cross-domain scenarios, which are \\textbf{Scene}, \\textbf{Weathers}, and \\textbf{Day-Night} adaptation. In addition, Due to the lack of foggy weather (a normal target data), we generate a foggy dataset (Foggy-nuscenes) in various dense degrees inspired by Foggy-Cityscapes \\cite{sakaridis2018semantic}. The Foggy-nuscenes dataset will be released for research. Besides, we also evaluate $M^{2}ATS$ in the continual changing scenario of increased foggy degrees.\n\n\\textbf{Scene Adaptation} We set Boston as the source scene data and realize UDA on the Singapore target domain. Since scene layouts are frequently changing in autonomous driving, the domain gap occurs in multiple scenes. \n\n\\textbf{Weathers Adaptation} The sunny weather is considered as source domain data, rainy and foggy weather is considered as target domain data. Various weather conditions are common phenomena in the real world, and multi-view 3d object detection should be reliable under such conditions. \n\n\\textbf{Day-Night Adaptation} We design daytime data as the source domain and realize UDA on the target domain (night data). Since the camera-based method has a tremendous domain gap from day to night, it is essential to explore the domain adaptation method in the day-night scenario. \n\n\\textbf{Continues Changing Adaptation} We set sunny weather data as the source domain and set continuously increased foggy degree data as the target domain. Specifically, we realize UDA from sunny to Foggy-1, Foggy-3, and Foggy-5 step by step with the degree of fog increased. Since the continual changing domain gap usually appears in autonomous driving, the domain shift is essential to be addressed. Moreover, we demonstrate the detailed information and generation method of the dataset in the appendix.\n\n\n\n\n\n\\subsubsection{Implementation details}\n$M^{2}ATS$ framework is built based on BEVDepth \\cite{li2022bevdepth}. According to previous work \\cite{li2022bevdepth, reading2021categorical, huang2021bevdet, li2022unifying}, ResNet-50 and ResNet-101 \\cite{he2016deep} serve as backbone to extract image features respectively. We adopt $256\\times 704$ as image input size and the same data augmentation methods as \\cite{li2022bevdepth}. We apply AdamW \\cite{loshchilov2017decoupled} optimizer with 2e-4 learning rate and without any decay. For training, the source domain pretrain and UDA experiments are trained for 24 and 12 epochs \\textbf{without CBGS} \\cite{zhu2019class}. During inference, our method infers without any test time augmentation or model ensemble. We report the evaluation metrics following previous 3d detection works\\cite{li2022bevdepth,huang2021bevdet}, including nuScenes Detection Score (NDS), mean Average Precision (mAP), as well as five True Positive (TP) metrics including mean Average Translation Error (mATE), mean Average Scale Error (mASE), mean Average Orientation Error (mAOE), mean Average Velocity Error (mAVE), mean Average Attribute Error (mAAE). All experiments are conducted on NVIDIA Tesla V100 GPUs. \n\n\n\\subsection{Main results}\n\\label{sec:4.2}\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{.\/images\/Vis_v3.pdf}\n\\centering\n\\vspace{-0.2cm}\n\\caption{ Visualizations on the benefits of our proposed method. (a) Qualitative results: The upper and bottom parts are visualization of BevDepth \\cite{li2022bevdepth} and our proposed method respectively. The results are visualized on the weather adaptation scenario. (b) Visualization of feature distributions using T-SNE \\cite{van2013barnes}. The \\textcolor{blue}{blue spots} denote the source features, while \\textcolor{red}{red spots} represent target features.}\n\\label{fig:vis}\n\\vspace{-0.2cm}\n\\end{figure*}\n\nWe compare our proposed method with other BEV perception methods to verify the superior performance of $M^{2}ATS$. Meanwhile, to further demonstrate our special design in addressing domain shift of LSS-based multi-view 3D object detection, we reproduce other promising 2D and mono-view 3D detection Domain Adaptation (DA) methods on BEVDepth \\cite{li2022bevdepth}, \\ie, SFA \\cite{wang2021exploring} and STM3D \\cite{li2022unsupervised}.\n\n\n\n\\noindent\\textbf{Scene Adaptation} As shown in Tab.\\ref{tab:scene}, $M^{2}ATS$ outperforms all the baseline methods, which obviously exceeds BEVDepth \\cite{li2022bevdepth} of R50 and R101 backbone by 3.4\\% and 2.4\\% NDS. It thus demonstrates that our proposed method can effectively address the multi-latent spaces domain shift caused by scene and environmental change. Compared with other SOTA DA methods, $M^{2}ATS$ outperforms SFA and STM3D by 2.7\\% and 2.5\\% NDS respectively. And it even improves mAP by 1.9\\% compared with 2nd place. The comparison further demonstrates that our proposed method is specially designed for LSS-based 3D object detection.\n\n\n\\noindent\\textbf{Weathers Adaptation} As shown in Tab. \\ref{tab:weather}, in Sunny to Foggy-3 scenario adaptation, $M^{2}ATS$ outperforms other baseline methods by a significant margin. It even exceeds BEVDepth by around 10\\% in both NDS and mAP (R50 setting). Moreover, compared with SFA and STM3D, $M^{2}ATS$ improves NDS by 3.8\\% and 3.2\\% respectively since it utilizes multi-space domain invariant features to realize a better representation in extreme weather data. In order to evaluate the generalization of $M^{2}ATS$, we also conduct experiments on Sunny to Rainy, $M^{2}ATS$ also increases 2.4\\% NDS compared with 2nd place.\n\n\n\n\n\\noindent\\textbf{Day-Night Adaptation} The Day-Night adaptation is the most challenging scenario for camera-based methods, $M^{2}ATS$ significantly improves the detection performance and solves the domain shift problem in the Night domain. As shown in Tab. \\ref{tab:day}, the tremendous domain gap makes baseline methods perform extremely poorly with only 0.062 NDS and 0.036 mAP under R101. While $M^{2}ATS$, especially with R101 as its backbone, can achieve 0.188 NDS and 0.127 mAP. Even compared with other domain adaptation methods, it also achieves a superior improvement of more than 4.0\\% and 6.2\\%. Since previous DA methods like STM3D and STA ignore the inaccuracy depth estimation in Night data, it thus can not effectively deal with Day-Night domain shift in the LSS-based method.\n\n\\noindent\\textbf{Continues Changing Adaptation} As shown in Tab. \\ref{tab:fog}, due to the page limitation, we only compare $M^{2}ATS$ with the baseline method. Along with the increased foggy degree, the baseline method shows an obvious performance degradation. However, $M^{2}ATS$ alleviates the gradually increased domain gap and outperforms the baseline method by 12.3\\% NDS in the final Foggy-5 domain.\n\n\n\\begin{table}[t]\n \\begin{center}\n \\caption{Results of different methods for continuous changing adaptation scenario on Nuscenes-Foggy, from Sunny to Foggy-1, Foggy-3, and Foggy-5 step by step. The metric is NDS}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{0.8mm}{\n \t\\begin{tabular}{c|c|ccc}\n \t\\hline\n \n Train on&Method(R50) & Foggy-1 & Foggy-3 & Foggy-5 \\\\\n \\hline\n \\hline\n \\multirow{2}{*}{Sunny}& BEVDepth \\cite{li2022bevdepth}& 0.2214 & 0.1592 & 0.096\\\\\n &Ours(BEVDepth) & \\textbf{0.2835} & \\textbf{0.2728} & \\textbf{0.2190} \\\\\n \\hline\n\t\t\\end{tabular}}\n \\label{tab:fog}\n \\vspace{-0.6cm}\n \\end{center}\n\\end{table}\n\n\\subsection{Ablation study}\n\\label{ablation}\nTo better reflect the role of each component in $M^{2}ATS$, we conduct ablation experiments to analyze how each component can deal with domain shift for LSS-based BEV perception. It should be noted that the ablation study is only conducted on \\textbf{Sunny-Rainy} weather adaptation. The ablation study of other scenarios is presented in the appendix. \n\n\n\n\\noindent\\textbf{The effectiveness of DAT and MFA.}\nAs shown in Tab. \\ref{tab:abl}, vanilla BEVDepth ($Ex_{0}$) can only achieve 26.8\\% NDS and 19.6\\% mAP when the scenario is transformed from sunny to the rainy domain. For DAT, it transfers multi-latent space knowledge to the student model via domain invariant multi-space features and more reliable pseudo labels, which are constructed by depth-aware information. As shown in $Ex_{1}$, the student model can learn pixel and instance level knowledge from DAT, NDS, and mAP are improved by 1.5\\% and 3.5\\% respectively. $Ex_{2}$ evaluates the benefits of mean-teacher mechanism, achieving further 0.4\\% mAP improvement. By gradually aligning the multi-space feature ($Ex_{3-5}$) in the student model, our method will get a 2\\% improvement in NDS, which demonstrates that it is essential to align the global-level representation of two domains in multi-latent spaces. When we combine DAT and MFA in $M^{2}ATS$ ($Ex_{6-7}$), NDS can be further enhanced to 30.5\\% while mAP can achieve 24.3\\%. We can come to the conclusion that NDS and mAP continuously increase with the addition of each component in DAT and MFA, demonstrating that each of these modules is necessary and effective. It also proves that DAT and MFA can compensate each other to address the domain shift in multi-latent spaces.\n\n\n\n\n \n \n\n\\begin{table}[t]\n \\begin{center}\n \\caption{Ablation studies on the Sunny to Rainy scenario. It shows the effectiveness of DAT and MFA in the framework. For DAT, it consists of three components, including depth-aware information(DA), mean-teacher(MT), and multi-latent space knowledge transfer(KT). For MFA, it concludes three-latent space alignments, which are Bev(BA), image(IA), and voxel(VA) feature alignment.}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{1.1mm}{\n \t\\begin{tabular}{c|ccc|ccc|cc}\n \t\\hline\n \n Name & DA & MT & KT& BA& IA & VA & \\cellcolor{lightgray} NDS \u2191&\\cellcolor{lightgray} mAP \u2191 \\\\\n \\hline\n \\hline\n $Ex_{0}$ & - & - & - & - & - & - & 0.268 & 0.196 \\\\\n \n \\hline\n $Ex_{1}$ & \\Checkmark & - & \\Checkmark & - & - & - & 0.283 & 0.231 \\\\\n \n $Ex_{2}$ & \\Checkmark & \\Checkmark & \\Checkmark & - & - & - & 0.286 & 0.235 \\\\\\hline\n \n $Ex_{3}$ & - & - & - & \\Checkmark & - & - &0.276 & 0.200\\\\\n $Ex_{4}$ & - & - & - & \\Checkmark & \\Checkmark & - & 0.282 & 0.204\\\\\n $Ex_{5}$ & - & - & - & \\Checkmark & \\Checkmark & \\Checkmark & 0.288 & 0.207 \\\\\\hline\n $Ex_{6}$ & \\Checkmark & -& \\Checkmark & \\Checkmark&\\Checkmark & \\Checkmark & 0.301 & 0.238 \\\\\n \n \n $Ex_{7}$ & \\Checkmark & \\Checkmark & \\Checkmark & \\Checkmark & \\Checkmark & \\Checkmark & 0.305 & 0.243 \\\\\n \n\t\t\\hline \n\t\t\\end{tabular}}\n \\label{tab:abl}\n \\vspace{-0.3cm}\n \\end{center}\n\\end{table}\n\n\n\\begin{table}[t]\n \\begin{center}\n \\caption{The ablation study on the effectiveness of each component in depth-aware information. Pred means depth prediction, and UG means adaptive uncertainty-guided depth selection.}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{1.3mm}{\n \t\\begin{tabular}{c|ccc|cc}\n \t\\hline\n Depth-aware: & Lidar & Pred & UG & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 \\\\\n \\hline\n \\hline\n $Ex_{2-1}$& \\Checkmark & - & - & 0.275 & 0.223 \\\\\n $Ex_{2-2}$ & \\Checkmark &\\Checkmark & - & 0.278 & 0.228 \\\\\n $Ex_{2}$ & \\Checkmark & \\Checkmark & \\Checkmark & 0.286 & 0.235 \\\\\n \\hline\n\t\t\\end{tabular}}\n \\label{tab:abl_da}\n \\vspace{-0.3cm}\n \\end{center}\n\\end{table}\n\n\\begin{table}[t]\n \\begin{center}\n \\caption{The ablation study on the effectiveness of each component in Multi-latent space Knowledge Transfer. PL means transferring instance-level pseudo labels. BEV, Voxel, and Image stand for transferring on corresponding latent spaces.}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{0.8mm}{\n \t\\begin{tabular}{c|cccc|cc}\n \t\\hline\n Latent Space: & PL & BEV & Voxel & Image & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 \\\\\n \\hline\n \\hline\n $Ex_{2-3}$& \\Checkmark& - & - & - & 0.280 & 0.213 \\\\\n $Ex_{2-4}$& \\Checkmark& \\Checkmark & - & - & 0.283 & 0.222 \\\\\n $Ex_{2-5}$& \\Checkmark & \\Checkmark &\\Checkmark& - & 0.285 & 0.230 \\\\\n $Ex_{2}$ & \\Checkmark & \\Checkmark & \\Checkmark&\\Checkmark & 0.286 & 0.235 \\\\\n\t\t\\hline \n\t\t\\end{tabular}}\n \\label{tab:abl_kt}\n \\vspace{-0.45cm}\n \\end{center}\n\\end{table}\n\n \n \n\n\n \n\\noindent\\textbf{Detailed ablation study of DAT}\nWe study the effectiveness of depth-aware information composition and multi-latent knowledge transferring in DAT.\nAs shown in Tab. \\ref{tab:abl_da}, only taking lidar ground truth to replace depth prediction ($Ex_{2-1}$) can improve 0.7\\% NDS and 2.7\\% mAP compared with $Ex_{0}$. The obviously increased mAP demonstrates that lidar data plays an important role in domain invariant voxel feature construction. However, due to the sparse property of lidar data, we utilize dense depth prediction to composite sparse lidar. In ($Ex_{2-2}$), NDS and mAP can achieve 27.9\\% and 22.8\\%, which only have limited improvement compared with $Ex_{2-1}$. Therefore, we introduce uncertainty guidance to adaptively select more reliable and task-relevant depth predictions. $Ex_{2}$ has obvious performance progress compared with $Ex_{2-1}$ and $Ex_{2-2}$, demonstrating the uncertainty-guided depth selection can reduce the domain shift caused by domain specific depth prediction. \nAs shown in Tab. \\ref{tab:abl_kt}, applying for knowledge transfer in different latent spaces can be beneficial to $M^{2}ATS$. With pseudo label, BEV, voxel, and image feature transferred between DAT and student model, NDS is gradually improved from 26.8\\% to 28.6\\%, and mAP is improved from 19.6\\% to 23.5\\%. The improved performance demonstrates that multi-space global-level alignment is introduced to ease different domain shifts in each latent space. It shows that pseudo label and three spaces knowledge are all essential for the student model to address multi-latent space domain gaps, which is constructed by depth-aware information.\n\n\n\n\n\n\\subsection{Qualitative analysis}\n\\label{sec:4.4}\nWe further present some visualization results of the prediction produced by the $M^{2}ATS$ and the baseline BEVDpeth~\\cite{li2022bevdepth}, as shown in Fig. \\ref{fig:vis} (a). It is quite clear that the BEVDpeth fails to locate the objects well, while $M^{2}ATS$ yields more accurate localization results as its predicted \\textcolor{green}{green box} overlaps better with the ground truth \\textcolor{red}{red box}. We can also observe that $M^{2}ATS$ can detect objects that baseline ignores, demonstrating the superiority of $M^{2}ATS$ in object detection and presenting great potential in deploying to real-world autonomous driving applications. The visualization in Fig. \\ref{fig:vis} (b) further verifies the explicit cross domain ability of $M^{2}ATS$. As a clear separation can be seen in the clusters of the \\textcolor{blue}{blue-source} and \\textcolor{red}{red-target} dots produced by BEVDepth, the features generated by $M^{2}ATS$ get closer together further demonstrates the ability of our proposed method in representing domain invariant features.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.98\\textwidth]{.\/images\/fog_v3.png}\n\\centering\n\\vspace{-0.65cm}\n\\caption{The visualization of Foggy-Nuscenes dataset. The first row is the original multi-view images in Nuscenes \\cite{caesar2020nuscenes}, and the last three rows demonstrate images of increased foggy degree.}\n\\label{fig:fog}\n\\vspace{-0.3cm}\n\\end{figure*}\n\n\n\\section{Conclusion and discussion of limitations}\nWe explore the UDA problem in Multi-View 3D object detection and propose a novel Multi-level Multi-space Alignment Teacher-Student ($M^{2}ATS$) framework to fully ease multi-latent space domain shift for LSS-based BEV perception. On the one hand, Depth-Aware Teacher (DAT) leverages depth-aware information to generate reliable pseudo labels and multi-space domain-invariant features, which are transferred to the student model. On the other hand, the Multi-space Feature Aligned (MFA) student model utilizes source and target domain data to align the global-level feature representation. $M^{2}ATS$ achieves SOTA performance in three challenge UDA and continual changing domain gap scenarios. For limitations, the teacher-student framework brings more computational cost and extra small parameters (discriminator) during training. However, the student model keeps the same forward time and computational cost with a baseline in inference. \n\n\\section{Appendix}\n\n\nIn the supplementary material, we first present more details of our generated Foggy Nuscenes dataset in Sec .\\ref{sec:1}, which aims at adding a foggy weather scene in Nuscenes \\cite{caesar2020nuscenes} and providing an open source dataset (Foggy-Nuscenes) for research in autonomous driving. \nSecondly, in Sec .\\ref{sec:2}, we show the details of each scenario and the partition of dataset.\nIn Sec .\\ref{sec:3}, we then provide additional and detailed cross domain training strategy.\nIn Sec .\\ref{sec:4}, the extra ablation studies are conducted on Day-night cross-domain scenario, which investigate the impact of each component in the Multi-level Multi-space Alignment Teacher-Student ($M^{2}ATS$) framework.\nFinally, we provide additional qualitative analysis on Day-night scenario to further evaluate the effectiveness of our proposed method in Sec~\\ref{sec:5}.\n\n\\begin{table*}[t]\n \\begin{center}\n \\caption{The partitioning and details of four UDA scenarios. Frames means the number of multi-view image frames in the datasets.}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{3mm}{\n \t\\begin{tabular}{c|c|c|c|c|c}\n \t\\hline\n UDA scenarios & Domain & Training sequences & Frames & Validation sequences & Frames \\\\\n \\hline\n \\hline\n \\multirow{2}{*}{Scene changing}&Source(Boston) & 857 & 15695 & 77 & 3090\\\\\n &Target(Singapore) & 693 & 12435 & 73 & 2929\\\\\n \\hline\n \\multirow{2}{*}{Weather changing}&Source(Sunny) & 1276 & 23070 & 126 & 5051\\\\\n &Target(Rainy) & 274 & 5060 & 24 & 968\\\\\n \\hline\n \\multirow{2}{*}{Weather changing}&Source(Nuscenes) & 1550 & 28130 & 150 & 6019\\\\\n &Target(Foggy-3) & 1550 & 28130 & 150 & 6019\\\\\n \\hline\n \\multirow{2}{*}{Day-night changing}&Source(Day) & 1367 & 24745 & 135 & 5417\\\\\n &Target(Night) & 183 & 3385 & 15 & 602\\\\\n \\hline\n \\multirow{4}{*}{Continual changing}&Source(Nuscenes) & 1550 & 28130 & 150 & 6019\\\\\n &Target-1(Foggy-1) & 1550 & 28130 & 150 & 6019\\\\\n &Target-2(Foggy-3) & 1550 & 28130 & 150 & 6019\\\\\n &Target-3(Foggy-5) & 1550 & 28130 & 150 & 6019\\\\\n \\hline\n\t\t\\end{tabular}}\n \\label{tab:data}\n \\end{center}\n \\vspace{-0.3cm}\n\\end{table*}\n\\subsection{Supplementary description of foggy Nuscenes dataset}\n\\label{sec:1}\nWe apply the fog simulation pipeline \\cite{sakaridis2018semantic} to the multi-view images provided in the Nuscenes dataset \\cite{caesar2020nuscenes}. specifically, we generate synthetic foggy images for all scenes of training, validation, and test set, which reserve original annotation of 3D object detection task. We utilize five different density degree of foggy simulator to construct Foggy-Nuscenes dataset, including Foggy-1, Foggy-2, Foggy-3, Foggy-4, and Foggy-5 (gradually increasing fog density). As shown in Fig .\\ref{fig:fog}, we adopt Foggy-1, Foggy-3, and Foggy-5 as the experimental datasets for weather adaptation and continual changing scenario, which have an obvious domain gap with original Nuscenes dataset \\cite{caesar2020nuscenes}. \n\n\n\n\n\n\\subsection{Additional details of UDA scenarios}\n\\label{sec:2}\nIn this paper, we utilize Nuscenes \\cite{caesar2020nuscenes} and Foggy-Nuscenes datasets to generate three classical and one continual changing UDA scenarios. As shown in Tab .\\ref{tab:data}, for Scene changing scenario, We set Boston as the source scene data and realize UDA on the Singapore target domain. The source domain contains 15.6k frames labeled data and target domain has 12.4k frames unlabeled data. For Weather changing scenario, the sunny weather is considered as source domain data, rainy and foggy weather is considered as target domain data. In the second row, the source domain has 1276 training sequences and rainy target domain has 274 training sequences. In the third row, we leverage all Nuscenes dataset as source domain data and set Foggy-3 as target domain data. As we can see, in Day-night changing scenario, we design daytime data as the source domain and realize UDA on the target domain (night data). Since the source domain data is way more larger than the target domain data and the camera can not capture night-time data clearly, which is considered as the most challenging adaptation scenarios. Finally, we set all Nuscenes sequence as the source domain data and set continuously increased foggy degree data as the target domain. The various target domains are of the same sequence and scene with source domain, but the different foggy degree on the frames bring significant domain gap. We introduce this continual changing scenario to demonstrate the continual domain adaptation ability of our proposed method.\n\\begin{table}[t]\n \\begin{center}\n \\caption{Ablation studies on the Day-night scenario. It shows the effectiveness of DAT and MFA in the framework. For DAT, it consists of three components, including depth-aware information(DA), mean-teacher(MT), and multi-latent space knowledge transfer(KT). For MFA, it concludes three-latent space alignments, which are Bev(BA), image(IA), and voxel(VA) feature alignment.}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{1.0mm}{\n \t\\begin{tabular}{c|ccc|ccc|cc}\n \t\\hline\n \n Name & DA & MT & KT& BA& IA & VA & \\cellcolor{lightgray} NDS \u2191&\\cellcolor{lightgray} mAP \u2191 \\\\\n \\hline\n \\hline\n $Ex_{0}$ & - & - & - & - & - & - & 0.050 & 0.012 \\\\\n \n \\hline\n $Ex_{1}$ & \\Checkmark & - & \\Checkmark & - & - & - & 0.103 & 0.034 \\\\\n \n $Ex_{2}$ & \\Checkmark & \\Checkmark & \\Checkmark & - & - & - & 0.104 & 0.041 \\\\\\hline\n \n $Ex_{3}$ & - & - & - & \\Checkmark & - & - &0.065 & 0.038\\\\\n $Ex_{4}$ & - & - & - & \\Checkmark & \\Checkmark & - & 0.071 & 0.042\\\\\n $Ex_{5}$ & - & - & - & \\Checkmark & \\Checkmark & \\Checkmark & 0.098 & 0.051 \\\\\\hline\n $Ex_{6}$ & \\Checkmark & -& \\Checkmark & \\Checkmark&\\Checkmark & \\Checkmark & 0.124 & 0.049 \\\\\n \n \n $Ex_{7}$ & \\Checkmark & \\Checkmark & \\Checkmark & \\Checkmark & \\Checkmark & \\Checkmark & 0.132 & 0.054 \\\\\n \n\t\t\\hline \n\t\t\\end{tabular}}\n \\label{tab:abl_ap}\n \\end{center}\n \\vspace{-0.3cm}\n\\end{table}\n\n\\subsection{Additional implementation details}\n\\label{sec:3}\nOur training process of cross domain can be divided into 2 stages: pretraining on source domain and transfer training from source to target domain. \nFirstly, in order to fully leverage the feature extraction ability of the model \\cite{li2022bevdepth}, we load the backbone of ImageNet \\cite{deng2009imagenet} pretrained parameters. Then we train the model on the source domain data in a supervised manner, which aims to obtain the source domain pretrained model parameters.\nIn cross domain phase, we load the integrated model parameters, which are pretrained on source domain, into the student model and conduct transferred training for 12 epochs. \n$M^{2}ATS$ framework adopts source domain and target domain data as input, and only leverages source domain annotation. During training, we alternate Depth-Aware Teacher knowledge transferred and Multi-space Feature Alignment training to update student model. Finally, we directly infer the student model on target domain data and achieve the result of our proposed method.\n\n\n\\subsection{Additional ablation study on day-night scenario}\n\\label{sec:4}\nThe $M^{2}ATS$ framework contains a Depth-Aware Teacher (DAT) and a Multi-space Feature Aligned (MFA) student model.\nTo evaluate each component of the proposed $M^{2}ATS$ framework, we conduct ablation experiments to analyze how each component can mitigate domain shift for LSS-based BEV perception. It should be noted that the ablation study is conducted on the most challenging \\textbf{Day-Night} scenario, in which other methods suffer tremendous performance degradation. \n\n\\begin{table}[t]\n \\begin{center}\n \\caption{The ablation study on the effectiveness of each component in depth-aware information. Pred means depth prediction, and UG means adaptive uncertainty-guided depth selection.}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{1.3mm}{\n \t\\begin{tabular}{c|ccc|cc}\n \t\\hline\n Depth-aware: & Lidar & Pred & UG & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 \\\\\n \\hline\n \\hline\n $Ex_{2-1}$& \\Checkmark & - & - & 0.068 & 0.028 \\\\\n $Ex_{2-2}$ & \\Checkmark &\\Checkmark & - & 0.084 & 0.032 \\\\\n $Ex_{2}$ & \\Checkmark & \\Checkmark & \\Checkmark & 0.104 & 0.041 \\\\\n \\hline\n\t\t\\end{tabular}}\n \\label{tab:abl_da_ap}\n \\end{center}\n \\vspace{-0.3cm}\n\\end{table}\n\n\\begin{table}[t]\n \\begin{center}\n \\caption{The ablation study on the effectiveness of each component in Multi-latent space Knowledge Transfer. PL means transferring instance-level pseudo labels. BEV, Voxel, and Image stand for transferring on corresponding latent space.}\n \\vspace{-0.2cm}\n \\setlength{\\tabcolsep}{0.8mm}{\n \t\\begin{tabular}{c|cccc|cc}\n \t\\hline\n Latent Space: & PL & BEV & Voxel & Image & \\cellcolor{lightgray} NDS \u2191& \\cellcolor{lightgray} mAP \u2191 \\\\\n \\hline\n \\hline\n $Ex_{2-3}$& \\Checkmark& - & - & - & 0.076 & 0.036 \\\\\n $Ex_{2-4}$& \\Checkmark& \\Checkmark & - & - & 0.096 & 0.039 \\\\\n $Ex_{2-5}$& \\Checkmark & \\Checkmark &\\Checkmark& - & 0.101 & 0.044 \\\\\n $Ex_{2}$ & \\Checkmark & \\Checkmark & \\Checkmark&\\Checkmark & 0.104 & 0.041 \\\\\n\t\t\\hline \n\t\t\\end{tabular}}\n \\label{tab:abl_kt_ap}\n \\end{center}\n \\vspace{-0.3cm}\n\\end{table}\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{.\/images\/vis_ap_v2.png}\n\\centering\n\\caption{Visualizations on the benefits of our proposed method. The upper and bottom parts are visualization of BevDepth \\cite{li2022bevdepth} and our proposed method respectively. The \\textcolor{red}{red box} is the Ground Truth, and the \\textcolor{green}{green box} is the predictions.}\n\\label{fig:vis_ap}\n\\vspace{-0.3cm}\n\\end{figure*}\n\\noindent\\textbf{The effectiveness of DAT and MFA.}\nAs shown in Tab. \\ref{tab:abl_ap}, due to the faint light of target domain (night-time), vanilla BEVDepth ($Ex_{0}$) can only achieve 5.0\\% NDS and 1.2\\% mAP when the scenario is transformed from source domain (daytime). For DAT, it transfers multi-latent space knowledge to the student model via pixel-level multi-space features and instance-level pseudo labels, which are constructed by depth-aware information. As shown in $Ex_{1}$, the student model can learn domain-invariant knowledge from DAT, NDS and mAP are \nthus improved by 5.3\\% and 2.2\\% respectively. $Ex_{2}$ evaluates the benefits of mean-teacher mechanism, which brings 0.7\\% mAP improvement. As we can see, depth-aware information plays an important role in cross domain transferred learning and greatly improves NDS compared with baseline. By gradually aligning the multi-space feature ($Ex_{3-5}$) in the MFA student model, our method get a 4.8\\% and 3.9\\% improvement in NDS and mAP, which demonstrates that it is essential to align the global-level representation of two domains in multi-latent spaces. The MFA student model has a great improvement on mAP compared with baseline, in which the global-level alignment focuses more on classification and location of objects. \nWhen we combine DAT and MFA in $M^{2}ATS$ ($Ex_{6-7}$), NDS can be further enhanced to 13.2\\% while mAP can achieve 5.4\\%. We can come to the conclusion that NDS and mAP continuously increase with the addition of each component in DAT and MFA, showing similar results with the previous ablation study on submission. It also proves that DAT and MFA can compensate each other to address the domain shift in multi-latent spaces.\n\n\n\n\\noindent\\textbf{Detailed ablation study of DAT}\nWe study the effectiveness of depth-aware information composition and multi-latent knowledge transferring in DAT.\nAs shown in Tab. \\ref{tab:abl_da_ap}, only taking lidar ground truth to replace depth prediction ($Ex_{2-1}$) can only improve 1.8\\% NDS and 1.6\\% mAP compared with BEVDepth ($Ex_{0}$). The obviously increased performance demonstrates that lidar data plays an important role in domain-invariant voxel feature construction. However, due to the sparse property of lidar data, we utilize dense depth prediction to composite sparse lidar data. In ($Ex_{2-2}$), NDS and mAP can achieve 8.4\\% and 3.2\\%, which only have limited improvement compared with $Ex_{2-1}$. Therefore, we introduce uncertainty guidance to adaptively select more reliable and task-relevant depth predictions to composite sparse lidar data. Due to the reliable composite depth-aware information, $Ex_{2}$ has obvious performance progress compared with $Ex_{2-2}$, which further improves 2.0\\% NDS and 0.9\\% mAP.\nThe results demonstrate the uncertainty-guided depth selection can reduce the domain shift caused by domain specific depth prediction. \nAs shown in Tab. \\ref{tab:abl_kt_ap}, applying knowledge transfer on different latent spaces can be beneficial to $M^{2}ATS$. With pseudo label, BEV, voxel, and image feature transferred between DAT and student model, NDS is gradually improved from 5.0\\% to 10.4\\%, and mAP is improved from 1.2\\% to 4.1\\%. The results show that pseudo label and the knowledge of three spaces are all essential for the student model to address multi-latent space domain gaps, which is constructed by depth-aware information. In conclusion, leveraging depth-aware information has a significant impact on addressing multi-space domain shift in BEV perception, which transfers domain-invariant knowledge from teacher model to student model.\n\n\n\\subsection{Additional qualitative analysis}\n\\label{sec:5}\nIn contrast with the visualization on submission, we further present some visualization results of the prediction produced by the $M^{2}ATS$ and the baseline BEVDpeth~\\cite{li2022bevdepth} on the most challenging scenario (day-night), as shown in Fig. \\ref{fig:vis_ap}. Due to the faint light of night-time data, we can not classify and locate objects even with the naked eye, not to mention camera-based methods. It is quite clear that the BEVDpeth has various inaccurate and missing detection, while $M^{2}ATS$ yields more accurate localization results as its predicted \\textcolor{green}{green box} overlaps better with the ground truth \\textcolor{red}{red box}.\nWe can also observe that $M^{2}ATS$ can detect objects that baseline ignores, demonstrating the superiority of $M^{2}ATS$ in object detection and presenting great potential in deploying to real-world autonomous driving applications. However, our proposed method still has some missing detection which inspires us to pay more attention on the BEV perception in night-time.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{S:introduction}\n\n\n\nAn increasing number of studies\n involve the regression analysis of $p$ continuous covariates and continuous or discrete univariate responses on $n$ subjects, with $p$ being much larger than $n$.\nThe development of effective clustering and sparse regression models for reliable predictions is especially challenging in these ``small $n$, large $p$'' problems.\nThe goal of the analysis is often three-pronged: {\\it{(i) Cluster identification:}} We wish to identify clusters of covariates with similar patterns for the subjects. For example, in biomedical studies where the covariates are gene expression levels, subsets of genes associated with distinctive between-subject patterns may correspond to different underlying biological processes; {\\it{(ii) Detection of sparse regression predictors:}} From the set of $p$ covariates, we wish to select a sparse subset of reliable predictors for the subject-specific responses and infer the nature of their relationship with the responses. In most genomic applications, just a few of the biological processes are usually relevant to a response variable of interest, and we need reliable and parsimonious regression models; and {\\it{(iii) Response prediction:}} Using the inferred regression relationship, we wish to predict the responses of $\\tilde{n}$ additional subjects for whom only covariate information is available. The reliability of an inference procedure is measured by its prediction accuracy for out-of-sample individuals.\n\nIn high-throughput regression settings with continuous covariates and continuous or discrete outcomes, this paper proposes a nonparametric Bayesian framework called \\textbf{VariScan} for simultaneous clustering, variable selection, and prediction.\n\\subsection{Motivating applications}\n\nOur methods and computational endeavors are motivated by recent high-throughput investigations in biomedical research, especially in cancer. Advances in array-based technologies allow simultaneous measurements of biological units (e.g.\\ genes) on a relatively small number of subjects. Practitioners wish to select important genes involved with the disease processes and to develop efficient prediction models for patient-specific clinical outcomes such as continuous survival times or categorical disease subtypes. The analytical challenges posed by such data include not only high-dimensionality but also the existence of considerable gene-gene correlations induced by biological interactions. In this article, we analyze gene expression profiles assessed using microarrays in patients with diffuse large B-cell lymphoma (DLBCL) \\citep{Rosenwald_etal_2002} and breast cancer \\citep{vantVeer_2002}. Both datasets are publicly available and have the following general characteristics: for individuals $i=1,\\ldots,n$, the data consist of mRNA expression levels $x_{i1},\\ldots,x_{ip}$ for $p$ genes, where $n>>p$, along with censored survival times denoted by $w_i$. More details, analysis results, and gains using our methods over competing approaches are discussed in Section~\\ref{S:benchmark_data}.\n\n\nThe scope and success of the proposed methodology and its associated theoretical results extend far beyond the examples we discuss in this paper. For instance, the technique is not restricted to biomedical studies; we have successfully applied VariScan in a variety of other high-dimensional~applications and achieved high inferential gains relative to existing~methodologies.\n\n\n\n\n\n\\subsection{Challenges in high-dimensional predictor detection}\nDespite the large number of existing methods related to clustering, variable selection and prediction, researchers continue to develop new methods to meet the challenges posed by newer applications and larger datasets. Predictor detection becomes particularly problematic in big datasets due to the pervasive collinearity of the covariates.\n\nFor a simple demonstration of this fact, consider a process that independently samples $n$-variate covariate column vectors $\\boldsymbol{x}_1\\ldots,\\boldsymbol{x}_p$, so that $p=200$ vectors with $n=10$ i.i.d.\\ elements are generated from a common normal distribution. Although the vectors are independently generated, extreme values of the pairwise correlations are observed in the sample, as shown in the histogram of Figure \\ref{F:toy_pair}. The proportion of extremely high or low correlations typically increases with $p$, and with greater correlation of the generated vectors under the true~process.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.25]{example_pairwise.pdf}\n\\caption{Pairwise sample correlations for $p=200$ vectors independently generated from a multivariate normal distribution with $n=10$ uncorrelated elements.}\n\\label{F:toy_pair}\n\\end{center}\n\\end{figure}\n\n\n\nMulticollinearity is common in high-dimensional problems because the $n$ - dimensional space of the covariate columns becomes saturated with the large number of covariates. This is disadvantageous for regression because a cohort of highly correlated covariates is weakly identifiable as regression predictors.\nFor {example,\n imagine that the $j^{th}$ and $k^{th}$ covariate columns have a sample correlation close to 1, but that neither covariate is really a predictor in a linear regression model. An alternative model in which \\textit{both} covariates are predictors with equal and opposite regression coefficients, has a nearly identical joint likelihood for all regression outcomes. Consequently, an inference procedure is often unable to choose between these competing models as the likely explanation for the data.\n\n In the absence of strong application-specific priors to guide model selection, collinearity makes it impossible to pick the true set of predictors in high-dimensional problems.\n Furthermore, collinearity causes unstable inferences and erroneous test case predictions \\citep{Weisberg_1985}. The problem is exacerbated if some of the regression outcomes are unobserved, as with categorical responses and survival~applications.\n\n\n\n\n\n\\subsection{Bidirectional clustering with adaptively nonlinear functional regression and prediction}\n\n\n Since the data in small $n$, large $p$ regression problems are informative only about the combined effect of a cohort of highly correlated covariates, we address the issue of collinearity using clustering approaches. Specifically, VariScan utilizes the sparsity-inducing property of Poisson-Dirichlet processes (PDPs) to first group the $p$ columns of the covariate matrix into $q$ latent clusters, where $q \\ll p$, with each cluster consisting of columns with similar patterns across the subjects. The data are allowed to direct the choice between a class of PDPs and their special case, a Dirichlet process, for selecting a suitable allocation scheme for the covariates. These partitions could provide meaningful insight into unknown biological processes (e.g.\\ signaling pathways) represented by the latent clusters.\n\n To flexibly capture the within-cluster pattern of the covariates, the $n$ subjects are allowed to group differently in each cluster via a nested Dirichlet process. This feature is motivated by genomic studies \\citep[e.g.,][]{Jiang_Tang_Zhang_2004} which have demonstrated that subjects or biological samples often group differently under different biological processes. In essence, this modeling framework specifies a random, bidirectional (covariate, subject) nested clustering of the high-dimensional covariate~matrix.\n\nClustering downsizes the small $n$, large $p$ problem to a ``small $n$, small $q$'' problem, facilitating an effective stochastic search of the indices $\\mathcal{S}^* \\subset\\{1,\\ldots,q\\}$ of potential \\textit{cluster predictors}. If necessary, we could then infer the indices $\\mathcal{S}\\subset\\{1,\\ldots,p\\}$ of the covariate predictors. This feature differentiates the VariScan procedure from black-box nonlinear prediction methods. In addition, the technique is capable of detecting functional relationships through elements such as nonlinear functional kernels and basis functions such as splines or wavelets. An adaptive mixture of linear and nonlinear elements in the regression relationship aims to achieve a balance between model parsimony and~flexibility. These aspects of VariScan define a joint model for the responses and covariates, resulting in an effective model-based clustering and variable selection procedure, improved posterior\ninferences and accurate test case~predictions.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{toy_VariScan2.pdf}\n\\vspace{-2.5 cm}\n\\caption{Stylized example illustrating the basic methodology for reliable prediction for $n=10$ subjects and $p=25$ covariates allocated to $q=11$ number of PDP clusters. The column labels represent the covariate indices. The row labels are the subjects. See the text for further explanation.}\n\\label{F:toy_VariScan}\n\\end{center}\n\\end{figure}\n\nFigure \\ref{F:toy_VariScan} illustrates the key ideas of VariScan using a toy example with $n=10$ subjects and $p=25$ covariates. The plot in the upper left panel represents a heatmap of the covariates. When investigators are interested in discovering a sparse prediction model for additional subjects, the posterior analysis averages over all possible realizations of two basic steps, both of which are stochastic and may be stylistically described as follows:\n\n \\begin{enumerate}\n \\item\\textbf{Clustering} \\quad The column vectors are allocated in an unsupervised manner to $q=11$ number of PDP clusters. This is plotted in the upper right panel, where the columns are grouped via bidirectional clustering to reveal the similarities in the within-cluster~patterns.\n\n \\item\\textbf{Variable selection and regression} \\quad One covariate is stochastically selected from each cluster and is known as the \\textit{cluster representative}. The middle right panel displays the set of representatives, $\\{\\boldsymbol{x}_7,\\boldsymbol{x}_4,\\boldsymbol{x}_{11},\\boldsymbol{x}_5,\\boldsymbol{x}_{24},$\n$\\boldsymbol{x}_{17},\\boldsymbol{x}_9,\\boldsymbol{x}_{12},\\boldsymbol{x}_3,\n\\boldsymbol{x}_{15},\\boldsymbol{x}_{14}\\}$, for the 11 clusters. The regression predictors are stochastically selected from the random set of the cluster representatives. Some representatives are not associated with the response; the remaining covariates are outcome predictors and may have either a linear or nonlinear regression relationship. The linear predictors $\\{\\boldsymbol{x}_{24},\\boldsymbol{x}_{12},\\boldsymbol{x}_{3}\\}$ and non-linear predictors $\\{\\boldsymbol{x}_{11},\\boldsymbol{x}_{9}\\}$ are shown in the middle left panel. For a nonlinear function $h$, the regression equation for a subject is displayed in the lower panel for a zero-mean Gaussian error, $\\epsilon$. The subscripts of the $\\beta$ parameters are the cluster labels, e.g., covariate $\\boldsymbol{x}_{24}$ represents the fifth PDP cluster.\n\\end{enumerate}\nWhen out-of-the-bag prediction is not of primary interest, alternative variable selection strategies discussed in Section \\ref{S:predictors} may be applied.\n\n\n\n\n\\subsection{Existing Bayesian approaches and limitations}\\label{S:lit review}\n\n\n\nThere is a vast literature on Bayesian strategies for one or more of the three inferential goals mentioned at the beginning of Section \\ref{S:introduction}. A majority of Bayesian model-based clustering techniques rely on the celebrated Dirichlet process; see \\citet[chap.\\ 4]{muller2013bayesian} for a comprehensive review. A seminal paper by \\cite*{Lijoi_Mena_Prunster_2007b} advocated the use of Gibbs-type priors \\citep*{Gnedin_Pitman_2005, Lijoi_Mena_Prunster_2007a} for accommodating more flexible clustering mechanisms than those induced by the Dirichlet process. This work also demonstrated the practical utility of PDPs in genomic applications.\n\nAmong model-based clustering techniques based on Dirichlet processes, the approaches of \\cite{Medvedovic_Siva_2002}, \\cite{Dahl_2006}, and \\cite{Muller_Quintana_Rosner_2011} assume that it is possible to \\textit{globally} reshuffle the rows and columns of the covariate matrix to reveal the clustering pattern. More closely related to our clustering objectives is the nonparametric Bayesian local clustering (NoB-LoC) approach of \\cite{Lee_etal_2013}, which clusters the covariates \\textit{locally} using two sets of Dirichlet processes. Although some similarities exist between NoB-LoC and the clustering aspect of VariScan, there are major differences. Specifically, the VariScan framework can accommodate high-dimensional regression in addition to bidirectional clustering. Furthermore, VariScan typically produces more efficient inferences by its greater flexibility in modeling a larger class of clustering patterns via PDPs. The distinction becomes especially important for genomic datasets where PDP-based models are often preferred to Dirichlet-based models by log-Bayes factors on the order of thousands; see Section \\ref{S:benchmark_data} for an example. Moreover, the Markov chain Monte Carlo (MCMC) implementation of VariScan explores the posterior substantially faster due to its better ability to allocate outlying covariates to singleton clusters via augmented variable Gibbs sampling. From a theoretical perspective, contrary to widely held beliefs about the non-identifiability of mixture model clusters, we discover the remarkable theoretical property of VariScan that, as both $n$ and $p$ grow, a fixed set of covariates that (do not) co-cluster under the true VariScan model, also (do not) asymptotically co-cluster under its posterior.}\n\n\n\n{From a regression-based Bayesian viewpoint, perhaps the most ubiquitous approaches are based on Bayesian variable selection techniques in linear and non-linear regression models. See \\cite{david2002bayesian} for a comprehensive review. For Gaussian responses, the common linear methods include stochastic search variable selection \\citep{George_McCulloch_1993}, selection-based priors \\citep{Kuo_Mallick_1997} and shrinkage-based methods \\citep{Park_Casella_2008, xu2015bayesian, griffin2010inference}. Some of these methods have been extended to non-linear regression contexts \\citep{smith1996nonparametric} and to generalized linear models \\citep{dey2000generalized,meyer2002predictive}. However, most of the {afore-mentioned regression} methods are based on {strong} parametric assumptions and do not explicitly account for the multicollinearity commonly observed in high-dimensional settings. Nonparametric approaches typically assume priors on the error residuals \\citep{hanson2002modeling,kundu2014bayes} or on the regression coefficients using random effect representations \\citep{bush1996semiparametric, maclehose2010bayesian}. For nonparametric mean function estimations, they are typically based on basis function expansions such as wavelets\\citep{morris2006wavelet} and splines \\citep{baladandayuthapani2005spatially}. We take a fundamentally different approach in this article by defining a nonparametric joint model, first on the covariates and then via an adaptive nonlinear prediction model on the responses.}\n\n\n\nThe rest of the paper is organized as follows. We introduce the VariScan model in Section \\ref{S:VariScan_model}. \nSome theoretical results for the VariScan procedure are presented in Sections \\ref{S:clusters}. Through simulations in Section \\ref{S:simulation2} and \\ref{S:simulation}, we demonstrate the accuracy of the clustering mechanism and compare the prediction reliability of VariScan with that of several established variable selection procedures using artificial datasets. In Section \\ref{S:benchmark_data}, we analyze the motivating gene expression microarray\ndatasets for leukemia and breast cancer to demonstrate the effectiveness of VariScan and compare its prediction accuracy with those of competing methods. Additional supplementary materials contain the theorem proofs, as well as additional simulation and data analyses~results.\n\n\\bigskip\n\n\n\\section{VariScan Model}\\label{S:VariScan_model}\n\nIn this section, we layout the detailed construction of the Variscan model components, which involves two major steps. First, we utilize the sparsity-inducing property of Poisson-Dirichlet processes to perform a directional nested clustering of the covariate matrix (Section \\ref{S:covariates}), and second, we describe the choice of the cluster-specific predictors and nonlinearly relate them to Gaussian regression outcomes of the subjects (Section \\ref{S:predictors}). Subsequently, Section \\ref{S:justifications} provides details of the model justifications and generalizations to discrete and survival outcomes.\n\n\\subsection{Covariate clustering model}\\label{S:covariates}\n\n\n\nFirst, each of the $p$ covariate matrix columns, $\\boldsymbol{x}_{1},\\ldots,\\boldsymbol{x}_{p}$, is assigned to one of $q$ latent clusters, where $q \\ll p$, and where the assignments and $q$ are unknown.\nThat is, for $j=1,\\ldots,p$ and $k=1,\\ldots,q$, an \\textbf{allocation variable}\n$c_j$ equals $k$ if the $j^{th}$ covariate is assigned to the $k^{th}$ cluster.\n\nWe posit that the $q$ clusters are associated with \\textbf{latent vectors} $\\boldsymbol{v}_1,\\ldots,\\boldsymbol{v}_q$ of length $n$. The covariate columns assigned to a latent cluster are essentially contaminated versions of these cluster's latent vector and thus induces high correlations among covariates belonging to a cluster.\nIn practice, however, a few individuals within each cluster may have highly variable covariates. We model this aspect by associating a larger error variance with those individuals. This is achieved via a Bernoulli variable,~$z_{ik}$, for which the value $z_{ik}=0$ indicates a high variance:\n\\begin{align*}\nx_{ij} \\mid z_{i k}, c_j = k &\\stackrel{indep}\\sim\n \\begin{cases}\n N(v_{ik}, \\tau_1^2) \\quad &\\text{if $z_{ik}=0$}\\\\\n N(v_{ik}, \\tau^2) \\quad&\\text{if $z_{ik}=1$}\\\\\n \\end{cases}\n\\end{align*}\n where\n$\\tau_1^2$ and $\\tau^2$ are variance parameters with inverse Gamma priors and $\\tau_1^2$ is much greater than $\\tau^2$. It is assumed that the support of $\\tau$ is bounded below by a small, positive constant, $\\tau_*$. Although not necessary from a methodological perspective, this restriction guarantees the asymptotic result of Section \\ref{S:clusters}.\nThe indicator variables for the individual--cluster combinations are apriori modeled {\\it i.i.d} as:\n\\begin{equation*}\nz_{ik} \\stackrel{iid}\\sim \\text{Ber}(\\xi), \\qquad\\text{$i=1,\\ldots,n$ and $k=1,\\ldots,q$,}\n\\end{equation*}\nwhere $\\xi \\sim \\text{beta}(\\iota_1,\\iota_0)$. The condition $\\iota_1 \\gg \\iota_0$ guarantees that prior probability $P(z_{ik} = 1)$ is nearly equal to 1, and so only a small proportion of the individuals have highly variable covariates within each cluster.\n\n\\bigskip\n\n\\noindent \\textbf{Allocation variables.} As an appropriate model for the covariate-to-cluster allocations that accommodates a wide range of allocation patterns, we rely on the partitions induced by the \\textit{two-parameter Poisson-Dirichlet process}, $\\mathbb{PDP}\\bigl(\\alpha_1, d\\bigr)$, with discount parameter $0 \\le d < 1$ and precision or mass parameter $\\alpha_1>0$. In genomic applications, for example, these partitions may allow the discovery of unknown biological processes represented by the latent clusters.\n We defer additional details of the empirical and theoretical justifications of using PDP processes until Section~\\ref{S:justifications}.\n\nThe PDP was introduced by \\cite{Perman_etal_1992} and later investigated by \\cite{Pitman_1995} and \\cite{Pitman_Yor_1997}. Refer to \\cite{Lijoi_Prunster_2010} for a detailed discussion of different classes of Bayesian nonparametric~models, including Gibbs-type priors \\citep*{Gnedin_Pitman_2005, Lijoi_Mena_Prunster_2007a} such as Dirichlet processes and PDPs. \\cite*{Lijoi_Mena_Prunster_2007b} were the first to implement Gibbs-type priors for more flexible clustering mechanisms than Dirichlet process partitions.\n\n\nThe PDP-based allocation variables are apriori exchangeable and evolve as follows. Since the cluster allocations labels are arbitrary, we may assume without loss of generality that $c_1=1$, i.e., the first covariate is assigned to the first cluster. Subsequently, for covariates $j=2,\\ldots,p$, suppose there exist\n $q^{(j-1)}$ distinct clusters among $c_1,\\ldots,c_{j-1}$, with the $k^{th}$ cluster containing $n_{k}^{(j-1)}$ number of covariates. The predictive probability that the $j^{th}$ covariate is assigned to the $k^{th}$ cluster is~then\n\\begin{align*}\nP\\left(c_j = k \\mid c_1, \\ldots, c_{j-1} \\right) \\propto\n \\begin{cases}\n n_{k}^{(j-1)} - d\n \\quad &\\text{if $k = 1,\\ldots,q^{(j-1)}$}\\\\\n \\alpha_1 + q^{(j-1)} \\cdot d \\quad &\\text{if $k = q^{(j-1)} + 1$}\\\\\n \\end{cases}\n\\end{align*}\nwhere the event $c_j=q^{(j-1)} + 1$ in the second line corresponds to the $j^{th}$ covariate opening a new cluster.\n When $d = 0$, we obtain the well known P\\`{o}lya urn scheme for Dirichlet processes \\citep{Ferguson_1973}.\n\n In general, exchangeability holds for all product partition models \\citep{Barry_Hartigan_1993, Quintana_Iglesias_2003} and species sampling models \\citep{Ishwaran_James_2003}, of which PDPs are a special case. The number of clusters, $q$, stochastically increases as $\\alpha_1$ and $d$ increase. For $d$ fixed, the $p$ covariates are each assigned to $p$ singleton clusters in the limit as $\\alpha_1 \\to \\infty$. \n\n A PDP achieves dimension reduction in the number of covariates because $q$, the random number of clusters, is asymptotically equivalent to\n\\begin{align}\n \\begin{cases}\n \\alpha_1 \\cdot \\log p \\qquad &\\text{if $d = 0$} \\\\ \n T_{d, \\alpha_1} \\cdot p^d\\qquad &\\text{if $0 < d < 1$}\\\\\n \\end{cases}\\label{q}\n\\end{align}\nfor a random variable $T_{d, \\alpha_1} >0$ as $p\\to \\infty$. This implies that the number of Dirichlet process clusters (i.e., when $d=0$) is asymptotically of a smaller order than the number of PDP clusters when $d>0$. This property was effectively utilized by \\cite*{Lijoi_Mena_Prunster_2007b} in species prediction problems and applied to gene discovery settings. The use of Dirichlet processes to achieve\n dimension reduction has precedence in the literature; see \\cite{Medvedovic_etal_2004}, \\cite{Kim_etal_2006}, \\cite{Dunson_etal_2008} and \\cite{Dunson_Park_2008}. \n\n\n\nThe PDP discount parameter $d$ is given the mixture prior $\\frac{1}{2}\\delta_0 + \\frac{1}{2} U(0,1)$, where $\\delta_0$ denotes the point mass at 0. Posterior inferences of this parameter allows us to flexibly choose between Dirichlet processes and more general~PDPs for the best-fitting clustering mechanism.\n\n\n\\bigskip\n\n\\noindent \\textbf{Latent vectors.} The hierarchical prior for the covariates is completed by specifying a \\textit{base distribution} $G^{(n)}$ in $\\mathcal{R}^n$ for the latent vectors $\\boldsymbol{v}_1,\\ldots,\\boldsymbol{v}_q$. Consistent with our goal of developing a flexible and scalable inference procedure capable of fitting large datasets, we impose additional lower-dimensional structure on the $n$-variate base distribution. Specifically, since the $n$ subjects are exchangeable, base distribution $G^{(n)}$ is assumed to be the $n$-fold product measure of a univariate distribution, $G$. This allows the individuals and clusters to communicate through the $nq$ number of latent vector elements:\n\\begin{equation}\nv_{ik} \\stackrel{iid}\\sim G \\qquad \\text{for } i=1,\\ldots,n, \\text{ and } k=1,\\ldots,q. \\label{v}\n\\end{equation}\nThe unknown, univariate distribution, $G$, is itself given a nonparametric Dirichlet process prior, allowing the latent vectors to flexibly capture the within-covariate patterns of the subjects:\n\\begin{equation}\nG \\sim \\mathcal{DP}(\\alpha_2)\\label{G}\n\\end{equation}\nfor mass parameter $\\alpha_2>0$ and univariate base distribution, $N(\\mu_2, \\tau_2^2)$. Being a realization of a Dirichlet process, distribution $G$ is discrete and allows the subjects to group differently in different PDP clusters. \nThe number of distinct values among the $v_{ik}$'s is asymptotically equivalent to $\\alpha_2 \\cdot \\log n q$, facilitating further dimension reduction and scalability of inference as $n$ approaches hundreds or thousands of individuals, as commonly encountered in genomic datasets.\n\n\n\\smallskip\n\n\n\\subsection{Prediction and regression model}\\label{S:predictors}\n\nNow, suppose there are $n_{k}$ covariates allocated to the $k^{th}$ cluster. \nWe posit that each cluster elects from among its member covariates a \\textit{representative}, denoted by $\\boldsymbol{u}_k$. A subset of the $q$ cluster representatives, rather than the covariates, feature in an additive regression model that can accommodate nonlinear functional relationships. The cluster representatives may be chosen in several different ways depending on the application. Some possible options include:\n\\begin{enumerate}\n\\item[\\textit{(i)}] Select with apriori equal probability one of the $n_k$ covariates belonging to the $k^{th}$ cluster as the representative. Let $s_k$ denote the index of the covariate chosen as the representative, so that $c_{s_k}=k$ and $\\boldsymbol{u}_k=\\boldsymbol{x}_{s_k}$.\n\\item[\\textit{(ii)}] Set latent vector $\\boldsymbol{v}_k$ of Section \\ref{S:covariates} as the cluster representative.\n\\end{enumerate}\nOption \\textit{(i)} is preferable when practitioners are mainly interested in identifying the effects of individual regressors, as in gene selection applications in cancer survival times (as noted in the introduction). Option \\textit{(ii)} is preferable when the emphasis is less on covariate selection and more on identifying clusters of candidate variables (e.g., genomic pathways) that are jointly associated with the responses.\n\n\nThe regression predictors are selected from among the $q$ cluster representatives, with their parent clusters constituting the set of \\textit{cluster predictors}, $\\mathcal{S}^* \\subset\\{1,\\ldots,q\\}$.\nExtensions of the spike-and-slab approaches \\citep{George_McCulloch_1993, Kuo_Mallick_1997, Brown_etal_1998} are applied to relate the regression outcomes to the cluster representatives as:\n\\begin{align}\ny_i &\\stackrel{indep}\\sim N\\left( \\eta_i,\\, \\sigma_i^2\\right), \\quad\\text{where} \\notag\\\\\n\\eta_i &= \\beta_0 + \\sum_{k=1}^q \\gamma_{k}^{(1)} \\beta_k^{(1)} u_{ik} + \\sum_{k=1}^q \\gamma_{k}^{(2)} h(u_{ik},\\boldsymbol{\\beta}_k^{(2)}) \\label{eta_i}\n\\end{align}\nand $h$ is a nonlinear function. Possible options for the nonlinear function $h$ in equation (\\ref{eta_i}) include reproducible kernel Hilbert spaces \\citep{mallickJRSSB2005}, nonlinear basis smoothing splines \\citep{Eubank1999}, and wavelets. Alternatively, due to their interpretability as a linear model, order-$r$ splines with $m$ number of knots \\citep{deBoor_1978, Hastie_Tibshirani_1990,Denison_etal_1998} are especially attractive and computationally tractable.\n\nThe linear predictor $\\eta_i$ in expression (\\ref{eta_i}) implicitly relies on a vector of cluster-specific indicators, $\\boldsymbol{\\gamma}=(\\boldsymbol{\\gamma}_1,\\ldots,\\boldsymbol{\\gamma}_q)$, where the triplet of indicators, $\\boldsymbol{\\gamma}_k=(\\gamma_{k}^{(0)}, \\gamma_{k}^{(1)}, \\gamma_{k}^{(2)})$, add to 1 for each cluster $k$. If $\\gamma_{k}^{(0)}=1$, the cluster representative and none of the covariates belonging to cluster~$k$ are associated with the responses. If $\\gamma_{k}^{(1)}=1$, the cluster representative appears as a simple linear regressor in equation (\\ref{eta_i}); $\\gamma_{k}^{(2)}=1$ implies a nonlinear regressor. \nThe number of linear predictors, non-linear predictors, and non-predictors are respectively, $q_1 =\\sum_{j=1}^q \\gamma_j^{(1)}$, $q_2 =\\sum_{j=1}^q \\gamma_j^{(2)}$, and $q_0 =q-q_1 - q_2$.\nFor a simple illustration of this concept, consider again the toy example of Figure \\ref{F:toy_VariScan}, where one covariate is nominated from each cluster as the~representative. Of the $q=11$ cluster representatives, $q_1=3$ are linear predictors, $q_2=2$ are non-linear predictors, and the remaining $q_0=6$ representatives are non-predictors.\n\n\n\nFor nonlinear functions $h$ having a linear representation (e.g., splines), let $\\boldsymbol{U}_{\\boldsymbol{\\gamma}}$ be a matrix of $n$ rows consisting of the intercept column and the independent regressors based on the cluster representatives. For example, if we use order-$r$ splines with $m$ number of knots in equation (\\ref{eta_i}), then the number of columns, $\\text{col}(\\boldsymbol{U}_{\\boldsymbol{\\gamma}})=q_1 + (m+r)\\cdot q_2 + 1$. With $[\\cdot]$ denoting densities of random variables, the\n prior,\n\\begin{equation}\n[\\boldsymbol{\\gamma}] \\propto \\omega_0^{q_0}\\omega_1^{q_1}\\omega_2^{q_2}\\cdot \\mathcal{I}\\biggl(\\text{col}(\\boldsymbol{U}_{\\boldsymbol{\\gamma}}) < n\\biggr), \\label{gamma}\n\\end{equation}\nwhere the probabilities $\\boldsymbol{\\omega}=(\\omega_{0}, \\omega_{1}, \\omega_{2})$ are given the Dirichlet distribution prior, $\\boldsymbol{\\omega} \\sim \\mathcal{D}_3(1,1,1)$.\nThe truncated prior for $\\boldsymbol{\\gamma}$ is designed to ensure model sparsity and prevent overfitting, as explained below. Conditional on the variances of the regression outcomes in equation (\\ref{eta_i}), we postulate a weighted g~prior for the regression coefficients:\n\\begin{equation}\n\\boldsymbol{\\beta}_{\\boldsymbol{\\gamma}} | \\boldsymbol{\\Sigma} \\sim N_{|\\mathcal{S}^*|+1}\\biggl(\\boldsymbol{0}, \\sigma_\\beta^2({\\boldsymbol{U}_{\\boldsymbol{\\gamma}}}'\\boldsymbol{\\Sigma}^{-1}\\boldsymbol{U}_{\\boldsymbol{\\gamma}})^{-1}\\biggr),\\label{Zellner}\n\\end{equation}\nwhere matrix $\\boldsymbol{\\Sigma}=\\text{diag}(\\sigma_1^2,\\ldots,\\sigma_n^2)$.\n\nA schematic representation of the entire hierarchical model involving both the clustering and prediction components is shown in Figure \\ref{F:dag}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.1]{dag_2016.pdf}\n\\caption{Directed acyclic graph of the VariScan model in which the cluster representatives are chosen from the set of co-clustered covariates. Circles represent stochastic model parameters, solid rectangles represent data and deterministic variables, and dashed rectangles represent model constants. Solid (dashed) arrows represent stochastic (deterministic) relationships. }\n\\label{F:dag}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\bigskip\n\n\n\\subsection{Model justification and generalizations}\\label{S:justifications}\n\nIn this section, we discuss the justification, consequences, and generalizations of different aspects of the Variscan model. In particular, we investigate the appropriateness of PDPs in this application as a tool for covariate clustering. We also discuss the choice of basis functions for the nonlinear prediction model and consider generalizations to discrete and survival outcomes.\n\n\\bigskip\n\n\\noindent \\textbf{Empirical justification of PDPs.} We conducted an exploratory data analysis (EDA) of the gene expression levels in the DLBCL data set of \\citet*{Rosenwald_etal_2002}.\nRandomly selecting a set of $p=500$ probes for $n=100$ randomly chosen individuals, we iteratively applied the k-means procedure until the covariates were grouped into fairly concordant clusters with a small overall value of $\\tau^2$.\nThe allocation pattern depicted in Figure \\ref{F:eda_barchart} is atypical of Dirichlet processes which, as is well known among practitioners, are usually associated with relatively small number of clusters and exponentially decaying cluster sizes. Instead,\nthe large number of clusters ($\\hat{q}=161$) and the predominance of small clusters suggest a non-Dirichlet model for the covariate-cluster assignments. More specifically, a PDP is favored due to the slower, power law decay in the cluster sizes typically associated with these models.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{eda_barchart.pdf}\n\\caption{Barchart of cluster sizes obtained by exploratory data analysis.}\n\\label{F:eda_barchart}\n\\end{center}\n\\end{figure}\n\n\\bigskip\n\n\\noindent \\textbf{Theoretical justifications for a PDP model.}\n\\cite{Sethuraman_1994} derived the \\emph{stick-breaking representation} for a Dirichlet process, and then \\cite {Pitman_1995} extended it to PDPs. These stick-breaking representations have the following consequences for the induced partitions. Let $\\mathbb{N}$ be the set of natural numbers.\n Subject to a one-to-one transformation of the first $q$ natural numbers into $\\mathbb{N}$, the allocation variables $c_1,\\ldots,c_p$ may be regarded as i.i.d.\\ samples from a discrete distribution $F_{\\alpha_1,d}$ on $\\mathbb{N}$ with stick-breaking probabilities, $\\pi_1 = V_1$ and $\\pi_h = V_h \\prod_{t=1}^{h-1}(1-V_t)$ for $h = 2,3,\\ldots$, where $V_h \\stackrel{indep}\\sim \\text{beta}(1-d, \\alpha_1+hd)$.\nThis implies that for large values of $p$ and for clusters $k=1,\\ldots,q$, the frequencies $n_k^{(p)}\/p$ are approximately equal to $\\pi_{h_k}$ for some distinct integers $h_1,\\ldots,h_q$.\n\n\nAs previously mentioned, the VariScan model assumes that the base distribution $G^{(n)}$ of the PDP is the $n$-fold product measure of a univariate distribution, $G$, which follows a Dirichlet process with mass parameter $\\alpha_2$. This bidirectional clustering structure has some interesting consequences. Let $\\{\\phi_h\\}_{h=1}^{\\infty}$ be the stick-breaking probabilities associated with this nested Dirichlet process. For two or more of the $q$ PDP clusters, the latent vectors are identical with a probability bounded above by ${q \\choose 2} \\cdot \\bigl(\\sum_{h=1}^\\infty \\phi_h^2\\bigr)^n$. Applying the asymptotic relationship of $p$ and $q$ given in expression (\\ref{q}), we find that the upper bound tends to 0 as the dataset grows, provided $p$ grows at a slower-than-exponential rate as $n$. In fact, for $n$ as small as 50 and $p$ as small as 250, in simulations as well as in data analyses, we found all the latent vectors associated with the PDP clusters to be distinct. Consequently, from a practical standpoint, the VariScan allocations may be regarded as clusters with unique characteristics in even moderate-sized datasets.\n\nTheorem \\ref{Theorem:stick-breaking} below provides formal expressions for the first and second moments of the random log-probabilities of the discrete distribution $F_{\\alpha_1,d}$. In conjunction with equation (\\ref{q}), this result justifies the use of PDPs when the observed number of clusters is large or the cluster sizes decay slowly. Part~\\ref{T:DP_lim} provides an explanation for the fact that Dirichlet process allocations typically consist of a small number of clusters, only a few of which are large, with exponential decay in the cluster sizes. Part~\\ref{T:PDP_lim} suggests that in PDPs with $d>0$ (i.e., non-Dirichlet realizations), there is a slower, power law decay of the cluster sizes as $d$ increases. Part~\\ref{T:PDP_DP} indicates that for every $\\alpha_1$ and $d>0$, a PDP realization $F_{\\alpha_1,d}$ is thicker tailed compared to a Dirichlet process realization, $F_{\\alpha_1,0}$. See Section \\ref{S_sup:stick-breaking proof} of the Appendix for a proof.\n\n\n\nIt should be noted that the differential allocation patterns of PDPs and Dirichlet processes are well known, and has been previously emphasized by several papers, including \\cite*{Lijoi_Mena_Prunster_2007a} and \\cite*{Lijoi_Mena_Prunster_2007b}. However, it is difficult to come across a formal proof for this differential behavior. Although the theorem is primarily of interest when the base measure is non-atomic, it is relevant in this application because of the empirically observed uniqueness of the latent vectors in high-dimensional~settings due to VariScan's nested structure.\n\n\\smallskip\n\n\\begin{theorem}\\label{Theorem:stick-breaking}\nConsider the process $\\mathbb{PDP}\\bigl(\\alpha_1, d \\bigr)$ with mass parameter $\\alpha_1>0$ and discount parameter $0 \\le d < 1$. Let $\\psi(x)=d\\log \\Gamma(x)\/dx$ denote the digamma function and $\\psi_1(x)=d^2\\log \\Gamma(x)\/dx^2$ denote the trigamma function.\n\\begin{enumerate}\n\n \n\\item\\label{T:PDP} For $0 < d < 1$, the distribution $F_{\\alpha_1,d} \\in \\mathbb{N}$ is a realization of a PDP with stick-breaking probabilities $\\pi_h$, where $h \\in \\mathbb{N}$. \n However, $F_{\\alpha_1,d}$ is not a Dirichlet process realization because $d \\neq 0$. Then\n \\begin{enumerate}\n \\item\\label{T:PDP_E} $E(\\log \\pi_h)=\\psi(1-d)-\\psi(\\alpha_1)+\\frac{1}{d}\\bigl(\\psi(\\alpha_1\/d)-\\psi(\\alpha_1\/d+h)\\bigr)$. This implies that $\\lim_{h \\to \\infty}E(\\log \\pi_h)=-\\infty$.\n\n \\item\\label{T:PDP_V} $\\text{Var}(\\log \\pi_h)=\\psi_1(1-d)-\\psi_1(\\alpha_1)+\\frac{1}{d^2}\\bigl(\\psi_1(\\alpha_1\/d)-\\psi_1(\\alpha_1\/d+h)\\bigr)$. Unlike a Dirichlet process realization, $\\lim_{h \\to \\infty} \\text{Var}(\\log \\pi_h)$ is finite regardless of $d>0$.\n\n \\item\\label{T:PDP_lim} For any $\\alpha_1>0$ and as $h\\to\\infty$,\n $\\log \\pi_h\/\\log h^{-1\/d} \\stackrel{p}\\rightarrow 1\n $ for non-Dirichlet process realizations.\n\n \\end{enumerate}\n\n\\item\\label{T:DP} For $d=0$, the distribution $F_{\\alpha_1,0} \\in \\mathbb{N}$ is a Dirichlet process realization with stick-breaking probabilities $\\pi_h^*$ based on $V_h^* \\stackrel{iid}\\sim \\text{beta}(1, \\alpha_1)$ for $h \\in \\mathbb{N}$. Then\n \\begin{enumerate}\n \\item\\label{T:DP_E} $E(\\log \\pi_h^*)=\\psi(1)-\\psi(\\alpha_1)-h\/\\alpha_1$. Thus, $\\lim_{h \\to \\infty}E(\\log \\pi_h^*) = -\\infty$.\n\n \\item\\label{T:DP_V} $\\text{Var}(\\log \\pi_h^*)=\\psi_1(1)-\\psi_1(\\alpha_1)+h\/\\alpha_1^2$. Thus, $\\lim_{h \\to \\infty} \\text{Var}(\\log \\pi_h^*)=\\infty$.\n\n \\item\\label{T:DP_lim} As $h\\to\\infty$,\n $\n \\sqrt{h}\\left(\\frac{1}{h}\\log (\\pi_h^*) + 1\/\\alpha_1\\right) \\stackrel{L}\\rightarrow N(0, 1\/\\alpha_1^2)$. This implies that as $h\\to\\infty$, the random stick-breaking Dirichlet process probabilities, $\\pi_h^*$, are stochastically equivalent to $e^{-h\/\\alpha_1}$.\n \\end{enumerate}\n\n\n \\item\\label{T:PDP_DP} As $h\\to\\infty$,\n $\\sqrt{h}\\left(\\frac{1}{h}\\log (\\pi_h^*\/\\pi_h) + 1\/\\alpha_1\\right) \\stackrel{L}\\rightarrow N(0, 1\/\\alpha_1^2)$. That is, as $h\\to\\infty$, the ratios of the Dirichlet process and non-Dirichlet process stick-breaking random probabilities, $\\pi_h^*\/\\pi_h$, are stochastically equivalent to $e^{-h\/\\alpha_1}$ for every $d>0$. \n\\end{enumerate}\n\\end{theorem}\n\n\n\\smallskip\n\n\n\\begin{remark}\nBy Lemma~1 of \\cite{Ishwaran_James_2003}, $\\lim_{h \\to \\infty}E(\\log \\pi_h^*) = -\\infty$ in Part~\\ref{T:DP_E} of Theorem \\ref{Theorem:stick-breaking} implies that $\\sum_{h=1}^{\\infty} \\pi_h^*=1$ almost surely for a Dirichlet process. A similar comment applies in Part~\\ref{T:PDP_E} for a PDP.\n\\end{remark}\n\n\\bigskip\n\n\\noindent \\textbf{Empirical justification of nested Dirichlet process model for the latent vector elements.}\nFor the DLBCL dataset, Figure \\ref{F:DP barchart} presents a summary of the VariScan model estimates for the 14,000 latent vector elements with estimated Bernoulli indicators $\\hat{z}_{ik}=1$. More than 87\\% of the $n\\hat{q}=16,500$ latent vector elements were estimated to have $\\hat{z}_{ik}=1$, implying that a relatively small proportion of covariate values for the DLBCL dataset can be regarded as random noise having no clustering structure. Further details about the inference procedure are provided in Section \\ref{S:inference}. In Figure \\ref{F:DP barchart}, the small number of clusters corresponding to the large number of latent vector elements, and the sharp decline in the cluster sizes compared with Figure \\ref{F:eda_barchart}, are consistent with Dirichlet process allocation patterns. Similar results were obtained for the breast cancer data and for other genomic datasets that we have analyzed.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.31]{dahl_DP_barchart.pdf}\n\\caption{For the DLBCL dataset, least-squares Dirichlet process configuration of the more than 14,000 latent vector elements with Bernoulli indicators equal to 1.}\n\\label{F:DP barchart}\n\\end{center}\n\\end{figure}\n\n\n\\bigskip\n\n\n\\noindent \\textbf{Choice of basis functions: model parsimony versus flexibility.} \\label{S:parsimony_v_flexibility}\nThe reliability of inference and prediction rapidly deteriorates as the number of cluster predictors and additive nonlinear components in equation (\\ref{eta_i}) increases beyond a threshold value and approaches the number of subjects, $n$. The restriction in the prior~(\\ref{gamma}) prevents over-fitting. It ensures that the matrix $\\boldsymbol{U}_{\\boldsymbol{\\gamma}}$, consisting of the independent regression variables, has fewer columns than rows, and is a sufficient condition for the existence of $({\\boldsymbol{U}_{\\boldsymbol{\\gamma}}}'\\boldsymbol{\\Sigma}^{-1}\\boldsymbol{U}_{\\boldsymbol{\\gamma}})^{-1}$ and the least-squares estimate of $\\boldsymbol{\\beta}_{\\boldsymbol{\\gamma}}$ in equation (\\ref{eta_i}). In spline-based models, the relatively small number of subjects also puts a constraint on the order of the splines, often necessitating the use of linear splines with $m=1$ knot per cluster in equation (\\ref{eta_i}). In the applications presented in this paper, we fixed the knot for each covariate at the sample median.\n\n\nUnusually small values of $\\sigma_i^2$ in equation (\\ref{eta_i}) correspond to over-fitted models, whereas unusually large values correspond to under-fitted models. Any parameters that determine $\\sigma_1^2,\\ldots,\\sigma_n^2$ are key, and their priors must be carefully chosen.\n For instance, linear regression assumes that $\\sigma_i^2=\\sigma^2$. We have found that non-informative priors for $\\sigma^2$ do not work well because the optimal model sizes for variable selection are unknown.\n Additionally, we have found that it is helpful to restrict the range of $\\sigma^2$ based on reasonable goals for inference precision.\nIn the examples discussed in this paper, we assigned the following truncated prior:\n$\\sigma^{-2} \\sim \\chi^2_{\\nu} \\cdot \\mathcal{I}\\left(0.95^{-1}\/\\text{Var}(\\hat{\\boldsymbol{y}}) < \\sigma^{-2} < 0.5^{-1}\/ \\text{Var}(\\hat{\\boldsymbol{y}})\\right)\n$,\nwhere the degrees of freedom $\\nu$ were appropriately chosen and the vector $\\hat{\\boldsymbol{y}}$ relied on EDA estimates of latent regression outcomes from a previous study or the training set individuals. The support for $\\sigma^{-2}$\napproximately corresponds to the constraint, $0.5 < R^2 < 0.95$, quantifying the effectiveness of~regression.\nAs Sections \\ref{S:simulation} and \\ref{S:benchmark_data} demonstrate, the aforementioned strategies often result in high reliability of the response predictions.\n\n\n\\bigskip\n\n\\noindent \\textbf{Generalizations for discrete or survival outcomes} In a general investigation, the subject-specific responses may be discrete or continuous, and\/or may be censored. \nIn such cases, the responses, denoted by $w_1,\\ldots,w_n$, can be modeled as deterministic transformations of random variables~$R_i$ from an exponential family distribution. The Laplace approximation \\citep{Harville_1977} transforms each $R_i$ into a Gaussian \\textit{regression outcome},~$y_i$, that can be modeled using our Variscan model proposed above.\nThe details of the calculation are as follows.\nFor a set of functions~$f_i$, we assume that $w_i = f_i(R_i)$ and density function\n$[R_i \\mid \\varrho_{i}, \\varsigma ] = r(R_i,\\varsigma)\\cdot \\exp\\left(\n\\frac{R_i\\,\\varrho_{i}-b(\\varrho_{i})}{a(\\varsigma)}\\right)$,\nwhere $r(\\cdot)$ is a non-negative function, $\\varsigma$ is a dispersion parameter, $\\varrho_{i}$ is the canonical parameter, and $[\\cdot]$ represents densities with respect to a dominating measure.\nThe Laplace approximation relates the $R_i$'s to Gaussian regression outcomes:\n$y_{i} = \\eta_{i} + \\frac{\\partial \\eta_{i}}{\\partial \\mu_{i}}\\cdot(R_{i} - \\mu_{i})$ is approximately $N\\left(\\eta_{i},\\sigma_i^2\\right)$\nwith precision $\\sigma_i^{-2}=\\{b^{''}(\\mu_{i})\\}^{-1}\\left(\\partial\n\\mu_{i}\/\\partial \\eta_{i}\\right)^2$.\nFor an appropriate link function $g(\\cdot)$, the mean $\\eta_{i}$ equals $g(\\mu_{i})$.\nGaussian, Poisson, and binary responses are applicable in this setting.\nAccelerated failure time (AFT) censored outcomes \\citep{Buckley_James_1979, Cox_Oakes_1984} also fall into this modeling framework.\n\n\nThe idea of using a Laplace-type approximation for exponential families is not new. Some early examples in Bayesian settings include \\cite{Zeger_Karim_1991}, \\cite{Chib_Greenberg_1994}, and \\cite{Chib_Greenberg_Winkelmann_1998}.\nFor linear regression, the approximation is exact\nwith $y_{i} = R_{i}$.\nThe Laplace approximation\nis not restrictive even when it is approximate; for example, MCMC proposals for the model parameters can be filtered through a Metropolis-Hastings step to obtain samples from the target posterior. Alternatively, inference strategies relying on normal mixture representations through auxiliary variables could be used to relate the $R_i$'s to the $y_i$'s. For instance, \\cite{Albert_Chib_1993} used truncated normal sampling to obtain a probit model for binary responses, and \\cite{HolmesHeld2006}\n utilized a scale mixture representation of the normal distribution \\citep{Andrews_Mallows_1974,West_1987} to implement logistic regression using latent variables.\n\n\n\\bigskip\n\n \\section{Posterior inference}\\label{S:inference}\n\n\n Starting with an initial configuration obtained by a na\\\"{i}ve, preliminary analysis, the model parameters are iteratively updated by MCMC methods.\n Due to the intensive nature of the posterior inference, the analysis is performed in two stages, with cluster detection followed by predictor discovery:\n\n \\smallskip\n\n \\begin{enumerate}\n \\item[\\textbf{Stage 1}] Focusing only on the covariates and ignoring the responses:\n\n \\smallskip\n\n \\begin{enumerate}\n \\item[\\textit{Stage 1a}] The allocation variables, latent vector elements, and binary indicators are iteratively updated until the MCMC chain converges. Monte Carlo estimates are computed for the posterior probability of clustering for each pair of covariates. Applying the technique of \\cite{Dahl_2006}, these pairwise probabilities are used to compute a point estimate, called the \\textit{least-squares allocation}, for the allocation~variables.\n Further details of the MCMC procedure are provided in Sections \\ref{S:MCMC_c} and \\ref{S:MCMC_v} of the Appendix.\n\n\n \\smallskip\n\n \\item[\\textit{Stage 1b}] Conditional on the least-squares allocation being the true clustering of the covariates, a second MCMC sample of the latent vector elements and binary indicators is generated. Again applying the technique of \\cite{Dahl_2006}, we compute a point estimate, called the \\textit{least-squares configuration}, for the latent vector elements and binary indicators.\n \\end{enumerate}\n\n \\bigskip\n\n \\item[\\textbf{Stage 2}] Conditional on the least-squares allocation and least-squares configuration, and focussing on the responses, the cluster predictors and latent regression outcomes, if any, are generated to obtain a third MCMC sample. The MCMC sample is post-processed to predict the responses for out-of-the-bag test set individuals. The interested reader is referred to Sections \\ref{S:MCMC_gamma}, \\ref{S:MCMC_y} and \\ref{S:test_case_prediction} of the Appendix for details.\n \\end{enumerate}\n\nAs a further benefit of a coherent model for the covariates, VariScan is able to perform model-based imputations of any missing subject-specific covariates as part of the MCMC procedure.\n\n\n\\bigskip\n\n\n\n\n\\section{Clustering consistency}\\label{S:clusters}\n\n\nAs mentioned in Section \\ref{S:covariates}, the latent vectors associated with two or more PDP clusters may be identical under the VariScan model, but this probability becomes vanishingly small as $n$ grows. Consequently, for practical purposes, the VariScan allocations may be interpreted as distinct, identifiable clusters in even moderately large datasets. In order to study the reliability of VariScan's clustering procedure in our targeted Big Data applications, we\n make large-sample theoretical comparisons between the VariScan model's cluster allocations and the true allocations of a hypothetical covariate generating process.\n\n In the general problem of using mixture models to allocate $p$ objects to an unknown number of clusters, the problem of non-identifiability and redundancy of the detected clusters has been extensively documented in Bayesian and frequentist applications \\citep[e.g., see][]{Fruhwirth-Schnatter_2006}. Some partial solutions are available in the Bayesian literature. For example, in finite mixture models, rather than assuming exchangeability of the mixture component parameters, \\cite{Petralia_etal_2012} regard them as draws\nfrom a repulsive process, leading to fewer, better separated and more interpretable\nclusters. \\cite{Rousseau_Mengersen_2011} show that a carefully chosen prior leads to asymptotic emptying of the\nredundant components in over-fitted finite mixture models.\nThe underlying strategy of these procedures is that they focus on detecting the correct number of clusters rather than the correct allocation of the $p$~objects.\n\nIn contrast to the non-identifiability of the detected clusters in fixed $n$ settings, Theorem \\ref{Thm:consistency} establishes the interesting fact that, when $p$ and $n$ are both large, a fixed set of covariates that (do not) co-cluster under the true process, also (do not) asymptotically co-cluster under the posterior. The key intuition is that, as with most mixture model applications, when $n$-dimensional objects are clustered and $n$ is small, it is possible for the clusters to be erroneously placed too close together even if $p$ is large. On the other hand, if $n$ is allowed to grow with $p$, then objects in $\\mathcal{R}^n$ eventually become well separated. \nConsequently, for $n$ and $p$ large enough, the VariScan method is able to infer the true clustering for a fixed subset of the $p$ covariate columns. In the sequel, using synthetic datasets in Section~\\ref{S:simulation2}, we exhibit the high accuracy of VariScan's clustering-related inferences.\n\n \\bigskip\n\n \\paragraph{\\textbf{The true model.}} The VariScan model's exchangeability assumption for the $p$ covariates stems from our belief in the existence of a true, unknown de Finetti density in $\\mathcal{R}^n$ from which the column vectors arise as a random sample. In particular, for any given $n$ and $p$, we make the following assumptions about the true covariate-generating process:\n\n\\begin{enumerate}[(a)]\n\\item\\label{true:first} The column vectors $\\boldsymbol{x}_1,\\ldots,\\boldsymbol{x}_p$ are a random sample of size $p$ from an $n$-variate distribution $P_0^{(n)}$ convolved with $n$-variate, independent-component Gaussian errors.\n \\item The true distribution $P_0^{(n)}$ is discrete in the space $\\mathcal{R}^n$. Let the $n$-dimensional atoms of $P_0^{(n)}$ be denoted by $\\boldsymbol{v}^{(0)}_t=(v^{(0)}_{1t},\\ldots,v^{(0)}_{nt})'$ for positive integers $t$.\n \\item\\label{true.alloc} Due to the discreteness of distribution $P_0^{(n)}$, there exist true allocation variables, $c_1^{(0)},\\ldots,c_p^{(0)}$, mapping the $p$ covariates to distinct atoms of $P_0^{(n)}$. For subjects $i=1,\\ldots,n$, and columns $j=1,\\ldots,p$, the covariates are then distributed as\n \\begin{equation}\n x_{ij} \\mid c_j^{(0)} \\stackrel{indep}\\sim N(v^{(0)}_{i\\, c_j^{(0)}}, \\tau_0^2), \\label{true.lik.x}\n \\end{equation}\n\\item The $n$-variate atoms of distribution $P_0^{(n)}$ are i.i.d.\\ realizations of the $n$-fold product measure of a univariate distribution, $G_0$. Consequently, the atom elements are $v^{(0)}_{it}\\stackrel{i.i.d.}\\sim G_0$ for $i=1,\\ldots,n$, and $j=1,\\ldots,p$.\n\n \\item\\label{true:last} The true distribution $G_0$ is non-atomic and has compact support on the real line.\n\\end{enumerate}\n\n\n\\bigskip\n\nLet $\\mathcal{L}=\\{j_1,\\ldots, j_L\\} \\subset \\{1,\\ldots,p\\}$ be a fixed subset of $L$ covariate indexes. Given a vector of inferred allocations $\\boldsymbol{c}=(c_1,\\ldots,c_p)$, we quantify the inference accuracy by the \\textit{proportion of correctly clustered covariate pairs}:\n \\begin{equation}\n \\varkappa_{\\mathcal{L}}(\\boldsymbol{c}) = \\frac{1}{{L \\choose 2}} \\sum_{j_1 \\neq j_2 \\in \\mathcal{L}} \\mathcal{I}\\biggl(\\mathcal{I}(c_{j_1}=c_{j_2})=\\mathcal{I}(c_{j_1}^{(0)}=c_{j_2}^{(0)})\\biggr). \\label{varkappa}\n \\end{equation}\n A value near 1 indicates the high accuracy of inferred allocations $\\boldsymbol{c}$ for the set $\\mathcal{L}$. Notice that the measure $\\varkappa_{\\mathcal{L}}(\\boldsymbol{c})$ is invariant to permutations of the clusters labels. This is desirable because the labels are arbitrary.\n\n\n\n\n\\bigskip\n\n\\begin{theorem}\\label{Thm:consistency}\nDenote the covariate matrix by $\\boldsymbol{X}_{np}$. In addition to assumptions (\\ref{true:first})--(\\ref{true:last}) about the true covariate-generating process, suppose that the true standard deviation $\\tau_0$ in equation (\\ref{true.lik.x}) is bounded below by $\\tau_*$, the small, positive constant postulated in Section~\\ref{S:covariates} as a lower bound for the Variscan model parameters, $\\tau_1$ and $\\tau$.\n\nLet $\\mathcal{L}=\\{j_1,\\ldots, j_L\\} \\subset \\{1,\\ldots,p\\}$ be a fixed subset of $L$ covariate indexes. Then there exists an increasing sequence of numbers $\\{p_n\\}$ such that, as $n$ grows and provided $p>p_n$, the VariScan clustering inferences for the covariate subset $\\mathcal{L}$ are aposteriori consistent. That is,\n\\[\n\\lim_{\\substack{n \\to \\infty \\\\ p> p_n}} P\\bigl[\\varkappa_{\\mathcal{L}}(\\boldsymbol{c})=1 \\mid \\boldsymbol{X}_{np}\\bigr] \\to 1.\n\\]\n\n\n\\end{theorem}\n\n\\bigskip\n\nSee Section \\ref{S:proof of Thm:consistency} of the Appendix for a proof.\nThe result relies on non-trivial extensions, in several directions, of the important theoretical insights provided by \\citep{Ghosal_Ghosh_Ramamoorthi_1999}.\nSpecifically, it extends Theorem 3 of \\cite{Ghosal_Ghosh_Ramamoorthi_1999} to densities on $\\mathcal{R}^n$ arising as convolutions of vector locations with errors distributed as zero-mean finite normal mixtures. \n\\bigskip\n\n\\section{Simulation studies}\n\n\\smallskip\n\n\\subsection{Cluster-related inferences}\\label{S:simulation2}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{d_cred.pdf}\n\\caption{ 95\\% posterior credible intervals for the discount parameter, $d$ for different values of $\\tau_0$. The true value, $d_0$, is shown by the red dashed line.}\n\\label{F:d}\n\\end{figure}\n\n We investigate VariScan's accuracy as a clustering procedure using artificial datasets for which the true clustering pattern is known. We simulated the covariates for $n=50$ subjects and $p=250$ genes from a discrete distribution convolved with Gaussian noise, and compared the co-clustering posterior probabilities of the $p$ covariates with the truth. The parameters of the true model were chosen to approximate match the corresponding estimates for the DLBCL dataset of \\cite{Rosenwald_etal_2002}. Specifically, for each of 25 synthetic datasets, and for the true model's parameter $\\tau_0$ in Theorem~\\ref{Thm:consistency} belonging to the range $[0.60, 0.96]$, we generated the following quantities to obtain the matrix $\\boldsymbol{X}$ in Step \\ref{X_step} below:\n\n\\begin{enumerate}\n\n \\item \\textbf{True allocation variables:} We generated $c_1^{(0)},\\ldots,c_p^{(0)}$ as the partitions induced by a PDP with true discount parameter $d^{(0)}=0.33$ and mass parameter $\\alpha_1=20$. The true number of clusters, $Q_0$, was thereby computed for this non-Dirichlet allocation.\n\n\\item \\textbf{Latent vector elements:} For $i=1,\\ldots,n$ and $k=1,\\ldots,Q_0$, elements $v_{ik}^{(0)} \\stackrel{iid}\\sim G_0$, where $G_0 \\sim \\mathcal{DP}(\\alpha_2)$,\nwith mass $\\alpha_2=10$ and uniform base distribution $U_0$ on the interval $[1.4,2.6]$.\n\n\\item\\label{X_step} \\textbf{Covariates:} $x_{ij} \\stackrel{indep}\\sim N(v_{ic_j}^{(0)}, \\tau_0^2)$ for $i=1,\\ldots,n$ and $j=1,\\ldots,p$.\n\n\\end{enumerate}\n\n No responses were generated in this study. \n Applying the general technique of \\cite{Dahl_2006} developed for Dirichlet process models, we computed a point estimate for the allocations, called the \\textit{least-squares configuration}, and denoted by $\\hat{c}_1,\\ldots,\\hat{c}_p$. For the full set of covariates, we estimated the accuracy of the least-squares allocation by the \\textit{estimated proportion of correctly clustered covariate pairs},\n \\[\n \\hat{\\varkappa} = \\frac{1}{{p \\choose 2}} \\sum_{j_1 \\neq j_2 \\in \\{1,\\ldots,p\\}} \\mathcal{I}\\biggl(\\mathcal{I}(\\hat{c}_{j_1}=\\hat{c}_{j_2})=\\mathcal{I}(c_{j_1}^{(0)}=c_{j_2}^{(0)})\\biggr).\n \\]\n A high value of $\\hat{\\varkappa}$ is indicative of VariScan's high clustering accuracy for all $p$ covariates.\n\n For each value of $\\tau_0$, the second column of Table \\ref{T:varkappa} displays the percentage $\\hat{\\varkappa}$ averaged over the 25 independent replications. We find that, for each $\\tau_0$, significantly less than 5 pairs were incorrectly clustered out of the ${250 \\choose 2}=$ 31,125 different covariate pairs, and so $\\hat{\\varkappa}$ was significantly greater than 0.999. The posterior inferences appear to be robust to large noise levels, i.e., large values of $\\tau_0$. For every dataset, $\\hat{q}$, the estimated number of clusters in the least-squares allocation was exactly equal to $Q_0$, the true number of~clusters. Recall that the non-atomicity of true distribution $G_0$ is a sufficient condition of Theorem~\\ref{Thm:consistency}. Although the condition is not satisfied in this setting, we nevertheless obtained highly accurate clustering-related inferences for the full set of $p=250$~covariates.\n\n\n\n Accurate inferences were also obtained for the PDP discount parameter, $d \\in [0,1)$. Figure \\ref{F:d} plots the 95\\% posterior credible intervals for $d$ against different values of $\\tau_0$. The posterior inferences are substantially more precise than the prior and each interval contained the true value,~$d_0=0.33$. Furthermore, in spite of being assigned a prior probability of 0.5, there is no posterior mass allocated to Dirichlet process models.\nThe ability of VariScan to discriminate between PDP and Dirichlet process models was evaluated using the log-Bayes factor, $\n \\log\\left(P[d>0|\\boldsymbol{X}]\/P[d=0|\\boldsymbol{X}]\\right)$. With $\\Theta^*$ representing all the parameters except $d$, and applying Jensen's inequality, the log-Bayes factor exceeds $E\\left(\\log\\left(\\frac{P[d>0|\\boldsymbol{X},\\Theta^*]}{p[d=0|\\boldsymbol{X},\\Theta^*]} \\right) \\mid \\boldsymbol{X} \\right)$, which (unlike the log-Bayes factor) can be estimated using just the post--burn-in MCMC sample. For each $\\tau_0$, the third column of Table \\ref{T:varkappa} displays 95\\% posterior credible intervals for this\n lower bound.\n The Bayes factors are significantly greater than $e^{10}=22,026.5$ and are overwhelmingly in favor of PDP~allocations, i.e., the true model.\n\n\n {\\small\n\n\\begin{table}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1}\n\\begin{tabular}{ c | c |c }\n\\hline\\hline\n \\textbf{True $\\tau_0$} &\\textbf{Percent $\\hat{\\varkappa}$} &\\textbf{95\\% C.I.\\ for lower }\\\\\n & &\\textbf{bound of log-BF} \\\\\n\\hline\n0.60 &99.984 (0.000) &(11.05, 11.10)\\\\\n0.66 &99.978 (0.000) &(11.17, 11.25)\\\\\n0.72 &99.976 (0.000) &(10.89, 10.98)\\\\\n0.78 &99.973 (0.001) &(10.23, 10.31)\\\\\n0.84 &99.971 (0.000) &(10.86, 10.93)\\\\\n0.90 &99.960 (0.000) &(11.88, 11.94)\\\\\n0.96 &99.941 (0.001) &(10.49, 10.56)\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{For different values of simulation parameter $\\tau_0$, column 2 displays the proportion of correctly clustered covariate pairs, with the standard errors for the 25 independent replications shown in parentheses. Column 3 presents 95\\% posterior credible intervals for the lower bound of the log-Bayes factor of PDP models relative to Dirichlet process models. See the text for further explanation.}\\label{T:varkappa}\n\\end{table}\n}\n\n\n\n\n\n\n\n\\subsection{Prediction accuracy}\\label{S:simulation}\n\n\nWe evaluate the operating characteristics of our methods using a simulation study based\non the DLBCL dataset of \\cite{Rosenwald_etal_2002}. To generate the simulated data, we selected $p=500$ genes from the original gene expression dataset of 7,399 probes, as detailed below:\n\n\n\n\\begin{enumerate}\n\n\\item Select $10$ covariates with pairwise correlations less than 0.5 as the true predictor set, $\\mathcal{S} \\subset\\{1,\\ldots,500\\}$, so that $|\\mathcal{S}|=10$.\n\n\n\\item For each value of $\\beta^*\\in \\{0.2, 0.6, 1.0\\}$:\n\n\\begin{enumerate}\n\n \\item For subjects $i=1,\\ldots,100$, generate failure times $t_i$ from distribution $ \\mathcal{E}_i$, denoting the exponential distribution with mean~$\\exp(\\beta^* \\sum_{j\\in \\mathcal{S}} x_{ij})$. Note that the model used to generate the outcomes differs from VariScan assumption~(\\ref{eta_i}) for the log-failure times.\n\n\\item For 20\\% of individuals, generate their censoring times as follows: $u_i \\sim$ $\\mathcal{E}_i \\cdot \\mathcal{I}(u_i < t_i)$. Set the survival times of these individuals to $w_i=\\log u_i$ and their failure statuses to $\\delta_i=0$.\n\n\\item For the remaining individuals, set $w_i = \\log t_i$ and $\\delta_i=1$.\n\n\\end{enumerate}\n\n\\item Randomly assign 67 individuals to the training set and the remaining 33 individuals to the test set.\n\n\\item Assuming the AFT survival model, apply the VariScan procedure with linear splines and $m=1$ knot per spline. Choose a single covariate from each cluster as the representative as described in Section \\ref{S:predictors}. Make posterior inferences using the training data and predict the outcomes for the test~cases.\n\n\\end{enumerate}\n\nWe analyzed the\n same set of simulated data using six other techniques for gene selection with survival outcomes: lasso \\citep{Tibshirani_1997}, adaptive lasso \\citep{Zou_2006}, elastic net \\citep{Zou_Trevor_2005}, $L_2$-boosting \\citep{Hothorn_Buhlmann_2006}, random survival forests \\citep{Ishwaran_etal_2010}, and supervised principal components \\citep{Bair_Tibshirani_2004}, which have been implemented in the R packages glmnet, mboost, randomSurvivalForest, and superpc. The ``RSF-VH'' version of the random survival forests procedure was chosen because of its success in high-dimensional~problems. The selected techniques are excellent examples of the three categories of approaches for small $n$, large $p$ problems (variable\nselection, nonlinear prediction, and regression\nbased on lower-dimensional projections) discussed in Section~\\ref{S:introduction}.\nWe repeated this procedure over fifteen independent replications.\n\n\n\n\n\n\nWe compared the prediction errors of the methods using the \\textit{concordance error rate}, which is defined as $1-C$, where $C$ denotes the c index of \\cite{Harrell_etal_1982}. Let the set of ``usable'' pairs of subjects be $\\mathcal{U} = \\{(i,j): w_i < w_j, \\delta_i=1\\} \\cup \\{(i,j): w_i = w_j, \\delta_i\\neq \\delta_j\\}$. The concordance error rate of a procedure is \\citep{May_etal_2004}:\n $\n 1 - C = \\frac{1}{|\\mathcal{U}|}\\sum_{(i,j) \\in \\mathcal{U}} \\mathcal{I}(\\tilde{w}_i \\ge \\tilde{w}_j) - \\frac{1}{2|\\mathcal{U}|}\\sum_{(i,j) \\in \\mathcal{U}} \\mathcal{I}(\\tilde{w}_i = \\tilde{w}_j)\n$,\nwhere $\\tilde{w}_i$ is the predicted response of subject $i$. For example, for the VariScan procedure applied to analyze AFT survival outcomes, the predicted responses are $\\tilde{w}_i=\\exp(\\tilde{y}_i)$, where\n $\\tilde{y}_i$ are the predicted regression outcomes. \n\n \\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.31]{new2_sim_boxplot_1.pdf}\n\\includegraphics[scale=0.31]{new2_sim_boxplot_2.pdf}\n\\includegraphics[scale=0.31]{new2_sim_boxplot_3.pdf}\n\\caption{Side-by-side boxplots comparing the percentage concordance error rates of the different techniques in the simulation study.}\n\\label{F:C2_simulation}\n\\end{center}\n\\end{figure}\n\nThe concordance error rate measures a procedure's probability of incorrectly ranking the failure times of two randomly chosen individuals. The accuracy of a procedure is inversely related to its concordance error rate. The measure is especially useful for comparisons because it does not rely on the survivor function, which is estimable by VariScan, but not by some of the other procedures.\n Figure~\\ref{F:C2_simulation} depicts boxplots of the concordance error rates of the procedures sorted by increasing order of prediction accuracy. \n We find that as $\\beta^*$ increases, the concordance error rates progressively decrease for most procedures, including VariScan. For larger $\\beta^*$, the error rates for VariScan are significantly lower than the error rates for the other~methods.\n\nIn order to facilitate a more systematic evaluation, we have plotted in Figure~\\ref{F:sims} the error rates versus model sizes for the different methods, thereby providing a joint examination of model parsimony and prediction. To aid a visual interpretation, we did not include the supervised principal components method, since it performs the worst in terms of prediction and detects models that are two to four fold larger than $L_2$-boosting, which typically produces the largest models among the depicted methods. The three panels correspond to increasing effect size, $\\beta^*$. A few facts are evident from the plots. VariScan seems to balance sparsity and prediction the best for all values of $\\beta^*$, with its performance increasing appreciably with $\\beta^*$. Penalization approaches such as lasso, adaptive lasso, and elastic net produce sparser models but have lower prediction accuracies. $L_2$-boosting is comparable to Variscan in terms of prediction accuracy, but detects larger models for the lower effect sizes (left and middle panel); Variscan is the clear winner for the largest effect size (right panel). Additionally, especially for the largest $\\beta^*$, we observe substantial variability between the simulation runs for the penalization approaches, as reflected by the large standard errors. Further simulation study comparisons of VariScan and the competing approaches are presented in Section \\ref{SA:prediction_simulation} of the Appendix.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{plot_sims.pdf}\n\\caption{Plot of concordance error rates versus model sizes for the competing methods along with the standard errors (shown by whiskers).\nThe left, middle and right respectively correspond to effect size $\\beta^*$ equal to 0.2, 0.6, and 1.}\n\\label{F:sims}\n\\end{center}\n\\end{figure}\n\n\\textbf{Nonlinearity measure.} Unlike some existing approaches, VariScan is able to measure the degree of nonlinearity in the relationships between the responses and covariates. For example, we could define \\textit{nonlinearity measure} $\\mathcal{N}$ as the posterior expectation,\n\\begin{equation}\n\\mathcal{N} = E\\bigl( \\frac{\\omega_2}{\\omega_1+\\omega_2} | \\boldsymbol{w},\\boldsymbol{X}\\bigr). \\label{N}\n\\end{equation}\nThis represents the posterior odds that a hypothetical, new cluster is a non-linear predictor in equation~(\\ref{eta_i}) rather than a simple linear regressor. \nA value of $\\mathcal{N}$ close to 1 corresponds to predominantly nonlinear associations between the responses and their predictors.\n\nAveraging over the 15 independent replications of the simulation, as $\\beta^*$ varied over the set $\\{0.2, 0.6, 1.0\\}$, the estimates of the nonlinearity measure $\\mathcal{N}$ defined in equation~(\\ref{N}), were 0.72, 0.41, and 0.25, respectively. The corresponding standard errors were 0.04, 0.07, and 0.06. This indicates that on the scale of the simulated log--failure times, simple linear regressors are increasingly preferred to linear splines as the signal-to-noise ratio, quantified by $\\beta^*$, increases. Such interpretable measures of nonlinearity are not provided by the competing methods.\n\n\\bigskip\n\n\\begin{center}\n\\section{Analysis of benchmark data sets} \\label{S:benchmark_data}\n\\end{center}\n\nReturning\n to the two publicly available datasets of Section \\ref{S:introduction}, we chose $p=500$ probes for further analysis. For the DLBCL dataset of \\citet*{Rosenwald_etal_2002}, we randomly selected 100 out of the 235 individuals who had non-zero survival times. Of the individuals selected, 50\\% had censored failure times. For the breast cancer dataset of \\citet*{vantVeer_2002}, we analyzed the 76 individuals with non-zero survival times, of which 44 individuals (57.9\\%) had censored failure times.\n\nWe performed 50 independent replications of the three steps that follow. \\textit{(i)} We randomly split the data into training and test sets in a 2:1 ratio. \\textit{(ii)} We analyzed the survival times and $p=500$ gene expression levels of the training cases using the techniques VariScan, lasso, adaptive lasso, elastic net, $L_2$-boosting, random survival forests, and supervised principal components. \\textit{(iii)}~The different techniques were used to predict the test~case outcomes. For the VariScan procedure, a single covariate from each cluster was chosen to be the cluster representative.\n\n\nThe number of clusters for the least-squares allocation of covariates, $\\hat{q}$, computed in Stage 1a of the analysis, were 165 and 117 respectively for the DLBCL and the breast cancer datasets. The nonlinearity measure $\\mathcal{N}$ estimates were 0.97 and 0.75 respectively with small standard errors. This indicates that the responses in both datasets, but especially in the DLBCL dataset, have predominantly nonlinear relationships with the predictors. In spite of being assigned a prior probability of 0.5, the estimated posterior probability of the Dirichlet process model (corresponding to discount parameter $d=0$) was exactly 0 for both datasets, justifying the PDP-based allocation scheme.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.31]{d.pdf}\n\\includegraphics[scale=0.31]{q.pdf}\n\\includegraphics[scale=0.31]{dahl_covariate_barchart.pdf}\n\\caption{Posterior summaries for the DLBCL dataset. The top panels and the lower panel summarize the least-squares covariate-to-cluster PDP allocation of the 500 genes. \n}\n\\label{F:clustering}\n\\end{center}\n\\end{figure}\n\nFor the DLBCL data, the upper panel of Figure \\ref{F:clustering} displays the estimated posterior density of the PDP's discount parameter $d$. The estimated posterior probability of the event $[d=0]$ is exactly zero, implying that a non-Dirichlet process clustering mechanism is strongly favored by the data, as suggested earlier by the EDA. The middle panel of Figure \\ref{F:clustering} plots the estimated posterior density of the number of clusters. The a posteriori large number of clusters (for $p=500$ covariates) is suggestive of a PDP model with $d>0$ (i.e.\\ a non-Dirichlet process model).\nThe lower panel of Figure \\ref{F:clustering} summarizes the cluster sizes of the least-squares allocation \\citep{Dahl_2006}. The large number of clusters ($\\hat{q}=165$) and the multiplicity of small clusters are very unusual for a Dirichlet process, justifying the use of the more general PDP~model.\n\n\n\nThe effectiveness of VariScan as a model-based clustering procedure can be shown as follows.\nFor each of the $\\hat{q}=165$ clusters in the least-squares allocation of Stage 1a, we computed the correlations between its member covariates and the latent vector for individuals with $\\hat{z}_{ik}=1$. The cluster-wise median correlations are plotted in Figure \\ref{F:corr}. The plots reveal fairly good within-cluster concordance regardless of the cluster size. Figure \\ref{F:heatmap_cluster} displays heatmaps for the DLBCL covariates that were allocated to column clusters having more than 10 members.\nThe panels display the covariates before and after bidirectional clustering of the subjects and probes, with the lower panel of Figure \\ref{F:heatmap_cluster} illustrating the within-cluster patterns revealed by VariScan. For each column cluster in the lower panel, the uppermost~rows represent the covariates of any subjects that do not follow the cluster structure and which are better modeled as random noise (i.e., covariates with $\\hat{z}_{ik}=0$).\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{dahl_median_corr.pdf}\n\\includegraphics[scale=0.3]{corr_size.pdf}\n\\caption{For the DLBCL dataset, median pairwise correlations for the $\\hat{q}=165$ PDP clusters in the least-squares allocation of Stage 1a.}\n\\label{F:corr}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{raw_cluster.pdf}\n\\includegraphics[scale=0.5]{VariScan_cluster.pdf}\n\\caption{Heatmaps of DLBCL covariates that were assigned to latent column clusters with more than 10 members. The panels display the covariates before and after bidirectional local clustering by VariScan. The vertical lines in the bottom panel mark the covariate-clusters. The color key for both panels is displayed at the top of the plot.}\n\\label{F:heatmap_cluster}\n\\end{center}\n\\end{figure}\n\n\nComparing the test case predictions with the actual survival times, boxplots of numerical summaries of the concordance error rates for all the methods are presented in Figure \\ref{F:C}. \nThe success of VariScan appears to be robust to the different censoring rates of survival datasets. Although $L_2$-boosting had comparable error rates for the DLBCL dataset, VariScan had the lowest error rates for both datasets. Further data analysis results and comparisons are available in Section \\ref{S_sup:benchmark data} of the~Appendix.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.25]{new_DLBCL_boxplot.pdf}\n\\includegraphics[scale=0.25]{new_vantVeer_boxplot.pdf}\n\\caption{Side-by-side boxplots of percentage concordance error rates for the benchmark datasets.}\n\\label{F:C}\n\\end{center}\n\\end{figure}\n\n\n\n\nFor subsequent biological interpretations, we selected genes having high probability of being selected as predictors (with the upper percentile decided by the model size). We then analyzed these genes for their role in cancer progression by cross-referencing with the existing literature. For the breast cancer dataset, our survey indicated several prominent genes related to breast cancer development and progression, such as TGF-B2 \\citep{pmid17261761}, ABCC3, which is known to be up-regulated in primary breast cancers, and LAPTM4B, which is related to breast carcinoma relapse with metastasis \\citep{pmid20098429}. For the DLBCL dataset, we found several genes related to DLBCL progression, such as the presence of multiple chemokine ligands (CXCL9 and CCL18), interleukin receptors of IL2 and IL5 \\citep{pmid16418498}, and BNIP3, which is down-regulated in DLBCL and is a known marker associated with positive survival \\citep{pmid18288132}. A detailed functional\/mechanistic analysis of the main set of genes for both datasets is provided in Section \\ref{S_sup:benchmark data} of the Appendix.\\\\\n\n\\bigskip\n\n\\section{Conclusions}\\label{S:conclusion}\n\nUtilizing the sparsity-inducing property of PDPs, VariScan offers an efficient technique for clustering, variable selection, and prediction in high-dimensional regression problems. The covariates are grouped into a smaller number of clusters consisting of covariates with similar across-subject patterns. We theoretically demonstrate how a PDP allocation can be differentiated from a Dirichlet process allocation in terms of the relative sizes of the latent clusters. We provide a theoretical explanation for the impressive ability of VariScan to aposteriori detect the true covariate clusters for a general class of models.\n\nIn simulations and real data analysis, we show that VariScan makes highly accurate cluster-related inferences. The technique consistently outperforms established methodologies such as random survival forests, $L_2$-boosting, and supervised principal components, in terms of prediction accuracy. \nIn the analyses of benchmark microarray datasets, we identified several genes having known implications in cancer development and progression, which further engenders our hypothesis.\n\nThe VariScan methodology focusses on continuous covariates as a proof of concept, achieving simultaneous clustering, variable selection, and prediction in high-throughput regression settings and possessing appealing theoretical and empirical properties. Generalization to count, categorical, and ordinal covariates is possible. It is important to investigate the dependence structures and theoretical properties associated with the more general framework. This will be the focus of our group's future~research.\n\nDue to the intensive nature of the MCMC inference, we performed these analyses in two stages, with cluster detection followed by predictor discovery. We are currently working on implementing VariScan's MCMC procedure in a parallel computing framework using graphical processing units \\citep{suchard2010understanding}. We plan to make the software available as an R package for general purpose use in the near future. The single-stage analysis will allow the regression and clustering results to be interrelated, as implied by the VariScan model. We anticipate being able to dramatically speed up the calculations by multiple orders of magnitude, which will allow for single-stage inferences of user-specified datasets on ordinary desktop and laptop~computers.\n\n\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLet $A$ and $B$ be two closed \nconvex nonempty sets in a Hilbert space $X$. The (2-set) convex feasibility problem asks to find a point in the intersection of $A$ and $B$ (or, when $A \\cap B=\\emptyset$, a pair of points, one in $A$ and the other in $B$, that realizes the distance between $A$ and $B$). The relevance of this problem is due to the fact that many mathematical and concrete problems in applications can be formulated as a convex feasibility\nproblem. As typical examples, we mention solution of convex inequalities, partial differential equations, minimization of convex nonsmooth functions, medical imaging, computerized tomography and image reconstruction. \n\nThe method of alternating projections is the simplest iterative procedure for finding a solution of the convex feasibility problem and it goes back to von Neumann \\cite{vonNeumann}: let us denote by $P_A$ and $P_B$ the projections on the sets $A$ and $B$, respectively, and, given a starting point $c_0\\in X$, consider the {\\em alternating projections sequences} $\\{c_n\\}$ and $\\{d_n\\}$ given by $$d_n=P_{B}(c_{n-1})\\ \\ \\text{and}\\ \\ c_n=P_{A}(d_n)\\ \\ \\ \\ \\ (n\\in\\N).$$\nIf the sequences $\\{c_n\\}$ and $\\{d_n\\}$ converge in norm, we say that the method of alternating projections converges.\nOriginally, von Neumann proved that the method of alternating projection converges when $A$ and $B$ are closed subspace. Then, for two generic convex sets, the weak convergence of the alternating projection sequences was proved by Bregman in 1965 (\\cite{Bregman}). Nevertheless, the problem whether the alternating projections algorithm converges in norm for each couple of convex sets remained open till the example given by Hundal in 2004 (\\cite{Hundal}). This example shows that the alternating projections do not converge in norm when $A$ is a suitable convex cone and $B$ is a hyperplane touching the vertex of $A$. Moreover, this example emphasizes the importance of finding sufficient conditions ensuring the norm convergence of the alternating projections algorithm. In the literature, conditions of this type were studied (see, e.g., \\cite{BauschkeBorwein93,BorweinSimsTam}), even before the example by Hundal.\nHere, we focus on those conditions based on the notions of regularity, introduced in \\cite{BauschkeBorwein93}. Indeed, in the present paper, we investigate the relationships between regularity of the couple $(A,B)$ (see Definition~\\ref{def: regularity} below) and ``stability'' properties of the alternating projections method in the following sense. Let us suppose that \n$\\{A_n\\}$ and $\\{B_n\\}$\nare two sequences of closed convex sets such that $A_n\\rightarrow\nA$ and $B_n\\rightarrow B$ for the Attouch-Wets {variational} convergence (see Definition~\\ref{def:AW}) and let us introduce the definition of {\\em perturbed alternating projections sequences}.\n\n\n\\begin{definition}\\label{def:perturbedseq} Given $a_0\\in X$, the {\\em perturbed alternating projections sequences} $\\{a_n\\}$ and $\\{b_n\\}$, w.r.t. $\\{A_n\\}$ and $\\{B_n\\}$ and with starting point $a_0$, are defined inductively by\n\t$$b_n=P_{B_n}(a_{n-1})\\ \\ \\ \\text{and}\\ \\ \\ a_n=P_{A_n}(b_n) \\ \\ \\ \\ \\ \\ \\ \\ \\ (n\\in\\N).$$ \n\\end{definition}\n\n\\noindent Our aim is to find some conditions on the limit sets $A$ and $B$ such that, for each choice of the sequences $\\{A_n\\}$ and $\\{B_n\\}$ and for each choice of the starting point $a_0$, \nthe corresponding perturbed alternating projections sequences $\\{a_n\\}$ and $\\{b_n\\}$ satisfy $\\mathrm{dist}(a_n,A\\cap B)\\to 0$ and $\\mathrm{dist}(b_n,A\\cap B)\\to 0$. If this is the case, we say that the couple $(A,B)$ is {\\em $d$-stable}.\nIn particular, we show that the regularity of the couple $(A,B)$ implies not only the norm convergence of the alternating projections sequences for the couple $(A,B)$ (as already known from \\cite{BauschkeBorwein93}), but also that the couple $(A,B)$ is $d$-stable. This result \n might be interesting also in applications since real data are often affected by some uncertainties. Hence stability of the convex feasibility problem with respect to data perturbations is a desirable property, also in view of computational developments. \n\nLet us conclude the introduction by a brief description of the structure of the paper. In Section \\ref{SEction notations}, we list some notations and definitions, and we recall some well-known facts about the alternating projections method. Section~\\ref{Section regularity} is devoted to various notions of regularity and their relationships. It is worth pointing out that in this section we provide a new and alternative proof of the convergence of the alternating projections algorithm under regularity assumptions. This proof well illustrates the main geometrical idea behind the proof of our main result Theorem~\\ref{teo: mainHilbert}, stated and proved in Section \\ref{Section Main Result}. This result shows that {\\em a regular couple $(A,B)$ is $d$-stable whenever $A\\cap B$ (or a suitable substitute if $A \\cap B=\\emptyset$) is bounded}. \nCorollaries~\\ref{Corollary:stronglyexp}, \\ref{corollary:corpilur}, and \\ref{cor:sottospazisommachiusa} simplify and generalize some of the results obtained in \\cite{DebeMigl}, since there we considered only the case where $A \\cap B\\neq\\emptyset$ whereas, in the present paper, we encompass also the situation where the intersection of $A$ and $B$ is empty. We conclude the paper with Section 5, where we discuss the necessity of the assumptions of our main result and we state a natural open problem: suppose that $A \\cap B$ is bounded, is regularity equivalent to $d$-stability? \t\n\n\n\n\n\n\n\\section{Notation and preliminaries }\\label{SEction notations}\n\nThroughout all this paper, $X$ denotes a nontrivial real normed space with\nthe topological dual $X^*$. We\ndenote by $B_X$ and $S_X$ the closed unit ball and the unit sphere of $X$, respectively. \nFor $x,y\\in X$, $[x,y]$ denotes the closed segment in $X$ with\nendpoints $x$ and $y$, and $(x,y)=[x,y]\\setminus\\{x,y\\}$ is the\ncorresponding ``open'' segment. \nFor a subset $A$ of $X$, we denote by $\\inte(A)$, $\\conv(A)$ and\n$\\cconv(A)$ the interior, the convex hull and the closed convex\nhull of $A$, respectively.\nLet us recall that a body is a closed convex setsin $X$ with nonempty interior.\n\n We denote by $$\\textstyle \\mathrm{diam}(A)=\\sup_{x,y\\in A}\\|x-y\\|,$$\nthe (possibly infinite) diameter of $A$. For $x\\in X$, let\n$$\\dist(x,A) =\\inf_{a\\in A} \\|a-x\\|.$$ Moreover, given $A,B$\nnonempty subsets of $X$, we denote by $\\dist(A,B)$ the usual\n``distance'' between $A$ and $B$, that is,\n$$ \\dist(A,B)=\\inf_{a\\in A} \\dist(a,B).$$\n\nNow, we recall two notions of convergence for sequences of sets (for a wide overview about this topic see, e.g., \\cite{Beer}). By $\\C(X)$ we denote the family of all nonempty closed subsets of\n\t$X$.\nLet us introduce the (extended) Hausdorff metric $h$ on\n$\\C(X)$. For $A,B\\in\\C(X)$, we define the excess of $A$ over $B$\nas\n$$e(A,B) = \\sup_{a\\in A} \\mathrm{dist}(a,B).$$\n\\noindent Moreover, if $A\\neq\\emptyset$ and $B=\\emptyset$ we put\n$e(A,B)=\\infty$, if $A=\\emptyset$ we put $e(A,B)=0$. Then, we define\n\n$$h(A,B)=\\max \\bigl\\{ e(A,B),e(B,A) \\bigr\\}.$$\n\n\\begin{definition} A sequence $\\{A_j\\}$ in $\\C(X)$ is said to\n\tHausdorff converge to $A\\in\\C(X)$ if $$\\textstyle \\lim_j h(A_j,A)\n\t= 0.$$\n\\end{definition}\n\n\n\nAs the second notion of convergence, we consider the so called Attouch-Wets convergence (see,\ne.g., \\cite[Definition~8.2.13]{LUCC}), which can be seen as a\nlocalization of the Hausdorff convergence. If $N\\in\\N$ and\n$A,B\\in\\C(X)$, define\n\\begin{eqnarray*}\n\te_N(A,B) &=& e(A\\cap N B_X, B)\\in[0,\\infty),\\\\\n\th_N(A,B) &=& \\max\\{e_N(A,B), e_N(B,A)\\}.\n\\end{eqnarray*}\n\n\n\\begin{definition}\\label{def:AW} A sequence $\\{A_j\\}$ in $\\C(X)$ is said to\n\tAttouch-Wets converge to $A\\in\\C(X)$ if, for each $N\\in\\N$,\n\t$$\\textstyle \\lim_j h_N(A_j,A)= 0.$$\n\\end{definition}\n\nWe conclude this section by recalling some well known results about distance between two convex sets and projection of a point onto a convex set.\n\nSuppose that $A$ and $B$ are closed convex nonempty subsets of $X$, we denote\n\\begin{eqnarray*}\n\tE&=&\\{a\\in A;\\, d(a,B)=d(A,B)\\},\\\\\n\tF&=&\\{b\\in B;\\, d(b,B)=d(A,B)\\}.\n\\end{eqnarray*}\n\n{If $C$ is a nonempty closed convex subset of $X$, the projection of a point $x$ onto $C$ is denoted by $P_Cx$.} \nWe say that $v=P_{\\overline{B-A}}(0)$ is the {\\em displacement vector} for the couple $(A,B)$.\nIt is clear that if $A\\cap B\\neq\\emptyset$ then $E=F=A\\cap B$ and the displacement vector for the couple $(A,B)$ is null.\nWe {recall} the following fact, where, given a map $T:X\\rightarrow X$, $\\mathrm{Fix}(T)$ denotes the set of fixed points of $T$.\n \n\n\\begin{fact}[{\\cite[Fact~1.1]{BauschkeBorwein93}}]\\label{fact: BB93} Suppose that $X$ is a Hilbert space and that $A,B$ are closed convex nonempty subsets of $X$. Then we have:\n\t\\begin{enumerate}\n\t\t\\item $\\|v\\|=d(A,B)$ and $E+v=F$;\n\t\t\\item $E=\\mathrm{Fix}(P_A P_B)=A\\cap(B-v)$ and $F=\\mathrm{Fix}(P_B P_A)=B\\cap(A+v)$;\n\t\t\\item $P_B e=P_F e=e+v$ ($e\\in E$) and $P_A f=P_E f=f-v$ ($f\\in F$).\n \t\\end{enumerate}\n\t\\end{fact} \n\n\n\\section{Notions of regularity for a couple of convex sets} \\label{Section regularity}\n{\nIn this section we introduce some notions of regularity for a couple of nonempty closed convex sets $A$ and $B$. This class of notions was originally introduced in \\cite{BauschkeBorwein93}, in order to obtain some conditions ensuring the norm convergence of the alternating projections algorithm (see, also, \\cite{BorweinZhu}).\nHere we list three different type of regularity: (i) and (ii) are exactly as they appeared in \\cite{BauschkeBorwein93}, whereas (iii) is new.\n}\n\n\\begin{definition}\\label{def: regularity} Let $X$ be a Hilbert space and $A,B$ closed convex nonempty subsets of $X$. Suppose that $E,F$ are nonempty. We say that the couple $(A,B)$ is:\\begin{enumerate}\n\t\t\\item {\\em regular} if for each $\\epsilon>0$ there exists $\\delta>0$ such that $\\mathrm{dist}(x,E)\\leq \\epsilon$, whenever $x\\in X$ satisfies\n\t\t$$\\max\\{\\mathrm{dist}(x,A),\\mathrm{dist}(x,B-v)\\}\\leq\\delta;$$\n\\item {\\em boundedly regular} if for each bounded set $S\\subset X$ and for each $\\epsilon>0$ there exists $\\delta>0$ such that $\\mathrm{dist}(x,E)\\leq \\epsilon$, whenever $x\\in S$ satisfies\n$$\\max\\{\\mathrm{dist}(x,A),\\mathrm{dist}(x,B-v)\\}\\leq\\delta;$$\n\\item {\\em linearly regular for points bounded away from $E$} if for each $\\epsilon>0$ there exists $K>0$ such that \n$$\\mathrm{dist}(x,E)\\leq K\\max\\{\\mathrm{dist}(x,A),\\mathrm{dist}(x,B-v)\\},$$\nwhenever $\\mathrm{dist}(x,E)\\geq \\epsilon$.\n\t\\end{enumerate} \n\\end{definition}\n\nThe following proposition shows that (i) and (iii) in the definition above are equivalent. The latter part of the proposition is a generalization of \\cite[Theorem~3.15]{BauschkeBorwein93}.\n\n\\begin{proposition}\\label{prop: regular-largedistances} Let $X$ be a Hilbert space and $A,B$ closed convex nonempty subsets of $X$. Suppose that $E, F$ are nonempty. Let us consider the following conditions. \\begin{enumerate}\n\t\t\\item The couple $(A,B)$ is regular.\n\t\t\\item The couple $(A,B)$ is boundedly regular.\n\t\t\\item The couple $(A,B)$\n\t\tis linearly regular for points bounded away from $E$.\n\t\\end{enumerate}\nThen $(iii)\\Leftrightarrow(i)\\Rightarrow(ii)$. Moreover, if $E$ is bounded, then $(ii)\\Rightarrow(i)$.\n\\end{proposition}\n\n\\begin{proof}\nThe implications $(iii)\\Rightarrow(i)\\Rightarrow(ii)$ are trivial. Let us prove that $(i)\\Rightarrow(iii)$. Suppose on the contrary that there exists $\\epsilon>0$ and a sequence $\\{x_n\\}\\subset X$ such that $\\mathrm{dist}(x_n,E)>\\epsilon$ ($n\\in\\N$) and \n$$\\textstyle \\frac{\\max\\{\\mathrm{dist}(x_n,A),\\mathrm{dist}(x_n,B-v)\\}}{\\mathrm{dist}(x_n,E)}\\to 0.$$\nFor each $n\\in\\N$, let $e_n\\in E$, $a_n\\in A$, and $b_n\\in B$ be such that $\\|e_n-x_n\\|=\\mathrm{dist}(x_n,E)$, $\\|a_n-x_n\\|=\\mathrm{dist}(x_n,A)$, and $\\|b_n-v-x_n\\|=\\mathrm{dist}(x_n,B-v)$. Put $\\lambda_n=\\frac{\\epsilon}{\\|e_n-x_n\\|}\\in(0,1)$ and define $z_n=\\lambda_n x_n+(1-\\lambda_n)e_n$, \n$a'_n=\\lambda_n a_n+(1-\\lambda_n)e_n\\in A$, and $b'_n=\\lambda_n b_n+(1-\\lambda_n)(e_n+v)\\in B$.\nBy our construction, it is clear that\n$$\\textstyle \\frac{\\mathrm{dist}(z_n,A)}{\\epsilon}\\leq\\frac{\\|z_n-a'_n\\|}{\\epsilon}=\\frac{\\|x_n-a_n\\|}{\\|e_n-x_n\\|}\\ \\ \\ \\text{and}\\ \\ \\ \\frac{\\mathrm{dist}(z_n,B-v)}{\\epsilon}\\leq\\frac{\\|b'_n-v-z_n\\|}{\\epsilon}=\\frac{\\|b_n-v-x_n\\|}{\\|e_n-x_n\\|}.$$\nHence, $\\mathrm{dist}(z_n,E)=\\epsilon$ and $\\max\\{\\mathrm{dist}(z_n,A),\\mathrm{dist}(z_n,B-v)\\}\\to 0$. This contradicts (i) and the proof is concluded.\n\nNow, suppose that $E$ is bounded, and let us prove that $(ii)\\Rightarrow(i)$. Suppose on the contrary that there exists $\\epsilon>0$ and a sequence $\\{x_n\\}\\subset X$ such that $\\mathrm{dist}(x_n,E)>\\epsilon$ ($n\\in\\N$) and \n$$\\textstyle \\max\\{\\mathrm{dist}(x_n,A),\\mathrm{dist}(x_n,B-v)\\}\\to 0.$$\nFor each $n\\in\\N$, let $e_n\\in E$, $a_n\\in A$, and $b_n\\in B$ be such that $\\|e_n-x_n\\|=\\mathrm{dist}(x_n,E)$, $\\|a_n-x_n\\|=\\mathrm{dist}(x_n,A)$, and $\\|b_n-v-x_n\\|=\\mathrm{dist}(x_n,B-v)$. Put $\\lambda_n=\\frac{\\epsilon}{\\|e_n-x_n\\|}\\in(0,1)$ and define $z_n=\\lambda_n x_n+(1-\\lambda_n)e_n$, \n$a'_n=\\lambda_n a_n+(1-\\lambda_n)e_n\\in A$, and $b'_n=\\lambda_n b_n+(1-\\lambda_n)(e_n+v)\\in B$.\nBy our construction, it is clear that\n$$\\textstyle {\\mathrm{dist}(z_n,A)}\\leq{\\|z_n-a'_n\\|}\\leq{\\|x_n-a_n\\|}$$\nand\n$$ {\\mathrm{dist}(z_n,B-v)}\\leq{\\|b'_n-v-z_n\\|}\\leq{\\|b_n-v-x_n\\|}.$$\nHence, $\\mathrm{dist}(z_n,E)=\\epsilon$ and $\\max\\{\\mathrm{dist}(z_n,A),\\mathrm{dist}(z_n,B-v)\\}\\to 0$. Moreover, since $E$ is bounded $\\{z_n\\}$ is a bounded sequence. This contradicts (ii) and the proof is concluded.\n\\end{proof}\n\nThe following theorem follows by \\cite[Theorem~3.7]{BauschkeBorwein93}. \n\n\\begin{theorem}\nLet $X$ be a Hilbert space and $A,B$ closed convex nonempty subsets of $X$. Suppose that the couple $(A,B)$ is regular. Then the alternating projections method converges.\n\\end{theorem}\n\nWe present an alternative proof of this theorem containing a simplified version of the argument that we will use in our main result Theorem~\\ref{teo: mainHilbert}. For the sake of simplicity we present a proof in the case $A\\cap B$ is nonempty. The proof in the general case is similar.\n\n\\begin{proof} \nBy \\cite[Theorem~3.3, (iv)]{BauschkeBorwein93}, it is sufficient to prove that $\\mathrm{dist}(c_n,A\\cap B)\\to 0$.\n\tLet us recall that, by the definition of the sequences $\\{c_n\\}$ and $\\{d_n\\}$, we have \n\t\\begin{enumerate}\n\t\t\\item[($\\alpha$)] $\\mathrm{dist}(c_n,A\\cap B)\\leq\\mathrm{dist}(d_n,A\\cap B)$ and $\\mathrm{dist}(d_{n+1},A\\cap B)\\leq\\mathrm{dist}(c_n,A\\cap B).$\n\t\\end{enumerate}\n\tLet $\\epsilon>0$, by the equivalence $(i)\\Leftrightarrow(iii)$ in Proposition~\\ref{prop: regular-largedistances}, there exists $K>0$ such that $$\\mathrm{dist}(x,A\\cap B)\\leq K\\max\\{\\mathrm{dist}(x,A),\\mathrm{dist}(x,B)\\},$$\n\twhenever $\\mathrm{dist}(x,A\\cap B)\\geq \\epsilon$. Observe that {$K\\geq 1$} and define $\\eta=\\sqrt{1-\\frac1{K^2}}$.\n\t{Then a computation, based on some trigonometric considerations in the plane defined by $c_n,d_n$ and $d_{n+1}$,} shows that the following condition holds for each $n\\in\\N$: \n\t\\begin{enumerate}\n\t\t\\item[($\\beta$)] if $\\mathrm{dist}(c_n,A\\cap B)\\geq\\epsilon$\t then $\\textstyle \\mathrm{dist}(c_n,A\\cap B)\\leq\\eta\\,\\mathrm{dist}(d_n,A\\cap B);$\n\t\tif $\\mathrm{dist}(d_{n+1},A\\cap B)\\geq\\epsilon$ then $\\textstyle \\mathrm{dist}(d_{n+1},A\\cap B)\\leq\\eta\\,\\mathrm{dist}(c_n,A\\cap B).$\n\t\\end{enumerate}\n\t {By} taking into account ($\\alpha$), ($\\beta$), and the fact that $\\eta<1$, we have that eventually $\\mathrm{dist}(c_n,A\\cap B)\\leq\\epsilon$ and $\\mathrm{dist}(d_n,A\\cap B)\\leq\\epsilon$. The proof is concluded.\t \n\\end{proof}\n\n\\section{{Regularity and perturbed alternating projections}}\\label{Section Main Result}\n{\n\tThis section is devoted to prove our main result. Indeed, here we show that if a couple $(A,B)$ of convex closed sets is regular then not only the alternating projections method converges but also the couple $(A,B)$ satisfies certain ``stability'' properties with respect to perturbed projections sequences. In the present section, if not differently stated, $X$ denotes a Hilbert space. If $u,v\\in X\\setminus\\{0\\}$, we denote as usual\n\t$$\\textstyle\\cos(u,v)=\\frac{\\langle u,v\\rangle}{\\|u\\|\\|v\\|},$$\n\twhere $\\langle \\cdot,\\cdot\\rangle$ denotes the inner product in $X$. \n\t\nLet us start by making precise the word ``stability'' by introducing\n} \nthe following two notions of stability for a couple $(A,B)$ of convex closed subsets of $X$.\n\n\n\n\\begin{definition}\\label{def: stability}\n\tLet $A$ and $B$ be closed convex subsets of $X$ such that $E, F$ are nonempty. We say that the couple $(A,B)$ is {\\em stable} [{\\em $d$-stable}, respectively] if for each choice of sequences $\\{A_n\\},\\{B_n\\}\\subset\\C(X)$ converging {with respect to} the Attouch-Wets convergence to $A$ and $B$, respectively, and for each choice of the starting point $a_0$, the corresponding perturbed alternating projections sequences $\\{a_n\\}$ and $\\{b_n\\}$ converge in norm [satisfy \n\t$\\mathrm{dist}(a_n, E)\\to0$ and $\\mathrm{dist}(b_n, F)\\to0$, respectively].\n\\end{definition} \n\n\n\\begin{remark}\n\tWe remark that the couple $(A,B)$ is {\\em stable} if and only\t\n\tif for each choice of sequences $\\{A_n\\},\\{B_n\\}\\subset\\C(X)$ converging {with respect to} the Attouch-Wets convergence to $A$ and $B$, respectively, and for each choice of the starting point $a_0$, there exists $e\\in E$ such that the perturbed alternating projections sequences $\\{a_n\\}$ and $\\{b_n\\}$ satisfy\n $a_n\\to e$ and $b_n\\to e+v$ in norm.\n\\end{remark}\n\n\\begin{proof}\n\t\tLet us start by proving that if $a_n\\to e$ then $e\\in E$. It is not difficult to prove that, since $$a_{n+1}= P_{A_n}P_{B_n}a_n=P_{A}P_{B}a_n+(P_{A_n}P_{B_n}-P_{A}P_{B})a_n$$ and since $A_n\\to A,B_n\\to B$ for the Attouch-Wets convergence, we have $e=P_AP_B e$. By Fact~\\ref{fact: BB93}, (ii), we have that $e\\in E$. Similarly, it is easy to see that \n\t$$b_{n+1}=P_{B_n}a_n=P_{B}a_n+(P_{B_n}-P_{B})a_n\\to P_B e=e+v,$$\n\tand the proof is concluded.\n\\end{proof}\n\n\n\nIt is clear that if the couple $(A,B)$ is stable,\nthen it is $d$-stable. Moreover, if $E,F$ are singletons then also the converse implication holds true.\n{The following basic assumptions will be considered in the sequel of the paper.}\n\n\n\\begin{BA}\\label{ba}\n\tLet $A,B$ be closed convex non\\-empty subsets of $X$. Suppose that:\n\t\\begin{enumerate}\n\t\t\\item $E,F$ are nonempty and bounded;\n\t\t\\item $\\{A_n\\}$ and $\\{B_n\\}$ are sequences of closed convex sets such that $A_n\\rightarrow\n\t\tA$ and $B_n\\rightarrow B$ for the Attouch-Wets convergence.\n\t\\end{enumerate} \n\\end{BA}\n\n\nNow, let us prove a chain of lemmas and propositions that we shall use in the proof of our main result, Theorem~\\ref{teo: mainHilbert} below. \n\n\n\\begin{lemma}\\label{lemma:2dimensional}\n\tLet $G$ be a closed convex subset of $X$. Suppose that there exist $\\epsilon,K>0$ such that $\\epsilon B_X\\subset G\\subset K B_X$. Then, if $u,w\\in\\partial G$ and $\\cos(u,w)=\\theta>0$, we have \n\t$$\\textstyle \\|u-w\\|^2\\leq K^2(\\frac{K^2}{\\epsilon^2}+1)\\frac{1-\\theta^2}{\\theta^2}.$$\n\\end{lemma}\n\n\\begin{proof}\n\tWithout any loss of generality we can suppose that $X=\\R^2$ and $u=(\\|u\\|,0)$. Let us denote $w=(x,y)$, with $x,y\\in\\R$, and suppose that $u,w\\in\\partial G$. \n\t\n\tWe claim that \n\t$\\textstyle |y|\\geq|\\frac{\\epsilon}{\\|u\\|}(x-\\|u\\|)|.$\n\tIndeed, since $\\epsilon B_X\\subset G$, an easy convexity argument shows that\n\tif $u\\in \\partial G$ and $w$ is such that $|y|<|\\frac{\\epsilon}{\\|u\\|}(x-\\|u\\|)|$ then we would have $w\\notin \\partial G $. \n\t\n\t \n{Now,} suppose that $\\cos(u,w)=\\frac{x}{\\|w\\|}=\\theta>0$. We have $y^2=\\frac{1-\\theta^2}{\\theta^2}x^2$ and hence, by our claim,\n$$\\textstyle (x-\\|u\\|)^2\\leq\\frac{1-\\theta^2}{\\theta^2}\\frac{\\|u\\|^2}{\\epsilon^2} x^2.$$\nHence,\n$$\\textstyle \\|u-w\\|^2=(x-\\|u\\|)^2+y^2\\leq x^2(\\frac{\\|u\\|^2}{\\epsilon^2}+1)\\frac{1-\\theta^2}{\\theta^2}\\leq K^2(\\frac{K^2}{\\epsilon^2}+1)\\frac{1-\\theta^2}{\\theta^2}.$$\n\\end{proof}\n\n\n\n\\begin{proposition}\\label{prop:eventuallycos<1}\n Let Basic assumptions~\\ref{ba} be satisfied and, for each $n\\in\\N$, let $a_n\\in A_n$ and $b_n\\in B_n$. Suppose that the couple $(A,B)$ is regular. \nLet $\\epsilon>0$, then there exist $\\eta\\in(0,1)$ and $n_1\\in\\N$ such that for each $n\\geq n_1$ we have:\n\\begin{enumerate}\n\t\\item if $\\mathrm{dist}(a_n,E)\\geq2\\epsilon$\tand $\\mathrm{dist}(b_n,F)\\geq2\\epsilon$ then $\\cos\\bigl(a_n-e,b_n-(e+v)\\bigr)\\leq\\eta,$ whenever $e\\in E+\\epsilon B_X$.\n\\item if $\\mathrm{dist}(a_n,E)\\geq2\\epsilon$\tand $\\mathrm{dist}(b_{n+1},F)\\geq2\\epsilon$ then $\\cos\\bigl(b_{n+1}-f,a_n+v-f\\bigr)\\leq\\eta,$ whenever $f\\in F+\\epsilon B_X$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} Let us prove that there exist $\\eta\\in(0,1)$ such that eventually (i) holds, the proof that there exist $\\eta\\in(0,1)$ such that eventually (ii) holds is similar. \n\tSuppose that this is not the case, then there exist sequences $\\{e_k\\}\\subset E+\\epsilon B_X$, $ \\{\\theta_k\\}\\subset (0,1)$ and an increasing sequence of the integers $\\{n_k\\}$ such that $\\mathrm{dist}(a_{n_k},E)\\geq2\\epsilon$, $\\mathrm{dist}(b_{n_k},F)\\geq2\\epsilon$, and\n\t$$\\cos\\bigl(a_{n_k}-e_k,b_{n_k}-(e_k+v)\\bigr)=\\theta_k\\to 1.$$\n\tLet $G=E+2\\epsilon B_X$ and observe that $G$ is a bounded body in $X$. Since $e_k\\in\\inte G$ and $a_{n_k}\\not\\in\\inte G$, there exists a unique point $a'_k\\in[e_k,a_{n_k}]\\cap\\partial G$. Similarly, there exists a unique point $b'_k\\in[e_k,b_{n_k}-v]\\cap\\partial G$.\n\tMoreover, it is clear that $$\\cos\\bigl(a'_{k}-e_k,b'_{k}-e_k\\bigr)=\\theta_k.$$\n\tLemma~\\ref{lemma:2dimensional} implies that $\\|a'_{k}-b'_{k}\\|\\to0$. \n\t Since $G$ is bounded and $A_{n_k}\\to A,\\,B_{n_k}\\to B$ for the Attouch-Wets convergence, there exist sequences $\\{a''_k\\}\\subset A$ and $\\{b''_k\\}\\subset B-v$ such that $\\|a''_k-a'_k\\|\\to 0$ and $\\|b''_k-b'_k\\|\\to 0$. Hence, $\\|a''_{k}-b''_{k}\\|\\to0$ and eventually $\\mathrm{dist}(a''_k, E)\\geq \\epsilon$, a contradiction since the couple $(A,B)$ is regular.\n\\end{proof}\n\n\\begin{lemma}\\label{prop: eventuallycos0}\n\t Let Basic assumptions~\\ref{ba} be satisfied, suppose that the couple $(A,B)$ is regular, and let $\\delta,\\epsilon>0$. For each $n\\in\\N$, let $a_n,x_n\\in A_n$ and $b_n,y_n\\in B_n$ be such that $\\mathrm{dist}(x_n, E)\\to 0$ and $\\mathrm{dist}(y_n, F)\\to 0$. \n\tThen there exists $n_2\\in\\N$ such that for each $n\\geq n_2$ we have:\n\t\\begin{enumerate}\n\t\t\\item if $\\mathrm{dist}(a_n,E)\\geq2\\epsilon$, $\\mathrm{dist}(b_n,F)\\geq2\\epsilon$, and $a_n=P_{A_n}b_n$ then \n$$\\cos\\bigl(x_n-a_n,b_n-(a_n+v)\\bigr)\\leq\\delta;$$\t\n\t\t\\item if $\\mathrm{dist}(a_n,E)\\geq2\\epsilon$, $\\mathrm{dist}(b_{n+1},F)\\geq2\\epsilon$, and $b_{n+1}=P_{B_{n+1}}a_n$ then \n$$\\cos\\bigl(y_{n+1}-b_{n+1},a_n+v-b_{n+1}\\bigr)\\leq\\delta.$$\t\n\n\\end{enumerate}\n\t\\end{lemma}\n\n\\begin{proof} Let us prove that eventually (i) holds, the proof that eventually (ii) holds is similar. {Since $\\mathrm{dist}(a_n,E)\\geq 2 \\varepsilon$ and $\\mathrm{dist}(x_n,E)\\rightarrow 0$ we have that eventually $x_n-a_n\\neq0$}. By Proposition~\\ref{prop:eventuallycos<1}, there exists $\\eta\\in(0,1)$ and $n_1\\in\\N$ such that\n\t$$\\cos\\bigl(a_n-e,b_n-(e+v)\\bigr)\\leq\\eta,$$\n\twhenever $n\\geq n_1$ and $e\\in E+\\epsilon B_X$. Since $\\mathrm{dist}(a_n,E)\\geq2\\epsilon$\tand $\\mathrm{dist}(b_n,F)\\geq2\\epsilon$, it is not difficult to see that there exists a constant $\\eta'>0$ such that $\\|b_n-(a_n+v)\\|\\geq\\eta'$, whenever $n\\geq n_1$. In particular, eventually $\\cos\\bigl(x_n-a_n,b_n-(a_n+v)\\bigr)$ is well-defined. \n\t If $v=0$, the thesis is trivial since \n\t$$\\langle x_n-a_n,b_n-a_n\\rangle\\leq0,$$\n\twhenever $n\\in\\N$. \n\t\n\tSuppose that $v\\neq 0$. \n\t\tWe claim that, if $v$ denotes the displacement vector for the couple $(A,B)$, eventually we have $$\\textstyle \\langle v,a_n-x_n\\rangle\\leq \\delta\\eta' \\|a_n-x_n\\|.$$\n\tTo prove our claim observe that, since $\\mathrm{dist}(x_n, E)\\to 0$, we can suppose without any loss of generality that $\\mathrm{dist}(x_n, E)\\leq\\epsilon$ ($n\\in\\N$). Moreover, we can consider a sequence $\\{x'_n\\}\\subset E$ such that $\\|x'_n-x_n\\|\\to0$. Let $G=E+2\\epsilon B_X$ and observe that $G$ is a bounded body in $X$. Since $x_n\\in\\inte G$ and $a_{n}\\not\\in\\inte G$, there exists a unique point $a'_n\\in[x_n,a_{n}]\\cap\\partial G$. \n\tSince $G$ is bounded and $A_{n}\\to A$ for the Attouch-Wets convergence, there exists a sequence $\\{a''_n\\}\\subset A$ such that $\\|a''_n-a'_n\\|\\to 0$. Since $\\{x'_n\\}\\subset E$, it is clear that \n\t$\\langle v, a''_n-x'_n\\rangle\\leq 0$ and hence eventually\n\t$$\\langle v, a''_n-x'_n\\rangle-\\delta\\eta'\\|a''_n-x'_n\\|\\leq -\\delta\\eta'\\epsilon.$$\t\n\tSince $\\|x'_n-x_n\\|\\to0$ and $\\|a''_n-a'_n\\|\\to0$, eventually we have\n\t$$\\langle v, a'_n-x_n\\rangle-\\delta\\eta'\\|a'_n-x_n\\|\\leq0.$$\n\tBy homogeneity and by our construction the claim is proved.\n\t\n\tNow, by our claim, since $a_n=P_{A_n}b_n$ and $x_n\\in A_n$ ($n\\in\\N$), we have \n\t $$\\textstyle \\langle x_n-a_n,b_n-(a_n+v)\\rangle=\\langle x_n-a_n,b_n-a_n\\rangle+\\langle a_n-x_n,v\\rangle\\leq \\delta\\eta' \\|a_n-x_n\\|.$$\n\tEventually, since $\\|b_n-(a_n+v)\\|\\geq\\eta'$, we have \n\t \t$$\\cos\\bigl(x_n-a_n,b_n-(a_n+v)\\bigr)\\leq \\frac{\\delta\\eta'}{\\|b_n-(a_n+v)\\|}\\leq \\delta.$$\n\t\n\\end{proof}\n{\n\\noindent Now, we need a simple geometrical result whose proof is a simple application of the law of cosines combined with the triangle inequality. The details of the proof are left to the reader.} \n\n\\begin{fact}\\label{fact: quasiortho}\nLet $\\eta,\\eta'\\in(0,1)$ be such that $\\eta<\\eta'$. If $\\delta\\in(0,1)$ satisfies $\\frac{\\delta+\\eta}{1-\\delta}\\leq \\eta'$ and if $x,y\\in X$ are linearly independent vectors such that $\\cos(x,y)\\leq \\eta$ and $\\cos(y-x,-x)\\leq \\delta$ then $\\|x\\|\\leq \\eta'\\|y\\|$. \n\\end{fact}\n\nLet us recall that, given a normed space $Z$, the {\\em modulus of convexity of $Z$} is the function $\\delta_Z:[0,2]\\to [0,1]$ defined by \n$$\n\\delta_Z(\\eta)=\\inf \\left\\lbrace 1-\\left\\| \\dfrac{x+y}{2} \\right\\|:x,y \\in B_X, \\|x-y\\|\\geq \\eta\\right\\rbrace. \n$$\nMoreover, we say that $Z$ is {\\em uniformly rotund} if $\\delta_Z(\\eta)>0$, whenever $\\eta \\in (0,2] $.\n\n\\begin{lemma}\\label{lemma: unifrotund} \n\tLet $Z$ be a uniformly rotund {normed} space. For each $\\rho\\geq 0$ and $M>0$ there exists $\\epsilon'>0$ such that if $C$ is a convex set such that $\\rho-\\epsilon'\\leq\\|c\\|\\leq\\rho+\\epsilon'$, whenever $c\\in C$, then $\\mathrm{diam}(C)\\leq M$.\n\\end{lemma}\n\n\\begin{proof}\nIn the case $\\rho= 0$ the proof is trivial, so we can suppose $\\rho>0$. We claim that if we take $\\epsilon'>0$ such that $\\varepsilon'\\left(2-\\delta_Z(M) \\right) < \\rho \\delta_Z(M)$ the thesis follows. To see this, suppose that $C$ is a convex set such that $\\rho-\\epsilon'\\leq\\|c\\|\\leq\\rho+\\epsilon'$, whenever $c\\in C$, and suppose on the contrary that there exist $c_1,c_2 \\in C$ such that $\\|c_1-c_2\\|> M$. By the definition of $\\delta_Z$ and since $\\frac{c_1+c_2}{2} \\in C$, we have \n\t$$\n\\rho - \\varepsilon'\\leq\\left\\| \\dfrac{c_1+c_2}{2}\\right\\| \\leq \\left( \\rho + \\varepsilon'\\right) \\left( 1- \\delta_Z(M)\\right). \n$$\n Therefore, we have $\\varepsilon'\\left(2-\\delta_Z(M) \\right) \\geq \\rho \\delta_Z(M)$, a contradiction. \n\\end{proof}\n\n{Since it is well known that a Hilbert space is a uniformly rotund space, the previous lemma allows us to prove the following proposition. }\n\n\\begin{proposition}\\label{prop: normprojectionsmall} Let Basic assumptions~\\ref{ba} be satisfied. For each $M>0$ there exist $\\theta\\in(0,M)$ and $n_0\\in\\N$ such that if $n\\geq n_0$ we have:\n\t\\begin{enumerate}\n\t\t\\item if $b_n\\in B_n$, $a_n=P_{A_n}b_n$, and $\\mathrm{dist}(b_n, F)\\leq \\theta$ then $$\\mathrm{dist}(a_n, E)\\leq 2M;$$\n\t\t\\item if $a_n\\in A_n$, $b_{n+1}=P_{B_{n+1}}a_n$, and $\\mathrm{dist}(a_n, E)\\leq \\theta$ then $$\\mathrm{dist}(b_{n+1}, F)\\leq 2M.$$\n\\end{enumerate} \n\\end{proposition}\t\n\n\n\\begin{proof} Let $M>0$ and $\\rho=\\|v\\|$, {where $v$ is the displacement vector}. \n Let $\\epsilon'\\in(0,3M)$ be given by Lemma~\\ref{lemma: unifrotund}. Put $\\theta=\\epsilon'\/3$, since Basic assumption~\\ref{ba} are satisfied, there exists $n_0\\in \\N$ such that if $n\\geq n_0$ we have:\n\t\\begin{enumerate}\n\t\t\\item[(a)] if $w\\in A_n$ then $\\mathrm{dist}(w,F)\\geq \\rho-3\\theta$;\n\t\t\\item[(b)] if $e\\in E$, there exists $x\\in A_n$ such that $\\|e-x\\|\\leq\\theta$.\n\t\\end{enumerate} \nNow, let $n\\geq n_0$, $b_n\\in B_n$, $a_n=P_{A_n}b_n$, and $\\mathrm{dist}(b_n, F)\\leq\\theta$. Let $f_n\\in F$ be such that $\\|f_n-b_n\\|\\leq\\theta$ and put $e_n=f_n-v\\in E$. By (b), there exists $x_n\\in A_n$ such that $\\|x_n-e_n\\|\\leq \\theta$. Hence, since $a_n=P_{A_n}b_n$ \nand $\\|e_n-f_n\\|=\\rho$, we have\n\\begin{eqnarray*}\n \\|a_n-f_n\\|&\\leq& \\|a_n-b_n\\|+\\|f_n-b_n\\|\\\\\n&\\leq& \\|x_n-b_n\\|+\\|f_n-b_n\\|\\\\\n&\\leq& \\|x_n-e_n\\|+ \\rho +2\\|f_n-b_n\\|\\leq\\rho+3\\theta.\n\\end{eqnarray*} \n\tLet us consider the convex set $C=[x_n-f_n,a_n-f_n]$. Observe that, since $$\\|x_n-f_n\\|\\leq\\|e_n-x_n\\|+\\|e_n-f_n\\|\\leq\\rho+\\theta,$$ we have that \n\t$\\|c\\|\\leq\\rho+3\\theta$, whenever $c\\in C$.\n\tMoreover, since $C\\subset A_n$ and $f_n\\in F$, by (a) we have $\\|c\\|\\geq\\rho-3\\theta$, whenever $c\\in C$. Hence, we\n\t can apply Lemma~\\ref{lemma: unifrotund} to the set $C$ and we have $\\|a_n-x_n\\|=\\mathrm{diam}(C)\\leq M$. Then\n\t$$\\mathrm{dist}(a_n, E)\\leq \\|a_n-e_n\\|\\leq \\|a_n-x_n\\|+\\|e_n-x_n\\|\\leq M+\\theta\\leq 2M.$$\n\t\tThe proof that eventually (ii) holds is similar. \n\\end{proof}\n\n{\nWe are now ready to state and prove the main result of this paper.}\n\n\n\\begin{theorem}\\label{teo: mainHilbert}\n\t Let $A,B$ be closed convex nonempty subsets of $X$ such that $E$ and $F$ are bounded. Suppose that the couple $(A,B)$ is regular, then the couple $(A,B)$ is $d$-stable.\n\t\n\\end{theorem}\n\n\\begin{proof}\nLet $a_0\\in X$ and let $\\{a_n\\}$ and $\\{b_n\\}$ {be} the corresponding perturbed alternating projections sequences, { i.e,\n$$a_n=P_{A_n}(b_n) \\quad \\text{and} \\quad b_n=P_{B_n}(a_{n-1}).$$ \nFirst of all, we remark that it is enough to prove that $\\mathrm{dist}(a_n, E)\\to 0$ since the proof that $\\mathrm{dist}(b_n, F)\\to 0$ follows by the symmetry of the problem. \nTherefore our aim is to prove that for each $M>0$, eventually we have $$\\mathrm{dist}(a_n, E)\\leq M.$$}\n \n\n \nBy applying Proposition~\\ref{prop: normprojectionsmall} twice, there exists $0<\\epsilonn_4$.\n\\end{proof}\n\n\n{If the intersection of $A$ and $B$ is nonempty, we obtain, as an immediate consequence of Theorem \\ref{teo: mainHilbert}, the following result.}\n\\begin{corollary} \n\t Let $A,B$ be closed convex nonempty subsets of $X$ such that $A\\cap B$ is bounded and nonempty. If the couple $(A,B)$ is regular then the perturbed alternating projections sequences $\\{a_n\\}$ and $\\{b_n\\}$ satisfy $\\mathrm{dist}(a_n,A\\cap B)\\to 0$ and $\\mathrm{dist}(b_n,A\\cap B)\\to 0$\n\t\\end{corollary}\n\n{We conclude this section by putting in evidence some relationships between the results of \\cite{DebeMigl} and Theorem \\ref{teo: mainHilbert}.}\n\t{First of all, we briefly recall some notions.}\t\n{\t\\begin{definition}[{see, e.g., \\cite[Definition~7.10]{FHHMZ}}]\\label{def:strexp} Let $A$ be a nonempty subset of a normed space $Z$. A point $a\\in A$\n\t\tis called a strongly exposed point of $A$ if there exists a\n\t\tsupport functional $f\\in Z^*\\setminus\\{0\\}$ for $A$ at $a$ $\\bigl($i.e.,\n\t\t$f (a) = \\sup f(A)$$\\bigr)$, such that $x_n\\to a$ for all sequences\n\t\t$\\{x_n\\}$ in $A$ such that $\\lim_n f(x_n) = \\sup f(A)$. In this\n\t\tcase, we say that $f$ strongly exposes $A$ at $a$.\n\t\\end{definition}}\n\n\n\n\t\\begin{definition}[{see, e.g., \\cite[Definition~1.3]{KVZ}}]\n\t\tLet $A$ be a body in a normed space $Z$. We say that $x\\in\\partial A$ is an\n\t\t{\\em LUR (locally uniformly rotund) point} of $A$ if for each\n\t\t$\\epsilon>0$ there exists $\\delta>0$ such that if $y\\in A$\n\t\tand $\\dist(\\partial A,(x+y)\/2)<\\delta$ then $\\|x-y\\|<\\epsilon$. \n\t\\end{definition}\n\tWe say that $A$ is an {\\em LUR body} if each point in\n\t$\\partial A$ is an LUR point of $A$. The following lemma shows that each LUR point is a strongly exposed point. \t\n\t\\begin{lemma}[{\\cite[Lemma~4.3]{DebeMiglMol}} ]\\label{slicelimitatoselur} Let $A$ be a body in a normed space $Z$\n\t\tand suppose that $a\\in\\partial A$ is an LUR point of $A$. Then, if\n\t\t$f\\in S_{Z^*}$ is a support functional for $A$ in $a$, $f$\n\t\tstrongly exposes $A$ at $a$.\n\t\\end{lemma} \n\nFirst, we show that a more general variant of the assumptions of one of the main results in \\cite{DebeMigl}, namely \\cite[Theorem~3.3]{DebeMigl}, imply that the couple $(A,B)$ is regular. It is interesting to remark that here we consider also the case in which $A$ and $B$ do not intersect.\n\t\t \t\n\t\\begin{proposition} \\label{prop:stronglyexpregular}\n\t\tLet $A,B$ be nonempty closed convex subsets of $X$. Let us suppose that there exist $e \\in A \\cap (B-v)$ and a linear continuous functional $x^*\\in S_{X^*}$ such that\n\t\t$$ \\inf x^*(B-v)=x^*(e)=\\sup x^*(A)$$\n\t\tand such that $x^*$ strongly exposes $A$ at $e$. Then the couple $(A,B)$ is regular. \n\t\\end{proposition}\n\\begin{proof} {There is no loss of generality in assuming $e=0$.}\n\tIt is a simple matter to see that $E=\\{0\\}$. Now, suppose on the contrary that that $(A,B)$ is not regular. Therefore there exist sequences $\\{x_n\\}\\subset X$, $\\{a_n\\}\\subset A$, $\\{b_n\\}\\subset B$, and a real number $\\bar{\\varepsilon}>0$ such that \n\t\\begin{equation} \\label{dist e}\n\t\t\\mathrm{dist}(x_n,E)=\\|x_n\\|>\\bar{\\varepsilon}, \n\t\\end{equation}\n\t\tand such that\n\\begin{equation}\\label{eq: notregular}\n\\mathrm{dist}(x_n,A)=\\|x_n -a_n\\|\\rightarrow 0, \\quad \\mathrm{dist}(x_n,B-v)=\\|x_n -b_n+v\\|\\rightarrow 0.\n\\end{equation}\t\n\t\n\t\t By (\\ref{eq: notregular}) and since $ {\\inf x^*(B-v)}=0=\\sup x^*(A)$, it holds $\\lim_n x^*(x_n)=0$ and hence $\\lim_n x^*(a_n)=0$. Since $x^*$ strongly exposes $A$ at $e$, the last equality implies that $\\|a_n\\|\\rightarrow 0$. We conclude that $\\|x_n\\|\\rightarrow 0$, contrary to (\\ref{dist e}).\n\\end{proof}\n\nBy combining the previous proposition and Theorem~\\ref{teo: mainHilbert}, we obtain the following corollary generalizing \\cite[Theorem~3.3]{DebeMigl}.\n\n\\begin{corollary}\\label{Corollary:stronglyexp}\n\tLet $A,B$ be\n\tnonempty closed convex subsets of $X$. \n\tLet us suppose that there exist $e \\in A \\cap (B-v)$ and a linear continuous functional $x^*\\in S_{X^*}$ such that\n\t$$ \\inf x^*(B-v)=x^*(e)=\\sup x^*(A)$$\n\tand such that $x^*$ strongly exposes $A$ at $e$.\n\tThen, \n\tthe couple $(A,B)$ is stable.\n\\end{corollary} \n\n\n\n Moreover, in \\cite{DebeMigl}, the authors proved the following sufficient condition for the stability of a couple $(A,B)$.\n\n\\begin{theorem}[{\\cite[Theorem~4.2]{DebeMigl}}]\\label{theorem:corpilur} Let $X$ be a Hilbert space and $A,B$\n\tnonempty closed convex subsets of $X$. \n\tSuppose that $\\inte(A\\cap B)\\neq\\emptyset$, then the couple $(A,B)$ is stable.\n\\end{theorem}\n\nBy combining Corollary~\\ref{Corollary:stronglyexp} and Theorem~\\ref{theorem:corpilur}, we obtain the following sufficient condition for the stability of the couple $(A,B)$ generalizing \\cite[Corollary~4.3, (ii)]{DebeMigl}.\n\t\n\\begin{corollary}\\label{corollary:corpilur} Let $X$ be a Hilbert space, suppose that \t$A,B$ are bodies in $X$ and that $A$ is LUR. Then the couple $(A,B)$ is stable.\n\\end{corollary}\n\n\\begin{proof}\n\tIf $\\inte(A\\cap B)\\neq\\emptyset$, {the thesis follows by applying Theorem~\\ref{theorem:corpilur}.} If $\\inte(A\\cap B)=\\emptyset$,\n\tsince $A$ and $B$ are bodies, we have $\\mathrm{int}(A)\\cap B=\\emptyset$. Since $A$ is LUR the intersection $A \\cap (B-v)$ reduces to a singleton $\\{e\\}$. By the Hahn-Banach theorem, there exists a linear functional $x^*\\in X^*$ such that $$ \\inf x^*(B-v)=x^*(e)=\\sup x^*(A).$$ Since $A$ is an LUR body, by Lemma \\ref{slicelimitatoselur}, we have that $x^*$ strongly exposes $A$ at $e$. We are now in position to apply Corollary \\ref{Corollary:stronglyexp} and conclude the proof.\n\t\\end{proof}\n\n\n\t\n\tFinally, we show that \\cite[Theorem~5.2]{DebeMigl}, follows by Theorem~\\ref{teo: mainHilbert}.\n\\begin{corollary}\\label{cor:sottospazisommachiusa}\n\tLet $U,V$ be closed subspaces of $X$ such that $U\\cap V=\\{0\\}$ and $U+V$ is closed. Then the couple $(U,V)$ is stable.\n\\end{corollary} \n\\begin{proof}\n\tSince $U+V$ is closed, by \\cite[Corollary~4.5]{BauschkeBorwein93}, the couple $(U,V)$ is regular. By Theorem~\\ref{teo: mainHilbert}, the couple $(U,V)$ is $d$-stable. Since $U\\cap V$ is a singleton, the couple $(U,V)$ is stable. \n\\end{proof}\n \n\n\n\n\\section{Final remarks, examples, and an open problem}\n\n\n\nKnown examples show that the hypothesis about regularity of the couple $(A,B)$, in Theorem~\\ref{teo: mainHilbert}, is necessary. To see this, it is indeed sufficient to consider any couple $(A,B)$ of sets such that $A\\cap B$ is a singleton and such that, for a suitable starting point, the method of alternating projections does not converge (see \\cite{Hundal} for such a couple of sets). \n\nA natural question is whether, in the same theorem, the hypothesis about regularity of the couple $(A,B)$ can be replaced by the weaker hypothesis that ``for any starting point the method of alternating projections converges''. The answer to previous question is negative; indeed, in \\cite[Theorem~5.7]{DebeMigl}, the authors provided an example of a couple $(A,B)$ of closed subspaces of a Hilbert space such that $A\\cap B=\\{0\\}$ and such that the couple $(A,B)$ is not stable (and hence not $d$-stable since $A\\cap B$ is a singleton).{It is interesting to observe} that, by the classical Von Neumann result \\cite{vonNeumann}, the method of alternating projections converges for this couple of sets.\n\nThe next example shows that, if we consider closed convex sets $A,B\\subset X$ such that $A\\cap B$ is nonempty and bounded, the regularity of the couple $(A,B)$ does not imply in general that $(A,B)$ is stable. In particular, we cannot replace $d$-stability with stability in the statement of Theorem~\\ref{teo: mainHilbert}.\n\n\\begin{example}[{\\cite[Example~4.4]{DebeMigl}}] \\label{ex: notconverge}\n\t\tLet $X=\\R^2$ and let us consider the following compact convex subsets of $X$:\n\t\t\\begin{eqnarray*}\n\t\t\tA&=&\\textstyle \\conv\\{(1,1),(-1,1),(1,0),(-1,0)\\};\\\\ \n\t\t\t\t\tB&=&\\textstyle\\conv\\{(1,-1),(-1,-1),(1,0),(-1,0)\\}.\n\t\t\\end{eqnarray*} \n\tThen the couple $(A,B)$ is regular (see \\cite[Theorem~3.9]{BauschkeBorwein93}) but not stable (see \\cite[Example~4.4]{DebeMigl}). \n\\end{example} \n\n{Now, the following example shows that, even in finite dimension, the hypothesis concerning the boundedness of the sets $E,F$ cannot be dropped in the statement of Theorem~\\ref{teo: mainHilbert}. } \n\n\\begin{example}\\label{ex: EnotBounded}\nLet $A,B$ be the subsets of $\\R^3$ defined by\n$$A=\\{(x,y,z)\\in\\R^3;\\, z=0, y\\geq 0\\},\\ \\ \\ B=\\{(x,y,z)\\in\\R^3;\\, z=0\\},$$\nthen the following conditions hold:\n\\begin{enumerate}\n\t\\item[(a)] $A\\cap B$ coincides with $A$ (and hence $(A,B)$ is regular); \n\\item[(b)] $A\\cap B$ is not bounded;\n\\item[(c)] the couple $(A,B)$ is not $d$-stable.\n\\end{enumerate}\n\\end{example}\n\n\nThe proof of (a) and (b) is trivial. To prove (c), we need the following lemma, whose elementary proof is left to the reader.\n\n\\begin{lemma}\\label{lemma: example fin-dim} Let $A,B$ be defined as in Example~\\ref{ex: EnotBounded}. For each $n\\in\\N$ and $x_0\\geq1$, let $P_{n,x_0}^1,P_{n,x_0}^2,P_{n,x_0}^3\\in\\R^3$ be defined by \n$$\\textstyle\nP_{n,x_0}^1=(x_0+n x_0,-1,0),\\ \\ \nP_{n,x_0}^2=(x_0+n x_0+\\frac{1}{n x_0},0,0),\\ \\ \nP_{n,x_0}^3=(0,\\frac1n,\\frac1n).\n$$\t\nLet $t_{n,x_0}$ be the line in $\\R^3$ containing the points $P_{n,x_0}^1$ and $P_{n,x_0}^3$, and let $r_{n,x_0}$ be the ray in $\\R^3$ with initial point $P_{n,x_0}^1$ and containing the points $P_{n,x_0}^2$. Let $A_{n,x_0},B_{n,x_0}$ be the closed convex subsets of $\\R^3$ defined by\n\t$$A_{n,x_0}=\\conv(t_{n,x_0}\\cup r_{n,x_0}),\\ \\ \\ B_{n,x_0}=\\{(x,y,z)\\in\\R^3;\\, z=0\\}.$$\nThen the following conditions hold.\n\\begin{enumerate}\n\t\\item \n\tfor each $N\\in\\N$,\n\t$\\textstyle \\lim_n h_N(A_{n,x_0},A)= 0$, uniformly with respect to $x_0\\geq1$;\n\t\\item For each $n\\in\\N$ and $x_0\\geq 1$, the alternating projections sequences, relative to the sets $A_{n,x_0},B_{n,x_0}$ and starting point $(x_0,0,0)$, converge to $P^1_{n,x_0}$. \n\\end{enumerate}\t\n\t\n\\end{lemma}\n\n\\begin{proof}[Sketch of the proof of Example~\\ref{ex: EnotBounded}, (c)] \nFix the starting point $a_0=(1,0,0)$ and let $A_{1,1},B_{1,1}$, be defined by Lemma~\\ref{lemma: example fin-dim}. Observe that, if we consider the points $a^1_k=(P_{A_{1,1}} P_{B_{1,1}})^k a_0$ ($k\\in\\N$), by Lemma~\\ref{lemma: example fin-dim}, (ii), there exists $N_1\\in\\N$ such that $\\mathrm{dist}(a^1_{N_1}, A\\cap B)\\geq\\frac12$.\nDefine $A_n=A_{1,1}$ and $B_n=B_{1,1}=B$, whenever $1\\leq n\\leq N_1$. Then define $A_{N_1+1}=A$ and $B_{N_1+1}=B$, and observe that $a^2_0:=P_A P_B a^1_{N_1}=(x_1,0,0)$ for some $x_1\\geq1$. Similarly, if we consider the points $a^2_{k}=(P_{A_{2,x_1}} P_{B_{2,x_1}})^k a^2_{0}$ ($k\\in\\N$), then there exists $N_2\\in\\N$ such that $\\mathrm{dist}(a^2_{N_2}, A\\cap B)\\geq\\frac12$. \nDefine $A_n=A_{2,x_1},B_n=B_{2,x_1}=B$, whenever $N_1+1< n\\leq N_2$. Then define $A_{N_2+1}=A,B_{N_2+1}=B$, and observe that $a^3_{0}:=P_A P_B a_{N_2}^{2}=(x_2,0,0)$ for some $x_2\\geq1$.\n\n Then, proceeding inductively, we can construct sequences $\\{A_n\\}$ and $\\{B_n\\}$ such that, by Lemma~\\ref{lemma: example fin-dim}, (i), $A_n\\to A$ and $B_n\\to B$ for the Attouch-Wets convergence.\n Moreover, by our construction, \n it is easy to see that the corresponding perturbed alternating projections sequences $\\{a_n\\}$ and $\\{b_n\\}$, with starting point $a_0$, are such that \n $$\\textstyle \\limsup_n \\mathrm{dist}(a_n,A\\cap B)\\geq \\frac12.$$\n This proves that the couple $(A,B)$ is not $d$-stable.\n\\end{proof}\n\n\n\n\nFinally, we conclude with an open problem asking whether the inverse of Theorem~\\ref{teo: mainHilbert} holds true.\n\n\\begin{problem} \nLet $A,B$ be closed convex nonempty subsets of $X$ such that $E$ and $F$ are nonempty and bounded. Suppose that the couple $(A,B)$ is $d$-stable. Does the couple $(A,B)$ is regular?\t\n\\end{problem}\n\n\n\n\\section*{Acknowledgements.}\nThe research of the authors is partially\nsupported by GNAMPA-INdAM, Progetto GNAMPA 2020. {The second author is also partially supported by the Ministerio de Ciencia, Innovaci\u00f3n y Universidades (MCIU), Agencia Estatal de Investigaci\u00f3n (AEI) (Spain) and\tFondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-\tB-I00 (MCIU\/AEI\/FEDER, UE)}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpska b/data_all_eng_slimpj/shuffled/split2/finalzzpska new file mode 100644 index 0000000000000000000000000000000000000000..52385cf9574156b01a33c063feb14741746f9bd0 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzpska @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\\label{sec:intro}\n\nIn recent years, modern TTS systems have largely moved to sequence-to-sequence (Seq2Seq) models, e.g. \\cite{WangY2017_Tacotron,ShenJ2018_Tacotron2,PingW2018_DeepVoice3}, where the alignment between the phonetic or orthographic input sequence and the acoustic output sequence is learned during training and inferred during synthesis. One advantage of this approach is that it leads to a more end-to-end system, in the best case avoiding the need for pre-aligned training data, separate phonetic transcription, or a separate duration model at synthesis. For singing synthesis, not requiring pre-aligned training data is particularly attractive, as many existing tools (e.g. forced alignment with a HMM model) do not yield sufficiently accurate results on expressive singing, often requiring manual correction.\n\nA common approach for Seq2Seq models in TTS is to use a content-based attention mechanism, e.g. \\cite{GehringJ2017_ConvAttention}, sometimes additionally using location-based information, e.g. \\cite{GravesA2013_GMMAttention}. As these mechanisms require access to acoustic information at inference, they are normally used in combination with an autoregressive decoder. Recently, some systems have been proposed that use a feed-forward decoder and an alternative attention mechanism that does not rely on access to acoustic information \\cite{PengK2019_ParaNet,RenY2019_FastSpeech}. These notably provide faster, parallelizable inference, and are reported to produce more robust alignments with fewer mispronounced, repeated or skipped phonemes.\n\nIn the case of singing synthesis, this feed-forward approach is interesting as it avoids the exposure bias problem \\cite{RanzatoMA2016_ExposureBias}, caused by the discrepancy between teacher forced training and fully autoregressive inference. This problem can be especially noticeable in long sustained vowels were prediction errors tend to accumulate over time. Additionally, in our experience reaching similar quality results compared to non-Seq2Seq systems can be quite challenging with standard content-based attention mechanisms.\n\nTo facilitate evaluation of different systems, we only model timbre in this work, and assume F0 to be given. Although we use ground truth F0 extracted from recordings, it is feasible to predict F0 from the input score with an external model, or possibly predict it jointly. Related to this, we use WORLD vocoder features \\cite{MoriseM2016_WORLD} as the output of our system, rather than the more commonly used mel-spectrogram features, as this allows exact control over the synthesized F0. For the best quality results, Seq2Seq systems are typically combined with a neural vocoder, e.g. \\cite{TamamoriA2017_WaveNetVocoder,PrengerR2019_WaveGlow,WangX2019_NSFVocoder}, which can work well from both vocoder or mel-spectrogram features. However, in order to get a better idea of the performance of our model on its own, we do not use this approach in the experiments presented here.\n\nThe principal contributions of this paper are:\n\\begin{inparaenum}\n \\item Propose a singing synthesizer based on the feed-forward Transformer with a practical Seq2Seq mechanism using an external duration model.\n \\item Evaluate the quality of this feed-forward model compared to a baseline autoregressive model.\n \\item Evaluate the importance of self-attention.\n \\item Evaluate the importance of the accuracy of the duration model used.\n\\end{inparaenum}\n\n\n\\section{Proposed system}\\label{sec:proposed_system}\n\nIn singing synthesis, the alignment between the input phonetic sequence and the output acoustic sequence is strongly constrained by the given musical score. This is a notable difference from TTS, which is generally only weakly constrained by the (average) speech rate. Exploiting this fact, we propose to first generate an approximate initial alignment using note timings and a phoneme duration model. Once the input sequence is roughly aligned to the target output timesteps, we assume that the network is able to gradually refine the alignment through a series of transformations, until reaching something close to the target. Note that this approach is quite different from the approach using content-based attention, as here the initial alignment doesn't use any content at all.\n\nAn important point here is that we assume that the accuracy of the phoneme duration model is not critical to the end results. We assume that the decoder is powerful enough to be able to recover from errors in the initial alignment, to a certain degree. At the same time, the initial alignment can never hugely deviate from the true alignment, as it is heavily constrained by the note timings. To see if this assumption is correct, we purposely use a very simplistic duration model, based on average phoneme durations computed on a different dataset whose segmentation was corrected by hand. While language dependent, in this case the phoneme duration model is not singer dependent and the values could simply be copied from a table, without the need for any data with phonetic timings.\n\n\\subsection{Model architecture}\nThe input to our system is a musical score, consisting of a sequence of notes. Each note consists of an onset, duration, pitch, and a sequence of phonemes, typically corresponding to a syllable. In this work, we define the note onset as the vowel onset, and note end as the onset of the following vowel or silence. Additionally, we provide an external F0 to our system, in order to capture the affect of pitch on timbre. The output of our system is sequence of harmonic and aperiodic vocoder features, which in this case are simply concatenated.\n\nThe main components of our proposed system, as depicted in \\cref{fig:model_architecture}, are the encoder, the aligner and the decoder. The encoder takes the input phonetic sequence and computes a sequence of hidden states corresponding to each phoneme and their local context. The aligner provides a hard alignment by repeating these states according to the predicted phoneme durations, obtaining a sequence of the same length as the output acoustic sequence. Next, some additional conditioning signals derived from F0 and position are added. The decoder, based on the Transformer model \\cite{VaswaniA2017_AttentionIsAllYouNeed}, finally transforms the sequence of encoder hidden states to the target output sequence, through a series of self-attention and convolutional layers.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=17.8cm]{figures\/model_arch.pdf}\n \\caption{A diagram of the complete model architecture. On the left is the full system, composed of encoder, aligner and decoder, which themselves are composed of different higher level blocks. On the right, these higher level blocks (sub-layer, gated linear unit (GLU) and attention) are shown in detail.}\n \\label{fig:model_architecture}\n\\end{figure*}\n\n\\subsection{Encoder}\nOur encoder is based on the encoder proposed in \\cite{PingW2018_DeepVoice3}. First, embeddings are computed from each input phoneme. Then, a series of convolutional blocks with gated linear units (GLUs) \\cite{DauphinYN2017_GLU} allows encoding information about the phonetic context of each phoneme, e.g. corresponding to triphones or pentaphones. Finally a residual shortcut connection from the monophone embeddings is added to the local context output of the convolutional blocks.\n\n\\subsection{Duration model}\nAs noted, we purposely choose to use a very simplistic phoneme duration model in this work. It consists of a simple look-up table, populated with average phoneme durations computed from a dataset of a different singer with manually corrected phonetic segmentation. A simple heuristic is used to ensure that the sum of predicted phoneme durations matches the target note duration.\n\nAs we assume the note onset to correspond to the vowel onset, we first shift all onset consonants of each note to the preceding note (or silence). Then, we look up the sequence of average phoneme durations for each note, $[d_1,d_2,\\dotsc,d_N]$, where $N$ is the corresponding number of phonemes, and $d_1$ corresponds to the duration of the vowel. In order to match the target note duration, $d_n$, we use the predicted consonant durations and fill the remaining duration with the vowel. However, we also ensure at least half of the note's duration is occupied by the vowel by fixing $r_v=0.5$. The scaling factor for the consonants, $r_c$, then becomes,\n\\begin{equation}\nr_c = \n \\begin{cases}\n 1 & \\text{ for } N=1,\\\\\n \\min\\left(1, \\cfrac{d_n - \\rint{r_v d_n}}{\\sum_{i=2}^{N} d_i}\\right) & \\text{ otherwise. }\n \\end{cases}\n\\end{equation}\nAnd, the adjusted phoneme durations, $[\\hat{d}_1,\\hat{d}_2,\\dotsc,\\hat{d}_N]$,\n\\begin{equation}\n\\hat{d}_i =\n \\begin{cases}\n d_n - \\sum\\limits_{j=2}^{N} \\max\\left(1, \\rint{r_c d_j}\\right) & \\text{ for } i=1,\\\\\n \\max\\left(1, \\rint{r_c d_i}\\right) & \\text{ for } i=2,3,\\dotsc,N.\n \\end{cases}\n\\end{equation}\nNote that all durations here are in integer number of frames, and that there are corrections for rounding errors and zero frame durations.\n\n\\subsection{F0 and position conditioning}\nContinuous log F0 is encoded as a low dimensional vector between zero and one by evaluating several triangular basis functions whose centers are placed at frequencies appropriate for the training data's pitch range.\n\nTransformers typically use an additive trigonometric positional encoding to give the self-attention blocks a sense of the position of their inputs, and provide a linear inductive bias along early on in training. However, in our case we found that a simple $K$-dimensional cyclical encoding of the normalized frame position within each note, $p \\in [0,1] \\subset \\mathbb{R}$, gave slightly better results,\n\\begin{equation}\nv = \\frac{1}{2}\\cos\\left(2\\pi p - 2\\pi\\frac{k-1}{K}\\right) + \\frac{1}{2}\\quad\\text{for }k=1 \\ldots K.\n\\end{equation}\n\n\n\\subsection{Decoder}\n\nOur decoder is based on a feed-forward variant of the Transformer model \\cite{VaswaniA2017_AttentionIsAllYouNeed}, similar to \\cite{RenY2019_FastSpeech}. Each layer consists of a self-attention sub-layer block and a convolutional sub-layer block. Both sub-layers blocks have layer normalization \\cite{BaJ2016_LayerNorm}, dropout and a residual shortcut connection.\n\nFollowing \\cite{VaswaniA2017_AttentionIsAllYouNeed}, our self-attention blocks use the scaled dot product as a scoring function. Additionally, similar to \\cite{SperberM2018_GaussianBiasSelfAttention}, we bias the scores with a Gaussian along the diagonal to favor a more localized self-attention,\n\\begin{equation}\n\\Attention(Q,K,V) = \\softmax\\left(\\frac{QK^{\\transpose}}{\\sqrt{d_{\\text{model}}}} + M\\right) V,\n\\end{equation}\n\\begin{equation}\nM_{j,k} = \\frac{-(j-k)^2}{2\\sigma^2},\n\\end{equation}\nwhere $d_{\\text{model}}$ is the dimensionality of the input vectors, $M \\in \\mathbb{R}^{T \\times T}$ for sequence length $T$, and $\\sigma$ is a learned scale parameter. To reduce memory and computational requirements for the self-attention layers, we may use a reduction factor $r \\geq 1$, which means $r$ frames are predicted per output timestep \\cite{WangY2017_Tacotron,PingW2018_DeepVoice3}. While the use of multi-head attention is typical for NLP applications, we did not find this improved results in our case.\n\nFor the convolutional blocks we use GLUs, which for our case outperform the 2-layer convolutional network with central ReLU activation typically used in Transformer architectures.\n\n\n\\section{Experiments}\\label{sec:experiments}\n\nFor the experiments in this work, we train a model on a proprietary dataset of 41 pop songs performed by a professional English male singer. From this dataset 35 songs were used for training (\\durhm{1}{26} total), 4 for validation and 2 for testing.\n\nOur proposed system uses 64-dimensional input features similar to \\cite{BlaauwM2017_NPSS_MDPI}, extracted with a \\SI{10}{\\milli\\second} hop time. A reduction factor, $r=2$, is used. We use 256-dimensional phoneme embeddings, and an encoder with a single 3x1 GLU block with 64 channels. F0 is coarse coded to a 4-dimensional vector, as is the position within the note, albeit with a cyclical encoding. The decoder consists of 6 layers with (single head) self-attention and 3x1 GLU blocks, all with 256 channels. The final output projection is to $64r$ channels. Dropout probability is set to 0.1 throughout the model. The learned standard deviation of the Gaussian bias of the self-attention blocks is initialized to 30. Initialization of convolutional layers follows \\cite{GehringJ2017_ConvAttention}. We use the Adam optimizer with $\\beta_1=0.9$, $\\beta_2=0.98$, $\\epsilon=\\num{1e-9}$, and a batch size of 32. We follow the learning rate schedule from \\cite{VaswaniA2017_AttentionIsAllYouNeed}, with a 4000 step warm-up, a base learning rate of \\num{1e-3}, and a total of \\SI{50}{\\kilo\\nounit} updates. Additionally, we use Polyak averaging with a decay of 0.995 for validation and testing. The objective that we optimize is a simple L1 loss between output and target features.\n\nWe compare our proposed feed-forward model, which we label \\mlabel{FFT-NPSS}, to an autoregressive baseline model roughly following \\cite{BlaauwM2017_NPSS_MDPI}, labeled \\mlabel{AR-NPSS}. To study the importance of the accuracy of the approximate initial alignment, we train a version of our model, which uses ground truth phonetic durations rather than predictions by the simple averages duration model. Note that the baseline \\mlabel{AR-NPSS} is a non-Seq2Seq model, so it is also trained on ground truth phonetic durations. To study the importance of self-attention in the model we train a version of our model without attention blocks as well.\n\nWe ran a MOS listening test with 18 participants, which each rated a random subset of 12 out of 20 phrases. Per test 6 stimuli were presented; the 4 systems mentioned previously, and visible and hidden references consisting of a WORLD re-synthesis of the target recording. All systems are presented and rated together to encourage a comparison between them.\n\n\\begin{table}\n \\centering\n \\caption{Mean Opinion Score (MOS) ratings on a 1--5 scale with their respective 95\\% confidence intervals.}\n \\label{tab:mos}\n \\input{mos_table.tex}\n\\end{table}\n\nThe results of our listening test are shown in \\cref{tab:mos}. We can see that the \\mlabel{FFT-NPSS} system using ground truth phoneme durations performs best, but it is closely followed by the proposed Seq2Seq variant using a simple averages duration model. This shows that the initial alignment provided by the duration model has some importance, but it is not critical. Additionally, our proposed system outperforms the baseline autoregressive \\mlabel{AR-NPSS} system, possibly due to avoiding issues related to exposure bias. Finally, the variant of the \\mlabel{FFT-NPSS} system without self-attention layers performed worst overall, showing that self-attention is an important component for this kind of Seq2Seq mechanism, in our observations especially in terms of providing a coherent timbre over time. While all systems are still rated considerably below the reference WORLD re-synthesis, we expect that this gap would be reduced if we combine our system with a neural vocoder. Some sound examples, both with and without neural vocoder, are available online\\footnote{\\url{https:\/\/mtg.github.io\/singing-synthesis-demos\/transformer\/}}.\n\n\n\\section{Relation to prior work}\\label{sec:prior_work}\n\nOur work is most closely related to the recently proposed FastSpeech model for TTS \\cite{RenY2019_FastSpeech}. This model is also based on the feed-forward Transformer and an initial alignment obtained from a duration model. However, in this case the duration model is trained with the help of a teacher model based on an autogressive Transformer \\cite{LiN2019_TransformerTTS}, which is also used for generating the target mel-spectrogram features. We wanted to avoid the need to train an autoregressive teacher model, as we found this generally challenging for the case of singing voice. Additionally, we apply some modifications to the architecture, such as the use of GLU convolutional blocks, alternative positional encoding and a Gaussian bias for the self-attention layers.\n\nThe ParaNet model \\cite{PengK2019_ParaNet} proposes a different approach to feed-forward TTS. Here, standard content-based encoder-decoder attention is used, but the model is trained trained with the help of attention distillation with an autoregressive teacher model based on \\cite{PingW2018_DeepVoice3}. Besides the reasons mentioned above, we found that the hard alignment used in our approach makes it easier to obtain a quality similar to non-Seq2Seq models, compared to the soft alignment of encoder-decoder attention.\n\nIn singing synthesis, the only Seq2Seq system we are aware of is \\cite{LeeJ2019_KoreanSS}. This model is based on the DCTTS model \\cite{TachibanaH2018_DCTTS}, using content-based encoder-decoder attention, with autoregressive decoder. Similar to our approach, there is an initial alignment of the input states to the output timesteps. However, relying on the fact that the Korean syllable structure has at most one onset and one coda consonant, the first and last frame of the note are assigned to each consonant respectively, and the remaining frames are assigned to the vowel. After which, learning the attention alignment can be facilitated by using diagonally guided attention \\cite{TachibanaH2018_DCTTS}.\n\nNon-Seq2Seq singing synthesizers include those based on autoregressive models \\cite{BlaauwM2017_NPSS_MDPI,BousF2019_SpecEnvSingSynth,YiYH2019_DAR_SS}, feed-forward CNN \\cite{NakamuraK2019_TechnoSpeechCNN}, and feed-forward GAN-based approaches \\cite{HonoY2019_GAN_SS,ChandnaP2019_WGANSing}.\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe presented a singing synthesizer based on the Transformer model, with a practical Seq2Seq mechanism allowing feed-forward operation. This approach allows training models without the need for pre-aligned training data, which can be cumbersome to prepare for singing data. Compared to a baseline autoregressive model, the proposed model allows for faster inference, avoids issues related to exposure bias, and rates as good or slightly better in listening tests. The use of self-attention resulted to be a key factor in obtaining good quality results, especially in terms of producing coherent timbre. As our model relies on an initial alignment provided by a duration model, we compared a very simplistic duration model to ground truth durations, to see the importance of the initial alignment's accuracy. In listening tests, using ground truth durations was rated highest, but the difference was relatively small. While not shown due to lack of space, in our observations the model can recover from errors in the initial alignment, most likely thanks to the decoder's non-causal convolutions and self-attention layers. For example, while the duration of phrase-final consonants tends to be systematically underpredicted by average durations, in the output of the synthesizer these phonemes have durations close to the target.\\thanksanywhere{This work was funded by TROMPA H2020 No 770376.}\n\n\n\n\n\\clearpage\n\n\n\\bibliographystyle{IEEEbib}\n{\\small","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nPerturbation theory provides a useful tool in the studies \nof the weak coupling limit of field theories on a lattice \\cite{Capitani}.\nTo simplify perturbative computations it is helpful to derive analytical \nexpressions for the Feynman integrals.\nTypical integrals emerging in the one-loop approximation have the form\n\\begin{equation}\\label{GenOneLoopInt}\n\\int_{BZ} {dk \\over (2\\pi)^4} {{\\cal P}(\\hat k_{\\mu_0},\\widehat {(k-p_1)}_{\\mu_1},...,\\widehat {(k-p_1-...-p_n)}_{\\mu_1} ) \\over D_{latt}(k) D_{latt}(p_1-k) ... D_{latt}(p_1+...+p_n-k) }\n\\end{equation}\nwhere $\\displaystyle \\hat p_\\mu = {2\\over a} \\sin \\left( {p_\\mu a \\over 2}\\right)$,\n$\\ D_{latt}(k)= \\sum_{\\mu=1}^4 \\hat k_\\mu^2 +m^2 $ (in the boson case), symbol ${\\cal P} $ in the numerator\nmeans ``some polynomial of'' and symbol $BZ$ indicates that integration is performed \nover the Brillouin zone, that is the domain $\\displaystyle -\\;{\\pi\\over a}< k_\\mu < {\\pi\\over a}$.\n\nSuch integrals cannot be calculated analytically at finite values of $a$.\nIn the limit $a\\to 0$ they can be evaluated by the Kawai--Nakayama--Seo method \\cite{Kawai}.\n\nThe above integrand can be represented in the form \n\\begin{equation}\\label{Kawai001}\nI(k,\\tilde p,m^2;a)=I(k,0,0;a)\\ +\\ (I(k,\\tilde p,m^2;a)-I(k,0,0;a)),\n\\end{equation}\nwhere $\\tilde p$ is the set of all relevant external momenta, $m$ is the mass.\n$I(k,\\tilde p,m^2;a)-I(k,0,0;a)$ has a smooth continuum limit ($pa\\to 0$ and $ma\\to 0$)\nand involves no ultraviolet (UV) divergencies.\nIt can be computed with some continuum regularization\nsuch as dimensional (DR) or with fictitious mass\\footnote{The regularization by fictitious mass\nis obtained by adding the term $\\mu_R^2 = 2\\mu_B^2$ to the denominator of each boson or fermion propagator.} (FMP).\n\nBoth $I(k,0,0;a)$ and $(I(k,\\tilde p,m^2;a)-I(k,0,0;a))$\ninvolve infrared (IR) divergencies.\n\nThough these IR divergencies cancel each other, IR regularization is needed:\n$$\nI(k,\\tilde p,m^2;a) = \\lim_{\\mu_R^2 \\to 0} I(k,\\tilde p,m^2;a,\\mu_R^2).\n$$\nIn this work, we use the infrared regularization introduced below in the formulas\n(\\ref{ScalarBosonPropLatt}) for bosons and (\\ref {DenomFermPropLatt}) for fermions.\nThus the sought-for integral \nis represented as the sum of the integral over the Euclidean momentum space\n(which is readily calculated by well-known method)\nand the \"zero-momentum\" integral over the Brillouin zone.\nCalculation of the latter integrals forms the subject of the present study.\n\nIn recent years, considerable study was given to computations with \nrather complicated actions (see, for example \\cite{Holger1} and \\cite{Holger2}). \nIn so doing, one is confronted with an integrand involving products of \n$(\\hat k_\\mu)^n$ at large values of $n$.\nAlgorithms for computation of such integrals with both\nbosonic and fermionic denominators were proposed in \\cite{BCP} and \\cite{Melnikov}.\n\nAn outline of this paper is as follows.\nIn this work, the algorithm proposed in \\cite{BCP} is employed to obtain \na comprehensive set of the integrals needed in computations\nof various matrix elements. For the reader's convenience, a detailed exposition of the Burgio--Caracciolo--Pelissetto\n(BCP) algorithm is given. In Section~1, we deal with the bosonic case.\nMaking use of the BCP algorithm, we derive an explicit form of the recursion relations:\nformulas (\\ref{FMR_BasBosInt_FinPart_min}), (\\ref{FMR_BasBosInt_FinPart_min_vspom}),\nand (\\ref{RRforJfuncBosNeg}). \nIn Section~2 we describe computations in the fermionic case.\nIn this Section, we also begin with the exposition of the BCP algorithm \nand use it to find an explicit form of the recursion relations for the \nfunctions $B(p,q)$ and $J(p,q)$ related to the functions ${\\cal F}_\\delta(p,q)$ used in \\cite{BCP}.\nThese relations are presented in the Appendices. As a matter of fact,\nthey provide a computer program for a calculation of the general fermionic integrals (\\ref{eq:GenFermInt}).\nThe results obtained in this way are discussed in the Conclusions.\n\n\\section{Boson Integrals}\\label{BosonIntegrals}\n\nWe compute the bosonic `zero-momentum' integrals of the type (we set $a=1$)\n\\begin{equation}\\label{BosIntInitDef}\nF(q,n_1,n_2,n_3,n_4) = \\lim_{\\delta\\to 0} \\int dk\\;{(\\cos k_1 )^{n_1} (\\cos k_2 )^{n_2} (\\cos k_3 )^{n_3} (\\cos k_4 )^{n_4} \\over \\Delta_B^{(q+\\delta)}}\n\\end{equation}\nwhere\n\\begin{equation}\\label{ScalarBosonPropLatt}\n\\Delta_B = 4+\\mu_B^2-\\cos(k_1 )-\\cos(k_2 )-\\cos(k_3 )-\\cos(k_4 )\n\\end{equation}\nis the scalar boson propagator,\n$\\mu_B$ is the infrared regulator mass, and $\\delta$ is an infinitesimal\nparameter needed for an additional intermediate regularization.\nAll integrals in one-loop calculations with the boson propagators can be reduced to the integrals of this type.\n\nSince $F(q;n_1,n_2,n_3,n_4) $ is symmetric in the arguments \n$ n_1, n_2, n_3,$ and $ n_4$, we consider only the case\n\\begin{equation} \nn_1 \\geq n_2 \\geq n_3 \\geq n_4.\n\\end{equation}\n\n\\subsection{Formulas of Reduction}\nA computation of the massless\\footnote{Here $\\mu_B$ is the regulator mass, \nthus we consider the massless limit at the end of computations.} boson integrals \nover the Brillouin zone is based on the following algorithm:\n\\begin{eqnarray}\\label{ReducFormulasGT2}\n&\\mbox{ if} &\\qquad n_4\\geq 2\\qquad \\mbox{ then} \\qquad F(q,n_1,n_2,n_3,n_4) = F(q,n_1,n_2,n_3,n_4-2)\\\\ \\nonumber\n&& -\\; {1\\over q-1+\\delta} \\left((n_4-1) F(q-1,n_1,n_2,n_3,n_4-1) -\n(n_4-2) F(q-1,n_1,n_2,n_3,n_4-3) \\right),\\\\ \\nonumber\n & \\mbox{else if}& \\quad n_3\\geq 2 \\qquad \\mbox{ then} \\qquad F(q,n_1,n_2,n_3,0) = F(q,n_1,n_2,n_3-2,0)\\\\ \\nonumber\n && -\\; {1\\over q-1+\\delta} \\left((n_3-1) F(q-1,n_1,n_2,n_3-1,0) -\n(n_3-2) F(q-1,n_1,n_2,n_3-3,0) \\right),\\\\ \\nonumber\n& \\mbox{ else if} &\\quad n_2\\geq 2 \\qquad \\mbox{ then} \\qquad F(q,n_1,n_2,0,0) = F(q,n_1,n_2-2,0,0)\\\\ \\nonumber\n&& -\\; {1\\over q-1+\\delta} \\left((n_2-1) F(q-1,n_1,n_2-1,0,0) -\n(n_2-2) F(q-1,n_1,n_2-3,0,0) \\right),\\\\ \\nonumber\n& \\mbox{ else if} &\\quad n_1\\geq 2 \\qquad \\mbox{ then} \\qquad F(q,n_1,0,0,0) = F(q,n_1-2,0,0,0)\\\\ \\nonumber\n&& -\\; {1\\over q-1+\\delta} \\left((n_1-1) F(q-1,n_1-1,0,0,0) -\n(n_1-2) F(q-1,n_1-3,0,0,0) \\right). \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\\label{ReducFormulasOne}\nF(q,n_1,n_2,n_3,1) &=& (4+\\mu_B^2) F(q,n_1,n_2,n_3,0) - F(q-1,n_1,n_2,n_3,0) \\\\ \\nonumber\n &-& F(q,n_1+1,n_2,n_3,0) - F(q,n_1,n_2+1,n_3,0) - F(q,n_1,n_2,n_3+1,0), \\\\ \\nonumber\nF(q,n_1,n_2,1,0) &=& {1\\over 2} \\left( (4+\\mu_B^2) F(q,n_1,n_2,0,0) - F(q-1,n_1,n_2,0,0)\\right. \\\\ \\nonumber\n && \\left. - F(q,n_1+1,n_2,0,0) - F(q,n_1,n_2+1,0,0)\\right), \\\\ \\nonumber\nF(q,n_1,1,0,0) &=& {1\\over 3} \\left( (4+\\mu_B^2) F(q,n_1,0,0,0) - F(q-1,n_1,0,0,0)\\right. \\\\ \\nonumber\n && \\left. - F(q,n_1+1,0,0,0) \\right), \\\\ \\nonumber\nF(q,1,0,0,0) &=& {1\\over 4} \\left( (4+\\mu_B^2) F(q,0,0,0,0) - F(q-1,0,0,0,0) \\right). \\nonumber\n\\end{eqnarray}\nThe above identities can be obtained using integration by parts \\cite{BCP};\norder $O(\\delta^2)$ terms should be omitted. \n\nThus we obtain an expression for each integral $F(q,n_1,n_2,n_3,n_4)$\nin terms of the functions\\footnote{In what follows, $\\displaystyle G(q, \\mu_B^2) = lim_{\\delta\\to 0} G_\\delta (q,\\mu_B^2)$.}\n\\begin{equation}\nG_\\delta (q,\\mu_B^2)=\\int {dk \\over (2\\pi)^4 } {1\\over (\\Delta_B)^{q+\\delta}},\n\\end{equation}\nit has the form\n\\begin{eqnarray}\\label{BosFtoGgen}\nF(q,n_1,n_2,n_3,n_4) &=& \\sum_{r=q-n_1-n_2-n_3-n_4}^{q} a_{qr}(\\delta,\\mu_B^2,\\tilde n) \nG_\\delta (r,\\mu_B^2) \\\\ \\nonumber\n&=&\n\\sum_{r=q-n_1-n_2-n_3-n_4}^{0} a_{qr}(\\delta,0,\\tilde n) G_\\delta (r,0) \\;+\\;\n\\sum_{r=1}^{q} a_{qr}(0,\\mu_B^2,\\tilde n) G(r,\\mu_B^2)\\;+\\; {\\cal O}(\\mu_B^2), \\nonumber\n\\end{eqnarray}\nwhere $\\tilde n$ is short-hand notation for $n_1,n_2,n_3,n_4$.\nIn this sum, the terms with $r\\leq 0$ and those with $r>0$ should be considered separately:\n\\begin{itemize}\n\\item The coefficients $a_{qr}(\\delta,\\mu_B^2,\\tilde n)$ at $r\\leq 0$ involve the pole $\\displaystyle {1\\over \\delta}$:\n\\[\na_{qr}(\\delta,\\mu_B^2,\\tilde n) = a_{qr}^{(sing)}(\\mu_B^2,\\tilde n)\\;{1\\over \\delta} + a_{qr}^{(reg)}(\\mu_B^2,\\tilde n) + O(\\delta),\n\\]\nso that $ G_\\delta (r,0)$ must be expanded to the order ${\\cal O}(\\delta)$.\nSince $ G_\\delta (r,\\mu_B^2)$ has no infrared divergencies at $r \\leq 0$\nand the coefficients $a_{qr}(\\delta,\\mu_B^2,\\tilde n)$ are polynomials in $\\mu_B^2$, \nthe values of $a_{qr}^{(sing)}$ and $a_{qr}^{(reg)}$ should be evaluated at $\\mu_B=0$.\n\n\\item At $r > 0$, the coefficients $a_{qr}(\\delta,\\mu_B^2,\\tilde n)$ involve no \npoles in $\\delta$ and, therefore, $\\delta$ can be safely set to zero. However, at $r>0$,\n$G_\\delta (r,\\mu_B^2)$ involves infrared divergencies, so that\nthe $\\mu_B$ dependence of the coefficients $a_{qr}$ should be kept.\n\\end{itemize}\n\nFrom these properties it follows that we should compute the quantities \nthat appear in the right-hand sides of the formulas\n\\begin{eqnarray}\\label{DefJ}\n\\mbox{for} \\quad r\\leq 0, \\qquad && G_\\delta (r,\\mu_B^2) = {\\cal B}_{-r} + J(r)\\delta +O(\\delta^2)\\;+\\; {\\cal O}(\\mu_B^2), \\\\[1mm] \\nonumber\nr=1 \\qquad && G (1,\\mu_B^2) = J(1)\\;+\\; {\\cal O}(\\mu_B^2), \\\\[1mm] \\nonumber\nr=2 \\qquad && G (2,\\mu_B^2) = J(2) + (\\ln \\mu_B^2 + C)\\;+\\; {\\cal O}(\\mu_B^2), \\\\[1mm] \\nonumber\n\\mbox{for} \\quad r > 2, \\qquad && G (r,\\mu_B^2) = J(r) + D_{r0}(\\ln \\mu_B^2 + C) + \\sum_{k=1}^{r-2} D_{rk}\/(\\mu_B)^{2k}\\;+\\; {\\cal O}(\\mu_B^2),\n\\end{eqnarray}\nwhere $C=0.577...$ is the Euler-Mascheroni constant, $D_{rn}$ and $J(r)$ are some constants to be determined and \n${\\cal B}_r$ are given by \n\\begin{equation}\\label{Bdirect}\n{\\cal B}_r=\\lim_{\\mu_B\\to 0}\\ \\int_{BZ} {dk\\over (2\\pi)^4}\\; \\Delta_B^r\n\\end{equation}\n(note that $r>0$), some of them can be found in Appendix~1.\n\n\\subsection{Computation of the Divergent Part \\\\ (Fictitious Mass Regularization)}\n\nWe consider the representation of $G_\\delta(q,\\mu_B^2)$\nin terms of the modified Bessel function:\n\\begin{equation} \\label{GdeltaReprBessel}\nG_\\delta (q,\\mu_B^2)=\\int {dk \\over (2\\pi)^4 } {1\\over (\\Delta_B)^{q+\\delta}}\n={1\\over \\Gamma(q+\\delta)} \\int_0^\\infty t^{q-1+\\delta}\\; dt\\ \\left[ e^{-4t-\\mu_B^2 t} I_0^4(t)\\right]\n\\end{equation}\nand divide the domain of integration into two parts:\n$\\int_0^\\infty = \\int_0^1 + \\int_1^\\infty $. The integral over\nthe segment $[0,1]$ converges. The divergent part arises from the\nlatter integral and can be isolated by subtracting $q-1$ terms of the\nasymptotic expansion at $z\\to \\infty$ of the function \n\\begin{equation}\\label{InfeldAsExp0}\n\\exp(-4z) I_0^4(z) \\simeq {1\\over (2\\pi z)^2}\n\\left(1+ {b_1\\over z} + {b_2\\over z^2} + ... \\right);\n\\end{equation}\n$b_i$ at $i\\leq 20$ are given in Appendix~1.\nWe isolate the divergent part $\\bar G_{div}^{M} (q,\\delta,\\mu_B)$ as follows:\n\\begin{eqnarray} \\label{G_FMRdef1}\nG_\\delta (q,\\mu_B^2)= \\bar G_{div}^{M} (q,\\delta,\\mu_B) + \\bar J_\\delta(q) &=& {1\\over \\Gamma(q+\\delta)} \\left\\{ \n\\int_0^1 t^{q-1+\\delta}\\; dt\\ \\left[ e^{-4t-\\mu_B^2 t} I_0^4(t)\\right]\\right. + \\\\ \\nonumber\n&& + \\int_1^\\infty t^{q-1+\\delta}\\; dt\\ e^{-\\mu_B^2 t} \\left[ e^{-4t} I_0^4(t)- {1\\over (2\\pi t)^2} \n\\sum_{n=0}^{q-2} {b_n \\over t^n} \\right] \\\\ \\nonumber\n&& + \\int_1^\\infty t^{q-1+\\delta}\\; dt\\ \\left. {1\\over (2\\pi t)^2}\\; e^{-\\mu_B^2 t} \\\n\\sum_{n=0}^{q-2} {b_n \\over t^n} \\right\\}, \\nonumber\n\\end{eqnarray}\nwhere the first and second lines are designated by $\\bar J_\\delta(q)$ and\nthe third---by $\\bar G_{div}^{M}(q,\\delta,\\mu_B)$:\n\\begin{eqnarray}\\label{FMR_DivPart_nonmin}\n&& \\bar G_{div}^{M}(q,\\delta,\\mu_B) =\n{1\\over (2\\pi )^2\\; \\Gamma(q+\\delta)} \\; \\sum_{n=0}^{q-2}\\left( \\int_0^\\infty - \\int_0^1 \\right)\\;\ndt \\ b_n t^{q-3-n+\\delta}\\; e^{-\\mu_B^2 t} \\\\ \\nonumber\n&& = {1\\over (2\\pi)^2 \\Gamma(q)} \n\\left[ \\; - \\; b_{q-2} l_C\n+ \\sum_{k=1}^{q-2} b_{q-k-2} \\left({\\Gamma (k) \\over (\\mu_B^2)^k} \n- {1\\over k} \\right)\\right] + \\\\ \\nonumber\n&+& {\\delta \\over (2\\pi)^2 \\Gamma(q)} \\ \\left[ b_{q-2} \\left( {1\\over 2} l_C^2 + {\\pi^2\\over 12}\n+\\psi(q) l_C \\right)\\right.\n\\left. +\\sum_{n=1}^{q-2} b_{q-2-n} \\left( {\\Gamma(n)\\over (\\mu_B^2)^n }\n(\\psi(n) - \\psi(q) - l_C + C) + {1\\over n^2} \\right) \\right], \\nonumber\n\\end{eqnarray}\nwhere $\\displaystyle \\psi(n)=\\sum_{k=1}^{n-1} {1\\over k} \\ - C$ and $l_C=(\\ln \\mu_B^2 +C)$.\n\nIn the case of purely boson integrals, $O(\\delta)$ terms can be omitted;\nhowever, they are needed for a computation of the divergent part of the integrals\n(\\ref{eq:GenFermInt}) involving fermion denominators.\n\nWe can also isolate the divergent part in the so called ``minimal way''\n\\begin{equation}\\label{FMR_DivPart_min}\nG_{div}^{M}(q,\\delta,\\mu_B) = {1\\over (2\\pi)^2 \\Gamma(q)} \\left\\{\n\\left[-\\; b_{q-2} l_C + \\sum_{k=1}^{q-2} b_{q-k-2} {\\Gamma (k) \\over (\\mu_B^2)^k} \n\\right] \\ + \\right. \n\\end{equation}\n\\[\n+ \\delta \\left. \\left[ b_{q-2} \\left( {1\\over 2} l_C^2 \n+ \\psi(q) l_C \\right) +\\sum_{n=1}^{q-2} b_{q-2-n} {\\Gamma(n)\\over (\\mu_B^2)^n }\n\\left(\\psi(n) - \\psi(q) - l_C + C \\right) \\right] \\right\\}; \\nonumber\n\\]\n\n\\begin{equation}\\label{DeltaJ}\n\\!\\! \\bar G_{div}^{M}(q,\\delta,\\mu_B) = G_{div}^{M}(q,\\delta,\\mu_B) - {1\\over (2\\pi)^2 \\Gamma(q)} \n \\sum_{k=1}^{q-2} {b_{q-k-2} \\over k}\n + {\\delta \\over (2\\pi)^2 \\Gamma(q)} \\ \\left[ b_{q-2} {\\pi^2\\over 12}\n+\\sum_{n=1}^{q-2} {b_{q-2-n} \\over n^2}\\right].\n\\end{equation}\n\nNote that $J(q)$ that appears in (\\ref{DefJ}) is connected with the quantity \n$\\bar J(q)=\\lim_{\\delta\\to 0} \\bar J_\\delta(q)$ (see (\\ref{G_FMRdef1})) by the relations\n\\begin{equation}\nG (q,\\mu_B^2) = \\lim_{\\delta\\to 0} G_\\delta (q, \\mu_B^2) = \n\\bar G_{div}^{M}(q,0,\\mu_B^2) + \\bar J(q) = G_{div}^{M} (q,0,\\mu_B^2) + J(q),\n\\end{equation}\nso that\n\\begin{equation}\n\\bar J(q) - J(q) = {1\\over (2\\pi)^2 \\Gamma(q)} \\sum_{k=1}^{q-2} {b_{q-k-2} \\over k}\n\\end{equation}\nand the coefficients $D_{rk}$ from the formula (\\ref{DefJ}) are determined from the\nthe equation (\\ref{FMR_DivPart_min}):\n\\begin{equation}\nD_{rk} = {b_{r-k-2} \\Gamma(k)\\over (2\\pi)^2 \\Gamma(r)} \\ \\ \\mbox{at} \\ \\ 1\\leq k\\leq r-2;\n\\qquad D_{r0} = -\\;{ b_{r-2} \\over (2\\pi)^2 \\Gamma(r)}.\n\\end{equation}\n\n\\subsection{Computation of the Finite Parts \\\\ (Fictitious Mass Regularization)}\n\nUsing the reduction formulas (\\ref{ReducFormulasGT2}) and (\\ref{ReducFormulasOne}), \nwe obtain expressions for the integrals\n$F(q;n_1,n_2,n_3,n_4) \\equiv F(q,\\tilde n)$ \nin terms of the quantities $J(r), {\\cal B}_r$, and $D_{rn}$ determined above (see formula (\\ref{BosFtoGgen})).\n\nThe next step is to use recursion relations \nmaking it possible to express $J(r)$ at $r\\geq 4$\nin terms of the basic boson constants $J(1)$, $J(2)$, and $J(3)$\nand at $r\\leq 0$ in terms of $J(1)$, $J(2)$, $J(3)$, and $J(0)$.\nThe recursion relations are obtained by making use of the trivial identity\n\\begin{equation}\n\\Delta_B-4 - \\mu_B^2 +\\sum_{\\mu=1}^{4} \\cos(k_\\mu)=0.\n\\end{equation}\nInserting this identity in the integrals\n\\begin{eqnarray}\nF(q,1,1,1,1) &=& \\int dk\\;{\\cos(k_1) \\cos(k_2) \\cos(k_3) \\cos(k_4) \\over \\Delta_B^{(q+\\delta)}} \\nonumber\n\\end{eqnarray}\nwe arrive at\n\\begin{equation}\n(4+\\mu_B^2) F(q,1,1,1,1)= 4 F(q,2,1,1,1)+4 F(q-1,1,1,1,1),\n\\end{equation}\nNow we express $F(q,1,1,1,1)$ etc. in terms of \nthe values $J(r)$ and thus obtain the sought for relations between them.\nWith these relations, $J(r)$ at $r\\leq 0$ and $r\\geq 4$ is readily \nexpressed in terms of $J(1)$, $J(2)$, and $J(3)$.\nWe consider FMR with the finite part (\\ref{FMR_DivPart_min}) defined in the ``minimal'' way.\nGiven\n\\begin{equation}\\label{FMR_DivPart_min0}\nG_{div}^{M}(q,\\mu_B^2) = {1\\over (2\\pi)^2 \\Gamma(q)} \n\\left[b_{q-2} l_C + \\sum_{k=1}^{q-2} b_{q-k-2} {\\Gamma (k) \\over (\\mu_B^2)^k} \\right],\n\\end{equation}\nwe derive the recursion relations for $J(q)\\quad (q>0)$ as follows:\n\n\\begin{eqnarray}\\label{FMR_BasBosInt_FinPart_min}\nJ(q) &=& {1 \\over 384 (q-1) (q-2)^2 (q-3)} \\\\ \\nonumber\n&& \\left\\{ 16 (q-2) (q-3) \\left[12 + 25 (q-2) (q-3) \\right] J(q-1) \\right. \\\\ \\nonumber\n&& +\\; 4(q-3)^2 \\left[ -17 - 35(q-3)^2 \\right]\\; J(q-2) \\\\ \\nonumber\n&& +\\; 4\\;\\left[1+ 5(q-3)^3(q-4)-5(q-3)(q-4)^2 \\right] \\;J(q-3)\\\\ \\nonumber\n&& \\left. -(q-4)^4\\;J(q-4) \\right\\}\\\\ \\nonumber\n&& +\\; {1\\over (q-2)}\\; D(q)\\\\ \\nonumber\n&& -\\; {25\\over 24 (q-1) (q-2)}\\; (2q-5)\\; D(q-1)\\\\ \\nonumber\n&& + \\; {1\\over 96 (q-1)(q-2)^2}\\; \\left[17+105(q-3)^2\\right] \\; D(q-2)\\\\ \\nonumber\n&& +\\; {5\\over 96 (q-1)(q-2)^2(q-3)}\\; \\left[-1-4(q-3)^2(q-4)+2(q-4)^2\\right]\\; D(q-3)\\\\ \\nonumber\n&& + \\; {5\\over 384 (q-1) (q-2)^2 (q-3)}\\; (q-4)^3 \\; D(q-4);\\nonumber\n\\end{eqnarray}\nwhere $\\displaystyle D(q)={b_{q-2}\\over (q-1)\\!}$; this being so, $D(q)$\nsatisfies the recurrent relation\n\\begin{eqnarray}\\label{FMR_BasBosInt_FinPart_min_vspom}\n D(q) &=& {1\\over 384 (q-1) (q-2)^2 (q-3)} \\\\ \\nonumber\n&& \\left\\{ 16 (q-2)(q-3) \\left[12+25(q-2)(q-3) \\right]\\; D(q-1) \\right. \\\\ \\nonumber\n&& +\\; 4(q-3)^2 \\left[ -17 - 35(q-3)^2\\right]\\; D(q-2)\\\\ \\nonumber\n&& +\\; 4 \\left[1+ 5(q-3)^3(q-4)-5(q-3)(q-4)^2 \\right]\\; D(q-3)\\\\ \\nonumber\n&& \\left. -\\;(q-4)^4\\;D(q-4)\\right\\};\n\\end{eqnarray}\nwith the initial conditions\n\n\\begin{eqnarray}\\label{J0123boson}\nJ(0)&=& J_0; \\\\ \\nonumber\nJ(1)&=& 2 Z_0; \\\\ \\nonumber\nJ(2)&=& {F_0\\over (2\\pi)^2}; \\\\ \\nonumber\nJ(3)&=& {Z_1\\over 32}\\;+\\; {1\\over (2\\pi)^2} {F_0\\over 4}\n\\;-\\; {1\\over (2\\pi)^2}\\,{13 \\over 48} \\;-\\; {1\\over 128}; \\nonumber\n\\end{eqnarray}\nand\n\n\\begin{equation}\\label{IniCondforDIVboson}\nD(1)=0; \\qquad \\qquad\nD(2)= {1\\over (2\\pi)^{2}}; \\qquad \\qquad\nD(3)= {1\\over (2\\pi)^{2}}\\ {1\\over 4}. \n\\end{equation}\n\nRecurrent relations for $J(q)$ at $q<0$ can be\nderived by the same token, they have the form\n\\begin{eqnarray}\\label{RRforJfuncBosNeg}\nJ(q)&=& -\\; {1\\over q^4}\\ \\left[-4 \\left( 1+5(q+1)q+5(q+1)^2 q^2\\right) J(q+1) \\right. \\\\ \\nonumber\n\t&+& 4 (q+1)^2 (17+35(q+1)^2) J(q+2) \\\\ \\nonumber\n\t&-& 16 (q+2) (q+1) (25 (q+2) (q+1)+12) J(q+3) \\\\ \\nonumber\n\t&+& 384 (q+3) (q+2)^2 (q+1) J(q+4) \\\\[1mm] \\nonumber\n\t&+&{2 q^3 (3+5 (q+3)^2 (q+1)) \\over (q+4)(q+3)(q+2)(q+1)}\\; {\\cal B}_{-q} \\\\[1mm] \\nonumber\n\t&+& 4 \\;{ (-40 q^6-330 q^5-985 q^4-1376 q^3 -1015 q^2-410 q-70)\\over (q+4)(q+3)(q+2)(q+1)}{\\cal B}_{-q-1}\\\\[1mm] \\nonumber\n\t&+& 8 \\;{ (q+1) (105 q^4+788 q^3+1998 q^2+2052 q +788)\\over (q+4)(q+3)(q+2)}{\\cal B}_{-q-2}\\\\[1mm] \\nonumber\n\t&+& 32 \\;{ (q+2) (q+1) (31 (q+4)^2 -81 (q+3)^2)\\over (q+4)(q+3)} {\\cal B}_{-q-3} \\\\[1mm] \\nonumber\n\t&+&\\left. 768\\; {(q+3) (q+2) (q+1)\\over(q+4)} {\\cal B}_{-q-4} \\right] \\qquad \\mbox{for}\\quad q\\leq -3. \\nonumber\n\\end{eqnarray}\nThe values of ${\\cal B}_q$ can be computed either directly by the formula \n(\\ref{Bdirect}) or with the use of the recurrent relations\n\n\\begin{eqnarray}\\label{RecRelForBbosonic}\n{\\cal B}_{q}&=& {1\\over q^4}\\left[ -\\; 384 (q-1) (q-2)^2 (q-3) \\;{\\cal B}_{q-4} \\right. \\\\ \\nonumber\n &&\\qquad +\\; 16 (q-1) (q-2) \\big(25 (q-1) (q-2) + 12\\big) \\;{\\cal B}_{q-3} \\\\ \\nonumber\n &&\\qquad -\\; 4 (q-1)^2 \\big( 35 (q-1)^2 + 17\\big) \\;{\\cal B}_{q-2} \\\\ \\nonumber\n &&\\qquad \\left. +\\; 4 (q^5 - (q-1)^5) \\; {\\cal B}_{q-1} \\right] \\nonumber\n\\end{eqnarray}\n\n\n\nwith the initial conditions\n\n\\begin{equation}\\label{IniCondForBbosonic}\n {\\cal B}_{0} = 1; \\qquad\n {\\cal B}_{1} = 4; \\qquad\n {\\cal B}_{2} = 18;\\qquad\n {\\cal B}_{3} = 88.\n\\end{equation}\n\n\nThe values of $J(-1), J(-2)$, and $J(-3)$ can be determined \nin the same way, the respective identities have the form\n\\begin{eqnarray} \n J(-4) &=& -9\/16 (13\/9-781\/36 J(-3)+83 J(-2)-108 J(-1)+ 32 J(0) ); \\\\ \\nonumber\n J(-3) &=& 16\/27 (-11\/2+211\/12 J(-2)-157\/3 J(-1)+124\/3 J(0)+16 J(1));\\\\ \\nonumber\n J(-2) &=& -3 \\left(-3\/2+{8\\over (2\\pi )^2} -31\/12 J(-1)+13\/3 J(0)+ 4 J(1)\\right);\\\\ \\nonumber\n J(-1) &=& 144 \\left(-1\/36-{ 13\\over 9 (2\\pi)^{2}}+1\/36 J(0)+4\/3 J(2)-16\/3 J(3)\\right); \\nonumber\n\\end{eqnarray}\n\nThe integrals (\\ref{BosIntInitDef}) can also be expressed \nin therms of the quantities $l_C$, and\n\\begin{eqnarray}\\label{BasBosConstNum}\nZ_0 &\\approx& 0.154933390231060214084837208 \\\\ \\nonumber\nZ_1 &\\approx& 0.107781313539874001343391550 \\\\ \\nonumber\nF_0 &\\approx& F_0^C -\\ln 2 = 4.369225233874758 -\\ln 2 \\nonumber\n\\end{eqnarray}\ndetermined from the relations\\footnote{In the review \\cite{Capitani}, the constant $F_0^C \\approx 4.369225233874758$ \nis designated by $F_0$.}\n\\begin{eqnarray}\nF(1,0,0,0,0) &=& 2Z_0+O(\\mu_B^2) \\\\ \\nonumber\nF(2,0,0,0,0) &=& - \\; {l_C\\over (2\\pi)^2}\\;+\\;{F_0\\over (2\\pi)^2} \\; +O(\\mu_B^2)\\\\ \\nonumber\nF(3,0,0,0,0) &=& {1\\over (2\\pi)^2}\\left({1\\over 2\\;\\mu_B^2} - \\; {l_C\\over 4} - {13\\over 48} + {F_0 \\over 4} \\right)\n-{1\\over 128}\\;+\\;{Z_1\\over 32} +O(\\mu_B^2). \\nonumber\n\\end{eqnarray}\nThis being so, the initial conditions for the recurrent relations\nare given by the formula (\\ref{J0123boson}).\n\nIt should be noted that $J_0$ does not appear \nin the ultimate expressions for the integrals of the type (\\ref{BosIntInitDef}),\ntherefore, its numerical value is not needed.\n\n\n\\subsection{Dimensional Regularization \\label{sec:DRbos}}\n\nFirst we introduce the quantity $\\bar J(q;\\tilde n)$\nanalogous to $\\bar J_\\delta(q)$ defined in (\\ref{G_FMRdef1}):\n\\begin{eqnarray} \\label{defJbasic1}\n \\bar J(q;\\tilde n) &=& \\lim_{\\mu_B \\to 0} \\lim_{\\delta \\to 0}{1\\over \\Gamma(q+\\delta)} \\left\\{ \n\\int_0^1 t^{q-1+\\delta}\\; dt\\ \\left[ e^{-(4+\\mu_B^2) t} {{\\cal T}(\\tilde n)} \\right]\\right. + \\\\ \\nonumber\n&+& \\left. \\int_1^\\infty t^{q-1+\\delta}\\; dt\\ e^{-\\mu_B^2 t} \\left[ e^{-4t} {{\\cal T}(\\tilde n)} - {1\\over (2\\pi t)^2} \n\\sum_{k=0}^{q-2} {b_k (\\tilde n) \\over t^k} \\right]\\right\\}, \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation} \n{\\cal T}(\\tilde n)= \\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_1} I_0(t)\\right]\\ \n\\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_2} I_0(t)\\right]\\ \n\\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_3} I_0(t)\\right]\\ \n\\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_4} I_0(t)\\right].\n\\end{equation}\nIt represents the finite part of the general boson integral (\\ref{BosIntInitDef})\nprovided that the divergent part is defined by the formula \nsimilar to (\\ref{FMR_DivPart_nonmin}).\nWe omit here $O(\\delta)$ terms because they are only needed for the calculation of fermion integrals\nin the fictitious mass regularization. Thus we set $\\delta=0$.\nThen it should be noted that\n\\begin{eqnarray}\n \\bar J(q;\\tilde n) &=& \\lim_{\\epsilon \\to 0} {1\\over \\Gamma(q)} \\left\\{ \n\\int_0^1 t^{q-1}\\; dt\\ \\left[ e^{-(4-2\\epsilon)t} I_0^{-2\\epsilon}(t) {{\\cal T}(\\tilde n)} \\right]\\right. + \\\\ \\nonumber\n&+& \\int_1^\\infty t^{q-1}\\; dt\\, \\left[ e^{-(4-2\\epsilon)t} I_0^{-2\\epsilon}(t) {{\\cal T}(\\tilde n)} - {1\\over (2\\pi t)^{2-\\epsilon}} \n\\sum_{k=0}^{q-2} {\\tilde b_k(\\tilde n) \\over t^k} \\right], \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\tilde b_k (\\tilde n) = b_k(\\tilde n) -2\\epsilon d_k (\\tilde n),\n\\end{equation}\nwhere \n$b_k(\\tilde n)$ are the coefficients of the asymptotic expansion at $t \\to \\infty$\n\\begin{equation}\n(2\\pi t)^2\\; e^{-4t} {{\\cal T}(\\tilde n)} \\;\\simeq \\sum_{k=1}^\\infty {b_k (\\tilde n) \\over t^k} \n\\end{equation}\nand $d_k(\\tilde n)$ are the coefficients of the asymptotic expansion\n\\begin{equation}\\label{d_coeff_def}\n(2\\pi t)^2 \\; e^{-4t} {{\\cal T}(\\tilde n)}\\;\\ln \\left[ e^{-t} I_0(t) \\sqrt{2\\pi t}\\right] \\simeq \n\\sum_{n=1}^\\infty {d_n (\\tilde n) \\over t^n} .\n\\end{equation}\n\\vskip 1mm\nNow we {\\bf define} the general boson integral (\\ref{BosIntInitDef}) in the \ndimensional regularization by the formula\n\\begin{equation}\nF(q;\\tilde n) = \\bar J(q;\\tilde n) + F^{DR}_{div}(q;\\tilde n),\n\\end{equation}\nwhere\n\\begin{equation} \nF^{DR}_{div}(q;\\tilde n) = \\int_1^\\infty t^{q-1}\\; dt\\ {1\\over (2\\pi t)^{2-\\epsilon}} \\\n\\sum_{k=0}^{q-2} {\\tilde b_k(\\tilde n) \\over t^k}\n\\end{equation}\nand the dimensional regularization implies that\n\\begin{equation}\n\\int_1^\\infty dt\\ t^{n+\\epsilon} =0 \\quad\\mbox{at}\\ \\ n\\neq -1, \\qquad \\qquad \n\\int_1^\\infty {dt \\over t^{1-\\epsilon}} = -\\ {1\\over \\epsilon}.\n\\end{equation}\nThis being so,\n\\begin{equation} \nF^{DR}_{div}(q;\\tilde n) = {1\\over (2\\pi )^{2}} {1\\over \\Gamma(q)} \n\\left\\{ -\\ {1\\over \\epsilon}\\; b_{q-2}(\\tilde n) - \\ln(2\\pi)\\, b_{q-2}(\\tilde n) +2d_{q-2}(\\tilde n) \\right\\}.\n\\end{equation}\nNow we isolate the \"canonical\" divergent part in the dimensional regularization; \nthat is, \n\\begin{equation}\n\\bar F^{DR}_{div}(q;\\tilde n, \\mu^2) =-\\; {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[ {1\\over \\epsilon}\\;-C+\\ln\\left({4\\pi\\over\\mu^2}\\right)\\right] b_{q-2}(\\tilde n).\n\\end{equation}\nwhere $\\mu$ is the parameter of dimensional regularization\\footnote{In this subsection it is considered that intergation in (\\ref{BosIntInitDef}) is performed over the $4-2\\epsilon$ dimensional space; the integral\nunder consideration should be multiplied by $\\mu^{2\\epsilon}$};\nthe \"canonical\" divergent part is needed to compensate for the \ninfrared divergent part in the respective continuum integral\nwith nonvanishing external momenta. We see that\n\\begin{eqnarray}\\label{GDR-barGDR}\nF^{DR}_{div}(q;\\tilde n,) &=& \\bar F^{DR}_{div}(q;\\tilde n, \\mu^2) \\\\ \\nonumber\n&+& {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[\\left(-C+\\ln {2\\over\\mu^2}\\right) b_{q-2}(\\tilde n)+2d_{q-2}(\\tilde n) \\right]. \\nonumber\n\\end{eqnarray}\nThe respective finite parts can be determined by the formula\n\\begin{equation}\\label{2repr-of-FDR}\nF^{DR}(q;\\tilde n) = \\bar J(q;\\tilde n) + F^{DR}_{div}(q;\\tilde n) = \\bar J^{DR}(q;\\tilde n) + \\bar F^{DR}_{div}(q;\\tilde n),\n\\end{equation}\n\\vskip 1mm\nNow we express $\\bar J^{DR}(q;\\tilde n)$ in terms of the quantity $J(q;\\tilde n)$\nwhich can be calculated in the fictitious mass regularization by the method described below.\nFirst we note that $\\bar J(q;\\tilde n)$ is connected with $\\bar J^{DR}(q;\\tilde n)$ by\nthe relation\n\\begin{equation}\n\\bar J^{DR}(q;\\tilde n)= \\bar J(q;\\tilde n) + {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[\\left(-C+\\ln {2}\\right) b_{q-2}(\\tilde n)+2d_{q-2}(\\tilde n) \\right].\n\\end{equation}\nThe relation between $\\bar J(q;\\tilde n)$ and $J(q;\\tilde n)$ is\nderived from the formula\n\\begin{equation}\nF^{FMR}(q;\\tilde n) = \\bar J(q;\\tilde n) + \\bar F^M_{div}(q;\\tilde n) = J(q;\\tilde n) + F^{M}_{div}(q;\\tilde n),\n\\end{equation}\n(it is the definition of $J(q;\\tilde n)$). From the formula analogous to (\\ref{DeltaJ}) it follows that\n\\begin{eqnarray}\\label{FFMR-barFFMR}\n\\bar J(q;\\tilde n) &=& J(q;\\tilde n) + (F^M_{div}(q;\\tilde n) - \\bar F^{M}_{div}(q;\\tilde n))\\\\ \\nonumber\n&=& J(q;\\tilde n)\\; + \\;{1\\over (2\\pi)^2 \\Gamma(q)} \\sum_{k=1}^{q-2} {b_{q-k-2} (\\tilde n)\\over k},\n\\end{eqnarray}\nwhere $\\bar F^{M}_{div}(q;\\tilde n)$ and $ F^{M}_{div}(q;\\tilde n)$ are natural\nanalogs of the quantities $\\bar G^{M}_{div}(q)$ and $ G^{M}_{div}(q)$ introduced \nabove. Combining formula (\\ref{FFMR-barFFMR}) with (\\ref{GDR-barGDR}) and (\\ref{2repr-of-FDR}), \nwe arrive at\n\\begin{eqnarray}\nF^{DR}(q;\\tilde n) &=& \\bar F^{DR}_{div}(q;\\tilde n;\\mu)\\;+\\; \\bar J^{DR}(q;\\tilde n;\\mu),\\ \\ \\ \\mbox{where} \\\\ \\nonumber\n\\bar F^{DR}_{div}(q;\\tilde n, \\mu) &=& -\\ {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[ {1\\over \\epsilon}\\;-C+\\ln\\left({4\\pi\\over\\mu^2}\\right)\\right] b_{q-2}(\\tilde n), \\\\ \\nonumber\n\\bar J^{DR}(q;\\tilde n;\\mu) &=& J(q;\\tilde n)\\; + \\;{1\\over (2\\pi)^2 \\Gamma(q)} \\sum_{k=1}^{q-2} {b_{q-k-2}(\\tilde n) \\over k} \\\\ \\nonumber\n&& + {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[(-C+\\ln 2) b_{q-2}(\\tilde n)+2d_{q-2}(\\tilde n)\\right], \\nonumber\n\\end{eqnarray}\n where $J(q;\\tilde n)$ can be calculated as follows:\nusing the relations (\\ref{ReducFormulasGT2}) and \n(\\ref{ReducFormulasOne}), $F(q,\\tilde n)$ is transformed to \na linear combination of the quantities $G_\\delta(r,\\mu_B^2)$ (\\ref{BosFtoGgen})\nand the substitutions (\\ref{DefJ}) are employed. \nIn the resulting expression, $\\mu_B^{-1}$ and $l_C$\nare formally set equal to zero, all that remains represents\nthe sought-for $J(q;\\tilde n)$.\n\nIt should be noticed that $\\displaystyle {1\\over \\epsilon}$ appears in $F^{DR}(q;\\tilde n)$\nonly in the combination $\\displaystyle {1\\over \\epsilon} -\\ln 2 - F_0 + \\ln \\left({4\\pi\\over \\mu^2}\\right)$\n(the Euler-Mascheroni constant $C$ cancels in the total expression).\n\n\n\n\n\\section{Fermion Integrals}\n\n\nHere we consider the integrals (remember that $a=1$)\n\\begin{equation}\\label{eq:GenFermInt}\nF(p,q;\\tilde n)=\\lim_{\\delta \\to 0}\\int {d^4k\\over (2\\pi)^4} \n{\\cos^{n_1}(k_1) \\cos^{n_2}(k_2) \\cos^{n_3}(k_3) \\cos^{n_4}(k_4) \\over \\Delta_B^q\n\\Delta_F^{p+\\delta}}\n\\end{equation}\nwhere $p>0$, $\\delta$ is a regularization parameter, and \n\\begin{equation}\\label{DenomFermPropLatt}\n\\Delta_F = 10-4\\;\\sum_{\\mu=1}^4 \\cos(k_\\mu) + \\sum_{1\\leq \\mu < \\nu \\leq 4} \\cos(k_\\mu) \\cos(k_\\nu) + \\mu_B^2\n\\end{equation}\nis the denominator of the fermionic propagator.\nMaking use of the recursion relations \n\\begin{eqnarray}\\label{RecFermnnnn}\n\\hspace*{-5mm} F(p,q,...,l,...) &=& F(p,q,...,l-2,...) \\\\ \\nonumber\n&&\\hspace*{-22mm} + \\ \\mu_B^2 \\left(F(p,q,...,l-1,...)-F(p,q,...,l-3,...)\\right) \\\\ \\nonumber\n&&\\hspace*{-22mm} -\\ \\left(F(p,q-1,...,l-1,...)-F(p,q-1,...,l-3,...)\\right) \\\\ \\nonumber\n&&\\hspace*{-22mm} -\\ {q\\over p-1+\\delta} \\; \\left(F(p-1,q+1,...,l-1,...) - F(p-1,q+1,...,l-3,...)\\right) \\\\ \\nonumber\n&&\\hspace*{-22mm} -\\ {1\\over p-1+\\delta}\\; \\left( (l-2) F(p-1,q,...,l-2,...) - (l-3)F(p-1,q,...,l-4,...)\\right);\n\\end{eqnarray} \n\\begin{eqnarray}\\label{RecFerm21} \nF(p,q;n1,n2,n3,2)&=&F(p,q-2,n1,n2,n3,0)-2\\,\\mu_B^2\\,F(p,q-1,n1,n2,n3,0)\\\\ \\nonumber\n&&\t-2\\,F(p-1,q,n1,n2,n3,0)+(4+2\\,\\mu_B^2+\\mu_B^4)\\,F(p,q,n1,n2,n3,0)\\\\ \\nonumber\n&&\t-F(p,q,n1+2,n2,n3,0)-F(p,q,n1,n2+2,n3,0) \\\\ \\nonumber\n&& -F(p,q,n1,n2,n3+2,0), \\\\ \\nonumber\nF(p,q,n1,n2,n3,1)&=&(\\mu_B^2+4)\\,F(p,q,n1,n2,n3,0)\\\\ \\nonumber\n&&\t-F(p,q-1,n1,n2,n3,0)-F(p,q,n1+1,n2,n3,0)\\\\ \\nonumber\n&&\t-F(p,q,n1,n2+1,n3,0)-F(p,q,n1,n2,n3+1,0)\\\\ \\nonumber\nF(p,q;n1,n2,2,0) &=& {1\\over 2} \\left( F(p,q-2,n1,n2,0,0) - 2 \\mu_B^2 F(p,q-1;n1,n2,0,0)\\right. \\\\ \\nonumber\n&& - 2 F(p-1,q;n1,n2,0,0) + (4+2 \\mu_B^2+\\mu_B^4) F(p,q,n1,n2,0,0) \\\\ \\nonumber\n&& \\left. - F(p,q;n1+2,n2,0,0)-F(p,q;n1,n2+2,0,0)\\right); \\\\ \\nonumber\nF(p,q;n1,n2,1,0) &=& {1\\over 2} \\left( (\\mu_B^2+4) F(p,q;n1,n2,0,0) - F(p,q-1;n1,n2,0,0)\\right. \\\\ \\nonumber\n&& \\left. - F(p,q;n1+1,n2,0,0) - F(p,q;n1,n2+1,0,0) \\right); \\\\ \\nonumber\nF(p,q;n1,2,0,0) &=& {1\\over 3} \\Big( F(p,q-2;n1,0,0,0) - 2 \\mu_B^2 F(p,q-1;n1,0,0,0) \\\\ \\nonumber\n&& - 2 F(p-1,q;n1,0,0,0) + (4 + 2 \\mu_B^2 + \\mu_B^4) F(p,q;n1,0,0,0)\\\\ \\nonumber\n&& - F(p,q;n1+2,0,0,0) \\Big); \\\\ \\nonumber\nF(p,q;n1,1,0,0) &=& {1\\over 3} \\Big( (\\mu_B^2+4) F(p,q;n1,0,0,0) - F(p,q-1;n1,0,0,0)\\\\ \\nonumber\n&& - F(p,q;n1+1,0,0,0)\\Big); \\\\ \\nonumber\nF(p,q;2,0,0,0) &=& {1\\over 4} \\Big( F(p,q-2;0,0,0,0) - 2 \\mu_B^2 F(p,q-1;0,0,0,0)\\\\ \\nonumber\n&& - 2 F(p-1,q,0,0,0,0) + ( 4 + 2 \\mu_B^2 + \\mu_B^4) F(p,q;0,0,0,0)); \\\\ \\nonumber\nF(p,q;1,0,0,0) &=& {1\\over 4} \\Big( (\\mu_B^2 + 4) F(p,q;0,0,0,0) -F (p,q-1;0,0,0,0)\\Big);\\nonumber\n\\end{eqnarray} \n\nwe can express the quantities (\\ref{eq:GenFermInt}) in terms of the integrals\n\\begin{equation}\\label{eq:BasFermInt}\nG_\\delta(p,q;\\mu_B^2)=\\int {d^4k\\over (2\\pi)^4} {1 \\over \\Delta_B^q \\Delta_F^{p+\\delta}}\n\\end{equation}\nas follows:\n\\begin{equation}\\label{FtoGfirst}\nF(p,q;n_1, n_2, n_3, n_4)=\\sum_{r=p-n}^{p} \\ \\sum_{s=q+2p-n-2r}^{q+p-r} C_{pq;\\tilde n}^{rs}(\\mu_B^2,\\delta) G_\\delta(r,s;\\mu_B^2),\n\\end{equation}\nwhere $n=n_1+n_2+n_3+n_4$. The coefficients $C_{pq;n_1 n_2 n_3 n_4}^{rs}$ \nare polynomials in $\\mu_B^2$ (however, at $r+s\\leq 2$ $C_{pq;\\tilde n}^{rs}(\\mu_B^2,\\delta)$\ncan be replaced by $C_{pq;\\tilde n}^{rs}(0,\\delta)$); at $r\\leq 0$ they involve\nthe singularity $\\displaystyle {1\\over \\delta}$; that is, they can be represented in the form\n\\begin{equation} \nC_{pq;n_1 n_2 n_3 n_4}^{rs} = {1\\over \\delta} S_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2) + R_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2)\n+ O(\\delta),\n\\end{equation}\nwhere $S_{pq;n_1 n_2 n_3 n_4}^{rs}=0$ at $r\\leq 0$ or $p\\leq 0$.\nA straightforward calculation of both $S_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2)$ \nand $R_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2)$ by employing the above relations is rather simple.\n\nTo compute the basic integrals $G_\\delta(p,q;\\mu_B^2)$, we consider the cases\n$p>0$ and $p\\leq 0$ separately. At $p>0$, only zeroth order of the expansion \nof $G_\\delta(p,q)$ in a power series in $\\delta$ gives a nonvanishing contribution, \nwhereas at $p\\leq 0$ one should also keep the term linear in $\\delta$.\n\nIt is convenient\\footnote{It should be noted that the functions $G_\\delta(p,q)$ are related to the functions ${\\cal F}_\\delta(p,q)$ used in \\cite{BCP} by the formulas ${\\cal F}_\\delta(p,q)= 2^{-p-q} G_\\delta(p,q)$.} to represent $G_\\delta(p,q)$ in the form \n\\begin{eqnarray}\\label{GdeltaExpansion}\nG_\\delta(p,q)&=& D(p,q;\\mu_B^2) + B(p,q) + \\delta \\;(L(p,q;\\mu_B^2)+J(p,q)) + O(\\delta^2) , \\qquad p\\leq 0; \\\\ \\nonumber\nG_\\delta(p,q)&=& D(p,q;\\mu_B^2) + J(p,q) + O(\\delta), \\qquad p > 0. \\nonumber\n\\end{eqnarray}\nwhere the quantities $B(p,q)$, $D(p,q;\\mu_B^2)$, $L(p,q;\\mu_B^2)$, and $J(p,q)$ \nare defined as follows:\\\\\n$D(p,q;\\mu_B^2)+\\delta L(p,q;\\mu_B^2)$ is \nthe divergent part of $G_\\delta(p,q;\\mu_B^2)$ at $p\\leq 0$ (up to terms $O(\\delta^2)$),\\\\\n$D(p,q;\\mu_B^2)$ is \nthe divergent part of $G_\\delta(p,q;\\mu_B^2)$ at $p > 0$ (up to terms $O(\\delta)$),\\\\\n$B(p,q)+ \\delta \\;J(p,q)$ and $J(p,q)$ are the respective finite parts\\footnote{Note\nthat $J(p,q)$ designates the finite part in the order $O(1)$ at $p>0$ and in the order\n$O(\\delta$ at $p\\leq 0$}.\n\nThe finite and divergent parts are unambiguously fixed by the requirement that \n$D(p,q;\\mu_B^2)$ and $L(p,q;\\mu_B^2)$ can be represented in the form\n\\begin{eqnarray}\\label{DPcoeffOne}\nD(p,q;\\mu_B^2) &=& D_{0}(p,q) (\\ln \\mu_B^2 +C) + \\sum_{r=1}^{p+q-2} {D_{r}(p,q) \\over (\\mu_B^2)^r} \\\\ \\nonumber\nL(p,q;\\mu_B^2) &=& {1\\over 2} L_{0}^{(2)}(p,q) (\\ln \\mu_B^2 +C)^2 + L_{0}^{(1)}(p,q) (\\ln \\mu_B^2 +C) \\\\ \\nonumber\n&& + \\sum_{r=1}^{p+q-2} {L_{r}^{(2)}(p,q) (\\ln \\mu_B^2 + C) \\over (\\mu_B^2)^r} \n+\\sum_{r=1}^{p+q-2} {L_{r}^{(1)}(p,q) \\over (\\mu_B^2)^r}. \\nonumber\n\\end{eqnarray}\n\nIn the domain $p\\leq 0$ we also use the quantities\n\\begin{equation}\\label{BBandJJat_pleq0}\n{\\cal B}(p,q;\\mu_B^2)= B(p,q)+ D(p,q;\\mu_B^2) \\qquad \\mbox{and} \\qquad\n{\\cal J}(p,q;\\mu_B^2) = L(p,q;\\mu_B^2)+J(p,q). \\nonumber\n\\end{equation}\nAt $p> 0$, \n\\begin{equation}\\label{BBandJJat_pgeq0}\n{\\cal B}(p,q;\\mu_B^2)= B(p,q) = 0 \\qquad \\mbox{and} \\qquad\n{\\cal J}(p,q;\\mu_B^2) = D(p,q;\\mu_B^2)+J(p,q). \\nonumber\n\\end{equation}\nNote that, at $q < 2-p,\\ \\ $ $D(p,q,\\mu_B^2)=L(p,q,\\mu_B^2)=0$, thus\n${\\cal B}(p,q;\\mu_B^2)=B(p,q)$ and ${\\cal J}(p,q;\\mu_B^2)=J(p,q)$ and one can use both\ndesignations.\n\n\n\\subsection{Divergent Part in the Fictitious Mass Regularization \\label{DPFIFMR}}\n\n\nFirst we note that (symbol ${\\cal D\\!P}$ means `divergent part of')\n\\begin{eqnarray}\\label{CalcDPcoeff}\np\\leq 0, q\\geq2-p \\qquad && D (p,q;\\mu_B^2) = {\\cal D\\!P} \\int {dk\\over (2\\pi)^4} \\; {\\Delta_F^{-p} \\over \\Delta_B^{q}}, \\\\ \\nonumber\np = 0, q\\geq2 \\qquad && L (0,q;\\mu_B^2) = {d\\over d\\delta}\\left|_{\\delta=0} {\\cal D\\!P}\n\\int {dk\\over (2\\pi)^4}\\; { 1 \\over \\Delta_B^{q+\\delta}} \\right. \\\\ \\nonumber\n&& +\\sum_{l=1}^{q-2} {(-1)^l\\over l} \\; \n{\\cal D\\!P} \\int {dk\\over (2\\pi)^4}\\; {\\Delta^l \\over \\Delta_B^{l+q}} , \\\\ \\nonumber\np>0, q\\geq2-p \\qquad && D (p,q;\\mu_B^2) = \\sum_{l=0}^{p+q-2} {(-1)^l\\; (p+l-1)!\\over l! (p-1)!} \n{\\cal D\\!P} \\int {dk\\over (2\\pi)^4} \\; {\\Delta^l \\over \\Delta_B^{p+q+l}}, \\nonumber\n\\end{eqnarray}\n\nThe divergent parts of $\\delta$-independent integrals \nin formulas (\\ref{CalcDPcoeff}) can be \ncalculated as follows. First one employs the recursion relations \n(\\ref{ReducFormulasGT2}) and (\\ref{ReducFormulasOne})\nfor boson integrals to transform the integrand to a linear combination \nof the basic boson integrals (\\ref{GdeltaReprBessel}) and then \nevaluates the divergent part of each integral by the formula \n\\begin{equation}\n{\\cal D\\!P} \\int {dk\\over (2\\pi)^4} \\; {1 \\over \\Delta_B^{q}}\n= {1\\over (2\\pi)^2 \\Gamma(q)} \n\\left[-\\; b_{q-2} l_C + \\sum_{k=1}^{q-2} b_{q-k-2} {\\Gamma (k) \\over (\\mu_B^2)^k} \n\\right] \n\\end{equation}\n(see derivation of the formula (\\ref{FMR_DivPart_min})), according to the MS prescription\nin the FMR.\nThe $\\delta$-dependent divergent parts that appears in formula (\\ref{CalcDPcoeff})\ncan determined by the same token, however, with the use of the formula\n\\begin{eqnarray} \n&& {d\\over d\\delta}\\left|_{\\delta=0} {\\cal D\\!P} \n\\int {dk\\over (2\\pi)^4}\\; { 1 \\over \\Delta_B^{q+\\delta}} \\right. = \\\\ \\nonumber\n&& ={1\\over (2\\pi)^2 \\Gamma(q)}\n \\left[ b_{q-2} \\left( {1\\over 2} l_C^2 \n+ \\psi(q) l_C \\right) +\\sum_{n=1}^{q-2} b_{q-2-n} {\\Gamma(n)\\over (\\mu_B^2)^n }\n\\left(\\psi(n) - \\psi(q) - l_C + C \\right) \\right]; \\nonumber\n\\end{eqnarray}\n(see (\\ref{FMR_DivPart_min})). \nThe divergent parts at $p\\leq 0$ can be obtained by the recursion relations, see below.\n\nThe divergent parts $D(p,q;\\mu_B^2)$ and $L(p,q;\\mu_B^2)$ introduced in the\nformula (\\ref{GdeltaExpansion}) are presented in Appendix~2 at $p+q\\leq 8$;\nat other values it can be readily calculated by the above formulas.\n\n\\subsection{Finite Parts}\n\nGiven the divergent part, we use the recurrent relations (\\ref{RecFermnnnn})\nand (\\ref{RecFerm21}) to express any integral of the type (\\ref{eq:GenFermInt})\nin terms of the functions $B(p,q)$ and $J(p,q)$, which, in their turn, can be found\nby making use of the two types of the recursion relations \\cite{BCP}.\n\nThe relations of the first type (the so called $T$-identities) can be obtained \nby inserting the expression $ \\Delta_B-4 - \\mu_B^2 +\\sum_{\\mu=1}^{4} \\cos(k_\\mu)$\nwhich is identically equals zero, in the integrand\n\\begin{eqnarray}\nF(q,1,1,1,1) &=& \\int dk\\;{\\cos(k_1) \\cos(k_2) \\cos(k_3) \\cos(k_4) \\over\\Delta_F^{(p+\\delta)}\\; \\Delta_B^{q}} \\nonumber\n\\end{eqnarray}\nThe relations of the second type ($S$-identities) can be obtained by inserting \n$$\n10-4\\;\\sum_{\\mu=1}^4 \\cos(k_\\mu) + \\sum_{1\\leq \\mu < \\nu \\leq 4} \\cos(k_\\mu) \\cos(k_\\nu) + \\mu_B^2 - \\Delta_F,\n$$\nwhich is also identically equals zero, in the same integrand.\nIn so doing, we arrive at\n\\begin{eqnarray} \\label {TandSidentities}\n(4+\\mu_B^2) F(p,q,1,1,1,1) - F(p,q-1,1,1,1,1) - 4 F(p,q,2,1,1,1)) &=& 0, \\\\ \\nonumber\n(4+2 \\mu_B^2+\\mu_B^4) F(p+1,q,1,1,1,1) - 2 \\mu_B^2 F(p+1,q-1,1,1,1,1)&& \\\\ \\nonumber\n- 4 F(p+1,q,3,1,1,1) + F(p+1,q-2,1,1,1,1)-2 F(p,q,1,1,1,1) &=& 0.\n\\end{eqnarray}\nThen we express the functions $F(p,q;\\tilde n)$ emerging here in terms of $B(p,q)$\nand $J(p,q)$ and obtain the sought for identities. Due to the terms singular in $\\delta$,\nthe cases $p=1,2,3$ should be considered separately and, \nin the case $p\\leq 0$, one should remember that only the order $O(\\delta)$ part is nontrivial.\n\nIn what follows, we indicate how to use the explicit form of the derived recurrent relations\nfor the ${\\cal B}$ and ${\\cal J}$ functions presented in Appendices 3-11 in order to compute\\footnote{For more detail, \nsee http:\/\/www.lattice.itep.ru\/$\\sim$pbaivid\/lattpt\/ or contact me via e-mai\n}\n$B(p,q)$ and $J(p,q)$ at $p<9$ and any values of $q$.\nNote that some of the relations deal with $B(p,q)$ and $J(p,q)$,\nwhereas the other---with ${\\cal B}(p,q)$ and ${\\cal J}(p,q)$.\nIn the appendices, $B(p,q)$ and $J(p,q)$ are designated by\n{\\tt B(p,q)} and {\\tt J(p,q)}, respectively, whereas ${\\cal B}(p,q)$ and ${\\cal J}(p,q)$\nare designated by {\\tt BB(p,q)} and {\\tt JJ(p,q)}.\n\nBoth the divergent and the finite parts as well as the recurrent relations were \nobtained using the FORM \\cite{Vermaseren} and (partially) REDUCE \\cite{HEARN} packages.\n\n\\subsubsection{${\\cal B}$ functions}\n\nFirst we compute the ${\\cal B}$ functions defined by the formula\n(\\ref{BBandJJat_pleq0}) (see also (\\ref {GdeltaExpansion})). \n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p=0$, $q>0$}\n\\end{itemize}\n\\begin{equation}\n{\\cal B}(0,q,\\mu_B^2) = G_{div}^{M}(q,\\mu_B^2)+J(q), \n\\end{equation}\nwhere $G_{div}^{M}(q)$ is defined by the equation (\\ref{FMR_DivPart_min0})\n($G_{div}^{M}(q)\\neq 0$ only at $q\\geq 2$)\\\\\nand $J(q)$ at $q\\geq 4$---by the recurrent relations\\footnote{Note that $J(q)$\nhas nothing to do with $J(p,q)$ or $J(0,q)$.} (\\ref{FMR_BasBosInt_FinPart_min}).\nThe initial conditions are provided by $J(q)$ (and, therefore, ${\\cal B}(0,q,\\mu_B^2)$) \nat $1 \\leq q \\leq 3$ (see the formulas (\\ref{J0123boson}), (\\ref{IniCondforDIVboson}), \nand (\\ref{BasBosConstNum})). \n\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p=0$, $q\\leq0$}\n\\end{itemize}\n\\begin{equation}\n{\\cal B}(0,q,\\mu_B^2) = {\\cal B}_{-q}, \n\\end{equation}\nwhere ${\\cal B}_{-q}$ are defined in the formula (\\ref{Bdirect})\nand can also be determined from the recurrent relations (\\ref{RecRelForBbosonic})\nwith the initial conditions (\\ref{IniCondForBbosonic}) \n(some values are given in Appendix~1).\n\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p\\leq -1$, $q\\leq 0$}\n\\end{itemize}\nIn this domain, ${\\cal B}(p,q;\\mu_B^2)$ functions\ninvolve no divergencies: ${\\cal B}(p,q;\\mu_B^2) = B(p,q) = {\\cal B}(p,q;0)$. Provided that\n$B(0,q) = {\\cal B}_{-q}$ at $q\\leq 0$ are known, $B(p,q)$ at $p<0, q\\leq 0$\ncan be determined by the recurrent relations \nfor the domain $p<0, q \\geq 4$ (see below), however, with $\\mu_B=0$,\nand, therefore, with $B(p,q)$ instead of ${\\cal B}(p,q)$.\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions over the strip $p\\leq -1$, $1\\leq q \\leq 3$}\n\\end{itemize}\n\nIn this domain, ${\\cal B}(p,q) = B(p,q)$ unless $(p,q)=(-1,3)$.\n\n\\noindent ${\\cal B}$ functions at $-2\\leq p \\leq -1$, $1\\leq q \\leq 3$ can be \ncalculated by the formula \n\\begin{equation}\n{\\cal B}(p,q)=\\int dk\\ { \\Delta_F^{-p}(k,\\mu_B^2) \\over \\Delta^q_B(k,\\mu_B^2)},\n\\end{equation}\nthe result is as follows:\n\\begin{eqnarray}\\label{Binput}\nB(-1,1) &=& 1 + 12 \\;Z_1;\\\\ \\nonumber\nB(-1,2) &=& 4 \\;Z_0;\\\\ \\nonumber\nB(-1,3) &=& {1 \\over (2\\pi)^2}\\ \\left({1\\over 2} + F_0 \\right) \\; + \\; {1\\over 2} \\;Z_0,\n\\qquad \\qquad \\left[{\\cal B} (-1,3) = B(-1,3) - {l_C \\over (2\\pi)^2}\\right]; \\\\[1mm] \\nonumber\nB(-2,1) &=& 188\/3 - \\;{1 \\over (2\\pi)^2}\\; {736\\over 9} - {632\\over 3} \\;Z_1 - {224\\over 3} \\;Z_0;\\\\ \\nonumber\nB(-2,2) &=& - 11 + \\;{24 \\over (2\\pi)^2} + 114 \\;Z_1 + 24 \\;Z_0;\\\\ \\nonumber\nB(-2,3) &=& {3\\over 4} - 3 \\;Z_1 + 6 \\;Z_0. \\nonumber\n\\end{eqnarray}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{picture}(400,400)(0,0)\n\\LongArrow(0,200)(400,200)\n\\LongArrow(200,0)(200,400)\n\\SetColor{Orange}\n\\Line(40,400)(400,40)\n\\SetColor{Blue}\n\\Vertex(260,280){2}\n\\Vertex(240,300){2}\n\\Vertex(260,300){2}\n\\Vertex(220,320){2}\n\\Vertex(240,320){2}\n\\Vertex(260,320){2}\n\\Vertex(200,340){2}\n\\Vertex(220,340){2}\n\\Vertex(240,340){2}\n\\Vertex(260,340){2}\n\\Vertex(200,360){2}\n\\Vertex(220,360){2}\n\\Vertex(240,360){2}\n\\Vertex(260,360){2}\n\\Vertex(200,380){2}\n\\Vertex(220,380){2}\n\\Vertex(240,380){2}\n\\Vertex(260,380){2}\n\\Vertex(20,220){2}\n\\Vertex(40,220){2}\n\\Vertex(60,220){2}\n\\Vertex(80,220){2}\n\\Vertex(100,220){2}\n\\Vertex(20,240){2}\n\\Vertex(40,240){2}\n\\Vertex(60,240){2}\n\\Vertex(80,240){2}\n\\Vertex(100,240){2}\n\\Vertex(20,260){2}\n\\Vertex(40,260){2}\n\\Vertex(60,260){2}\n\\Vertex(80,260){2}\n\\Vertex(100,260){2}\n\\Vertex(200,20){2}\n\\Vertex(220,20){2}\n\\Vertex(240,20){2}\n\\Vertex(260,20){2}\n\\Vertex(200,40){2}\n\\Vertex(220,40){2}\n\\Vertex(240,40){2}\n\\Vertex(260,40){2}\n\\Vertex(200,60){2}\n\\Vertex(220,60){2}\n\\Vertex(240,60){2}\n\\Vertex(260,60){2}\n\\SetColor{Green}\n\\Vertex(20,20){2}\n\\Vertex(40,20){2}\n\\Vertex(60,20){2}\n\\Vertex(80,20){2}\n\\Vertex(100,20){2}\n\\Vertex(120,20){2}\n\\Vertex(140,20){2}\n\\Vertex(160,20){2}\n\\Vertex(180,20){2}\n\\Vertex(20,40){2}\n\\Vertex(40,40){2}\n\\Vertex(60,40){2}\n\\Vertex(80,40){2}\n\\Vertex(100,40){2}\n\\Vertex(120,40){2}\n\\Vertex(140,40){2}\n\\Vertex(160,40){2}\n\\Vertex(180,40){2}\n\\Vertex(20,60){2}\n\\Vertex(40,60){2}\n\\Vertex(60,60){2}\n\\Vertex(80,60){2}\n\\Vertex(100,60){2}\n\\Vertex(120,60){2}\n\\Vertex(140,60){2}\n\\Vertex(160,60){2}\n\\Vertex(180,60){2}\n\\Vertex(20,80){2}\n\\Vertex(40,80){2}\n\\Vertex(60,80){2}\n\\Vertex(80,80){2}\n\\Vertex(100,80){2}\n\\Vertex(120,80){2}\n\\Vertex(140,80){2}\n\\Vertex(160,80){2}\n\\Vertex(180,80){2}\n\\Vertex(20,100){2}\n\\Vertex(40,100){2}\n\\Vertex(60,100){2}\n\\Vertex(80,100){2}\n\\Vertex(100,100){2}\n\\Vertex(120,100){2}\n\\Vertex(140,100){2}\n\\Vertex(160,100){2}\n\\Vertex(180,100){2}\n\\Vertex(20,120){2}\n\\Vertex(40,120){2}\n\\Vertex(60,120){2}\n\\Vertex(80,120){2}\n\\Vertex(100,120){2}\n\\Vertex(120,120){2}\n\\Vertex(140,120){2}\n\\Vertex(160,120){2}\n\\Vertex(180,120){2}\n\\Vertex(20,140){2}\n\\Vertex(40,140){2}\n\\Vertex(60,140){2}\n\\Vertex(80,140){2}\n\\Vertex(100,140){2}\n\\Vertex(120,140){2}\n\\Vertex(140,140){2}\n\\Vertex(160,140){2}\n\\Vertex(180,140){2}\n\\Vertex(20,160){2}\n\\Vertex(40,160){2}\n\\Vertex(60,160){2}\n\\Vertex(80,160){2}\n\\Vertex(100,160){2}\n\\Vertex(120,160){2}\n\\Vertex(140,160){2}\n\\Vertex(160,160){2}\n\\Vertex(180,160){2}\n\\Vertex(20,180){2}\n\\Vertex(40,180){2}\n\\Vertex(60,180){2}\n\\Vertex(80,180){2}\n\\Vertex(100,180){2}\n\\Vertex(120,180){2}\n\\Vertex(140,180){2}\n\\Vertex(160,180){2}\n\\Vertex(180,180){2}\n\\Vertex(20,200){2}\n\\Vertex(40,200){2}\n\\Vertex(60,200){2}\n\\Vertex(80,200){2}\n\\Vertex(100,200){2}\n\\Vertex(120,200){2}\n\\Vertex(140,200){2}\n\\Vertex(160,200){2}\n\\Vertex(180,200){2}\n\\Vertex(20,280){2}\n\\Vertex(40,280){2}\n\\Vertex(60,280){2}\n\\Vertex(80,280){2}\n\\Vertex(100,280){2}\n\\Vertex(120,280){2}\n\\Vertex(140,280){2}\n\\Vertex(160,280){2}\n\\Vertex(180,280){2}\n\\Vertex(20,300){2}\n\\Vertex(40,300){2}\n\\Vertex(60,300){2}\n\\Vertex(80,300){2}\n\\Vertex(100,300){2}\n\\Vertex(120,300){2}\n\\Vertex(140,300){2}\n\\Vertex(160,300){2}\n\\Vertex(180,300){2}\n\\Vertex(20,320){2}\n\\Vertex(40,320){2}\n\\Vertex(60,320){2}\n\\Vertex(80,320){2}\n\\Vertex(100,320){2}\n\\Vertex(120,320){2}\n\\Vertex(140,320){2}\n\\Vertex(160,320){2}\n\\Vertex(180,320){2}\n\\Vertex(20,340){2}\n\\Vertex(40,340){2}\n\\Vertex(60,340){2}\n\\Vertex(80,340){2}\n\\Vertex(100,340){2}\n\\Vertex(120,340){2}\n\\Vertex(140,340){2}\n\\Vertex(160,340){2}\n\\Vertex(180,340){2}\n\\Vertex(20,360){2}\n\\Vertex(40,360){2}\n\\Vertex(60,360){2}\n\\Vertex(80,360){2}\n\\Vertex(100,360){2}\n\\Vertex(120,360){2}\n\\Vertex(140,360){2}\n\\Vertex(160,360){2}\n\\Vertex(180,360){2}\n\\Vertex(20,380){2}\n\\Vertex(40,380){2}\n\\Vertex(60,380){2}\n\\Vertex(80,380){2}\n\\Vertex(100,380){2}\n\\Vertex(120,380){2}\n\\Vertex(140,380){2}\n\\Vertex(160,380){2}\n\\Vertex(180,380){2}\n\\Vertex(280,20){2}\n\\Vertex(300,20){2}\n\\Vertex(320,20){2}\n\\Vertex(340,20){2}\n\\Vertex(360,20){2}\n\\Vertex(380,20){2}\n\\Vertex(280,40){2}\n\\Vertex(300,40){2}\n\\Vertex(320,40){2}\n\\Vertex(340,40){2}\n\\Vertex(360,40){2}\n\\Vertex(380,40){2}\n\\Vertex(280,60){2}\n\\Vertex(300,60){2}\n\\Vertex(320,60){2}\n\\Vertex(340,60){2}\n\\Vertex(360,60){2}\n\\Vertex(380,60){2}\n\\Vertex(280,80){2}\n\\Vertex(300,80){2}\n\\Vertex(320,80){2}\n\\Vertex(340,80){2}\n\\Vertex(360,80){2}\n\\Vertex(380,80){2}\n\\Vertex(280,100){2}\n\\Vertex(300,100){2}\n\\Vertex(320,100){2}\n\\Vertex(340,100){2}\n\\Vertex(360,100){2}\n\\Vertex(380,100){2}\n\\Vertex(280,120){2}\n\\Vertex(300,120){2}\n\\Vertex(320,120){2}\n\\Vertex(340,120){2}\n\\Ve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\n\\Text(240,197)[tr]{2} \n\\Text(260,197)[tr]{3} \n\\Text(280,197)[tr]{4} \n\\Text(300,197)[tr]{5} \n\\Text(320,197)[tr]{6} \n\\Text(340,197)[tr]{7} \n\\Text(360,197)[tr]{8}\n\\Text(380,197)[tr]{9} \n\\Text(198,224)[tr]{1} \n\\Text(198,264)[tr]{3} \n\\Text(198,284)[tr]{4} \n\\Text(198,304)[tr]{5} \n\\Text(198,324)[tr]{6} \n\\Text(198,344)[tr]{7} \n\\Text(198,364)[tr]{8} \n\\Text(198,184)[tr]{-1} \n\\Text(198,164)[tr]{-2} \n\\Text(198,144)[tr]{-3} \n\\Text(198,124)[tr]{-4} \n\\Text(198,104)[tr]{-5} \n\\Text(198,84)[tr]{-6} \n\\Text(196,399)[tr]{\\Large $q$} \n\\Text(400,197)[tr]{\\Large $p$} \n\\end{picture}\n\\end{center}\n\\caption{Values of $(p,q)$ at which the \nfunctions $G(p,q;\\mu_B^2)$ are calculated are shown by dots.\nThe 12 fermionic basic constants are determined from\nthe functions associated with red dots; yellow dots \nshow the constants $X_0 \\div X_3$ that appear in the \nexpressions for $G(p,q;\\mu_B^2)$ but cancel in the\nexpressions for Feynman integrals. Functions shown by gray\nare calculated explicitly and given in the formulas (\\ref{Binput}),\n(\\ref{Ydef1}), (\\ref{Ydef2}), (\\ref{eq:JcrossDOWN}), (\\ref{eq:JcrossLEFT}), \nand (\\ref{eq:JcrossUP}), which provide the initial conditions for the recurrent \nrelations. The order of implementation\nof the recurrent relations is as follows:\n1. ${\\cal B}$ functions are calculated at $p=0$, then at $p<0,q\\leq 0$, then \nover the left strip, then at $p<0, q\\geq 4$.\n2. ${\\cal J}$ functions are calculated in the ``down'' strip, then\nlower-left domain (shown by green), then ``left'' strip, then ``up'' strip\nthen up-left domain, then up-right domain, then down-right domain. }\n\\end{figure}\n\nAt $p\\leq 3$, ${\\cal B}(p,q;\\mu_B^2)$\nfunctions involve no divergencies: ${\\cal B}(p,q;\\mu_B^2) = B(p,q) = {\\cal B}(p,q;0)$. \nThey can be determined by the recurrent relations presented in Appendix~3.\nThe initial conditions are provided by the formulas (\\ref{Binput})\nand $B$ functions calculated previously.\n\nTo express $B(r,1)$ or $B(r,2)$ or $B(r,3)$ at $r\\leq 3$ in terms of $F_0, Z_0$, and $Z_1$, \none should apply these relations beginning with $p=r$ and ending with $p=-3$.\nIn so doing, one must know $B(p,q)$ at $q=0,-1,-2$ and $r\\leq p \\leq 0$\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p\\leq -1$, $q\\geq 4$}\n\\end{itemize}\n${\\cal B}$ functions in this domain \nare computed by the recurrent relations presented in the Appendix~4.\nThe initial conditions for these relations are provided by ${\\cal B}(0,q;\\mu_B^2)$ \nas well as ${\\cal B}(p,q;\\mu_B^2)$ at $p\\leq -1, q\\leq 3$ presented above.\n\n\n\\subsubsection{${\\cal J}$ functions.}\n\nHere, the quantities $J(p,q)$ introduced in (\\ref{GdeltaExpansion})\nare expressed in terms of the boson constants $Z_1$, $Z_0$, $F_0$ \nintroduced in the previous Section, and the quantities $Y_0 \\div Y_{11}$ defined by the relations\n\\begin{eqnarray}\\label{Ydef1}\nY_4={J(1,0)\\over 2}, & \\qquad Y_5 = J(1,-1), & \\qquad Y_6 = 2 J(1,-2), \\\\ \\nonumber\nY_7={J(2,-1)\\over 2}, & \\qquad Y_8 = J(2,-2), & \\qquad Y_9 = {J(3,-2)\\over 2}, \\\\ \\nonumber\nY_{10} = J(3,-3), & \\qquad Y_{11}= 2 J(3,-4), & \\qquad Y_0= {J(2,0)\\over 4}\\; - \\; {F_0 \\over 16\\pi^2}, \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{Ydef2}\nY_1 &=& {1 \\over 48} - {1 \\over 4}\\; Z_0 - {1 \\over 24}\\; J(-1,2) + \n{1 \\over 12}\\; J(0,1) + {1 \\over 12}\\; J(1,0); \\\\ \\nonumber\nY_2 &=& { 1 \\over 6 } - {1\\over \\pi^2} - Z_0 - { 1 \\over 6 }\\; J(-1,2) + \n{1 \\over 3}\\; J(0,1) -{ 1 \\over 24} \\; J(1,-2) - {1 \\over 12} \\; J(1,-1) - \\\\ \\nonumber\n&& - \n{17 \\over 8}\\; J(1,0) + 4\\; J(1,1) - {1 \\over 48}\\; J(2,-2) + {25 \\over 6}\\; J(2,-1)\n - 4\\; J(2,0); \\\\ \\nonumber\nY_3 &=& - {1\\over 384\\pi^2} - F_0\\; {1\\over 128\\pi^2} + {1 \\over 96}\\; Z_0 - \n{1 \\over 48}\\; J(-1,3) + {1 \\over 192} \\; J(0,1) + {1 \\over 48}\\; J(0,2) + {1 \\over 48}\\; J(1,1);\n\\end{eqnarray}\nand the quantities \n\\begin{eqnarray}\nX_0= J(-1,1), && X_1= J(-1,3), \\\\ \\nonumber\nX_2= J(0,0), && X_3=J(0,2). \\nonumber\n\\end{eqnarray}\nas well. The numerical values of the quantities $X_0, X_1, X_2$, and $X_3$\nare not needed because expressions for the integrals (\\ref{eq:GenFermInt}) \nin terms of $G(p,q)$ etc. do not involve them. As for the other constants,\nthey are well known (the table below is taken from the review \\cite{Capitani},\nnotation ${\\cal F}(p,q)$ is borrowed from there):\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n$Y_0$ & $-$ 0.01849765846791657356 \\\\\n$Y_1$ & \\hphantom{$-$} 0.00376636333661866811 \\\\\n$Y_2$ & \\hphantom{$-$} 0.00265395729487879354 \\\\\n$Y_3$ & \\hphantom{$-$} 0.00022751540615147107 \\\\\n$Y_4= {\\cal F}(1,0)$ & \\hphantom{$-$} 0.08539036359532067914 \\\\\n$Y_5= {\\cal F}(1,-1)$ & \\hphantom{$-$} 0.46936331002699614475 \\\\\n$Y_6= {\\cal F}(1,-2)$ & \\hphantom{$-$} 3.39456907367713000586 \\\\\n$Y_7= {\\cal F}(2,-1)$ & \\hphantom{$-$} 0.05188019503901136636 \\\\\n$Y_8= {\\cal F}(2,-2)$ & \\hphantom{$-$} 0.23874773756341478520 \\\\\n$Y_9= {\\cal F}(3,-2)$ & \\hphantom{$-$} 0.03447644143803223145 \\\\\n$Y_{10}= {\\cal F}(3,-3)$ & \\hphantom{$-$} 0.13202727122781293085 \\\\\n$Y_{11}= {\\cal F}(3,-4)$ & \\hphantom{$-$} 0.75167199030295682254 \\\\\n\\hline\n\\end{tabular}\n\\caption{New constants appearing in the general fermionic case.}\n\\label{tab:fermionicconstants}\n\\end{center}\n\\end{table}\n\n\n\\begin{itemize} \n{\\item Thus ${\\cal J}$ functions over the domain\n$\n{\\cal A} = \\left\\{ (p,q): 0 \\leq p \\leq 3,\\ -6 \\leq p \\leq 6-p \\right\\} \\cup $\\\\ \n$\\cup \\left\\{ (p,q): - 4\\leq p \\leq -1,\\ 1\\leq q \\leq 3 \\right\\}\n$\ncalculated by the procedure indicated in \\cite{BCP} can \nbe represented in terms of the above constants as follows:}\n\\end{itemize}\n\\begin{eqnarray}\\label{eq:JcrossDOWN}\n&& J(0,0) = \\; X_2; \\\\ \\nonumber && \n J(0,-1) = 4 \\; X_2 + 315 \\; Y_{10} - 1218 \\; Y_9 - 134 \\; Y_8 + 804 \\; Y_7 - 2 \\; Y_6 \n+ {25 \\over 2} \\; Y_5 - 114 \\; Y_4 + {15 \\over (2\\pi)^2 }; \\\\ \\nonumber && \n J(0,-2) = - 443\/12 + 18 \\; X_2 + 525\/16 \\; Y_{11} + 5265\/4 \\; Y_{10} - 7661\/2 \\; Y_9- 8173\/12 \\; Y_8 \\\\ \\nonumber && \n\\quad + 2085 \\; Y_7 - 7\/48 \\; Y_6 + 1657\/24 \\; Y_5 - 339\/2 \\; Y_4 + 35\/4 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(0,-3) = - 1549\/27 + 88\\; X_2 + 595\/4\\; Y_{11} + 92395\/6 \\; Y_{10} - 487883\/9\\; Y_9 - 197384\/27 \\; Y_8 \\\\ \\nonumber && \n\\quad + 103330\/3 \\; Y_7 - 6569\/108 \\; Y_6 + 57949\/108 \\; Y_5 - 12829\/3 \\; Y_4 + 9029\/18 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(0,-4) = - 1024087\/576 + 917\/2 \\; X_2 + 1919221\/768 \\; Y_{11} + 7343317\/192 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 20889553\/288 \\; Y_9 - 14398667\/576 \\; Y_8 + 3075865\/144 \\; Y_7 + 325123\/2304 \\; Y_6 \\\\ \\nonumber && \n\\quad + 2538989\/1152 \\; Y_5 + 501607\/96 \\; Y_4 - 910433\/576 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(0,-5) = 22816157\/32400 + 2514 \\; X_2 + 14234765\/4032 \\; Y_{11} + 6270202009\/8400 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 171277520509\/63000 \\; Y_9 - 11273402747\/32400 \\; Y_8 + 33415313987\/18900 \\; Y_7 \\\\ \\nonumber && \n\\quad - 402215297\/129600 \\; Y_6 + 52200958189\/2268000 \\; Y_5 - 4825909001\/21000 \\; Y_4 \\\\ \\nonumber && \n\\quad + 3567631667\/126000 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(0,-6) = - 59060175671\/583200 + 14376 X_2 - 456897556151\/2268000\\; (2pi)^{-2} \\\\ \\nonumber && \n\\quad + 290748296317\/1814400 Y_{11} - 46846789343\/252000 Y_{10} + 1099242273317\/226800\\; Y_9 \\\\ \\nonumber && \n\\quad - 1709413667089\/4082400\\; Y_8 - 8051167629571\/1701000\\; Y_7 + 41141268191\/2332800\\; Y_6 \\\\ \\nonumber && \n\\quad + 1776831395699\/40824000\\; Y_5 + 61514640887\/54000\\ Y_4; \\\\ \\nonumber && \n J(1,0) = 2 \\; Y_4; \\\\ \\nonumber && \n J(1,-1) = \\; Y_5; \\\\ \\nonumber && \n J(1,-2) = 1\/2 \\; Y_6; \\\\ \\nonumber && \n J(1,-3) = 52\/3 + 683\/2 \\; Y_{10} - 1342 \\; Y_9 - 439\/3 \\; Y_8 + 924 \\; Y_7\\\\ \\nonumber && \n\\quad - 13\/3 \\; Y_6 + 89\/12 \\; Y_5 - 132 \\; Y_4 + 17 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(1,-4) = - 761\/9 + 683\/12 \\; Y_{11} - 10807\/3 \\; Y_{10} + 48538\/3 \\; Y_9 + 12539\/9 \\; Y_8 \\\\ \\nonumber && \n\\quad - 35284\/3 \\; Y_7 + 1481\/36 \\; Y_6 - 887\/18 \\; Y_5 + 1874 \\; Y_4 - 803\/3 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(1,-5) = 34745\/27 - 52147\/63 \\; Y_{11} + 6069519\/140 \\; Y_{10} - 27087257\/140 \\; Y_9 \\\\ \\nonumber && \n\\quad - 450260\/27 \\; Y_8 + 88135577\/630 \\; Y_7 - 42845\/108 \\; Y_6 + 2730857\/3780 \\; Y_5 \\\\ \\nonumber &&\n\\quad - 3125867\/140 \\; Y_4 + 541505\/168 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(1,-6) = - 3470071\/324 + 123774917\/10800 \\; Y_{11} - 15164760473\/31500 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 34075577647\/15750 \\; Y_9 + 10210300231\/56700 \\; Y_8 - 36997574549\/23625 \\; Y_7 \\\\ \\nonumber && \n\\quad + 5205091\/1296 \\; Y_6 - 4977314411\/567000 \\; Y_5 + 1327628299\/5250 \\; Y_4 \\\\ \\nonumber && \n\\quad - 232435501\/6300 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(2,0) = 4 \\; Y_0 + F_0 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(2,-1) = 2 \\; Y_7; \\\\ \\nonumber && \n J(2,-2) = \\; Y_8; \\\\ \\nonumber && \n J(2,-3) = - \\; Y_{10} + 90 \\; Y_9 - 2 \\; Y_8 - 84 \\; Y_7 + 5\/2 \\; Y_5 + 18 \\; Y_4 - 3 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(2,-4) = - 13\/3 - 1\/4 \\; Y_{11} + \\; Y_{10} - 326 \\; Y_9 + 13\/3 \\; Y_8 + 300 \\; Y_7 + 31\/12 \\; Y_6 \\\\ \\nonumber && \n\\quad - 19\/6 \\; Y_5 - 66 \\; Y_4 + 13 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(2,-5) = 1012\/9 + 53\/63 \\; Y_{11} + 519787\/210 \\; Y_{10} - 316251\/35 \\; Y_9 - 3217\/3 \\; Y_8 \\\\ \\nonumber && \n\\quad + 214918\/35 \\; Y_7 - 343\/9 \\; Y_6 + 17621\/420 \\; Y_5 - 29051\/35 \\; Y_4 + 3859\/42 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(2,-6) = - 48007\/36 + 8466287\/15120 \\; Y_{11} - 272218199\/6300 \\; Y_{10} + 593087111\/3150 \\; Y_9 \\\\ \\nonumber && \n\\quad + 64920421\/3780 \\; Y_8 - 30568297\/225 \\; Y_7 + 70867\/144 \\; Y_6 - 22014731\/37800 \\; Y_5 \\\\ \\nonumber && \n\\quad + 7424029\/350 \\; Y_4 - 3722933\/1260 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(3,0) = - 35\/256 \\; Y_{11} - 57\/64 \\; Y_{10} - 301\/2304 \\; Y_9 + 85\/96 \\; Y_8 + 1405\/1152 \\; Y_7\\\\ \\nonumber && \n\\quad - 1\/384 \\; Y_5 - 461\/768 \\; Y_4 - 2 \\; Y_0 + (1433\/1536 - 1\/2 F_0) \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(3,-1) = 155\/96 \\; Y_9 - 83\/48 \\; Y_7 + 11\/32 \\; Y_4 + 4 \\; Y_0 + (F_0 - 31\/64 )\\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(3,-2) = 2 \\; Y_9; \\\\ \\nonumber && \n J(3,-3) = \\; Y_{10}; \\\\ \\nonumber && \n J(3,-4) = 1\/2 \\; Y_{11}; \\\\ \\nonumber && \n J(3,-5) = - 85\/42 \\; Y_{11} - 1649\/140 \\; Y_{10} + 15548\/35 \\; Y_9 + 7\/6 \\; Y_8 - 14264\/35 \\; Y_7 \\\\ \\nonumber && \n\\quad + 5609\/840 \\; Y_5 + 2943\/35 \\; Y_4 - 106\/7 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(3,-6) = - 224\/9 + 2812\/945 \\; Y_{11} + 47669\/1575 \\; Y_{10} - 595726\/225 \\; Y_9 + 6854\/945 \\; Y_8 \\\\ \\nonumber && \n\\quad + 3804196\/1575 \\; Y_7 + 101\/9 \\; Y_6 - 173539\/9450 \\; Y_5 - 90298\/175 \\; Y_4 + 33203\/315 \\; (2\\pi)^{-2}; \\nonumber \n\\end{eqnarray} \n\\begin{eqnarray}\\label{eq:JcrossLEFT}\n&& J(-1,1) = \\; X_0; \\\\ \\nonumber && \n J(-2,1) = - 1135\/9 - 158\/9 \\; X_0 + 448\/3 \\; X_3 + 722\/9 \\; X_2 - 448\/3 \\; X_1 + 1295\/12 \\; Y_{11} \\\\ \\nonumber && \n\\quad + 56237\/9 \\; Y_{10} - 61066\/3 \\; Y_9 - 83036\/27 \\; Y_8 + 108068\/9 \\; Y_7 - 283\/18 \\; Y_6 \\\\ \\nonumber && \n\\quad + 15217\/54 \\; Y_5 - 1274 \\; Y_4 - 19456\/3 \\; Y_3 - 32\/3 \\; Y_2 + 1472\/3 \\; Y_1 + 896\/3 \\; Y_0 \\\\ \\nonumber && \n\\quad + 865\/9 \\; (2\\pi)^{-2} + 64\/9 F_0 \\; (2\\pi)^{-2} + 536\/3 \\; Z_1 + 5488\/27 \\; Z_0; \\\\ \\nonumber && \n J(-3,1) = 11845003\/15000 + 105068\/375 \\; X_0 - 371584\/125 \\; X_3 - 200897\/375 \\; X_2 \\\\ \\nonumber && \n\\quad + 371584\/125 \\; X_1 + 204127\/160 \\; Y_{11} + 19090453\/600 \\; Y_{10} - 133042333\/1500 \\; Y_9 \\\\ \\nonumber && \n\\quad - 175667117\/9000 \\; Y_8+ 39223253\/750 \\; Y_7 - 214589\/12000 \\; Y_6 + 8996881\/18000 \\; Y_5 \\\\ \\nonumber && \n\\quad - 207851\/60 \\; Y_4 + 15388672\/125 \\; Y_3 + 27568\/125 \\; Y_2 - 1398848\/125 \\; Y_1 \\\\ \\nonumber && \n\\quad - 640512\/125 \\; Y_0 + 6221759\/15000 \\; (2\\pi)^{-2} - 153472\/375 F_0 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 2696248\/625 \\; Z_1 - 24764192\/5625 \\; Z_0; \\\\ \\nonumber && \n J(-4,1) = - 3329832752387\/64827000 - 117549296\/25725 \\; X_0 + 446453248\/8575 \\; X_3 \\\\ \\nonumber && \n\\quad + 566923184\/25725 \\; X_2 - 446453248\/8575 \\; X_1 + 737380289\/14112 \\; Y_{11} \\\\ \\nonumber && \n\\quad + 242892361651\/123480 \\; Y_{10} - 5552574235627\/926100 \\; Y_9 \\\\ \\nonumber && \n\\quad - 640505149891\/617400 \\; Y_8 + 1563368068291\/463050 \\; Y_7 - 23776321823\/7408800 \\; Y_6 \\\\ \\nonumber && \n\\quad + 321309701383\/3704400 \\; Y_5 - 89421147379\/308700 \\; Y_4 - 17930338304\/8575 \\; Y_3 \\\\ \\nonumber && \n\\quad - 92940608\/25725 \\; Y_2 + 1782287616\/8575 \\; Y_1 + 2138417152\/25725 \\; Y_0 \\\\ \\nonumber && \n\\quad + 991072905679\/64827000 \\; (2\\pi)^{-2} + 4988416\/525 F_0 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad + 25248697568\/300125 \\; Z_1 + 73082913664\/900375 \\; Z_0; \\\\ \\nonumber && \n J(-1,2) = 2\/3 - 8 \\; X_3 + 8\\; X_1 - 1\/24\\; Y_8 + 50\/3 \\; Y_7 - 1\/24\\; Y_6 - 1\/6\\; Y_5 - 35\/6 \\; Y_4 \\\\ \\nonumber && \n\\quad + 384 \\; Y_3 - 2 \\; Y_2 - 16 \\; Y_1 - 32 \\; Y_0 + 4 (F_0 - 1) \\; (2\\pi)^{-2} - 10 \\; Z_0; \\\\ \\nonumber && \n J(-2,2) = 135\/4 + 19\/2 \\; X_0 - 48 \\; X_3 - 41\/2 \\; X_2 + 48 \\; X_1 - 105\/16\\; Y_{11} - 1851\/4 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 3075\/2 \\; Y_9 + 827\/4 \\; Y_8 - 841 \\; Y_7 + 11\/8 \\; Y_6 - 269\/8 \\; Y_5 + 175\/2 \\; Y_4 + 2304 \\; Y_3 \\\\ \\nonumber && \n\\quad + 12 \\; Y_2 - 96 \\; Y_1 - 96 \\; Y_0 - 45\/4 \\; (2\\pi)^{-2} - 114 \\; Z_1 - 76 \\; Z_0; \\\\ \\nonumber && \n J(-3,2) = - 29723\/16 - 2365\/9 \\; X_0 + 7712\/3 \\; X_3 + 32881\/36 \\; X_2 - 7712\/3 \\; X_1 \\\\ \\nonumber && \n\\quad + 56805\/64 \\; Y_{11} + 6355039\/144 \\; Y_{10} - 5035139\/36 \\; Y_9 - 4682939\/216 \\; Y_8 \\\\ \\nonumber && \n\\quad + 1411847\/18 \\; Y_7 - 4543\/48 \\; Y_6 + 2051831\/864 \\; Y_5 - 68044\/9 \\; Y_4 - 330752\/3 \\; Y_3\\\\ \\nonumber && \n\\quad - 252 \\; Y_2 + 26992\/3 \\; Y_1 + 13952\/3 \\; Y_0 + (54773\/72 + 2336\/9 F_0) \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad + 13706\/3 \\; Z_1 + 111704\/27 \\; Z_0; \\\\ \\nonumber && \n J(-4,2) = 492009959\/12000 + 87748\/15 \\; X_0 - 324224\/5 \\; X_3 - 275752\/15 \\; X_2 + 324224\/5 \\; X_1 \\\\ \\nonumber && \n\\quad - 3344999\/384 \\; Y_{11} - 240354739\/480 \\; Y_{10} + 5720706571\/3600 \\; Y_9 \\\\ \\nonumber && \n\\quad + 329440181\/1440 \\; Y_8 - 302115271\/360 \\; Y_7 + 2092337\/1920 \\; Y_6 - 518442119\/14400 \\; Y_5 \\\\ \\nonumber && \n\\quad + 285938617\/3600 \\; Y_4 + 2650112 \\; Y_3 + 75952\/15 \\; Y_2 - 1259712\/5 \\; Y_1 - 318976\/3 \\; Y_0 \\\\ \\nonumber && \n\\quad - (423392089\/36000 + 159872\/15 F_0) \\; (2\\pi)^{-2} - 44526248\/375 \\; Z_1 - 40322528\/375 \\; Z_0; \\\\ \\nonumber && \n J(-1,3) = \\; X_1; \\\\ \\nonumber && \n J(-2,3) = 1\/12 - 1\/4 \\; X_0 - 12 \\; X_3 + \\; X_2 + 12 \\; X_1 + 105\/2 \\; Y_{10} - 203 \\; Y_9 - 1075\/48 \\; Y_8 \\\\ \\nonumber && \n + 159 \\; Y_7 - 19\/48 \\; Y_6 + 23\/12 \\; Y_5 - 103\/4 \\; Y_4 + 576 \\; Y_3 - 7 \\; Y_2 - 60 \\; Y_1 \\\\ \\nonumber && \n - 48 \\; Y_0 - 7\/2 \\; (2\\pi)^{-2} + 6 F_0 \\; (2\\pi)^{-2} + 12 \\; Z_1 - 19 \\; Z_0; \\\\ \\nonumber && \n J(-3,3) = 871\/2 + 629\/9 \\; X_0 - 1696\/3 \\; X_3 - 1778\/9 \\; X_2 + 1696\/3 \\; X_1 - 175\/2 \\; Y_{11} \\\\ \\nonumber && \n - 50840\/9 \\; Y_{10} + 167119\/9 \\; Y_9 + 279089\/108 \\; Y_8 - 91946\/9 \\; Y_7 + 187\/12 \\; Y_6 \\\\ \\nonumber && \n - 10154\/27 \\; Y_5 + 9446\/9 \\; Y_4 + 77824\/3 \\; Y_3 + 116 \\; Y_2 - 4784\/3 \\; Y_1 - 3136\/3 \\; Y_0 \\\\ \\nonumber && \n - 1739\/6 \\; (2\\pi)^{-2} - 304\/9 F_0 \\; (2\\pi)^{-2} - 1502 \\; Z_1 - 28456\/27 \\; Z_0; \\\\ \\nonumber && \n J(-4,3) = - 40263778\/1875 - 2997542\/1125 \\; X_0 + 10588096\/375 \\; X_3 + 20165461\/2250 \\; X_2 \\\\ \\nonumber && \n - 10588096\/375 \\; X_1 + 602637\/80 \\; Y_{11} + 161298737\/450 \\; Y_{10} \\\\ \\nonumber && \n - 5056485013\/4500 \\; Y_9 - 4717685677\/27000 \\; Y_8 + 1381579093\/2250 \\; Y_7 \\\\ \\nonumber && \n - 8668453\/12000 \\; Y_6 + 284825729\/13500 \\; Y_5 - 50984761\/900 \\; Y_4 \\\\ \\nonumber && \n - 444755968\/375 \\; Y_3 - 347064\/125 \\; Y_2 + 39247712\/375 \\; Y_1 + 17916928\/375 \\; Y_0 \\\\ \\nonumber && \n + 182925853\/15000 \\; (2\\pi)^{-2} + 4427968\/1125 F_0 \\; (2\\pi)^{-2} + 38504584\/625 \\; Z_1 \\\\ \\nonumber && \n + 860640608\/16875 \\; Z_0; \\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:JcrossUP}\n&& J(0,1) = 1\/12 - 4 \\; X_3 + 4 \\; X_1 - 1\/48 \\; Y_8 + 25\/3 \\; Y_7 - 1\/48 \\; Y_6 - 1\/12 \\; Y_5 - 59\/12 \\; Y_4 \\\\ \\nonumber && \n\\quad + 192 \\; Y_3 - \\; Y_2 + 4 \\; Y_1 - 16 \\; Y_0 + (2 F_0 - 2) \\; (2\\pi)^{-2} - 2 \\; Z_0; \\\\ \\nonumber && \n J(0,2) = \\; X_3; \\\\ \\nonumber && \n J(0,3) = 91\/1024 + 1\/384 \\; X_0 + 1\/4 \\; X_3 - 1\/96 \\; X_2 - 35\/768 \\; Y_{11} - 27\/32 \\; Y_{10} + 109\/72 \\; Y_9 \\\\ \\nonumber && \n\\quad + 19229\/36864 \\; Y_8 + 383\/2304 \\; Y_7 - 11\/4096 \\; Y_6 - 443\/9216 \\; Y_5 - 1243\/3072 \\; Y_4 \\\\ \\nonumber && \n\\quad - 3\/2 \\; Y_3 - 33\/256 \\; Y_2 + 17\/64 \\; Y_1 - 33\/16 \\; Y_0 - 13\/72 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 5\/48 F_0 \\; (2\\pi)^{-2} - 1\/4 \\; Z_1 - 49\/192 \\; Z_0; \\\\ \\nonumber && \n J(0,4) = 80441\/2211840 + 31\/27648 \\; X_0 + 17\/288 \\; X_3 - 31\/6912 \\; X_2 + 1\/288 \\; X_1 \\\\ \\nonumber && \n\\quad + 161\/24576 \\; Y_{11} - 17767\/92160 \\; Y_{10} + 23\/32 \\; Y_9 + 161077\/2949120 \\; Y_8 \\\\ \\nonumber && \n\\quad - 10667\/552960 \\; Y_7 - 14369\/8847360 \\; Y_6 - 16271\/737280 \\; Y_5 - 343291\/2211840 \\; Y_4 \\\\ \\nonumber && \n\\quad - 1379\/1440 \\; Y_3 - 14369\/184320 \\; Y_2 + 9409\/46080 \\; Y_1 - 8609\/11520 \\; Y_0 \\\\ \\nonumber && \n\\quad +( 539\/17280 F_0 - 18031\/86400 ) \\; (2\\pi)^{-2} - 237\/2560 \\; Z_1 - 965\/9216 \\; Z_0; \\\\ \\nonumber && \n J(0,5) = 7632781\/594542592 + 523\/1327104 \\; X_0 + 209\/13824 \\; X_3 - 523\/331776 \\; X_2 \\\\ \\nonumber && \n\\quad + 25\/13824 \\; X_1 + 77651\/42467328 \\; Y_{11} - 1253633\/17694720 \\; Y_{10} + 56974703\/ \\\\ \\nonumber && \n\\quad 159252480 \\; Y_9 + 782803057\/35672555520 \\; Y_8 + 34653559\/2229534720 \\; Y_7 - \\\\ \\nonumber && \n\\quad 10633541\/11890851840 \\; Y_6 - 80793859\/8918138880 \\; Y_5 - 291120383\/2972712960 \\; Y_4 \\\\ \\nonumber && \n\\quad - 262597\/645120 \\; Y_3 - 10633541\/247726080 \\; Y_2 + 8290501\/61931520 \\; Y_1 - 5150021\/15482880 \\; Y_0\\\\ \\nonumber && \\quad + (43529\/5806080 F_0 - 8543128133\/78033715200 )\\; (2\\pi)^{-2} \\\\ \\nonumber &&\n\\quad - 569479\/20643840 \\; Z_1 - 2292133\/61931520 \\; Z_0; \\\\ \\nonumber && \n J(0,6) = 1133309347\/237817036800 + 1429\/10616832 \\; X_0 + 2371\/552960 \\; X_3 - 1429\/2654208 \\; X_2 \\\\ \\nonumber && \n\\quad + 401\/552960 \\; X_1 + 10571051\/10192158720 \\; Y_{11} - 639402031\/29727129600 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 48006407249\/267544166400 \\; Y_9 + 5666277343\/1223059046400 \\; Y_8 + 2239163267\/107017666560 \\; Y_7 \\\\ \\nonumber && \\quad - 470661551\/951268147200 \\; Y_6 - 1645375871\/428070666240 \\; Y_5 - 40565004503\/713451110400 \\; Y_4 \\\\ \\nonumber && \\quad - 23258381\/154828800 \\; Y_3 - 470661551\/19818086400 \\; Y_2 + 406642351\/4954521600 \\; Y_1 \\\\ \\nonumber && \n \\quad - 153477551\/1238630400 \\; Y_0 \n+( 1169977\/232243200 F_0 - 1234692078509\/18728091648000)\\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 117647549\/14863564800 \\; Z_1 - 202376477\/14863564800 \\; Z_0; \\\\ \\nonumber &&\n J(1,1) = - 1\/48 + 1\/192 \\; Y_8 - 25\/12 \\; Y_7 + 1\/192 \\; Y_6 + 1\/48 \\; Y_5 + 59\/48 \\; Y_4\\\\ \\nonumber && \n\\quad + 1\/4 \\; Y_2 - \\; Y_1 + 4 \\; Y_0 + (1 + F_0) \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(1,2) = - 19\/1536 - 35\/768 \\; Y_{11} - 19\/64 \\; Y_{10} - 173\/288 \\; Y_9 + 5497\/18432 \\; Y_8 \\\\ \\nonumber && \n\\quad - 293\/1152 \\; Y_7 + 19\/6144 \\; Y_6 + 53\/4608 \\; Y_5 + 187\/512 \\; Y_4 + \\; Y_3 + 19\/128 \\; Y_2 \\\\ \\nonumber && \n\\quad - 19\/32 \\; Y_1 + 3\/8 \\; Y_0 + 307\/576 \\; (2\\pi)^{-2} + 1\/96 \\; Z_0; \\\\ \\nonumber && \n J(1,3) = - 5491\/737280 + 721\/73728 \\; Y_{11} + 1957\/30720 \\; Y_{10} - 21361\/55296 \\; Y_9 \\\\ \\nonumber && \n\\quad - 542443\/8847360 \\; Y_8 - 266629\/552960 \\; Y_7 + 5959\/2949120 \\; Y_6 + 18289\/2211840 \\; Y_5 \\\\ \\nonumber && \n\\quad + 274901\/737280 \\; Y_4 + 149\/480 \\; Y_3 + 5959\/61440 \\; Y_2 - 5959\/15360 \\; Y_1 \\\\ \\nonumber && \n\\quad + 1799\/3840 \\; Y_0 + (438551\/2764800 + 11\/160 F_0) \\; (2\\pi)^{-2} - 13\/5120 \\; Z_1 + 35\/3072 \\; Z_0; \\\\ \\nonumber && J(1,4) = - 135181\/27525120 - 1519\/3538944 \\; Y_{11} - 4123\/1474560 \\; Y_{10}- 2532143\/13271040 \\; Y_9 \\\\ \\nonumber && \n\\quad + 12187051\/2972712960 \\; Y_8 - 12605351\/37158912 \\; Y_7 + 1307897\/990904320 \\; Y_6 \\\\ \\nonumber && \n\\quad + 783523\/148635648 \\; Y_5 + 18292649\/82575360 \\; Y_4 - 533\/161280 \\; Y_3 \\\\ \\nonumber && \n\\quad + 1307897\/20643840 \\; Y_2 - 1307897\/5160960 \\; Y_1 + 93817\/1290240 \\; Y_0 \\\\ \\nonumber && \n\\quad + (346641047\/2167603200 - 1\/1680 F_0) \\; (2\\pi)^{-2} - 22817\/15482880 \\; Z_1 + 146059\/15482880 \\; Z_0; \\\\ \\nonumber && \n J(1,5) = - 32271257\/9512681472 + 9401131\/2038431744 \\; Y_{11} + 178621489\/5945425920 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 1051530899\/53508833280 \\; Y_9 - 49631503063\/1712282664960 \\; Y_8 \\\\ \\nonumber && \n\\quad - 5652301063\/15288238080 \\; Y_7 + 167849953\/190253629440 \\; Y_6 \\\\ \\nonumber && \n\\quad + 1548254101\/428070666240 \\; Y_5 + 26366380697\/142690222080 \\; Y_4 \\\\ \\nonumber && \n\\quad - 4481117\/30965760 \\; Y_3 + 167849953\/3963617280 \\; Y_2 - 167849953\/990904320 \\; Y_1 \\\\ \\nonumber && \n\\quad + 9544673\/247726080 \\; Y_0 + (260638894247\/3745618329600 + 1339\/215040 F_0) \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 541139\/990904320 \\; Z_1 + 806521\/110100480 \\; Z_0; \\\\ \\nonumber && \n J(2,1) = 23\/4608 - 35\/384 \\; Y_{11} - 19\/32 \\; Y_{10} - 173\/144 \\; Y_9 + 3619\/6144 \\; Y_8 \\\\ \\nonumber && \n\\quad + 2827\/1152 \\; Y_7 - 23\/18432 \\; Y_6 - 31\/4608 \\; Y_5 - 3949\/4608 \\; Y_4 + \\; Y_3 - 23\/384 \\; Y_2 \\\\ \\nonumber && \n\\quad + 23\/96 \\; Y_1 - 19\/24 \\; Y_0 + ( 127\/288 - 1\/4 F_0) \\; (2\\pi)^{-2} - 1\/96 \\; Z_0; \\\\ \\nonumber && \n J(2,2) = 7043\/1105920 + 343\/12288 \\; Y_{11} + 931\/5120 \\; Y_{10} - 3731\/4608 \\; Y_9 \\\\ \\nonumber && \n\\quad - 806147\/4423680 \\; Y_8 + 339959\/276480 \\; Y_7 - 6467\/4423680 \\; Y_6 - 5879\/1105920 \\; Y_5 \\\\ \\nonumber && \n\\quad - 365953\/1105920 \\; Y_4 + 221\/240 \\; Y_3 - 6467\/92160 \\; Y_2 + 6467\/23040 \\; Y_1 \\\\ \\nonumber && \n\\quad + 4373\/5760 \\; Y_0 + (23\/160 F_0 - 25657\/76800) \\; (2\\pi)^{-2} - 1\/480 \\; Z_1 - 7\/512 \\; Z_0; \\\\ \\nonumber && \n J(2,3) = 4611371\/743178240 - 73045\/3538944 \\; Y_{11} - 39653\/294912 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 12722017\/13271040 \\; Y_9 + 393115913\/2972712960 \\; Y_8 + 103603669\/61931520 \\; Y_7 \\\\ \\nonumber && \n\\quad - 4248887\/2972712960 \\; Y_6 - 504563\/82575360 \\; Y_5 - 79536305\/148635648 \\; Y_4 \\\\ \\nonumber && \n\\quad + 155081\/161280 \\; Y_3 - 4248887\/61931520 \\; Y_2 + 4248887\/15482880 \\; Y_1 \\\\ \\nonumber && \n\\quad + 66473\/3870720 \\; Y_0 + 493568683\/6502809600 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 1093\/26880 F_0 \\; (2\\pi)^{-2} - 10069\/5160960 \\; Z_1 - 74813\/5160960 \\; Z_0; \\\\ \\nonumber && \n J(2,4) = 196602193\/35672555520 + 3693635\/509607936 \\; Y_{11} + 14035813\/297271296 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 12442566511\/13377208320 \\; Y_9 - 20639077343\/428070666240 \\; Y_8 \\\\ \\nonumber && \n\\quad + 36967799143\/26754416640 \\; Y_7 - 181900981\/142690222080 \\; Y_6 \\\\ \\nonumber && \n\\quad - 530928403\/107017666560 \\; Y_5 - 2755058051\/7134511104 \\; Y_4 + 2424601\/2580480 \\; Y_3 \\\\ \\nonumber && \n\\quad - 181900981\/2972712960 \\; Y_2 + 181900981\/743178240 \\; Y_1 + 46110859\/185794560 \\; Y_0 \\\\ \\nonumber && \n\\quad - 79216495853\/936404582400 \\; (2\\pi)^{-2} + 4141\/215040 F_0 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 408367\/247726080 \\; Z_1 - 125077\/9175040 \\; Z_0; \\\\ \\nonumber && \n J(3,1) = - 2473\/2211840 + 1687\/24576 \\; Y_{11} + 4579\/10240 \\; Y_{10} + 28153\/27648 \\; Y_9 \\\\ \\nonumber && \n\\quad - 3930323\/8847360 \\; Y_8 - 866969\/552960 \\; Y_7 + 2797\/8847360 \\; Y_6 + 5689\/2211840 \\; Y_5 \\\\ \\nonumber && \n\\quad + 1137263\/2211840 \\; Y_4 - 211\/480 \\; Y_3 + 2797\/184320 \\; Y_2 - 2797\/46080 \\; Y_1 \\\\ \\nonumber && \n\\quad + 12317\/11520 \\; Y_0 + ( 23\/80 F_0 - 323129\/460800) \\; (2\\pi)^{-2} - 3\/5120 \\; Z_1 + 11\/3072 \\; Z_0; \\\\ \\nonumber && \n J(3,2) = - 326465\/148635648 - 206605\/3538944 \\; Y_{11} - 112157\/294912 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 2493365\/2654208 \\; Y_9 + 225122965\/594542592 \\; Y_8 - 27027877\/37158912 \\; Y_7 \\\\ \\nonumber && \n\\quad + 336725\/594542592 \\; Y_6 + 19049\/16515072 \\; Y_5 + 6739903\/148635648 \\; Y_4 \\\\ \\nonumber && \n\\quad - 22955\/32256 \\; Y_3 + 336725\/12386304 \\; Y_2 - 336725\/3096576 \\; Y_1 - 483563\/774144 \\; Y_0 \\\\ \\nonumber && \n\\quad + (429664553\/1300561920 - 331\/2688 F_0) \\; (2\\pi)^{-2} - 95\/344064 \\; Z_1 + 6823\/1032192 \\; Z_0; \\\\ \\nonumber && \n J(3,3) = - 69355327\/23781703680 + 4168243\/339738624 \\; Y_{11} + 79196617\/990904320 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 3687114839\/2972712960 \\; Y_9 - 22463493247\/285380444160 \\; Y_8 \\\\ \\nonumber && \n\\quad - 1869533095\/1189085184 \\; Y_7 + 70582891\/95126814720 \\; Y_6 \\\\ \\nonumber && \n\\quad + 45684329\/14269022208 \\; Y_5 + 362000483\/880803840 \\; Y_4 - 1586311\/1720320 \\; Y_3 \\\\ \\nonumber && \n\\quad + 70582891\/1981808640 \\; Y_2 - 70582891\/495452160 \\; Y_1 + 7231211\/123863040 \\; Y_0 \\\\ \\nonumber && \n\\quad - 131572132177\/624269721600 \\; (2\\pi)^{-2} + 12241\/215040 F_0 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 34099\/165150720 \\; Z_1 + 1460153\/165150720 \\; Z_0; \\nonumber \n\\end{eqnarray}\n\n\\begin{itemize} \n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $0\\leq p\\leq 3$, $q\\leq -7$}\n\\end{itemize}\n\nProvided that ${\\cal J}(p,q;\\mu_B^2)$ over the domain ${\\cal A}$ (at $q\\leq 0$)\nand $B(0,q)$ at $q\\leq 0$ are known, ${\\cal J}$ functions over this strip \ncan be found by the recurrent relations in Appendix~5.\nNote that there are no divergent terms here.\n\n\\begin{itemize} \n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ p\\leq -1$, $q\\leq 0$}\n\\end{itemize}\n\nGiven ${\\cal J}$ functions over the strip $0\\leq p \\leq 3,\\ q\\leq 0$\nand ${\\cal B}$ functions at $p\\leq 0, q \\leq 0$,\n${\\cal J}$ functions at $ p\\leq -1$, $q\\leq 0$ can be\ndetermined using the recurrent relations\nin Appendix~8, however, with $\\mu_B=0$ (in this domain, ${\\cal J}(p,q;\\mu_B^2)=J(p,q)$).\n\n\\begin{itemize} \n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ p\\leq -5$, $1 \\leq q\\leq 3$}.\n\\end{itemize}\n\nThey can be determined using the recurrent relations presented in Appendix~6.\nThe initial conditions are provided by the formulas (\\ref{eq:JcrossLEFT})\nand $B$ functions determined previously.\nIn calculation of $J(p,q)$ one needs $B(r,s)$ at $r\\geq p$ and $-3\\leq q \\leq 3$.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ 0\\leq p\\leq 3$ and $q\\geq 6-p$.}\n\\end{itemize}\n\nIn this domain, the divergent part $D(p,q)$ is calculated separately,\nthe results are partially presented in (\\ref{DivPart_PP_min_domain});\nthus we calculate the functions $J(p,q)$ by the formulas\n\n\\begin{equation}\nJ(n,q+1-n) = { 1 \\over (2q-1)(2q-3)} \\sum_{k=0}^3 M(n,k,q) Z_k(q),\n\\end{equation}\n\nwhere $M(n,k,q)$ are given by \n\\begin{eqnarray}\nM(0,0,q) &=&(q + 1)\/32\/(q - 1); \\\\ \\nonumber\nM(0,1,q) &=& -(8q-3)(q-2)\/32\/q\/(q - 1); \\\\ \\nonumber\nM(0,2,q) &=&-(q-2)\/4\/q\/(q - 1); \\\\ \\nonumber\nM(0,3,q) &=&-(q-2)\/2\/q\/(q-1)^2; \\\\ \\nonumber\nM(1,0,q) &=&(q + 1)\/32; \\\\ \\nonumber\nM(1,1,q) &=&3\/32; \\\\ \\nonumber\nM(1,2,q) &=& -(q-2)\/4\/q; \\\\ \\nonumber\nM(1,3,q) &=& -(q-2)\/2\/q\/(q - 1); \\\\ \\nonumber\nM(2,0,q) &=& q(q + 1)\/32; \\\\ \\nonumber\nM(2,1,q) &=& 3q\/32; \\\\ \\nonumber\nM(2,2,q) &=&3\/16\/q; \\\\ \\nonumber\nM(2,3,q) &=& -(q-2)\/2\/(q - 1); \\\\ \\nonumber\nM(3,0,q) &=& q(q+1)^2\/64; \\\\ \\nonumber\nM(3,1,q) &=&3q(q + 1)\/64; \\\\ \\nonumber\nM(3,2,q) &=&3(q + 1)\/32\/q; \\\\ \\nonumber\nM(3,3,q) &=& - (8q^2-19q+3)\/16\/q\/(q - 1); \\\\ \\nonumber\n\\end{eqnarray}\nand $Z_k(q)$ can be determined from the recurrent relations\npresented in Appendix~7. The initial conditions are provided by the formulas \n(\\ref{eq:JcrossDOWN}), (\\ref{eq:JcrossLEFT}), and (\\ref{eq:JcrossUP});\n${\\cal B}(0,q)$ at $q>0$ and $D(p,q,r)$ at $0\\leq p \\leq 3, q>1-p, 0\\leq r\\leq p+q-2$ are also needed.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ p\\leq -1$, $q\\geq 4$.}\n\\end{itemize}\n\nThe respective recurrent relations can be found in Appendix~8,\nthe initial conditions are provided by the ${\\cal B}(p,q)$ \nfunctions at $p\\leq 0$ and $q\\geq -2$ and ${\\cal J}(p,q)$ functions\nat $0\\leq p\\leq 3$ and $-2\\leq q \\leq 3$.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q)$ functions in the domain $p \\geq 4, q\\geq 0 $ }\n\\end{itemize}\nThe recurrent relations in Appendix~9 give expressions for the \nfinite part $J(p,q)$, the divergent part should be found by the \nprocedure described in the beginning of this Section,\nsee also (\\ref{DivPart_PP_min_domain}).\n\nIt should be also noted that the formulas in Appendix~9 {\\bf are valid \nonly in the case $p>4$,} in order to use them at $p=4$ the quantities $J(0,q)$\nthat appear in the right-hand part should be replaced by $B(0,q)$.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q)$ functions in the domain $p \\geq 4, q < 0 $ }\n\\end{itemize}\nThe explicit expressions for the finite parts $J(p,q)$ at $q=-1, -2$ and \n$p\\leq 9$ are presented in the Appendix~10, the recurrent relations\nvalid at $p>4, q<-2$ are given in the Appendix~11.\nTo employ these relations at $p=4$, one should replace $J(0,q)$ that\nappears in the right-hand side by the function $B(0,q)$.\n\nThe divergent parts are given by the formula (\\ref{DivPart_PM_up_to_9}).\n\nTherewith, it should be noted that the constants $X_0 \\div X_3$ \nthat appear in some expressions for $J(p,q)$ at $p\\leq 0$ cancel \nin the expressions for the integrals (\\ref{eq:GenFermInt}) at $p>0$\nand thus their numerical values are not needed.\n\n\\vspace*{-1mm}\n\\subsection{Dimensional Regularization}\n\nIn the dimensional regularization, \neach integral $F(p,q;n1,n2,n3,n4)$ (see (\\ref{eq:GenFermInt})) is associated with\nthe respective boson integral $B_F(p,q;n1,n2,n3,n4)$ that has the same divergent part. \n$B_F(p,q;n1,n2,n3,n4)$ is determined by the procedure \nsimilar to that indicated in subsection (\\ref{DPFIFMR}).\nFor example, at $p>0, q\\geq2-p$\n\\begin{equation}\\label{ABIexample1}\nB_F (p,q;n1,n2,n3,n4) = \\sum_{l=0}^{p+q-2} {(-1)^l\\; (p+l-1)!\\over l! (p-1)!} \n\\int {dk\\over (2\\pi)^4} \\; {\\Delta^l \\cos^{n_1}(k_1)\\ .... \\cos^{n_4}(k_4)\\ \\over \\Delta_B^{p+q+l}}. \\\\ \\nonumber\n\\end{equation}\nThen we compute $B_F(p,q;n1,n2,n3,n4)$ in the dimensional regularization as \nit is described in subsection (\\ref{sec:DRbos}) and \n$F(p,q;n1,n2,n3,n4)-B_F(p,q;n1,n2,n3,n4)$ (which is convergent) in the fictitious mass regularization.\nThe sum of these quantities provides the sought for result.\\\\[1mm]\n\n\\vspace*{-1mm}\n\\section{Conclusions}\n\nThe BCP algorithm described above and the explicit formulas obtained with it and presented\nin the Appendices make it a straightforward matter to express an integral \nof the type (\\ref{eq:GenFermInt}) at $p\\leq 9$ and arbitrary values of \n$n_1, n_2, n_3, n_4$ and $q$ in terms of the constants $F_0, Z_0, Z_1$ and $Y_0 \\div Y_{11}$.\nIn fact, these formulas provide a computer program,\nwhich can easily be realized with various packages. \nSuch program was written in FORM and performed, some of \nthe results are presented on the web: \\ \\ \n{\\tt http:\/\/www.lattice.itep.ru\/$\\sim$pbaivid\/lattpt\/ }\n\nThese are\n\\begin{itemize} \n\\item the values of the functions $J(p,q)$ and $B(p,q)$\nat $-26 \\leq p \\leq 0,\\ \\ -56 - 2p \\leq q \\leq 34 $ \nand the values of $J(p,q)$ at $1\\leq p \\leq 9, \\ \\ -28 \\leq q \\leq 33 - p$;\n\\item the expressions for the integrals of the type (\\ref{eq:GenFermInt})\nat some particular values of $p$ and $q$ and $n_1\\leq 6$;\n\\item the program for the computation of the integrals \n(\\ref{eq:GenFermInt}) at $0\\leq p,q \\leq 9$ and $n_\\mu^{max}\\leq 6$\nthat can be readily used by anyone.\n\\end{itemize} \n\nI hope that this work will facilitate using the BCP algorithm in practical computations.\n\n{\\large \\bf Acknowledgments:} I am grateful to H.Perlt, A.Schiller, and V. Bornyakov for stimulating discussions,\nto P.Buividovich for the help with the presentation on the web,\nand to the Leipzig University, where this study was started, for hospitality.\nThis work was supported in part by the grant for scientific schools no. NSh-679.2008.2\nand by the Russian Foundation for Basic Research (RFBR grant no. 07-02-0237).\n\n\\newpage\n\n{\\Large \\bf Appendix 1.}\\\\[2mm]\n\n\nSome values used in the text are listed below.\n\nThe integrals defined in (\\ref{Bdirect}) at $4\\leq q\\leq 12$\n(see also (\\ref{IniCondForBbosonic}))\n\\begin{eqnarray}\\label{BBosonicTable1}\n&& {\\cal B}_{4} = 917\/2; \\\\ \\nonumber\n&& {\\cal B}_{5} = 2514; \\\\ \\nonumber\n&& {\\cal B}_{6} = 14376; \\\\ \\nonumber\n&& {\\cal B}_{7} = 85152; \\\\ \\nonumber\n&& {\\cal B}_{8} = 16628949\/32; \\\\ \\nonumber\n&& {\\cal B}_{9} = 26026877\/8; \\\\ \\nonumber\n&& {\\cal B}_{10} = 333148183\/16; \\\\ \\nonumber\n&& {\\cal B}_{11} = 543325293\/4; \\\\ \\nonumber\n&& {\\cal B}_{12} = 14415564199\/16; \\nonumber\n\\end{eqnarray}\n\nThe coefficients introduced in formula (\\ref{InfeldAsExp0}) are\n\\begin{eqnarray}\n&&\tb_{0}=1; \\\\ \\nonumber\n&&\tb_{1}=1\/2; \\\\ \\nonumber\n&&\tb_{2}=3\/8; \\\\ \\nonumber\n&&\tb_{3}=13\/32; \\\\ \\nonumber\n&&\tb_{4}=77\/128; \\\\ \\nonumber\n&&\tb_{5}=297\/256; \\\\ \\nonumber\n&&\tb_{6}=5727\/2048; \\\\ \\nonumber\n&&\tb_{7}=66687\/8192; \\\\ \\nonumber\n&&\tb_{8}=912303\/32768; \\\\ \\nonumber\n&&\tb_{9}=3586545\/32768; \\\\ \\nonumber\n&&\tb_{10}=127448505\/262144; \\\\ \\nonumber\n&&\tb_{11}=2523924765\/1048576; \\\\ \\nonumber\n&&\tb_{12}=110207056005\/8388608; \\\\ \\nonumber\n&&\tb_{13}=657259273755\/8388608; \\\\ \\nonumber\n&&\tb_{14}=68022530602425\/134217728; \\\\ \\nonumber\n&&\tb_{15}=1897008475419225\/536870912; \\\\ \\nonumber\n&&\tb_{16}=56719614296927925\/2147483648; \\\\ \\nonumber\n&&\tb_{17}=226232753142332475\/1073741824; \\\\ \\nonumber\n&&\tb_{18}=15346146376168947675\/8589934592; \\\\ \\nonumber\n&&\tb_{19}=275641831899783381375\/17179869184; \\\\ \\nonumber\n&&\tb_{20}=41819089838429396989125\/274877906944; \\nonumber\n\\end{eqnarray}\nThe coefficients $c_q(n_1,n_2,n_3,n_4)$ used \nin the dimensional regularization are introduced in (\\ref{d_coeff_def}):\n\\begin{eqnarray}\n c_0(n_1,n_2,n_3,n_4) &=& 0, \\\\ \\nonumber\n c_1(n_1,n_2,n_3,n_4) &=& 1\/8, \\nonumber\n\\end{eqnarray}\nthese equations at $q=0,1$ are valid for all $n_\\mu$.\nThe coefficients $c_q=c_q(0,0,0,0)$ at $q\\leq 10$ are\n\\begin{eqnarray}\n c_2(0,0,0,0) &=& 1\/8; \\\\ \\nonumber\n c_3(0,0,0,0) &=& 55\/384; \\\\ \\nonumber\n c_4(0,0,0,0) &=& 5\/24; \\\\ \\nonumber\n c_5(0,0,0,0) &=& 1973\/5120; \\\\ \\nonumber\n c_6(0,0,0,0) &=& 54583\/61440; \\\\ \\nonumber\n c_7(0,0,0,0) &=& 8558131\/3440640; \\\\ \\nonumber\n c_8(0,0,0,0) &=& 4727509\/573440; \\\\ \\nonumber\n c_9(0,0,0,0) &=& 652905649\/20643840; \\\\ \\nonumber\n c_{10}(0,0,0,0) &=& 2276619691\/16515072; \\nonumber\n\\end{eqnarray}\nAnd the values some other of the coefficients $c_q(n_1,n_2,n_3,n_4)$:\n\\begin{eqnarray}\n c_2(1,0,0,0) &=& 1\/16; \\\\ \\nonumber\n c_2(1,1,0,0) &=& 0; \\\\ \\nonumber\n c_2(1,1,1,0) &=& - 1\/16; \\\\ \\nonumber\n c_2(1,1,1,1) &=& - 1\/8; \\\\ \\nonumber\n c_2(2,0,0,0) &=& 0; \\\\ \\nonumber\n c_2(2,1,0,0) &=& - 1\/16; \\\\ \\nonumber\n c_2(2,1,1,0) &=& - 1\/8; \\\\ \\nonumber\n c_2(2,1,1,1) &=& - 3\/16; \\\\ \\nonumber\n c_3(1,0,0,0) &=& 25\/384; \\\\ \\nonumber\n c_3(1,1,0,0) &=& 7\/384; \\\\ \\nonumber\n c_3(1,1,1,0) &=& 1\/384; \\\\ \\nonumber\n c_4(1,0,0,0) &=& 27\/256; \\\\ \\nonumber\n c_4(1,1,0,0) &=& 19\/384; \\\\ \\nonumber\n c_4(1,1,1,0) &=& 19\/768; \\\\ \\nonumber\n c_4(1,1,1,1) &=& 1\/64; \\\\ \\nonumber\n c_4(2,0,0,0) &=& 55\/384; \\\\ \\nonumber\n c_5(1,0,0,0) &=& 1143\/5120; \\\\ \\nonumber\n c_5(1,1,0,0) &=& 1999\/15360; \\\\ \\nonumber\n c_5(2,0,0,0) &=& 1433\/5120; \\\\ \\nonumber\n c_6(1,0,0,0) &=& 7085\/12288; \\\\ \\nonumber\n c_6(1,1,0,0) &=& 23387\/61440; \\\\ \\nonumber\n c_6(2,0,0,0) &=& 40867\/61440; \\nonumber\n\\end{eqnarray}\n\\pagebreak \n\n{\\Large \\bf Appendix 2.}\n\n\\begin{eqnarray}\\label{DPpartQ_eq_0_dlt}\n L(0,2) &=& ( \\;l_C + 1\/2 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,3) &=& ( - 3\/4 \\mu_B^{-2} - 1\/2 \\; \\mu_B^{-2} \\;l_C + 5\/8 \\;l_C + 1\/8 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,4) &=& ( ( - 1\/6 \\; \\mu_B^{-4} - 1\/12 \\; \\mu_B^{-2} + 137\/960 ) \\;l_C \\\\ \\nonumber \n && - 5\/36 \\; \\mu_B^{-4} - 31\/144 \\; \\mu_B^{-2} + 1\/32 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,5) &=& ( ( - 1\/12 \\; \\mu_B^{-6} - 1\/48 \\; \\mu_B^{-4} - 1\/64 \\; \\mu_B^{-2} + 15527\/322560 ) \\;l_C \\\\ \\nonumber \n && - 7\/144 \\; \\mu_B^{-6} - 101\/2880 \\; \\mu_B^{-4} - 151\/3840 \\; \\mu_B^{-2}+ 13\/1536 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,6) &=& ( ( - 1\/20 \\; \\mu_B^{-8} - 1\/120 \\; \\mu_B^{-6} - 1\/320 \\; \\mu_B^{-4} - 13\/3840 \\; \\mu_B^{-2} + 172241\/12902400 ) \\;l_C \\\\ \\nonumber \n && - 9\/400 \\; \\mu_B^{-8} - 77\/7200 \\; \\mu_B^{-6} - 709\/134400 \\; \\mu_B^{-4} - 3953\/403200 \\; \\mu_B^{-2} \\\\ \\nonumber \n && + 77\/30720 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,7) &=& ( (- 1\/30 \\; \\mu_B^{-10} - 1\/240 \\; \\mu_B^{-8} - 1\/960 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 13\/23040 \\; \\mu_B^{-4} - 77\/92160 \\; \\mu_B^{-2} + 457867\/94617600 ) \\;l_C \\\\ \\nonumber \n&& - 11\/900 \\; \\mu_B^{-10} - 439\/100800 \\; \\mu_B^{-8} - 53\/38400 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 5371\/4838400 \\; \\mu_B^{-4} - 183101\/77414400 \\; \\mu_B^{-2} + 33\/40960 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,8) &=& ( ( - 1\/42 \\; \\mu_B^{-12} - 1\/420 \\; \\mu_B^{-10} - 1\/2240 \\; \\mu_B^{-8} - 13\/80640 \\;\\mu_B^{-6}\\\\ \\nonumber \n && - 11\/92160 \\; \\mu_B^{-4} - 33\/143360 \\; \\mu_B^{-2} + 51135377\/31791513600 )\\;l_C \\\\ \\nonumber \n && - 13\/1764 \\; \\mu_B^{-12} - 743\/352800 \\; \\mu_B^{-10} - 2789\/5644800 \\; \\mu_B^{-8} \\\\ \\nonumber \n && - 3371\/13547520 \\;\\mu_B^{-6} - 28441\/121651200 \\; \\mu_B^{-4} - 10159\/14450688 \\; \\mu_B^{-2} \\\\ \\nonumber \n && + 1909\/6881280 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\\label{DivPart_PP_min_domain}\n D(0,2) &=& - \\;l_C \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,3) &=& ( 1\/2 \\; \\mu_B^{-2} - 1\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,4) &=& ( 1\/6 \\; \\mu_B^{-4} + 1\/12 \\; \\mu_B^{-2} - 1\/16 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,5) &=& ( 1\/12 \\; \\mu_B^{-6} + 1\/48 \\; \\mu_B^{-4} + 1\/64 \\; \\mu_B^{-2} - 13\/768 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,6) &=& ( 1\/20 \\; \\mu_B^{-8} + 1\/120 \\; \\mu_B^{-6} + 1\/320 \\; \\mu_B^{-4} + 13\/3840 \\; \\mu_B^{-2} - 77\/15360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,7) &=& ( 1\/30 \\; \\mu_B^{-10} + 1\/240 \\; \\mu_B^{-8} + 1\/960 \\; \\mu_B^{-6} + 13\/23040 \\; \\mu_B^{-4}\\\\ \\nonumber \n&& + 77\/92160 \\; \\mu_B^{-2} - 33\/20480 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,8) &=& ( 1\/42 \\; \\mu_B^{-12} + 1\/420 \\; \\mu_B^{-10} + 1\/2240 \\; \\mu_B^{-8} + 13\/80640 \\; \\mu_B^{-6}\\\\ \\nonumber \n&& + 11\/92160 \\; \\mu_B^{-4} + 33\/143360 \\; \\mu_B^{-2} - 1909\/3440640 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(1,1) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(1,2) &=& ( 1\/2 \\; \\mu_B^{-2} ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(1,3) &=& ( 1\/6 \\; \\mu_B^{-4} + 1\/48 \\; \\mu_B^{-2} - 11\/160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(1,4) &=& ( 1\/12 \\;\\mu_B^{-6} + 1\/120 \\; \\mu_B^{-4} + 7\/480 \\; \\mu_B^{-2} + 1\/1680 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n D(1,5) &=& ( 1\/20 \\; \\mu_B^{-8} + 1\/240 \\; \\mu_B^{-6} + 3\/1120 \\; \\mu_B^{-4} + 19\/21504 \\; \\mu_B^{-2} - 1339\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(1,6) &=& ( 1\/30 \\; \\mu_B^{-10} + 1\/420 \\; \\mu_B^{-8} + 23\/26880 \\; \\mu_B^{-6} + 13\/53760 \\; \\mu_B^{-4} \\\\ \\nonumber \n && + 181\/215040 \\; \\mu_B^{-2} - 67\/630784 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(1,7) &=& ( 1\/42 \\;\\mu_B^{-12} + 1\/672 \\;\\mu_B^{-10} + 29\/80640 \\;\\mu_B^{-8} + 277\/3225600 \\;\\mu_B^{-6} \\\\ \\nonumber \n&& + 15163\/141926400 \\;\\mu_B^{-4} + 46523\/567705600 \\; \\mu_B^{-2} - 52001\/75694080 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(2,1) &=& ( 1\/2 \\; \\mu_B^{-2} + 1\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(2,2) &=& ( 1\/6 \\; \\mu_B^{-4} - 1\/24 \\; \\mu_B^{-2} - 23\/160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(2,3) &=& ( 1\/12 \\;\\mu_B^{-6} - 1\/240 \\; \\mu_B^{-4} + 1\/40 \\;\\mu_B^{-2} + 1093\/26880 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n D(2,4) &=& ( 1\/20 \\;\\mu_B^{-8} + 13\/3360 \\;\\mu_B^{-4} - 37\/8960 \\;\\mu_B^{-2} - 4141\/215040 \\;l_C )\/(2\\pi)^{2};\\\\ \\nonumber \n D(2,5) &=& ( 1\/30 \\; \\mu_B^{-10} + 1\/1680 \\; \\mu_B^{-8} + 29\/26880 \\; \\mu_B^{-6} - 1\/3072 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 31\/15360 \\; \\mu_B^{-2} + 34689\/6307840 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(2,6) &=& ( 1\/42 \\;\\mu_B^{-12} + 1\/1680 \\; \\mu_B^{-10} + 11\/26880 \\; \\mu_B^{-8} - 17\/537600 \\;\\mu_B^{-6} \\\\ \\nonumber\n&& + 4747\/23654400 \\; \\mu_B^{-4} - 36473\/94617600 \\; \\mu_B^{-2} - 128103\/50462720 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,1) &=& ( 1\/6 \\; \\mu_B^{-4} - 5\/48 \\; \\mu_B^{-2} - 23\/80 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,2) &=& ( 1\/12 \\; \\mu_B^{-6} - 1\/60 \\; \\mu_B^{-4} + 3\/64 \\; \\mu_B^{-2} + 331\/2688 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,3) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/240 \\; \\mu_B^{-6} + 3\/448 \\;\\mu_B^{-4} - 19\/1344 \\; \\mu_B^{-2} - 12241\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,4) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/840 \\; \\mu_B^{-8} + 23\/13440 \\;\\mu_B^{-6} - 19\/13440 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 481\/86016 \\; \\mu_B^{-2} + 35879\/1576960 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,5) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/3360 \\; \\mu_B^{-10} + 1\/1680 \\;\\mu_B^{-8} - 53\/215040 \\;\\mu_B^{-6} \\\\ \\nonumber \n&& + 4801\/9461760 \\; \\mu_B^{-4} - 33797\/18923520 \\; \\mu_B^{-2} - 492689\/50462720 \\;l_C ) \/ (2\\pi)^{2} ; \\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\n D(4,1) &=& ( 1\/12 \\; \\mu_B^{-6} - 7\/240 \\; \\mu_B^{-4} + 77\/960 \\;\\mu_B^{-2} + 7201\/26880 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n D(4,2) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/120 \\; \\mu_B^{-6} + 5\/448 \\;\\mu_B^{-4} - 85\/2688 \\; \\mu_B^{-2} - 29671\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(4,3) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/336 \\; \\mu_B^{-8} + 37\/13440 \\;\\mu_B^{-6} - 19\/5760 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 17267\/1290240 \\; \\mu_B^{-2} + 51223\/788480 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(4,4) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/840 \\; \\mu_B^{-10} + 37\/40320 \\; \\mu_B^{-8} - 11\/17920 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& + 16853\/14192640 \\; \\mu_B^{-4} - 923\/177408 \\; \\mu_B^{-2} - 2284033\/75694080 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(5,1) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/80 \\; \\mu_B^{-6} + 29\/1680 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& - 2117\/35840 \\; \\mu_B^{-2} - 30869\/107520 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(5,2) &=& ( 1\/30 \\;\\mu_B^{-10} - 1\/210 \\;\\mu_B^{-8} + 113\/26880 \\;\\mu_B^{-6} - 1009\/161280 \\;\\mu_B^{-4} \\\\ \\nonumber \n&& + 35873\/1290240 \\; \\mu_B^{-2} + 482633\/3153920 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(5,3) &=& ( 1\/42 \\;\\mu_B^{-12} - 1\/480 \\; \\mu_B^{-10} + 37\/26880 \\; \\mu_B^{-8} - 767\/645120 \\; \\mu_B^{-6} \\\\ \\nonumber\n&& + 69689\/28385280 \\;\\mu_B^{-4} - 116749\/9461760 \\; \\mu_B^{-2} - 5912603\/75694080 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(6,1) &=& ( 1\/30 \\; \\mu_B^{-10} - 11\/1680 \\; \\mu_B^{-8} + 163\/26880 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 3407\/322560 \\; \\mu_B^{-4} + 66911\/1290240 \\; \\mu_B^{-2} + 400193\/1261568 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(6,2) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/336 \\; \\mu_B^{-10} + 53\/26880 \\; \\mu_B^{-8} - 467\/230400 \\;\\mu_B^{-6} \\\\ \\nonumber \n && + 162457\/35481600 \\; \\mu_B^{-4} - 1214401\/47308800 \\; \\mu_B^{-2} - 26852377\/151388160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(7,1) &=& ( 1\/42 \\; \\mu_B^{-12} - 13\/3360 \\; \\mu_B^{-10} + 109\/40320 \\; \\mu_B^{-8} - 10267\/3225600 \\; \\mu_B^{-6}\\\\ \\nonumber \n&& + 53257\/6758400 \\; \\mu_B^{-4} - 27585673\/567705600 \\; \\mu_B^{-2} - 18404583\/50462720 \\;l_C ) \/ (2\\pi)^{2};\\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\\label{DivPart_PM_up_to_9}\n D(2,0) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,0) &=& ( 1\/2 \\; \\mu_B^{-2} + 1\/2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,-1) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(4,0) &=& ( 1\/6 \\; \\mu_B^{-4} - 1\/6 \\; \\mu_B^{-2} - 1\/2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(4,-1) &=& ( 1\/2 \\; \\mu_B^{-2} + 3\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(4,-2) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(5,0) &=& ( 1\/12 \\; \\mu_B^{-6} - 1\/24 \\; \\mu_B^{-4} + 1\/8 \\; \\mu_B^{-2} + 95\/192 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(5,-1) &=& ( 1\/6 \\; \\mu_B^{-4} - 11\/48 \\; \\mu_B^{-2} - 25\/32 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(5,-2) &=& ( 1\/2 \\; \\mu_B^{-2} + \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(5,-3) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(6,0) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/60 \\; \\mu_B^{-6} + 1\/40 \\; \\mu_B^{-4} - 19\/192 \\; \\mu_B^{-2} - 1027\/1920 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(6,-1) &=& ( 1\/12 \\;\\mu_B^{-6} - 13\/240 \\;\\mu_B^{-4} + 29\/160 \\;\\mu_B^{-2} + 3163\/3840 \\;l_C )\/(2\\pi)^{2}; \\\\ \\nonumber \n D(6,-2) &=& ( 1\/6 \\; \\mu_B^{-4} - 7\/24 \\; \\mu_B^{-2} - 181\/160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(6,-3) &=& ( 1\/2 \\; \\mu_B^{-2} + 5\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(6,-4) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(7,0) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/120 \\; \\mu_B^{-8} + 1\/120 \\; \\mu_B^{-6} - 19\/1152 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 1027\/11520 \\; \\mu_B^{-2} + 3067\/5120 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(7,-1) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/48 \\; \\mu_B^{-6} + 11\/320 \\; \\mu_B^{-4} - 1181\/7680 \\; \\mu_B^{-2} - 14099\/15360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(7,-2) &=& ( 1\/12 \\; \\mu_B^{-6} - 1\/15 \\; \\mu_B^{-4} + 239\/960 \\;\\mu_B^{-2} + 2447\/1920 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n D(7,-3) &=& ( 1\/6 \\; \\mu_B^{-4} - 17\/48 \\; \\mu_B^{-2} - 31\/20 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(7,-4) &=& ( 1\/2 \\; \\mu_B^{-2} + 3\/2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(7,-5) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(8,0) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/210 \\; \\mu_B^{-10} + 1\/280 \\; \\mu_B^{-8} - 19\/4032 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& + 1027\/80640 \\; \\mu_B^{-4} - 3067\/35840 \\; \\mu_B^{-2} - 74609\/107520 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(8,-1) &=& ( 1\/30 \\; \\mu_B^{-10} - 17\/1680 \\; \\mu_B^{-8} + 37\/3360 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 3923\/161280 \\; \\mu_B^{-4} + 9281\/64512 \\; \\mu_B^{-2} + 150949\/143360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(8,-2) &=& (1\/20 \\; \\mu_B^{-8} - 1\/40 \\; \\mu_B^{-6} + 61\/1344 \\; \\mu_B^{-4} - 405\/1792 \\; \\mu_B^{-2} - 79489\/53760 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(8,-3) &=& ( 1\/12 \\; \\mu_B^{-6} - 19\/240 \\; \\mu_B^{-4} + 21\/64 \\;\\mu_B^{-2} + 10037\/5376 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(8,-4) &=& ( 1\/6 \\; \\mu_B^{-4} - 5\/12 \\; \\mu_B^{-2} - 163\/80 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(8,-5) &=& ( 1\/2 \\; \\mu_B^{-2} + 7\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(8,-6) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(9,-1) &=& ( 1\/42 \\; \\mu_B^{-12} - 19\/3360 \\; \\mu_B^{-10} + 41\/8960 \\;\\mu_B^{-8} - 4303\/645120 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& + 50513\/2580480 \\; \\mu_B^{-4} - 163217\/1146880 \\; \\mu_B^{-2} - 8535263\/6881280 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(9,-2) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/84 \\; \\mu_B^{-8} + 379\/26880 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 791\/23040 \\; \\mu_B^{-4} + 5087\/23040 \\; \\mu_B^{-2} + 501267\/286720 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(9,-3) &=& ( 1\/20 \\; \\mu_B^{-8} - 7\/240 \\; \\mu_B^{-6} + 13\/224 \\; \\mu_B^{-4} - 6841\/21504 \\; \\mu_B^{-2} - 487141\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(9,-4) &=& ( 1\/12 \\; \\mu_B^{-6} - 11\/120 \\; \\mu_B^{-4} + 67\/160 \\;\\mu_B^{-2} + 8807\/3360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(9,-5) &=& ( 1\/6 \\; \\mu_B^{-4} - 23\/48 \\; \\mu_B^{-2} - 83\/32 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(9,-6) &=& ( 1\/2 \\; \\mu_B^{-2} + 2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(9,-7) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\nonumber \n\\end{eqnarray}\n\n\\pagebreak[1]\n{\\Large \\bf Appendix 3.}\n\\begin{verbatim}\nB(p,1) = ( 1536*B(p+2,2)*(80*p^4 + 596*p^3 + 1628*p^2 + 1935*p + 846)\n + 64*B(p+1,2)*(520*p^4 + 3404*p^3 + 8106*p^2+ 8413*p+3245)\n + 48*B( p,2)*(40*p^4 + 216*p^3 + 394*p^2 + 290*p + 75)\n + 73728*B(p+4,1)*(8*p^5 + 92*p^4 + 410*p^3 + 885*p^2 + 927*p + 378)\n + 1536*B(p+3,1)*(320*p^5 + 3220*p^4 + 12772*p^3 + 24943*p^2 + 23993*p + 9110)\n + 64*B(p+2,1)*(1080*p^5 + 8260*p^4 + 24470*p^3 + 35247*p^2 + 24823*p + 6920)\n + 48*B(p+1,1)*( - 40*p^5 - 480*p^4 - 1946*p^3 - 3508*p^2 - 2911*p - 915)\n + 1152*B(p+5,-3)*(4*p^5 + 48*p^4 + 221*p^3 + 489*p^2 + 522*p + 216)\n + 12*B(p+4,-3)*( - 724*p^5 - 6504*p^4 - 22567*p^3 - 37962*p^2 - 31033*p - 9858)\n + 9*B(p+3,-3)*( - 128*p^5 - 828*p^4 - 1924*p^3 - 1961*p^2 - 867*p - 130)\n + 6*B(p+2,-3)*(32*p^5 + 180*p^4 + 296*p^3 + 15*p^2 - 328*p - 195)\n + 7296*B(p+5,-2)*(4*p^5 + 48*p^4 + 221*p^3 + 489*p^2 + 522*p + 216)\n + 4*B(p+4,-2)*( - 13756*p^5 - 133560*p^4 - 508645*p^3 - 953406*p^2\n - 881659*p - 322086)\n + 2*B(p+3,-2)*( - 3588*p^5 - 25448*p^4 - 67995*p^3 - 85874*p^2\n - 52149*p - 12410)\n + 4*B(p+2,-2)*(244*p^5 + 1600*p^4 + 3635*p^3 + 3080*p^2 + 216*p - 585)\n + 3072*B(p+5,-1)*(4*p^5 + 48*p^4 + 221*p^3 + 489*p^2 + 522*p + 216)\n + 32*B(p+4,-1)*( - 5908*p^5 - 62136*p^4 - 256375*p^3 - 518814*p^2\n - 515197*p - 201018)\n + 8*B(p+3,-1)*( - 1508*p^5 - 9252*p^4 - 19927*p^3 - 16451*p^2 - 2040*p + 2084)\n + 2*B(p+2,-1)*(1712*p^5 + 13228*p^4 + 37256*p^3 + 46893*p^2 + 26223*p + 5070)\n + 3*B(p+1,-1)*( - 64*p^5 - 472*p^4 - 1072*p^3 - 422*p^2 + 1160*p + 975)\n + 73728*B(p+5,0)*( - 4*p^5 - 48*p^4 - 221*p^3 - 489*p^2 - 522*p - 216)\n + 768*B(p+4,0)*( - 400*p^5 - 4432*p^4 - 19118*p^3 - 40113*p^2 - 41003*p - 16386)\n + 8*B(p+3,0)*(10100*p^5 + 96976*p^4 + 370679*p^3 + 704416*p^2\n + 663875*p + 247102)\n + 4*B(p+2,0)*(3220*p^5 + 22680*p^4 + 62143*p^3 + 84494*p^2 + 57851*p + 16040)\n + 4*B(p+1,0)*( - 492*p^5 - 3268*p^4 - 7609*p^3 - 7163*p^2 - 2243*p - 120)\n)\/12\/(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15);\n\nB(p,2) = ( 2304*B(p+2,2)*(2*p^2 + 7*p + 6)\n + 32*B(p+1,2)*(4*p^2 + 12*p + 11)\n + 768*B(p+3,1)*( - 10*p^3 - 59*p^2 - 110*p - 64)\n + 128*B(p+2,1)*( - 60*p^3 - 299*p^2 - 495*p - 271)\n + 24*B(p+1,1)*( - 40*p^3 - 144*p^2 - 172*p - 69)\n + 6*B(p+4,-3)*( - 10*p^3 - 61*p^2 - 115*p - 66)\n + 12*B(p+3,-3)*p*(10*p^2 + 31*p + 22)\n + 38*B(p+4,-2)*( - 10*p^3 - 61*p^2 - 115*p - 66)\n + B(p+3,-2)*(742*p^3 + 2795*p^2 + 3167*p + 1090)\n + 18*B(p+2,-2)*p*(2*p^2 + 5*p + 3)\n + 16*B(p+4,-1)*( - 10*p^3 - 61*p^2 - 115*p - 66)\n + 4*B(p+3,-1)*(590*p^3 + 2719*p^2 + 4057*p + 1958)\n + 6*B(p+2,-1)*(20*p^3 + 50*p^2 + 84*p + 59)\n + 384*B(p+4,0)*(10*p^3 + 61*p^2 + 115*p + 66)\n + 4*B(p+3,0)*(1178*p^3 + 6705*p^2 + 12285*p + 7174)\n + 2*B(p+2,0)*( - 302*p^3 - 591*p^2 + 69*p + 416)\n + 6*B(p+1,0)*( - 10*p^3 - 23*p^2 - 10*p + 3)\n)\/48\/(2*p + 3)\/(p+1);\n\nB(p,3) = ( 4*B(p+1,2)*( - 20*p^2 - 46*p - 23) \n + 192*B(p+3,1)*( - 2*p^3 - 11*p^2 - 19*p - 10) \n + 2*B(p+2,1)*( - 138*p^3 - 589*p^2 - 827*p - 376)\n + 3*B(p+1,1)*( - 4*p^3 - 14*p^2 - 14*p - 3) \n + 3*B(p+4,-3)*( - p^3 - 6*p^2 - 11*p - 6) \n + 6*B(p+3,-3)*p*(p^2 + 3*p + 2) \n + 19*B(p+4,-2)*( - p^3 - 6*p^2 - 11*p - 6) \n + 2*B(p+3,-2)*(19*p^3 + 70*p^2 + 77*p + 26) \n + 8*B(p+4,-1)*( - p^3 - 6*p^2 - 11*p - 6)\n + B(p+3,-1)*(124*p^3 + 559*p^2 + 809*p + 374)\n + 3*B(p+2,-1)*( - 2*p^3 - 8*p^2 - 5*p + 1)\n + 192*B(p+4,0)*(p^3 + 6*p^2 + 11*p + 6)\n + 2*B(p+3,0)*(89*p^3 + 472*p^2 + 793*p + 410)\n + 2*B(p+2,0)*( - 37*p^3 - 119*p^2 - 122*p - 40)\n)\/12\/(2*p + 3);\n\\end{verbatim}\n\n\n{\\Large \\bf Appendix 4.}\n\nHere and below, the symbol {\\tt BB(p,q)} in any Appendix\ndesignates ${\\cal B}(p,q)$, whereas {\\tt B(p,q)} designates $B(p,q)$.\n\n\\begin{verbatim}\nBB(p,q)= (( 24*BB(p+1,q-1)*(5*p+3)*(q-2)*(q-3) \n + 2*BB(p+1,q-2)*(q-3)*(3+p*(20*p +21*q-28)-3*(q-5)*(2*q-5))\n + 24*BB(p+2,q-2)*(p+1)*(q-3)*(11*p+14) \n + 4*BB(p+2,q-3)*(p+1)*(10*p*(3*p+2*q+2) -15*q^2+121*q-175)\n + BB(p+2,q-4)*(p+1)*(2*p*(5*p-4*q+26) -3*(7*q-20)*(q-4))\n + 24*BB(p+3,q-3)*(p+2)*(p+1)*(15*p+31) \n - 2*BB(p+3,q-4)*(p+2)*(p+1)*(73*p+ 66*q - 85) \n - 4*BB(p+3,q-5)*(p+2)*(p+1)*(10*p + 10*q - 23)\n - 6*BB(p+3,q-6)*(p+2)*(p+1)*(p + q - 3) \n + (-192*BB(p+4,q-4)+20*BB(p+4,q-5) +3*BB(p+4,q-6))*(p+3)*(p+2)*(p+1))\/12 \n+muB^2*( - 24*BB(p+1,q)*(q-1)*(q-2)*(q-3) \n + 6*BB(p+1,q-1)*(-7*p+4*q-13)*(q-2)*(q-3)\n - 120*BB(p+2,q-1)*(p+1)*(q-2)*(q-3)\n + 2*BB(p+2,q-2)*(p+1)*(-80*p+39*q-204)*(q-3) \n + BB(p+2,q-3)*(p+1)*(2*p*(-5*p+8*q-38)+3*(21*q^2 -131*q+203))\n + 264*BB(p+3,q-2)*(p+2)*(p+1)*(-q+3)\n + 2*BB(p+3,q-3)*(p+2)*(p+1)*( - 47*p + 92*q - 383)\n + 2*BB(p+3,q-4)*(p+2)*(p+1)*(30*p + 64*q - 183)\n + 6*BB(p+3,q-5)*(p+2)*(p+1)*(3*p + 4*(q-3))\n + (- 168*BB(p+4,q-3) + 106*BB(p+4,q-4) + 31*BB(p+4,q-5) \n + 6*BB(p+4,q-6))*(p+3)*(p+2)*(p+1))\/12 \n+muB^4*( - 12*BB(p+1,q)*(q-1)*(q-2)*(q-3) \n - 18*BB(p+2,q-1)*(p+1)*(q-2)*(q-3) \n + BB(p+2,q-2)*(p+1)*(- 8*p - 63*q + 165)*(q-3) \n - 12*BB(p+3,q-2)*(p+2)*(p+1)*(q-3) \n + 2*BB(p+3,q-3)*(p+2)*(p+1)*( - 10*p - 68*q + 209) \n - 18*BB(p+3,q-4)*(p+2)*(p+1)*(p + 2*(q-3)) \n - (6*BB(p+4,q-3)+ 61*BB(p+4,q-4)\n + 18*BB(p+4,q-5))*(p+3)*(p+2)*(p+1))\/12 \n+muB^6*( 7*BB(p+2,q-1)*(p+1)*(q-2)*(q-3) \n + 16*BB(p+3,q-2)*(p+2)*(p+1)*(q-3) \n + 2*BB(p+3,q-3)*(p+2)*(p+1)*(p + 4*(q-3)) \n + (9*BB(p+4,q-3)+6*BB(p+4,q-4))*(p+3)*(p+2)*(p+1))\/4 \n+ muB^8*( - BB(p+3,q-2)*(p+2)*(p+1)*(q-3) \n - BB(p+4,q-3)*(p+3)*(p+2)*(p+1))\/2)\n\/(-2)\/(q-1)\/(q-2)\/(q-3);\n\\end{verbatim}\n\n{\\Large \\bf Appendix 5.}\n\n\\begin{verbatim}\nJ(0,p) = ( 6144*J(3,3+p)*( - 4428972*p^9 - 96411618*p^8\n - 738214452*p^7 - 2743289766*p^6 - 5415151530*p^5\n - 5585181043*p^4 - 2539249652*p^3 - 67504115*p^2 + 233419148*p + 30299280)\n + 256*J(3,2+p)*(139751460*p^9 + 3070483974*p^8\n + 23818829328*p^7 + 89626132062*p^6 + 178818682872*p^5\n + 185987859019*p^4 + 85026655094*p^3 + 2165966651*p^2\n - 7934451500*p - 1033526160)\n + 32*J(3,1+p)*( - 53064072*p^9 - 1189331640*p^8\n - 9510540174*p^7 - 36782676126*p^6 - 74957974878*p^5\n - 78555090042*p^4 - 34288910369*p^3 + 1761870029*p^2\n + 4642726412*p + 613660240)\n + 72*J(3,p)*( - 682164*p^9 - 21229552*p^8 - 194740397*p^7\n - 812923675*p^6 - 1742215447*p^5 - 1924011565*p^4\n - 966215776*p^3 - 94303632*p^2 + 58837408*p + 8169600)\n + 18*J(3,-1+p)*( - 69084*p^9 + 722493*p^8 + 15902908*p^7\n + 89008228*p^6 + 226205991*p^5 + 276568093*p^4\n + 134439291*p^3 - 7926512*p^2 - 20882848*p - 2804160)\n + 3072*J(2,4+p)*(4428972*p^10 + 122985450*p^9\n + 1316684160*p^8 + 7172576478*p^7 + 21874890126*p^6\n + 38076090223*p^5 + 36050335910*p^4 + 15303002027*p^3\n + 171605542*p^2 - 1430814168*p - 181795680)\n + 128*J(2,3+p)*( - 113259276*p^10 - 3205411506*p^9\n - 35040435228*p^8 - 194508285894*p^7 - 602752373652*p^6\n - 1062870299215*p^5 - 1016468592718*p^4 - 434436290355*p^3\n - 4442421788*p^2 + 41237682352*p + 5258528640)\n + 32*J(2,2+p)*(130442076*p^10 + 3413592018*p^9\n + 34321240080*p^8 + 177812525358*p^7 + 522305372877*p^6\n + 884026310037*p^5 + 817641485963*p^4 + 335486857709*p^3\n - 4288900710*p^2 - 34908660968*p - 4428920480)\n + 8*J(2,1+p)*(93154860*p^10 + 2376526194*p^9\n + 22985820378*p^8 + 114170267016*p^7 + 322016440995*p^6\n + 526074477498*p^5 + 474729514373*p^4 + 196318386058*p^3\n + 4895262676*p^2 - 16149703888*p - 2017957440)\n + 36*J(2,p)*(2556162*p^10 + 61577829*p^9\n + 565688878*p^8 + 2669754107*p^7 + 7157734171*p^6\n + 11142549507*p^5 + 9630402307*p^4 + 3844859343*p^3\n + 101897232*p^2 - 294481376*p - 35482560)\n + 18*J(2,-1+p)*( - 115758*p^10 - 3068601*p^9\n - 28796323*p^8 - 133726220*p^7 - 342937513*p^6\n - 494864596*p^5 - 374444902*p^4 - 102361975*p^3\n + 32325936*p^2 + 22856992*p + 2804160)\n + 192*J(1,4+p)*(8878356*p^10 + 259596072*p^9\n + 2924912646*p^8 + 16568071920*p^7 + 51995667138*p^6\n + 92411299412*p^5 + 88851864277*p^4 + 38204834002*p^3\n + 531726629*p^2 - 3565245012*p - 454489200)\n + 16*J(1,3+p)*( - 9079560*p^11 - 430232670*p^10\n - 7372801566*p^9 - 61656609141*p^8 - 287230567893*p^7\n - 786492878195*p^6 - 1266898022654*p^5 - 1131366942039*p^4\n - 454714067077*p^3 + 2641691135*p^2 + 44246230980*p + 5521098480)\n + 4*J(1,2+p)*(59152572*p^11 + 1593555408*p^10\n + 16912911906*p^9 + 95861230578*p^8 + 324080159613*p^7\n + 679347181198*p^6 + 874596683962*p^5 + 642957167296*p^4\n + 207762312811*p^3 - 18032380112*p^2 - 24531833312*p - 2885392320)\n + 3*J(1,1+p)*( - 1971756*p^11 - 49219953*p^10\n - 469272444*p^9 - 2325010713*p^8 - 6687694463*p^7\n - 11583569599*p^6 - 11836532108*p^5 - 6263432419*p^4\n - 618308369*p^3 + 977759600*p^2 + 412312224*p + 42062400)\n + 36*J(1,p)*p*(1593*p^10 + 20446*p^9 + 98415*p^8\n + 200300*p^7 + 49659*p^6 - 470802*p^5 - 694935*p^4\n - 123400*p^3 + 456948*p^2 + 373456*p + 88320)\n + 54*J(1,-1+p)*p^2*(378*p^9 + 4783*p^8 + 23383*p^7\n + 54276*p^6 + 53436*p^5 - 8397*p^4 - 62973*p^3\n - 49606*p^2 - 14224*p - 1056)\n + 559872*J(0,5+p)*p*( - 7*p^10 - 151*p^9 - 1386*p^8\n - 7002*p^7 - 20859*p^6 - 35475*p^5 - 26276*p^4\n + 13252*p^3 + 41616*p^2 + 29376*p + 6912)\n + 23328*J(0,4+p)*p*( - 16*p^11 - 145*p^10 + 657*p^9\n + 15080*p^8 + 91206*p^7 + 283311*p^6 + 476125*p^5\n + 331666*p^4 - 199140*p^3 - 545816*p^2 - 368832*p - 84096)\n + 1944*J(0,3+p)*p*(200*p^11 + 2865*p^10 + 14981*p^9\n + 25008*p^8 - 73738*p^7 - 437223*p^6 - 855551*p^5\n - 635826*p^4 + 311548*p^3 + 911928*p^2 + 602560*p + 133248)\n + 1944*J(0,2+p)*p*( - 70*p^11 - 1015*p^10 - 5959*p^9\n - 17811*p^8 - 26156*p^7 - 6696*p^6 + 36010*p^5\n + 49732*p^4 + 13471*p^3 - 20274*p^2 - 17296*p - 3936)\n + 486*J(0,1+p)*p*(40*p^11 + 539*p^10 + 2915*p^9\n + 8077*p^8 + 11771*p^7 + 6819*p^6 - 4591*p^5\n - 11321*p^4 - 9191*p^3 - 4018*p^2 - 944*p - 96)\n )\/972\/p^5\/(p^7 + 12*p^6 + 54*p^5 + 108*p^4\n + 69*p^3 - 72*p^2 - 124*p - 48)\n +(\n 16*B(0,p+5)*(9569448*p^11 + 263172276*p^10\n + 2796333210*p^9 + 14979144042*p^8 + 45346896150*p^7\n + 81451195130*p^6 + 86698023173*p^5 + 50610254573*p^4\n + 11013851543*p^3 - 3191145997*p^2 - 1994622412*p - 245554896)\n + 4*B(0,p+4)*( - 61193772*p^11 - 1412170434*p^10\n - 12221193447*p^9 - 55077683067*p^8 - 145501643868*p^7\n - 235023215404*p^6 - 229902193451*p^5 - 124704441431*p^4\n - 24368641382*p^3 + 8307539200*p^2 + 4697326336*p + 557574080)\n + 3*B(0,p+3)*(2924316*p^11 + 54602601*p^10\n + 399544548*p^9 + 1560379173*p^8 + 3620624261*p^7\n + 5121679611*p^6 + 4177323684*p^5 + 1389082017*p^4\n - 624082689*p^3 - 825833082*p^2 - 322097816*p - 46392864)\n + 18*B(0,p+2)*( - 25866*p^11 - 271739*p^10\n - 1238954*p^9 - 3192049*p^8 - 4973923*p^7\n - 4312062*p^6 - 536457*p^5 + 3801002*p^4 + 5378048*p^3\n + 3758272*p^2 + 1396000*p + 217728)\n)\/486\/(p+2)^2\/(p+1)^4\/p^5\/(p-1);\n\nJ(3, - 6 + q) = ( 8*B(0,q-1)*(2*q^3 - 7*q^2 - 3*q + 18)\n + 2*B(0,q-2)*( - 13*q^3 + 109*q^2 - 298*q + 264)\n + 1536*J(3,q-3)*( - q + 3)\n + 64*J(3,q-4)*(31*q - 87)\n + 8*J(3,q-5)*( - 6*q + 13)\n + 768*J(2,q-2)*q*(q - 3)\n + 32*J(2,q-3)*( - 25*q^2 + 61*q + 30)\n + 8*J(2,q-4)*(27*q^2 - 127*q + 148)\n + 4*J(2,q-5)*(12*q^2 - 65*q + 84)\n + 6*J(2,q-6)*(q^2 - 6*q + 8)\n + 48*J(1,q-2)*(2*q^2 - 3*q - 9)\n + 4*J(1,q-3)*( - 2*q^3 - 22*q^2 + 150*q - 201)\n + J(1,q-4)*(13*q^3 - 130*q^2 + 418*q - 420)\n)\/6\/(q - 2);\n\nJ(2, - 6 + q) = ( 16*B(0,q-1)*( - 2034*q^4 + 13237*q^3\n - 18362*q^2 - 27483*q + 55062)\n + 4*B(0,q-2)*(13197*q^4 - 150272*q^3 + 634577*q^2 - 1175522*q + 803760)\n + 12*B(0,q-3)*(q^4 - 46*q^2 + 165*q - 162)\n + 9*B(0,q-4)*( - 7*q^4 + 95*q^3 - 470*q^2 + 1000*q - 768)\n + 3072*J(3,q-3)*(1017*q^2 - 6110*q + 9177)\n + 128*J(3,q-4)*( - 31698*q^2 + 184307*q - 267447)\n + 16*J(3,q-5)*(8463*q^2 - 44580*q + 56333)\n + 1536*J(2,q-2)*q*( - 1017*q^2 + 6110*q - 9177)\n + 64*J(2,q-3)*(25677*q^3 - 140330*q^2 + 160103*q + 88998)\n + 16*J(2,q-4)*( - 28941*q^3 + 222222*q^2 - 561953*q + 468596)\n + 16*J(2,q-5)*( - 5571*q^3 + 47436*q^2 - 132356*q + 120561)\n + 96*J(1,q-2)*( - 54*q^4 - 1494*q^3 + 7171*q^2 + 3216*q - 29475)\n + 8*J(1,q-3)*(2709*q^4 + 9326*q^3 - 193019*q^2 + 616413*q - 583827)\n + 2*J(1,q-4)*( - 14142*q^4 + 181553*q^3 - 859787*q^2 + 1768658*q - 1324320)\n + 24*J(1,q-5)*(11*q^4 - 141*q^3 + 664*q^2 - 1356*q+1008)\n + 18*J(1,q-6)*(q^4 - 14*q^3 + 71*q^2 - 154*q + 120)\n)\/12294\/(q-2)\/(q-3)\/(q-4);\n\nJ(1, - 6 + q) = ( 8392704*J(3,q-1)*(738162*q^7 - 13826958*q^6 + 107438526*q^5\n - 446532922*q^4 + 1064669687*q^3 - 1443484631*q^2\n + 1020689473*q - 289691337)\n + 349696*J(3,q-2)*( - 23305518*q^7 + 431941470*q^6\n - 3310148214*q^5 + 13505408608*q^4 - 31391328545*q^3\n + 41045263193*q^2 - 27524008819*q + 7242283425)\n + 64*J(3,q-3)*(5997560976*q^7 - 107967926286*q^6\n + 794611952772*q^5 - 3056894462088*q^4\n + 6480621271814*q^3 - 7204980963707*q^2\n + 3391870943690*q - 197859183171)\n + 288*J(3,q-4)*q*(71372070*q^6 - 1276658510*q^5\n + 9289274641*q^4 - 35007711040*q^3 + 71462127680*q^2\n - 73822280850*q + 29283876009)\n + 36*J(3,q-5)*q*( - 24179544*q^6 + 434455572*q^5\n - 3159380738*q^4 + 11827334899*q^3 - 23816366035*q^2\n + 24112882577*q - 9374746731)\n + 4196352*J(2,q)*(-738162*q^8 + 12350634*q^7 - 79784610*q^6 + 231655870*q^5\n - 171603843*q^4 - 685854743*q^3 + 1866279789*q^2\n -1751687609*q + 579382674)\n + 174848*J(2,q-1)*(18876546*q^8 - 305743830*q^7\n + 1860247722*q^6 - 4615222660*q^5 - 558487079*q^4\n + 27753054687*q^3 - 58619476249*q^2 + 49578489187*q - 15063949524)\n + 64*J(2,q-2)*( - 14810167224*q^8 + 270992616750*q^7 - 2041549790688*q^6\n + 8136869079315*q^5 - 18296821447304*q^4 + 22800932361744*q^3\n - 14321555406258*q^2 + 3748781983207*q - 395718366342)\n + 16*J(2,q-3)*( - 10824070743*q^8 + 207455511240*q^7 - 1660383188328*q^6\n + 7176976971729*q^5 - 18064650946958*q^4 + 26521521351577*q^3\n - 21365258254495*q^2 + 7986599358662*q - 791436732684)\n + 36*J(2,q-4)*q*( - 541141251*q^7 + 10676099500*q^6\n - 88342419106*q^5 + 396420066920*q^4 - 1038067530681*q^3\n + 1577944291122*q^2 - 1278897041426*q + 420807674922)\n + 36*J(2,q-5)*q*(15986898*q^7 - 324636501*q^6\n + 2767299563*q^5-12801558868*q^4 + 34573795993*q^3\n - 54197750750*q^2 + 45244769538*q - 15277905873)\n + 262272*J(1,q)*( - 1476324*q^8 + 22486782*q^7\n - 118088346*q^6+140996162*q^5 + 996391080*q^4\n - 4565718547*q^3 + 8063013471*q^2 - 6565443637*q + 2027839359)\n + 21856*J(1,q-1)*(1479240*q^9 - 718824*q^8 - 290505888*q^7\n + 3040697422*q^6 - 14276034470*q^5 + 36530607805*q^4\n - 52418121878*q^3 + 40019451544*q^2 - 13989602036*q + 1448456685)\n + 8*J(1,q-2)*q*( - 6672113604*q^8 + 143144560506*q^7 - 1317848051532*q^6\n + 6794557026194*q^5 - 21434683657627*q^4 + 42303073328518*q^3\n - 50886066609614*q^2 + 33974478049038*q - 9569982531879)\n + 6*J(1,q-3)*q*(221313528*q^8 - 5165620026*q^7 + 51969200474*q^6\n - 293881218901*q^5 + 1019347369493*q^4 - 2213379274564*q^3\n + 2923819741196*q^2 - 2132296331629*q + 649364820429)\n + 27*J(1,q-4)*q*( - 91046*q^8 + 2191855*q^7 - 22772339*q^6\n + 133314169*q^5 - 480172079*q^4 + 1085700520*q^3\n -1495620096*q^2 + 1136307096*q-358858080)\n + 324*J(1,q-5)*q*( - 3304*q^8 + 86533*q^7 - 972784*q^6\n + 6115624*q^5 - 23438146*q^4 + 55819747*q^3\n - 80159286*q^2 + 62865696*q - 20314080)\n + 63732096*J(0,q)*q*( - q^9 + 23*q^8 - 228*q^7\n + 1278*q^6 - 4461*q^5 + 10047*q^4 - 14582*q^3\n + 13132*q^2 - 6648*q + 1440)\n + 2655504*J(0,q-1)*q*(25*q^9 - 625*q^8 + 6812*q^7\n - 42490*q^6 + 167117*q^5 - 429505*q^4 + 720358*q^3\n - 758580*q^2 + 453528*q - 116640)\n + 663876*J(0,q-2)*q*( - 35*q^9 + 945*q^8 - 11182*q^7\n + 76062*q^6 - 327403*q^5 + 923145*q^4 - 1700348*q^3 + 1964808*q^2\n - 1284552*q + 358560)\n + 663876*J(0,q-3)*q*(5*q^9 - 145*q^8 + 1845*q^7 - 13500*q^6\n + 62476*q^5 - 189120*q^4 + 373040*q^3 - 459875*q^2 + 318994*q - 93720)\n + 165969*J(0,q-4)*q*( - q^9 + 31*q^8 - 421*q^7 + 3281*q^6 - 16130*q^5\n + 51704*q^4 - 107584*q^3 + 139264*q^2 - 100864*q + 30720)\n)\/25515\/q\/(q-1)\/(q-2)\/(q-3)\/(q-4)\/(q-4)\/(q-5)\/(q-5)\/(7*q-24)\n +(87424*B(0,q+1)*( - 1476324*q^6 + 16581486*q^5 - 47333430*q^4\n - 98082016*q^3 + 746063306*q^2 - 1287219275*q + 675946453)\n + 43712*B(0,q)*(4862637*q^6 - 78129165*q^5 + 502175106*q^4\n - 1641458936*q^3 + 2853167747*q^2 - 2474965235*q + 830365446)\n + 24*B(0,q-1)*( - 187645068*q^6 + 3303988746*q^5 - 23822963042*q^4\n + 90075021407*q^3 - 188498675784*q^2 + 207202955323*q - 93562325614)\n + 108*B(0,q-2)*( - 2349349*q^6 + 47038978*q^5 - 386433111*q^4\n + 1668058574*q^3 - 3990763652*q^2 + 5015869656*q - 2585112480)\n + 324*B(0,q-3)*(125620*q^6 - 2697750*q^5 + 23873044*q^4\n - 111347697*q^3 + 288317245*q^2 - 392099154*q + 218030520)\n + 1701*B(0,q-4)*( - 631*q^6 + 14513*q^5 - 137798*q^4\n + 690128*q^3 - 1917728*q^2 + 2792192*q - 1653760)\n)\/51030\/(q-2)\/(q-4)\/(q-4)\/(q-5)\/(q-5)\/(7*q-24);\n\\end{verbatim}\n\n\n\n{\\Large \\bf Appendix 6.}\n\\begin{verbatim}\nJ(p,1)= -( J(p+1,0)*(123*p^3 + 325*p^2 + 141*p + 8)\n\/(3*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+2,0)*( - 1610*p^4 - 7315*p^3 - 12784*p^2 - 10287*p - 3208)\n\/(3*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+3,0)*(2*( - 10100*p^5 - 96976*p^4 - 370679*p^3 - 704416*p^2 - 663875*p\n - 247102))\/(3*(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15))+\nJ(p+4,0)*(64*(400*p^5 + 4432*p^4 + 19118*p^3 + 40113*p^2\n + 41003*p + 16386))\/(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15)+\nJ(p+5,0)*(6144*(2*p^4 + 21*p^3 + 79*p^2 + 126*p + 72))\n\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+1,-1)*(16*p^3 + 54*p^2 - 8*p - 65)\/(4*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+2,-1)*( - 856*p^4 - 4474*p^3 - 7443*p^2 - 4839*p\n - 1014)\/(6*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+3,-1)*(2*(1508*p^5 + 9252*p^4 + 19927*p^3 + 16451*\np^2 + 2040*p - 2084))\/(3*(32*p^5 + 176*p^4 + 336*p^3\n + 280*p^2 + 106*p + 15))+\nJ(p+4,-1)*(8*(5908*p^5 + 62136*p^4 + 256375*p^3 + \n518814*p^2 + 515197*p + 201018))\/(3*(32*p^5\n + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15))+\nJ(p+5,-1)*(256*( - 2*p^4 - 21*p^3 - 79*p^2 - 126*p - \n72))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+2,-2)*( - 61*p^3 - 156*p^2 - 56*p + 39)\n\/(3*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+3,-2)*(1794*p^4 + 8239*p^3 + 13400*p^2 + 9437*p + \n2482)\/(6*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+4,-2)*(13756*p^5 + 133560*p^4 + 508645*p^3 +\n953406*p^2 + 881659*p + 322086)\/(3*(32*p^5 +\n 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15))+\nJ(p+5,-2)*(608*( - 2*p^4 - 21*p^3 - 79*p^2 - 126*p -\n72))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+2,-3)*( - 8*p^3 - 13*p^2 + 8*p + 13)\/(2*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+3,-3)*(3*(64*p^4 + 254*p^3 + 327*p^2 + 163*p + 26)\n)\/(4*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+4,-3)*(724*p^5 + 6504*p^4 + 22567*p^3 + 37962*p^2\n + 31033*p + 9858)\/(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15)+\nJ(p+5,-3)*(96*( - 2*p^4 - 21*p^3 - 79*p^2 - 126*p - 72\n))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+1,1)*(4*(10*p^3 + 80*p^2 + 129*p + 61))\/(8*p^3 + 12*p^2 + 6*p + 1)+\nJ(p+2,1)*(16*( - 540*p^4 - 2780*p^3 - 5285*p^2 - \n4411*p - 1384))\/(3*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+3,1)*(128*( - 320*p^5 - 3220*p^4 - 12772*p^3 - \n24943*p^2 - 23993*p - 9110))\/(32*p^5 +\n 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15)+\nJ(p+4,1)*(6144*( - 4*p^4 - 40*p^3 - 145*p^2 - 225*p\n- 126))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p,2)*(4*( - 10*p^2 - 14*p - 5))\/(8*p^3 + 12*p^2 + 6*p + 1)+\nJ(p+1,2)*(16*( - 260*p^3 - 1052*p^2 - 1423*p - 649))\/\n(3*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+2,2)*(128*( - 40*p^3 - 238*p^2 - 457*p - 282))\/(\n16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nB(p+1,0)*( - 100*p^4 + 579*p^3 + 2723*p^2 + 2859*p + \n677)\/(6*(8*p^5 + 36*p^4 + 58*p^3 + 43*p^2 + 15*p + 2))+\nB(p+2,0)*(2920*p^5 + 14282*p^4 + 21517*p^3 + 7182*p^2\n - 8141*p - 5168)\/(6*(16*p^6 + 128*p^5 + 384*\np^4 + 548*p^3 + 391*p^2 + 135*p + 18))+\nB(p+3,0)*(4*(960*p^6 - 5788*p^5 - 128500*p^4 - 629377*\np^3 - 1383907*p^2 - 1447282*p - 585016))\n\/(3*(32*p^7 + 400*p^6 + 1952*p^5 + 4744*p^4\n + 6098*p^3 + 4117*p^2 + 1377*p + 180))+\nB(p+4,0)*(3968*(2*p^3 + 13*p^2 + 27*p + 18))\/(16*p^5 +\n 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p+1,-1)*(44*p^3 + 244*p^2 + 440*p + 259)\/(4*(8*p^5 +\n 36*p^4 + 58*p^3 + 43*p^2 + 15*p + 2))+\nB(p+2,-1)*(1600*p^5 + 8748*p^4 + 7904*p^3 - 25623*p^2\n - 49512*p - 23133)\/(12*(16*p^6 + 128*p^5 + \n384*p^4 + 548*p^3 + 391*p^2 + 135*p + 18))+\nB(p+3,-1)*( - 400*p^6 - 1728*p^5 - 6008*p^4 - 71868*p^3\n - 306657*p^2 - 490625*p - 262964)\/(3*(32*p^7\n + 400*p^6 + 1952*p^5 + 4744*p^4 + 6098*p^3 + 4117*p^2 + 1377*p + 180))+\nB(p+4,-1)*(5920*(2*p^3 + 13*p^2 + 27*p + 18))\/(16*p^5\n+ 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p+2,-2)*(60*p^3 - 163*p^2 - 777*p - 554)\n\/(6*(8*p^4 + 28*p^3 + 30*p^2 + 13*p + 2))+\nB(p+3,-2)*( - 120*p^4 + 6946*p^3 + 27377*p^2 + 31658*p\n + 11347)\/(12*(16*p^5 + 96*p^4 + 192*p^3 + 164*p^2 + 63*p + 9))+\nB(p+4,-2)*(2816*(2*p^3 + 13*p^2 + 27*p + 18))\/(16*p^5 \n+ 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p+2,-3)*(3*( - 8*p^2 - 21*p - 13))\n\/(2*(8*p^4 + 28*p^3 + 30*p^2 + 13*p + 2))+\nB(p+3,-3)*(6*(22*p^3 + 83*p^2 + 95*p + 34))\n\/(16*p^5 + 96*p^4 + 192*p^3 + 164*p^2 + 63*p + 9)+\nB(p+4,-3)*(576*(2*p^3 + 13*p^2 + 27*p + 18))\n\/(16*p^5 + 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p,1)*(30*p^3 + 90*p^2 + 85*p + 29)\n\/(4*p*(4*p^4 + 16*p^3 + 21*p^2 + 11*p + 2))+\nB(p+1,1)*( - 2400*p^6 - 17780*p^5 - 43716*p^4 -\n32887*p^3 + 23271*p^2 + 43533*p + 15615)\n\/(6*(16*p^7 + 144*p^6 + 512*p^5 + 932*p^4 + 939*p^3\n + 526*p^2 + 153*p + 18))+\nB(p+2,1)*(2*( - 3600*p^7 - 67792*p^6 - 501064*p^5 - \n1910808*p^4 - 4105397*p^3 - 5021161*p^2\n - 3276852*p - 894496))\/(3*(32*p^8 + 464*p^7\n + 2752*p^6 + 8648*p^5 + 15586*p^4 + 16313*p^3\n + 9611*p^2 + 2934*p + 360))+\nB(p+3,1)*(64*( - 1000*p^5 - 10760*p^4 - 45112*p^3 - \n92251*p^2 - 92195*p - 36122))\/(32*p^7 + 400*p^6 + 1952*p^5 + 4744*p^4\n + 6098*p^3 + 4117*p^2 + 1377*p + 180)+\nB(p+4,1)*(6144*( - 2*p^3 - 13*p^2 - 27*p - 18))\n\/(16*p^5 + 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p,2)*( - 440*p^4 - 1476*p^3 - 1390*p^2 - 255*p +\n 87)\/(3*p*(8*p^5 + 36*p^4 + 58*p^3 + 43*p^2+15*p+2))+\nB(p+1,2)*(4*( - 160*p^5 - 2736*p^4 - 12072*p^3 - \n21842*p^2 - 17255*p - 4765))\/(3*(16*p^7 \n+ 144*p^6 + 512*p^5 + 932*p^4 + 939*p^3 + 526*p^2 + 153*p + 18))+\nB(p+2,2)*(128*(40*p^5 + 404*p^4 + 1558*p^3\n + 2847*p^2 + 2426*p + 750))\/(16*p^7 + 208*p^6\n + 1064*p^5 + 2728*p^4 + 3701*p^3 + 2605*p^2 + 898*p + 120)+\nB(p-1,3)*( - 16*p - 29)\/(p*(4*p^3 + 12*p^2 + 9*p + 2))+\nB(p,3)*(4*(88*p^3 + 452*p^2 + 762*p + 447))\n\/(16*p^6 + 128*p^5 + 384*p^4 + 548*p^3 + 391*p^2 + 135*p + 18) +\nB(p+1,3)*(768*(4*p^5 + 42*p^4 + 167*p^3 + 308*p^2 +\n 251*p + 63))\/(16*p^8 + 224*p^7 + 1272*p^6\n + 3792*p^5 + 6429*p^4 + 6306*p^3 + 3503*p^2 + 1018*p + 120) );\n\nJ(p,2)= - (J(p+1,0)*(5*p - 1)\/8 + \nJ(p+2,0)*(302*p^3 + 591*p^2 - 69*p - 416)\n \/(24*(2*p^2 + 5*p + 3))+ \nJ(p+3,0)*( - 1178*p^3 - 6705*p^2 - 12285*p - 7174)\n \/(12*(2*p^2 + 5*p + 3))+ \nJ(p+4,0)*(8*( - 10*p^3 - 61*p^2 - 115*p - 66))\/(2*p^2 + 5*p + 3)+ \nJ(p+2,-1)*( - 20*p^3 - 50*p^2 - 84*p - 59)\/(8*(2*p^2 + 5*p + 3))+ \nJ(p+3,-1)*( - 590*p^3 - 2719*p^2 - 4057*p - 1958)\n \/(12*(2*p^2 + 5*p + 3))+ \nJ(p+4,-1)*(10*p^3 + 61*p^2 + 115*p + 66)\/(3*(2*p^2 + 5*p + 3))+ \nJ(p+2,-2)*( - 3*p)\/8+ \nJ(p+3,-2)*( - 742*p^3 - 2795*p^2 - 3167*p - 1090)\n \/(48*(2*p^2 + 5*p + 3))+ \nJ(p+4,-2)*(19*(10*p^3 + 61*p^2 + 115*p + 66))\/(24*(2*p^2 + 5*p + 3))+ \nJ(p+3,-3)*(p*( - 10*p^2 - 31*p - 22))\/(4*(2*p^2 + 5*p + 3))+ \nJ(p+4,-3)*(10*p^3 + 61*p^2 + 115*p + 66)\/(8*(2*p^2 + 5*p + 3))+ \nJ(p+1,1)*(20*p^2 + 42*p + 23)\/(2*(p + 1))+ \nJ(p+2,1)*(8*(60*p^3 + 299*p^2 + 495*p + 271))\n \/(3*(2*p^2 + 5*p + 3))+ \nJ(p+3,1)*(16*(10*p^3 + 59*p^2 + 110*p + 64))\/(2*p^2 + 5*p + 3)+ \nJ(p+1,2)*(2*( - 4*p^2 - 12*p - 11))\/(3*(2*p^2 + 5*p + 3))+ \nJ(p+2,2)*(48*( - p - 2))\/(p + 1)+ \nB(p+1,0)*(5*p^2 + 20*p + 9)\/(8*(p^2 + 3*p + 2))+ \nB(p+2,0)*( - 438*p^4 - 1663*p^3 - 1857*p^2 - 1002*p - 542)\n \/(24*(2*p^4 + 15*p^3 + 40*p^2 + 45*p + 18))+ \nB(p+3,0)*( - 144*p^3 - 1246*p^2 - 2905*p - 1978)\n \/(6*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+2,-1)*( - 40*p^4 - 300*p^3 - 708*p^2 - 528*p - 53)\n \/(8*(2*p^4 + 15*p^3 + 40*p^2 + 45*p + 18))+ \nB(p+3,-1)*(5*(12*p^3 - 292*p^2 - 985*p - 706))\n \/(24*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+2,-2)*(3*( - p^2 - 4*p - 2))\/(8*(p^2 + 3*p + 2))+ \nB(p+3,-2)*(18*p^3 - 1643*p^2 - 5213*p - 3710)\n \/(48*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+3,-3)*(3*( - 10*p^2 - 31*p - 22))\n \/(4*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+1,1)*(120*p^5 + 1240*p^4 + 4674*p^3 + 8128*p^2 + 6639*p + 2082)\n \/(8*(2*p^5 + 17*p^4 + 55*p^3 + 85*p^2 + 63*p + 18))+ \nB(p+2,1)*(540*p^4 + 6980*p^3 + 26255*p^2 + 38886*p + 20048)\n \/(12*(2*p^4 + 15*p^3 + 40*p^2 + 45*p + 18))+ \nB(p+3,1)*(8*(10*p^2 + 31*p + 22))\/(2*p^3 + 11*p^2 + 18*p + 9)+ \nB(p,2)*1\/(p^2 + 3*p + 2)+ \nB(p+1,2)*( - 200*p^4 - 1168*p^3 - 2588*p^2 - 2525*p - 854)\n \/(6*(2*p^5 + 17*p^4 + 55*p^3 + 85*p^2 + 63*p + 18))+ \nB(p,3)*( - 20*p^2 - 60*p - 57)\n \/(2*(2*p^4 + 13*p^3 + 29*p^2 + 27*p + 9)) );\n\nJ(p,3)= ( 12*B(p,3)*(4*p^3 + 21*p^2 + 36*p + 21)\n + 4*B(p+1, 2)*(20*p^4 + 92*p^3 + 125*p^2 + 36*p - 23)\n + 192*B(p+3,1)*( - p^4 - 6*p^3 - 13*p^2 - 12*p - 4)\n + 2*B(p+2,1)*( - 239*p^4 - 1382*p^3 - 2873*p^2 - 2556*p - 826)\n + 3*B(p+1,1)*( - 10*p^4 - 60*p^3 - 133*p^2 - 132*p - 51)\n + 18*B(p+3,-3)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 88*B(p+3,-2)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 185*B(p+3,-1)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 3*B(p+2,-1)*( - 4*p^4 - 34*p^3 - 97*p^2 - 108*p - 41)\n + 124*B(p+3,0)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 2*B(p+2,0)*( - 103*p^4 - 570*p^3 - 1123*p^2 -948*p -292)\n + 4*J(p+1,2)*( - 20*p^5 - 166*p^4 - 519*p^3 - 764*p^2 - 529*p - 138)\n + 192*J(p+3,1)*( - 2*p^6 - 23*p^5 - 107*p^4 - 257*p^3\n\t - 335*p^2 - 224*p - 60)\n + 2*J(p+2,1)*( - 138*p^6 - 1417*p^5 - 5879*p^4 - 12645*p^3\n\t - 14887*p^2 - 9098*p - 2256)\n + 3*J(p+1,1)*( - 4*p^6 - 38*p^5 - 142*p^4 - 265*p^3\n\t - 256*p^2 - 117*p - 18)\n + 3*J(p+4,-3)*( - p^6 - 12*p^5 - 58*p^4 - 144*p^3 - 193*p^2 - 132*p - 36)\n + 6*J(p+3,-3)*p*(p^5 + 9*p^4 + 31*p^3\t + 51*p^2 + 40*p + 12)\n + 19*J(p+4,-2)*( - p^6 - 12*p^5 - 58*p^4 - 144*p^3\n\t - 193*p^2 - 132*p - 36)\n + 2*J(p+3,-2)*(19*p^6 + 184*p^5 + 706*p^4 + 1372*p^3\n\t + 1423*p^2 + 748*p + 156)\n + 8*J(p+4,-1)*( - p^6 - 12*p^5 - 58*p^4 - 144*p^3\n\t - 193*p^2 - 132*p - 36)\n + J(p+3,-1)*(124*p^6 + 1303*p^5 + 5527*p^4 + 12121*p^3\n\t + 14497*p^2 + 8968*p + 2244)\n + 3*J(p+2,-1)*( - 2*p^6 - 20*p^5 - 75*p^4 - 129*p^3 - 97*p^2 - 19*p + 6)\n + 192*J(p+4,0)*(p^6 + 12*p^5 + 58*p^4 + 144*p^3\n\t + 193*p^2 + 132*p + 36)\n + 2*J(p+3,0)*(89*p^6 + 1006*p^5 + 4604*p^4 + 10894*p^3\n\t + 14015*p^2 + 9268*p + 2460)\n + 2*J(p+2,0)*( - 37*p^6 - 341*p^5 - 1243*p^4 - 2303*p^3\n \t- 2296*p^2 - 1172*p - 240)\n)\/12\/(p+1)\/(p+2)\/(p+3)\/(2*p+3);\n\\end{verbatim}\n\n{\\Large \\bf Appendix 7.}\n\nHere $B(0,q,n)$ is the coefficient of the expansion \n\\[\n{\\cal B}(0,q;\\mu_B^2)=\\sum_{n=0}^{q-2} {B(0,q,n)\\over (\\mu_B^2)^n } + D_0(0,q)(\\ln \\mu_B^2 + C),\n\\]\n(see (\\ref{DPcoeffOne}) and (\\ref{BBandJJat_pleq0}))\nand $D(p,q,r)$ is nothing but $D_{r}(p,q)$ defined in formula (\\ref{DPcoeffOne}).\n\n\\begin{verbatim}\nZ_0(q) = (( - 414720*q^2 + 5184000*q + 105460920\/(q-1) + 22917120\/(q-2)\n + 1462140\/(q-3) - 337161300\/(q+1) + 5257440)*B(0,q - 4,0)\n + (2903040*q^2 - 30274560*q - 350758440\/(q-1) - 49733280\/(q-2)\n - 1462140\/(q-3) + 1236495780\/(q+1) - 26287200)*B(0,q - 3,1)\n + (8294400*q^2 - 84602880*q + 426474720\/(q-1) - 12792960\/(q-2)\n - 4267440\/(q-3) - 1713368880\/(q+1) + 420923520)*B(0,q - 3,0)\n + ( - 8709120*q^2 + 72783360*q + 388058400\/(q-1) + 26816160\/(q-2)\n - 1679831040\/(q+1) + 52574400)*B(0,q - 2,2)\n + ( - 50595840*q^2 + 413061120*q - 681805440\/(q-1) + 32651520\/(q-2)\n + 4409120640\/(q+1) - 1598987520)*B(0,q - 2,1)\n + ( - 58060800*q^2 + 443750400*q - 1814227200\/(q-1) - 29468160\/(q-2)\n + 8956742400\/(q+1) - 2136844800)*B(0,q - 2,0)\n + (14515200*q^2 - 91238400*q - 142760880\/(q-1) + 1003590000\/(q+1)\n - 52574400)*B(0,q - 1,3)\n + (128563200*q^2 - 787968000*q + 234077760\/(q-1) - 4178364480\/(q+1)\n + 2271421440)*B(0,q - 1,2)\n + (306892800*q^2 - 1758412800*q + 1767352320\/(q-1) - 17987028480\/(q+1)\n + 6567989760)*B(0,q - 1,1)\n + (165888000*q^2 - 729907200*q + 388177920\/(q-1) - 2986813440\/(q+1)\n + 1305262080)*B(0,q - 1,0)\n + (414720*q^2 + 829440*q)*B(0,q + 3,7)\n + (9123840*q^2 + 18247680*q)*B(0,q + 3,6)\n + (74649600*q^2 + 149299200*q)*B(0,q + 3,5)\n + (282009600*q^2 + 564019200*q)*B(0,q + 3,4)\n + (491028480*q^2 + 982056960*q)*B(0,q + 3,3)\n + (318504960*q^2 + 637009920*q)*B(0,q + 3,2)\n + ( - 2903040*q^2 + 207360*q)*B(0,q + 2,6)\n + ( - 53913600*q^2 + 1658880*q)*B(0,q + 2,5)\n + ( - 356659200*q^2 - 20736000*q)*B(0,q + 2,4)\n + ( - 1011916800*q^2 - 265420800*q)*B(0,q + 2,3)\n + ( - 1141309440*q^2 - 915701760*q)*B(0,q + 2,2)\n + ( - 318504960*q^2 - 955514880*q)*B(0,q + 2,1)\n + (8709120*q^2 - 18662400*q + 5257440\/(q+1) - 5257440)*B(0,q + 1,5)\n + (132710400*q^2 - 273715200*q - 336216960\/(q+1) + 336216960)*B(0,q + 1,4)\n + (680140800*q^2 - 1277337600*q - 2232368640\/(q+1) + 2232368640)*B(0,q + 1,3)\n + (1343692800*q^2 - 1891123200*q - 2815395840\/(q+1)\n + 2815395840)*B(0,q + 1,2)\n + (809533440*q^2 + 212336640*q + 3530096640\/(q+1) - 3530096640)*B(0,q + 1,1)\n + (955514880*q + 4140564480\/(q+1) - 4140564480)*B(0,q + 1,0)\n + ( - 14515200*q^2 + 61171200*q - 228350880\/(q+1) + 26287200)*B(0,q,4)\n + ( - 174182400*q^2 + 713318400*q + 1824491520\/(q+1) - 1429574400)*B(0,q,3)\n + ( - 646963200*q^2 + 2463436800*q + 11243888640\/(q+1) - 6663513600)*B(0,q,2)\n + ( - 779673600*q^2 + 2322432000*q + 6705192960\/(q+1) - 4209131520)*B(0,q,1)\n + ( - 159252480*q^2 - 278691840*q - 11515944960\/(q+1) + 5381406720)*B(0,q,0)\n + (3034260\/(q-1) + 706560\/(q-2) + 32940\/(q-3) + 604860\/(q-5)\n - 3963900\/(q+1))*D(3,q - 8,1)\n + ( - 22299660\/(q-1) - 4327200\/(q-2) - 164700\/(q-3) - 2903328\/(q-4)\n - 1814580\/(q-5) + 28191708\/(q+1))*D(3,q - 7,2)\n + (2175210\/(q-1) + 974320\/(q-2) + 1169730\/(q-3) - 1752336\/(q-4)\n + 3125110\/(q-5) - 12880514\/(q+1))*D(3,q - 7,1)\n + (70144380\/(q-1) + 11037600\/(q-2) + 5773140\/(q-3) + 8709984\/(q-4)\n + 1814580\/(q-5) - 85867524\/(q+1))*D(3,q - 6,3)\n + ( - 36941910\/(q-1) - 11226160\/(q-2) + 1505790\/(q-3) - 14098512\/(q-4)\n - 6149410\/(q-5) + 112114682\/(q+1))*D(3,q - 6,2)\n + ( - 59868660\/(q-1) + 3139840\/(q-2) + 3407220\/(q-3) - 11231232\/(q-4)\n + 10685860\/(q-5) + 86076892\/(q+1))*D(3,q - 6,1)\n + ( - 122426100\/(q-1) - 19848480\/(q-2) - 16660620\/(q-3) - 8709984\/(q-4)\n - 604860\/(q-5) + 145025724\/(q+1))*D(3,q - 5,4)\n + (98060730\/(q-1) + 17506320\/(q-2) + 24591150\/(q-3) + 28615152\/(q-4)\n + 2721870\/(q-5) - 303376182\/(q+1))*D(3,q - 5,3)\n + (252260220\/(q-1) - 25022880\/(q-2) + 30365100\/(q-3) - 48235296\/(q-4)\n - 604860\/(q-5) - 349490604\/(q+1))*D(3,q - 5,2)\n + (239320080\/(q-1) + 52675200\/(q-2) + 9501840\/(q-3) + 87594624\/(q-4)\n - 16936080\/(q-5) - 306353424\/(q+1))*D(3,q - 5,1)\n + (129869280\/(q-1) + 25993440\/(q-2) + 16495920\/(q-3) + 2903328\/(q-4)\n - 146231568\/(q+1))*D(3,q - 4,5)\n + ( - 112768560\/(q-1) - 32463600\/(q-2) - 49388040\/(q-3) - 12764304\/(q-4)\n + 425803704\/(q+1))*D(3,q - 4,4)\n + ( - 543929760\/(q-1) - 33583200\/(q-2) + 64962720\/(q-3) + 1550304\/(q-4)\n + 828260736\/(q+1))*D(3,q - 4,3)\n + (350288640\/(q-1) - 32133120\/(q-2) - 93375360\/(q-3) + 81593856\/(q-4)\n - 526175616\/(q+1))*D(3,q - 4,2)\n + ( - 841824000\/(q-1) + 77160960\/(q-2) - 173082240\/(q-3) + 8418816\/(q-4)\n + 13071744\/(q+1))*D(3,q - 4,1)\n + ( - 85719600\/(q-1) - 19195200\/(q-2) - 5476680\/(q-3)\n + 87167160\/(q+1))*D(3,q - 3,6)\n + (66969720\/(q-1) + 42256800\/(q-2) + 22104900\/(q-3)\n - 350303580\/(q+1))*D(3,q - 3,5)\n + (662632560\/(q-1) - 9288000\/(q-2) + 9513720\/(q-3)\n - 1072048680\/(q+1))*D(3,q - 3,4)\n + ( - 897347520\/(q-1) - 51736320\/(q-2) - 171905760\/(q-3)\n + 1579946400\/(q+1))*D(3,q - 3,3)\n + ( - 347765760\/(q-1) + 318689280\/(q-2) - 67564800\/(q-3)\n + 886268160\/(q+1))*D(3,q - 3,2)\n + (1177804800\/(q-1) - 349470720\/(q-2) + 99671040\/(q-3)\n + 1938539520\/(q+1))*D(3,q - 3,1)\n + (33372000\/(q-1) + 5633280\/(q-2) - 27393120\/(q+1))*D(3,q - 2,7)\n + ( - 23814000\/(q-1) - 17400960\/(q-2) + 173510640\/(q+1))*D(3,q - 2,6)\n + ( - 389590560\/(q-1) - 33592320\/(q-2) + 729384480\/(q+1))*D(3,q - 2,5)\n + (592565760\/(q-1) + 186209280\/(q-2) - 1316528640\/(q+1))*D(3,q - 2,4)\n + (1726202880\/(q-1) + 196715520\/(q-2) - 2554398720\/(q+1))*D(3,q - 2,3)\n + ( - 2683514880\/(q-1) - 91791360\/(q-2) + 5111009280\/(q+1))*D(3,q - 2,2)\n + ( - 1648926720\/(q-1) + 139345920\/(q-2) - 2179768320\/(q+1))*D(3,q - 2,1)\n + ( - 5974560\/(q-1) + 2656800\/(q+1))*D(3,q - 1,8)\n + (4801680\/(q-1) - 49314960\/(q+1))*D(3,q - 1,7)\n + (108125280\/(q-1) - 232264800\/(q+1))*D(3,q - 1,6)\n + ( - 136183680\/(q-1) + 456347520\/(q+1))*D(3,q - 1,5)\n + ( - 1061475840\/(q-1) + 1740510720\/(q+1))*D(3,q - 1,4)\n + (173352960\/(q-1) - 4119275520\/(q+1))*D(3,q - 1,3)\n + (3423928320\/(q-1) - 10377953280\/(q+1))*D(3,q - 1,2)\n + (2229534720\/(q-1) - 1592524800\/(q+1))*D(3,q - 1,1)\n + 414720\/(q+1)*D(3,q,9) + 6428160\/(q+1)*D(3,q,8)\n + 21150720\/(q+1)*D(3,q,7) - 75479040\/(q+1)*D(3,q,6)\n - 296939520\/(q+1)*D(3,q,5) + 1074954240\/(q+1)*D(3,q,4)\n + 4804116480\/(q+1)*D(3,q,3) + 4459069440\/(q+1)*D(3,q,2)\n + ( - 55219230\/(q-1) - 8610480\/(q-2) - 247050\/(q-3) - 1451664\/(q-4)\n - 907290\/(q-5) + 102412674\/(q+1) + 1866240)*D(2,q - 7,1)\n + (172115550\/(q-1) + 21855600\/(q-2) + 3215970\/(q-3) + 4354992\/(q-4)\n + 907290\/(q-5) - 327435642\/(q+1) - 7464960)*D(2,q - 6,2)\n + ( - 183874230\/(q-1) - 31308000\/(q-2) - 3285090\/(q-3) - 2452320\/(q-4)\n - 2117010\/(q-5) + 549853290\/(q+1) - 21841920)*D(2,q - 6,1)\n + ( - 297787050\/(q-1) - 31999440\/(q-2) - 8659710\/(q-3) - 4354992\/(q-4)\n - 302430\/(q-5) + 590587782\/(q+1) + 17418240)*D(2,q - 5,3)\n + (477897210\/(q-1) + 66902640\/(q-2) + 7588890\/(q-3) + 9710640\/(q-4)\n + 1512150\/(q-5) - 1517830410\/(q+1) + 75479040)*D(2,q - 5,2)\n + (246864120\/(q-1) - 42446400\/(q-2) - 7535160\/(q-3) + 20407872\/(q-4)\n - 6048600\/(q-5) - 286075752\/(q+1) + 33315840)*D(2,q - 5,1)\n + (309786840\/(q-1) + 29772720\/(q-2) + 8412660\/(q-3) + 1451664\/(q-4)\n - 654424524\/(q+1) - 26127360)*D(2,q - 4,4)\n + ( - 663575040\/(q-1) - 71833200\/(q-2) - 18628200\/(q-3) - 7107984\/(q-4)\n + 2314426344\/(q+1) - 149022720)*D(2,q - 4,3)\n + ( - 710805600\/(q-1) + 40844160\/(q-2) - 36370800\/(q-3) + 28281600\/(q-4)\n + 1218275280\/(q+1) - 73681920)*D(2,q - 4,2)\n + (1596136320\/(q-1) - 11470080\/(q-2) - 28428480\/(q-3) - 89899776\/(q-4)\n - 4522732224\/(q+1) + 201000960)*D(2,q - 4,1)\n + ( - 194803920\/(q-1) - 16395840\/(q-2) - 2771280\/(q-3)\n + 452746080\/(q+1) + 26127360)*D(2,q - 3,5)\n + (518633640\/(q-1) + 48381600\/(q-2) + 13039380\/(q-3)\n - 2107865820\/(q+1) + 183859200)*D(2,q - 3,4)\n + (1012815360\/(q-1) + 40938240\/(q-2) - 50103360\/(q-3)\n - 2232649920\/(q+1) + 76723200)*D(2,q - 3,3)\n + ( - 2834017920\/(q-1) + 23185920\/(q-2) + 140374080\/(q-3)\n + 10110626880\/(q+1) - 742348800)*D(2,q - 3,2)\n + (731934720\/(q-1) - 143769600\/(q-2) + 135060480\/(q-3)\n - 1296829440\/(q+1) + 511488000)*D(2,q - 3,1)\n + (69033600\/(q-1) + 3964320\/(q-2) - 188030880\/(q+1) - 17418240)*D(2,q - 2,6)\n + ( - 219587760\/(q-1) - 17556480\/(q-2) + 1152483120\/(q+1)\n - 145152000)*D(2,q - 2,5)\n + ( - 728297280\/(q-1) + 29093760\/(q-2) + 2024331840\/(q+1)\n - 28339200)*D(2,q - 2,4)\n + (2019525120\/(q-1) - 57300480\/(q-2) - 10609781760\/(q+1)\n + 1346457600)*D(2,q - 2,3)\n + (1086750720\/(q-1) - 235653120\/(q-2) - 4727715840\/(q+1)\n - 199065600)*D(2,q - 2,2)\n + ( - 2711162880\/(q-1) + 273162240\/(q-2) + 19434885120\/(q+1)\n - 5240954880)*D(2,q - 2,1)\n + ( - 10711440\/(q-1) + 41089680\/(q+1) + 7464960)*D(2,q - 1,7)\n + (40821840\/(q-1) - 357097680\/(q+1) + 71608320)*D(2,q - 1,6)\n + (183746880\/(q-1) - 876052800\/(q+1) - 13409280)*D(2,q - 1,5)\n + ( - 500601600\/(q-1) + 5522446080\/(q+1) - 1304985600)*D(2,q - 1,4)\n + ( - 908236800\/(q-1) + 7611770880\/(q+1) - 818380800)*D(2,q - 1,3)\n + (1542205440\/(q-1) - 20560711680\/(q+1) + 7406346240)*D(2,q - 1,2)\n + ( - 145981440\/(q-1) - 16137584640\/(q+1) + 7843184640)*D(2,q - 1,1)\n + ( - 207360\/(q+1) + 207360)*D(2,q + 1,9)\n + ( - 2488320\/(q+1) + 2488320)*D(2,q + 1,8)\n + (3732480\/(q+1) - 3732480)*D(2,q + 1,7)\n + (131880960\/(q+1) - 131880960)*D(2,q + 1,6)\n + (343388160\/(q+1) - 343388160)*D(2,q + 1,5)\n + ( - 806215680\/(q+1) + 806215680)*D(2,q + 1,4)\n + ( - 3808788480\/(q+1) + 3808788480)*D(2,q + 1,3)\n + ( - 3503554560\/(q+1) + 3503554560)*D(2,q + 1,2)\n + ( - 2864160\/(q+1) - 1866240)*D(2,q,8)\n + (53412480\/(q+1) - 20183040)*D(2,q,7)\n + (136045440\/(q+1) + 15068160)*D(2,q,6)\n + ( - 1351987200\/(q+1) + 651110400)*D(2,q,5)\n + ( - 3259975680\/(q+1) + 1014681600)*D(2,q,4)\n + (7019274240\/(q+1) - 4265533440)*D(2,q,3)\n + (18685624320\/(q+1) - 9900195840)*D(2,q,2)\n + (5414584320\/(q+1) - 3503554560)*D(2,q,1)\n + (5806080*q + 525346245\/(q-1) + 60022440\/(q-2) + 271485\/(q-3)\n + 3629160\/(q-4) + 756075\/(q-5) - 1441292205\/(q+1) + 56987280)*D(1,q -5,1)\n + ( - 20321280*q - 1095709320\/(q-1) - 90395640\/(q-2) - 7076160\/(q-3)\n - 3629160\/(q-4) + 3353470920\/(q+1) - 170961840)*D(1,q - 4,2)\n + ( - 88542720*q + 128255400\/(q-1) + 20111040\/(q-2) + 4887540\/(q-3)\n - 3279168\/(q-4) - 2442027132\/(q+1) + 755926560)*D(1,q - 4,1)\n + (40642560*q + 1140907140\/(q-1) + 66553680\/(q-2) + 6895170\/(q-3)\n - 4146890790\/(q+1) + 284936400)*D(1,q - 3,3)\n + (269982720*q + 169401240\/(q-1) + 995040\/(q-2) + 5354100\/(q-3)\n + 3574083060\/(q+1) - 1710309600)*D(1,q - 3,2)\n + (385689600*q - 2864138400\/(q-1) - 61000320\/(q-2) + 5040720\/(q-3)\n + 16590738000\/(q+1) - 3805142400)*D(1,q - 3,1)\n + ( - 50803200*q - 595411200\/(q-1) - 21235200\/(q-2) + 2887281600\/(q+1)\n - 284936400)*D(1,q - 2,4)\n + ( - 457228800*q - 378488160\/(q-1) - 15764640\/(q-2) - 2653613760\/(q+1)\n + 2041070400)*D(1,q - 2,3)\n + ( - 1026432000*q + 2902803840\/(q-1) + 28934400\/(q-2) - 25745139840\/(q+1)\n + 7837274880)*D(1,q - 2,2)\n + ( - 120268800*q + 479473920\/(q-1) + 80524800\/(q-2) - 2808011520\/(q+1)\n + 539619840)*D(1,q - 2,1)\n + (40642560*q + 125452800\/(q-1) - 1090694160\/(q+1) + 170961840)*D(1,q - 1,5)\n + (464486400*q + 170800560\/(q-1) + 1167529680\/(q+1)\n - 1351296000)*D(1,q - 1,4)\n + (1451520000*q - 940357440\/(q-1) + 18635400000\/(q+1)\n - 7927856640)*D(1,q - 1,3)\n + (481075200*q - 833817600\/(q-1) + 9558397440\/(q+1)\n - 3023516160)*D(1,q - 1,2)\n + ( - 2202992640*q + 830822400\/(q-1) - 27323412480\/(q+1)\n + 13092710400)*D(1,q - 1,1)\n - 725760*q*D(1,q + 2,8)\n - 13893120*q*D(1,q + 2,7)\n - 85017600*q*D(1,q + 2,6)\n - 120268800*q*D(1,q + 2,5)\n + 550748160*q*D(1,q + 2,4)\n + 1897758720*q*D(1,q + 2,3)\n + 1592524800*q*D(1,q + 2,2)\n + (5806080*q - 8141040\/(q+1) + 8141040)*D(1,q + 1,7)\n + (95800320*q + 66152160\/(q+1) - 66152160)*D(1,q + 1,6)\n + (485222400*q + 769979520\/(q+1) - 769979520)*D(1,q + 1,5)\n + (481075200*q + 1112209920\/(q+1) - 1112209920)*D(1,q + 1,4)\n + ( - 2202992640*q - 3838924800\/(q+1) + 3838924800)*D(1,q + 1,3)\n + ( - 4591779840*q - 9196830720\/(q+1) + 9196830720)*D(1,q + 1,2)\n + ( - 1592524800*q - 3822059520\/(q+1) + 3822059520)*D(1,q + 1,1)\n + ( - 20321280*q + 189367200\/(q+1) - 56987280)*D(1,q,6)\n + ( - 283046400*q - 362849760\/(q+1) + 468715680)*D(1,q,5)\n + ( - 1150848000*q - 6244715520\/(q+1) + 3941015040)*D(1,q,4)\n + ( - 721612800*q - 6432307200\/(q+1) + 3339394560)*D(1,q,3)\n + (3304488960*q + 18636963840\/(q+1) - 12536709120)*D(1,q,2)\n + (3490283520*q + 19292774400\/(q+1) - 13052067840)*D(1,q,1)\n + (2903040*q^2 - 27786240*q - 41109120\/(q-1) - 3663360\/(q-2) + 285120\/(q-3)\n - 8389440\/(q+1) + 81907200)*D(0,q - 3,1)\n + ( - 8709120*q^2 + 66562560*q + 43856640\/(q-1) + 1451520\/(q-2)\n + 74960640\/(q+1) - 163814400)*D(0,q - 2,2)\n + ( - 50595840*q^2 + 396472320*q + 249246720\/(q-1) + 14929920\/(q-2)\n + 498908160\/(q+1) - 999475200)*D(0,q - 2,1)\n + (14515200*q^2 - 82944000*q - 15344640\/(q-1) - 119439360\/(q+1)\n + 163814400)*D(0,q - 1,3)\n + (128563200*q^2 - 754790400*q - 135613440\/(q-1) - 1170754560\/(q+1)\n + 1530316800)*D(0,q - 1,2)\n + (306892800*q^2 - 1924300800*q - 282839040\/(q-1) - 3881779200\/(q+1)\n + 4227655680)*D(0,q - 1,1)\n + (414720*q^2 + 829440*q)*D(0,q + 3,7)\n + (9123840*q^2 + 18247680*q)*D(0,q + 3,6)\n + (74649600*q^2 + 149299200*q)*D(0,q + 3,5)\n + (282009600*q^2 + 564019200*q)*D(0,q + 3,4)\n + (491028480*q^2 + 982056960*q)*D(0,q + 3,3)\n + (318504960*q^2 + 637009920*q)*D(0,q + 3,2)\n + ( - 2903040*q^2 - 207360*q)*D(0,q + 2,6)\n + ( - 53913600*q^2 - 1658880*q)*D(0,q + 2,5)\n + ( - 356659200*q^2 + 20736000*q)*D(0,q + 2,4)\n + ( - 1011916800*q^2 + 265420800*q)*D(0,q + 2,3)\n + ( - 1141309440*q^2 + 915701760*q)*D(0,q + 2,2)\n + ( - 318504960*q^2 + 955514880*q)*D(0,q + 2,1)\n + (8709120*q^2 - 16174080*q - 16381440\/(q+1) + 16381440)*D(0,q + 1,5)\n + (132710400*q^2 - 257126400*q - 265420800\/(q+1) + 265420800)*D(0,q + 1,4)\n + (680140800*q^2 - 1443225600*q - 1575106560\/(q+1) + 1575106560)*D(0,q + 1,3)\n + (1343692800*q^2 - 3483648000*q - 4283228160\/(q+1)\n + 4283228160)*D(0,q + 1,2)\n + (809533440*q^2 - 3450470400*q - 5202247680\/(q+1) + 5202247680)*D(0,q + 1,1)\n + ( - 14515200*q^2 + 54950400*q + 74649600\/(q+1) - 81907200)*D(0,q,4)\n + ( - 174182400*q^2 + 680140800*q + 970444800\/(q+1) - 1040947200)*D(0,q,3)\n + ( - 646963200*q^2 + 2712268800*q + 4494735360\/(q+1) - 4476487680)*D(0,q,2)\n + ( - 779673600*q^2 + 3914956800*q + 8944680960\/(q+1) - 7697203200)*D(0,q,1)\n + (1517130\/(q-1) + 353280\/(q-2) + 16470\/(q-3) + 302430\/(q-5)\n - 1981950\/(q+1))*J(3,q - 8)\n + (6662520\/(q-1) + 1568960\/(q-2) + 626040\/(q-3) - 150336\/(q-4)\n + 2016200\/(q-5) - 13488184\/(q+1))*J(3,q - 7)\n + ( - 144679680\/(q-1) - 25155840\/(q-2) + 607680\/(q-3) - 1002240\/(q-4)\n - 19355520\/(q-5) + 177696960\/(q+1))*J(3,q - 6)\n + (129669120\/(q-1) + 65802240\/(q-2) - 44904960\/(q-3)\n + 9621504\/(q-4) + 215824896\/(q+1))*J(3,q - 5)\n + (2016645120\/(q-1) - 189112320\/(q-2) + 113909760\/(q-3)\n - 4272721920\/(q+1))*J(3,q - 4)\n + ( - 4910284800\/(q-1) + 159252480\/(q-2) + 10590289920\/(q+1))*J(3,q - 3)\n + (7585650\/(q-1) + 1413120\/(q-2) + 49410\/(q-3) + 302430\/(q-5)\n - 13873650\/(q+1) - 207360)*J(2,q - 8)\n + (29684340\/(q-1) + 5413440\/(q-2) + 1285020\/(q-3) - 150336\/(q-4)\n + 604860\/(q-5) - 84893004\/(q+1) + 2764800)*J(2,q - 7)\n + ( - 34566480\/(q-1) + 12218240\/(q-2) + 3280320\/(q-3) - 300672\/(q-4)\n - 4032400\/(q-5) + 14409632\/(q+1) - 5944320)*J(2,q - 6)\n + ( - 311880960\/(q-1) + 21319680\/(q-2) + 2508480\/(q-3) + 2004480\/(q-4)\n + 19355520\/(q-5) + 728736960\/(q+1) - 19353600)*J(2,q - 5)\n + ( - 546186240\/(q-1) - 48936960\/(q-2) + 51586560\/(q-3) - 9621504\/(q-4)\n + 1238552064\/(q+1) - 165335040)*J(2,q - 4)\n + ( - 498216960\/(q-1) + 199065600\/(q-2) - 113909760\/(q-3)\n - 3636817920\/(q+1) + 1293926400)*J(2,q - 3)\n + (4817387520\/(q-1) - 159252480\/(q-2) - 5613649920\/(q+1)\n - 1751777280)*J(2,q - 2)\n + ( - 725760*q - 100585665\/(q-1) - 14945280\/(q-2) - 90495\/(q-3)\n - 756075\/(q-5) + 256898475\/(q+1) - 8141040)*J(1,q - 6)\n + (12441600*q - 88154460\/(q-1) - 10180320\/(q-2) - 4797900\/(q-3)\n + 375840\/(q-4) + 604860\/(q-5) + 650604780\/(q+1) - 137954880)*J(1,q - 5)\n + ( - 60134400*q + 913299840\/(q-1) + 20880000\/(q-2) - 2079360\/(q-3)\n - 300672\/(q-4) - 4009344768\/(q+1) + 724688640)*J(1,q - 4)\n + (280108800\/(q-1) - 11750400\/(q-2) - 9106560\/(q-3) - 1342730880\/(q+1)\n + 256711680)*J(1,q - 3)\n + (550748160*q - 391219200\/(q-1) - 36495360\/(q-2) + 11039016960\/(q+1)\n - 4306452480)*J(1,q - 2)\n + ( - 796262400*q - 905748480\/(q-1) - 1403412480\/(q+1)\n + 2826731520)*J(1,q - 1)\n + ( - 414720*q^2 + 4769280*q + 12597120\/(q-1) + 2211840\/(q-2) - 285120\/(q-3)\n - 5400000\/(q+1) - 16381440)*J(0,q - 4)\n + (8294400*q^2 - 81285120*q - 111352320\/(q-1) - 17971200\/(q-2)\n + 1140480\/(q-3) - 33557760\/(q+1) + 244684800)*J(0,q - 3)\n + ( - 58060800*q^2 + 485222400*q + 236390400\/(q-1) + 32071680\/(q-2)\n + 976527360\/(q+1) - 1326274560)*J(0,q - 2)\n + (165888000*q^2 - 1260748800*q - 49766400\/(q-1) - 4439162880\/(q+1)\n + 3413975040)*J(0,q - 1)\n + ( - 159252480*q^2 + 1552711680*q + 6210846720\/(q+1) - 4379443200)*J(0,q)\n ) \/ 19906560;\n\nZ_1(q) = (( - 41148*q - 194304\/(q-1) - 26292\/(q-2) + 246888)*B(0,q - 3,0)\n + (164592*q + 414900\/(q-1) + 26292\/(q-2) - 740664)*B(0,q - 2,1)\n + ( - 72000*q - 318960\/(q-1) + 34320\/(q-2) + 324000)*B(0,q - 2,0)\n + ( - 246888*q - 220596\/(q-1) + 740664)*B(0,q - 1,2)\n + (98208*q + 178656\/(q-1) - 294624)*B(0,q - 1,1)\n + (1127232*q + 1319616\/(q-1) - 3381696)*B(0,q - 1,0)\n - 41148*q*B(0,q + 1,4)\n - 45792*q*B(0,q + 1,3)\n + 848448*q*B(0,q + 1,2)\n + 2377728*q*B(0,q + 1,1)\n + 1327104*q*B(0,q + 1,0)\n + (164592*q - 246888)*B(0,q,3)\n + (19584*q - 29376)*B(0,q,2)\n + ( - 2003328*q + 3004992)*B(0,q,1)\n + ( - 2529792*q + 3794688)*B(0,q,0)\n + ( - 8832\/(q-1) - 1476\/(q-2) - 5892\/(q-4))*D(3,q - 7,1)\n + (54468\/(q-1) + 7380\/(q-2) + 17676\/(q-3) + 17676\/(q-4))*D(3,q - 6,2)\n + (2482\/(q-1) - 11622\/(q-2) + 12150\/(q-3) - 30442\/(q-4))*D(3,q - 6,1)\n + ( - 139860\/(q-1) - 32436\/(q-2) - 53028\/(q-3) - 17676\/(q-4))*D(3,q - 5,3)\n + (40598\/(q-1) + 20394\/(q-2) + 81390\/(q-3) + 59902\/(q-4))*D(3,q - 5,2)\n + (37216\/(q-1) - 35580\/(q-2) + 76032\/(q-3) - 104092\/(q-4))*D(3,q - 5,1)\n + (197292\/(q-1) + 67788\/(q-2) + 53028\/(q-3) + 5892\/(q-4))*D(3,q - 4,4)\n + ( - 74622\/(q-1) - 101910\/(q-2) - 169770\/(q-3) - 26514\/(q-4))*D(3,q - 4,3)\n + ( - 137916\/(q-1) - 92724\/(q-2) + 278604\/(q-3) + 5892\/(q-4))*D(3,q - 4,2)\n + ( - 469008\/(q-1) - 116208\/(q-2) - 507120\/(q-3) + 164976\/(q-4))*D(3,q - 4,1)\n + ( - 164916\/(q-1) - 60408\/(q-2) - 17676\/(q-3))*D(3,q - 3,5)\n + (60978\/(q-1) + 163356\/(q-2) + 76230\/(q-3))*D(3,q - 3,4)\n + (475212\/(q-1) - 144264\/(q-2) - 2772\/(q-3))*D(3,q - 3,3)\n + ( - 351744\/(q-1) + 64896\/(q-2) - 498240\/(q-3))*D(3,q - 3,2)\n + (1008576\/(q-1) + 993024\/(q-2) - 92736\/(q-3))*D(3,q - 3,1)\n + (78048\/(q-1) + 19152\/(q-2))*D(3,q - 2,6)\n + ( - 36576\/(q-1) - 69480\/(q-2))*D(3,q - 2,5)\n + ( - 448848\/(q-1) - 73296\/(q-2))*D(3,q - 2,4)\n + (733248\/(q-1) + 630720\/(q-2))*D(3,q - 2,3)\n + (1196928\/(q-1) + 450432\/(q-2))*D(3,q - 2,2)\n + ( - 705024\/(q-1) - 483840\/(q-2))*D(3,q - 2,1)\n - 16200\/(q-1)*D(3,q - 1,7)\n + 11556\/(q-1)*D(3,q - 1,6)\n + 204552\/(q-1)*D(3,q - 1,5)\n - 266112\/(q-1)*D(3,q - 1,4)\n - 1772928\/(q-1)*D(3,q - 1,3)\n - 1548288\/(q-1)*D(3,q - 1,2)\n + (108198\/(q-1) + 11070\/(q-2) + 8838\/(q-3) + 8838\/(q-4) - 56700)*D(2,q - 6,1)\n + ( - 276030\/(q-1) - 30978\/(q-2) - 26514\/(q-3) - 8838\/(q-4)\n + 170100)*D(2,q - 5,2)\n + (256764\/(q-1) + 58086\/(q-2) + 12708\/(q-3) + 20622\/(q-4)\n - 218592)*D(2,q - 5,1)\n + (378366\/(q-1) + 48654\/(q-2) + 26514\/(q-3) + 2946\/(q-4)\n - 283500)*D(2,q - 4,3)\n + ( - 560046\/(q-1) - 91206\/(q-2) - 56898\/(q-3) - 14730\/(q-4)\n + 522720)*D(2,q - 4,2)\n + ( - 110904\/(q-1) + 96552\/(q-2) - 129432\/(q-3) + 58920\/(q-4)\n - 180144)*D(2,q - 4,1)\n + ( - 295938\/(q-1) - 37584\/(q-2) - 8838\/(q-3) + 283500)*D(2,q - 3,4)\n + (605046\/(q-1) + 101820\/(q-2) + 42534\/(q-3) - 665280)*D(2,q - 3,3)\n + (530352\/(q-1) + 80592\/(q-2) - 168480\/(q-3) - 181440)*D(2,q - 3,2)\n + ( - 825312\/(q-1) + 180864\/(q-2) + 532512\/(q-3) + 1518336)*D(2,q - 3,1)\n + (125892\/(q-1) + 11052\/(q-2) - 170100)*D(2,q - 2,5)\n + ( - 336048\/(q-1) - 51912\/(q-2) + 475200)*D(2,q - 2,4)\n + ( - 752208\/(q-1) + 156240\/(q-2) + 777024)*D(2,q - 2,3)\n + (1178112\/(q-1) - 340992\/(q-2) - 2460672)*D(2,q - 2,2)\n + ( - 177408\/(q-1) - 767232\/(q-2) - 1442304)*D(2,q - 2,1)\n + ( - 22824\/(q-1) + 56700)*D(2,q - 1,6)\n + (80244\/(q-1) - 180576)*D(2,q - 1,5)\n + (246960\/(q-1) - 713232)*D(2,q - 1,4)\n + ( - 468288\/(q-1) + 1347840)*D(2,q - 1,3)\n + ( - 582912\/(q-1) + 3303936)*D(2,q - 1,2)\n + (497664\/(q-1) + 110592)*D(2,q - 1,1)\n - 8100*D(2,q,7) + 28512*D(2,q,6) + 212976*D(2,q,5) - 139968*D(2,q,4)\n - 1575936*D(2,q,3) - 1603584*D(2,q,2)\n + ( - 136404*q - 761517\/(q-1) - 42759\/(q-2) - 22095\/(q-3) - 7365\/(q-4)\n + 842490)*D(1,q - 4,1)\n + (341010*q + 1163655\/(q-1) + 64854\/(q-2) + 22095\/(q-3)\n - 1684980)*D(1,q - 3,2)\n + (144684*q + 297576\/(q-1) - 56556\/(q-2) + 21816\/(q-3) - 366696)*D(1,q - 3,1)\n + ( - 454680*q - 797388\/(q-1) - 36348\/(q-2) + 1684980)*D(1,q - 2,3)\n + ( - 489456*q - 575604\/(q-1) - 9540\/(q-2) + 1253448)*D(1,q - 2,2)\n + (853056*q + 1411440\/(q-1) - 15888\/(q-2) - 3081312)*D(1,q - 2,1)\n + (341010*q + 208434\/(q-1) - 842490)*D(1,q - 1,4)\n + (689544*q + 302484\/(q-1) - 1304568)*D(1,q - 1,3)\n + ( - 1067616*q - 716640\/(q-1) + 2909088)*D(1,q - 1,2)\n + ( - 1528128*q - 794304\/(q-1) + 3600576)*D(1,q - 1,1)\n + 22734*q*D(1,q + 1,6) + 108972*q*D(1,q + 1,5)\n - 41472*q*D(1,q + 1,4) - 931392*q*D(1,q + 1,3)\n - 1520640*q*D(1,q + 1,2) - 663552*q*D(1,q + 1,1)\n + ( - 136404*q + 168498)*D(1,q,5) + ( - 444816*q + 443376)*D(1,q,4)\n + (482688*q - 816768)*D(1,q,3)\n + (2328192*q - 2827584)*D(1,q,2) + (815616*q - 1472256)*D(1,q,1)\n + ( - 8640*q^2 + 60480*q + 53568\/(q-1) - 1728\/(q-2) - 129600)*D(0,q - 2,1)\n + (17280*q^2 - 86400*q - 25920\/(q-1) + 129600)*D(0,q - 1,2)\n + (138240*q^2 - 691200*q - 207360\/(q-1) + 1036800)*D(0,q - 1,1)\n + ( - 1728*q^2 - 1728*q)*D(0,q + 2,5)\n + ( - 34560*q^2 - 34560*q)*D(0,q + 2,4)\n + ( - 241920*q^2 - 241920*q)*D(0,q + 2,3)\n + ( - 691200*q^2 - 691200*q)*D(0,q + 2,2)\n + ( - 663552*q^2 - 663552*q)*D(0,q + 2,1)\n + (8640*q^2 - 8640*q)*D(0,q + 1,4)\n + (138240*q^2 - 138240*q)*D(0,q + 1,3)\n + (725760*q^2 - 725760*q)*D(0,q + 1,2)\n + (1382400*q^2 - 1382400*q)*D(0,q + 1,1)\n + ( - 17280*q^2 + 51840*q - 43200)*D(0,q,3)\n + ( - 207360*q^2 + 622080*q - 518400)*D(0,q,2)\n + ( - 725760*q^2 + 2177280*q - 1810944)*D(0,q,1)\n + ( - 4416\/(q-1) - 738\/(q-2) - 2946\/(q-4))*J(3,q - 7)\n + ( - 12376\/(q-1) - 7656\/(q-2) + 1656\/(q-3) - 19640\/(q-4))*J(3,q - 6)\n + (377376\/(q-1) + 37632\/(q-2) + 11040\/(q-3) + 188544\/(q-4))*J(3,q - 5)\n + ( - 1218816\/(q-1) + 232704\/(q-2) - 105984\/(q-3))*J(3,q - 4)\n + (1216512\/(q-1) - 552960\/(q-2))*J(3,q - 3)\n + ( - 17664\/(q-1) - 2214\/(q-2) - 2946\/(q-4) + 8100)*J(2,q - 7)\n + ( - 45960\/(q-1) - 16788\/(q-2) + 1656\/(q-3) - 5892\/(q-4)\n + 38016)*J(2,q - 6)\n + ( - 12400\/(q-1) - 38784\/(q-2) + 3312\/(q-3) + 39280\/(q-4)\n + 84816)*J(2,q - 5)\n + (60672\/(q-1) - 60864\/(q-2) - 22080\/(q-3) - 188544\/(q-4)\n - 265536)*J(2,q - 4)\n + (1200384\/(q-1) - 221184\/(q-2) + 105984\/(q-3) + 36864)*J(2,q - 3)\n + ( - 1714176\/(q-1) + 552960\/(q-2) + 110592)*J(2,q - 2)\n + (22734*q + 186816\/(q-1) + 14253\/(q-2) + 7365\/(q-4) - 168498)*J(1,q - 5)\n + ( - 8928*q - 18564\/(q-1) + 48420\/(q-2) - 4140\/(q-3) - 5892\/(q-4)\n - 25560)*J(1,q - 4)\n + ( - 226656*q - 636624\/(q-1) + 14304\/(q-2) + 3312\/(q-3) + 988992)*J(1,q - 3)\n + (158976*q + 25344\/(q-1) + 41472\/(q-2) - 663552)*J(1,q - 2)\n + (857088*q + 290304\/(q-1) - 635904)*J(1,q - 1)\n + (1728*q^2 - 15552*q - 27648\/(q-1) + 1728\/(q-2) + 43200)*J(0,q - 3)\n + ( - 34560*q^2 + 241920*q + 214272\/(q-1) - 6912\/(q-2) - 518400)*J(0,q - 2)\n + (241920*q^2 - 1209600*q - 359424\/(q-1) + 1810944)*J(0,q - 1)\n + ( - 691200*q^2 + 2073600*q - 1714176)*J(0,q)\n) \/ 41472;\n\nZ_2(q) = ((900*q^2 - 3764*q + 1964\/(q-1) + 1964)*B(0,q - 2,0)\n + ( - 2700*q^2 + 7528*q - 1964\/(q-1) - 1964)*B(0,q - 1,1)\n + ( - 1776*q^2 + 3616*q - 368\/(q-1) - 368)*B(0,q - 1,0)\n - 900*q^2*B(0,q + 1,3) - 3408*q^2*B(0,q + 1,2)\n - 3072*q^2*B(0,q + 1,1) + (2700*q^2 - 3764*q)*B(0,q,2)\n + (5184*q^2 - 6112*q)*B(0,q,1) + (1152*q^2 - 192*q)*B(0,q,0)\n + (156\/(q-1) + 468\/(q-3) + 312)*D(3,q - 6,1)\n + ( - 780\/(q-1) - 624\/(q-2) - 1404\/(q-3) - 1560)*D(3,q - 5,2)\n + (554\/(q-1) - 504\/(q-2) + 2418\/(q-3) + 1108)*D(3,q - 5,1)\n + (1716\/(q-1) + 1872\/(q-2) + 1404\/(q-3) + 3120)*D(3,q - 4,3)\n + ( - 1986\/(q-1) - 2648\/(q-2) - 4758\/(q-3) - 4896)*D(3,q - 4,2)\n + (1604\/(q-1) - 3072\/(q-2) + 8268\/(q-3) + 2824)*D(3,q - 4,1)\n + ( - 2028\/(q-1) - 1872\/(q-2) - 468\/(q-3) - 3120)*D(3,q - 3,4)\n + (2894\/(q-1) + 5768\/(q-2) + 2106\/(q-3) + 6480)*D(3,q - 3,3)\n + (316\/(q-1) - 9072\/(q-2) - 468\/(q-3) - 4376)*D(3,q - 3,2)\n + (6480\/(q-1) + 16576\/(q-2) - 13104\/(q-3) + 10400)*D(3,q - 3,1)\n + (1248\/(q-1) + 624\/(q-2) + 1560)*D(3,q - 2,5)\n + ( - 2176\/(q-1) - 2616\/(q-2) - 3484)*D(3,q - 2,4)\n + ( - 2416\/(q-1) - 240\/(q-2) - 2536)*D(3,q - 2,3)\n + (12416\/(q-1) + 17664\/(q-2) + 21248)*D(3,q - 2,2)\n + ( - 21888\/(q-1) + 5376\/(q-2) - 19200)*D(3,q - 2,1)\n + ( - 312\/(q-1) - 312)*D(3,q - 1,6) + (636\/(q-1) + 636)*D(3,q - 1,5)\n + (3144\/(q-1) + 3144)*D(3,q - 1,4)\n + ( - 9504\/(q-1) - 9504)*D(3,q - 1,3) + ( - 21504\/(q-1) - 21504)*D(3,q - 1,2)\n + (936*q - 1170\/(q-1) - 312\/(q-2) - 702\/(q-3) - 1560)*D(2,q - 5,1)\n + ( - 2340*q + 2418\/(q-1) + 936\/(q-2) + 702\/(q-3) + 3120)*D(2,q - 4,2)\n + (4496*q - 4362\/(q-1) - 336\/(q-2) - 1638\/(q-3) - 5076)*D(2,q - 4,1)\n + (3120*q - 2574\/(q-1) - 936\/(q-2) - 234\/(q-3) - 3120)*D(2,q - 3,3)\n + ( - 8544*q + 6642\/(q-1) + 1896\/(q-2) + 1170\/(q-3) + 7980)*D(2,q - 3,2)\n + (6176*q - 4824\/(q-1) + 4832\/(q-2) - 4680\/(q-3) - 3968)*D(2,q - 3,1)\n + ( - 2340*q + 1404\/(q-1) + 312\/(q-2) + 1560)*D(2,q - 2,4)\n + (8096*q - 4832\/(q-1) - 1464\/(q-2) - 5564)*D(2,q - 2,3)\n + ( - 544*q + 656\/(q-1) + 5760\/(q-2) + 3536)*D(2,q - 2,2)\n + ( - 22144*q - 3264\/(q-1) - 18048\/(q-2) - 12288)*D(2,q - 2,1)\n + (936*q - 312\/(q-1) - 312)*D(2,q - 1,5) + ( - 3824*q + 1452\/(q-1)\n + 1452)*D(2,q - 1,4)\n + ( - 5984*q - 480\/(q-1) - 480)*D(2,q - 1,3) + (27904*q - 4800\/(q-1)\n - 4800)*D(2,q - 1,2)\n + (16896*q + 18432\/(q-1) + 18432)*D(2,q - 1,1)\n - 156*q*D(2,q,6) + 720*q*D(2,q,5)\n + 3216*q*D(2,q,4) - 6528*q*D(2,q,3) - 26880*q*D(2,q,2) - 18432*q*D(2,q,1)\n + (2010*q^2 - 9136*q + 5025\/(q-1) + 780\/(q-2) + 585\/(q-3)\n + 5610)*D(1,q - 3,1)\n + ( - 4020*q^2 + 13704*q - 5220\/(q-1) - 780\/(q-2) - 5610)*D(1,q - 2,2)\n + ( - 1428*q^2 - 2316*q + 2532\/(q-1) - 864\/(q-2) + 2100)*D(1,q - 2,1)\n + (4020*q^2 - 9136*q + 1870\/(q-1) + 1870)*D(1,q - 1,3)\n + (4716*q^2 - 3096*q - 12\/(q-1) - 12)*D(1,q - 1,2)\n + ( - 15696*q^2 + 27232*q - 1840\/(q-1) - 1840)*D(1,q - 1,1)\n + 402*q^2*D(1,q + 1,5) + 1644*q^2*D(1,q + 1,4)\n - 1776*q^2*D(1,q + 1,3) - 16128*q^2*D(1,q + 1,2) - 18432*q^2*D(1,q + 1,1)\n + ( - 2010*q^2 + 2284*q)*D(1,q,4)\n + ( - 4860*q^2 + 2836*q)*D(1,q,3)\n + (11328*q^2 - 13856*q)*D(1,q,2)\n + (37248*q^2 - 19776*q)*D(1,q,1)\n + (78\/(q-1) + 234\/(q-3) + 156)*J(3,q - 6)\n + (472\/(q-1) - 96\/(q-2) + 1560\/(q-3) + 944)*J(3,q - 5)\n + ( - 5312\/(q-1) - 640\/(q-2) - 14976\/(q-3) - 10624)*J(3,q - 4)\n + (3072\/(q-1) + 6144\/(q-2) + 6144)*J(3,q - 3)\n + ( - 156*q + 234\/(q-1) + 234\/(q-3) + 312)*J(2,q - 6)\n + ( - 944*q + 1100\/(q-1) - 96\/(q-2) + 468\/(q-3) + 1208)*J(2,q - 5)\n + ( - 2864*q + 2048\/(q-1) - 192\/(q-2) - 3120\/(q-3) + 912)*J(2,q - 4)\n + (1536*q + 5056\/(q-1) + 1280\/(q-2) + 14976\/(q-3) + 10688)*J(2,q - 3)\n + (23808*q - 7680\/(q-1) - 6144\/(q-2) - 10752)*J(2,q - 2)\n + ( - 402*q^2 + 2284*q - 1675\/(q-1) - 585\/(q-3) - 1870)*J(1,q - 4)\n + ( - 72*q^2 + 2576*q - 2364\/(q-1) + 240\/(q-2) + 468\/(q-3) - 2088)*J(1,q - 3)\n + (6144*q^2 - 11648*q + 448\/(q-1) - 192\/(q-2) + 352)*J(1,q - 2)\n + ( - 19200*q^2 + 9984*q + 1152\/(q-1) + 1152)*J(1,q - 1)\n) \/ 1152;\n\nZ_3(q) = (( - 26*q^3 + 88*q^2 - 62*q)*B(0,q - 1,0)\n + ( - 26*q^3 + 26*q^2)*B(0,q + 1,2)\n + ( - 64*q^3 + 64*q^2)*B(0,q + 1,1)\n + (52*q^3 - 114*q^2 + 62*q)*B(0,q,1)\n + (16*q^3 + 24*q^2 - 40*q)*B(0,q,0)\n + ( - 12*q - 24\/(q-2) - 12)*D(3,q - 5,1)\n + (48*q + 72\/(q-2) + 36)*D(3,q - 4,2)\n + ( - 72*q - 124\/(q-2) - 62)*D(3,q - 4,1)\n + ( - 72*q - 72\/(q-2) - 36)*D(3,q - 3,3)\n + (200*q + 244\/(q-2) + 122)*D(3,q - 3,2)\n + ( - 324*q - 424\/(q-2) - 212)*D(3,q - 3,1)\n + (48*q + 24\/(q-2) + 12)*D(3,q - 2,4)\n + ( - 160*q - 108\/(q-2) - 54)*D(3,q - 2,3)\n + (184*q + 24\/(q-2) + 12)*D(3,q - 2,2)\n + (1056*q + 672\/(q-2) + 336)*D(3,q - 2,1)\n - 12*q*D(3,q - 1,5) + 38*q*D(3,q - 1,4) + 92*q*D(3,q - 1,3)\n - 832*q*D(3,q - 1,2) - 1536*q*D(3,q - 1,1)\n + ( - 30*q^2 + 54*q + 36\/(q-2) + 18)*D(2,q - 4,1)\n + (60*q^2 - 96*q - 36\/(q-2) - 18)*D(2,q - 3,2)\n + ( - 176*q^2 + 254*q + 84\/(q-2) + 42)*D(2,q - 3,1)\n + ( - 60*q^2 + 84*q + 12\/(q-2) + 6)*D(2,q - 2,3)\n + (240*q^2 - 336*q - 60\/(q-2) - 30)*D(2,q - 2,2)\n + ( - 312*q^2 + 568*q + 240\/(q-2) + 120)*D(2,q - 2,1)\n + (30*q^2 - 36*q)*D(2,q - 1,4) + ( - 144*q^2 + 182*q)*D(2,q - 1,3)\n + ( - 8*q^2 - 200*q)*D(2,q - 1,2)\n + (1472*q^2 - 1536*q)*D(2,q - 1,1) + ( - 6*q^2 + 6*q)*D(2,q,5)\n + (32*q^2 - 32*q)*D(2,q,4)\n + (104*q^2 - 104*q)*D(2,q,3) + ( - 352*q^2 + 352*q)*D(2,q,2)\n + ( - 768*q^2 + 768*q)*D(2,q,1)\n + ( - 52*q^3 + 208*q^2 - 186*q - 30\/(q-2) - 15)*D(1,q - 2,1)\n + (78*q^3 - 234*q^2 + 171*q)*D(1,q - 1,2)\n + (74*q^3 + 96*q^2 - 194*q)*D(1,q - 1,1)\n + (13*q^3 - 13*q^2)*D(1,q + 1,4)\n + (58*q^3 - 58*q^2)*D(1,q + 1,3)\n + (64*q^3 - 64*q^2)*D(1,q + 1,2)\n + ( - 52*q^3 + 104*q^2 - 52*q)*D(1,q,3)\n + ( - 124*q^3 + 90*q^2 + 34*q)*D(1,q,2)\n + ( - 16*q^3 - 344*q^2 + 360*q)*D(1,q,1)\n + ( - 6*q - 12\/(q-2) - 6)*J(3,q - 5)\n + ( - 48*q - 80\/(q-2) - 40)*J(3,q - 4)\n + (1984*q + 768\/(q-2) + 384)*J(3,q - 3)\n + (6*q^2 - 12*q - 12\/(q-2) - 6)*J(2,q - 5)\n + (48*q^2 - 68*q - 24\/(q-2) - 12)*J(2,q - 4)\n + (216*q^2 - 152*q + 160\/(q-2) + 80)*J(2,q - 3)\n + ( - 800*q^2 - 1248*q - 768\/(q-2) - 384)*J(2,q - 2)\n + (13*q^3 - 65*q^2 + 67*q + 30\/(q-2) + 15)*J(1,q - 3)\n + ( - 8*q^3 - 128*q^2 + 144*q - 24\/(q-2) - 12)*J(1,q - 2)\n + (96*q^2 + 240*q)*J(1,q - 1)\n) \/ 96;\n\\end{verbatim}\n\n\n\n{\\Large \\bf Appendix 8.}\n\nHere and below, the symbol {\\tt JJ(p,q)} in any Appendix\ndesignates ${\\cal J}(p,q)$, whereas {\\tt J(p,q)} designates $J(p,q)$.\nIn computations at $q\\leq 0$, $\\mu_B=${\\tt muB}$=0$ and\n{\\tt JJ(p,q)} should be replaced with {\\tt J(p,q)}.\n\n\n\\begin{verbatim}\nJJ(p,q)=(( 24*BB(p,q)*(1\/(p+3)+1\/(p+2)+1\/(p+1))*( - q^3 + 6*q^2 - 11*q + 6) \n + 24*BB(p+1,q-1)*(1\/(p+3)+1\/(p+2))*( - 5*(p+1)*q^2\n + 25*(p+1)*q - 30*(p+1) + 2*q^2 - 10*q + 12)\n + 48*BB(p+1,q-1)\/(p+1)*(q^2 - 5*q + 6) \n + 2*BB(p+1,q-2)\/(p+3)*( - 8*(p+2)*(p+1)*q + 24*(p+2)*(p+1)\n - 21*(p+1)*q^2 + 163*(p+1)*q - 300*(p+1) + 6*q^3 - 42*q^2 + 96*q - 72)\n + 2*BB(p+1,q-2)\/(p+2)*( - 12*(p+3)*(p+1)*q + 36*(p+3)*(p+1)\n - 21*(p+1)*q^2 + 163*(p+1)*q - 300*(p+1) + 6*q^3 - 42*q^2 + 96*q - 72) \n + 12*BB(p+1,q-2)\/(p+1)*(q^3 - 7*q^2 + 16*q - 12) \n + 24*BB(p+2,q-2)\/(p+3)*(p+1)*( - 11*(p+2)*q + 33*(p+2) + 8*q - 24) \n + 192*BB(p+2,q-2)\/(p+2)*(p+1)*(q - 3) \n + 4*BB(p+2,q-3)\/(p+3)*(p+1)*( - 20*(p+2)*q + 118*(p+2) + 15*q^2 - 81*q + 83) \n + 4*BB(p+2,q-3)\/(p+2)*(p+1)*(12*(p+3) + 15*q^2 - 81*q + 83)\n + BB(p+2,q-4)\/(p+3)*(p+1)*( - 4*(p+2)*q + 46*(p+2) + 21*q^2 - 172*q + 352) \n + BB(p+2,q-4)\/(p+2)*(p+1)*(12*(p+3)*q-48*(p+3)+21*q^2 - 172*q + 352) \n + 336*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1) \n + 4*BB(p+3,q-4)\/(p+3)*(p+2)*(p+1)*(33*q - 152) \n + 4*BB(p+3,q-5)\/(p+3)*(p+2)*(p+1)*(10*q - 53) \n + 6*BB(p+3,q-6)\/(p+3)*(p+2)*(p+1)*(q - 6) \n + 24*JJ(p+1,q-1)*(5*(p+1)*q^2 - 25*(p+1)*q + 30*(p+1)\n - 2*q^2 + 10*q - 12) \n + 2*JJ(p+1,q-2)*(12*(p+3)*(p+1)*q - 36*(p+3)*(p+1)\n + 8*(p+2)*(p+1)*q - 24*(p+2)*(p+1) + 21*(p+1)*q^2\n - 163*(p+1)*q + 300*(p+1) - 6*q^3 + 42*q^2-96*q+72)\n + 24*JJ(p+2,q-2)*(p+1)*(11*(p+2)*q - 33*(p+2) - 8*q + 24) \n + 4*JJ(p+2,q-3)*(p+1)*(30*(p+3)*(p+2)-12*(p+3)+20*(p+2)*q\n - 118*(p+2) - 15*q^2 + 81*q - 83) \n + JJ(p+2,q-4)*(p+1)*(10*(p+3)*(p+2) - 12*(p+3)*q+48*(p+3)\n + 4*(p+2)*q - 46*(p+2) - 21*q^2 + 172*q - 352) \n + 24*JJ(p+3,q-3)*(p+2)*(p+1)*(15*(p+3) - 14) \n + 2*JJ(p+3,q-4)*(p+2)*(p+1)*( - 73*(p+3) - 66*q + 304) \n + 4*JJ(p+3,q-5)*(p+2)*(p+1)*( - 10*(p+3) - 10*q + 53)\n + 6*JJ(p+3,q-6)*(p+2)*(p+1)*( - (p+3) - q + 6) \n - 192*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1) \n + 20*JJ(p+4,q-5)*(p+3)*(p+2)*(p+1) \n + 3*JJ(p+4,q-6)*(p+3)*(p+2)*(p+1)\n )\/12+muB^2*(\n 24*BB(p+1,q)*(1\/(p+3)+1\/(p+2)+1\/(p+1))*(q^3 - 6*q^2 + 11*q - 6) \n + 6*BB(p+1,q-1)*(1\/(p+3)+1\/(p+2))*(7*(p+1)*q^2-35*(p+1)*q\n + 42*(p+1) - 4*q^3 + 26*q^2 - 54*q + 36) \n + 12*BB(p+1,q-1)\/(p+1)*( - 2*q^3 + 13*q^2 - 27*q + 18)\n + 120*BB(p+2,q-1)*(1\/(p+3)+1\/(p+2))*(p+1)*(q^2 - 5*q + 6) \n + 2*BB(p+2,q-2)\/(p+3)*(p+1)*(68*(p+2)*q - 204*(p+2)\n - 39*q^2 + 149*q - 96) \n + 2*BB(p+2,q-2)\/(p+2)*(p+1)*(12*(p+3)*q - 36*(p+3) - 39*q^2 + 149*q - 96) \n + BB(p+2,q-3)\/(p+3)*(p+1)*(8*(p+2)*q - 58*(p+2) - 63*q^2 + 449*q - 805) \n + BB(p+2,q-3)\/(p+2)*(p+1)*( - 24*(p+3)*q + 84*(p+3)\n - 63*q^2 + 449*q - 805) \n + 264*BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*(q - 3) \n + 4*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1)*( - 46*q + 121) \n + 2*BB(p+3,q-4)\/(p+3)*(p+2)*(p+1)*( - 64*q + 273) \n + 6*BB(p+3,q-5)\/(p+3)*(p+2)*(p+1)*( - 4*q + 21) \n + 24*JJ(p+1,q)*( - q^3 + 6*q^2 - 11*q + 6) \n + 6*JJ(p+1,q-1)*( - 7*(p+1)*q^2 + 35*(p+1)*q\n - 42*(p+1) + 4*q^3 - 26*q^2 + 54*q - 36) \n + 120*JJ(p+2,q-1)*(p+1)*( - q^2 + 5*q - 6) \n + 2*JJ(p+2,q-2)*(p+1)*(-12*(p+3)*q+36*(p+3) - 68*(p+2)*q\n + 204*(p+2) + 39*q^2 - 149*q + 96)\n + JJ(p+2,q-3)*(p+1)*( - 10*(p+3)*(p+2) + 24*(p+3)*q\n -84*(p+3)-8*(p+2)*q+58*(p+2) + 63*q^2 - 449*q + 805)\n + 264*JJ(p+3,q-2)*(p+2)*(p+1)*( - q + 3)\n + 2*JJ(p+3,q-3)*(p+2)*(p+1)*( - 47*(p+3) + 92*q - 242) \n + 2*JJ(p+3,q-4)*(p+2)*(p+1)*(30*(p+3) + 64*q - 273) \n + 6*JJ(p+3,q-5)*(p+2)*(p+1)*(3*(p+3) + 4*q - 21) \n - 168*JJ(p+4,q-3)*(p+3)*(p+2)*(p+1) \n + 106*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1) \n + 31*JJ(p+4,q-5)*(p+3)*(p+2)*(p+1) \n + 6*JJ(p+4,q-6)*(p+3)*(p+2)*(p+1)\n )\/12+muB^4*(\n 12*BB(p+1,q)*(1\/(p+3)+1\/(p+2)+1\/(p+1))*(q^3 - 6*q^2 + 11*q - 6) \n + 18*BB(p+2,q-1)*(1\/(p+3)+1\/(p+2))*(p+1)*(q^2 - 5*q + 6) \n + BB(p+2,q-2)\/(p+3)*(p+1)*( - 4*(p+2)*q + 12*(p+2) + 63*q^2 - 382*q + 579) \n + BB(p+2,q-2)\/(p+2)*(p+1)*(12*(p+3)*q - 36*(p+3) + 63*q^2 - 382*q + 579) \n + 12*BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*(q - 3) \n + 2*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1)*(68*q - 239) \n + 18*BB(p+3,q-4)\/(p+3)*(p+2)*(p+1)*(2*q - 9) \n + 12*JJ(p+1,q)*( - q^3 + 6*q^2 - 11*q + 6) \n + 18*JJ(p+2,q-1)*(p+1)*( - q^2 + 5*q - 6) \n + JJ(p+2,q-2)*(p+1)*(-12*(p+3)*q + 36*(p+3) + 4*(p+2)*q\n - 12*(p+2) - 63*q^2 + 382*q - 579) \n + 12*JJ(p+3,q-2)*(p+2)*(p+1)*( - q + 3) \n + 2*JJ(p+3,q-3)*(p+2)*(p+1)*( - 10*(p+3) - 68*q + 239) \n + 18*JJ(p+3,q-4)*(p+2)*(p+1)*( - (p+3) - 2*q + 9) \n - 6*JJ(p+4,q-3)*(p+3)*(p+2)*(p+1) \n - 61*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1) \n - 18*JJ(p+4,q-5)*(p+3)*(p+2)*(p+1)\n )\/12+muB^6*(\n 7*BB(p+2,q-1)*(1\/(p+3)+1\/(p+2))*(p+1)*( - q^2 + 5*q - 6) \n + 16*BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*( - q + 3) \n + 2*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1)*( - 4*q + 15) \n + 7*JJ(p+2,q-1)*(p+1)*(q^2 - 5*q + 6) \n + 16*JJ(p+3,q-2)*(p+2)*(p+1)*(q - 3) \n + 2*JJ(p+3,q-3)*(p+2)*(p+1)*((p+3) + 4*q - 15) \n + 9*JJ(p+4,q-3)*(p+3)*(p+2)*(p+1) \n + 6*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1)\n )\/4+muB^8*(\n BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*(q - 3) \n + JJ(p+3,q-2)*(p+2)*(p+1)*( - q + 3) \n - JJ(p+4,q-3)*(p+3)*(p+2)*(p+1)\n )\/2)\/(-2)\/(q-1)\/(q-2)\/(q-3);\n\nJJ(-3,q) = ( 6*BB(0,q - 6)\n + ( - 18*muB^2 + 40)*BB(0,q - 5)\n + (18*muB^4 - 60*muB^2 + 146)*BB(0,q - 4)\n + ( - 6*muB^6 + 20*muB^4 + 94*muB^2 - 360)*BB(0,q - 3)\n + ( - 21*q^2 + 160*q - 314)*BB(-1,q - 4)\n + ((63*q^2 - 425*q + 731)*muB^2 - 60*q^2 + 324*q - 500)*BB(-1,q - 3)\n + (( - 63*q^2 + 370*q - 543)*muB^4 + (78*q^2 - 322*q + 264)*muB^2\n - 192*q + 576)*BB(-1,q - 2)\n + ((21*q^2 - 105*q + 126)*muB^6 + ( - 18*q^2 + 90*q - 108)*muB^4\n + ( - 120*q^2 + 600*q - 720)*muB^2)*BB(-1,q - 1)\n + (9*q^3 - 21*q^2 - 158*q + 420)*BB(-2,q - 2)\n + (( - 18*q^3 + 75*q^2 - 33*q - 90)*muB^2\n + 156*q^2 - 780*q + 936)*BB(-2,q - 1)\n + ((9*q^3 - 54*q^2 + 99*q - 54)*muB^4\n + (18*q^3 - 108*q^2 + 198*q - 108)*muB^2)*BB(-2,q)\n + ( - 18*q^3 + 108*q^2 - 198*q + 108)*BB(-3,q)\n + ( - 6*muB^2 - 3)*JJ(1,q - 6)\n + (18*muB^4 - 31*muB^2 - 20)*JJ(1,q - 5)\n + ( - 18*muB^6 + 61*muB^4 - 106*muB^2 + 192)*JJ(1,q - 4)\n + (6*muB^8 - 27*muB^6 + 6*muB^4 + 168*muB^2)*JJ(1,q - 3)\n + (6*q - 36)*JJ(0,q - 6)\n + (( - 24*q + 126)*muB^2 + 40*q - 212)*JJ(0,q - 5)\n + ((36*q - 162)*muB^4 + ( - 128*q + 546)*muB^2\n + 132*q - 608)*JJ(0,q - 4)\n + (( - 24*q + 90)*muB^6 + (136*q - 478)*muB^4\n + ( - 184*q + 484)*muB^2 + 336)*JJ(0,q - 3)\n + ((6*q - 18)*muB^8 + ( - 48*q + 144)*muB^6 + (12*q - 36)*muB^4\n + (264*q - 792)*muB^2)*JJ(0,q - 2)\n + ( - 21*q^2 + 168*q - 306)*JJ(-1,q - 4)\n + ((63*q^2 - 441*q + 747)*muB^2 - 60*q^2 + 244*q + 140)*JJ(-1,q - 3)\n + (( - 63*q^2 + 378*q - 567)*muB^4\n + (78*q^2 - 162*q - 216)*muB^2 - 456*q + 1368)*JJ(-1,q - 2)\n + ((21*q^2 - 105*q + 126)*muB^6 + ( - 18*q^2 + 90*q - 108)*muB^4\n + ( - 120*q^2 + 600*q - 720)*muB^2)*JJ(-1,q - 1)\n + (6*q^3 - 246*q + 576)*JJ(-2,q - 2)\n + (( - 12*q^3 + 36*q^2 + 48*q - 144)*muB^2\n + 144*q^2 - 720*q + 864)*JJ(-2,q - 1)\n + ((6*q^3 - 36*q^2 + 66*q - 36)*muB^4\n + (12*q^3 - 72*q^2 + 132*q - 72)*muB^2)*JJ(-2,q))\n\/ (12*q^3 - 72*q^2 + 132*q - 72);\n\nJJ(-2,q) = ((21*q^2 - 168*q + 316)*BB(0,q - 4)\n + (( - 63*q^2 + 441*q - 757)*muB^2 + 60*q^2 - 244*q - 20)*BB(0,q - 3)\n + ((63*q^2 - 378*q + 567)*muB^4\n + ( - 78*q^2 + 162*q + 216)*muB^2 + 456*q - 1368)*BB(0,q - 2)\n + (( - 21*q^2 + 105*q - 126)*muB^6 + (18*q^2 - 90*q + 108)*muB^4\n + (120*q^2 - 600*q + 720)*muB^2)*BB(0,q - 1)\n + ( - 42*q^2 + 342*q - 648)*BB(-1,q - 2)\n + ((42*q^2 - 210*q + 252)*muB^2 - 120*q^2 + 600*q - 720)*BB(-1,q - 1)\n + (6*muB^2 + 3)*JJ(2,q - 6)\n + ( - 18*muB^4 + 31*muB^2 + 20)*JJ(2,q - 5)\n + (18*muB^6 - 61*muB^4 + 106*muB^2 - 192)*JJ(2,q - 4)\n + ( - 6*muB^8 + 27*muB^6 - 6*muB^4 - 168*muB^2)*JJ(2,q - 3)\n + ( - 6*q + 30)*JJ(1,q - 6)\n + ((24*q - 108)*muB^2 - 40*q + 172)*JJ(1,q - 5)\n + (( - 36*q + 144)*muB^4 + (128*q - 486)*muB^2 - 132*q + 462)*JJ(1,q - 4)\n + ((24*q - 84)*muB^6 + ( - 136*q + 458)*muB^4\n + (184*q - 578)*muB^2 + 24)*JJ(1,q - 3)\n + (( - 6*q + 18)*muB^8 + (48*q - 144)*muB^6\n + ( - 12*q + 36)*muB^4 + ( - 264*q + 792)*muB^2)*JJ(1,q - 2)\n + ( - 21*q^2 + 160*q - 304)*JJ(0,q - 4)\n + ((63*q^2 - 425*q + 721)*muB^2 - 60*q^2 + 324*q - 380)*JJ(0,q - 3)\n + (( - 63*q^2 + 370*q - 543)*muB^4 + (78*q^2 - 322*q + 264)*muB^2\n - 192*q + 576)*JJ(0,q - 2)\n + ((21*q^2 - 105*q + 126)*muB^6 + ( - 18*q^2 + 90*q - 108)*muB^4\n + ( - 120*q^2 + 600*q - 720)*muB^2)*JJ(0,q - 1)\n + (12*q^3 - 42*q^2 - 110*q + 384)*JJ(-1,q - 2)\n + (( - 24*q^3 + 114*q^2 - 114*q - 36)*muB^2\n + 168*q^2 - 840*q + 1008)*JJ(-1,q - 1)\n + ((12*q^3 - 72*q^2 + 132*q - 72)*muB^4\n + (24*q^3 - 144*q^2 + 264*q - 144)*muB^2)*JJ(-1,q))\n\/ (24*q^3 - 144*q^2 + 264*q - 144);\n\nJJ(-1,q) = ( ( - 42*q^2 + 262*q - 408)*BB(0,q - 2)\n + ((42*q^2 - 210*q + 252)*muB^2 - 120*q^2 + 600*q - 720)*BB(0,q - 1)\n + ( - 12*muB^2 - 6)*JJ(3,q - 6)\n + (36*muB^4 - 62*muB^2 - 40)*JJ(3,q - 5) \n + ( - 36*muB^6 + 122*muB^4 - 212*muB^2 + 384)*JJ(3,q - 4)\n + (12*muB^8 - 54*muB^6 + 12*muB^4 + 336*muB^2)*JJ(3,q - 3)\n + (6*q - 24)*JJ(2,q - 6)\n + (( - 24*q + 90)*muB^2 + 40*q - 132)*JJ(2,q - 5)\n + ((36*q - 126)*muB^4 + ( - 128*q + 426)*muB^2 + 132*q - 316)*JJ(2,q - 4)\n + (( - 24*q + 78)*muB^6 + (136*q - 438)*muB^4\n + ( - 184*q + 672)*muB^2 - 384)*JJ(2,q - 3)\n + ((6*q - 18)*muB^8 + ( - 48*q + 144)*muB^6\n + (12*q - 36)*muB^4 + (264*q - 792)*muB^2)*JJ(2,q - 2)\n + (21*q^2 - 152*q + 282)*JJ(1,q - 4)\n + (( - 63*q^2 + 409*q - 675)*muB^2 + 60*q^2 - 404*q + 660)*JJ(1,q - 3)\n + ((63*q^2 - 362*q + 519)*muB^4\n + ( - 78*q^2 + 482*q - 744)*muB^2 - 72*q + 216)*JJ(1,q - 2)\n + (( - 21*q^2 + 105*q - 126)*muB^6 + (18*q^2 - 90*q + 108)*muB^4\n + (120*q^2 - 600*q + 720)*muB^2)*JJ(1,q - 1)\n + (12*q^3 - 84*q^2 + 192*q - 144)*JJ(0,q - 2)\n + (( - 24*q^3 + 156*q^2 - 324*q + 216)*muB^2\n + 48*q^2 - 240*q + 288)*JJ(0,q - 1)\n + ((12*q^3 - 72*q^2 + 132*q - 72)*muB^4\n + (24*q^3 - 144*q^2 + 264*q - 144)*muB^2)*JJ(0,q))\n \/ (24*q^3 - 144*q^2 + 264*q - 144);\n\\end{verbatim}\n\n\n\n{\\Large \\bf Appendix 9.}\n\n\\begin{verbatim}\nJ(p,q) = ( 24*J(p-4,q+4)*(q^3 + 6*q^2 + 11*q + 6) \n+ 2*J(p-3,q+2)*( - 6*q^3 + 21*q^2*(p-3) - 30*q^2\n + 12*q*(p-1)*(p-3) + 8*q*(p-2)*(p-3) + 5*q*(p-3)\n - 48*q + 12*(p-1)*(p-3) + 8*(p-2)*(p-3) - 16*(p-3) - 24) \n+ 24*J(p-3,q+3)*(5*q^2*(p-3) - 2*q^2 + 15*q*(p-3) - 6*q + 10*(p-3) - 4) \n+ J(p-2,q)*(p-3)*( - 21*q^2 - 12*q*(p-1) + 4*q*(p-2)\n + 4*q + 10*(p-1)*(p-2) - 30*(p-2)) \n+ 4*J(p-2,q+1)*(p-3)*( - 15*q^2 + 20*q*(p-2) - 39*q\n + 30*(p-1)*(p-2) - 12*(p-1) - 38*(p-2) + 1) \n+ 24*J(p-2,q+2)*(p-3)*(11*q*(p-2) - 8*q + 11*(p-2) - 8) \n+ 6*J(p-1,q-2)*(p-2)*(p-3)*( - q - (p-1) + 2) \n+ 4*J(p-1,q-1)*(p-2)*(p-3)*( - 10*q - 10*(p-1) + 13) \n+ 2*J(p-1,q)*(p-2)*(p-3)*( - 66*q - 73*(p-1) + 40) \n+ 24*J(p-1,q+1)*(p-2)*(p-3)*(15*(p-1) - 14) \n+ (3*J(p,q-2) + 20*J(p,q-1))*(p-1)*(p-2)*(p-3) \n+ 6*D(p-3,q+3,1)*(4*q^3 - 7*q^2*(p-3) + 22*q^2\n - 21*q*(p-3) + 38*q - 14*(p-3) + 20) \n+ 24*D(p-3,q+4,1)*( - q^3 - 6*q^2 - 11*q - 6) \n+ D(p-2,q+1,1)*(p-3)*(63*q^2 + 24*q*(p-1) - 8*q*(p-2)\n + 55*q - 10*(p-1)*(p-2) + 12*(p-1) + 26*(p-2) + 17) \n+ 2*D(p-2,q+2,1)*(p-3)*(39*q^2-12*q*(p-1) - 68*q*(p-2)\n + 163*q - 12*(p-1) - 68*(p-2) + 124) \n+ 120*D(p-2,q+3,1)*(p-3)*( - q^2 - 3*q - 2) \n+ 6*D(p-1,q-1,1)*(p-2)*(p-3)*(4*q + 3*(p-1) - 5) \n+ 2*D(p-1,q,1)*(p-2)*(p-3)*(64*q + 30*(p-1) - 17) \n+ 2*D(p-1,q+1,1)*(p-2)*(p-3)*(92*q - 47*(p-1) + 126) \n- 264*D(p-1,q+2,1)*(p-2)*(p-3)*(q + 1) \n+ (6*D(p,q-2,1) + 31*D(p,q-1,1) + 106*D(p,q,1)\n - 168*D(p,q+1,1))*(p-1)*(p-2)*(p-3) \n- 12*D(p-3,q+4,2)*(q + 1)*(q + 2)*(q + 3)\n+ D(p-2,q+2,2)*(p-3)*(-63*q^2 - 12*q*(p-1) + 4*q*(p-2)\n - 122*q - 12*(p-1) + 4*(p-2) - 59) \n+ 18*D(p-2,q+3,2)*(p-3)*( - q^2 - 3*q - 2) \n+ 18*D(p-1,q,2)*(p-2)*(p-3)*( - 2*q - (p-1) + 1) \n+ 2*D(p-1,q+1,2)*(p-2)*(p-3)*( - 68*q - 10*(p-1) - 33) \n- 12*D(p-1,q+2,2)*(p-2)*(p-3)*(q + 1) \n- (18*D(p,q-1,2) + 61*D(p,q,2) + 6*D(p,q+1,2))*(p-1)*(p-2)*(p-3) \n+ 3*(p-3)*( 7*D(p - 2,q + 3,3)*(q + 1)*(q + 2)\n- 2*D(p - 1,q + 2,4)*(q+1)*(p-2) \n+ 16*D(p - 1,q + 2,3)*(q+1)*(p-2) \n+ 2*D(p - 1,q + 1,3)*(4*q+p)*(p-2) \n+(- 2*D(p,q + 1,4) + 9*D(p,q + 1,3)\n + 6*D(p,q,3))*(p-1)*(p-2)\n))\/192\/(p-1)\/(p-2)\/(p-3);\n\\end{verbatim}\n\n{\\Large \\bf Appendix 10.}\n\\begin{eqnarray}\\hspace*{-12 mm}\n&& J(4,-1) = - 523385\/2976768 \\; Y_{11} - 104677\/91392 \\; Y_{10} - 17130665\/26790912 \\; Y_9 \\\\ \\nonumber && \n + 523385\/459648 \\; Y_8 + 11014687\/5515776 \\; Y_7 - 104677\/31256064 \\; Y_5 \\\\ \\nonumber && \n - 53544991\/62512128 \\; Y_4-3 \\; Y_0 + 11316787\/7354368 \\; (2\\pi)^{-2}-3\/4 \\; F_0 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(5,-1) = 881852823875\/3876752130048 \\; Y_{11} + 176370564775\/119023091712 \\; Y_{10} \\\\ \\nonumber && \n + 28894631548079\/34890769170432 \\; Y_9 - 881852823875\/598616137728 \\; Y_8 \\\\ \\nonumber && \n - 17115981580297\/7183393652736 \\; Y_7 + 176370564775\/40705897365504 \\; Y_5 \\\\ \\nonumber && \n + 86132421399433\/81411794731008 \\; Y_4 + 25\/8 \\; Y_0 \\\\ \\nonumber && \n + (25\/32\\; F_0 - 20532421134805\/9577858203648 ) (2\\pi)^{-2};\\\\ \\nonumber && \n J(6,-1) = - 28542210469686553\/93496923371077632 \\; Y_{11} \\\\ \\nonumber && \n - 28542210469686553\/14352597885911040 \\; Y_{10} - 2087037111021330529\/4207361551698493440 \\; Y_9 \\\\ \\nonumber && \n + 28542210469686553\/14437024932298752 \\; Y_8 + 2217244724616422759\/866221495937925120 \\; Y_7 \\\\ \\nonumber && \n - 28542210469686553\/4908588476981575680 \\; Y_5 - 12694496481519981287\/9817176953963151360 \\; Y_4 \\\\ \\nonumber && \n - 3163\/960 \\; Y_0 + (3108094517023801819\/1154961994583900160 - 3163\/3840 \\; F_0 ) (2\\pi)^{-2}; \\\\ \\nonumber && \n J(7,-1) = 318815426788929230845117\/821909762068921372901376 \\; Y_{11} \\\\ \\nonumber && \n + 318815426788929230845117\/126170358212334421278720 \\; Y_{10} \\\\ \\nonumber &&\n + 11542347686773160604861797\/36985939293101461780561920 \\; Y_9 \\\\ \\nonumber && \n - 318815426788929230845117\/126912536790054035521536 \\; Y_8 \\\\ \\nonumber && \n - 22134243295873177135745923\/7614752207403242131292160 \\; Y_7 \\\\ \\nonumber && \n + 318815426788929230845117\/43150262508618372077322240 \\; Y_5 \\\\ \\nonumber && \n + 135786390066815073864737539\/86300525017236744154644480 \\; Y_4 + 14099\/3840 \\; Y_0 \\\\ \\nonumber && \n - 34028993605943331376331719\/10153002943204322841722880 \\; (2\\pi)^{-2} + 14099\/15360 \\; F_0 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(8,-1) = - 9235026172226013567109728738337\/18732046758141338739230512250880 \\; Y_{11} \\\\ \\nonumber && \n - 9235026172226013567109728738337\/2875533493574328315232754073600 \\; Y_{10} \\\\ \\nonumber && \n - 55876713828271512392254716439529\/842942104116360243265373051289600 \\; Y_9 \\\\ \\nonumber && \n + 9235026172226013567109728738337\/2892448396477706717087064391680 \\; Y_8 \\\\ \\nonumber && \n + 582625738707581471281711807374367\/173546903788662403025223863500800 \\; Y_7 \\\\ \\nonumber && \n - 9235026172226013567109728738337\/983432454802420283809601893171200 \\; Y_5 \\\\ \\nonumber && \n - 3802005063193739791001963690889631\/1966864909604840567619203786342400 \\; Y_4 \\\\ \\nonumber && \n - 150949\/35840 \\; Y_0 + 965476530505339393064578836261427\/ \\\\ \\nonumber && \n 231395871718216537366965151334400 \\; (2\\pi)^{-2} - 150949\/143360 \\; F_0 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(9,-1) = \\\\ \\nonumber && \n 10189310117043029375946591507272718551\/16263612564540505685529672268508037120 \\; Y_{11} \\\\ \\nonumber && \n + 10189310117043029375946591507272718551\/2496607191925077627164642672797286400 \\; Y_{10} \\\\ \\nonumber && \n - 150087551903588632763072238483785668913\/731862565404322755848835252082861670400 \\; Y_9 \\\\ \\nonumber && \n - 10189310117043029375946591507272718551\/2511293116583460436736199394401976320 \\; Y_8 \\\\ \\nonumber && \n - 598626882055818268392045217522764699881\/150677586995007626204171963664118579200 \\; Y_7 \\\\ \\nonumber && \n + 10189310117043029375946591507272718551\/853839659638376548490307794096671948800 \\; Y_5 \\\\ \\nonumber && \n + 4095937462034066852819132682453485174633\/1707679319276753096980615588193343897600 \\; Y_4 \\\\ \\nonumber && \n + 8535263\/1720320 \\; Y_0 + 8535263\/6881280 \\; F_0 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n - 1051450108870580974979233285323814292021\/200903449326676834938895951552158105600 \\; (2\\pi)^{-2} ; \\\\ \\nonumber \n&& J(4,-2) = 775\/124032 \\; Y_{11} + 155\/3808 \\; Y_{10} + 2315881\/1116288 \\; Y_9 - 775\/19152 \\; Y_8 \\\\ \\nonumber && \n - 535775\/229824 \\; Y_7 + 155\/1302336 \\; Y_5 + 1268159\/2604672 \\; Y_4 + 4 \\; Y_0 \\\\ \\nonumber && \n - 256019\/306432 \\; (2\\pi)^{-2} + \\; F_0 \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(5,-2) = - 35198207035\/161531338752 \\; Y_{11} - 7039641407\/4959295488 \\; Y_{10} \\\\ \\nonumber && \n - 1590044736319\/1453782048768 \\; Y_9 + 35198207035\/24942339072 \\; Y_8 \\\\ \\nonumber && \n + 824164813817\/299308068864 \\; Y_7 - 7039641407\/1696079056896 \\; Y_5 \\\\ \\nonumber && \n - 3786323755001\/3392158113792 \\; Y_4 - 4 \\; Y_0 + (886011822725\/399077425152 - \\; F_0 )\\; (2\\pi)^{-2}; \\\\ \\nonumber && \n J(6,-2) = 11638328158211917\/35061346264154112 \\; Y_{11} \\\\ \\nonumber && \n + 11638328158211917\/5382224207216640 \\; Y_{10} + 1835049614849615189\/1577760581886935040 \\; Y_9 \\\\ \\nonumber && \n - 11638328158211917\/5413884349612032 \\; Y_8 - 1118098215139974931\/324833060976721920 \\; Y_7 \\\\ \\nonumber && \n + 11638328158211917\/1840720678868090880 \\; Y_5 + 5661105033256044691\/3681441357736181760 \\; Y_4 \\\\ \\nonumber && \n + 181\/40 \\; Y_0 + (181\/160 \\; F_0 - 1412418713096881687\/433110747968962560 )\\; (2\\pi)^{-2} ; \\\\ \\nonumber && \n J(7,-2) = - 16095494538985786829003\/34246240086205057204224 \\; Y_{11} \\\\ \\nonumber && \n - 16095494538985786829003\/5257098258847267553280 \\; Y_{10} \\\\ \\nonumber && \n - 1220878670422477200084451\/1541080803879227574190080 \\; Y_9 \\\\ \\nonumber && \n + 16095494538985786829003\/5288022366252251480064 \\; Y_8 \\\\ \\nonumber && \n + 1258484886934090324074869\/317281341975135088803840 \\; Y_7 \\\\ \\nonumber && \n - 16095494538985786829003\/1797927604525765503221760 \\; Y_5 \\\\ \\nonumber && \n - 1435246543741057501042993\/719171041810306201288704 \\; Y_4 - 2447\/480 \\; Y_0 \\\\ \\nonumber && \n + (1817967522495901075073489\/423041789300180118405120 - 2447\/1920 \\; F_0 )\\;(2\\pi)^{-2}; \\\\ \\nonumber && \n J(8,-2) = 97496623093573646692443907729\/156100389651177822826920935424 \\; Y_{11} \\\\ \\nonumber && \n + 97496623093573646692443907729\/23962779113119402626939617280 \\; Y_{10} \\\\ \\nonumber && \n + 703823889446906512753183459525\/1404903506860600405442288418816 \\; Y_9 \\\\ \\nonumber && \n - 97496623093573646692443907729\/24103736637314222642392203264 \\; Y_8 \\\\ \\nonumber && \n - 6768519700158561511510642407343\/1446224198238853358543532195840 \\; Y_7 \\\\ \\nonumber && \n + 97496623093573646692443907729\/8195270456686835698413349109760 \\; Y_5 \\\\ \\nonumber && \n + 41525287801127597648514834574639\/16390540913373671396826698219520 \\; Y_4 \\\\ \\nonumber && \n + 79489\/13440 \\; Y_0 + 79489\/53760 \\; F_0 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n - 2132153450344177531160317231847\/385659786197027562278275252224 \\; (2\\pi)^{-2} ; \\\\ \\nonumber && \n J(9,-2) = - 184720162027043790012174961950856949\/225883507840840356743467670395944960 \\; Y_{11} \\\\ \\nonumber && \n - 184720162027043790012174961950856949\/34675099887848300377286703788851200 \\; Y_{10} \\\\ \\nonumber && \n - 1182390536568106894749732117915566173\/10164757852837816053456045167817523200 \\; Y_9 \\\\ \\nonumber && \n + 184720162027043790012174961950856949\/34879071063659172732447213811138560 \\; Y_8 \\\\ \\nonumber && \n + 11666341875097592955381754264025444939\/2092744263819550363946832828668313600 \\; Y_7 \\\\ \\nonumber && \n - 184720162027043790012174961950856949\/11858884161644118729032052695787110400 \\; Y_5 \\\\ \\nonumber && \n - 76075734513749258577040135726310510027\/23717768323288237458064105391574220800 \\; Y_4 \\\\ \\nonumber && \n - 501267\/71680 \\; Y_0 - 501267\/286720 \\; F_0 \\; (2\\pi)^{-2} \\\\ \\nonumber && \n + 19693107617724772864308858550443517679\/2790325685092733818595777104891084800 \\; (2\\pi)^{-2}; \\nonumber \n\\end{eqnarray}\n\\newpage\n\n{\\Large \\bf Appendix 11.}\n\\begin{verbatim}\nJ(p,q) = -( 6*D(p - 1,q + 4,4)*( - p^2*q - 3*p^2 + 5*p*q + 15*p - 6*q - 18) \n+ 6*D(p,q + 3,4)*( - p^3 + 6*p^2 - 11*p + 6)\n + 21*D(p-2,q+5,3)*(p*q^2 + 7*p*q + 12*p - 3*q^2 - 21*q - 36) \n + 48*D(p - 1,q + 4,3)*(p^2*q + 3*p^2 - 5*p*q - 15*p + 6*q + 18) \n + 6*D(p-1,q+3,3)*(p-2)*(p-3)*(p+4*q+8) \n + 27*D(p,q + 3,3)*(p^3 - 6*p^2 + 11*p - 6)\n + 18*D(p,q+ 2,3)*(p^3 - 6*p^2 + 11*p - 6) \n + 12*D(p-3,q+6,2)*( - q^3 - 12*q^2 - 47*q - 60) \n + 18*D(p-2,q+5,2)*( - p*q^2 - 7*p*q - 12*p + 3*q^2 + 21*q + 36) \n + D(p - 2,q + 4,2)*( - 8*p^2*q - 24*p^2\n - 63*p*q^2 - 346*p*q - 471*p + 189*q^2 + 1110*q + 1629) \n + 12*D(p-1,q+4,2)*( - p^2*q - 3*p^2 + 5*p*q + 15*p - 6*q - 18) \n + 2*D(p-1,q+3,2)*( - 10*p^3 - 68*p^2*q - 109*p^2\n + 340*p*q + 735*p - 408*q - 954) \n + 6*D(p,q + 3,2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 18*D(p - 1,q + 2,2)*( - p^3 - 2*p^2*q + 3*p^2 + 10*p*q + 4*p - 12*q - 12) \n + 61*D(p,q + 2,2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 18*D(p,q + 1,2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 24*D(p-3,q+6,1)*( - q^3 - 12*q^2 - 47*q - 60) \n + 6*D(p-3,q+5,1)*( - 7*p*q^2 - 49*p*q - 84*p\n + 4*q^3 + 67*q^2 + 321*q + 468) \n + 120*D(p - 2,q + 5,1)*( - p*q^2 - 7*p*q - 12*p + 3*q^2 + 21*q + 36) \n + 2*D(p - 2,q + 4,1)*( - 80*p^2*q - 240*p^2\n + 39*p*q^2 + 707*p*q + 1770*p - 117*q^2 - 1401*q - 3150) \n + 264*D(p - 1,q + 4,1)*( - p^2*q - 3*p^2 + 5*p*q + 15*p - 6*q - 18) \n + D(p - 2,q + 3,1)*( - 10*p^3 + 16*p^2*q + 130*p^2\n + 63*p*q^2 + 251*p*q - 21*p - 189*q^2 - 897*q - 837) \n + 2*D(p - 1,q + 3,1)*( - 47*p^3 + 92*p^2*q\n + 592*p^2 - 460*p*q - 2067*p + 552*q + 2142) \n + 168*D(p,q + 3,1)*( - p^3 + 6*p^2 - 11*p + 6) \n + 2*D(p - 1,q + 2,1)*(30*p^3 + 64*p^2*q - 69*p^2\n - 320*p*q - 225*p + 384*q + 486) \n + 106*D(p,q + 2,1)*(p^3 - 6*p^2 + 11*p - 6) \n + 6*D(p - 1,q + 1,1)*(3*p^3 + 4*p^2*q - 15*p^2 - 20*p*q + 18*p + 24*q) \n + 31*D(p,q + 1,1)*(p^3 - 6*p^2 + 11*p - 6) \n + 6*D(p,q,1)*(p^3 - 6*p^2 + 11*p - 6) \n + 24*J(p - 4,q + 6)*(q^3 + 12*q^2 + 47*q + 60) \n + 24*J(p - 3,q + 5)*(5*p*q^2 + 35*p*q + 60*p - 17*q^2 - 119*q - 204) \n + 2*J(p - 3,q + 4)*(20*p^2*q + 60*p^2 + 21*p*q^2\n + p*q - 186*p - 6*q^3 - 129*q^2 - 423*q - 270) \n + 24*J(p - 2,q + 4)*(11*p^2*q + 33*p^2 - 63*p*q - 189*p + 90*q + 270) \n + 4*J(p - 2,q + 3)*(30*p^3 + 20*p^2*q - 190*p^2\n - 15*p*q^2 - 199*p*q + 231*p + 45*q^2 + 417*q + 207) \n + 24*J(p - 1,q + 3)*(15*p^3 - 104*p^2 + 235*p - 174) \n + J(p - 2,q + 2)*(10*p^3 - 8*p^2*q - 106*p^2\n - 21*p*q^2 - 52*p*q + 240*p+63*q^2+228*q-36)\n + 2*J(p - 1,q + 2)*( - 73*p^3 - 66*p^2*q + 346*p^2\n + 330*p*q - 343*p - 396*q - 114) \n + 192*J(p,q + 2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 4*J(p-1,q+1)*( - 10*p^3 - 10*p^2*q + 53*p^2 + 50*p*q - 75*p - 60*q + 18) \n + 20*J(p,q+1)*(p^3 - 6*p^2 + 11*p - 6) \n + 6*J(p-1,q)*( - p^3 - p^2*q + 6*p^2 + 5*p*q - 11*p - 6*q + 6) \n )\/3\/(p-1)\/(p-2)\/(p-3);\n\\end{verbatim}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}\nThe solar atmosphere contains a wide variety of chromospheric ejections that cover a large range\nof scales: from the smallest ones with maximum size of a few megameters, such as penumbral \nmicrojets \\citep[e.g.,][]{Katsukawa:2007wk, Drews:2017}, or spicules \\citep[][among others]\n{Hansteen+DePontieu2006, de-Pontieu:2007kl, Pereira:2012dz}; up to ejections that can reach, \nin extreme cases, several tens of megameters, like surges \\citep[e.g.,][]{canfield1996, Kurokawa2007, \nGuglielmino:2010lr,YangH:2014} and macrospicules \\citep{Bohlin1975, Georgakilas, Murawski2011, \nKayshap2013}. Surges, in particular, are often associated with magnetic flux emergence from the solar \ninterior. They are typically observed as darkenings in images taken in the {H$\\alpha$}\\ blue\/red wings with line-of-sight \n(LOS) velocities of a few to several tens of km s$^{-1}$, and are usually related to other explosive \nphenomena like EUV and X-ray jets, UV bursts and Ellerman bombs \\citep[see]\n[hereafter NS2017, and references therein]{Nobrega-Siverio:2017a}. Although \nobservationally known for several decades now, the understanding of surges has progressed \nslowly and various aspects like, e.g., their impact on\n the transition region (TR) and corona concerning the mass and energy budget, are still poorly known.\n\nFrom the theoretical point of view, the first explanation of the surge phenomenon came through 2.5D \nnumerical models \\citep{Shibata1992a,Yokoyama:1995uq,Yokoyama:1996kx}, where a cold ejection was \nidentified next to a hot jet as a consequence of a magnetic reconnection process between the magnetic \nfield in plasma emerged from the interior and the preexisting coronal field. \\cite{Nishizuka:2008zl} used \na similar numerical setup to associate the surge with jet-like features seen in {\\ion{Ca}{2}}\\ H+K observations by \nmeans of morphological image comparisons. Further 2.5D models that include\nthe formation of a cool chromospheric ejection are those of \\cite{jiang2012}\n(canopy-type coronal magnetic field), and \\cite{YangL:2013, YangL:2018}, who\nstudy the cool jets resulting from the interaction between moving magnetic features at\nthe base of their experiment and the preexisting ambient field in the\natmosphere. Turning to three dimensional models, in the magnetic flux emergence experiment of\n\\cite{Moreno-Insertis:2013aa}, a dense wall-like surge appeared surrounding the emerged \nregion with temperatures from $10^4$~K to a few times $10^5$~K and speeds around $50$ km s$^{-1}$. \n\\cite{MacTaggart2015} found similar velocities for the surges in their 3D model of flux emergence in \nsmall-scale active regions. The availability of a radiation-MHD code like Bifrost \n\\citep{Gudiksen:2011qy} has opened up the\npossibility of much more detailed modeling of the cool ejections than\nbefore. Bifrost has a realistic treatment of the material properties of the plasma,\ncalculates the radiative transfer in the photosphere and chromosphere \nand includes the radiative and heat conduction entropy sources in the corona. Using that code, \\cite{Nobrega-Siverio:2016}, hereafter NS2016,\nargued that entropy sources play an important \nrole during the surge formation and showed that a relevant fraction of the surge could not be obtained in \nprevious and more idealized experiments. \n\n\nThe realistic treatment of surges may require an even larger degree\n of complication. The solar atmosphere is a highly dynamical environment;\nthe evolution sometimes occurs on short timescales that bring different\natomic species out of equilibrium ionization, thus complicating both the\nmodeling and the observational diagnostics\n\\citep[e.g.,][]{Griem:1964,Raymond:1978,Joselyn:1979, Hansteen:1993}. For\nhydrogen, for instance, using 2D numerical experiments,\n\\cite{Leenaarts:2007sf, Leenaarts:2011qy} found that the temperature\nvariations in the chromosphere can be much larger than for statistical\nequilibrium (SE), which has an impact on, e.g., its coolest regions (the\nso-called cool pockets). For helium, \\citet{golding2014,golding:2016}\ndescribed how nonequilibrium (NEQ) ionization leads to higher temperatures\nin wavefronts and lower temperatures in the gas between shocks. For heavy\nelements, \\cite{Bradshaw:2003,Bradshaw:2006, Bradshaw:2011, Reep:2017}\nshowed, through 1D hydrodynamic simulations, that there are large departures\nfrom SE balance in cooling coronal loops, nanoflares and other impulsive\nheating events that affects the EUV emissivity. Through 3D\n experiments, \\cite{Olluri:2013fu} found that deduced electron densities for {\\ion{O}{4}}\\ can be up to\nan order of magnitude higher when NEQ effects are taken into\naccount. Also in 3D, \\cite{olluri:2015} discussed the importance of the NEQ ionization of \ncoronal and TR lines to reproduce \nabsolute intensities, line widths, ratios, among others, \nobserved by, e.g., \\cite{Chae:1998, Doschek:2006, Doschek:2008qy}. \n\\cite{De-Pontieu2015} were able to explain the correlation between\nnon-thermal line broadening and intensity of TR lines only\nwhen including NEQ ionization in their 2.5D numerical\nexperiments. \\cite{Martinez-Sykora:2016obs} studied the statistical\nproperties of the ionization of silicon and oxygen in different solar\ncontexts: quiet Sun, coronal hole, plage, quiescent active region, and\nflaring active region, finding similarities with the observed intensity\nratios only if NEQ effects are taken into account. Given their highly\ntime-dependent nature and the relevance of the heating and cooling mechanisms\nin their evolution, surges are likely to be affected by NEQ ionization. \nMotivated by this fact, NS2017 included the NEQ\nionization of silicon to compare synthetic {\\ion{Si}{4}}\\ spectra of two 2.5D\nnumerical experiments with surge observations obtained by \nthe \\textit{Interface Region Imaging Spectrograph}\n\\citep[\\textit{IRIS},][]{De-Pontieu:2014vn} and the Swedish 1-m Solar Telescope\n\\citep[SST,][]{Scharmer:2003ve}. The results showed that the experiments were\nable to reproduce major features of the observed surge; nonetheless, the\ntheoretical aspects to understand the enhanced {\\ion{Si}{4}}\\ emissivity within the\nnumerical surge and its properties were not addressed in that publication. \n\n\n\n\n\\begin{figure*}\n\\epsscale{1.18}\n\\plotone{figure1.pdf}\n\\caption{\nLeft: Horizontal averages for the initial stratification of $\\rho$, $P_{g}$, and $T$ normalized to their \nphotospheric values at $z = 0$~Mm, namely, $\\rho_{ph} = 3.1\\times10^{-7}$~g cm$^{-3}$,\n$P_{g_{ph}} = 1.1 \\times 10^{5}$~erg cm$^{-3}$ and $T_{ph}= 5.7\\times10^3$~ K. Solid black line represents the stratification for the vertical coronal field experiment; the red dotted, for the slanted one. The horizontal and vertical dotted lines mark the reference normalization values at $z=0$ Mm. Right: 2D maps for the initial temperature with magnetic field lines in black for the vertical experiment (top) and slanted experiment (bottom). The maps only show temperatures below $2\\times10^4$ K (although the range varies from 1660 K up to $\\sim$ 1 MK) and heights between $-2.6 \\leq z \\leq 8.0$~Mm (the top of the domain reaches $z=30$ Mm). The solar surface is roughly at $z=0$ Mm (white dashed horizontal line). \\label{figure1}}\n\\end{figure*} \n\nThe aim of the present paper is to provide theoretical explanations\nconcerning the relevance of the NEQ ionization for surges and the\ncorresponding impact on the emissivity of TR lines. We use\n2.5D numerical experiments carried out with the Bifrost code\n\\citep{Gudiksen:2011qy} including the module developed by \\cite{olluri:2013aa}\nthat solves the time-dependent rate equations to calculate the ionization\nstates of different elements, thus allowing for departures from SE. Here we apply this module to determine the\nionization levels of silicon and oxygen. We conclude that consideration of \nNEQ is necessary to get the proper population levels of the ions and, consequently,\nthe right emissivity to interpret observations. A statistical\nanalysis of temperature is provided to constrain the plasma properties\ninvolved in the emissivity of relevant lines of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ within the\nsurges. Through detailed Lagrange tracing, we are able to determine the\norigin of the emitting plasma and the role of the optically thin radiation\nand thermal conduction to explain the departure of SE of the relevant\nions. Furthermore, we compute synthetic profiles to understand previous\nobservational results and predict future ones,\nhighlighting the surge regions that are more likely to be detected\nand addressing the importance of the angle of LOS.\n\nThe layout of the paper is as follows. Section \\ref{sec:2} describes the physical and numerical models. \nSection \\ref{sec:3} explains the general features of the time evolution of the experiments. In Section \n\\ref{sec:4}, we show the main results of the paper splitting the section in a) the relevance of the NEQ \nionization of {\\ion{Si}{4}}\\,, and also {\\ion{O}{4}}, in surges (Section \\ref{sec:4.1}); b) the consequences of the NEQ \nionization for the surge plasma emitting in those TR lines, analyzing its properties and \ncompare them with a generic quiet TR (Section \\ref{sec:4.2}); and c) the origin of the NEQ plasma, \naddressing the role of the entropy sources (Section \\ref{sec:4.3}). In Section\n\\ref{sec:5}, we have calculated absolute intensities and synthetic spectral \nprofiles for diagnostic purposes and comparison with observations, emphasizing also the importance of \nthe surge geometry and LOS. Finally, Section \\ref{sec:6} contains a summary and conclusions.\n\n\n\\Needspace{5\\baselineskip}\n\\section{The physical and numerical model}\\label{sec:2}\nWe have run two 2.5D numerical flux emergence experiments in which surges are a natural \nconsequence of magnetic reconnection processes. Those two experiments \nwere also used by NS2017 and \\cite{Rouppe2017} to compare the synthetic profiles with \nthe complex profiles observed with \\textit{IRIS}\\ and SST.\n\nThis section is divided into two parts: (1) the numerical code, and (2) the description of the \nmodel underlying our experiments.\n\n\\Needspace{5\\baselineskip}\n\\subsection{The numerical code}\\label{sec:2.1}\nThe two experiments have been carried out with the 3D radiation-MHD (R-MHD)\nBifrost code \\citep{Gudiksen:2011qy, Carlsson:2012uq, Hayek:2010ac}, which treats \nthe radiative transfer from the photosphere to the corona and thermal conduction \nin a self-consistent manner (see also NS2016 for further details of this\ncode applied to surge experiments). Furthermore, we have enabled in the code a module \ndeveloped by \\citep{olluri:2013aa} to follow the NEQ ionization \nstates of elements with atomic \nnumber greater than 2. This module solves the rate equations for those elements using \nthe temperature, mass density, electronic number density $n_e$ and velocity values of the simulation without modifying the results of the R-MHD calculation, so there is no feedback, e.g., on the energy equation terms such as the optically thin losses (see the discussion in Section \\ref{sec:6.1}). In particular, we have employed it to calculate the NEQ ionization fraction for silicon and oxygen, using abundances from \\cite{Asplund:2009}, 7.52 and 8.69, respectively, in the customary astronomical scale\nwhere 12 corresponds to hydrogen. \n\n\\Needspace{5\\baselineskip}\n\\subsection{Description of the models}\\label{sec:2.2}\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Physical domain and initial condition}\n\nIn the two experiments, we began with a statistically stationary 2D snapshot that spans\n from the uppermost layers of the solar interior to the corona, and whose physical \n domain is $0.0$~Mm $\\leq x \\leq$ $32.0$~Mm and $-2.6$~Mm $\\leq z \\leq$ \n $30.0$~Mm, where $z=0$~Mm corresponds to the solar surface. The grid is \n uniform in the $x$-direction with $\\Delta x=31$~km, but it is nonuniform in the \n vertical direction in order to better resolve the photosphere and chromosphere:\n the vertical grid spacing is $20$ km from the photosphere to the transition\n region, and increases gradually in the corona up to $147$ km at the top of the domain.\n\nThe left panel in Figure~\\ref{figure1} contains the horizontal averages for the initial \ndensity, $\\rho$, gas pressure, $P_g$, and temperature, $T$, for both experiments\nnormalized to photospheric values, namely, $\\rho_{ph} = 3.1\\times10^{-7}$~g cm$^{-3}$,\n$P_{g_{ph}} = 1.1 \\times 10^{5}$~erg cm$^{-3}$ and $T_{ph}= 5.7\\times10^3$~ K. \nThe corona has a temperature around $1$ MK and a magnetic field with a strength \nof $10$ G, with the difference that one of the experiments \n(hereafter \\textit{the vertical experiment}) has a vertical \nmagnetic field in the corona \nwhile in the other (\\textit{the slanted experiment}), the magnetic field in the corona is inclined \n$30$\\degree\\ with respect to the vertical direction (see magnetic field lines superimposed in black \nin the 2D temperature maps for the initial snapshot in Figure~\\ref{figure1}). \n\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Chemical elements calculated in NEQ and their spectral lines}\n\nWe have used the NEQ module of\n \\cite{olluri:2013aa} mentioned in the \\nameref{sec:introduction} \nto compute the nonequilibrium ionization of silicon \nin both numerical experiments. Furthermore, in the vertical experiment we also \ncalculate the NEQ ionization of oxygen, with the goal of predicting \nfuture observational results. Once the NEQ populations are obtained, we are able \nto compute the emissivity using\n\\begin{eqnarray}\n\t\\epsilon_{\\lambda} & = & \\frac{h\\, c}{4\\, \\pi\\, \\lambda}\\, n_u\\, A_{ul},\n \\label{eq:emissivity}\n\\end{eqnarray}\nwhere $h$ is the Planck's constant, $c$ the light speed, $\\lambda$ is the\nwavelength of the spectral line, $n_u$ the population density of the upper\nlevel of the transition (i.e., the number density of emitters), and $A_{ul}$ the Einstein coefficient for\nspontaneous de-excitation given by\n\\begin{eqnarray}\n\tA_{ul} & = & \\frac{8\\, \\pi^2\\, e^2\\, }{m_e\\, c} \\frac{1}{\\lambda^2}\\, \\frac{g_l}{g_u}\\, f_{lu},\n\\label{eq:einstein}\n\\end{eqnarray}\nwhere $e$ is the electron charge, $m_e$ the electron mass, $g_l$ and $g_u$ the statistical \nweights of the lower and upper states respectively, and $f_{lu}$ the oscillator strength. \nThe units used in this paper for the emissivity $\\epsilon$ are erg \ncm$^{-3}$ sr$^{-1}$ s$^{-1}$. For the sake of compactness, we will refer to it in the following \nas $\\epsilon_{_{CGS}}$.\n\nSince we are interested in understanding the response of the TR to \nchromospheric phenomena like surges, we have chosen the following \\textit{IRIS}\\ lines: \n{\\ion{Si}{4}}\\ 1402.77 $\\textup{\\AA}$, which is the weakest of the two silicon resonance \nlines; and {\\ion{O}{4}}\\ 1401.16 $\\textup{\\AA}$, the strongest of the forbidden oxygen lines \nthat \\textit{IRIS}\\ is able to observe. The corresponding formation temperature peaks in \nstatistical equilibrium, $T_{_{SE}}$, and other relevant parameters to calculate\nthe Einstein coefficient (Equation \\ref{eq:einstein}) and the corresponding emissivity \n(Equation \\ref{eq:emissivity}) of these lines are shown in Table \\ref{table1}.\nUnder optically thin conditions, {\\ion{Si}{4}}\\ 1393.76 \n$\\textup{\\AA}$ is twice stronger than 1402.77 $\\textup{\\AA}$, so the results we obtain in\nthis paper can also be applied to {\\ion{Si}{4}}\\ 1393.76 $\\textup{\\AA}$. Furthermore, the study of\n{\\ion{Si}{4}}\\ 1402.77 $\\textup{\\AA}$ can provide theoretical support to our previous paper NS2017. \nIn turn, the choice of the 1401.16 $\\textup{\\AA}$ line for\noxygen is because the {\\ion{O}{4}}\\ lines \nare faint and require longer exposure times to be observed \\citep{De-Pontieu:2014vn}. \nThus, in order to make any prediction that could be corroborated in future \\textit{IRIS}\\ \nanalysis, we focus on the strongest of the oxygen lines, which has a better chance \nto be detected. For simplicity, hereafter we refer to the {\\ion{Si}{4}}\\ 1402.77 $\\textup{\\AA}$ \nand {\\ion{O}{4}}\\ 1401.16 $\\textup{\\AA}$ emissivities as the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities, \nrespectively.\n\n\n\\setcounter{table}{0}\n\\begin{deluxetable}{c|ccccc}\n\\caption{Relevant parameters for the studied emission lines}\\label{table1}\n\\tablehead{\n\\colhead{Line} & \\colhead{$T_{_{SE}}$ (K)} & \\colhead{$g_u$} & \\colhead{$g_l$} & \\colhead{$f_{lu}$} & \\colhead{$n_u\/\\epsilon_{\\lambda}$}}\n\\startdata\n{\\ion{Si}{4}}\\ 1402.77 \\AA & $10^{4.9}$ & $2$ & $2$ & $2.7\\times10^{-1}$ & $974$\\\\\n\\hline\n{\\ion{O}{4}}\\ 1401.16 \\AA & $10^{5.2}$ & $6$ & $4$ & $5.1\\times10^{-7}$ & $7.62\\times10^{8}$ \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\Needspace{5\\baselineskip} \n\\subsubsection{Boundary conditions}\nWe are imposing periodicity at the side boundaries; \nfor the vertical direction, characteristic conditions are implemented at the\ntop, whereas an open boundary is maintained at the bottom keeping a fixed value\nof the entropy of the incoming plasma. Additionally, in order to\nproduce flux emergence, we inject a twisted magnetic tube through the bottom\nboundary following the method described by \\cite{Martinez-Sykora:2008aa}. \nThe parameters of the tube (specifically, the initial location of\nthe axis, $x_0$ and $z_0$; the field strength there, $B_0$; the tube radius\n$R_0$; and the amount of field line twist $q$) are identical in both\nexperiments and given in Table \\ref{table2}. The total axial magnetic flux is\n$\\Phi_0 = 6.3 \\times 10^{18}$~Mx, which is in the range of \nan ephemeral active region \\citep{Zwaan:1987yf}. Details about this kind of\nsetup are provided in the paper by NS2016.\n\n\\begin{table}[h!]\n\\renewcommand{\\thetable}{\\arabic{table}}\n\\centering\n\\caption{Parameters of the initial twisted magnetic tube for both experiments} \\label{table2}\n\\begin{tabular}{ccccc}\n\\tablewidth{0pt}\n\\hline\n\\hline\n$x_0$ (Mm) & $z_0$ (Mm) & $R_0$ (Mm) & $q$ (Mm$^{-1}$) & $B_0$ (kG) \\\\\n\\hline\n\\decimals\n15.0 & -2.8 & 0.10 & 2.4 & 20 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{figure*}\n\\epsscale{1.18}\n\\plotone{figure2.pdf}\n\\caption{Image (taken from NS2017) showing 2D \ntemperature maps for the context of the surge experiments and the regions of interest. \nA) The vertical coronal magnetic field experiment at $t=65.0$ minute. \nB) The slanted coronal magnetic case \nat $t=64.3$ minute. Additionally, magnetic field lines (green), \ntemperature contours for the $T_{_{SE}}=7.9 \\times 10^{4}$ of {\\ion{Si}{4}}\\ K (blue), and \nfor $T=1.2 \\times 10^{6}$ K (red) are added. \\label{figure2}}\n\\end{figure*} \n\n\n\\Needspace{5\\baselineskip}\n\\section{General features of the time evolution of the experiments}\\label{sec:3}\nThe numerical experiments start with the injection of the twisted magnetic tube \nthrough the bottom boundary ($t=0$ minute). Within the convection zone, the tube rises\nwith velocities of $\\lesssim 2$ km s$^{-1}$ and suffers deformations due to the\nconvection flows, mainly in the regions where the downflows are located. The twisted tube \ncontinues rising until it reaches the surface. There, the magnetized plasma accumulates\nuntil it develops a buoyancy instability ($t \\approx 40$ minute) in a similar way as\nexplained by NS2016.\n\n\nThe subsequent phases of evolution are characterized by the emergence and \nexpansion of the magnetized plasma into the solar atmosphere, producing a dome-like \nstructure of cool and dense matter ($t \\sim 50$ minute). During the \nexpansion process, the dome interior becomes rarefied due to gravitational flows. \nSimultaneously, the magnetic field of the emerged plasma collides with the preexisting \ncoronal ambient field and, as a consequence, non-stationary magnetic reconnection occurs,\nforming and ejecting several plasmoids. Our vertical experiment \nhas recently been used by \\cite{Rouppe2017} to show that the {\\ion{Si}{4}}\\ spectral synthesis of those\nplasmoids is able to reproduce the highly broadened line profiles,\noften with non-Gaussian and triangular shapes, seen in \\textit{IRIS}\\ observations.\n\nAs an indirect consequence of the magnetic reconnection, a surge is obtained\nin both experiments. This is illustrated in Figure \\ref{figure2} through temperature \nmaps with overlying magnetic field lines for each experiment: panel A, the vertical \nexperiment at $t=65.0$ minute, and panel B, the slanted one at $t=64.3$ minute. Those\nare representative instants when the surge is clearly distinguishable as an elongated \nstructure detached from the dome. For later\nreference, different regions have been marked in the figure that\n will be seen below to correspond to prominent features of the surge\nin terms of NEQ ionization and brightness in the spectra: the internal footpoint, which is located at the base of a wedge created by the detachment process that separates the surge from the dome (NS2016); the external footpoint, which is\njust the external boundary of the surge, and the flanks and top of the crests.\nAlthough not directly discussed in this paper, another region is probably worth mentioning, namely the \nhot jet: in the vertical experiment, it is shown clearly through the red temperature contour of \n$T=1.2 \\times 10^{6}$ K; in the slanted one, the temperatures of the high-speed collimated ejection \nare not distinguishable from the rest of the corona. The difference between both\nexperiments may be due to the fact that the slanted case has a denser emerged\ndome and, perhaps, the entropy sources in it are less efficient in \nheating the plasma that passes through the magnetic reconnection site.\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\section{The role of the nonequilibrium (NEQ) ionization}\\label{sec:4}\nThe importance of the NEQ ionization is studied in this section from a triple\nperspective: (a) the comparison of the NEQ number densities with those calculated under the SE approximation\n(Section~\\ref{sec:4.1}); (b) the consequences for the emissivity of the\nplasma (Section~\\ref{sec:4.2}); and (c) the key mechanisms that cause the\ndeparture from statistical equilibrium in the surge plasma\n(Section~\\ref{sec:4.3}).\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{The SE and NEQ number densities}\\label{sec:4.1}\nThe results of the current paper are obtained by solving the equation rates for\nthe relevant ionization states of Si and O, i.e., taking into account\nnonequilibrium effects using the \\citet{olluri:2013aa} module mentioned in\nearlier sections: the number densities of emitters $n_u$ thus calculated will be\nindicated with the symbol $n_{_{NEQ}}$. In order to test the accuracy of the SE\napproximation, we have also calculated the $n_u$ that would be obtained \nimposing statistical equilibrium in the \\citet{olluri:2013aa} module: those will be \nindicated with the symbol $n_{_{SE}}$. The accuracy or otherwise of the SE \napproximation is measured here through the following ratio:\n\n\\begin{eqnarray}\n\tr & = & \\frac{n_{_{SE}} - n_{_{NEQ}}} { n_{_{SE}} + n_{_{NEQ}} },\n\\label{eq:ratio}\n\\end{eqnarray}\nThe parameter $r$ varies between -1 and 1; its meaning is as follows:\n\n\n\\begin{enumerate}[a)]\n\n\\item If $r \\approxeq 0$, the number density of emitters obtained imposing SE would be approximately\nequal to the one allowing NEQ rates ($n_{_{SE}} \\approxeq n_{_{NEQ}}$), so in those regions the SE approximation to calculate the state of ionization would be valid.\n\n\\item If clearly $r < 0$, this means that $n_{_{SE}} < n_{_{NEQ}}$, so the approximation of SE ionization would underestimate the real \npopulation. As $r$ becomes more negative, the NEQ\neffects would be more prominent and the SE approximation \nwould become less accurate. In the extreme case ($r = -1$), the assumption of SE would mistakenly result in an absence of ions in the ionization state of interest!\n\n\\item On the other hand, if $r > 0$, it follows that $n_{_{SE}} > n_{_{NEQ}}$, so the computation \nof the ionization in SE would be wrong again, but in this case because it would overestimate the real population. When $r = 1$, SE would give as a result a totally fictitious population, since\nthe full NEQ calculation indicates that there are no ions!\n\n\\end{enumerate}\n\n\n\\begin{figure}\n\\epsscale{1.05}\n\\plotone{figure3.pdf}\n\\caption{2D maps of the ratio $r$ from Equation (\\ref{eq:ratio}) for the\n {\\ion{Si}{4}}\\ population for (A) the slanted experiment at $t=64.3$ minute and (B)\n the vertical one at $t=65.0$ minute. Panel C shows $r$ for the\n {\\ion{O}{4}}\\ population in the vertical experiment at $t=65.0$ minute. \n A gray color mask is overplotted where the emissivity obtained from\n the NEQ computation is $\\epsilon_{_{CGS}} < 10^{-10}$. Below each 2D map,\n the median M of $\\vert r \\vert$ in the high-emissivity region (i.e., outside\n of the mask) is shown. Solid and dashed\n vertical lines delimit the ETR and QTR regions, respectively. The\n horizontal line in the M panels marks the average value of M within the\n QTR. \\label{figure3}}\n\\end{figure} \n\n \nThe ratio $r$ is plotted in Figure~\\ref{figure3} for\n the two experiments described in this paper. The upper panels in each block\n contain 2D maps of $r$, namely, for (A) {\\ion{Si}{4}}\\ in the slanted experiment at\n $t=64.3$ minute; (B) {\\ion{Si}{4}}\\ in the vertical experiment at $t=65$ minute;\n and (C) {\\ion{O}{4}}\\ also in the vertical experiment at $t=65$ minute. To limit\n the diagram to the relevant regions, a grey mask is overplotted and only those pixels with emissivity obtained from\n the NEQ computation above a threshold ($\\epsilon_{_{CGS}} \\geq 10^{-10}$) are being shown. \n The bottom panel in each block contains a line plot for the median M of the absolute value of $r$\n in the regions not covered by the mask in each column. Using\n the absolute value of the ratio elucidates the areas where NEQ ionization is \n important, either because SE underestimates ($r < 0$) or overestimates \n ($r > 0$) the real number density of emitters $n_u$. In the figure, two regions can be clearly distinguished:\n \n \n\n\n1. The \\textit{Quiet Transition Region} (hereafter QTR). We define it as \nthe transition region that has not been perturbed by the flux emergence and subsequent surge \nand\/or jet phenomena. The horizontal extent of the QTR is marked in the figure with dashed vertical lines \nand corresponds to the region located between $0.0 \\leq x \\leq 2.0$ and $22.0 \\leq x \\leq 32.0$ Mm, \nfor the slanted experiment; and $0.0 \\leq x \\leq 3.0$ and $26.5 \\leq x \\leq 32.0$ Mm, for the vertical one. \nIn this domain, $r$ mostly shows negative values (blue color in the image) in a thin \nlayer in the transition region ($z \\sim 2$ Mm). The corresponding M value is on average \nbetween $0.2$ and $0.3$ (horizontal dashed line in red in the panels), which indicates that both {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ \nsuffer significant departures from statistical equilibrium.\n\n\n2. The second region corresponds to the main result of this section: the emerged \ndomain, namely, the dome and surge, are severely affected by the NEQ ionization both for silicon \nand oxygen (see the dark blue color which corresponds to $r\\approx-1$, and corresponding M value close to 1). \nThe value $r\\approx-1$ is found either in cold regions with $T \\sim 2 \\times 10^4$ K and\nalso in hot domains ($T \\sim 5 \\times 10^5$ K). Also, on the left of \nthe surge, we also find some regions where $r > 0$ (red), especially in the slanted experiment,\nwhich indicates that the SE approximation is overestimating the real population. Since our main goal is \nto study the surge, we focus on its surroundings, and, in particular, on the domain marked in the figure with solid lines, i.e., $8.5 \\leq x \\leq 21.0$ Mm, for the slanted experiment, and $11.0 \\leq x \\leq 25.5$ Mm, for the vertical one. In the following, we refer to this range as the \\textit{Enhanced Transition Region} (ETR). \n In the associated 1D panels, we see that M shows larger values than in the QTR; in fact, the median reaches values close to one in many places of the ETR . There are some specific locations within the ETR where M shows substantially lower values, e.g., $x=16.2$ Mm or $x=18.1$ Mm in the B and C panels . Looking at the 2D panels, we realize that in those locations, part of the TR has $r$ close to zero (white patch above the blue line). Consequently, the median in that vertical columns decreases. Note, however, that even in those locations the M values in the ETR are larger than, or at least comparable to, the largest ones found in the QTR. This finding highlights the relevance of including the NEQ calculation for eruptive phenomena like surges, since without it, the calculated {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ populations would be totally erroneous. This would translate into wrong emissivity values and therefore mistaken synthesis diagnostics. \n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\epsscale{1.0}\n\\plotone{figure4.pdf}\n\\caption{\nMaps of the 2D emissivity $\\epsilon$ for A) {\\ion{Si}{4}}\\ in the slanted experiment, \nB) {\\ion{Si}{4}}\\ in the vertical one, and C) {\\ion{O}{4}}\\ also for the vertical experiment. Diamonds have been superimposed on the region of high emissivity to mark the position of $z_{max}$ (see Equation \\ref{eq:H}). \nA color scale at the right column of the figure contains the translation from emissivity to number densities of emitters $n_u$. In each of the blocks, secondary panels for $\\epsilon$, $T$, and $n_e$ have been inserted that use $H$ as vertical scale. Additionally, a panel of the ratio of the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities, $R_{\\epsilon}$, is added for B and C blocks. All the maps only show places where $\\epsilon_{_{CGS}} \\geq 10^{-8}$. The instants in the panels and the vertical lines are\nthe same as in Figure \\ref{figure3}. The accompanying animation shows the time evolution of the\nthree experiments from the early stages of the surge until the its decay phase.\\\\\n(An animation of the figure is available)}\\label{figure4}\n\\end{figure*} \n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Characterizing the plasma in NEQ}\\label{sec:4.2}\nOnce we have studied the NEQ effects on the two different domains of our \nexperiments (ETR, QTR), we now turn to the associated question of the \nemissivity, in particular, for the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ lines. To that end, we start by showing 2D\nmaps of the emissivity in Figure \\ref{figure4} (top panel in each block) for the same \ninstants as in Figure \\ref{figure3}. In this case, we have constrained the\nmaps to values of $\\epsilon_{_{CGS}} > 10^{-8}$, just to \nfocus on the layer with the largest emission, which is the natural candidate to be observed. \nSince the emitting layer is really thin, we are adding small 2D maps at the bottom of each block containing a blow-up of the emissivity, $\\epsilon$, and, additionally, of the temperature, $T$; electronic number density, $n_e$; and the ratio between the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities, $R_{\\epsilon}$. More precisely, for each vertical column we define a height coordinate $H$ centered at the position [called $z_{max}(x)$ in the following] of the \nmaximum emissivity in that column:\n\\begin{eqnarray}\n\tH & = & z - z_{max},\n\\label{eq:H}\n\\end{eqnarray}\nand use it, instead of $z$, in the maps. For\n clarity, in the top panel we have indicated the location of $z_{max}$ at\n selected columns using symbols. Since emissivities can be converted into number densities of emitters $n_u$ via simple multiplication with a constant factor\n (Equation~\\ref{eq:emissivity} and Table~\\ref{table1}), a\n color bar with $n_u$ both for {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ has been added in the figure. \n\n\n By comparing the two emissivity panels in each block of\n Figure \\ref{figure4} (see also associated movie), we find that the region of high emissivity\n at the footpoints and crests of the surge covers a larger vertical range\n than in other regions. This is mainly caused by the varying mutual angle of the vertical\n with the local tangent to the TR, so, in some sense, it is a line-of-sight\n (LOS) effect: full details of different LOS effects are discussed in Section\n \\ref{sec:5.2}. Inspecting the lower panels of $\\epsilon$ of each block, some locations (e.g.,\n the internal footpoint, $x \\approx 15$ Mm, in {\\ion{O}{4}}\\ at $t=65.0$ minute)\n are seen to have enhanced emissivity by a factor 2 or 3 in comparison to the\n maximum values usually seen at positions of the QTR and ETR; nonetheless, \n this behavior is sporadic as seen in the accompanying movie. \n \n We also notice that both {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ show similar values of emissivity, in spite of the \n huge contrast in the corresponding number density of emitters (see second \n color scale at the right-top corner of the image). This is due to the difference in the \n oscillator strengths $f_{lu}$, which for {\\ion{O}{4}}\\ is six orders of magnitude weaker than for \n {\\ion{Si}{4}}\\ (see Table \\ref{table1}). In panels B4 and C4, we have plotted \n the emissivity ratio of {\\ion{Si}{4}}\\ to {\\ion{O}{4}}, $R_{\\epsilon}$, finding that the typical values \n in the locations with the highest emissivity within the ETR are around 2 \n (although it can reach up to factors around 5), while in the QTR\n the average $R_{\\epsilon}$ is close to 1. On the other hand, in the locations with low emissivity \n and high temperature, especially in the \n QTR, we appreciate that $R_{\\epsilon}$ is lower than unity, which is not\n surprising since {\\ion{O}{4}}\\ can be found at higher temperatures. Note\n that this is a ratio of emissivities and does not correspond to the intensity ratio commonly\n used for density diagnostics (e.g., \\citealp{Hayes:1987,Feldman:2008,Polito2016}). \n\n \n \n \n We cannot find in the emissivity maps the same sort of drastic contrast \n between QTR and ETR that we found for the $r$ parameter in the previous \n section; nonetheless, we do appreciate differences between both\n regions in terms of temperature and electronic number density: the range of $T$ and \n $n_e$ in the ETR is larger than in the QTR. This is especially evident in the hot and \n low-density part, where the $T$ and $n_e$ of the ETR reach values around 1 MK and \n $10^9$ cm$^{-3}$, respectively (Note that the $n_e$ provided is obtained from local \n thermodynamic equilibrium (LTE) since the ionization of the main contributors for electrons, such as \n hydrogen and helium, are computed in LTE according to the equation-of-state table of Bifrost). \n In order to further explore those differences, we\n resort to a statistical study of the values of emissivity and temperature\n in the different regions (QTR, ETR) and for the two ions, which is presented in the\n following. The statistics is based on all plasma elements with\n $\\epsilon_{_{CGS}} \\geq 10^{-8}$ in the time span between surge formation\n ($t = 55.0$ minute) and decay ($t = 70.7$ minute). The resulting sample\n contains $4 \\times 10^6$ elements. Figure~\\ref{figure5} shows the\n corresponding Joint Probability Density Functions (JPDFs) for \n emissivity $\\epsilon$ and temperature $T$ for the vertical\n experiment. For {\\ion{Si}{4}}\\ we could also show results for the statistical distributions \n for the slanted experiment, but the resulting JPDFs are very similar to those \n presented here. This similarity suggests that,\n although the vertical and slanted experiments differ in \n terms of magnetic configuration, size of the emerged dome, and shape of the \n surge (compare the two panels of Figure \\ref{figure2}), the results described below\n could be applicable to different surge scenarios. In the following, \n we explain the results first for {\\ion{Si}{4}}\\ (Section \\ref{sec:4.2.1}), and then \n for {\\ion{O}{4}}\\ (Section \\ref{sec:4.2.2}).\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Plasma emitting in {\\ion{Si}{4}}}\\label{sec:4.2.1}\n\\begin{figure}\n\\epsscale{1.18}\n\\plotone{figure5.pdf}\n\\caption{JPDFs of emissivity and temperature in the QTR and ETR in\n the vertical experiment for {\\ion{Si}{4}}\\ (top row) and {\\ion{O}{4}}\\ (bottom row) for\n the time range $t=55.0$ minute to $t=70.7$ minute. The size of the sample is $4 \\times\n 10^6$ elements. The white lines are isocontours of probability\n equal to $10^{-4}$ in the ETR distribution. The areas\n marked by ovals and arrows are discussed in the text.\\label{figure5}}\n\\end{figure} \n\n\nWe start analyzing the QTR and ETR distributions for the {\\ion{Si}{4}}\\ emissivity\n(see top row of Figure \\ref{figure5}). The main result is that in the region\nwith the largest emissivity values the ETR is more densely populated than the\nQTR (see the region marked by a pink oval around $\\epsilon_{_{CGS}} \\sim\n10^{-5.6}$), i.e., the boundaries of the surges are more likely to show\nsignal in {\\ion{Si}{4}}\\ observations than the QTR. This helps explain why, in the\n\\textit{IRIS}\\ observations of our previous paper NS2017, we could detect the surge\nas an intrinsically brighter structure than the rest of the TR. Furthermore,\nboth the QTR and ETR have the greatest values of emissivity in the\ntemperature range between $10^{5.0}$ and $10^{5.1}$ K. This differs from what\none would expect in SE, where the maximum emissivity is located at the peak\nformation temperature ($T_{_{SE}} = 10^{4.9}$ K, see Table \\ref{table1}),\nagain an indicator of the importance of taking into account NEQ effects. Additionally,\nthe distribution both for QTR and ETR is more spread in temperature than what one would expect from a\ntransition region distribution computed in SE (see, e.g., Figure 15 of the paper by \\citealp{olluri:2015}). The\nmass density found both for QTR and ETR around the maximum {\\ion{Si}{4}}\\ emissivity\nis $\\rho \\sim 6.3 \\times 10^{-15}$ g cm$^{-3}$.\n\n\nAs part of the analysis, we have also found other features worth mentioning:\n\n\n\\begin{figure*}\n\\epsscale{1.18}\n\\plotone{figure6.pdf}\n\\caption{Density map showing the basic distribution of the (approximately) 6000 Lagrange \ntracers used in the text, distributed into two parts drawn with yellow and\nred dots corresponding to the two populations discussed in\nSection~\\ref{sec:4.3}. The accompanying animation shows the evolution of the\nLagrange tracers in the two experiments from \nearly stages of the surge formation ($t \\approx 55.0$ minute) until the its decay \nphase ($t \\approx 70.0$ minute).\\\\\n(An animation of this figure is available.)}\n\\label{figure6}\n\\end{figure*} \n\n\\begin{itemize}\n\n\\item The ETR has a broader temperature distribution than the QTR. \nIn order to illustrate this fact, all the panels of Figure \\ref{figure5} contain isolines \nin white for the probability $10^{-4}$ in the ETR. \nThe comparison of those contours with the QTR distribution shows that\nthe ETR has a wider distribution\nin temperature, specially above $10^{5.5}$ K. Although not shown in the figure, the mass density values for \n most of the emitting plasma (more precisely: the mass density values with probability\n above $10^{-4}$) are constrained to similar ranges \n for both the QTR and ETR: approximately at [$2.0 \\times 10^{-15}$, $7.9 \\times 10^{-14}$] g cm$^{-3}$, \n for the vertical experiment; and at [$1.0 \\times 10^{-15}$, $6 .0 \\times 10^{-14}$] g cm$^{-3}$, for the slanted one.\n\n\n\n\\item A secondary probabilty maximum is located $\\epsilon_{_{CGS}} \\sim 10^{-7.2}$ and \nT $\\sim 10^{4.3}$ K (see the arrows\nin the panels). This corresponds to the temperature of the second ionization of helium \naccording to the LTE equation-of-state of Bifrost: the energy deposited\nin the plasma is used to ionize the element instead of heating the plasma.\nIncluding the NEQ ionization of helium should scatter the density probability in temperatures, as shown by \n\\cite{golding:2016} in the TR of their numerical experiments\n(the equivalent to our QTR); nevertheless, the NEQ computation of helium and \na detailed discussion of their effects are out of scope of this paper. \n\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Plasma emitting in {\\ion{O}{4}}}\\label{sec:4.2.2}\nFocusing now on the statistical properties of the {\\ion{O}{4}}\\ emission (see bottom row \nof Figure \\ref{figure5}), we see that, like for {\\ion{Si}{4}}, the probability distributions for \nboth QTR and ETR differ from what we could expect for a SE distribution, \nsince they are centered at temperatures between $10^{4.9}$ and $10^{5.0}$ K\ninstead of $T_{_{SE}} = 10^{5.2}$ K (see Table \\ref{table1}). Furthermore,\nalso here, the ETR and QTR distributions are broader in temperature than what one would \nexpect from SE. Further noteworthy features of the plasma emitting in {\\ion{O}{4}}\\ are:\n\n\\begin{itemize}\n\n\\item The QTR exhibits larger probability than the ETR ($> 10^{-3}$) in\nthe maximum values of the emissivities ($\\epsilon_{_{CGS}} = 10^{-5.8}$); \nnevertheless, this fact changes around $\\epsilon_{_{CGS}} = 10^{-6.0}$,\nwhere the ETR shows larger emissivity (compare the region within the colored oval). \nDue to this complex behavior, we need to integrate the emissivity to know whether the \nETR can be detected as a brighter structure compared to the QTR. In Section \n\\ref{sec:5} we discuss this fact analyzing the obtained synthetic profiles.\n\n\n\\item The ETR shows emissivity in {\\ion{O}{4}}\\ in a larger range of temperatures than \nthe QTR, which is akin to the result for {\\ion{Si}{4}}\\ described in Section\n\\ref{sec:4.2.1}. This difference in the ranges is apparent mainly in hot coronal\n temperatures comparing the probability contours of the ETR (in white) \n with the QTR distribution.\n\n\\item We find the same secondary probability maximum as in the {\\ion{Si}{4}}\\\npanels at the temperature of the second ionization of helium (see the pinks arrows).\n\n\\item Comparing the {\\ion{O}{4}}\\ panels with the {\\ion{Si}{4}}\\ ones, we see that the {\\ion{O}{4}}\\ \ndistribution is more populated in hot temperatures, and correspondingly, \nlower densities. This is something we could expect since the ionization of this particular\noxygen ion occurs at higher temperatures than {\\ion{Si}{4}}.\n\n\\end{itemize}\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\n\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Lagrange tracing: how the entropy sources affect the NEQ ionization}\\label{sec:4.3}\nWe focus now on the role of the entropy sources in the emissivity and \nNEQ ionization. To that end, we follow in time $ \\approx 6000$ plasma elements of the \nETR through Lagrange tracing. In the following, we explain the set up for the Lagrange elements\n(Section \\ref{sec:4.3.1}) and the results obtained from their tracing (Section \\ref{sec:4.3.2}).\n\n\n\\subsubsection{The choice of the Lagrange elements}\\label{sec:4.3.1}\n\nThe Lagrange elements are selected at a given instant, corresponding to an\n intermediate evolutionary stage when the surge is clearly \n distinguishable as a separate structure from the dome. The\n selected instants are $t=64.3$ minute for the slanted experiment and\n $t=65.0$ minute for the vertical experiment, which are the same times\n used for Figures \\ref{figure2}, \\ref{figure3}, and \\ref{figure4}. In order to focus on \n the domain in and near the surge, we limit the selection to\n the rectangular areas: $12.0 \\leq x \\leq 25.2$, $2.2 \\leq z \\leq\n 15.0$ (vertical experiment); and $7.0 \\leq x \\leq 19.0$, $5.0 \\leq z \\leq\n 15.0$, and $19.0 \\leq x \\leq 20.5$, $1.5 \\leq z \\leq 15.0$ (slanted\n experiment). On those rectangles we lay a grid with uniform spacing $\\Delta\n x = \\Delta z =40$~ km: the Lagrange elements are chosen among the\n pixels in that grid. A further criterion is then introduced: we are\n interested in studying the origin and evolution of the plasma elements with\n strong emission in {\\ion{Si}{4}}\\ and {\\ion{O}{4}}. Thus, a lower bound in the {\\ion{Si}{4}}\\ and\n {\\ion{O}{4}}\\ emissivity is established, namely $\\epsilon_{_{CGS}} > 10^{-10}$,\n discarding all the pixels with emissivities below that value at the\n instants mentioned in the previous bullet point. The resulting choice of\n Lagrange tracers is shown in Figure \\ref{figure6} as red and yellow dots\n (the colors serve to distinguish the populations described below). Once the distribution is \n set, we then follow the tracers backward in\n time for 10 minutes, to study their origin, and forward in time for 5.7\n minutes, to see the whole surge evolution until the decay phase, with a\n high temporal cadence of $0.2$~seconds (see the accompanying animation to\n Figure~\\ref{figure6}).\n\n\n\n\\subsubsection{Plasma populations and role of the entropy sources}\\label{sec:4.3.2}\nStudying the time evolution of the Lagrange tracers, in particular their\nthermal properties, one can distinguish two populations that are the source of \nthe {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emission: one originating in the emerged dome (yellow plasma population in\nFigure~\\ref{figure6}) and the other one originating in the corona (red\npopulation). By carefully inspecting the tracers of each population, we find that\ntheir behavior is well defined: the major difference between the elements within\nthe same population is not the nature or order of the physical events described below, \nbut rather the starting time of the evolution for each tracer. Figure~\\ref{figure7} contains the time evolution of\ndifferent quantities as measured following a representative Lagrange element of\n each population, namely temperature, $T$, (green); {\\ion{Si}{4}}\\ emissivity,\n$\\epsilon_{_{Si\\ IV}}$, (dark blue); {\\ion{O}{4}}\\ emissivity, $\\epsilon_{_{O\\ IV}}$,\n(light blue); characteristic time of the optically thin losses,\n$\\tau_{thin}$, (black); and characteristic time for the thermal conduction,\n$\\tau_{Spitz}$ (red). \n\n\n\n\n\\begin{figure}\n\\epsscale{1.15}\n\\plotone{figure7.pdf}\n\\caption{Time evolution of key physical quantities for representative\n Lagrange tracers in the vertical experiment of Figure~\\ref{figure5}. Top:\n Lagrange element coming from the emerged dome (yellow population in\n Figure~\\ref{figure6}). Bottom: Lagrange originating in the corona (red\n population in Figure~\\ref{figure6}). The curves show (left\n ordinate axis) the logarithm of temperature $T$ (green); of the\n characteristic time of the optically thin losses $\\tau_{thin}$ (black);\n and of the characteristic time for the thermal conduction $\\tau_{Spitz}$\n (red); and (right ordinate axis) the logarithm of the {\\ion{Si}{4}}\\ emissivity\n (dark blue); and of the {\\ion{O}{4}}\\ emissivity (light blue). All quantities are\n in CGS units. \\label{figure7}}\n\\end{figure} \n\n\\textit{The first population} (top panel of Figure~\\ref{figure7}, corresponding to the\n elements marked in yellow in Figure~\\ref{figure6}), starts as\n cool and dense plasma coming from the emerged dome with extremely low\n emissivity (see the curves for the temperature in green, and for the\n emissivities in dark and light blue). At some point\n that plasma approaches the reconnection site and passes through the current\n sheet, thereby suffering strong Joule and viscous heating and\n quickly reaching TR temperatures. The sharp\n spike in the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity (blue curves) around $t\\sim\n 62.5$ minute corresponds to this phase: the temperature increase leads to\n the appearance of those ionic species, but, as the plasma continues being\n heated, it reaches high temperatures (maximum around $10^6$ K) and the\n number densities $n_u$ of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ decrease again. At those high\n temperatures the entropy sinks become efficient, with short\n characteristic times: see the red ($\\tau_{Spitz}$) and black\n ($\\tau_{thin}$) curves. The plasma thus enters a phase of gradual\n cooling, going again through TR temperatures, renewed\n formation of the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ ions, and increase in the corresponding\n emissivity (broad maximum in the blue curves in the right half of the\n panel). The plasma elements, finally, cool down to chromospheric\n temperatures, with the emissivity decreasing again to very low\n values. \n\n\n\n\n\n The defining feature of \\textit{the second population} (bottom panel of\n Figure~\\ref{figure7}, red dots in Figure~\\ref{figure6}) is that it\n originates in the corona as apparent in the temperature curve\n (green). This population starts at heights far above the reconnection\n site, with standard coronal temperature and density. During the magnetic\n reconnection process, its associated field line changes connectivity,\n becoming attached to the cool emerged region. Consequently, a steep\n temperature gradient arises along the field line, so the thermal\n conduction starts to cool down the plasma; given the temperature range,\n also the optically thin losses contribute to the cooling, although to a\n lesser extent (see the $\\tau_{Spitz}$ and $\\tau_{thin}$ curves around\n $t\\sim 61$ minute, in red and black, respectively). The temperature drops\n to values around T $\\sim 10^{5.5}$ K, which, according to the\n JPDFs of Figure~\\ref{figure5}, makes it sufficiently likely that the\n {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities from the Lagrange element are high. This\n explains the large increase, by a few orders of magnitude, in the blue\n emissivity curves around $t\\sim 61$ minute \n (although a small factor $\\sim 4$ is due to the simultaneous \n increase in the mass density, which is reflected in a linear fashion in the emissivity). \n This cooling to TR\n temperatures, however, is short lived: as the plasma element itself passes\n near the current sheet, it can\n be heated because of the Joule and viscous terms and the temperature\n climbs again to values where the emissivities are low: hence the sharp\n spike in the blue curves between $t\\sim 61$ and $t\\sim 61.5$ minute. There\n ensues a phase of gradual cooling from $t \\sim 64$ onward, similar to what\n happened to the previous population, with characteristic cooling times of\n a few to several hundred seconds (see red and black curves), passage\n through TR temperatures, broad maximum in the emissivity\n curves and with the plasma finally reaching chromospheric temperatures. \n\n\n\n\\begin{figure}\n\\epsscale{1.18}\n\\plotone{figure8.pdf}\n\\caption{\nJPDFs for the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity of the Lagrange tracers\n over $15.7$ minutes versus the \ncharacteristic time of the optically thin losses ($\\tau_{thin}$, left column) and the thermal conduction \n($\\tau_{Spitz}$, right column). \\label{figure8}}\n\\end{figure} \n\n\n\n\\begin{figure*}\n\\epsscale{1.21}\n\\plotone{figure9.pdf}\n\\caption{Synthetic profiles for the vertical experiment at different times for various LOS ($\\theta$). \nA) 2D map of the $\\epsilon_{_{CGS}}$ for {\\ion{O}{4}}\\ to show the context; B) synthetic spectral \nintensity for {\\ion{Si}{4}}, and ; C) synthetic spectral \nintensity for {\\ion{O}{4}}. To identify the LOS of each panel, we have added the symbols $0$, $-$, and\n$+$ respectively to $\\theta = 0, -15$ and $15$ \\degree. In the $\\theta \\neq 0$ rows, $z_P$ and $x_P$ are, respectively, the vertical and horizontal \ncoordinates of the rotated figures. The animation of this figure shows \nthe time evolution of the surge from its origin ($t = 61$ minute) up to its decay phase \n($t = 70.7$ minute) in the vertical experiment for the three LOS.\\\\\n(An animation of this figure is available.)} \\label{figure9}\n\\end{figure*} \n\nIn our previous paper (NS2016) we found that surges were constituted by four\ndifferent populations according to their thermal evolution. In the current\npaper, we see that only two of them, labelled Populations B and D in the\nNS2016 paper, are behind the enhanced emissivity of TR lines\nlike those from silicon and oxygen discussed here. The other two populations\ndescribed by NS2016 (A and C) keep cool chromospheric temperatures during\ntheir evolution and do not play any role for the TR elements.\n\n\n\n\nUsing the Lagrange tracing method developed here, we can produce conclusive\nevidence of enhanced {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity and occurrence of fast evolution \ndue to short-time scales in the entropy sources associated with heat conduction \nor optically thin radiative cooling. Figure~\\ref{figure8} contains double PDFs for $\\epsilon{_{CGS}}$\nversus $\\tau_{thin}$ (left panels) and $\\tau_{Spitz}$ (right panels) using\nas statistical sample the values of those quantities for all Lagrange\ntracers along their evolution. The choice of the ionic species ({\\ion{Si}{4}}, {\\ion{O}{4}}) and\nexperiment (slanted, vertical) in the panels is as in Figures \\ref{figure3} and \\ref{figure4}.\nThe figure clearly shows that when the entropy sources act on short time\nscales, the ({\\ion{Si}{4}}, {\\ion{O}{4}}) emissivities are large. In fact, the maximum\nvalues of $\\epsilon{_{CGS}}$ correspond to characteristic cooling times\nbetween $20-100$~s for $\\tau_{thin}$ and between $4-40$~s for\n$\\tau_{Spitz}$. Those\n changes are fast enough for the ionization levels of those elements to be\n far from statistical equilibrium.\n\n\n\n\\Needspace{5\\baselineskip}\n\\section{Observational consequences}\\label{sec:5}\nIn the paper by NS2017, different observed {\\ion{Si}{4}}\\ signatures\nwithin the surge were analyzed. Moreover, counterparts to the observational features were\nidentified in the synthetic spectral profiles obtained from the numerical\nmodel; however, a theoretical analysis to understand the origin of the\nspectral features and the reason for the\nbrightness in the various regions of the surge was not addressed. In the\nfollowing subsection a theoretical \nstudy is carried out trying to quantify the impact of the NEQ ionization of silicon\nand oxygen on the spectral and total intensities and the observational consequences thereof\n(Section \\ref{sec:5.1}). Then, given the involved geometry of the\n surge, the particular LOS for the (real or synthetic) observation turns out to be\ncrucial for the resulting total intensity and spectra. This is studied in Section \\ref{sec:5.2}.\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Synthetic profiles}\\label{sec:5.1}\nFigure \\ref{figure9} contains the synthetic profiles obtained by integrating the \nemissivity along the line of sight for different wavelengths in the {\\ion{Si}{4}}\\ $1402.77$ \n\\AA\\ and {\\ion{O}{4}}\\ $1401.16$ \\AA\\ spectral region and for the vertical experiment. The \nthree rows of the figure correspond to different inclination angles $\\theta$ for the \nLOS: from top to bottom, 0\\degree, -15\\degree\\ and 15\\degree, respectively. The \npanels in each row contain A) the context of the experiment through a 2D map of \nthe {\\ion{O}{4}}\\ emissivity; B) the synthetic spectral intensity for {\\ion{Si}{4}}\\ with the spectral \ndimension in ordinates and in the form of Doppler shifts from the central \nwavelength; and C) the corresponding synthetic spectral intensity for {\\ion{O}{4}}. \nThose spectra are obtained taking into account the Doppler shift due to the plasma \nvelocity and applying a spatial and spectral PSF (Gaussian) degradation \nas explained in detail by \\cite{Martinez-Sykora:2016obs}, their section 3.1, and by \nNS2017, their section 2.2. In this way, we will be able to directly\ncompare the results with \\textit{IRIS}\nobservations. In order to ease the identification of the LOS in each panel, \nwe have added to the labels on it, the symbols $0$, $-$, and $+$ respectively to $\\theta = 0, -15$ and $15$ \\degree. In the middle and bottom rows of the image, $x_P$ and \n$z_P$ are, respectively, the horizontal and vertical coordinates of the rotated figures. \n\nTo extend the analysis, it is also of interest to\nconsider the total emitted intensity for each vertical column, i.e.,\n\\begin{eqnarray}\n\tI_{\\epsilon} (x) & = & \\int_{z_0}^{z_f}{ \\epsilon\\; dz}\\;,\n\\label{eq:ie}\n\\end{eqnarray}\nwhich, following Equation \\ref{eq:emissivity}, is equal to the column density \nof the emitting species along the LOS except for a constant factor). Equation (\\ref{eq:ie})\nhas been calculated separately for {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ and with the\nemissivities obtained assuming either NEQ or SE ionization, to\nbetter gauge the importance of disregarding the NEQ \neffects. The results are shown in the middle and bottom\npanel of Figure \\ref{figure10} for {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ , respectively. The top\npanel of the image contains the 2D map of the emissivity for {\\ion{Si}{4}}\\ for\ncontext identification. Combining Figures \\ref{figure9} and \\ref{figure10},\nwe are able to discern and describe characteristic features of the spectral\nprofiles.\n\n\n\\begin{itemize}\n\n\\item \nThe most prominent feature in the synthetic profiles of Figure \\ref{figure9}\nis the brightening associated with the location of the internal footpoint of\nthe surge. In the corresponding movie, we can see how that footpoint is\nformed at around $t=64$ minute, as the surge detaches from the emerged\ndome. During those instants, the associated synthetic profiles are\ncharacterized by large intermittent intensities and\nbidirectional behavior with velocities of tens of km s$^{-1}$, as apparent,\ne.g., in the B0 and C0 panels at $x \\in [15,16]$ Mm. In\n{\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ (B and C panels), we find that the internal footpoint is\nusually the brightest region, although there are some instants in which the\nbrightest points can be located in the crests or the external footpoint. This\nis a potentially important result from the observational point of view\nbecause it can help us to unravel the spatial geometry of the surge in future\nobservational studies: if strong brightenings are detected in {\\ion{Si}{4}}\\ and also\nin {\\ion{O}{4}}\\ within the surge, it would be reasonable to think that they\ncorrespond to the internal footpoint of the surge. In this region, \nthe intensity ratio between {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ ranges between 2 and 7, approximately. Note\nthat, in general, the intensity ratio values vary depending on the observed region \nand features \\citep{Martinez-Sykora:2016obs}.\n\n\nIn Section \\ref{sec:5.2} we will see that LOS effects play a major role in causing \nthe large brightness of the internal footpoint (and other bright features) compared to \nthe rest of the surge. Here we consider the parallel question of the role of NEQ: what would be obtained for the intensity of the internal footpoint if SE were assumed? Comparing the values for {\\ion{Si}{4}}\\ in Figure\n\\ref{figure10} (middle panel, $13.8 \\le x \\le 16.1$), there is roughly a\nfactor $2$, in the average, between the NEQ and SE intensities; for {\\ion{O}{4}}\\ (bottom\npanel), there is no major difference in the intensity between both\ncalculations. One could conclude that while NEQ is important for\n the {\\ion{Si}{4}}\\ diagnostics, SE could be applied for the\n {\\ion{O}{4}}\\ case; nonetheless, even in the latter case, although the\n NEQ and SE intensities are similar, one would make a mistake in the\n determination of derived quantities like the number densities of emitters (Section\n\\ref{sec:4.1}) and temperatures (Section \\ref{sec:4.2}).\n\n \nFurther distinctive brightenings in the spectral profiles appear at the site \nof the crests and of the external footpoint of the surge. The brightness of those\n regions is clear in the {\\ion{Si}{4}}\\ profiles (B0, B$-$ and B$+$\npanels of Figure \\ref{figure9}) through their large intensity, and is sometimes\ncomparable or greater than that of the internal footpoint (see, e.g., the\nlocations at $x \\sim 13$, $18$, $23$ and $25$ Mm in the top row, or, $x \\sim\n14, 19$ and $25$ Mm in the bottom row). In the {\\ion{O}{4}}\\ profiles (C0, C$-$ and C$+$ panels),\nalthough faint in comparison with {\\ion{Si}{4}}\\ (around a factor 5 less intense), \nmost of those features are still slightly brighter than the rest of the surge. This difference in intensities\nbetween {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ can also be used to understand the observations: if\nstrong signals are observed in {\\ion{Si}{4}}\\ associated with some moderate signal in\n{\\ion{O}{4}}, it could indicate that we are detecting the crests or the external\nfootpoint. Concerning the NEQ\/SE comparison of the intensity (Figure\n\\ref{figure10}), the crests and footpoints show the same behavior as the\ninternal footpoint.\n\n\n\n\\item The intensity of the rest of the surge is small in comparison\n with that of the footpoints and crests just described, so we\n wonder whether one could see it as a bright structure in actual\n observations and distinguish it from the rest of the TR.\n In the middle panel of Figure \\ref{figure10}, comparing the $I_{\\epsilon}$\n values for {\\ion{Si}{4}}\\ within the ETR against those in the QTR, we can see that\n all the regions in the surge have a higher intensity than the\n QTR; outside of the brightest features, the excess emission of\n {\\ion{Si}{4}}\\ in the surge may be just a factor 2 or 3 above the QTR, but that\n can provide enough contrast to tell the two regions apart\n observationally, as found in the NS2017 paper. Note, importantly,\n that there is a large difference between the NEQ and SE results\n for {\\ion{Si}{4}}\\ in the surge, up to a factor 10, so\n the SE assumption would seriously underestimate the\n intensity. In fact, in most of the places {\\ion{Si}{4}}\\ would be similar or fainter\n than {\\ion{O}{4}}\\ if SE were valid as shown also, e.g. by \\cite{Dudik:2014} and \\cite{Dzifcakova:2017}.\n On the other hand, for {\\ion{O}{4}}\\ (bottom panel), the prominent features are the footpoints and crests\n of the surge, while the other parts have $I_{\\epsilon}$\n comparable to the QTR. As a consequence and from an observational point of view, while in {\\ion{Si}{4}}\\ we\n could find enhanced emission in the whole surge (which is compatible with\n the NS2017 observation), for {\\ion{O}{4}}\\ only the brightest regions would stand\n out, namely, the internal footpoints and, to a lesser extent, the crests and external footpoints. \n \n The underlying reason for the enhanced brightness of all those features, \n mainly the footpoints and crests of the surges, is not just the presence of \n additional numbers of ions due to NEQ effects: the complex geometry of\n the surge TR has important consequences when integrating the emissivity\n along the line of sight to obtain intensities . This is discussed in the\n next section. \n \n \n\\end{itemize}\n\n\n\n\\begin{figure}\n\\epsscale{1.21}\n\\plotone{figure10.pdf}\n\\caption{Top panel: 2D map of the {\\ion{Si}{4}}\\ emissivity. The\nvertically integrated intensity $I_{\\epsilon}$ is shown both for {\\ion{Si}{4}}\\ (middle panel) and\n{\\ion{O}{4}}\\ (bottom panel) assuming NEQ ionization (light blue curve) and SE (purple).\nSolid and dashed lines are overplotted in the image to \ndelimit the ETR and QTR regions as in previous figures. Dotted vertical lines are\nsuperimposed in the middle and bottom panel correspondingly to the cuts shown\nin Figure \\ref{figure11}.} \\label{figure10}\n\\end{figure} \n\n\\Needspace{5\\baselineskip}\n\\subsection{The role of the LOS}\\label{sec:5.2}\nThe observation of\n TR lines generated in the surge strongly depends on the particular\n LOS. We show this here through two different effects:\n\n\\begin{enumerate}[a)]\n\n\\item The alignment of the LOS with respect to the orientation of the surge's\n TR. We can appreciate this effect, e.g., in A0 panel of Figure \\ref{figure9}. \n There, considering,\n e.g., the crests situated at $x =13.5$, $x=18$ or $x=23$ Mm, we see that a\n vertical LOS grazing the left side of the crest will include contributions\n from a much larger length of the TR than if the crossing were perpendicular\n or nearly so. The same can be said of the external footpoint at $x=24.8$ Mm\n and also of the internal footpoint around $x = 16$ Mm. This effect clearly\n enhances the brightenings seen in those values of $x$ in panels B0 and C0.\n Further evidence can be found by checking the $I_{\\epsilon}$ curves in the\n middle and bottom panels of Figure \\ref{figure10}; in fact, since\n the effect is purely geometrical, the contribution to brightness can be seen both in the\n NEQ and SE curves in the two panels of the figure. \n Varying the angle of the LOS, we can reach enhancement factors \n between 2 and 4; nonetheless, discerning which part of that factor is exclusively due to the \n LOS is complicated, since variations in the angle of integration imply integrations along\n slightly different emitting layers. Additionally, the\n inclination of the LOS with respect to the surge's TR may be important for\n the apparent horizontal size of the brightenings. This can be seen through\n comparison of the three rows of Figure \\ref{figure9}: considering the size\n of the brightening associated with the internal footpoint, we see that it\n covers a larger horizontal range in the $\\theta=0$\\degree\\ and $\\theta=15$\\degree\\ cases (top and bottom rows), than in the $\\theta=-15$\\degree\\ case\n (middle row), since the good alignment of the latter is lost in the former.\n\n\\item The multiple crossings of the TR by individual lines of sight. \nGiven that the TR of the surge is folded, there are horizontal ranges in\nwhich the LOS crosses it more than once (typically three times, in some\nlimited ranges even five times). Given the optically thin\napproximation, the emitted intensity in those lines of sight may be\na few, or several, times larger than the value in a single crossing. \n\n\\end{enumerate}\n\n\nTo further illustrate those two effects, we use Figure\n \\ref{figure11}. The top panel contains a 2D map of the {\\ion{Si}{4}}\\ emissivity\nin which vertical cuts in different regions of interest are overplotted\nthrough colored and labelled lines. The corresponding {\\ion{Si}{4}}\\ emissivity\ndistribution along those cuts is shown in panel B. Additionally, a similar\nplot but for the {\\ion{O}{4}}\\ distribution is shown in panel C. Those vertical\ncuts are also shown in panels B and C of Figure \\ref{figure10} for comparison\npurposes. The light and dark\ngreen cuts (numbers 3 and 5) are typical examples of the effect of\nthe tangency between the LOS and the surge's TR at the crests: note the\nenhanced width of the maximum on the right in those two cuts due to tangency\neffects. Those cuts and\nthe one in black (number 2) further show the effect of\nmultiple crossings of the TR by the LOS. In particular, the LOS\ndrawn in black crosses the folded TR near the internal footpoint four times\n($5.5 < z < 8$ Mm), and an additional time at the the top of the surge ($z\n\\sim 11$ Mm). For the sake of completeness, we have added a dashed\nline for the crests (3 and 5) in the middle and lower panels showing the\nemissivity if SE had been assumed: the thickness of the high-emissivity TR\nwould be much smaller. Finally, in contrast to all\nthe foregoing cases, the rest of the surge (pink line, label 4) and the QTR\n(red and blue lines, labels 1 and 6, respectively) show a simpler geometry,\nthere is no TR-LOS alignment and there is just a single crossing.\n\n\n\n\n\\begin{figure}\n\\epsscale{1.21}\n\\plotone{figure11.pdf}\n\\caption{Illustration of the multiple crossings of the transition\n region by a single LOS. A) 2D map of the {\\ion{Si}{4}}\\ emissivity with colored\n and labelled lines overplotted in regions of interest. The lines\n corresponds to different vertical cuts used in panels B and C. B)\n {\\ion{Si}{4}}\\ emissivity versus the height $Z$ for the different vertical cuts\n shown in the 2D map. C) Like panel B but for the {\\ion{O}{4}}\\ \n emissivity. Additionally, for the cuts labelled 3 and 5 we have added as a dashed line the\n corresponding emissivity if SE had been assumed.\n} \\label{figure11}\n\\end{figure} \n\n\\Needspace{5\\baselineskip}\n\\section{Discussion}\\label{sec:6}\nIn this paper, we have carried out two 2.5D radiative-MHD numerical experiments \nof magnetic flux emergence through granular convection and into the solar\natmosphere. The experiments were performed with Bifrost, including an \nextra module of the code that computes the nonequilibrium ionization (NEQ)\nof silicon and oxygen. The time evolution of the two experiments leads to the \nformation of a cool and dense surge. We have studied the relevance\nof the NEQ ionization for the presence of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ in the\nperiphery of the surge \nand how it affects the corresponding emissivities. The properties\n of the surge plasma emitting in {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ were then\ncharacterized and compared \nwith those of the general TR plasma outside of the surge. We\nhave also analyzed the role of the heat \nconduction and optically-thin radiative cooling in the NEQ ionization. \nFurthermore, through forward modelling, \nwe have understood different features\nof the synthetic spectral profiles of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}, explaining the importance of the\nshape of the transition region surrounding the surge\ncombined with the different possible angles of the LOS and providing\npredictions for future observational studies. \n\n\nIn the following, we first address the implications of the\nimportance of the NEQ ionization in \nnumerical experiments of eruptive phenomena in which heating and \ncooling are key mechanisms. (Section \\ref{sec:6.1}). We then discuss \ntheir relevance for present and future observations (Section\n\\ref{sec:6.2}). \n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{On the importance of the nonequilibrium (NEQ) ionization}\\label{sec:6.1}\n\nThe main result of this paper is that the envelope of the emerged\ndomain, more specifically, \nof the dome and surge, are strongly affected by NEQ ionization (Section \\ref{sec:4.1}). \nFocusing on the boundaries of the surge, comparing the number densities of emitters\ncomputed via detailed solution of the NEQ rate equations with those obtained \nassuming statistical equilibrium (SE) we have concluded that the SE assumption \nwould produce an erroneous result in the population of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}, mainly \nbecause it leads to a gross underestimate of the number density of emitters. The transition \nregion outside of the flux emergence site is also affected by NEQ, but to a smaller \nextent. \n\nThe above result has consequences in the corresponding emissivity (Section\n\\ref{sec:4.2}) and therefore, in the interpretation of the observations. By\nmeans of statistical analysis, we have shown that the boundaries of the surge\nhave greater values of the {\\ion{Si}{4}}\\ emissivity than the region outside of the\nflux emergence site. Correspondingly, we have given the name\n\\textit{enhanced transition region} (ETR) to the former and \\textit{quiet\n transition region} (QTR) to the latter (Section \\ref{sec:4.2.1}). This\ndifference is part of the explanation of why the surge is a brighter\nstructure than the rest of the transition region in the \\textit{IRIS}\\ observations\nby NS2017. Furthermore, the joint probability distributions for emissivity\nand temperature are not centered at the peak formation temperature of\n{\\ion{Si}{4}}\\ or {\\ion{O}{4}}\\ in SE (see $T_{_{SE}}$ in Table \\ref{table1} and Section\n\\ref{sec:4.2.2}). This reinforces earlier results\n (e.g., \\citealp{olluri:2015}) about the inaccuracies inherent in the process of deducing\n temperature values from observations in transition region lines using SE\n considerations.\n\nIn Section \\ref{sec:4.3} we have found that there are two different\npopulations concerning the thermal evolution that leads to the {\\ion{Si}{4}}\\ and\n{\\ion{O}{4}}\\ emissivity. They have very different origins (one in the emerged plasma\ndome, the other in coronal heights) but both are characterized by the\nfact that they go through a period of rapid thermal change caused by the\noptically thin losses and thermal conduction: the maximum values of the\n{\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity in them are related to short characteristic\ncooling times: $20-100$ s for $\\tau_{thin}$ and around $4-40$ s for\n$\\tau_{Spitz}$. Those characteristic times are compatible with the\ntheoretical results by \\cite{Smith:2010}, who found that for typical\ndensities of the active corona and the transition region, the solar plasma\ncan be affected by NEQ effects if changes occur on timescales shorter than\nabout $10-100$ s. Those results highlight the role of optically thin losses\nand thermal conduction because a) they provide the physical mechanism to\ndiminish the entropy and, consequently, obtain plasma with the adequate\ntemperatures to form ions of {\\ion{Si}{4}}\\ or {\\ion{O}{4}}\\ ($\\sim 10^5$ K ); and b) they\nare fast enough to produce important departures from SE. On the other hand,\nthe ion populations calculated through the present NEQ module in Bifrost are\nnot used in the energy equation of the general R-MHD calculation (see Section\n\\ref{sec:2.1}), so this could underestimate the effects of the entropy sinks\nin the experiments. In fact, \\cite{Hansteen:1993} found deviations of more\nthan a factor two in the optically thin losses when considering\nnonequilibrium effects in his loop model, so $\\tau_{thin}$ could be even more\nefficient.\n\n\n\nOur results indicate that surges, although traditionally described as\nchromospheric phenomena, show important emission in transition region lines\ndue to the NEQ ionization linked to the quick action of the cooling\n processes, so the response of the transition region is intimately tied to\nthe surge dynamics and energetics. In fact, the same statement may apply for\nother eruptive phenomena, in which impulsive plasma heating and cooling\noccurs \\citep[see][for a review of NEQ processes in the solar\n atmosphere]{Dudik:2017rv}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Understanding observations and predictions for the future}\\label{sec:6.2}\n\n From the number density of emitters and emissivity\n results of Section~\\ref{sec:4} we gather that calculating heavy element\n populations directly through the rate equations instead of via the\n assumption of statistical equilibrium can be important to understand the\n observations of surges (and of other fast phenomena which reach TR\n temperatures in their periphery like spicules \\citealt{DePontieu:2017l1},\n or UV bursts \\citealt{Hansteen:2017ib}). In that section, analyzing the {\\ion{Si}{4}}\\\n and {\\ion{O}{4}}\\ emissivities of the plasma elements, we find that the ratio between them is approximately 2 in the regions with the highest emission within the ETR. Even though the intensity ratio is more commonly used (\\citealp{Hayes:1987,Feldman:2008,Polito2016}, among others), the emissivity ratio can also be a valuable tool to understand the behavior of the ions in different regions of the Sun. In particular, we see that in the ETR this ratio is larger than in the transition region that has not been perturbed by the\nflux emergence and subsequent surge and\/or jet phenomena (QTR).\n \n To provide theoretical support to the NS2017\nobservations and predictions for future ones, in Section \\ref{sec:5} we have\n therefore computed the synthetic profiles of {\\ion{Si}{4}}\\ 1402.77\n\\AA\\ and {\\ion{O}{4}}\\ 1401.16 \\AA, taking into account the Doppler shift because of\nthe plasma velocity and degrading the spatial and spectral resolutions to the\n\\textit{IRIS}\\ ones. A line-of-sight\n integration of the emissivities has also been carried out, to provide a\n measure for the total intensity emitted by the different regions of the\n surge. The strongest brightenings in {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ have been located at\n the site of the internal footpoint, followed by the crests and the external\n footpoint (Figure \\ref{figure9}, Section \\ref{sec:5.1}). The intensity\n ratio between {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ in those regions is, approximately, 5 (although it\n can range from 2 and 7). Those values are between those for a coronal hole \n and a quiescent active region obtained by \\cite{Martinez-Sykora:2016obs}, which\n is consistent since we are mimicking an initial stratification similar to a coronal\n hole in which a total axial magnetic flux in the range of an \n ephemeral active region \\citep{Zwaan:1987yf} has been injected. The comparison\n of the total intensity for the NEQ and SE cases (Figure \\ref{figure10},\n Section \\ref{sec:5.1}) leads to a further indication of the importance of\nusing NEQ equations to determine the number density of {\\ion{Si}{4}}: the NEQ\ncalculation yields intensities coming out of the surge which are a factor\nbetween $2$ and $10$ larger than when SE is assumed. For {\\ion{O}{4}}, instead, the\nNEQ and SE calculations yield similar integrated intensities. For {\\ion{O}{4}}, therefore, \nthe NEQ calculations are important mainly to determine derived quantities like \nnumber densities of emitters $n_u$ (Section \\ref{sec:4.1}) and temperatures (Section \\ref{sec:4.2}).\nIn addition, we have found that for {\\ion{Si}{4}}\\ all the regions in the surge have\na greater intensity than the QTR: this can explain why the surge\ncan be observationally distinguished from the QTR, as found in\nthe NS2017 paper. \n\nThe high brightness of various features in the surge has been seen\n to arise in no small part from different LOS effects tied to the peculiarly irregular shape\n of its TR, and, in particular, to its varying inclination and the folds\n that develop in it (Section~\\ref{sec:5.2}). On the one hand, whenever LOS\n and tangent plane to the TR are not mutually orthogonal, the issuing\n intensity collects emissivity from a larger number of plasma elements in\n the TR (alignment effect); on the other hand, given that the surge's TR is\n variously folded, forming crests and wedges, the LOS crosses the emitting\n layer multiple times (multiple-crossing effect). The alignment and\n multiple-crossing effects are quite evident in the footpoints and\n crests. This explains their remarkable brightness and makes clear their\n potential as beacons to indicate the presence of those special features in\n surge-like phenomena when observed in TR lines like {\\ion{Si}{4}}\\ (Figure\n \\ref{figure11}). Additionally, the multiple crossings can also have an\nimpact on the observed Doppler shifts since we could be integrating various\nemitting layers with different dynamics. So, when confronted with a TR \nobservation of a region where a surge is\n taking place, the detection of strong brightenings in\n{\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ could help unravel the geometry of\nthe surge. Furthermore, since the internal footpoint of\nthe surge is close to the reconnection site, we might also find observational\nevidences of reconnection in the neighborhood. This provides theoretical\nsupport to the location of the brightenings in the \\textit{IRIS}\\ observations by\nNS2017. In addition, if strong signals are observed in {\\ion{Si}{4}}\\ related to some\nmoderate signal in {\\ion{O}{4}}, they could correspond to the crests and the external\nfootpoint of the surge; nonetheless, the rest of the surge structure could be\nonly differentiated from the transition region in {\\ion{Si}{4}}.\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\ \\vspace{-2mm} \n\\acknowledgments We gratefully acknowledge financial support\nby the Spanish Ministry of Economy and Competitiveness (MINECO) through\nprojects AYA2011-24808 and AYA2014-55078-P, as well as by NASA through \ngrants NNX16AG90G, NNH15ZDA001N, NNX17AD33G, and by NSF grant \nAST1714955 and contract NNG09FA40C (\\textit{IRIS}).\nWe also acknowledge the computer resources and assistance provided at the \nMareNostrum (BSC\/CNS\/RES, Spain) and TeideHPC (ITER, Spain) \nsupercomputers, where the calculations have been carried out, and at the\nPleiades cluster (computing projects s1061, s1472 and s1630 from the High \nEnd Computing division of NASA), where relevant code developments\nhave been made. Finally, the authors are grateful to Dr. Peter R. Young for his \nsuggestions during the \\textit{Hinode}-11\/\\textit{IRIS}-8 science meeting,\nand also to Dr.~Jaroslav Dud\\'ik for his constructive comments to improve the paper.\n\n \n\n\n\\bibliographystyle{apj} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\numberwithin{equation}{section}\n\nOur goal in this paper is to prove a concentration inequality for product spaces which is somewhat different in spirit when compared\nwith the well-known concentration inequalities discovered by Talagrand \\cite{Tal1,Tal2}. Roughly speaking, it asserts that under\nsome mild regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits\npseudorandom behavior.\n\nTo state this inequality we need to introduce some pieces of notation. Let $n$ be a positive integer and let\n$(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ be a finite sequence of probability spaces.\nBy $(\\bbo,\\bcalf,\\bbp)$ we denote their product. More generally, for every nonempty subset $I$ of $\\{1,\\dots,n\\}$ by\n$(\\bbo_I,\\bcalf_{\\!I},\\mathbf{P}_{\\!I})$ we denote the product of the spaces $\\langle(\\Omega_i,\\calf_i,\\mathbb{P}_i):i\\in I\\rangle$.\nIn particular, we have\n\\begin{equation} \\label{e1.1}\n\\boldsymbol{\\Omega}=\\prod_{i=1}^n\\Omega_i \\ \\text{ and } \\ \\boldsymbol{\\Omega}_I=\\prod_{i\\in I} \\Omega_i.\n\\end{equation}\n(By convention, $\\boldsymbol{\\Omega}_{\\emptyset}$ stands for the empty set.)\n\nNow let $f\\colon\\boldsymbol{\\Omega}\\to\\rr$ be an integrable random variable and $I\\subseteq \\{1,\\dots,n\\}$ such that $I$ and\n$I^{\\mathsf{c}}\\coloneqq \\{1,\\dots,n\\}\\setminus I$ are nonempty. For every $\\mathbf{x}\\in\\boldsymbol{\\Omega}_I$ let\n$f_{\\mathbf{x}}\\colon\\boldsymbol{\\Omega}_{I^{\\mathsf{c}}}\\to\\rr$ be the section of $f$ at $\\bx$, that is,\n$f_{\\mathbf{x}}(\\mathbf{y})=f\\big( (\\mathbf{x},\\mathbf{y})\\big)$ for every $\\by\\in\\bbo_{I^{\\mathsf{c}}}$.\nFubini's theorem asserts that the random variable $\\mathbf{x}\\mapsto \\ave(f_{\\mathbf{x}})$ is integrable and satisfies\n\\begin{equation} \\label{e1.2}\n\\int \\ave(f_{\\mathbf{x}}) \\, d\\mathbf{P}_{\\!I} = \\ave(f).\n\\end{equation}\nBeyond this basic information, not much can be said at this level of generality. This random variable is rather amorphous.\n\nHowever, our main result shows that if $f\\in L_p(\\bbo,\\bcalf,\\bbp)$ for some $p>1$ and $n$ is sufficiently large,\nthen one can find a set $I$ of coordinates of cardinality proportional to $n$, such that the random variable\n$\\bbo_I\\ni \\mathbf{x}\\mapsto \\ave(f_{\\mathbf{x}})$ is highly concentrated around its mean. Specifically, we have the following theorem.\n\\begin{thm} \\label{t1}\nLet $0<\\ee\\mik 1$ and $12$ is reduced to the case $p=2$. In other words, Theorem \\ref{t1}\nis valid for any $p>1$. Also notice that Theorem 1 can be reformulated as follows.\n\\begin{Th1'}\nLet $\\ee, p, n$ be as in Theorem \\emph{\\ref{t1}} and let $X_1,\\dots,X_n$ be a finite sequence of independent random variables defined\non a probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$. Let $Y$ be another random variable which can be expressed as\n$Y=F(X_1,\\dots,X_n)$ for some measurable function $F$, and assume that $\\ave(|Y|^p)\\mik 1$. Then there exists an interval $J$ of\\,\n$\\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq\\emptyset$ and satisfying \\eqref{e1.4}, such that for every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e1.6}\n\\mathbb{P}\\big( |\\ave(Y\\, |\\, \\mathcal{F}_I)-\\ave(Y)|\\mik \\ee \\big)\\meg 1-\\ee\n\\end{equation}\nwhere\\, $\\ave(Y\\, |\\, \\mathcal{F}_I)$ stands for the conditional expectation of\\, $Y$ with respect to the\n$\\sigma\\text{-algebra}$ $\\mathcal{F}_I\\coloneqq \\sigma\\big(\\{X_i: i\\in I\\}\\big)$.\n\\end{Th1'}\nWe proceed to discuss another consequence of Theorem \\ref{t1} which is of ``geometric\" nature. Let $\\bbo$ be as in Theorem \\ref{t1}\nand let $A$ be a measurable event of $\\bbo$. Also let $I\\subseteq\\{1,\\dots,n\\}$ such that $I$ and $I^{\\mathsf{c}}$ are nonempty, and\nobserve that if $f$ is the indicator function of $A$, then for every $\\bx\\in\\bbo_I$ the quantity $\\ave(f_{\\bx})$ is the probability of the section\n$A_{\\bx}=\\{\\by\\in\\bbo_{I^{\\mathsf{c}}}: (\\bx,\\by)\\in A\\}$ of $A$ at $\\bx$. Taking into account this remark, we obtain the following corollary.\n\\begin{cor} \\label{c2}\nLet $\\ee, p, n$ and $(\\bbo,\\bcalf,\\bbp)$ be as in Theorem \\emph{\\ref{t1}}. Then for every $A\\in\\bcalf$ there exists an interval $J$\nof\\, $\\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq\\emptyset$ and satisfying \\eqref{e1.4}, such that for every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e1.7}\n\\mathbf{P}_{\\!I} \\Big( \\big\\{ \\mathbf{x}\\in\\boldsymbol{\\Omega}_I: |\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx}) - \\bbp(A)|\\mik \\ee \\bbp(A)^{1\/p}\n\\big\\}\\Big) \\meg 1-\\ee.\n\\end{equation}\n\\end{cor}\nVersions of Corollary \\ref{c2} for subsets of the product of certain finite probability spaces were proved in \\cite{DKT1,DKT2}\nand were applied to combinatorial problems (we will briefly comment on these applications in Subsection 4.1, and for a more complete\nexposition we refer the reader to \\cite{DK}). Theorem \\ref{t1} was motivated by these results and was found in an effort to abstract\ntheir probabilistic features. We expect that Theorem \\ref{t1} will in turn facilitate further applications, possibly even beyond the\ncombinatorial context of \\cite{DKT1,DKT2}.\n\nWe also note that Corollary \\ref{c2} does not hold true for $p=1$ (thus, the range of $p$ in Theorem \\ref{t1} is optimal).\nTo see this, let $n$ be an arbitrary positive integer and for every $i\\in \\{1,\\dots,n\\}$ let $(\\Omega_i,\\calf_i,\\mathbb{P}_i)$\nbe a probability space with the property that there exists a measurable event $A_i$ of $\\Omega_i$ with $\\mathbb{P}_i(A_i)=1\/2$.\nAs above, we denote by $(\\bbo,\\bcalf,\\bbp)$ the product of the spaces\n$(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ and we set $A=A_1\\times\\cdots\\times A_n\\in \\bcalf$.\nNotice that if $I$ is a subset of $\\{1,\\dots,n\\}$ such that $I$ and $I^{\\mathsf{c}}$ are nonempty, then for every $\\bx\\in\\bbo_I$\nwe have $\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx})=0$ if $\\bx\\notin\\prod_{i\\in I} A_i$ while $\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx})=2^{-n+|I|}$\nif $\\bx\\in\\prod_{i\\in I} A_i$. It follows, in particular, that for every $\\bx\\in\\bbo_I$ we have\n$|\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx})-\\bbp(A)|\\meg \\bbp(A)$ and so if $p=1$, then the probability of no section of $A$ can approximate\nthe probability of $A$ with the desired accuracy.\n\nSome final remarks on the proof of Theorem \\ref{t1} which is based on a certain estimate for martingale difference sequences.\nMartingales are, of course, very useful tools for obtaining concentration inequalities (see, e.g., \\cite{L,MS} and the references\ntherein). However, the most interesting part of the argument is how one locates the desired interval $J$. This is achieved\nwith a variant of Szemer\\'{e}di's regularity lemma~\\cite{Sz}, especially as described by Tao in \\cite{Tao}.\n\n\\subsubsection*{Acknowledgments}\nWe would like to thank the anonymous referees for their comments and remarks, and for suggesting Theorem 1$'$.\n\n\n\\section{An estimate for martingale difference sequences}\n\nRecall that a finite sequence $(d_i)_{i=1}^n$ of random variables is said to be a \\emph{martingale difference sequence} if it is of the form\n\\begin{equation} \\label{e2.1}\nd_i=f_i-f_{i-1}\n\\end{equation}\nwhere $(f_i)_{i=1}^n$ is a martingale and $f_0=0$. Clearly, for any $p\\meg 1$, every martingale difference sequence in $L_p$ is a monotone basic\nsequence. Also notice that martingale difference sequences are orthogonal in $L_2$. Hence, for every martingale difference sequence\n$(d_i)_{i=1}^n$ in $L_2$ we have\n\\begin{equation} \\label{e2.2}\n\\Big( \\sum_{i=1}^n \\|d_i\\|^2_{L_2} \\Big)^{1\/2} = \\big\\| \\sum_{i=1}^n d_i\\big\\|_{L_2}.\n\\end{equation}\nWe will need the following extension of this basic fact.\n\\begin{prop} \\label{p3}\nLet $(\\Omega,\\calf,\\mathbb{P})$ be a probability space and $1

\\theta$. Set $\\ell=\\lfloor \\theta^{-2}(p-1)^{-1}\\rfloor+1$ and notice\nthat $\\lfloor (n-2)\/\\ell\\rfloor\\meg 1$. Moreover, for every $k\\in \\{1,\\dots,\\ell+1\\}$ let $i_k=(k-1)\\lfloor (n-2)\/\\ell\\rfloor+1$.\nWith these choices, for every $k\\in\\{1,\\dots,\\ell\\}$ we have $1\\mik i_k < i_{k+1}\\mik n-1$ and\n\\begin{equation} \\label{e3.5}\ni_{k+1}-i_k = \\Big\\lfloor \\frac{n-2}{\\ell}\\Big\\rfloor \\meg \\frac{n}{2\\ell} \\meg \\Big( \\frac{\\theta^2 (p-1)^2}{4} \\Big) n\n\\end{equation}\nwhich implies, by our assumption that the lemma is false, that\n\\begin{equation} \\label{e3.6}\n\\|\\ave(g \\, | \\, \\cala_{i_{k+1}})-\\ave(g \\, | \\, \\cala_{i_k})\\|_{L_p} >\\theta.\n\\end{equation}\nWe set $d_1=\\ave(g \\, | \\, \\cala_{i_1})$ and $d_{k+1}=\\ave(g \\, | \\, \\cala_{i_{k+1}})-\\ave(g \\, | \\, \\cala_{i_k})$\nfor every $k\\in\\{1,\\dots,\\ell\\}$, and we observe that the sequence $(d_k)_{k=1}^{\\ell+1}$ is a martingale difference sequence\nin $L_p(\\Omega,\\calf,\\mathbb{P})$. Therefore, by Proposition \\ref{p3}, we obtain that\n\\begin{eqnarray} \\label{e3.7}\n\\ \\ \\ \\ \\ 1 \\!\\!\\!\\! & < & \\sqrt{p-1}\\theta\\sqrt{\\ell} \\ \\stackrel{\\eqref{e3.6}}{<} \\sqrt{p-1} \\Big(\\sum_{k=1}^{\\ell}\n\\| \\ave(g \\, | \\, \\cala_{i_{k+1}})-\\ave(g \\, | \\, \\cala_{i_k})\\|_{L_p}^2\\Big)^{1\/2} \\\\\n& \\mik & \\sqrt{p-1} \\Big(\\sum_{k=1}^{\\ell+1} \\|d_k\\|_{L_p}^2\\Big)^{1\/2} \\stackrel{\\eqref{e2.3}}{\\mik}\n\\big\\| \\sum_{k=1}^{\\ell+1} d_k \\big\\|_{L_p} = \\|\\ave(g \\, | \\, \\cala_{i_{\\ell+1}})\\|_{L_p} \\mik \\|g\\|_{L_p} \\nonumber\n\\end{eqnarray}\nwhich contradicts, of course, our hypothesis that $\\|g\\|_{L_p}\\mik 1$.\n\nFinally, let $1\\mik i < j\\mik n$ and notice that for every $i\\mik l\\mik m \\mik j$ we have\n\\[ \\ave(g \\, | \\, \\cala_m)-\\ave(g \\, | \\, \\cala_l)\\! =\\! \\ave(\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\, | \\, \\cala_m) -\n\\ave(\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\, | \\, \\cala_l) \\]\nwhich yields that $\\|\\ave(g \\, | \\, \\cala_m)-\\ave(g \\, | \\, \\cala_l)\\|_{L_p}\\mik 2 \\|\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\|_{L_p}$.\nThe proof of Lemma \\ref{l5} is completed.\n\\end{proof}\nWe will also need the following lemma. In its proof, and in the rest of this paper, we will follow the common practice\nwhen proving inequalities and we will ignore measurability issues since they can be resolved with standard arguments.\n\\begin{lem} \\label{l6}\nLet $n$ be a positive integer and $(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ a finite sequence\nof probability spaces, and denote by $(\\bbo,\\bcalf,\\bbp)$ their product. Also let $I\\subseteq \\{1,\\dots,n\\}$ and assume that\n$I$ and $I^{\\mathsf{c}}$ are nonempty. Then for every $p\\meg 1$ and every $g,h\\in L_p(\\bbo,\\bcalf,\\bbp)$ we have\n\\begin{equation} \\label{e3.8}\n\\int \\|g_{\\bx}-h_{\\bx}\\|_{L_1}^p \\, d\\bbp_{\\!I} \\mik \\|g-h\\|_{L_p}^p.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nNotice first that, by Fubini's theorem,\n\\begin{equation} \\label{e3.9}\n\\|g-h\\|_{L_p}^p = \\int \\Big( \\int |g_{\\bx}-h_{\\bx}|^p \\, d\\bbp_{\\!I^{\\mathsf{c}}} \\Big) \\, d\\bbp_{\\!I}.\n\\end{equation}\nOn the other hand, by Jensen's inequality, for every $\\bx\\in\\bbo_I$ we have\n\\begin{equation} \\label{e3.10}\n\\|g_{\\bx}-h_{\\bx}\\|_{L_1}^p = \\Big( \\int |g_{\\bx}-h_{\\bx}| \\, d\\bbp_{\\! I^{\\mathsf{c}}} \\Big)^p \\mik\n\\int |g_{\\bx}-h_{\\bx}|^p \\, d\\bbp_{\\! I^{\\mathsf{c}}}\n\\end{equation}\nand so, taking the average over all $\\bx\\in \\bbo_I$ and using \\eqref{e3.9}, we obtain the desired estimate.\n\\end{proof}\nWe are ready to complete the proof of Theorem \\ref{t1}.\n\\begin{proof}[Proof of Theorem \\emph{\\ref{t1}}]\nWe fix $f\\in L_p(\\bbo,\\bcalf,\\bbp)$ with $\\|f\\|_{L_p}\\mik 1$ and we set\n\\begin{equation} \\label{e3.11}\n\\theta=\\ee^{\\frac{p+1}{p}}.\n\\end{equation}\nSince $n\\meg 2\/c(\\ee,p)$, by \\eqref{e1.3} and \\eqref{e3.11}, we see that $n\\meg 8\\, \\theta^{-2}(p-1)^{-1}$.\nHence, by Lemma \\ref{l5} applied to the random variable $f$ and the filtration $(\\cals_m)_{m=1}^n$,\nthere exist $i,j\\in\\{1,\\dots,n-1\\}$ satisfying \\eqref{e3.3} and such that\n\\begin{equation} \\label{e3.12}\n\\|\\ave(f \\, | \\, \\cals_j)-\\ave(f \\, | \\, \\cals_i)\\|_{L_p}\\mik \\theta.\n\\end{equation}\nWe set $J=\\{i+1,\\dots,j\\}$ and we claim that the interval $J$ is as desired. To this end notice, first, that $J^{\\mathsf{c}}\\neq\\emptyset$.\nMoreover, by \\eqref{e3.3} and the choice of $c(\\ee,p)$ and $\\theta$ in \\eqref{e1.3} and \\eqref{e3.11} respectively, we have\n\\begin{equation} \\label{e3.13}\n|J|=j-i\\meg \\big( 4^{-1} \\theta^2 (p-1) \\big)\\, n = c(\\ee,p) \\, n.\n\\end{equation}\nNext, let $I$ be a nonempty subset of $J$ and set\n\\begin{equation} \\label{e3.14}\ng=\\ave(f \\, | \\, \\cals_j) \\ \\text{ and } \\ h=\\ave(f \\, | \\, \\cals_i).\n\\end{equation}\nWe have the following claim.\n\\begin{claim} \\label{c7}\nFor every $\\bx\\in\\bbo_I$ we have $\\ave(g_{\\bx})=\\ave(f_{\\bx})$ and $\\ave(h_{\\bx})=\\ave(f)$.\n\\end{claim}\n\\begin{proof}[Proof of Claim \\emph{\\ref{c7}}]\nFix $\\bx\\in \\bbo_I$ and set $\\mathcal{I}=\\{1,\\dots,i\\}$ and $\\mathcal{J}=\\{1,\\dots,j\\}$.\n\nFirst we argue to show that $\\ave(g_{\\bx})=\\ave(f_{\\bx})$. Indeed, observe that $I\\subseteq J\\subseteq \\mathcal{J}$ and so,\nby \\eqref{e3.14} and Fubini's theorem, we see that for every $\\by\\in \\bbo_{\\mathcal{J}\\setminus I}$ the function\n$g_{(\\bx,\\by)}\\colon \\bbo_{\\mathcal{J}^{\\mathsf{c}}}\\to \\rr$ is constant and equal to $\\ave(f_{(\\bx,\\by)})$. Therefore,\n\\begin{eqnarray} \\label{e3.15}\n\\ave(g_{\\bx})=\\int g_{\\bx} \\, d\\bbp_{\\! I^{\\mathsf{c}}} & = &\n\\int \\Big(\\int g_{(\\bx,\\by)} \\, d\\bbp_{\\! \\mathcal{J}^{\\mathsf{c}}} \\Big) \\, d\\bbp_{\\! \\mathcal{J}\\setminus I} \\\\\n& = & \\int \\ave(f_{(\\bx,\\by)}) \\, d\\bbp_{\\! \\mathcal{J}\\setminus I} \\nonumber \\\\\n& = & \\int\\Big(\\int f_{(\\bx,\\by)} \\, d\\bbp_{\\! \\mathcal{J}^{\\mathsf{c}}}\\Big) \\, d\\bbp_{\\! \\mathcal{J}\\setminus I} \\nonumber \\\\\n& = & \\int f_{\\bx} \\, d\\bbp_{\\! I^{\\mathsf{c}}}= \\ave(f_{\\bx}). \\nonumber\n\\end{eqnarray}\n\nWe proceed to show that $\\ave(h_{\\bx})=\\ave(f)$. As above we notice that, by \\eqref{e3.14} and Fubini's theorem, for every\n$\\bz\\in \\bbo_{\\mathcal{I}}$ the function $h_{\\bz}\\colon\\bbo_{\\mathcal{I}^{\\mathsf{c}}}\\to \\rr$ is constant and equal to $\\ave(f_{\\bz})$.\nSince $\\mathcal{I}\\cap I=\\emptyset$, the function $h_{(\\bx,\\bz)}\\colon\\bbo_{(\\mathcal{I}\\cup I)^{\\mathsf{c}}}\\to \\rr$ is also constant\nand equal to $\\ave(f_\\bz)$. Hence,\n\\begin{eqnarray} \\label{e3.16}\n\\ave(h_{\\bx}) = \\int h_{\\bx} \\, d\\bbp_{\\! I^{\\mathsf{c}}} & = &\n\\int\\Big(\\int h_{(\\bx,\\bz)} \\, d\\bbp_{\\! (\\mathcal{I}\\cup I)^{\\mathsf{c}}}\\Big) \\, d\\bbp_{\\! \\mathcal{I}} \\\\\n& = & \\int \\ave(f_{\\bz}) \\, d\\bbp_{\\! \\mathcal{I}} = \\ave(f) \\nonumber\n\\end{eqnarray}\nand the proof of Claim \\ref{c7} is completed.\n\\end{proof}\nBy Claim \\ref{c7}, for every $\\bx\\in\\bbo_I$ we have\n\\begin{equation} \\label{e3.17}\n|\\ave(f_{\\bx})-\\ave(f)| = \\big| \\int (g_{\\bx}-h_{\\bx}) \\, d\\bbp_{\\! I^{\\mathsf{c}}}\\big| \\mik \\|g_{\\bx}-h_{\\bx}\\|_{L_1}\n\\end{equation}\nand so\n\\begin{equation} \\label{e3.18}\n\\int |\\ave(f_{\\bx})-\\ave(f)|^p \\, d\\bbp_{\\! I} \\mik \\int \\|g_{\\bx}-h_{\\bx}\\|^p_{L_1} \\, d\\bbp_{\\! I}.\n\\end{equation}\nIt follows by Lemma \\ref{l6}, \\eqref{e3.12}, \\eqref{e3.14} and the previous estimate that\n\\begin{equation} \\label{e3.19}\n\\int |\\ave(f_{\\bx})-\\ave(f)|^p \\, d\\bbp_{\\! I} \\mik \\theta^p.\n\\end{equation}\nTherefore, by Markov's inequality, we conclude that\n\\begin{equation} \\label{e3.20}\n\\bbp_{\\! I} \\big( \\{ \\bx\\in\\bbo_I: |\\ave(f_{\\bx})-\\ave(f)| \\meg \\theta^{\\frac{p}{p+1}} \\} \\big)\n\\mik \\frac{\\theta^p}{\\theta^{p^2\/(p+1)}} = \\theta^{\\frac{p}{p+1}}\n\\end{equation}\nwhich is equivalent to saying, by the choice of $\\theta$ in \\eqref{e3.11}, that\n\\begin{equation} \\label{e3.21}\n\\bbp_{\\! I} \\big( \\{ \\bx\\in\\bbo_I: |\\ave(f_{\\bx})-\\ave(f)| \\mik \\ee \\} \\big) \\meg 1-\\ee.\n\\end{equation}\nThe proof of Theorem \\ref{t1} is completed.\n\\end{proof}\n\n\n\\section{Comments}\n\n\\subsection*{4.1}\n\nFor every positive integer $n$ and every finite set $A$ with $|A|\\meg 2$ let\n\\begin{equation} \\label{e4.1}\nA^n=\\{(a_1,\\dots,a_n): a_1,\\dots, a_n\\in A\\}\n\\end{equation}\nand let $\\mathbb{P}$ be the uniform probability measure on the hypercube $A^n$. Moreover, for every nonempty\nsubset $I$ of $\\{1,\\dots,n\\}$ by $\\mathbb{P}_{\\! A^I}$ we denote the uniform probability measure on\n$A^I\\coloneqq \\{(a_i)_{i\\in I}: a_i\\in A \\text{ for every } i\\in I\\}$. We have the following lemma.\n\\begin{lem} \\label{l8}\nLet $k,m$ be positive integers with $k\\meg 2$ and $0<\\eta\\mik 1$. Also let $A$ be a set with $|A|=k$ and let $n$ be a positive integer with\n\\begin{equation} \\label{e4.2}\nn\\meg \\frac{16m k^{3m}}{\\eta^3}\n\\end{equation}\nThen for every subset $D$ of $A^n$ there exists an interval $I\\subseteq \\{1,\\dots,n\\}$ with $|I|=m$ such that for every $t\\in A^I$ we have\n\\begin{equation} \\label{e4.3}\n|\\mathbb{P}_{\\! A^{I^{\\mathsf{c}}}}(D_t)-\\mathbb{P}(D)|\\mik \\eta\n\\end{equation}\nwhere $D_t=\\{s\\in A^{I^{\\mathsf{c}}}: (t,s)\\in D\\}$ is the section of $D$ at $t$.\n\\end{lem}\nA simpler version of Lemma \\ref{l8} was proved in \\cite{DKT1} and was used as a tool in a proof of the density Hales--Jewett\ntheorem \\cite{FK}; closely related applications were also obtained in \\cite{DKT2} (see also \\cite{DK}). Of course, the main point\nin Lemma \\ref{l8} is that by demanding a large---but not necessarily dense---set $I$ of coordinates, one can upgrade Theorem \\ref{t1}\nand guarantee that the probability of \\textit{every} section of $D$ along elements of $A^I$ is essentially equal to the probability of $D$.\nWe proceed to the proof.\n\\begin{proof}[Proof of Lemma \\emph{\\ref{l8}}]\nWe view $A$ and $A^n$ as discrete probability spaces equipped with their uniform probability measures. Then notice that the\nprobability space $A^n$ is the product of $n$ many copies of $A$. Next we set $\\ee = \\eta k^{-m}2^{-1\/3}$ and we observe that,\nby \\eqref{e1.3} and \\eqref{e4.2}, we have $n\\meg m\\big(2\/c(\\ee,2)\\big)$. Hence, by Corollary \\ref{c2} applied to the set $D$,\nthe constant $\\ee$ and $p=2$, there exists an interval $J\\subseteq \\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq \\emptyset$ and\n$|J|\\meg 2m$, and satisfying \\eqref{e1.7} for every nonempty $I\\subseteq J$. We select an interval $I\\subseteq J$ with $|I|=m$\nand we claim that $I$ is as desired. Indeed, by the choice of $\\ee$, we have\n\\begin{equation} \\label{e4.4}\n\\mathbb{P}_{\\! A^I}\\big( \\{ t\\in A^I: |\\mathbb{P}_{\\! A^{I^{\\mathsf{c}}}}(D_t)-\\mathbb{P}(D)|\\mik \\ee \\} \\big) \\meg 1-\\ee\n\\meg 1- k^{-m}2^{-1\/3} > 1-\\frac{1}{|A^I|}\n\\end{equation}\nwhich implies that $|\\mathbb{P}_{\\! A^{I^{\\mathsf{c}}}}(D_t)-\\mathbb{P}(D)|\\mik \\ee$ for every $t\\in A^I$. Since $\\ee\\mik\\eta$\nwe conclude that the estimate in \\eqref{e4.3} is satisfied and the proof is completed.\n\\end{proof}\n\n\\subsection*{4.2}\n\nThere is a natural extension of Theorem \\ref{t1} which deals simultaneously with a family of random variables.\nAlthough in applications one usually encounters only finite families of random variables (see, e.g., \\cite{DKT2}),\nthe cleanest formulation of this extension is for stochastic processes indexed by the sample space of a probability\nspace $(T,\\Sigma,\\mu)$. Specifically, we have the following theorem.\n\\begin{thm} \\label{t9}\nLet $0<\\ee\\mik 1$ and $10$ there exists $\\delta>0$ such that for\nevery $x,y\\in X$ with $\\|x\\|_X=\\|y\\|_X=1$ and $\\|x-y\\|_X\\meg \\ee$ we have that $\\|(x+y)\/2\\|_X\\mik 1-\\delta$. A classical result due to\nJames \\cite{J} and, independently, V. Gurarii and N. Gurarii \\cite{GG}, implies that for every uniformly convex Banach space $X$ and\nevery $p>1$ there exist $q\\meg 2$ and a constant $C>0$ such that for every $X$-valued martingale difference sequence $(d_i)_{i=1}^n$ we have\n\\begin{equation} \\label{e4.13}\n\\Big( \\sum_{i=1}^n \\|d_i\\|_{L_p(X)}^q \\Big)^{1\/q} \\mik C \\big\\| \\sum_{i=1}^n d_i \\big\\|_{L_p(X)}.\n\\end{equation}\n(See, also, \\cite{Pi} for a proof and a detailed presentation of related material.) Using this estimate and arguing precisely as in Section 3,\nwe obtain the following vector-valued version of Theorem \\ref{t1}.\n\\begin{thm} \\label{t10}\nFor every uniformly convex Banach space $X$, every $0<\\ee\\mik 1$ and every $p>1$ there exists a constant $c(X,\\ee,p)>0$ with the following\nproperty. Let $n$ be a positive integer with $n\\meg c(X,\\ee,p)^{-1}$ and let $(\\bbo,\\bcalf,\\bbp)$ be the product of a finite sequence\n$(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ of probability spaces. If $f\\colon\\bbo\\to X$ is a random variable\nwith $\\|f\\|_{L_p(X)}\\mik 1$, then there exists an interval $J$ of $\\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq\\emptyset$ and\n\\begin{equation} \\label{e4.14}\n|J| \\meg c(X,\\ee,p)\\, n\n\\end{equation}\nsuch that for every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e4.15}\n\\mathbf{P}_{\\!I} \\big( \\{ \\mathbf{x}\\in\\boldsymbol{\\Omega}_I: \\|\\ave(f_{\\bx}) - \\ave(f)\\|_X\\mik \\ee \\}\\big) \\meg 1-\\ee.\n\\end{equation}\n\\end{thm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAnswer Set Programming (ASP) is a widely used problem solving\napproach. It offers declarative languages that can be used \nto formalize actions, planning, and agent policies, in an expressive setting\n(e.g.\\ direct and indirect action effects)\n\\cite{lif99c,bara-2003,DBLP:journals\/aim\/ErdemGL16}),\nand has led to dedicated action languages \\cite{lifschitz08}.\nThis and the availability of efficient solvers\nmakes ASP a convenient tool for representing and reasoning about actions.\n\nConsider a scenario in which a robot may be in an unknown grid-cell environment with\nobstacles and aim to find a missing person (Fig.~\\ref{fig:scenario}). \nIt acts according to a policy, which tells it\n\\begin{wrapfigure}{r}{3cm}\n\n\\centering\n\n\\vspace*{-.5\\baselineskip}\n\n\\includegraphics[height=1.7cm]{person-search-initial-4x4.png}\n\n\\vspace*{-.5\\baselineskip}\n\n\\caption{\\quad Missing person search in an unkown environment}\n\\label{fig:scenario}\n\\vspace*{-\\baselineskip}\n\n\\end{wrapfigure}\nwhere to move next,\ndepending on the current observations (free \/ blocked cells) and\npossible memory of past observations, until the person is found. To\nthis end, an action domain with a policy description, formalized in an\nASP program, is evaluated in each step. \nNaturally, we wonder whether the policy works, i.e.,\nthe person is always found, regardless of actual obstacle\nlocations. This can generate a large state space (for an $n{\\times}n$\ngrid, of size larger than $2^{n{\\times}n}$) and simple approaches such\nas searching for a run in which the policy fails\nquickly become infeasible.\n\nTo overcome this, we aim at using abstraction, which is a well-known\napproach to reduce problem complexity. In a deliberate loss of\ninformation, the problem is approximated to achieve a smaller or\nsimpler state space, at the price of spurious\ncounterexamples to the behavior \\cite{clarke03}. In planning, abstraction\nis mostly focused on relaxing the model, by omitting preconditions of\nactions and details of the domain model\n\\cite{giunchiglia1992theory,knoblock1994automatically,sacerdoti1974planning}.\nCartesian abstraction \\cite{seipp2013counterexample} refines\nin the spirit of \\cite{clarke03} failure states of abstract\ntrajectories, starting from a trivial abstraction; the classical\nplanning setting, however, disregards incomplete\ninitial states\n(a known source of complexity).\nThese works do not consider policies with background knowledge that can do decision-making with information beyond action effects.\n\nIn the area of ASP-based action languages, abstraction has not been\nconsidered so far, and neither in the broader ASP context.\nIn order to exploit abstraction for reasoning about action\ndescriptions and policies in ASP, we need an abstraction method for\nASP programs that offers the following features. First, information\nloss on both the model and the domain is possible. Second, \nrelationships and dependencies expressed in the program should be\nlargely preserved. And third, abstractions should be (semi-)\nautomatically computable. We address this challenge with the\nfollowing contributions.\n\n\\begin{itemize}\n\\item We introduce a method to abstract ASP programs in order to\n obtain an over-approximation of the answer sets of a program\n $\\Pi$. That is, a program $\\Pi'$ is constructed such that each answer\n set $I$ of $\\Pi$ is abstracted to some answer set $I'$ of $\\Pi'$; \n While this abstraction is many to one, {\\em spurious} answer sets of\n $\\Pi'$ not corresponding to any answer set of $\\Pi$ may exist. \n \n\n\n\\item For abstraction, we consider omission of literals and also domain abstraction,\n where domain elements are merged. Note that omitting is different\n from forgetting literals (see \\cite{DBLP:conf\/lpnmr\/Leite17} for an overview), as the latter aims at\n preserving information.\n \n The abstraction types \n can be combined and in principle iterated to build\n hierarchical abstractions.\n\n\\item The method largely preserves the structure of the rules and\n works modularly for non-ground programs. Thus, it is particularly attractive\n for abstraction of parameterized problems, as e.g.,\\ in the search\n scenario (grid size $n$). Furthermore, it respects built-in\n predicates such as equality ($=$), comparisons ($<,\\leq$) etc., and\n can be readily implemented, with little information on the underlying\n abstraction.\n\\item We illustrate the use of the abstraction method for reasoning about\n actions, in particular to find counterexamples to an \n agent policy. Here, it can be particularly useful to\n identify and explain ``essential'' aspects of failure.\n\\end{itemize}\n\nWhile abstraction for ASP programs is motivated by applications in reasoning\nabout actions, the approach is domain independent and can be\nutilized in other contexts as well.\n\n\\section{Preliminaries}\n\n\\leanparagraph{ASP} A logic program $\\Pi$ is a set of rules $r$ of the form\n\n\\smallskip\n\n\\centerline{$\\alpha_0 \\leftarrow \\alpha_1,\\dots,\\alpha_m,\\mi{not}\\\n\\alpha_{m+1},\\dots,\\mi{not}\\ \\alpha_n,\\ \\ 0\\,{\\leq}\\, m \\,{\\leq}\\, n,$}\n\n\\smallskip\n\n\\noindent where each $\\alpha_i$ is a first-order (function-free) atom and \n$\\mi{not}$\nis default negation; $r$ is a \\emph{constraint}\nif $\\alpha_0$ is falsity ($\\bot$, then omitted) and a \\emph{fact} if\n$n\\,{=}\\,0$. We also write $\\alpha_0 \\leftarrow B^+(r),\\mi{not}\\ B^-(r)$,\nwhere $B^+(r)$ (positive body) is the set $\\{\\alpha_1, \\dots,\n\\alpha_m\\}$ and $B^-(r)$ (negative body) the set\n$\\{\\alpha_{m+1},$ $\\dots,\\alpha_n\\}$, \nor $\\alpha_0 {\\leftarrow} B(r)$.\nRules with variables \nstand for the set of their ground instances. \nSemantically, \n$\\Pi$ induces a set of answer sets\n\\cite{gelfond1991classical}, which\nare Herbrand models (sets $I$ of ground atoms) of $\\Pi$ \njustified\nby the rules, \nin that $I$ is a minimal model of $f\\Pi^I=$ $\\{ r \\in \\Pi\n\\mid I \\models B(r)\\}$\n\\cite{FLP04}.\nThe set of answer sets of a program $\\Pi$ is denoted as $AS(\\Pi)$. \nNegative literals $\\neg \\alpha$ can be \nencoded \nusing atoms $\\mi{neg}\\_\\alpha$ and\nconstraints $\\leftarrow \\alpha,\\mi{neg}\\_\\alpha$.\n\n \n\nCommon syntactic extensions are \\emph{choice rules} of the form\n$\\{\\alpha\\} \\leftarrow B$, which stands for the rules $\\alpha \\leftarrow\nB, \\mi{not}\\ \\alpha'$ and $\\alpha' \\leftarrow B, \\mi{not}\\, \\alpha$, where $\\alpha'$ is a\nnew atom, and cardinality constraints and conditional\nliterals\n\\cite{simons2002extending}; in particular, $i_\\ell\\{\\,a(X)\\,{:}\\,b(X)\\,\\}i_u$ is true whenever at least $i_\\ell$ and at most $i_u$ instances of $a(X)$ subject to $b(X)$ are true.\n\n\\leanparagraph{Describing actions and states} \nASP is used to describe dynamic domains by a ``history\nprogram\" \\cite{lif99c}, \nwhose answer sets represent\npossible evolutions of the system over a \ntime interval. This is\nachieved by adding a time variable to the atoms, and\nintroducing action atoms that may cause changes\nover time. An action is defined by its preconditions and effects over the\natoms. For illustration, the following rule describes a \\emph{direct effect} of the\naction $\\mi{goTo(X,Y)}$ over the robot's location $rAt(X,Y)$\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\mi{rAt}(X,Y,T{+}1) \\leftarrow \\mi{goTo}(X,Y,T).\n\\end{array} \\eeq {eq:direct}\n\\vspace*{-1.5\\baselineskip}\n\n\\noindent Actions can also have \\emph{indirect effects} over the state (rules not\nmentioning actions); e.g.,\n \nthe robot location is visited:\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\mi{visited}(X,Y,T) \\leftarrow \\mi{rAt}(X,Y,T).\n\\end{array} \\eeq {eq:indirect}\n\\noindent Inertia laws (unaffectedness) can be elegantly expressed, e.g.\\\n\\begin{eqnarray}\n \\mi{rAt}(X,Y,T{+}1) \\! \\leftarrow \\!\\!\\!\\!\\! & \\mi{rAt}(X,Y,T), \\mi{not}\\, \\neg \\mi{rAt}(X,Y,T{+}1). \\nonumber\n\n\\end{eqnarray}\n\n\\noindent says that the robot location remains by default the same.\n\nOne can also give further restrictions on the state, e.g., the robot and\nan obstacle can never be\nin the same %\ncell.\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\bot \\leftarrow \\mi{rAt}(X,Y,T), \\mi{obsAt}(X,Y,T).\n\\end{array} \\eeq {eq:obs}\n\n\\noindent Constraints can also define \\emph{preconditions} of an action, e.g.,\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\bot \\leftarrow \\mi{goTo}(X,Y,T), \\mi{obsAt}(X,Y,T).\n\\end{array} \\eeq {eq:precond}\nDedicated action languages carry this idea further with special \nsyntax for such axioms \\cite{gelfondaction98}, and can be translated to ASP \\cite{giunchiglia2004nct}.\n\n\\leanparagraph{Describing a policy} In addition to defining actions as above, ASP can also be used for further reasoning about the actions by singling out some of them under certain conditions. A policy that singles out the actions to execute from the current state can be described with a set\nof rules,\nwhere rules of\nform $a {\\leftarrow} B$ choose an action $a$ when certain\nconditions $B$ are satisfied in the state. \nFurther rules may \ndescribe auxiliary literals that are used by $B$. \n\nThe rules below make the agent move towards some farthest point on the grid, unless the person is seen or caught. \nIn the latter case, the agent moves towards the person's location. \n\\begin{equation}\n \\begin{split}\n&1\\,\\{\\mi{goTo}(X1,Y1,T) : \\mi{farthest}(X,Y,X1,Y1,T)\\}\\,1\\\\%[-.5ex]\n& \\leftarrow \\mi{rAt}(X,Y,T), \\mi{not\\ seen}(T), \\mi{not\\ caught}(T).\\\\%[-.5ex]\n&\\mi{goTo}(X,Y,T) \\leftarrow \\mi{seen}(T), \\mi{not\\ caught}(T), \\mi{pAt}(X,Y,T). \n\\end{split}\n\\label{eq:pol_formula}\n\\raisetag{24pt}\n\\end{equation} \n\\vspace*{-0.25\\baselineskip}\n\n\\noindent The farthest point is determined by the agent's location and the\ncells considered at that state;\nit is thus an indirect effect of\nthe previous move.\nThis also applies to $\\mi{seen}$ and $\\mi{caught}$:\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{split}\n&\\mi{caught}(T) \\leftarrow \\mi{rAt}(X,Y,T), \\mi{pAt}(X,Y,T).\\\\%[-.5ex]\n&\\mi{seen}(T) \\leftarrow \\mi{seeReachable}(X,Y,T), \\mi{pAt}(X,Y,T).\n\\end{split}\n\\raisetag{26pt}\n \\eeq {eq:pol_formula2}\nNotice that above is on choosing single actions. For policies that choose a sequence of actions, the policy rules will be more involved, as the stages of the plan might have to be considered.\n\n\n\\section{Constructing an Abstract ASP Program}\\label{sec:auto_abs}\n\nOur aim is to over-approximate a given program through constructing a\nsimpler program by reducing the vocabulary and preserving the behavior\nof the original program (i.e., the\nresults of reasoning on the original program are not lost), \nat the cost of obtaining spurious\nsolutions. \n\n\n\\begin{defn}\nGiven two programs $\\Pi$ and $\\Pi'$ with $|{\\bf L}|{\\geq}|{\\bf L}'|$, $\\Pi'$ is an \\emph{abstraction} of $\\Pi$ if there exists a mapping $m : {\\bf L} \\rightarrow {\\bf L'}$ such that for $I\\in AS(\\Pi)$, $I'=\\{m(l) ~|~ l \\in I\\}$ is an answer set of $\\Pi'$.\n\\end{defn}\n\n\nWe consider two important base cases for an abstraction mapping $m$. Literal\nomission is about omitting certain literals from the program, while\ndomain abstraction is on clustering different constants in the domain\nand treating them as equal. \n\n\n\\begin{defn}\nGiven a program $\\Pi$ and its abstraction $\\Pi'$,\n\\begin{myenumerate}\n\\item $\\Pi'$ is a \\emph{literal omission abstraction of $\\Pi$} if a\n set $L \\subseteq {\\bf L}$ of literals is omitted and the rest is\n kept, i.e.,\n ${\\bf L}' = {\\bf L}\n \\setminus L$ and $m(l)=\\emptyset$ if $l\\in L$ and $m(l)=l$ otherwise.\n\\item $\\Pi'$ is a \\emph{domain abstraction of $\\Pi$} if there is a function $m_{d}\\,{:}\\,D\n\\,{\\rightarrow}\\, \\widehat{D}$ for a Herbrand domain $D$ and its abstraction $\\widehat{D}$, such that for $l\\,{=}\\,p(v_1,\\dots,v_n)$ we have $m(l)\\,{=}\\,p(m_{d}(v_1),$ \\ldots, $m_{d}(v_n))$.\n\\end{myenumerate}\n\\end{defn}\n\nIn the following sections, we show a systematic way of building an abstraction of a given ASP program.\nWhen constructing an abstract program for a given mapping, the aim is\nto ensure that every original answer set $I$ is mapped to some\nabstract answer set, while (unavoidably) some spurious abstract answer\nsets may be introduced. Thus, an over-approximation of the original program is achieved.\nThe abstraction types can be composed to obtain further abstractions.\n\nNotice that literal omission is different\nthan forgetting (see \\cite{DBLP:conf\/lpnmr\/Leite17} for an overview),\nas it ensures the over-approximation of the original program by making\nsure that all of the original answer sets are preserved in the\nabstract program, without resorting to language extensions such as\nnested logic programs that otherwise might be necessary. \n\n\\vspace*{-0.5\\baselineskip}\n\n\\subsection{Literal omission} \n\n\nGiven $L$, we build from $\\Pi$ a program $\\Pi_{\\overline{L}}^m$\nas follows. For every literal $l \\in ({\\bf L} \\setminus L) \\cup \\{\\bot \\}$ and rule\n$r: l \\leftarrow B(r)$ in $\\Pi$, \n\\begin{enumerate}[(1)]\n\\itemsep=2pt\n\\item if $ B(r) \\subseteq {\\bf L} \\setminus L$, we include $m(l)\n \\leftarrow m(B(r))$;\n\\item otherwise, if $l\\,{\\neq}\\,\\bot$ we include for every $l' \\in B(r) \\cap\n L$ the rule $0 \\{ m(l) \\} 1 \\leftarrow m(B(r)\\setminus \\{l'\\}).$\n\\end{enumerate}\nNotice that constraints are omitted in the constructed program if the body contains an omitted literal. If instead, the constraint gets shrunk, then for some interpretation $\\widehat{I}$, the body may fire in $\\Pi_{\\overline{L}}^m$, while it was not the case in $\\Pi$ for any $I \\in AS(\\Pi)$ s.t. $m(I)=\\widehat{I}$. Thus \n$I$ cannot be mapped to an abstract answer set of $\\Pi_{\\overline{L}}^m$, i.e., $\\Pi_{\\overline{L}}^m$ is not an over-approximation of $\\Pi$.\n\nOmitting non-ground literals means omitting\nall occurrences of the predicate.\nIf in a rule $r$, the omitted non-ground literal $p(V_1,\\dots,V_n)$ shares some arguments, $V_i$, with the head $l$, then $l$ is conditioned over $\\mi{dom}(V_i)$ (a special predicate to represent the Herbrand domain) in the constructed rule, so that all values of $V_i$ are considered.\n\n\\begin{exmp}\\label{ex:toy}\nConsider the following simple program $\\Pi$:\n\\vspace*{-.25\\baselineskip}\n\\begin{align}\n a(X_1,X_2) &\\leftarrow c(X_1), b(X_2). \\label{eq:1}\\\\\n d(X_1,X_2) &\\leftarrow a(X_1,X_2), X_1{\\leq}X_2.\\label{eq:2}\n\\end{align} \n\\vspace*{-1.25\\baselineskip}\n\nIn omitting $c(X)$, while rule \\eqref{eq:2} remains the same, rule \\eqref{eq:1} changes to \n$0\\{a(X_1,X_2)\\,{:}\\,\\mi{dom}(X_1)\\}1 \\leftarrow b(X_2)$.\nFrom\n$\\Pi$\nand the facts $c(1),b(2)$, we get the answer set $\\{c(1)$, $b(2)$,\n$a(1,2)$, $d(1,2)\\}$, and with $c(2),b(2)$ we get\n$\\{c(2)$, $b(2),$ $a(2,2)$, $d(2,2)\\}$.\n After omitting $c(X)$, the abstract answer\nsets with fact $b(2)$ become $\\{b(2),$ $a(1,2),$ $d(1,2)\\}$ and\n$\\{b(2)$, $a(2,2),$ $d(2,2)\\}$, which cover the original answers, so that all original answer sets can be mapped to\nsome abstract answer set.\n\\end{exmp}\n\nFor a semantical more fine-grained removal, e.g., removing $c(X)$ for $X{<}3$, rules may be split in cases, e.g., (\\ref{eq:1}) into $X_1{<}3$ and $X_1{\\geq} 3$, and\ntreated after renaming separately.\n\nThe following result shows that $\\Pi_{\\overline{L}}^m$ can be seen as an over-approximation of $\\Pi$\n\n\\begin{thm}\nFor every $I \\in AS(\\Pi)$ and set $L$ of literals,\n$I_{|\\overline{L}} \\in AS(\\Pi_{\\overline{L}}^m)$ where $I_{|\\overline{L}} =I\\setminus L$.\n\\end{thm}\n\n\\noindent By introducing choice rules for any rule that contains the omitted literal, all possible cases that could have been achieved by having the omitted literal in the rule are covered. Thus, the abstract answer sets cover the original answer sets.\n\n\\vspace*{-0.25\\baselineskip}\n\n\\subsection{Domain abstraction} \n\nAbstraction on the domain, $D$, divides it into equivalence classes, $\\widehat{D}=\\{\\hat{d}_1,\\dots,\\hat{d}_k\\}$, where some values of the variables are seen as equal. \nSuch an abstraction can be constructed by keeping the structure of the\nliterals, and having abstract rules similar to the original ones. The\noriginal rule may rely on certain built-in relations between the\nliterals' variables, e.g., $=,\\neq,<,\\leq$, such as \\eqref{eq:2};\nwe can automatically lift \nthem to the abstraction\n(discussed below), and \naim to use\n\\smallskip\n\n\\centerline{$d(\\widehat{X}_1,\\widehat{X}_2) \\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{\\leq}\\widehat{X}_2.$}\n\n\\smallskip\n\n\\noindent where $\\widehat{X}_1,\\widehat{X}_2$ are variables ranging over $\\widehat{D}$.\nHowever, due to the mapping, the lifted relations may create\nuncertainties which must be dealt with. \nE.g.\\\nfor a mapping $m_d(\\{1,2,3\\})=k$, the atom $a(k,k)$ can be true in\nthe abstract state because $a(3,2)$ is true in the\noriginal state. \n The original program can have answer sets $I$ that contain (i) $a(3,2),\\mi{not}\\ d(3,2)$, or (ii) $a(2,2),d(2,2)$. If we keep the structure of the original rule, in any abstract answer set $d(k,k)$ must hold if $a(k,k)$ holds; hence, no $I$ with (i) can be mapped to an abstract answer set. This would result in losing a possible answer set. \nWe can avoid this\nby using an altered rule\n\\smallskip\n\n\\centerline{$0\\{d(\\widehat{X}_1,\\widehat{X}_2)\\}1 \\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{\\leq}\\widehat{X}_2.$}\n\n\\smallskip\n\nA naive approach would \nabstract all rules by modifying the heads\nto choice rules. However,\nnegation in rule bodies may cause\na loss of\noriginal answer sets in the abstraction. Say we have a rule with negation in the body, $d(X,X) \\leftarrow \\mi{not}\\ a(X,X)$. If it is only changed to a choice rule in the abstract program, when $a(k,k)$ holds we will not have $d(k,k)$, while originally we can have $\\{d(2,2),a(3,2)\\}$. Such rules must be treated specially to catch the cases of obtaining $d(k,k)$ while $a(k,k)$ holds.\n\nFor a finer-grained and systematic approach,\nwe focus on rules of form $r: l \\leftarrow B(r), \\Gamma_{\\mi{rel}}(r)$\nwhere the variables \nin $B(r)$ are standardized apart\nand $\\Gamma_{\\mi{rel}}$ consists\nof built-in relation literals that\nimpose restrictions \non the variables in $B(r)$.\n\n\\begin{exmp}\n\\label{ex:standardize}\nThe rules (\\ref{eq:1}) and (\\ref{eq:2}) are standardized apart and they\nhave $\\Gamma_{\\mi{rel}}(r)=\\top$ (or a dummy $X\\,{=}\\,X$) and\n$\\Gamma_{\\mi{rel}}(r)=X_1\\leq X_2$, respectively. The rule $\\mi{c} \\leftarrow\n\\mi{r}(X,Y), \\mi{p}(X,Y)$ is rewritten to the rule $\\mi{c} \\leftarrow\n\\mi{r}(X_1,Y_1), \\mi{p}(X_2,Y_2),\\Gamma_{rel}$ with $\\Gamma_{rel}\\,{=}\\, (X_1\\,{=}\\,X_2,Y_1\\,{=}\\,Y_2)$.\n\\end{exmp}\nThe basic idea is as follows:\nwhen\nconstructing the abstract program, we either (i) just abstract each\nliteral in a rule, or (ii) in case of uncertainty due to abstraction,\nwe guess the rule head to catch possible cases. The uncertainty may\noccur \ndue to having relation restrictions over non-singleton equivalence\nclasses\n(i.e. $|m_d^{-1}(\\hat{d}_i)|>1$), \nor having negative literals that are mapped to non-singleton abstract literals.\n\n\nTo the best of our knowledge, this is the first such approach of abstracting ASP programs.\n\n\n\\leanparagraph{Abstracting the relations} For simplicity, we first\nfocus on binary relations, e.g., $=,<,\\leq,\\neq$, \nand $\\Gamma_{rel}(r)$ \nof the form $\\mi{rel}(X,c)$ or $\\mi{rel}(X,Y)$.\n\nIt is necessary to\nreason about the cases that can occur for the truth values of ${\\mi{rel}}(\\hat{d}_1,\\hat{d}_2)$, for $\\hat{d}_1,\\hat{d}_2 \\in \\widehat{D}$, in order to obtain minimal abstract models that cover the original answer sets. There are four cases to consider:\n\n\n\n\n\\noindent\\begin{tabular}{r@{}l}\n I & $\\phantom{\\neg} \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\forall x_1\\,{\\in}\\, \\hat{d}_1,\\forall x_2\\,{\\in}\\, \\hat{d}_2. \\mi{rel}(x_1,x_2)$ \\\\%[1ex]\n II & $\\neg \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\forall x_1\\,{\\in}\\, \\hat{d}_1,\\forall x_2\\,{\\in}\\, \\hat{d}_2. \\neg \\mi{rel}(x_1,x_2)$ \n \\\\%[1ex]\n III & $\\phantom{\\neg} \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\exists x_1\\,{\\in}\\, \\hat{d}_1,\\exists x_2\\,{\\in}\\, \\hat{d}_2. \\neg \\mi{rel}(x_1,x_2)$\\\\%[1ex]\n IV & $\\neg \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\exists x_1\\,{\\in}\\, \\hat{d}_1,\\exists x_2\\,{\\in}\\, \\hat{d}_2. \\mi{rel}(x_1,x_2)$\n\\end{tabular}\n\n\n\\noindent For $\\mi{rel}(\\hat{d}_1,\\hat{d}_2){=} \\top$, Case III is\nmore common in domain abstractions, while case I occurs e.g.,\\ for singleton\nmappings (i.e., $|\\hat{d}_1|=|\\hat{d}_2|=1$) or for negative relations such as $\\neq$. For $\\mi{rel}(\\hat{d}_1,\\hat{d}_2) {=}\\bot$, Case II is the common case, e.g., $=,\\leq$, whereas case IV may occur for negative relations or $<$.\n\n\\begin{exmp}\nConsider $\\mi{rel}(X,Y)=X\\leq Y$ and a mapping $m_d(\\{1\\})=\\hat{d}_1, m_d(\\{2,3\\})=\\hat{d}_k$ with an order $\\hat{d}_1 < \\hat{d}_k$ on the abstract values. Notice that case I occurs for $\\hat{d}_1 \\leq \\hat{d}_k$ and $\\hat{d}_1 \\leq \\hat{d}_1$, while case III occurs for $\\hat{d}_k \\leq \\hat{d}_k$. The latter is due to the possibility of having $3 \\leq 2$ which is false.\n\\end{exmp}\n\nThe cases that the equivalence classes have for a binary $\\mi{rel}$\ncan be computed by simple queries and represented by facts of form $\\mi{type}_{\\mi{rel}}^{\\mi{case}}(\\hat{d}_1,\\hat{d}_2)$\nfor each equivalence classes $\\hat{d}_1,\\hat{d}_2$.\n\n\\leanparagraph{Program abstraction} We start with a procedure for programs\nwith rules $r: l \\leftarrow B(r), \\mi{rel}(t_1,t_1')$\nwhere $|B^-(r)|{\\leq} 1$. \n\nFor any rule $r$ and $*{\\in} \\{+,-\\}$, let the set\n$S^{*}_{\\mi{rel}}(r)=\\{l_j \\in B^{*}(r) \\mid arg(l_j) \\cap\n\\{t_1,t_1'\\} \\neq \\emptyset\\}$ be the positive and negative literals, respectively, that share an argument with $\\mi{rel}(t_1,t_1')$.\nWe assume for simplicity that $B^-(r) \\subseteq S_{rel}(r)$\nand discuss how to handle rules not meeting this assumption later.\n\nWe build a program $\\Pi_{dom}^m$ according to the mapping $m$\nas follows. \nFor any rule $r: l \\leftarrow B(r), \\mi{rel}(t_1,t_1')$ in $\\Pi$, we\nadd:\n\\vspace*{-.125\\baselineskip}\n\n\\begin{enumerate}[\\hspace{1cm}\\quad(1)]\n\n\\item[(0)] If $B^+(r)\\setminus S^+_{\\mi{rel}}(r)\\neq \\emptyset$:\n\\vspace*{-.25\\baselineskip}\n\\begin{enumerate} [$~\\hspace{-1.5em}$(a)]\n\\item If $\\mi{rel}(t_1,t_1'){=} \\top:$~ $m(l) \\leftarrow m(B(r))$. \n\\end{enumerate}\n\\vspace*{-.25\\baselineskip}\n\n\\item If $S_{\\mi{rel}}^+(r)\\neq \\emptyset$:\n\\vspace*{-.35\\baselineskip}\n\\begin{enumerate} [$~\\hspace{-1.5em}$(a)]\n\\itemsep=3pt\n\\item $m(l) \\leftarrow m(B(r)), {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{I}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j)$.\n\\item $0\\{m(l)\\}1 \\leftarrow m(B(r)), {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{III}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j)$.\n\\item $0\\{m(l)\\}1 \\leftarrow m(B(r)), \\neg {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{IV}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j)$.\n\\end{enumerate}\n\\vspace*{-.25\\baselineskip}\n\\item If $l_i {\\in} S_{\\mi{rel}}^-(r)$:\n\\vspace*{-.2\\baselineskip}\n\\begin{enumerate} [$~\\hspace{-2em}$(a$'$)] \n\\itemsep=3pt\n\\item $m(l) {\\leftarrow} m(B(r)), {rel}(\\hat{t}_i,\\hat{t}_j).$\n\\item[(b$'$)] $0\\{m(l)\\}1 {\\leftarrow} m(B^{\\mi{shift}}_{l_i}(r)), {rel}(\\hat{t}_i,\\hat{t}_j), \\mi{type}^{\\textup{III}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j).$ \n\\item[(c$'$)] same as (c), if $S_{\\mi{rel}}^+(r){=} \\emptyset$;\\\\[1pt]\n$ 0\\{m(l)\\}1 {\\leftarrow} m(B^{\\mi{shift}}_{l_i}(r)), \\neg {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{IV}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j).$ \\\\[1pt]\n$~0\\{m(l)\\}1 {\\leftarrow} m(B^{\\mi{shift}}_{l_i}(r)), {rel}(\\hat{t}_i,\\hat{t}_j), \\mi{type}^{\\textup{IV}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j).$ \n\\end{enumerate}\n\\vspace*{-.3\\baselineskip}\n\\end{enumerate}\n}\n\\noindent where $B^{\\mi{shift}}_{l_i}(r){=}B^+(r)\\cup \\{l_i\\},\\mi{not}\\ B^-(r){\\setminus} \\{l_i\\}$.\n\nCase (0) is the special case of having positive literals that do not share arguments with $\\mi{rel}$. If $\\mi{rel}{=}\\top$, then it will not be processed by next steps. Thus, the abstraction of $r$ is added. The assumption on $B^-(r)$ about being included in $S_{rel}(r)$ prohibits the case $B^-(r){\\setminus} S^-_{\\mi{rel}}(r){\\neq} \\emptyset$. \n\n\nIf $\\mi{rel}(t_1,t'_1)$ shares arguments with a positive body literal, we add\nrules to grasp the possible cases resulting from the relation type. In\ncase of uncertainty, \nthe head is made a choice,\nand for case IV, we flip the relation, $\\neg\\mi{rel}$, to catch the\ncase of the relation holding true. If $\\mi{rel}(t_1,t'_1)$ shares arguments with\na negative body literal, we need to \ngrasp the\nuncertainty arising\nfrom negation. We do this by adding rules in which we shift the\nrelated literal to \nthe positive body, via $B^{\\mi{shift}}_{l_i}(r)$.\n\n(2-c$'$) deals with the special case of\na type IV relation and a negative literal, e.g., $b(X_1) \\leftarrow\n\\mi{not}\\ a(X_1,X_2), X_1 {\\neq} X_2$. If\n$r$ \nis abstracted only\nby keeping the same structure,\n$m(B(r))$ \nmight not be\nsatisfied by abstract literals that actually have corresponding\nliterals which satisfy\n$B(r)$. \nE.g., $a(2,3){=}\\bot$\nsatisfies $r$; this can only be reflected in the abstraction by\n$a(k,k){=}\\bot$ which actually\ndoes \nnot satisfy\n$m(B(r))$. \nThus, \nwhen building the abstract rules, rules \nfor all combinations of shifting the literal and flipping the relation need to be added.\n\nNotably, the construction of $\\Pi^m_{dom}$ is modular, rule by\nrule;\nfacts $p(\\vec{t})$ are simply lifted to abstract facts $p(m(\\vec{t}))$. \n\n\n\\begin{exmp}\\label{ex:toy_dom}\nConsider the rules from Example~\\ref{ex:toy} plus \n\\begin{align}\n e(X_1) &\\leftarrow \\mi{not}\\ a(X_1,X_2), X_1{=}X_2.\\footnotemark \\label{eq:3}\n\\end{align} \n\\footnotetext{In order to ensure safety, these rules can be extended with special built-in domain predicates which do not require to be standardized apart.}\n\\noindent over the domain $D=\\{1,2,3\\}$.\nSuppose $\\widehat{D}{=}\\{\\hat{d}_1,\\hat{d}_k\\}$ with mapping $m_{d}(1){=}\\hat{d}_1$, $m_{d}(\\{2,3\\}){=}\\hat{d}_k$. The abstract program constructed is as follows, in simplified form:\n\\vspace*{-.25\\baselineskip}\n\\begin{align}\n a(\\widehat{X}_1,\\widehat{X}_2) &\\leftarrow c(\\widehat{X}_1), b(\\widehat{X}_2) \\label{eq:11}\\\\%[-.6ex]\n d(\\widehat{X}_1,\\widehat{X}_2) &\\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1 {\\leq} \\widehat{X}_2, \\widehat{X}_1 {=} \\hat{d}_1,\\widehat{X}_2 {=} \\hat{d}_k \\!\\!\\label{eq:21}\\\\%[-.6ex]\n \\hspace{-0.45cm} 0\\{d(\\widehat{X}_1,\\widehat{X}_2)\\}1 & \\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1 {\\leq} \\widehat{X}_2,\\widehat{X}_1{=} \\hat{d}_k,\\widehat{X}_2 {=} \\hat{d}_k \\label{eq:22}\\\\%[-.6ex]\n e(\\widehat{X}_1) &\\leftarrow \\mi{not}\\ a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{=}\\widehat{X}_2 \\!\\!\\!\\!\\label{eq:31}\\\\%[-.6ex]\n0\\{e(\\widehat{X}_1)\\}1 &\\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{=}\\widehat{X}_2, \\widehat{X}_1{=} \\hat{d}_k,\\widehat{X}_2 {=} \\hat{d}_k \\label{eq:33}\n\\end{align} \n\\vspace*{-1\\baselineskip}\n\n\\noindent Here the $\\mi{type}^{case}_{\\mi{rel}}$ literals have been evaluated, and\nredundant rules are omitted.\nObserve that \\eqref{eq:11} is same as \\eqref{eq:1} as it has\n$\\mi{rel}{=}\\top$. \nFrom \\eqref{eq:2}, we get \\eqref{eq:21} for $\\hat{d}_1,\\hat{d}_k$ which have case I for $\\leq$, and \\eqref{eq:22} for $\\hat{d}_k,\\hat{d}_k$ that have case III. From \\eqref{eq:3}, we get \\eqref{eq:31} and \\eqref{eq:33} with shifting for case III.\n\nFor given facts $c(3),b(2)$,\n$I =\\{a(3,2),e(1),e(2),e(3),c(3),b(2)\\} \\in AS(\\Pi)$.\nAfter applying m,\nthe facts become $c(\\hat{d}_k),b(\\hat{d}_k)$ and\n$\\{a(\\hat{d}_k,\\hat{d}_k),e(1),e(\\hat{d}_k),c(\\hat{d}_k),b(\\hat{d}_k)\\} \\in AS(\\Pi^m_{dom})$, which covers\n$I$. Note that the choice rule \\eqref{eq:33} ensures\nthat $e(\\hat{d}_k)$ can still be obtained even \nw\\textbf{}hen $a(\\hat{d}_k,\\hat{d}_k)$ holds. It likewise covers\n$\\{c(2), b(3), a(2,3), d(2,3), e(1), e(2), e(3)\\} \\in AS(\\Pi)$ for the facts $c(2),b(3)$.\n\n\\end{exmp}\n\nWe prove that the abstraction procedure constructs a system $\\Pi_{dom}^m$ that over-approximates $\\Pi$.\n\n\\begin{thm}\nLet $m$ be a domain abstraction over $\\Pi$.\nThen for every\n$I \\in AS(\\Pi)$, $m(I) \\in AS(\\Pi_{dom}^m)$.\n\\end{thm}\n\n\\begin{proof}[Proof (sketch)]\nWith the \nrules (0a), (1a-1b), and (2a$'$), we ensure that\n$\\widehat{I}$ is a\nmodel\nof\n$\\Pi^m_{dom}$, as we either keep the structure\nof a rule $r$ or change it to a choice rule. The rules added in steps\n(1b-1c) and (2b$'$-2c$'$) \nserve to catch the cases that may\nviolate the minimality of the model due to a negative literal or a\nrelation over non-singleton equivalence classes. The rules (1b,2b$'$) deal with\nhaving a literal (resp.\\ relation literal) that is false in\n$I$ but thought to be true in the abstract model $\\hat{I}$, and (1c,2c$'$)\ndeal with \na literal (resp.\\ relation literal) that is thought to be false in \n$\\hat{I}$ but true in $I$.\n\\end{proof}\n\\vspace*{-0.4\\baselineskip}\n\n\n\\leanparagraph{General case} The construction can be applied to more general programs by focusing on two aspects: 1) $|B^-(r)|{>}1$: For multiple negative literals in the rule, the shifting must be applied to each negative literal. 2) $|\\Gamma_{\\mi{rel}}|{>}1$: To handle multiple relation literals, a straightforward approach is to view $\\Gamma_{\\mi{rel}} = \\mi{rel}(t_1,t'_1),$ \\ldots, $\\mi{rel}(t_k,t'_k)$ as \na literal of an $n$-ary relation $\\mi{rel}'(X_1,X'_1,\\ldots,X_k,X'_k)$, $n\\,{=}\\,2k$.\nThe abstract version of such built-ins $\\mi{rel}'$ and the type cases I-IV are readily lifted\n\nLet $\\Pi_{dom}^{*\\ m}$ be the program obtained from a program $\\Pi$\nwith the generalized abstraction procedure.\nThen we \nobtain:\n\\begin{thm}\nFor every $I \\in AS(\\Pi)\n, $m(I) \\in AS(\\Pi_{dom}^{*\\ m})$.\n\\end{thm}\n\n\nFor constraints, the\nsteps creating choice rules can be skipped \nas we cannot guess over $\\bot$. \nFurther simplifications and optimizations\ncan help to \navoid introducing too many spurious answer sets.\nSyntactic extensions can also be addressed. Rules with choice and cardinality constraints can be lifted with the same structure. For conditional literals with conditions over negative literals, additional rules with shifting will be necessary; otherwise, the condition can be lifted the same.\n\n\\section{Using Abstraction for Policy Refutation}\\label{abs_policy}\n\nAs an application case, we are interested in the problem of defining declarative policies for reactive agents and reasoning about their behavior, especially in non-deterministic environments with uncertainty in the initial state. In such environments, searching for a plan that reaches the main goal easily becomes troublesome. Therefore, we focus on defining policies that choose a sequence of actions from the current state with the current observations, in order to achieve some subgoal, and then checking the overall behavior of these policies. More details of such policies can be found in \\cite{zgs16jelia}.\n\n\\leanparagraph{Background}\nFormally, a \\emph{system} \n$\\ensuremath{A}\\,{=}\\,\\langle \\mathcal{S},\\mathcal{S}_0,\\mathcal{A},\\Phi\\rangle$\nconsists of a finite set $\\mathcal{S}$ of states, a set\n$\\mathcal{S}_0 \\,{\\subseteq}\\, \\mathcal{S}$ of initial states, a finite set\n$\\mathcal{A}$ of actions, and a non-deterministic transition relation\n$\\Phi: \\mathcal{S}\\,{\\times}\\,\\mathcal{A} \\rightarrow 2^{\\mathcal{S}}$. \n\n\nA sequence $\\sigma=a_1,a_2,\\dots,a_n$ of actions is \\emph{executable}, if\n\n\\smallskip\n\n\\centerline{$\\exists s_0,\\dots,s_n \\in \\mathcal{S}_0\\, \\forall\\, 0\\leq i] (x) -- node[midway,left]{$h_{st}$} (y);\n\\path (2,1.5) node(z) {$\\hat{s}'$}\n(2,0) node(t) {$s'$};\n\\draw[->] (z) -- node[midway,right]{$h^{-1}_{st}$} (t);\n\\draw[->] (y) -- node[midway,above]{$\\hat{a}$} (z);\n\\draw[->] (x) -- node[midway,below]{$\\sigma$} (t);\n\\node(k) at (1,0.75) {$h_{act}$};\n\\draw[dashed] (1,0) -- (k);\n\\draw[->,dashed] (k) -- (1,1.5);\n\\end{tikzpicture}}~~~\n\\subfigure[][Domain abstraction]{\\includegraphics[scale=.6]{abs.pdf}}\n\\vspace{-1.5em}\n\\end{figure}\n\n\n\\leanparagraph{Action abstraction} \nWe consider a set $\\hat{\\mathcal{A}}$ of abstract actions\nfor $\\Psi$, and an abstraction function\n$h_{act}{:} \\Sigma {\\rightarrow} \\hat{\\mathcal{A}}$ which maps \naction sequences $\\sigma$ to abstract actions $\\hat{a}$.\nSimilarly, \nwe denote $h_{act}(\\sigma)$ by $\\hat{\\sigma}$ and identify\n$\\hat{a}$ with $\\{ \\sigma \\,{\\in}\\, \\Sigma \\mid h_{act}(\\sigma)\\,{=}\\,\\hat{a}\\}$.\n\n\n\\begin{exmp}[ctd]\\label{ex:regions}\nFigure~\\ref{fig:abs_fig}(b) shows an example of a domain abstraction that maps the large domain into a smaller sized domain, with $h_{st|D}(\\mi{rAt}(X,Y)){=}\\mi{rAt}(\\lceil \\frac{X}{n\/2}\\rceil,\\lceil\\frac{Y}{n\/2}\\rceil){=}\\mi{rAt}(Rx,Ry),$ $Rx,Ry {\\in} \\{1,2\\}$. Similarly, $\\mi{goTo}(X,Y)$ is mapped to some $\\mi{goTo}(Rx,Ry)\n. For simplicity, we will refer to the abstract cells as regions $\\{nw,ne,sw,se\\}$.\n\\end{exmp}\n\\vspace*{-0.5\\baselineskip}\n\n\\leanparagraph{Abstract system with an abstracted policy}\nThe transitions of\n$\\ensuremath{\\widehat{A}}$ are defined over those \nin $\\ensuremath{A}$ \nchosen by the policy.\n\n\\begin{defn}\nFor a system\n$\\ensuremath{A}=\\langle \\mathcal{S},\\mathcal{S}_0,\\mathcal{A},\\Phi\\rangle$, a set\n$\\Sigma$ of plans \nover $\\mathcal{A}$, a transition function\n$\\Phi_\\Sigma$, and a policy $P$, an abstract system $\\widehat{A}=\\langle \\hat{\\mathcal{S}}, \\hat{\\mathcal{S}}_0, \\hat{\\mathcal{A}}, \\widehat{\\Phi}_P \\rangle$\nis \\emph{generated} by a state abstraction $h_{st}$ and \naction abstraction $h_{act}$, if \n\n\\vspace{1pt}\n\n\\begin{compactitem}[--]\n\\item $\\hat{\\mathcal{S}}_0 = \\{\\hat{s}_0 \\mid s_0 \\in \\mathcal{S}_0\\}$ are the initial abstract states, and \n\\item $\\widehat{\\Phi}_{P}: \\widehat{S} \\times \\hat{\\mathcal{A}} \\rightarrow \\widehat{S}$ is the abstract transition function according to the policy $P$, defined as\n\\smallskip\n\n\\centerline{$\n\\widehat{\\Phi}_{P}(\\hat{s},\\hat{a}) {=} \n\\{\n\\hat{s}' \\mid \\exists s'' \\in\n\\hat{s}, \\sigma \\in P(s'')\\cap \\hat{a}:\ns' \\in \\Phi_\\Sigma(s'',\\sigma)\\}.\n$}\n\\vspace*{-0.25\\baselineskip}\n\\end{compactitem}\n\\end{defn}\nNote that any abstract transition $\\hat{s},\\hat{a},\\hat{s}'$\nin $\\widehat{A}$\nmust \nstem from a transition $s,\\sigma,s'$ in $\\mathcal{A}$\nvia policy $P$,\ni.e., an abstract transition is introduced only if there is a corresponding original transition. This gives an over-approximation of the policy's behavior on the original system \\cite{clarke03}.\n\n\n\\subsubsection{Constructing an abstract system}\nWe can apply the abstraction method to ASP programs \nwith action\ndescriptions and policy rules, where we focus on \npolicies with single action plans, \nwith some particulars.\n\n\\begin{itemize}\n \\item\n It is possible to have\nthe mapping $m_a{:}{\\cal A}{\\rightarrow} \\hat{\\cal A}$\ncreate abstract\npredicates, i.e., $m_a(a(v_1,\\dots,v_n))=\\hat{a}(m_{d}(v_1),\\dots,$\n$m_{d}(v_n))$ where $\\hat{a} = \\widehat{a(v_1,\\dots,v_n)}$ depends also on the arguments of $a$ in order keep some possibly necessary details of the original actions in the abstract action. However, with action atoms occurring in transition descriptions\nonly positively in rule bodies, and in policy rules only in rule\nheads, no further treatment of these atoms is necessary.\n\n\\item For policy rules \n$a {\\leftarrow} B$ that select an action $a$, abstract rules\n$0\\{\\hat{a}\\}1 {\\leftarrow} \\widehat{B}$ \n(while correct) are undesirable as they allow to skip the action and would create a spurious trajectory.\nTo have an optimization over the abstraction, \nthis can be avoided by\nliterals $l$ in\n$B$ with singleton mappings, i.e.,\n$|m^{-1}(\\hat{l})|{=}1$, or with a non-singleton mapping\nwhere $l {\\in} B^{+}\\setminus S^+_{\\mi{rel}}$, or with cases \nthat allow for simplification of the choice rules.\n\n\\item Time arguments amount to a special sort, and we do not abstract\nover it (i.e., each time point $t$ is abstracted to itself).\nThus, time variables, terms etc.\\ simply remain unaffected.\n\n\\item For plan abstraction\n$h_{act}{:} \\Sigma {\\rightarrow} \\hat{\\cal A}$, dedicated atoms\n$\\sigma$ can describe plans with their effects, obtained \nfrom unfolding\nthe \neffect rules of the actions in $\\sigma$\nand their preconditions.\n\\end{itemize}\n\n\\begin{exmp}[ctd]\nBy omitting most of the details\nexcept the directly\naffected literals and the literals related with the goal condition,\nthe domain abstraction \nin Ex.\\ref{ex:regions} \nyields the following abstract rules\nfor \\eqref{eq:pol_formula}-\\eqref{eq:pol_formula2}:\n\\begin{align}\n&1\\{\\mi{goTo}(\\widehat{X},\\widehat{Y},T){:} \\widehat{X},\\widehat{Y} {\\in} \\widehat{D}\\}1 {\\leftarrow} \\mi{not}\\ \\mi{seen}(T), \\mi{not}\\ \\mi{caught}(T).\\nonumber\\\\%[-.5ex]\n&\\mi{goTo}(\\widehat{X},\\widehat{Y},T) \\leftarrow \\mi{seen}(T), \\mi{not}\\ \\mi{caught}(T), \\mi{pAt}(\\widehat{X},\\widehat{Y},T).\\nonumber\\\\%[-.5ex]\n&0\\{\\mi{caught}(T)\\}1 \\leftarrow \\mi{rAt}(\\widehat{X},\\widehat{Y},T), \\mi{pAt}(\\widehat{X},\\widehat{Y},T).\\label{eq:abs_rules}\\\\%[-.5ex]\n&0\\{\\mi{seen}(T)\\}1 \\leftarrow \\mi{pAt}(\\widehat{X},\\widehat{Y},T). \\nonumber\n\\end{align}\n\\end{exmp}\n\n\\subsection{Counterexample search} Recall \nour aim of over-approxi\\-mating the problem\nof checking whether \nobeying the policy $P$ always reaches the goal $\\mu$ \n(i.e., all paths starting from ${\\cal S}_0$\nreach a state that satisfies $\\mu$).\nFor policies where all\nstates have outgoing transitions, state abstractions that distinguish\nthe goal conditions can avoid false positives. \nThat is, if no ``bad'' abstract trajectory exists in which\n$\\mu$ is unachieved (a {\\em counterexample}), then no \n``bad'' original trajectory exists;\nand we can check the policy behavior on\nthe abstract system, as in \\cite{clarke03}.\n\nConcretely, we search for an abstract counterexample (cex)\ntrajectory in the abstract system $\\ensuremath{\\widehat{A}}$\nof length at most $n$, where\n$n$ is large enough. As the original space state $\\mathcal{S}$ is\nfinite, any path trajectory longer than $|\\mathcal{S}|+1$\nclearly must loop. If we cannot find such a counterexample trajectory, the policy works, cf.\\ \\cite{ClarkeKOS04}. \nOn the other hand, if a cex trajectory\n$\\hat{\\tau}=\\hat{s}_0,\\hat{a}_0,\\dots,\\hat{s}_n$ is found, we need to\ncheck whether $\\hat{\\tau}$ has a corresponding concrete trajectory $\\tau$ in $\\ensuremath{A}_\\Sigma$. \nThe counterexample is \\emph{spurious}, if no such \n$\\tau$ exists.\n\n\\begin{figure}[t]\n\\caption{}\n\\centering\n\\subfigure[][a counterexample trajectory]{\\includegraphics[scale=0.75]{abs_failure_notarget2_vfinal_new_1_new.pdf}}\n\\subfigure[][a corresponding concrete trajectory with failure]{\\includegraphics[scale=0.75]{abs_failure_notarget2_vfinal_new_new.pdf}}\n\\label{fig:failure}\n\n\\vspace*{-1.5\\baselineskip}\n\n\\end{figure}\n\n\n\\begin{exmp}[ctd]\\label{ex:spur}\nA 3-step counterexample (Figure~\\ref{fig:failure}(a)) to\nfinding the person\nis \n$\\hat{\\tau}\\,{=}\\,\\mi{rAt}(nw,0),\\mi{goTo}(\\mi{nw},0),\\mi{rAt}(\\mi{nw}$, $1),\\mi{goTo}(\\mi{ne},1),\\mi{rAt(ne},2),\\mi{goTo}(\\mi{ne},2),\\mi{rAt(ne},3)$.\nFigure~\\ref{fig:failure}(b) shows a corresponding trajectory\nin \n$\\ensuremath{A}_\\Sigma$;\nit fails at step 2 to find an\naction corresponding to\n$\\mi{goTo}(\\mi{ne})$, \nas the\npolicy \nwould move to $\\mi{nw}$. In fact, no corresponding trajectory \nwithout\nfailure can be found; so $\\hat{\\tau}$ is spurious.\n\\end{exmp}\n \nExample~\\ref{ex:spur} shows that, as expected, omitting most of the\ndetails of the problem makes it easy to encounter spurious\ntrajectories. We need to add back some of the details that the policy uses in order to reduce spurious transitions.\n\n\\begin{exmp}[ctd]\nLet us apply domain abstraction on $\\mi{farthest}(X,Y,X1,Y1)$, and\nfurther auxiliary literals such as $\\mi{farthestDist}(X,Y,D)$. We get\n$\\mi{farthest}(Rx,Ry,Rx1,Ry1)$ and $\\mi{farthestDist}(Rx,Ry,RD)$,\nwhere $RD \\in \\{0,1\\}$ tells if the distance is $